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arp 220 is the nearest ( @xmath3 77 mpc ) example of an ultraluminous infrared galaxy ( ulirg ) that supports star formation at extreme levels . it contains two nuclei separated by 350 pc , both surrounded by massive discs of dense molecular gas ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? radio detections of supernovae at a rate of 13 yr@xmath4 @xcite confirm that huge populations of massive stars are present with an implied star formation rate ( sfr ) of @xmath5 yr@xmath4 . although arp 220 could contain active galactic nuclei ( agns ) , particularly in the western nucleus , the observed supernova rates indicate that star formation provides a substantial fraction of the power radiated by the nuclei . the nuclei of arp 220 provide access to the high - intensity mode of star formation in dense molecular media that appears to have been more common in young galaxies . these types of environments are of special interest from a range of perspectives , including the information they can provide regarding the role of galactic winds , cosmic rays , and magnetic fields in feedback processes that influence galaxy evolution . previous investigations show that arp 220 is likely to be a hadronic cosmic ray calorimeter where all of the power in cosmic rays is absorbed within the nuclear starburst zones ( e.g. , * ? ? ? both nuclei also contain extremely intense far - infrared ( fir ) radiation fields ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and the west nucleus is optically thick in the fir to wavelengths of @xmath6 @xcite . the production of the observed radio synchrotron emission then requires magnetic fields of milligauss strength ( e.g. , * ? ? ? * ; * ? ? ? llllc physical parameters & east nucleus & west st & west cnd & references + distance & 77.0 mpc & 77.0 mpc & 77.0 mpc & + cmz radius & 70 pc & 90 pc & 30 pc & 1,2,3 + cmz disc scale height@xmath7 & 40 pc & 40 pc & 40 pc & 4 + molecular gas mass & @xmath8 @xmath9 & @xmath10 @xmath9 & @xmath8 @xmath9 & 2,5 + ionized gas mass@xmath11 & @xmath12 @xmath9 & @xmath13 @xmath9 & @xmath12 @xmath9 & + average ism density@xmath14 & @xmath157700 @xmath16 & @xmath153500 @xmath16 & @xmath1542 000 @xmath16 & + fir luminosity & @xmath17 @xmath18 & @xmath17 @xmath18 & @xmath19 @xmath18 & 2 + fir radiation field energy density@xmath20 & 40 000 ev @xmath16 & 27 000 ev @xmath16 & 440 000 ev @xmath16 & + dust temperature & 90 k & 50 k & 170 k & 2,6 + sn explosion rate ( @xmath21 ) & 0.7 yr@xmath4 & 0.7 yr@xmath4 & 1.3 yr@xmath4 & 7 + star formation rate ( sfr)@xmath20 & 65 @xmath9 yr@xmath4 & 65 @xmath9 yr@xmath4 & 120 @xmath9 yr@xmath4 & + sn explosion energy@xmath22 & 10@xmath23 erg & 10@xmath23 erg & 10@xmath23 erg & + sn energy in cosmic ray protons@xmath22 & 5 20% & 5 20% & 5 20% & + ratio of primary protons to electrons ( @xmath24/@xmath25 ) & 50 & 50 & 50 & + slope of primary cosmic ray source function & 2.1 2.3 & 2.1 2.3 & 2.1 2.3 & + + + + + + + + + + in this paper , we study cosmic ray interactions in the arp 220 starburst nuclear regions using an updated version of the @xcite models , hereafter yegz . we develop a model with two spatial zones to accurately represent the inner and outer regions of the western nucleus as defined by its molecular gas properties @xcite . we incorporate photopion energy losses and photon photon interactions to account for the extreme fir radiation field . we calculate the hadronic calorimetry fraction for each nucleus for the best - fitting radio models , and we predict the total @xmath2-ray and neutrino fluxes . in section 2 , we review the physical parameters which we selected for the models . section 3 details the basic assumptions of the models and our findings for the arp 220 starburst nuclei . we present concluding remarks in section 4 . due to its extreme properties , arp 220 has been extensively studied across the electromagnetic spectrum . the nuclei of arp 220 are of particular interest as they contain more than half of the total bolometric infrared luminosity of the galaxy ( @xmath26 ; e.g. , * ? ? ? * ; * ? ? ? * and references therein ) . as the nuclei are less than 100 pc in radius , the presence of an agn or a ` hot ' starburst is required to explain the extraordinarily large surface brightness in the western nucleus @xcite ; however , the existence of an agn has yet to be definitively established ( e.g. , * ? ? ? further , the submillimetre observations suggest that whether or not agns are present , they are not the main heating source of the dust ( e.g. , * ? ? ? estimates of the fir luminosities of the eastern and western nuclei range from @xmath27 to @xmath28 and from @xmath29 to @xmath30 , respectively ( e.g. , * ? ? ? * ; * ? ? ? the range on these luminosities is quite large due to uncertainty in the true sizes , inclinations , and opacities of the nuclei and their associated molecular disc . to keep our adopted fir luminosity in rough agreement with the observed supernova rate , we assume values of @xmath31 and @xmath32 for the eastern and western nuclei ( see table 1 ) . assuming similar ratios between the nuclei for the supernova rate and molecular gas content , we adopt values of @xmath33 yr@xmath4 , @xmath34 for the eastern nucleus and @xmath35 yr@xmath4 , @xmath36 for the western nucleus . while our assumed molecular gas masses favour conservative estimates , other estimates of the gas content suggest the masses are as high as @xmath37 @xcite . [ cols="<,^,^,^,^,^,^,^ " , ] co observations of the western nucleus imply a temperature gradient increasing towards the centre and indicate significant differences in the physical conditions between the two nuclei @xcite . @xcite model the western nucleus as two distinct dust sources a cooler ( 50 k ) ring surrounding a hotter ( 170 k ) , dense dust core . we use this two - zone model for the western nucleus and have adjusted our single - zone model to account for the differences in temperature and density between the two regions ( see section 3 ) . for the eastern nucleus , we assume a single dust temperature of 90 k @xcite . previously , we developed and tested a model for cosmic ray interactions in the central molecular zones ( cmzs ) of star - forming and starburst galaxies ( yegz ; * ? ? ? * ; * ? ? ? our single - zone model accounts for a variety of energy losses via interactions with the interstellar medium ( ism ) , magnetic fields , and radiation fields and for energy - independent advective escape via a galactic wind ( see fig . 1 ) . the resulting cosmic ray energy spectrum depends on both the total cosmic ray lifetime and a power - law injection spectrum which is directly proportional to the volume integrated supernova rate ( see yegz for further details ) . accounting for the production of secondary cosmic rays , we use our calculations of the population of energetic particles to predict the radio , @xmath2-ray , and neutrino spectra . for the @xmath2-ray spectrum , we include both leptonic ( bremsstrahlung , inverse compton ) and hadronic ( neutral pion decay ) emission mechanisms . for the radio spectrum , we incorporate the effects of free free emission and absorption @xcite . as in our previous models , we assume that the ionized gas in the nuclei acts as a foreground screen that some fraction ( @xmath38 ) of the emitted synchrotron radiation passes through . when the covering fraction is low ( @xmath39 ) , the radio spectrum flattens at low frequencies @xcite , and when the covering fraction is high ( @xmath40 ) , the radio spectrum turns down at low frequencies ( yegz ) . as noted above , the western nucleus in arp 220 is best modelled with two separate regions : an inner circumnuclear disc ( cnd ) with a surrounding torus ( st ) . we model the cosmic ray populations of the two regions independently , treating each region as a uniform slab . however , the effects of absorption ( free free and @xmath2@xmath2 ) on the resulting radio and @xmath2-ray emission must be considered more carefully . absorption occurs within each emission region , and in the case of the inner cnd , absorption also occurs as the emitted radiation moves through the external , st ( see the appendix for further details ) . we perform @xmath41 tests following the approach described in yegz @xcite . comparing against radio observations for each nucleus , we vary magnetic field strength ( @xmath42 ) , wind speed ( @xmath43 ) , ionized gas density ( @xmath44 ) , and absorption fraction ( @xmath38 ) . while magnetic field strength and wind speed both directly affect the total cosmic ray lifetimes , the ionized gas density and the absorption fraction only affect the emitted radio spectrum . the free free emission and absorption coefficients are both directly proportional to the square of @xmath44 , and so , the frequency at which the radio spectrum flattens or turns down and the amount of free free emission at high frequencies both increase with @xmath44 . observations in @xcite , @xcite and @xcite separate the integrated fluxes of the eastern and western nuclei from the total flux , allowing us to constrain parameters for each nucleus individually . as we do not have radio observations which are separable between the two regions of the western nucleus , we can not constrain the magnetic field strength in each region separately . we therefore assume that the ratio between the magnetic field strength of the inner and outer regions of the western nucleus is equal to the square root of the ratio of the average gas densities , @xmath45 @xcite . thus , our magnetic field strength determination for the innermost western nucleus is an estimate that is guided by milky way observations . when assuming the standard 10% cosmic ray acceleration efficiency , we find that agreement between the models and the observed radio data occurs only in a very narrow area of parameter space ( see fig . the best - fitting models for the nuclei have magnetic field strengths limited to 1.0 mg for the western nucleus ( 3.5 mg in the cnd , estimated from scaling ) and 2.0 2.5 mg for the eastern nucleus ( see figs 2 and 3 and table 2 ) . as seen in fig . 3 , the total radio emission in both nuclei flattens at low frequencies , and in the eastern nucleus , the radio spectrum may be turning over completely . this flattening of the radio spectra requires moderate to high absorption fractions of 50 100% in the eastern nucleus and low to moderate absorption fractions of 10 70% in the western nucleus . in addition to moderate absorption fractions in each nucleus , we also find a high contribution from thermal emission to the total radio spectrum ( see fig . 3 ) , particularly in the western cnd where the majority of the radio emission is thermal above @xmath155 ghz . in part , this unusually high fraction of thermal emission is due to the inability of the model to effectively fit for free free absorption and free free emission simultaneously as seen in the eastern nucleus and in previous work ( see yegz ; * ? ? ? the ability of the models to accurately fit the fraction of thermal emission is further strained by the complicated nature of the western nucleus and the lack of separable radio observations . thus , in this particular case , the fractions of thermal emission in the best - fitting models have limited significance and do not necessarily contradict observations by @xcite which indicate more modest amounts of thermal emission ( @xmath46 ) . the western nucleus arp 220 has a very complex structure which we greatly simplified . while our two - zone density distribution reproduces the observed peak column density of @xmath47 @xmath48 , the mass is lower than that estimated by @xcite who derive the western nucleus gas mass from observations by the atacama large millimeter array ( alma ) of the submillimetre dust luminosity . this yields a total gas mass of @xmath49 for the western nucleus or four times our adopted value . we therefore explored the effect of increased gas mass on our model by tripling the mass in the western torus and ran a limited suite models with fixed parameters . we set the spectral index to @xmath50 and ran @xmath41 tests over the entire range of magnetic field strengths and wind speeds but over a subset of the previously tested ionized gas densities and absorption fractions . we ran tests on the western nucleus for acceleration efficiencies of 5% and 10% . in comparing the results of these models with a larger gas mass , we find that none of the tested models are within @xmath51 of the best - fitting model at the lower assumed gas mass . further more , these results yield extremely short cosmic ray electron lifetimes such that the physical validity of the models are in question . in addition to the higher gas mass estimates , @xcite also propose a geometry where the molecular gas in both nuclei is confined to a thin ( @xmath5210 pc ) disc . the evolution of supernovae and cosmic ray interactions in this type of high molecular mass structure is beyond the scope of this study which is designed to estimate cosmic ray interaction rates in arp 220 but will need to be considered when arp 220 is detected in @xmath2-rays . in our earlier works , we demonstrated that for a given ism , the yegz models are highly sensitive to the total flux of cosmic rays ( yegz ; * ? ? ? * ) . this flux is primarily effected by the original energy input into cosmic rays and the advective time - scale , or escape fraction . the energy input into cosmic rays is determined by the supernova rate and the assumed acceleration efficiency ( @xmath53 ) . within the uncertainty in the supernova rate , we vary acceleration efficiency from 5 to 20% . as shown above , for the standard 10% efficiency , the resulting best - fitting models are highly constrained in magnetic field strength , and we find that this is also true for an acceleration efficiency of 20% ( see table 2 ) . however , for a lower acceleration efficiency of 5% , equivalent to a lower supernova rate , we find a much larger range of acceptable fits in the eastern nucleus with magnetic field strengths ranging from 4 to 7.5 mg and wind speeds spanning our entire tested range . as such , the best - fitting models for arp 220 are essentially independent of wind ( advection ) speed ( see figs 1 and 2 ) . in contrast , a galactic wind was a vital component in modelling the cosmic ray populations of the starburst galaxies m82 and ngc 253 such that an extremely limited range of wind speeds resulted in fits within @xmath51 of the best - fitting models . the wind speed determines the advective timescale for a galaxy and the fraction of cosmic rays which escape . thus , wind speed is intrinsically tied to the proton calorimetry fraction for a galaxy which is closely related to the total radio and @xmath2-ray emission from a galaxy . other models for arp 220 have assumed fixed advection time - scales , thus ensuring proton calorimetry with the high gas densities in arp 220 @xcite . while our models agree with others in finding that the starburst regions of m82 and ngc 253 are only @xmath1540 60% proton calorimeters , we find that arp 220 s nuclei are 65 100% ( eastern ) and 90 100% ( western cnd ) proton calorimeters ( see fig . lcccccc & supernova & average gas & cosmic ray & radiation field & magnetic field & magnetic field + & power & density & energy density & energy density & energy density & strength + & ( erg yr@xmath4 ) & ( @xmath16 ) & ( ev @xmath16 ) & ( ev @xmath16 ) & ( ev @xmath16 ) & ( @xmath54 g ) + milky way & @xmath55 & 1 & 1.4 & 0.3 & 0.9 & 6 + m82 & @xmath56 & 260 & 470 & 490 & 2200 & 300 + arp 220 east & @xmath57 & 7700 & 1100 & 40 000 & @xmath58 & 6500 + arp 220 west cnd & @xmath59 & 42 000 & 2500 & 440 000 & @xmath60 & 7000 + + despite the uncertainty in the calorimetry fraction and the total cosmic ray flux in the eastern nucleus , we can still use our best - fitting models to make a prediction on the emitted @xmath2-ray and neutrino fluxes from arp 220 . to calculate the possible @xmath2-ray flux , we apply the parameters of models within @xmath51 from our best - fitting radio model . combining each possible set of models from the eastern and western nucleus , we find that the resulting @xmath2-ray spectra peak around @xmath150.3 gev with a maximum flux of @xmath61 gev @xmath62 s@xmath4 ( see fig . 4 ) . while this is roughly an order of magnitude lower than previous upper limits for arp 220 @xcite and _ fermi _ s differential sensitivity for four years of observations , it is only a factor of @xmath152 - 3 times smaller than the flux level of the recently detected ngc 2146 @xcite . we also compared our @xmath2-ray flux with the differential sensitivity 50 h of observations with the future southern cta array ( see fig . 4 ) and find it to be only a factor of a few larger than our maximum flux . arp 220 may still be detectable by _ fermi _ within the next several years and is a good target for cta , especially for energies near 1 tev . in addition to making a prediction for the @xmath2-ray spectrum , we can use our same results from the radio emission to predict the neutrino flux from arp 220 . proton interactions are responsible for the creation of secondary pions , both neutral and charged . while the neutral pions decay into @xmath2-rays , the charged pions decay into a neutrino and a muon which further decays into a secondary electron or positron and two more neutrinos . the spectrum of the first neutrino from the decay of the charged pion is what we calculate here , as the calculation of the spectra of neutrinos produced during muon decay is more complex ( see * ? ? ? the flux of our maximum model is roughly @xmath63 gev @xmath62 s@xmath4 at 0.1 pev , and at this energy , the range of possible models spans an order of magnitude in flux ( see fig . current point source sensitivity limits for the northern sky for icecube are @xmath64 gev @xmath62 s@xmath4 , assuming a spectrum of @xmath65 @xcite . thus , it seems unlikely that arp 220 will be detected as a point source during a similar time frame by icecube . however , extreme ulirgs such as arp 220 should make a significant contribution to a diffuse neutrino background @xcite . in addition to accounting for @xmath2-ray and neutrino emission in arp 220 , we have also take into account the effects of @xmath2@xmath2 absorption due to the intense radiation fields in the nuclei @xcite . at tev energies and above , @xmath2-rays and infrared photons can interact to produce an electron / positron pair @xcite . the resulting electrons will be of tev energies and most of their energy will be lost to emission of synchrotron x - rays @xcite . beginning at @xmath152 5 tev , the opacity for @xmath2@xmath2 absorption in both nuclei is significantly greater than 1 . this results in a steepening of the predicted @xmath2-ray spectrum at high energies ( see fig . we find no such increase in slope in the neutrino flux as the steepening is an effect of interactions between the @xmath2-ray and the ambient radiation field and not the cosmic ray proton population . therefore , in the case of arp 220 and other such ulirgs , the tev @xmath2-ray flux is an unreliable indicator of neutrino flux . if the effects of spectral steepening by @xmath2@xmath2 absorption are accurate , then arp 220 is unlikely to be detected by cta or other ground based cherenkov telescopes above @xmath1510 tev . in applying the yegz models to arp 220 , we find that the central starburst regions of arp 220 are moderate to complete cosmic ray proton calorimeters . as such , the leptonic cosmic ray population is dominated by secondary electrons and positrons . the majority of these secondaries are produced at low energies ( e.g. , * ? ? ? * ) and are likely a major contributor to heating of the ism via ionization @xcite . based on our best - fitting models for the radio spectrum , we make predictions for both the @xmath2-ray and neutrino fluxes . our maximum @xmath2-ray spectrum is a factor of a 2 5 lower than previous predictions by @xcite and less than a factor of 2 lower than those by @xcite . while the predicted @xmath2-ray flux will likely be detected by _ fermi _ in the future , under our model assumptions arp 220 is unlikely to be detected as a high energy neutrino point source with the current icecube observatory . additionally , @xmath2@xmath2 absorption of the tev energy @xmath2-rays make the tev @xmath2-ray flux a poor indicator of the neutrino flux in ulirgs and other such systems with extremely intense infrared radiation fields . in addition , we find that milligauss strength magnetic fields are still necessary to reproduce the observed radio fluxes from the starburst nuclei , even having assumed larger supernova rates than previous models by factors of 2 5 @xcite . differences in assumed volume and radiation field energy density across the various models account for the similar best - fitting magnetic field strengths despite the range in assumed supernova rates . the origins of milligauss strength magnetic fields in extreme starbursts and their impact on the evolution of these systems merit further examination . while the energy density in both magnetic and radiation fields is up from starbursts like m82 by two to three orders of magnitude , the change in the ratio of their energy densities is up by less than an order of magnitude ( see table 3 ) . conversely , we see a much larger change in the ratio of magnetic field energy density to cosmic ray energy density . because the cosmic ray energy density depends on the particle energy loss rate , it does not increase at the same rate as the magnetic and radiation field energy densities ( yoast - hull , gallagher , zweibel , in preparation ) . thus , the magnetic fields exceed energy equipartition with the cosmic rays by more than two orders of magnitude ( see table 3 ) . this work was supported in part by nsf ast-0907837 , nsf phy-0821899 ( to the center for magnetic self - organization in laboratory and astrophysical plasmas ) , and nsf phy-0969061 ( to the icecube collaboration ) . part of this research was carried out during jsg s appointment as a jubileumsprofessor at the chalmers university of technology . we thank susanne aalto , kazushi sakamoto , dave sanders , nick scoville , and eskil varenius for conversations on arp 220 , justin vandenbroucke and reinhard schlickeiser for discussions regarding the modelling , and francis halzen for his help and support . additionally , we thank the referee for their helpful comments . our single - zone model uses a simple solution to the radiative transfer equation of ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) @xmath66 where @xmath67 is the radiative flux prior to absorption , @xmath68 is the radiative flux after absorption , and @xmath69 is the optical depth for either free - free absorption or @xmath2-@xmath2 absorption . this is still the solution for the eastern nucleus and the surrounding torus in the western nucleus . in the western cnd , we must account for a standard emission and absorption region with an additional , external absorbing region . this observed flux is given by @xmath70 where @xmath67 is still the radiative flux prior to absorption , @xmath71 is the optical depth for @xmath2-@xmath2 or free - 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the cores of arp 220 , the closest ultraluminous infrared starburst galaxy , provide an opportunity to study interactions of cosmic rays under extreme conditions . in this paper , we model the populations of cosmic rays produced by supernovae in the central molecular zones of both starburst nuclei . we find that @xmath0 of cosmic rays are absorbed in these regions due to their huge molecular gas contents , and thus , the nuclei of arp 220 nearly complete proton calorimeters . as the cosmic ray protons collide with the interstellar medium , they produce secondary electrons that are also contained within the system and radiate synchrotron emission . using results from @xmath1 tests between the model and the observed radio spectral energy distribution , we predict the emergent @xmath2-ray and high - energy neutrino spectra and find the magnetic field to be at milligauss levels . because of the extremely intense far - infrared radiation fields , the @xmath2-ray spectrum steepens significantly at tev energies due to @xmath2@xmath2 absorption . neutrinos cosmic rays galaxies : individual : arp 220 galaxies : starburst gamma rays : galaxies radio continuum : galaxies

introduction arp 220 physical properties models & results discussion and conclusions acknowledgements two zone models

arxiv

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in communication networks structure and dynamics are tightly coupled . the structure controls the flow of information and is itself shaped by the dynamical process of information exchanged between nodes . in order to reconcile structure and dynamics , a generic model , based on the local interaction between nodes , is considered for the communication in large social networks . in agreement with data from a large human organization , we show that the flow is non - markovian and controlled by the temporal limitations of individuals . we confirm the versatility of our model by predicting simultaneously the degree - dependent node activity , the balance between information input and output of nodes and the degree distribution . finally , we quantify the limitations to network analysis when it is based on data sampled over a finite period of time . limitations on the processing capacities of nodes and links have a profound impact on the flow of information in online communication networks @xcite , the spreading of diseases in human encounter networks @xcite , and in social networks @xcite , where links between interacting individuals can be highly volatile @xcite . it is often assumed that communication takes place in an unrestrained way on a set of established connections , thereby neglecting , that structure and dynamics are interdependent . here we consider the evolution of a network where links form as a result of non - markovian interaction between nodes . in a time - limited environment , communication demands prioritization which is evident from the analysis of correspondence patterns @xcite . hence , information flow on a network is a result of individuals choices which are influenced by the state of surrounding nodes . in natural @xcite and online @xcite social networks , the nodes activity is a non - trivial function of their degree . the activity level can be quantified by the number of social relationships simultaneously maintained by an individual . this number has been suggested to reflect basic cognitive capabilities of primates @xcite and humans @xcite . here we model a network of individuals acting under time constraints and compare with a complete dataset of email communication in a large organization . the model is discussed in the context of other communication networks . we predict the information processing capacity of individuals as well as the structure of the network that they form . we use representative communication data from a large social organization , the university of oslo . the data comprise a complete time - ordered list of @xmath0 emails between 5600 employees , 30 000 students and approximately @xmath1 people outside the organization over a period of three months ( sep - nov 2010 ) . the email content was not recorded and identities of individuals were encrypted . we limit the influence of unsolicited bulk emails by disregarding those simultaneously sent to more than five recipients . however , the results are not sensitive to the filtering of bulk emails @xcite . previous work on email data has considered static network structures @xcite . _ results _ we show that the communication is non - markovian by comparing random and directed information flow : ( i ) random flow is given by random walks on the network . the walker follows an empirical time - independent jump - probability @xmath2 from node @xmath3 to node @xmath4 . the sum is taken over all nodes and @xmath5 is the number of emails sent from @xmath3 to @xmath4 during the timespan of the data . ( ii ) directed flow is given by the chronological email exchange . starting from a random node @xmath3 , we wait for @xmath3 to send an email , say to @xmath4 . we then jump to @xmath4 and wait for the next message @xmath4 sends either back to @xmath3 or to a new node @xmath6 . repeating this , we obtain a finite trajectory within the timespan of the data . the number of unique nodes visited by the directed and random flow as function of the number of jumps are compared by averaging over trajectories originating from all nodes ( fig . [ fig : markov ] ) . on average , directed flow visits relatively fewer nodes than random flow , indicating a significant correlation between sent and received messages . '' . the solid line is a best fit by eq . ( [ eq : delta ] ) . the dotted lines mark the peak and the dashed diagonal line shows @xmath7 . inset : out - degree distribution for model and empirical data . the dashed line denotes the scale - break @xmath8 . mean degree is @xmath9 ( twitter data yields a mean degree of @xmath10 and a similar exponent for the degree distribution @xcite ) . note the double - log scales.,title="fig : " ] + our model requires nodes to perform a trade - off between replying to others and initiating new conversations . specifically , consider @xmath11 nodes , each initially connected to one other node . the nodes have a limited capacity and can send a maximum of @xmath12 messages in a timestep @xmath13 day . the dynamics follows from three possible actions for a node @xmath3 of out - degree @xmath14 : ( a ) @xmath3 processes received emails and if @xmath3 has sent less than @xmath12 messages , any received email is replied to with a probability proportional to the sender s degree . emails not replied to within @xmath15 are subsequently deleted . in total , @xmath16 replies are sent by this action . \(b ) if less than @xmath12 emails have been sent in ( a ) , the remaining capacity @xmath17 is available for sending messages , called @xmath18 , to previously established contacts . the probability of sending a message to a contact is given by a constant @xmath19 . hence , granted sufficient capacity on average @xmath20 messages are initiated by @xmath3 . nodes with low @xmath14 will generally not reach their full capacity . \(c ) nodes establish new contacts by sending requests with a probability @xmath21 . the probability that a request is sent to a node @xmath4 is proportional to the degree of @xmath4 , @xmath22 . a link is established between @xmath3 and @xmath4 , if @xmath4 in the next timestep according to ( a ) replies to @xmath3 . in reality , contacts might as well be established by face - to - face encounters , i.e. via channels not recorded explicitly in our data . the total number of messages @xmath23 sent by a node in @xmath15 is the sum @xmath24 . analogously , messages received by a node in the same timestep are termed @xmath25 . nodes have an average lifetime @xmath26 and are therefore removed from the network with a probability @xmath27 . for every node removed , a new node with a single random connection to an existing node is introduced . @xmath26 is estimated to be @xmath28 years from the known mean email user turnover time in the organization . the parameters @xmath19 , @xmath21 and @xmath12 are determined below . according to ( c ) , a link is established between @xmath3 and @xmath4 if one of the nodes sends a message to the other and receives a reply . the probability , @xmath29 , that a message is sent from @xmath3 to @xmath4 in @xmath15 is proportional to @xmath22 , @xmath30 where we in the approximation assume that @xmath31 . according to ( a ) , the mean number of requests that @xmath4 receives during a timestep is proportional to @xmath21 and @xmath22 . the probability for @xmath4 to reply to a request from nodes of degree @xmath6 is proportional to @xmath32 , where @xmath33 is a constant and @xmath34 is the number of nodes with degree @xmath6 . the number of replies written by @xmath4 is the product of eq . ( [ eq1 ] ) and the integral over nodes @xmath35 since nodes reply to requests and therefore establish new links with a probability proportional to the sender degree , @xmath36 , the mean degree @xmath37 of a node s contacts is @xmath38 , a number generally larger than the mean degree @xmath39 ( fig . [ fig : rec_degree_vs_k ] ) . ) and weighted by the number of messages sent to recipients ( @xmath40 ) . the horizontal line shows @xmath41 . the curves marked by `` @xmath42 '' and `` @xmath43 '' are analogous to the unweighted case but for half , respectively , one quarter of the observational period . dashed lines show projection of nodes with two values of @xmath6 for a varying observation window . note the double - log scale . ] consequently the average degree - increase of nodes of degree @xmath6 per timestep becomes @xmath44 . the factor of 2 reflects the symmetry of sending and replying . the rate of losing links is inversely proportional to @xmath26 , @xmath45 . hence , the net degree - growth rate becomes @xmath46 , where @xmath47 . as long as a node has sufficient capacity to reply to all requests its degree increases approximately exponentially , @xmath48 . the degree distribution follows from the consideration that during @xmath15 , a fraction of nodes @xmath34 of degree @xmath6 changes their degree , @xmath49 $ ] , and a fraction @xmath50 is removed . a continuum - limit approximation yields @xmath51-\frac{n(k)}{\tau}\;. \label{eq : time_evol}\ ] ] the steady - state solution has the form @xmath52 , where @xmath53 . the constant @xmath54 is fixed by integrating eq . ( [ eq : time_evol ] ) over @xmath6 and by demanding that the total number of nodes @xmath55 be constant . this yields @xmath56 . the condition @xmath57 bounds the power - law exponent : @xmath58 . the data yield @xmath59 ( fig . [ fig : sent_vs_rec ] inset ) . so far we have assumed that nodes have infinite capacity . as a node s degree increases , it receives more messages and this assumption becomes invalid . consider the number of messages received by @xmath3 per timestep . contact requests from other nodes amount to @xmath60 messages . the senders of these messages are drawn from a distribution @xmath61 . the probability for @xmath3 to receive a message from its contacts is proportional to @xmath19 and @xmath14 , hence @xmath62 . analogously , as defined in ( a ) , @xmath3 issues @xmath63 requests to recipients distributed according to @xmath64 ( where @xmath65 ) due to the weighting of probabilities by the recipient degree . in the same timestep @xmath3 sends @xmath66 messages to its contacts . finally we consider back - and - forth communication . for every message sent by @xmath3 to @xmath4 , a response is returned with a probability @xmath67 ( eq . [ eq : prob_send_reply ] ) . in steady - state , the number of messages sent is identical for all timesteps and therefore @xmath3 receives @xmath68 replies to messages sent in the previous timestep . @xmath16 is the number of messages @xmath3 sends in response to messages received from others which again is a sum over contributions from the actions ( a)-(c ) : @xmath69 the terms on the right are respectively , requests from any node in the network ( distributed as @xmath70 ) , messages from existing contacts ( distributed as @xmath71 ) , and back - and - forth messages ( distributed as @xmath72 ) . each iteration of back - and - forth communication acts as a shift in the distribution of recipients relative to the distribution of senders @xmath73 . the distribution @xmath72 accounts for all high - order shifts . to close the equations for @xmath74 and @xmath16 , we use that the reply probability for each iteration is reduced by a factor @xmath33 to approximate @xmath75 . inserting eq . ( [ eq : alpha2 ] ) , @xmath76 and @xmath77 in eq . ( [ eq5 ] ) yields @xmath78 where we introduce @xmath79 . summing over @xmath80 , @xmath81 and @xmath82 we get @xmath83 here the first three terms ( referred to as @xmath84 ) are messages sent to recipients selected according to @xmath71 and with mean degree @xmath37 . the other terms , @xmath85 , are messages to recipients distributed according to the higher order distribution @xmath86 which has a mean @xmath87 and contribute significantly only for large @xmath14 . the mean of the weighted recipient degree ( weighted by number of messages received ) is @xmath88 , which departs from @xmath37 when @xmath85 becomes appreciable ( fig . [ fig : rec_degree_vs_k ] ) . for low @xmath14 ( @xmath89 ) , the ratio of sent to received messages becomes @xmath90 . conversely , @xmath91 when @xmath92 , hence an average node has a `` balanced '' email account . when @xmath14 becomes larger than @xmath39 , @xmath3 will increasingly receive requests and responses to its messages ( fig . [ fig : sent_vs_rec ] ) . the _ dunbar number _ @xmath93 is the degree where @xmath23 reaches the capacity limit ( @xmath94 ) and @xmath95 is maximal . the scale break in the degree distribution ( @xmath8 ) , fig . [ fig : sent_vs_rec ] ( inset ) , and @xmath96 , fig . [ fig : dunbar ] , nearly coincide . in fact @xmath97 is related to @xmath98 because nodes beyond @xmath98 have a reduced probability to form new links . to determine @xmath97 , consider the evolution of the nodes degree in the limit where all capacity is used for replying , hence @xmath99 . using that @xmath100 , we get @xmath101 which in turn yields @xmath102 . @xmath97 is found by solving this implicit equation . @xmath98 then follows from eq . ( [ eq : delta ] ) . , @xmath103 ( eq . [ eq : delta ] ) and @xmath95 is a superposition of a term @xmath104 due to the final quadratic term and a decaying term @xmath105 from the constant . at @xmath106 , nodes limited to @xmath12 messages per day , hence @xmath107 . ] the parameters @xmath108 , @xmath109 and @xmath110 are determined by the data in fig . [ fig : sent_vs_rec ] . from @xmath21 and @xmath111 we obtain @xmath112 . larger @xmath12 increases the limit of @xmath23 . @xmath21 is constrained by the offset at low @xmath113 and @xmath19 effects the skewness of the curve which follows from analysis of eqs . ( [ eq : alpha2 ] ) and ( [ eq : delta ] ) . fig . [ fig : dunbar ] shows the model prediction of @xmath114 and the corresponding email data . we complement our analysis with numerical computations . using a large number of nodes , @xmath115 , we iterate actions ( a)-(c ) until steady - state is reached . while the mean - field prediction ( figs . [ fig : rec_degree_vs_k ] and [ fig : dunbar ] ) is close to the numerical solution , some differences exist , e.g. at small @xmath6 , @xmath116 is not a strict power - law in the numerical solution due to the discreteness of @xmath6 . further , the simulation gives a smooth peak in @xmath95 ( fig . [ fig : dunbar ] ) which is narrower than in the empirical data . this is due to slight overestimation of the repeated back - and - forth communication between well - connected nodes ( @xmath117 ) relative to the data . we have also simulated the information flow ( fig . [ fig : markov ] ) and achieve similar results . finally , the average local clustering coefficient of the empirical and simulated networks is relatively small , @xmath118 for both ( similar clustering coefficient @xmath119 @xcite and @xmath120 to @xmath121 have been reported for other communication networks @xcite ) . we further checked the robustness of the model to variations @xcite . _ discussion _ the data were recorded over three months and the communication network is therefore a finite - time projection of the real network . the projection reduces the number of links . more active links will more likely persist through the projection than less active links . fig . [ fig : rec_degree_vs_k ] shows the mean recipient degree @xmath122 as function of the sender degree @xmath14 for three observation time intervals . consider again eq . ( [ eq : delta ] ) and remember that recipients of the @xmath84 ( @xmath85 ) messages are distributed as @xmath71 ( @xmath86 ) . when observing only a single day , the probability for an out - link between @xmath3 to @xmath4 not to be active is @xmath123 . for @xmath124 days we obtain @xmath125 . to produce the projected curves in fig . [ fig : rec_degree_vs_k ] , @xmath126 is applied to both axes , @xmath6 and @xmath127 . averaging w.r.t . all recipients @xmath4 ( distributed as @xmath71 ) , the projected sender out - degree becomes @xmath128 . similarly one can consider the projection of the mean recipient degree leading to a similar reduction in the degree for finite - time data . for example , consider the data for the quarter period ( @xmath129 ) in fig . [ fig : rec_degree_vs_k ] . we have @xmath130 and therefore @xmath131 hence less than half the links persist . _ concluding remarks _ the finite capacity of agents in social networks induces an upper limit on the number of possible interactions @xcite . we propose a comprehensive model that reconciles structure and dynamics of networks with finite capacity agents that dynamically form or lose links . in agreement with a complete set of email data and results from other social networks @xcite , our model predicts a scale - free degree distribution up to a distinct scale - break induced by the capacity limit . further , as agents gain importance in the network , the per - link - activity first increases with node - degree , peaks at intermediate degrees and declines at large degrees . the model and data therefore support the hypothesis of a general limit on the number ( 150 - 250 ) of active social relations that an individual can maintain @xcite and is in agreement with empirical observations on social networks @xcite .

acknowledgments

arxiv

the author expresses special thanks to h. horiuchi and a. tohsaki for their valuable and helpful discussion and suggestions on the present work . the fruitful discussion with m. itoh is highly appreciated . thanks are also due to g. rpke , p. schuck , t. yamada , and b. zhou for their stimulating discussion . this work was partially performed with the financial support by hpci strategic program of japanese mext , jsps kakenhi grant number 25400288 , and riken incentive research projects . 99 k. wildermuth and y. c. tang , _ a unified theory of the nucleus _ ( vieweg , braunschweig , 1977 ) . f. hoyle , astrophys . . ser . * 1 * , 121 ( 1954 ) . c. w. cook _ _ , phys . rev . * 107 * , 508 ( 1957 ) . h. horiuchi , prog . phys . * 51 * , 1266 ( 1974 ) ; * 53 * , 447 ( 1975 ) . y. fukushima _ et al_. , suppl . of j. phys . japan , * 44 * , 225 ( 1978 ) ; m. kamimura , nucl . phys . a * 351 * , 456 ( 1981 ) . e. uegaki , s. okabe , y. abe , and h. tanaka , prog . * 57 * , 1262 ( 1977 ) ; * 62 * , 1621 ( 1979 ) . p. descouvemont and d. baye , phys . c * 36 * , 54 ( 1987 ) . h. morinaga , phys . rev . * 101 * , 254 ( 1956 ) ; phys . lett . * 21 * , 78 ( 1966 ) . a. tohsaki , h. horiuchi , p. schuck , and g. rpke , phys . * 87 * , 192501 ( 2001 ) . y. funaki , a. tohsaki , h. horiuchi , p. schuck , and g. rpke , phys . c * 67 * , 051306(r ) ( 2003 ) . et al . _ , a * 738 * , 268 ( 2004 ) . m. freer _ et al . c * 80 * , 041303(r ) ( 2009 ) . c * 84 * , 054308 ( 2011 ) . fynbo and m. freer , physics * 4 * , 94 ( 2011 ) . et al . _ , c * 84 * , 027304 ( 2011 ) . et al . _ , lett . * 110 * , 152502 ( 2013 ) . m. freer _ et al . c * 83 * , 034314 ( 2011 ) . r. bijker and f. iachello , phys . c * 61 * , 067305 ( 2000 ) ; ann . ( amsterdam ) * 298 * , 334 ( 2002 ) . d. j. mari - lambarri , r. bijker , m. freer , m. gai , tz . kokalova , d.j . parker , and c. wheldon , phys . * 113 * , 012502 ( 2014 ) . c. kurokawa and k. kat@xmath93 , phys . c * 71 * , 021301 ( 2005 ) ; nucl . a * 792 * , 87 ( 2007 ) . s. ohtsubo , y. fukushima , m. kamimura , and e. hiyama , prog . exp . phys . 2013 , 073d02 . y. kanada - enyo , prog . . phys . * 117 * , 655 ( 2007 ) . m. chernykh , h. feldmeier , t. neff , p. von neumann - cosel , and a. richter , phys . lett . * 98 * , 032501 ( 2007 ) . m. itoh _ _ , j. phys . : conf . ser . * 436 * , 012006 ( 2013 ) . b. zhou , y. funaki , a. tohsaki , h. horiuchi , and z. z. ren , arxiv : 1408.2920 . y. funaki , a. tohsaki , h. horiuchi , p. schuck , and g. rpke , eur . j. a * 24 * , 321 ( 2005 ) . y. funaki , h. horiuchi , w. von oertzen , g. rpke , p. schuck , a. tohsaki , and t. yamada , phys . c * 80 * , 064326 ( 2009 ) . b. zhou , y. funaki , h. horiuchi , z. z. ren , g. rpke , p. schuck , a. tohsaki , c. xu , and t. yamada , phys . * 110 * , 262501 ( 2013 ) . t. suhara , y. funaki , b. zhou , h. horiuchi , and a. tohsaki , phys . lett . * 112 * , 062501 ( 2014 ) . y. funaki , t. yamada , e. hiyama , b. zhou , and k. ikeda , arxiv : 1405.6067 . y. funaki , h. horiuchi , and a. tohsaki , prog . * 115 * , 115 ( 2006 ) . a. b. volkov , nucl . a * 74 * , 33 ( 1965 ) . y. fujiwara , h. horiuchi , k. ikeda , m. kamimura , k. kat@xmath93 , y. suzuki , and e. uegaki , suppl . * 68 * , 29 ( 1980 ) . a. m. lane and r. g. thomas , rev . * 30 * , 257 ( 1958 ) . y. funaki , t. yamada , h. horiuchi , g. rpke , p. schuck , and a. tohsaki , phys . * 101 * , 082502 ( 2008 ) . y. kanada - enyo , phys . c * 89 * , 024302 ( 2014 ) . s. ohkubo , y. hirabayashi , phys . b * 684 * , 127 ( 2010 ) . y. funaki , t. yamada , h. horiuchi , g. rpke , p. schuck , and a. tohsaki , suppl . . phys . * 196 * , 439 ( 2012 ) . t. yamada , y. funaki , h. horiuchi , k. ikeda , and a. tohsaki , prog . theor . phys . * 120 * , 1139 ( 2008 ) .

the excited states in @xmath0 are investigated by using an extended version of the so - called tohsaki - horiuchi- schuck - rpke ( thsr ) wave function , where both the @xmath1 condensate and @xmath2 cluster asymptotic configurations are included . a new method is also used to resolve spurious continuum coupling with physical states . we focus on the structures of the `` hoyle band '' states , @xmath3 , and @xmath4 states , which are recently observed above the hoyle state and of the @xmath5 and @xmath6 states , which are also quite recently identified in experiment . their resonance parameters and decay properties are reasonably reproduced . all these states have gaslike configurations of the @xmath1 clusters with larger root mean square radii than that of the hoyle state . the hoyle band is not simply considered to be the @xmath7 rotation as suggested by previous cluster model calculations , nor to be a rotation of a rigid - body triangle - shaped object composed of the @xmath1 particles . this is mainly due to the specificity of the hoyle state , which has the @xmath1 condensate structure and gives a rise to the @xmath5 state with a prominent @xmath8 structure as a result of very strong monopole excitation from the hoyle state . nuclear clustering is one of the fundamental degrees of freedom in nuclear excitation @xcite . the hoyle state , the second @xmath9 state at @xmath10 mev in @xmath0 , as a typical example of the cluster states , has a long history ever since it was predicted by f. hoyle @xcite and subsequently observed by cook _ et al . _ @xcite as a key state in a synthesis of @xmath0 in stellar evolution . the microscopic and semi - microscopic cluster models have clarified that the hoyle state has the structure of the @xmath11 particle loosely coupling in an @xmath12-wave with the @xmath13 core @xcite , not like a linear - chain structure of the @xmath1 particles proposed by morinaga in 1950 s @xcite . in the last decade , however , the aspect of the @xmath11 condensate , where the @xmath1 clusters occupy an identical @xmath12-orbit , has triggered a great interest , since the so - called tohsaki - horiuchi - schuck - rpke ( thsr ) wave function @xcite , which has the @xmath11 condensate character , was shown to be equivalent to the hoyle state wave function obtained by solving the equations of the full @xmath1 resonating group method ( rgm ) or generator coordinate method ( gcm ) @xcite . on the other hand , the excited states of the hoyle state have been highlighted by recent great developments in experimental studies . the second @xmath14 state @xmath15 , which had been predicted at a few mev above the hoyle state by the cluster model calculations , was recently confirmed by many experiments @xcite . the gcm and rgm calculations propose that the @xmath3 state is built on the hoyle state as a rotational member with a @xmath8 configuration . freer _ et al . _ quite recently reported a new observation of the @xmath16 state at @xmath17 mev , which they consider to compose the `` hoyle band '' @xcite , together with the @xmath18 and @xmath3 states . it is proposed that this band is formed by a rotation of a rigid @xmath1 cluster structure with an equilateral triangle shape based on @xmath19 symmetry @xcite , which is , however , not consistent with the picture of loosely coupled @xmath8 structure or the @xmath1 gaslike structure . besides the @xmath14 and @xmath16 states , a @xmath20 state at @xmath21 mev with a broad width , @xmath22 mev , has been known for a long time . however , quite recently itoh _ et al . _ decomposed the broad @xmath20 state into the @xmath5 and @xmath6 states at @xmath23 mev and @xmath24 mev , with the widths of @xmath25 mev and @xmath26 mev , respectively @xcite . this observation of the two @xmath20 states is consistent with theoretical prediction done by using the orthogonality condition model ( ocm ) combined with the complex scaling method ( csm ) and the analytical continuation of coupling constant ( accc ) method @xcite . this was later on confirmed by another theoretical calculation using the ocm and csm with higher numerical accuracy @xcite . on the other hand , in the antisymmetrized molecular dynamics ( amd ) @xcite , fermionic molecular dynamics ( fmd ) @xcite , and gcm calculations @xcite , the observed @xmath5 state seems to be missing . the @xmath5 state given by the amd and fmd , which may correspond to the observed @xmath6 state , is dominated by a linear - chain - like configuration of the @xmath1 clusters and is not inconsistent with the @xmath5 state obtained by the gcm calculation @xcite , or with the @xmath6 state in ref . @xcite , where @xmath27_0 $ ] configuration is dominant . it should also be mentioned that the authors in ref . @xcite claimed that the @xmath5 state has an @xmath12-wave dominant structure with more dilute density than that of the hoyle state . these are also consistent with the observed decay properties of the @xmath5 and @xmath6 states that the former only decays into the @xmath28_0 $ ] channel and the latter decays into the @xmath27_0 $ ] channel with a sizable partial @xmath11-decay width @xcite . in this letter , we investigate the structures of the positive parity excited states above the @xmath1 threshold by using an extended version of the thsr wave function @xcite so as to include @xmath29 asymptotic configurations with a treatment of resonances . in particular , we focus on the `` hoyle band '' ( the @xmath18 , @xmath3 , and @xmath4 states state at @xmath17 mev as the @xmath4 state , for simplicity , though it is located lower than the @xmath16 state at @xmath30 mev which forms the ground state rotational band . ] ) , and the @xmath5 and @xmath6 states , together with the corresponding experimental data , though we also obtained some other positive parity excited states . the extended version of the thsr wave function is written as follows : @xmath31 , \label{eq:1}\end{aligned}\ ] ] where the @xmath32 is an intrinsic wave function of the @xmath11 particle , where the @xmath33 configuration of the four nucleons is assumed with the size parameter @xmath34 , which is kept fixed at @xmath35 fm as almost the same value as at free space . @xmath36 is the jacobi coordinates between the @xmath1 particles and @xmath37 , for @xmath38 . this is a fully microscopic wave function and every nucleons are antisymmetrized by @xmath39 . @xmath40 is a usual angular - momentum - projection operator . this wave function is characterized by the parameters @xmath41 and @xmath42 , which correspond to the sizes of the @xmath43 core and the remaining @xmath11 particle center - of - mass ( c.o.m . ) motion , respectively . in the subsequent calculations , the axial symmetric deformation is assumed , i.e. @xmath44 @xmath45 , for simplicity . we should note that the case of @xmath46 results in the original thsr wave function , where the c.o.m . motions of the @xmath1 particles relative to the total c.o.m . position are condensed into a lowest energy @xmath47 orbit , reflecting the bosonic feature @xcite . thus this new thsr wave function is a natural extension of the original version , so that taking @xmath48 allows for the @xmath2 cluster structure , deviating from the identical @xmath1-cluster motion for @xmath46 . this new wave function still gives gaslike cluster structure , as the original thsr wave function does , not being featured by the relative distance parameter between the @xmath43 and @xmath11 clusters . it should also be mentioned that the thsr - type wave functions were recently shown to give the best description for various cluster states such as the @xmath49 inversion doublet band in @xmath50 @xcite , @xmath1- and @xmath51-linear - chain states @xcite , and @xmath52 cluster states in @xmath53 @xcite , etc . for the excited states above the @xmath1 threshold , it is well known that the application of the bound state approximation gives accidental mixing between spurious continuum states and resonances . by using the fact that the root mean square ( r.m.s . ) radii of spurious continuum states are calculated to be extremely large within the bound state approximation , we developed a new method to remove the spurious components @xcite . first we diagonalize the operator of mean square radius as follows : @xmath54 where @xmath55 is the total c.o.m . position . we then remove out of the present model space the eigenstates belonging to unphysically large eigenvalues . by taking the following bases , @xmath56 with @xmath57 satisfying @xmath58 , we diagonalize hamiltonian as follows : @xmath59 for hamiltonian , we adopt volkov no . 2 force @xcite , with the strength parameters slightly modified @xcite , as effective nucleon - nucleon interaction . the cutoff radius is now taken to be @xmath60 fm . for diagonalizing the operator of r.m.s . radius in eq . ( [ eq : cutoff1 ] ) , we adopt @xmath61 mesh points for the four - parameter set , @xmath62 , up to around @xmath63 fm . since the present extended thsr wave function can include @xmath2 asymptotic form by taking the large values of the two width parameters @xmath41 and @xmath42 , the @xmath2 continuum components , as well as the @xmath1 continuum components , can be successfully removed by imposing the cut off for the mean square radius @xmath58 . the more details will be shown in a forthcoming full paper . although we could not obtain the excited states except for the @xmath18 and @xmath3 states by using the original thsr wave function @xcite , we can now obtain the other observed @xmath5 , @xmath6 , and @xmath4 states by using the present extended thsr wave function with a treatment of resonances . since all these states are resonance states above the @xmath1 threshold , we then calculate the partial widths of the @xmath11 particle decaying into @xmath64_j$ ] channel , which we simply denote as @xmath65_j$ ] , based on the @xmath66-matrix theory @xcite , where we use the following relations , @xmath67_j}= 2p_l(ka ) \gamma^2_{[i , l]_j } , \ \gamma^2_{[i , l]_j}=\frac{\hbar^2}{2\mu a } |a{\cal y}_{[i , l]_j}(a)|^2,\label{eq : width}\ ] ] where @xmath68 is the penetrability calculated from the coulomb wave functions , and @xmath69 , @xmath70 , and @xmath71 are the wave number of the relative motion , the channel radius , and the reduced mass , respectively . @xmath72_j}(r)$ ] is the @xmath11 reduced width amplitude ( rwa ) defined below , @xmath73_j}(r)\hspace{-0.1cm}=\hspace{-0.1cm}\sqrt{\frac{12!}{4!8!}}\langle [ \phi_{i}(\be),y_{l}(\vc{\hat \xi}_2)]_{jm}\frac{\delta(\xi_2-r)}{\xi_2 ^ 2}\phi(\alpha ) | \psi^{(\lambda)}_{jm } \rangle , \label{eq : rwa}\ ] ] where @xmath74 is the eigenfunction in eq . ( [ eq : eigenwf ] ) , @xmath75 . [ cols="^,^,^,^,^,^,^,^ " , ] we show in table [ tab:3 ] the calculated @xmath12-factors of the @xmath76 components , which can be defined below , @xmath77}(j)=\int dr \big ( r{\cal y}_{[i , l]_j}(r ) \big)^2 . \label{eq : sfact}\ ] ] we can see that except for the @xmath6 state , all the states have the largest contribution from the @xmath78_j$ ] channel . this supports the idea of @xmath2 rotation for the hoyle band , where the @xmath43 core is in the @xmath20 ground state . on the other hand , the hoyle state is considered to be the @xmath1 condensate state , where the @xmath1 clusters mutually move in an identical @xmath12-wave . since the ground state of @xmath43 is composed of weakly interacting @xmath79 clusters coupled loosely in a relative @xmath12-wave , it is natural that the hoyle state , with the @xmath11 condensate structure , also has a large overlap with the @xmath8 structure . this is the same situation as for the @xmath51 condensate state in @xmath80 discussed in refs . @xcite , which has a large overlap with the @xmath81 structure . however , the @xmath1 condensate structure in the hoyle state is not the same as the usual @xmath8 rotation , in which the remaining @xmath11 cluster orbits outside the @xmath43 core . namely in the hoyle state , the remaining @xmath11 cluster also orbits inside the @xmath43 core and the independent @xmath1-cluster motion in an identical @xmath47-orbit is realized . as a result , the hoyle state gains an extra binding , and hence its energy position is considered to be made lower than the @xmath82 line , as shown in fig . [ fig:1 ] . the same effect is also argued in the study of the @xmath51 condensate and @xmath81 rotational band @xcite , where the @xmath51 condensate is mentioned as `` complete condensate '' and the @xmath81 state as `` local condensate '' . due to the existence of the `` complete condensate '' , a higher @xmath20 excited state , which has the prominent @xmath8 structure , with the remaining @xmath11 cluster orbiting outside the @xmath43 core , appears as a higher nodal state , that is the @xmath5 state . in fact , we can see in table [ tab:2 ] that the @xmath5 state is strongly connected with the hoyle state by a monopole excitation . the calculated strength @xmath83 @xmath84 is much larger than the other transitions , in spite of the fact that the @xmath85 strength between the hoyle and ground states @xmath86 @xmath84 is still strong enough as to be comparable to the single nucleon strength @xcite . _ j=[0,0]_0 $ ] channel , @xmath87_0}(r)$ ] in eq . ( [ eq : rwa ] ) , for the @xmath88 , @xmath18 , and @xmath5 states . ] in fig . [ fig:2 ] , the rwas of @xmath89_0 $ ] channel for the @xmath18 and @xmath5 states are shown together with that for the ground state . while the rwa for the ground state has two nodes , that for the @xmath5 state has four nodes and for the hoyle state the nodal behaviour almost disappears and only a remnant of three nodes can be seen as an oscillatory behaviour . since the outmost nodal position corresponds to a radius of repulsive core between the core @xmath43 and the @xmath11 cluster , due to the effect of the pauli principle , the disappearance of the nodes for the hoyle state indicates a dissolution of the @xmath43 core , and hence formation of the @xmath1 condensate . on the other hand , the @xmath5 state , which is excited from the hoyle state by the monopole transition , recovers the distinct nodal behaviour and , with one additional node , forms a higher nodal @xmath8 structure . in table [ tab:3 ] , the @xmath6 state is shown to have the component of @xmath90_0 $ ] channel dominantly , which gives a rise to non - negligible partial decay width into this channel , consistently with the experimental information , as mentioned above . we also mention that the @xmath3 state also includes non - negligible mixture from the @xmath91_2 $ ] channel and @xmath4 states from the @xmath90_4 $ ] channel and smaller amount from the @xmath92_4 $ ] channel . these mixtures also deviate the @xmath3 and @xmath4 states from a pure @xmath29 rotational structure . we will discuss this point of view in a forthcoming paper . in conclusion , the use of the extended thsr wave function allows us to obtain the wave functions of the hoyle band and @xmath5 and @xmath6 states , which are recently confirmed by experiments . the calculated @xmath11-decay widths and the decay properties of these states are in good agreement with the experimental data . all these states are shown to have large r.m.s . radii and hence gaslike @xmath1-cluster structures . the @xmath18 , @xmath3 , and @xmath4 states are not considered to form a simple @xmath8 rotational band , due to the specificity of the hoyle state with the @xmath1 condensate feature , which allows the @xmath5 state to have a prominent @xmath8 structure as a result of the strong monopole excitation .

acknowledgements

arxiv

i zw 18 remains the most metal - poor blue compact dwarf ( bcd ) galaxy known since its discovery by sargent & searle ( 1970 ) . later spectroscopic observations by searle & sargent ( 1972 ) , lequeux et al . ( 1979 ) , french ( 1980 ) , kinman & davidson ( 1981 ) , pagel et al . ( 1992 ) , skillman & kennicutt ( 1993 ) , martin ( 1996 ) , izotov & thuan ( 1998 ) , vlchez & iglesias - pramo ( 1998 ) , izotov & thuan ( 1999 ) and izotov et al . ( 1999 ) have confirmed its low metallicity with an oxygen abundance of only @xmath5 1/50 the solar value . zwicky ( 1966 ) described i zw 18 as a double system of compact galaxies , which are in fact two bright knots of star formation with an angular separation of 58 . these two star - forming regions ( fig . [ fig1 ] ) are referred to as the brighter northwest ( nw ) and fainter southeast ( se ) components . later studies by davidson , kinman & friedman ( 1989 ) and dufour & hester ( 1990 ) have revealed a more complex optical morphology with additional diffuse features . the most prominent diffuse feature , hereafter i zw 18c ( fig . [ fig1 ] ) , is a blue irregular star - forming region @xmath5 22 northwest of the nw component . dufour , esteban & castaeda ( 1996a ) , izotov & thuan ( 1998 ) and van zee et al . ( 1998 ) have shown i zw 18c to have a systemic radial velocity equal to that of the ionized gas in the nw and se components , thus establishing its physical association to i zw 18 . furthermore , van zee et al . ( 1998 ) have shown that i zw 18c is embedded in a common h i envelope with the nw and se components . many studies have been focused on the evolutionary state of i zw 18 . searle & sargent ( 1972 ) and hunter & thronson ( 1995 ) have suggested that it may be a young galaxy , recently undergoing its first burst of star formation . the latter authors concluded from _ hubble space telescope _ ( _ hst _ ) images that the colors of the diffuse unresolved component surrounding the se and nw regions are consistent with a population of b and early a stars , i.e. with no evidence for older stars . ongoing massive star formation in i zw 18 is implied by the discovery of a population of wolf - rayet stars in the nw component ( izotov et al . 1997a ; legrand et al . 1997 ) . from the analysis of color - magnitude diagram ( cmd ) based on _ hst _ images , dufour et al . ( 1996b ) concluded that star formation in i zw 18 began at least 30 50 myr ago and continuing to the present . martin ( 1996 ) and dufour et al . ( 1996b ) have discussed the properties of expanding superbubbles of ionized gas driven by supernova explosions and have inferred dynamical ages of 15 27 myr and 13 15 myr respectively . recently , aloisi , tosi & greggio ( 1999 ) have discussed the star formation history in i zw 18 using the same _ hst _ wfpc2 archival data ( i.e. those by hunter & thronson ( 1995 ) and dufour et al . ( 1996b ) ) . they compared observed cmds and luminosity functions with synthetic ones and concluded that there were two episodes of star formation in i zw 18 , the first one occuring over the last 0.5 1 gyr , an age more than 10 times larger than that derived by dufour et al . ( 1996b ) , and the second one with more intense activity taking place between 15 and 20 myr ago . no star formation has occurred within the last 15 myr . stlin ( 2000 ) from _ hst _ nicmos @xmath10 and @xmath11 observations concluded that a 1 5 gyr old stellar population is present . the component i zw 18c has not been studied in such detail mainly because of its faintness . its flux - calibrated optical spectra ( izotov & thuan 1998 ; van zee et al . 1998 ) reveal a blue continuum with weak balmer absorption features and faint h@xmath0 and h@xmath1 in emission . such spectral features suggest that the h ii region is probably ionized by a population of early b stars . dufour et al . ( 1996b ) found in a @xmath12 vs. @xmath13 cmd analysis of i zw 18c a well - defined upper stellar main sequence indicating an age of the blue stars of @xmath5 40 myr . however , numerous faint red stars were also present in the cmd , implying an age of 100 300 myr . those authors concluded that i zw 18c consists of an older stellar population with an age of several hundred myr , but which has experienced recently a modest starburst in its southeastern half as evidenced by the presence of blue stars and h@xmath0 emission . aloisi et al . ( 1999 ) estimated an age for i zw 18c not exceeding 0.2 gyr . we use here high signal - to - noise 4.5 m mmt , 4 m kpno and keck ii spectroscopy to study the evolutionary status of i zw 18c . we also discuss the nature of the extended emission in the outermost regions of i zw 18 . our motivation for this study is the following . until now , age estimates for the stellar populations in i zw 18c are based solely on _ hst _ cmds . while in principle cmd studies are powerful tools for studying stellar populations , they critically depend on the adopted distance to the galaxy and the interstellar extinction , which are a priori unknown . we use here distance - independent techniques based on spectroscopic observations to derive the age of the stellar populations in i zw 18c . concerning the outermost regions of i zw 18 , while gaseous emission is an important contributor to the total light in the vicinity of the star - forming regions , there was no evidence for extended stellar emission at distances as far as 20 from the h ii regions ( e.g. , dufour et al . 1996b ; izotov et al . however , in some recent papers ( e.g. , legrand 2000 ; legrand et al . 2000 ; kunth & stlin 2000 ) such an old extended stellar population has been postulated . we use deep mmt and keck ii spectroscopic observations to clarify the origin of the extended emission and estimate the contribution of the ionized gas to it . the observations and data reduction are described in sect . the properties of the stellar population in i zw 18c are discussed in sect . 3 . in sect . 4 we discuss the properties of the outlying regions of i zw 18 . our results are summarized in sect . the keck ii spectroscopic observations of i zw 18c were carried out on january 9 , 2000 with the low - resolution imaging spectrograph ( lris ) ( oke et al . 1995 ) , using the 300 groove mm@xmath14 grating which provides a dispersion 2.52 pixel@xmath14 and a spectral resolution of about 8 in first order . the slit was 1@xmath15180 , centered on the brightest central region ( region c ) of i zw 18c and oriented with a position angle p.a . = 80@xmath16 ( slit orientation `` 1 '' in fig . [ fig1 ] ) . no binning along the spatial axis has been done , yielding a spatial sampling of 02 pixel@xmath14 . the total exposure time was 60 min , broken into three 20 min exposures . all exposures were taken at airmass of 1.42 . the seeing was 09 . mmt spectroscopic observations of i zw 18 and i zw 18c were carried out in the nights of 1997 april 29 and 30 . a signal - to - noise ratio s / n @xmath9 50 was reached in the continuum of the bright nw region of i zw 18 . observations were made in the blue channel of the mmt spectrograph using a highly optimized loral 3072 @xmath15 1024 ccd detector . a 15 @xmath15 180 slit was used along with a 300 groove mm@xmath14 grating in first order and an l-38 second - order blocking filter . this yields a spatial resolution along the slit of 03 pixel@xmath14 , a scale perpendicular to the slit of 1.9 pixel@xmath14 , a spectral range 3600 7500 , and a spectral resolution of @xmath5 7 ( fwhm ) . to improve the signal - to - noise ratio , ccd rows were binned by a factor of 2 , yielding a final spatial sampling of 06 pixel@xmath14 . the total exposure time was 180 minutes broken up in six subexposures , 30 minutes each . all exposures were taken at airmasses @xmath6 1.1 1.2 . the seeing during the observations was 07 fwhm . the slit was oriented in a position angle p.a . = 41@xmath16 to permit observations of the nw and se regions in i zw 18 and the eastern region ( region e ) in i zw 18c simultaneously ( slit orientation `` 2 '' in fig . [ fig1 ] ) . the kitt peak 4 m observations have been obtained on 18 march 1994 with the ritchey - chrtien rc2 spectrograph used in conjunction with the t2 kb 2048@xmath152048 ccd detector . we use a 2@xmath15300 slit with the kpc10a grating ( 316 lines mm@xmath14 ) in first order , with a gg 385 order separation filter . this filter cuts off all second - order contamination for wavelengths blueward of 7400 which is the wavelength region of interest here . the above instrumental set - up gave a spatial scale along the slit of 0.69 arcsec pixel@xmath14 , a scale perpendicular to the slit of 2.7 pixel@xmath14 , a spectral range of 35007500 and a spectral resolution of @xmath5 5 . all exposures were taken at airmass of 1.1 . the seeing was 15 . the slit was oriented along the se nw direction at a position angle of 41@xmath17 , the same as that during the mmt observations ( slit orientation `` 2 '' in fig . [ fig1 ] ) . the total exposure time was 60 minutes and was broken up into 3 subexposures . the spectrophotometric standard stars feige 34 and hz 44 were observed for flux calibration during each of three sets of the observations . spectra of hg - ne - ar ( keck ii ) and he - ne - ar ( mmt and 4 m kpno ) comparison lamps were obtained before and after each observation to provide the wavelength calibration . data reduction of spectral observations was carried out using the iraf software package . this included bias subtraction , cosmic - ray removal and flat - field correction using exposures of a quartz incandescent lamp . after wavelength calibration , night - sky background subtraction , and correcting for atmospheric extinction , each frame was calibrated to absolute fluxes . one - dimensional spectra were extracted by summing , without weighting , different numbers of rows along the slit depending on the exact region of interest . we have extracted spectra of two regions in i zw 18c ( fig . [ fig1 ] ) : ( 1 ) the brightest region c ( keck ii observations ) and ( 2 ) the eastern region e ( all observations ) . the extracted spectra are shown in fig . additionally , spectra of outlying regions of i zw 18 at different distances from it have been extracted . one of the key problems discussed over the last three decades is the evolutionary status of i zw 18 : is this galaxy young or old ? the evolutionary status of i zw 18c has not been discussed in comparable detail . high signal - to - noise spectra of i zw 18c reveal blue continua and show only emission and absorption hydrogen balmer lines . heavy element emission lines are not detected in the spectra , which precludes a metallicity determination of i zw 18c . for the sake of simplicity we assume the heavy element mass fraction in i zw 18c to be @xmath18/50 , the same value as in i zw 18 . however , the spectra obtained for i zw 18c allow to study stellar populations and constrain their age with various techniques . a useful technique for determining the age of a galaxy is to fit its observed spectral energy distribution ( sed ) by theoretical seds calculated for various stellar population ages and star formation histories . this method ( alone or in combination with photometric data ) has been applied to some extremely metal - deficient bcds with @xmath19 = ( 1/20 1/40)@xmath18 ( e.g. , sbs 0335052 ( izotov et al . 1997b ; papaderos et al . 1998 ) , sbs 1415 + 437 ( thuan et al . 1999a ) , tol 1214277 ( fricke et al . it was shown that , after subtraction of ionized gas emission , the underlying stellar components of these galaxies are consistent with populations not older than a few hundred myr . however , the spectral energy distribution fitting method is subject to uncertainties in the extinction , resulting in an age overestimate , if the adopted extinction is too low . therefore , other methods are desirable to constrain the stellar population ages . we discuss in this section two such methods , one relying on the balmer nebular emission line equivalent widths and the other on the balmer stellar absorption line equivalent widths . on the assumption of a dust - free ionization - bounded h ii region , the strongest hydrogen recombination emission lines h@xmath0 and h@xmath1 provide an estimate of the age of the young stellar population when o and early b stars are still present . however , even if dust is present in h ii regions , the age estimate is quite robust . this is because the ionizing flux from such a young stellar population and hence the equivalent widths of the balmer emission lines have a very strong temporal evolution . therefore , the dating method based on the h@xmath0 and h@xmath1 emission lines is relatively insensitive to dust extinction . the h@xmath0 and h@xmath1 emission lines are detected in i zw 18c in both regions c and e ( fig . [ fig2 ] ) . their fluxes and equivalent widths are listed in table [ tab1 ] . the exception is the keck ii spectrum of region e , where h@xmath1 emission was not detected . this non - detection is probably due to the patchy distribution of the ionized gas . in fig . [ fig3]a we compare the measured h@xmath0 and h@xmath1 emission line equivalent widths with those predicted for an instantaneous burst as a function of age . the theoretical dependences have been kindly calculated for us by d. schaerer using the schaerer & vacca ( 1998 ) code with the @xmath19 = 0.0004 geneva evolutionary tracks from lejeune & schaerer ( 2001 ) . they are shown by solid lines . the age derived from different hydrogen nebular emission lines in the various spectra is in a narrow range around @xmath5 15 myr . hence , the gas in i zw 18c is likely to be ionized by early b stars . another method of stellar population age determination is based on the equivalent widths of absorption features . this method probes larger ages as compared to the previous method because the most prominent hydrogen absorption lines are formed in the longer - lived a stars . gonzalez delgado & leitherer ( 1999 ) and gonzalez delgado , leitherer & heckman ( 1999 ) have calculated a grid of synthetic profiles of stellar hydrogen balmer absorption lines for effective temperatures and gravities , characteristics of galaxies with active star formation . they developed evolutionary stellar population synthesis models , synthesizing the profiles of the hydrogen balmer absorption lines from h@xmath1 to h13 for an instantaneous burst with an age ranging from @xmath20 to @xmath21 yr . the calculations were made for a stellar initial mass function with salpeter slope and with mass cutoffs @xmath22 = 1 @xmath23 and @xmath24 = 80 @xmath23 . the h@xmath2 , h@xmath3 and higher order hydrogen absorption lines due the underlying stellar populations are clearly seen in the keck ii spectra of i zw 18c ( fig . [ fig2]a - [ fig2]b ) . some hydrogen absorption lines are also seen in the 4 m kpno and mmt spectra ( fig . [ fig2]c - [ fig2]d ) . however , the signal - to - noise ratio is not high enough in the latter two spectra to measure equivalent widths . although higher - order hydrogen balmer absorption lines are seen in the spectrum of i zw 18c , they are not suitable for age determination because of ( a ) the relatively low signal - to - noise ratio at short wavelengths and uncertainties in the placement of the continuum in the blue region caused by many blended absorption features , and ( b ) the weak dependence of their equivalent widths on age ( gonzalez delgado et al . 1999 ) . in table [ tab2 ] we show the equivalent widths of the h@xmath3 and h@xmath2 absorption lines measured in the spectra of regions c and e in i zw 18c . we need to correct the equivalent widths of the absorption lines for the contamination by nebular emission . for region c we use the intensity of the h@xmath1 emission line to calculate the intensity of the h@xmath3 emission line adopting case b at the electron temperature of @xmath5 20000k ( e.g. , aller 1984 ) . we do not use the h@xmath2 absorption line in the spectrum of this region because of the strong contamination by nebular emission . the h@xmath1 emission line is not definitely detected in the keck ii spectrum of region e. for this region , the intensity of h@xmath0 emission line is used to correct equivalent widths of the absorption lines for the same effect . the corrected equivalent widths of h@xmath3 and h@xmath2 absorption lines are shown in table [ tab2 ] . in fig . [ fig3]b we show by solid lines the predicted behaviour of the equivalent widths of the h@xmath2 and h@xmath3 absorption lines with age for an instantaneous burst at a metallicity @xmath19 = 1/20 @xmath18 ( gonzalez delgado et al . 1999 ) . the measured equivalent widths are shown for region c by filled circles and for region e by stars . their values are consistent with an age of @xmath5 15 myr . this age estimation is in excellent agreement with that obtained from the nebular emission line analysis implying that the light of i zw 18c is dominated by a young stellar population . however , the age we derive here is significantly lower than the value of 40 myr derived by dufour et al . ( 1996b ) for the brightest stars from _ hst _ cmds of i zw 18c . although our two age estimates are consistent with each other , there are a number of uncertainties which may affect the result . a major uncertainty is the unknown metallicity of the stars in i zw 18c . we have assumed for simplicity the metallicity to be equal to that of the ionized gas in i zw 18 . note , however , that a lower stellar metallicity would increase the age and vice versa . to estimate an age from the emission lines we have assumed a dust - free ionization - bounded h ii region . if the h ii region is density - bounded or dust is present , then some of the ionizing photons escape the h ii region or are absorbed . equivalent widths of the balmer emission lines give in this case an upper limit to the age . another source of uncertainty comes from the small number of massive stars in i zw 18c . our age estimates are based on models where the stellar initial mass function is well - behaved and can be approximated by an analytical function . however , small number statistics can introduce stochastic fluctuations at the high star mass end of the imf . recently cervio , luridiana & castander ( 2000 ) have analyzed how such stochastic effects influence the observed parameters of young stellar clusters with solar metallicity such as the h@xmath1 equivalent width and the number of the ionizing photons . the number of ionizing photons in i zw 18c derived from the h@xmath1 emission line ( table 1 ) is @xmath5 2 @xmath15 10@xmath25 s@xmath14 and 4 @xmath15 10@xmath25 s@xmath14 for the e and c regions respectively . with an equivalent width @xmath5 6 7 of the h@xmath1 emission line , this corresponds to the case of a 10@xmath26 @xmath27 stellar cluster ( cervio et al . for such a cluster the age variations at a fixed h@xmath1 equivalent width can be as high as 15 percent at the 90% confidence level . hence , the age of i zw 18c derived from the emission lines can lie in the range @xmath5 10 25 myr , with a central value of 15 myr . similarly , age estimates based on absorption lines can also be slightly modified by stochastic effects . however , calculations are not yet available in the literature . gonzalez delgado et al . ( 1999 ) do not calculate the temporal evolution of the equivalent widths of the balmer absorption lines for the heavy element mass fraction @xmath19 = @xmath18/50 . therefore , we use models with @xmath19 = @xmath18/20 . extrapolation to the metallicity of i zw 18 would result in a @xmath6 1 decrease of the equivalent widths at a fixed age , or an age increase of up to 25 myr . finally , the age determination depends on the star formation history in the galaxy which we consider next . our estimates for the stellar population age in i zw 18c in sect . [ ageem ] and [ ageab ] are based on the assumption of an instantaneous burst of star formation . now we discuss how that age changes if continuous star formation is considered . we adopt a constant star formation rate in the interval between the initial time @xmath28 when star formation starts and the final time @xmath29 when it stops . time is zero at the present epoch and increases into the past . using model equivalent width of the emission and absorption lines and spectral energy distributions for instantaneous bursts ( schaerer , private communication ; lejeune & schaerer 2001 ; gonzalez delgado et al . 1999 ) , we calculate the temporal evolution of the hydrogen emission and absorption line equivalent widths for continuous star formation . the results of calculations are presented in figure [ fig3 ] . by dashed , dot - dashed and dotted lines are shown the temporal dependences of the equivalent widths of the h@xmath1 and h@xmath0 emission lines ( fig . [ fig3]a ) , and of the h@xmath3 and h@xmath2 absorption lines ( fig . [ fig3]b ) for continuous star formation starting at time @xmath28 , as defined by the abscissa value , and stopping at @xmath29 = 5 , 8 and 12.5 myr , respectively . in other words , the equivalent widths of the above four lines in the spectrum of the stellar population formed between @xmath28 and @xmath29 have a value @xmath30 at time @xmath28 in fig . [ fig3]a and [ fig3]b . at a fixed @xmath30 , the general trend seen from fig . [ fig3 ] for continuous star formation is that the younger the youngest stars , the larger the time interval @xmath31 , and the older the oldest stars . another feature is that , at a fixed age @xmath29 of the youngest stars , the age @xmath28 of the oldest stars derived from the observed emission line equivalent widths , differs from that derived from the observed absorption line equivalent widths . in particular , in the model where star formation stopped 5 myr ago ( dashed lines ) , the age of the oldest stars derived from hydrogen emission lines exceeds 100 myr , while the age of the oldest stars derived from hydrogen absorption lines is only @xmath5 50 myr . this model seems to be excluded by consideration of the luminosity of the ionizing radiation . the most massive stars in the stellar population with age 5 myr would have masses as high as 40 @xmath27 ( meynet et al . the number of the ionizing photons produced by a single 40 @xmath27 star is equal to @xmath32(lyc ) @xmath33 1.5 @xmath15 10@xmath34 s@xmath14 ( vacca , garmany & shull 1996 ) , larger than that derived from the observed flux of the h@xmath1 emission line in i zw 18c ( table [ tab1 ] ) , assuming an ionization - bounded h ii region . there can be an upward correction factor of @xmath6 2 due to the extinction , but the corrected @xmath32(lyc ) would still be below the value for a single 40 @xmath27 star . these estimates can however be modified by massive star small number statistics caused by the stochastic nature of star formation . though smaller , the difference between the age of the oldest stars derived from the equivalent widths of emission lines ( 50 myr ) and that derived from the equivalent widths of absorption lines ( 40 myr ) , is present in the continuous star formation model with age of the youngest stars equal to 8 myr ( dot - dashed lines in fig . [ fig3 ] ) . however , this difference is small in the continuous star formation model which stopped 12.5 myr ago ( dotted lines ) . in this model , the age of the oldest stars should be @xmath5 25 myr to consistently explain the observed hydrogen line equivalent widths . hence , similarly to the case of an instantaneous burst , we conclude that the observations of i zw 18c are best reproduced by a short star formation episode which occurred continuously between @xmath5 10 myr and @xmath5 25 myr ago . uncertainties in the observations and models may extend this range to between @xmath5 10 myr and @xmath6 100 myr ago . a useful constraint on the stellar population age can be obtained from the spectral energy distribution . this method , as already noted , is subject to interstellar extinction . however , when used in conjunction with the methods discussed in sect . [ age ] it provides a powerful tool for studying stellar populations by allowing to derive simultaneously the age and the extinction of the same region . to fit the observed spectral energy distributions we use model seds calculated by d. schaerer using the schaerer & vacca ( 1998 ) code and the @xmath19 = 0.0004 geneva evolutionary tracks of lejeune & schaerer ( 2001 ) . the contribution of the ionized gas was also included . this contribution is small because the equivalent widths of hydrogen emission lines in i zw 18c are low . because the observed spectral energy distribution is extinction - dependent , the extinction can be obtained for regions with known ages as derived from the equivalent widths of the hydrogen emission and absorption lines . we consider the case of the 15 myr instantaneous burst stellar population discussed in sect . first assume @xmath35(h@xmath1 ) = 0 , where @xmath35(h@xmath1 ) = @xmath36(@xmath37)/1.47 ( aller 1984 ) . comparison of the observed keck ii spectra of regions c and e in i zw 18c with the theoretical seds ( bottom spectra in fig . [ fig4 ] ) shows that theoretical seds are bluer than the observed extinction - uncorrected spectra . evidently , interstellar extinction is present in i zw 18c and it modifies the observed sed . we derive @xmath35(h@xmath1 ) = 0.3 for region c and @xmath35(h@xmath1 ) = 0.1 for region e to achieve the best agreement between the extinction - corrected observed seds and the theoretical seds ( upper spectra in fig . [ fig4 ] ) . for comparison , we show in fig . [ fig4]a by a dotted line the theoretical sed for a 40 myr stellar population which does not provide as good a fit . a theoretical 15 myr stellar population sed also fits well the 4 m kpno and mmt spectra of region e extinction - corrected for @xmath35(h@xmath1 ) = 0.1 ( fig . [ fig5 ] ) . the theoretical 40 myr stellar population sed with @xmath35(h@xmath1 ) = 0.1 fits less well ( bottom solid line ) . some support for a larger extinction in region c than in region e comes from the observed h@xmath0-to - h@xmath1 flux ratios ( table [ tab1 ] ) . they are respectively equal to 4.5 and 3.8 , corresponding to @xmath35(h@xmath1 ) @xmath5 0.7 and 0.4 . however , correction for underlying stellar absorption results in lower extinction coefficients . we note that we have not corrected the observed seds for the effect of atmospheric refraction . if such an effect were to be important , it can selectively decrease the blue light relatively to the red light , leading us to derive erroneously high extinction for regions c and e. indeed one may suspect that such an effect would be important for the keck ii spectrum which was obtained with a narrow slit of 1 at an airmass of 1.4 . filippenko ( 1982 ) has shown that atmospheric dispersion can produce an offset as high as @xmath5 12 of the blue region near [ o ii ] @xmath383727 relative to the red region near h@xmath0 @xmath386563 at this airmass . however , his calculations have been done for an altitude of 2 km , while the keck ii spectrum was obtained at an altitude about twice that . furthermore , emission from the c component is extended and originates in a region significantly larger than the width of the slit , reducing the effect of the atmospheric dispersion . perhaps the best argument for such an effect not to be important comes from the comparison of our different spectra of the same region . although the keck , mmt and 4 m spectra of region e were obtained with different slit widths at different airmasses , they are all well fitted by the same 15 myr single stellar population model . we have thus reached two important conclusions for i zw 18c : ( 1 ) the stellar population responsible for its observed sed is very young , with an age of @xmath5 15 myr and ( 2 ) the region is characterized by a varying interstellar extinction implying the presence of non - uniformly distributed absorbing material . i zw 18c is not the only very metal - deficient object to have a clumpy dust distribution . earlier similar conclusions have been reached for the metal - deficient bcd sbs 0335052 by izotov et al . ( 1997b , 1999 ) and thuan et al . ( 1997 , 1999b ) . while the brightest and youngest star - forming region in sbs 0335052 is relatively free of dust , extinction is higher at the location of the fainter and older super star clusters . clumpy regions with large extinctions are clearly seen in the _ hst _ @xmath39 image of sbs 0335052 ( thuan et al . 1997 ) . we have arrived at the conclusion that the light from i zw 18c is dominated by stars @xmath5 15 myr old . is this conclusion consistent with the photometric data ? in this section , we compare the predicted colors for the young stellar population with integrated broad - band colors of i zw 18c obtained from ground - based and _ hst _ photometric observations . we also discuss the consistency between the properties of the stellar population in i zw 18c obtained from the spectroscopic data with those obtained from analysis of _ hst _ wfpc2 cmds . in table [ tab3 ] we show the observed integrated @xmath13 magnitude and colors of i zw 18c . the second column shows these quantities without correction for interstellar extinction . because the spectroscopic data imply the presence of extinction in i zw 18c , we also show in the third column the colors corrected for interstellar extinction with @xmath35(h@xmath1 ) = 0.3 or @xmath4 = 0.65 mag . these values are for the brightest region c. the extinction is lower in the fainter region e. the faint northwestern region of i zw 18c appears to be redder as compared to other regions ( dufour et al . 1996b ) , but the lack of spectroscopic data prevents us from determining the interstellar extinction in that region . we assume that the extinction derived for region c to be representative for the whole galaxy . we compare the observed integrated colors of i zw 18c with those predicted by instantaneous burst models for different ages . the first set of models shown in table [ tab3 ] is the same as the one used for fitting the seds with an heavy element mass fraction @xmath19 = @xmath18/50 and based on geneva stellar evolutionary tracks . another set of predicted colors based on the padua stellar evolution models has been calculated by tantalo et al . ( 1996 ) for a single stellar population and a heavy element mass fraction of @xmath18/50 . comparison of the two sets of models shows that colors based on the padua stellar evolution models are systematically redder at a fixed age as compared to those based on the geneva ones . consequently , the use of padua models results in younger ages as compared to geneva models . in the following we compare the observed colors to the modeled ones based on geneva tracks . it is seen from table [ tab3 ] that the colors uncorrected for extinction are well reproduced by the model with a 100 myr stellar population . however , with this age the predicted equivalent widths of the hydrogen emission lines are too small ( @xmath30(h@xmath1 ) @xmath6 0.1 , @xmath30(h@xmath0 ) @xmath6 0.3 ) as compared to the observed ones ( fig . [ fig3]a ) . on the other hand , the predicted equivalent widths of the hydrogen absorption lines ( @xmath30(h@xmath3 ) @xmath9 10 , @xmath30(h@xmath2 ) @xmath9 8 ) are too high ( fig . [ fig3]b ) . again , to put observations into agreement with models , interstellar extinction has to be invoked . indeed , all observed colors corrected for an extinction with @xmath35(h@xmath1 ) = 0.3 are in fair agreement with predicted ones for a 15 20 myr single stellar population . our conclusions do not change significantly in the case of continuous star formation . in fig . [ fig6]a [ fig6]c we show by solid lines the theoretical dependences on age of the ( @xmath40 ) , ( @xmath37 ) and ( @xmath39 ) colors in the case of constant continuous star formation , for different choices of @xmath28 and @xmath29 . the observed colors uncorrected for extinction ( dashed lines ) can be fitted by models with star formation starting at @xmath28 = 100 300 myr . however , these models predict too low an equivalent width for the h@xmath0 emission line and too large an equivalent width for the h@xmath3 absorption line ( fig . [ fig6]d [ fig6]e ) . furthermore , models with star formation stopping at @xmath29 @xmath9 40 myr are excluded for the whole range of @xmath28 ( fig . [ fig6]d [ fig6]e ) . to have the observed colors come into agreement with the observed equivalent widths of the balmer lines , a non - negligible extinction must be assumed . we show by dotted lines in fig . [ fig6]a [ fig6]c the extinction - corrected colors with two values of the reddening , @xmath36(@xmath37 ) = 0.1 and 0.15 . in the latter case , the colors are explained by models with constant star formation starting at an age @xmath28 @xmath5 30 100 myr ( filled and open circles ) and stopping at an age @xmath29 = 8 12 myr . observational uncertainties will only slightly increase this age range . we conclude that our broad - band photometric data are consistent with a young stellar population and a non - negligible interstellar extinction in i zw 18c . we emphasize that the ages derived above hold only for the brightest regions of the c component . we can not exclude the possibility that the age of the stellar population in regions of i zw 18c , not covered by our spectroscopic observations , may be larger . cmd analysis is a powerful tool for studying stellar populations . however , as already pointed out , this method is sensitive to the adopted extinction and distance . while the extinction can be derived from spectroscopic observations , the determination of the distance is more uncertain . a distance of @xmath5 10 mpc to i zw 18 has generally been adopted for analyzing the cmds ( hunter & thronson 1995 , dufour et al . 1996b and aloisi et al . this assumes that the observed heliocentric radial velocity of the galaxy @xmath5 740 km s@xmath14 is a pure hubble flow velocity , and a hubble constant @xmath41 = 75 km s@xmath14 mpc@xmath14 . adopting this distance would lead to a conflict with the well - observed ionization state of i zw 18c . at 10 mpc the brightest stars observed in i zw 18c would have absolute @xmath13 magnitudes fainter than 6 mag ( dufour et al . 1996b ; aloisi et al . in that case , comparison with evolutionary tracks implies that the most massive stars in i zw 18c ( called the c component by dufour et al . ( 1996b ) ) would have masses less than 9 @xmath27 . the age of the stellar population with such an upper mass limit is at least 40 myr ( e.g. , dufour et al . 1996b ) , larger than the one derived from the equivalent widths of hydrogen emission and absorption lines ( @xmath5 10 25 myr ) . if the upper stellar mass limit of 9 @xmath27 derived by dufour et al . ( 1996b ) and aloisi et al . ( 1999 ) for i zw 18c is correct , then ionized gas should not be present in it because of the absence of early b stars . but h@xmath0 and h@xmath1 are clearly observed . our derived age of @xmath5 15 myr implies that early b stars with masses as high as @xmath5 15 @xmath27 are present in i zw 18c , if an instantaneous burst of star formation is assumed . in the case of continuous star formation , the age of the youngest stars would be smaller and the upper mass limit larger to account for the presence of the ionized gas . we argue therefore that the stellar absolute magnitudes derived by dufour et al . ( 1996b ) and aloisi et al . ( 1999 ) from their cmds are too faint because they are based on too small an adopted distance . stlin ( 2000 ) assumed a distance of 12.6 mpc to analyze his _ hst _ nicmos cmd . however , even this distance is not enough to explain the ionization state of i zw 18c . an additional effect is due to extinction , with @xmath4 = 0.65 mag for the region c. correcting for extinction and increasing the distance by a factor of @xmath5 1.5 to @xmath5 15 mpc would make the most massive stars more luminous by a factor of @xmath5 4 and push the mass upper limit to @xmath5 15 @xmath27 . a stellar population with such an upper mass limit would provide enough ionizing photons to account for the observed emission lines in i zw 18c . furthermore , the age of the brightest stars in the cmds of i zw 18c would be @xmath5 15 myr , consistent with that derived from the hydrogen emission and absorption lines . ground - based and _ hst _ h@xmath0 and broad - band imaging have revealed filamentary structure around i zw 18 , inside a 15 radius ( e.g. , hunter & thronson 1995 ; stlin , bergvall & rnnback 1996 ; dufour et al . 1996b ) . because of the presence of ionizing young stars , the light in these outlying regions is likely to be dominated by the emission of the ionized gas . this conclusion is supported by spectroscopic observations . dufour et al . ( 1996a ) and izotov & thuan ( 1998 ) have detected h@xmath0 emission at distances as large as 20 from i zw 18 in the nw direction of slit `` 2 '' ( fig . [ fig1 ] ) . izotov et al . ( 1999 ) have found that at a distance of @xmath5 5 to the northwest of the brightest nw region of i zw 18 , the equivalent width of the h@xmath1 emission line is @xmath5 300 . in this case the contribution of the gaseous continuum near h@xmath1 is @xmath5 30 percent of the total continuum . the contribution of the gaseous continuum near h@xmath0 is even larger , being @xmath5 50 percent of the total continuum . therefore , when analyzing stellar populations with the use of photometric data , it is essential to correct broad - band colors for ionized gas emission . however , in some recent papers ( e.g. , kunth & stlin 2000 ) the extended emission around i zw 18 has been attributed to an old stellar population , while the contribution of the ionized gas is assumed to be not dominant . that this can not be true is seen in fig . [ fig7 ] where we show a map of the h@xmath0 equivalent width distribution as obtained from _ hst _ narrow - band and broad - band images . while the h@xmath0 equivalent width is small in the direction of the stellar clusters , it exceeds 1000 in the outer regions . in the following , we analyze mmt and keck ii spectroscopic observations of the outer regions around i zw 18 to clarify two issues : a ) how important is the contribution of the ionized gas in the outer regions of i zw 18 ? and b ) is stellar emission present at large distances ? in figure [ fig8 ] we show the mmt spectrum of the region with a high h@xmath1 equivalent width . the spectrum is extracted within an aperture 15 @xmath15 3 ( slit `` 2 '' ) , centered at a distance 5 to the northwest from the nw component of i zw 18 . it is characterized by strong emission lines . we refer to this region as the `` h@xmath0 arc '' ( square box in fig . [ fig1 ] ) . a synthetic spectrum with a 2 myr stellar population combined with ionized gas emission fits best the observed sed of this region . the observed and extinction - corrected emission - line intensities in the h@xmath0 arc together with their equivalent widths are listed in table [ tab4 ] . the ionic and elemental abundances have been derived following izotov et al . ( 1994 , 1997c ) . the extinction coefficient @xmath35(h@xmath1 ) and the absorption equivalent width @xmath30(abs ) for the hydrogen lines are obtained by an iterative procedure . they are shown in table [ tab4 ] together with the observed flux @xmath42 of the h@xmath1 emission line . the electron temperature @xmath43(o iii ) was determined from the [ o iii ] @xmath384363 / ( @xmath384959 + @xmath385007 ) flux ratio and the electron number density @xmath44(s ii ) from the [ s ii ] @xmath386717/@xmath386731 flux ratio . the ionic and elemental abundances are shown in table [ tab5 ] together with ionization correction factors ( icfs ) . they are in good agreement with the abundances derived by skillman & kennicutt ( 1993 ) , izotov & thuan ( 1998 ) , vlchez & iglesias - pramo ( 1998 ) and izotov et al . ( 1999 ) for the nw and se components of i zw 18 . we have shown that the contribution of the ionized gas is large in the h@xmath0 arc , at @xmath5 5 from the nw component of i zw 18 . a similar situation prevails at significantly larger distances , as evidenced by deep keck ii spectroscopic observations . the slit during these observations ( slit `` 1 '' in fig . [ fig1 ] ) crossed the outer regions of i zw 18 including the expanding supershell of ionized gas best seen in h@xmath0 images ( e.g. , hunter & thronson 1995 ; dufour et al . the latter feature located at @xmath5 15 from i zw 18 is labeled in figure [ fig1 ] as `` loop '' . we can thus study with deep spectroscopy the extended diffuse emission in i zw 18 all the way from i zw 18c to the bright star at the edge of fig . [ fig1 ] and located @xmath5 40 from i zw 18 . in fig . [ fig9]a [ fig9]b we show respectively the flux distributions along the slit of the continuum near h@xmath1 and of the line + continuum emission at the wavelength of the h@xmath1 emission line . the origin is taken to be at region c. the locations of the bright star and loop are marked . in fig . [ fig9]c we show the continuum - subtracted flux distribution of the h@xmath1 emission line . the negative values of the flux in some regions of i zw 18c and around the star are probably caused by underlying stellar h@xmath1 absorption . no appreciable continuum emission is seen in fig . [ fig9]a between i zw 18c and the star . however , nebular emission of h@xmath0 is present nearly everywhere between i zw 18c and the star ( fig . [ fig9]c [ fig9]d ) , suggesting that the contribution to the total flux of the nebular emission from ionized gas is important in the outermost regions , as far as 30 from i zw 18 . in fig . [ fig9]e we show the intensity distribution of the continuum at the wavelength of 4200 approximating the @xmath7 band . we also plot by dotted lines the surface brightness levels in steps of 1 mag arcsec@xmath8 . it is seen that the continuum surface brightness is fainter than 27 mag arcsec@xmath8 everywhere between i zw 18c and the star , including the loop region . however , we can not exclude the presence of stellar emission at the level of 28 mag arcsec@xmath8 , as postulated by legrand ( 2000 ) . the contamination by extended ionized gas emission makes the detection of such an extremely faint hypothetical stellar background problematic . in fig . [ fig10 ] we show the distributions along slit `` 2 '' of : a ) the continuum intensity at 4200 , b ) the continuum - subtracted flux and c ) the equivalent width of the h@xmath0 emission line . the distribution of h@xmath0 emission ( fig . [ fig10]b ) around the main body is more extended as compared to the continuum ( fig . [ fig10]a ) , the latter being confined in a region with radius less than 12 around the nw component . the equivalent width of h@xmath0 is very high to the northwest of the nw component ( fig . [ fig10]c ) and must be taken into account when photometric properties of the stellar population in i zw 18 are analyzed . we also point out that the continuum distribution of region e in i zw 18c is narrower than that of the h@xmath0 emission line . the maximum h@xmath0 equivalent width in region e ( fig . [ fig10]c ) is offset to the northwest by @xmath5 2 relative to the continuum distribution ( fig . [ fig10]a ) . in fig . [ fig11 ] we show the spectrum of the loop . despite its faintness , several emission lines are seen . however , the sensitivity in the blue region was not sufficient to detect the [ o ii ] @xmath383727 emission line . the continuum is very weak and can be significantly affected by uncertainties in the sky subtraction . this makes the measurements of line equivalent widths difficult . the fluxes and equivalent widths of the detected lines are given in table [ tab6 ] . the flux errors include uncertainties in the placement of the continuum level and in the fitting of the lines by gaussian profiles . however , these errors do not take into account the uncertainties in the sky subtraction which might be large . indeed , the loop flux in the continuum is only @xmath5 1% above the night sky flux , while that number is as high as 50% for the continuum flux in i zw 18c . even with these large uncertainties the emission line equivalent widths in the loop spectrum are very high . in particular , the equivalent width of the h@xmath1 emission line is 471 or about half of the value expected for pure gaseous emission at the electron temperature @xmath43 = 20000k . hence , half of the flux in the continuum comes from the ionized gas , emphasizing again the importance of the correction of the spectral energy distribution and broad - band colors for gaseous emission . this goes contrary to the assumption of kunth & stlin ( 2000 ) that the contribution of gaseous emission does not affect the colors of the outlying regions of i zw 18 . if errors in the night sky subtraction are @xmath5 1% , then the equivalent width of the h@xmath1 emission line in the loop spectrum is in the range @xmath5 250 1000 . within the uncertainties , the emission of the loop is quite consistent with pure gaseous emission . kunth & stlin ( 2000 ) have derived radial distributions of the surface brightness in the @xmath7 band and of the @xmath45 and @xmath46 colors of i zw 18 ( their fig . they find that the colors rise continuously with increasing radius and reach @xmath45 = 0.6 mag and @xmath46 = 1.6 mag at a radius of 10 . assuming a purely stellar emission , they conclude that the observed colors can be reproduced by a single stellar population model with a metallicity of 1/50 @xmath18 and an age of log @xmath47 = 9.1 @xmath48 0.1 ( @xmath47 in yr ) , irrespective of the imf ( bruzual & charlot 2000 , unpublished ) . however , this age estimate is rather uncertain and is dependent on the particular population synthesis model used . for example , tantalo et al . ( 1996 ) using padua stellar evolutionary tracks give values @xmath45 = 0.8 mag and @xmath46 = 1.7 mag for a 1 gyr single stellar population with a metallicity of 1/50 @xmath18 . the bluer colors derived by kunth & stlin ( 2000 ) would give an age of 100 300 myr according to tantalo et al . ( 1996) models . on the other hand , leitherer et al . ( 1999) models using the geneva stellar evolutionary tracks predict @xmath45 = 0.5 mag and @xmath46 = 1.1 mag for a 1 gyr single stellar population with a metallicity of 1/20 @xmath18 , bluer than those derived by kunth & stlin ( 2000 ) . leitherer et al . ( 1999) models do not go beyond 1 gyr , but to reproduce the colors derived by kunth & stlin ( 2000 ) , the age of the stellar population in the outer regions of i zw 18 , if present , must be older than 1 gyr . their models are calculated for a metallicity of 1/20 @xmath18 , but colors with a metallicity of 1/50 @xmath18 are expected to be bluer , further increasing the derived age . these age estimates are very uncertain . the models by leitherer et al . ( 1999 ) are less reliable for ages greater 100 myr because they do not include asymptotic giant branch ( agb ) star evolution . tantalo et al . ( 1996 ) do include agb star evolution , but the little known mass loss processes in the agb phase introduce uncertainties in the predicted colors ( girardi & bertelli 1998 ) . the next source of uncertainties comes from the photometric observations themselves . beyond a radius of @xmath5 5 from i zw 18 , the @xmath45 color profile derived by kunth & stlin ( 2000 ) increases monotonously while the @xmath46 color profile shows discontinuous jumps . these discontinuities are difficult to understand if the same stellar population is responsible for both colors . kunth & stlin ( 2000 ) do not show the uncertainties of their photometry . however , similar deep @xmath10-band photometry of another galaxy sbs 0335052 ( vanzi et al . 2000 ) with ukirt shows that at the @xmath10-band surface brightness of 24 25 mag arcsec@xmath8 the errors are already @xmath5 0.5 mag or more . new recent @xmath7 and @xmath10 photometric observations of i zw 18 ( papaderos et al . 2001 ) do not confirm the large reddening of the @xmath46 color observed by kunth & stlin ( 2000 ) between radii 6 and 8 , nor the discontinuous jumps . to investigate whether the @xmath45 and @xmath46 colors of the extended emission can be explained by pure gaseous emission , we calculate the spectral energy distribution of the ionized gas emission in the corresponding wavelength range . the contribution of the free - bound , free - free and two - photon continuum emission is taken into account for the spectral range from 0 to 5 @xmath49 m ( aller 1984 ; ferland 1980 ) . as for the electron temperature , we adopt the value of 19000k , which is the mean value between the electron temperatures in the nw and se components of i zw 18 . emission lines are superposed on the gaseous continuum sed with intensities derived from the observed spectrum of the loop at the distance of @xmath5 15 from i zw 18 ( table [ tab6 ] ) , in the spectral range @xmath383700 7500 . outside this range , the intensities of emission lines ( mainly hydrogen lines ) have been calculated from the extinction - corrected flux of h@xmath1 with reddening @xmath4 = 0.16 mag . the reddening in the loop was calculated from the observed h@xmath0/h@xmath1 flux ratio ( table [ tab6 ] ) , assuming an electron temperature @xmath43 = 20000k . we derive @xmath45 = 0.8 mag and @xmath46 = 0.9 mag . if instead of the relative intensities of the emission lines observed in the loop , we use those seen in the nw or se regions of i zw 18 ( izotov et al . 1999 ) , we obtain slightly bluer colors , @xmath45 = 0.6 mag and @xmath46 = 0.7 mag . the color difference is mainly due to a smaller contribution in the outer regions of some emission lines , e.g. [ ne iii ] @xmath383869 to the @xmath7 band . from this comparison we conclude that colors become redder at larger distances , even in the case of pure gaseous emission . while the @xmath45 color of gaseous emission is similar to the asymptotic value of @xmath5 0.7 mag derived by kunth & stlin ( 2000 ) at distances @xmath5 15 , the predicted @xmath46 color of gaseous emission is considerably bluer than the value they obtained . however , that value is consistent with @xmath46 @xmath5 0.6 mag derived by papaderos et al . ( 2001 ) in the 6 9 arcsec radius range . we note that the @xmath45 and @xmath46 colors are not ideal for constraining the existence of a possible extended low - surface - brightness 1 gyr underlying stellar population in i zw 18 , the latter being uncertain , and the former being very similar to the color of ionized gas . the @xmath50 color is more useful because it can discriminate better between gaseous and a 1 gyr stellar population emission . indeed , adopting the relative line intensities in the h@xmath0 arc or in the loop , the @xmath50 colors of the ionized gas emission are 0.1 mag and + 0.1 mag respectively . the expected @xmath50 color for a 1 gyr stellar population is much redder , @xmath5 + 1.2 mag ( tantalo et al . 1996 ) . observations give @xmath50 @xmath5 0 at radii 8 10 arcsec ( papaderos et al . 2001 ) , strongly suggesting that the emission in the outer parts of i zw 18 is gaseous in origin . we conclude that there is no convincing observational evidence for the presence of an extended underlying low - surface - brightness stellar component in i zw 18 . its existence , as postulated by kunth & stlin ( 2000 ) , legrand ( 2000 ) and legrand et al . ( 2000 ) , is neither supported by spectroscopic nor photometric observations . we use spectroscopic and photometric data to constrain the age of the stellar population in the c component of i zw 18 ( @xmath51 i zw 18c ) and to study the origin of the extended emission around i zw 18 . we have arrived at the following main conclusions : \1 . deep 4 m kpno , mmt and keck ii spectra of i zw 18c show h@xmath1 and h@xmath0 hydrogen lines in emission , and h@xmath3 and h@xmath2 hydrogen lines in absorption . using their equivalent widths we derive an age of the stellar population of @xmath5 15 myr if an instantaneous burst is assumed . if star formation is continuous , then the equivalent widths are best reproduced by a short star formation episode continuously occurring between @xmath5 10 myr and @xmath5 25 myr ago . uncertainties in the observations and models may extend this range to between @xmath5 10 myr and @xmath6 100 myr ago . spectral energy distributions of the central ( c ) and eastern ( e ) regions of i zw 18c are used to derive extinction . the equivalent widths of the hydrogen emission and absorption lines and the spectral energy distributions are modeled by a 15 myr single stellar population if the extinction coefficient @xmath35(h@xmath1 ) = 0.1 0.3 , corresponding to @xmath4 = 0.20 0.65 mag . \3 . with the usually assumed distance of @xmath5 10 mpc the stellar population age derived from _ color - magnitude diagrams is too large as compared to the young age derived from the spectroscopic data . one possible source of the difference is interstellar extinction . furthermore , to have agreement between the cmds and the ionization state of i zw 18c , the distance to the bcd should be increased to @xmath5 15 mpc . concerning the extended emission around i zw 18 , keck ii spectra show h@xmath0 emission as far as 30 from the main body . the equivalent widths of emission lines are particularly strong in the extended envelope ( @xmath30(h@xmath1 ) = 471 ) , implying a dominant contribution of the ionized gas emission in the outermost regions of i zw 18 . within the large uncertainties of the continuum level , the emission at @xmath5 15 from i zw 18 is consistent with pure ionized gas emission . we do not find evidence for an old extended low - surface - brightness stellar component in the outlying regions of i zw 18 down to the surface brightness level @xmath7 @xmath5 27 mag arcsec@xmath8 , contrary to suggestions by kunth & stlin ( 2000 ) . it will be very difficult to detect the extended stellar emission at @xmath7 @xmath5 28 mag arcsec@xmath8 postulated by legrand ( 2000 ) , because of the important ionized gas emission at large distances . and n.g.g . thank the universitts sternwarte of gttingen for warm hospitality . we are grateful to d. schaerer for making available his stellar evolutionary synthesis models in electronic form and for valuable comments on the manuscript . y.i.i . thanks the gttingen academy of sciences for a gauss professorship . we acknowledge the financial support of the volkswagen foundation grant no . i/72919 ( y.i.i . , n.g.g . , p.p . and k.j.f . ) , of dfg grant 436 ukr 17/1/00 ( n.g.g . ) , deutsche agentur fr raumfahrtangelegenheiten ( dara ) gmbh grants 50 or 9407 6 and 50 or 9907 7 ( k.j.f . and p.p . ) , and of the national science foundation grants ast-9616863 ( t.x.t . and y.i.i . ) and ast-9803072 ( c.b.f . ) . stlin , g. , bergvall , n. , & rnnback , j. 1996 . in d. kunth , b. guiderdoni , m. heydari - malayeri , t. x. thuan ( eds . ) . the interplay between massive star formation , the ism and galaxy formation . gif - sur - yvette : edition frontires , p. 605 lcrccrccrccr h@xmath1 & [email protected] & [email protected] & & & & & [email protected] & [email protected]&&[email protected] & [email protected] + h@xmath0&[email protected]&[email protected]&&[email protected]&[email protected] & & [email protected]&[email protected]&&[email protected]&[email protected] + lrrcrrrrcrrr @xmath13 & 19.20 & & & & & + @xmath40 & 0.50&0.72&&0.83&0.77&0.59&0.40&&0.81&0.75&0.37 + @xmath37 & 0.00&0.20&&0.17&0.15&0.08&0.01&&0.18&0.12 & 0.02 + @xmath39 & 0.10&0.17&&0.10&0.08 & 0.04 & 0.13&&0.16 & 0.08 & 0.21 + @xmath52 & & & & 0.15&0.10 & 0.25 & 0.47&&0.25 & 0.29 & 0.67 + lccr 3727 [ o ii ] & [email protected] & [email protected]&53 + 3868 [ ne iii ] & [email protected] & [email protected]&20 + 3889 he i + h8 & [email protected] & [email protected]&31 + 3968 [ ne iii ] + h7 & [email protected] & [email protected]&39 + 4101 h@xmath3 & [email protected] & [email protected]&49 + 4340 h@xmath2 & [email protected] & [email protected]&106 + 4363 [ o iii ] & [email protected] & [email protected]&13 + 4471 he i & [email protected] & [email protected]&9 + 4686 he ii & [email protected] & [email protected]&7 + 4861 h@xmath1 & [email protected] & [email protected]&292 + 4959 [ o iii ] & [email protected] & [email protected]&179 + 5007 [ o iii ] & [email protected] & [email protected]&551 + 5876 he i & [email protected] & [email protected]&40 + 6563 h@xmath0 & [email protected] & [email protected]&1683 + 6678 he i & [email protected] & [email protected]&21 + 6717 [ s ii ] & [email protected] & [email protected]&25 + 6731 [ s ii ] & [email protected] & [email protected]&15 + + @xmath35(h@xmath1 ) & + @xmath53(h@xmath1 ) & + @xmath30(abs ) & lc @xmath43(o iii)(k ) & 18200@xmath484000 + @xmath43(o ii)(k ) & 15100@xmath483200 + @xmath43(s iii)(k ) & 16800@xmath483400 + @xmath44(s ii)(@xmath54 ) & 10@xmath4810 + + o@xmath55/h@xmath55(@xmath1510@xmath56 ) & [email protected] + o@xmath57/h@xmath55(@xmath1510@xmath56 ) & [email protected] + o@xmath58/h@xmath55(@xmath1510@xmath56 ) & [email protected] + o / h(@xmath1510@xmath56 ) & [email protected] + 12 + log(o / h ) & [email protected] + + ne@xmath57/h@xmath55(@xmath1510@xmath56 ) & [email protected] + icf(ne ) & 1.35 + log(ne / o ) & [email protected] lrcr 3889 h@xmath59+he i & [email protected]&0.183 & 54 + 4101 h@xmath3 & [email protected]&0.228 & 87 + 4340 h@xmath2 & [email protected]&0.392 & 125 + 4861 h@xmath1 & [email protected]&1.000 & 471 + 4959 [ o iii ] & [email protected]&0.148 & 69 + 5007 [ o iii ] & [email protected]&0.487 & 229 + 6563 h@xmath0 & [email protected]&2.902&1396

long - slit keck ii , 4 m kitt peak , and 4.5 m mmt spectrophotometric data are used to investigate the stellar population and the evolutionary status of i zw 18c , the faint c component of the nearby blue compact dwarf galaxy i zw 18 . hydrogen h@xmath0 and h@xmath1 emission lines are detected in the spectra of i zw 18c , implying that ionizing massive stars are present . high signal - to - noise keck ii spectra of different regions in i zw 18c reveal h@xmath2 , h@xmath3 and higher order hydrogen lines in absorption . several techniques are used to constrain the age of the stellar population in i zw 18c . ages derived from two different methods , one based on the equivalent widths of the h@xmath0 , h@xmath1 emission lines and the other on h@xmath2 , h@xmath3 absorption lines are consistent with a 15 myr instantaneous burst model . we find that a small extinction in the range @xmath4 = 0.20 0.65 mag is needed to fit the observed spectral energy distribution of i zw 18c with that model . in the case of constant star formation , all observed properties are consistent with stars forming continuously between @xmath5 10 myr and @xmath6 100 myr ago . we use all available observational constraints for i zw 18c , including those obtained from _ hubble space telescope _ color - magnitude diagrams , to argue that the distance to i zw 18 should be as high as @xmath5 15 mpc . the deep spectra also reveal extended ionized gas emission around i zw 18 . h@xmath0 emission is detected as far as 30 from it . to a @xmath7 surface brightness limit of @xmath5 27 mag arcsec@xmath8 we find no observational evidence for extended stellar emission in the outermost regions , at distances @xmath9 15 from i zw 18 .

introduction observations and data reduction the stellar population in i zw 18c the extended emission in i zw 18 conclusions

arxiv

several atoms play basic roles in modern physics and , in fact , very different roles . a unit of time , the second , is defined via the hyperfine interval in the cesium atom , while the atomic mass unit and the avogadro number are defined via the mass of a carbon atom . these two atoms are significant for our system of units , si . in addition , there are some favorite atomic systems where the basic laws of nature find their expression in the most transparent way . these simple atoms , most of which consist of two bound particles , have been crucial for our understanding of key moments of modern physics and their study is still of essential interest and importance . the simplicity and harmony of the theory of bound systems have been tempting and challenging for a while . johannes kepler believed the solar planetary system to be governed by a harmony of discrete numbers via geometry , trying with the so - called _ platonic _ or _ regular solids_. he failed to verify that kind of the harmony and discovered instead some regularities in the planetary orbital motion known now as kepler s laws . his discovery became a milestone in the development of theory of gravitation and mechanics and eventually led to the establishment of new mechanics ( classical mechanics ) . three centuries after the kepler s time , a planetary model was suggested for atoms . meantime certain regularities in the spectrum of the hydrogen atom were discovered . those regularities , like the kepler s laws , led once again to the establishment of new mechanics ( quantum mechanics ) and simultaneously realized the kepler s dream of the harmony of numbers governing the orbital motion . by now we know that a quantity describing a classical object can be of an arbitrary value while in the quantum case only discrete values are possible for some quantities . and this is how certain integer numbers enter the basic equations of modern physics . working on a new mechanics , a new dynamics or a new model , one used to try first to apply it to some ` simple ' objects . the simplest object is a free particle . however , only a limited number of properties of a free particle can be derived _ ab initio _ and studied with a high accuracy . study of simple atoms opens a broad field for possible effects and applications . a two - body atomic system such as the hydrogen atom is a natural object to verify a new model or to test a new approach . studies of the properties of the hydrogen atom have already served to establish the so - called ` old quantum mechanics ' ( the bohr theory ) , modern nonrelativistic quantum mechanics , relativistic quantum mechanics ( based on the dirac equation ) and quantum electrodynamics ( qed ) , which was the first successful quantum field theory . perhaps , we should even say that qed is the only quantum field theory which is successful for a really broad range of problems from atomic spectra to scattering , from low energy , related to microwave radiation , to high energy phenomena with hard annihilation and bremsstrahlung , from nano- to giga- electronvolt . figure [ 00hydr ] shows several crucial contributions to hydrogen energy levels . we note here that one of reasons for choosing a non - relativistic equation by schrdinger over a relativistic klein - gordon - fock equation was an incorrect description by the latter of the fine structure effects in the latter . another remark on importance of the hydrogen atom for qed is that the anomalous magnetic moment of an electron was first discovered by rabi and his colleagues @xcite as an anomaly in the hyperfine structure of hydrogen . immediately that was interpreted as a possible anomaly related to a free electron and only afterwards was that confirmed by a direct experiment . a historic overview of the ` contribution ' of the hydrogen atom to modern physics can be found in @xcite . levels are labelled by the values of the principal quantum number @xmath1 , orbital moment @xmath2 , electron angular momentum @xmath3 and atomic angular momentum @xmath4 , where @xmath5 is the nuclear spin . the gross structure ( @xmath6 ) is well explained by the bohr theory ( so - called ` old quantum theory ' ) and schrdinger theory which also predicts the hyperfine structure ( @xmath7 ) . the fine structure ( @xmath8 ) is explained by the dirac theory while the lamb shift ( @xmath9 ) is a result of bound state qed effects . meanwhile effects of quantum electrodynamics for free particles are responsible for the @xmath10 anomaly in the hyperfine structure . [ 00hydr ] ] one can expect that the simplest atoms are the easiest for a theoretical prediction . that is true only in part . a simple atom usually possesses a simple spectrum which has relatively simple and transparent properties . the atomic energy levels are often perturbed during measurements by various external factors such as a residual magnetic or electric field . because of the simplicity of the spectra , there is a good chance to understand the influence of those external factors on simple atoms . as examples of the highest - precision experiments on simple atoms , we remind here that the hyperfine interval in the ground state of hydrogen was the most accurately measured physical quantity for a few decades and today the @xmath11 interval in the hydrogen atom is among the most precisely measured values . decade after decade , theorists and experimentalists investigated simple atoms . as a result their theory is the most advanced atomic theory and it has to compete with very sophisticated experiments . the theory of simple atoms goes now far beyond non - relativistic quantum mechanics with the reduced mass of an electron . one has also to take into account relativistic effects , recoil effects , quantum electrodynamics , effects of the nuclear spin and nuclear structure . in an early time of modern physics a list of simple atoms consisted of hydrogen only and later deuterium , neutral helium and helium ion were added . now the list is much longer and quite impressive . it also contains tritium , hydrogen - like ions of almost all elements ( and other few - electron ions ) . artificial two - body atoms are the easiest to produce and the list also includes muonium and positronium , muonic atoms , pionic , kaonic and antiprotonic atoms , exotic bound systems of two unstable particles ( such as pionium and @xmath12-atoms ) and antihydrogen . often accuracies of theory and experiment are not compatible . however , there is a broad range of effects , for which theory and experiment approach the same high level of accuracy . the study of such effects forms a field called _ precision tests of bound state qed _ , which is reviewed in part here . a number of sources have contributed to uncertainty of such tests , and the current accuracy of qed calculations for free particles and two - body atoms is not a limiting factor for qed tests . the accuracy of the tests is limited by one of the three other sources : * an experimental uncertainty ; * an inaccuracy of taking into account effects of the strong interactions ; * an uncertainty due to a determination of the fundamental constants . the latter enters consideration because theory is not in a position to give itself any quantitative predictions . it provides us with some expressions containing values of certain fundamental constants , such as the rydberg constant @xmath13 , the fine structure constant @xmath14 , the electron - to - proton mass ratio etc . to make a prediction , one needs first to determine the fundamental constants by extracting their values from some other experiments . thus , theory serves as a bridge between different experiments . that makes the determination of fundamental physical constants to be another important part of precision physics of simple atoms . the contemporary situation with the qed uncertainty being below the total uncertainty of any qed test is a result of significant theoretical progress for the two last decades . twenty or thirty years ago the qed uncertainty was often the dominant one . the _ precision tests of qed _ form a _ multidisciplinary _ field involving atomic , nuclear and particle physics , laser spectroscopy , frequency metrology , accelerator physics , advanced quantum mechanics , quantum field theory etc . those tests are rather not to test qed itself , but to check the overall consistency of the results and methods from different fields and in part to search for possible new physics beyond the standard model . in doing more and more advanced qed calculations , we also need to verify our approaches to deal with infrared and ultraviolet divergences , renormalization and bound state problem for few - body atoms in the case of high order terms of perturbation theory . as already mentioned , the simplest object to test any theory is a free particle . a study with free leptons provides us with a possibility of testing the qed lagrangian . the most accurate data for a free particle are related to the anomalous magnetic moments of an electron and a muon . in the former case the limitation of the accuracy of a comparison of theory versus experiment is due to a determination of the fine structure constant @xmath14 while the latter qed test is limited by effects of strong interactions entering via hadronic intermediate states for the vacuum polarization effects and by the experimental accuracy . the qed theory of the anomalous magnetic moment is quite advanced including many hundreds of multiloop diagrams ( up to the four - loop level for the electron and five - loop level for the muon ) . that differs from a so called _ bound state qed _ , a quantum electrodynamics theory of simple atoms , which deals mainly with relatively few one - loop and two - loop diagrams , but those are not for free particles , but for the particles bound by the coulomb field . three - loop contributions are rather below the uncertainty of most of the bound qed calculations and if necessary can be calculated neglecting binding effects . these two qed theories , the free qed and the bound state qed , are very different in their approaches , problems and applications and it is worth to consider their tests separately . the bound state problem makes all calculations more complicated . intending to perform a calculation within the free qed , one can in principle find needed basic expressions in a number of textbooks . on the contrary , the bound state qed is not a well - established theory and there are no published common prescriptions for the relativistic quantum bound problem . it involves different effective approaches to solve the two - body bound problem . precision tests of the bound state qed offer a number of different options : * the approaches for the bound problem can be essentially checked with the low-@xmath15 two - body atomic systems like hydrogen and deuterium , neutral helium and helium ions , muonium , positronium , etc . at low value of the nuclear charge @xmath15 the binding energy is of order of @xmath16 and it is essentially smaller than the energy related to the rest mass @xmath17 of the orbiting particle . that is a regime of a weak coupling in contrast to the high-@xmath15 physics . the weak coupling allows efficient use of a perturbation expansion over the coulomb strength @xmath18 . many contributions of higher - order in @xmath18 are calculated diagrammatically and thus are closely related to other qed diagrams ( e.g. , for scattering theory ) . * studies of high-@xmath15 ions are related to a strong coupling regime , however , it is unlikely to provide us with more information on bound state qed because of substantial contributions due to the nuclear structure . such an investigation is rather useful for testing different nuclear models . however , in some particular cases , atomic systems with not too high @xmath15 can give some important information on higher order terms of the qed @xmath19 expansion . importance of particular atomic systems also depends on the energy interval under study . for instance , the hyperfine structure ( hfs ) interval depends more strongly on the nuclear - structure effects than the lamb shift , and the related calculations involve more details of the nuclear structure . the accuracy of the nuclear - finite - size contribution is often claimed to be very high , however , their uncertainty is customarily not well verified . it is often estimated from a variation of the result for an energy shift after application of certain models of the charge distribution while the charge radius is fixed . however , to claim such a calculation as an _ ab initio _ result , one has to reconsider first the original scattering and spectroscopy data on the nuclear form factor and related nuclear theory and to check how strongly they constrain the shape of the distribution in general and its parameters and afterwards to question how much the nuclear - size correction varies within allowed shapes and parameters . that may be done only on a nucleus - by - nucleus basis . lack of such a consideration in study of high-@xmath15 study reduces importance of the bound - state qed calculations , which are in such a case one more confirmation of applicability of the phenomenological model of the charge distribution . * studies of few - electron atoms involve electron - electron interactions . in the case of high @xmath15 ions , the electron - electron interaction is proportional to @xmath14 , while the interaction between an electron and the nucleus is proportional to @xmath18 . if the ion is highly charged with only few electrons remaining , the electron - electron interaction can be treated as a perturbation . as a result , at high @xmath15 the electron must be treated relativistically , i.e. , no expansion over @xmath18 can be used , however , the treatment of the electron - electron interaction as a perturbation leads to the @xmath20 expansion . in the case of light atoms , electrons are essentially nonrelativistic but the electron - electron interaction is compatible with the nucleus - electron interaction . the few - electron atoms ( like , e.g. , the neutral helium atom or heavy lithium - like ions ) is the only case when the uncertainty of the qed calculations used to be a limiting factor for a comparison of theory versus experiment . * there are some other two - body atoms under investigation . they contain a muon or a hadron as an orbiting particle . the orbit in a muonic atom is much closer to the nucleus than in the case of a conventional atom with an orbiting electron , and the muonic energy levels are much more sensitive to the nuclear structure . in the case of a hadronic atom , the orbit lies even lower than in a muonic atom , while the interaction of the orbiting particle and the nucleus is dominated by the strong interaction , in contrast to the electromagnetic interaction in conventional ( i.e. , electronic ) and muonic atoms . different exotic or muonic atoms offer a unique opportunity to study certain particle properties by spectroscopic means with high precision . frequently high-@xmath15 spectroscopy is quoted as a qed test _ at a strong field_. however , that is not exactly the case . a value of macroscopic meaning , such as the electric field strength * e * , has not much sense inside an atom . other details are more important . for example , the strength of the field can be characterized by the average potential energy @xmath21 which increases with the nuclear charge @xmath15 and the mass of the orbiting particle @xmath22 and decreases with the principal quantum number @xmath1 . the strongest field is related to high-@xmath15 atoms with an orbiting particle , heavier than electron , at the ground state . muonic atoms have been studied for a while and with @xmath23 they offer a test at a field stronger than in electronic atoms and at shorter distances @xmath24 the distance @xmath25 ( or a related characteristic value of the momentum transfer @xmath26 ) is another important characteristic of the electric field inside an atom . we also note that a characteristic value of the potential @xmath27 , the distance @xmath28 and the strength of the field @xmath29 strongly depends on a particular quantity under study . what we present above in eqs . ( [ eaver ] ) and ( [ raver ] ) is related to the leading contributions to atomic energy levels . higher order corrections , such as qed corrections , may involve various short - distance effects with characteristic distance @xmath30 and momentum transfer of about @xmath31 . they correspond to higher field than long - distance effects . the case of _ strong fields _ at short distances in which a characteristic momentum transfer is higher than @xmath32 leads to an enhancement of polarization effects of the electron - positron vacuum . that makes the physics of muonic atoms quite different from that of conventional atoms . high-@xmath15 ions offer another opportunity a _ strong coupling _ regime with the binding energy comparable to @xmath17 which implies a completely relativistic consideration . the strong - coupling regime is very different from perturbative weak - coupling one . the calculations are more complicated . they may be of interest for study of methods to be applied in the case of strong interactions . some of the high-@xmath15 results are important also for high - accuracy low-@xmath15 physics . however , one has to remember that the strong coupling is rather not a problem of qed , but a problem of its application . in this sense , the stronger the coupling , the less ` simple ' the system . to study pure qed problems in more detail we should prefer a weak - coupling situation . in muonic atoms the coupling constant and other parameters can take quite different values . while for @xmath33 the states of interest are @xmath34 , the principal quantum number @xmath1 for medium @xmath15 may be higher than 1 or 2 , which used to be common for the high-@xmath15 experiments with conventional atoms . however , in both cases ( muonic / exotic atoms and high-@xmath15 ions ) , understanding the nuclear properties is needed for any theoretical predictions . for instance , let us look at studies of the @xmath35 lamb shift in hydrogen - like uranium . recently , the experimental result was improved @xcite . the experimental uncertainty allows to check one - loop radiative corrections , but not two - loop effects calculated few years ago @xcite . those two - loop corrections are an object of intensive theoretical study and are of great interest ( see sects . 5 , 7 and 13 ) . the finite - nuclear - size uncertainty for the lamb shift in u@xmath36 is estimated at a level approximately tenfold below the experimental uncertainty . however , the uncertainty of the result was obtained @xcite ( see also @xcite ) by comparison of two distributions of the nuclear charge , which were the homogenous spherical distribution and the fermi distribution . the value of the mean square radius was fixed as @xmath37fm for both distributions . however , this result was obtained in ref . @xcite from muonic uranium spectroscopy suggesting a modified fermi distribution , which is different from both distributions applied in @xcite . it was stated @xcite that the uncertainty presented there was of pure statistical nature , while the model - dependence had not been studied and related systematic error was not included . apparently , the characteristic atomic momentum in muonic uranium is much higher than in the conventional hydrogen - like uranium ion and muonic spectra are substantially more sensitive to the nuclear - shape effects than conventional ones . if one expects that a comparison of the homogenous spherical distribution and the fermi distribution leads to a plausible estimation of the finite - nuclear - size effects ( which should be verified ) , that should be applied first to muonic atoms to learn the systematic error of the mean square radius then , with a value of the radius related to each distribution , one should calculate the energy levels . that should substantially increase uncertainty . this example shows that how fragile qed tests with high-@xmath15 ions can be and how much work should be additionally done to develop them . a purpose of this paper is to give a brief review on _ precision _ physics of simple atoms related to the _ accurate _ tests of quantum electrodynamics for bound states and the determination of fundamental constants . because of that , we focus our considerations mainly on light hydrogen - like atoms ( hydrogen , deuterium , helium ion , muonium , positronium ) and some medium @xmath15 ions , where the nuclear structure and hadronic effects are not too significant and allow a crucial test of advanced qed theory with high order contributions . we distinguish here qed and bound state qed , which example is an application of qed to the simplest atoms . studying less simple atoms we deal not with just bound state qed , but with its realization for not - too - simple atoms . the additional problem may be related to strong field , strong coupling , crucial effects due to the nuclear structure , electron - electron interaction in few - electron atoms etc . definitely , a number of investigations of less simple atoms are of interest . however , dealing with them is to go beyond the simplest bound state qed . we note , that the light hydrogen - like atoms are the most attractive from a theoretical point of view . they involve neither electron - electron interactions , nor strong - coupling effects or so . we consider in the next section , what is the most favorite choice for experimental accuracy options in testing bound state qed for hydrogen - like atoms . that is related to the light atoms . they are also favorite in principle for theoretical accuracy , being the simplest atomic systems . that does not mean that study of other atoms are out of interest . first of all , what is important is not just an atomic system , but a certain transition there . as one can see in this review , certain transitions , or combinations of certain quantities , related to different transitions , may offer various theoretical or experimental advantages . because of the simplicity of simple atoms and multidisciplinary nature of the _ precision tests of the bound state qed _ we have tried to review the results as simply and briefly as possible in order to make the paper readable by non - experts . detailed reference information on crucial theoretical contributions is collected in tables . while considering qed , there is always the problem of selecting units . from a theoretical point of view one should like to simplify equations by applying relativistic units in which @xmath38 , or using the atomic units . meanwhile , from an experimental point of view the equations should be expressed in units convenient for measurements . in our paper we choose a compromise . all results are presented in the units of the si . however , we present , when possible , most of the results using natural constants as a kind of units . for example , to define the so - called fermi energy which is a result of the non - relativistic interactions of the electron and nuclear magnetic moments , we write for hydrogen @xmath39and thus the proton and electron magnetic moments are explicitly expressed in units of the bohr magneton . the other factors do not directly contain electron charge @xmath40 , but only the fine structure constant @xmath14 , which does not depend on a choice of ` macroscopic ' units ( in which it may be defined as @xmath41 , @xmath42 , @xmath43 depending on the definition of the unit for the electric charge ) . still , here we make an exception for numerical values of atomic energy , which are always expressed in frequency units , i.e. , in terms of @xmath44 . this is because it is widely preferred to write equations for energy , while the actually measured quantities are the transition frequencies , spectroscopic linewidths and decay rates . the most frequently used notations are explained in appendix [ s : not ] . more details on physics of hydrogen - like atoms can be found in : * various basic questions in books @xcite ; * an overall review with an extended comparison of theory and experiments related to sixties and early seventies in @xcite ; * minireviews on particular questions in @xcite ; * review on theory of light hydrogen - like atoms in @xcite ; * original results presented at _ hydrogen atom _ conferences and on international conferences of _ precision physics of simple atomic systems _ ( psas ) in @xcite . the books @xcite published in series _ lecture notes in physics _ , volumes 570 and 627 , are also available on - line . the recent psas conference on simple atoms took place in early august 2004 in brazil as a satellite meeting to the international conference on atomic physics ( icap ) . the coming psas meeting is scheduled for june 2006 in venice . a few problems related to our paper are not presented here in detail . * we consider the fundamental constants only in connection to simple atoms and qed . more detail on fundamental constants can be found in @xcite . * heavy few - electron ions , which are of a great interest for study of application of bound state qed to strong - coupling and few - electron systems , are reviewed in , e.g. , @xcite . * recent progress with exotic and muonic atoms is presented in detail in @xcite . study of such atoms are not of a big interest because of qed , on contrary , they deliver us a crucial information on other part of physics , namely , particle and nuclear physics . most of this review was ready before new results on the fundamental constants @xcite became available and through out the paper we compare the qed - related results on the fundamental constants with a previous set of the recommended constants @xcite . we also note that a substantial part of qed results under review appeared between the publications of two recommended sets ( their deadlines for collecting the input data were 1998 and 2002 ) and most of recent results have been accommodated in @xcite . we start our review with an introductory discussion of spectra of simple atoms and related basic qed phenomena and next consider qed tests with hydrogen ( the lamb shift and hyperfine structure ) and other light atoms . we discuss studies of pure leptonic atoms such as muonium and positronium . in addition to spectra we consider the magnetic moments of bound particles in various two - body atomic systems . the concluding sections of the paper are devoted to fundamental constants and problems of the bound state qed . let us discuss the spectrum of simple two - body atoms in more detail . the gross structure of atomic levels in a hydrogen - like atom comes from the schrdinger equation with the coulomb potential and the result is well known : @xmath45 where @xmath15 is the nuclear charge in units of the proton charge , @xmath46 is the reduced mass of the atomic orbiting particle ( mostly , an electron ) @xmath47 here , @xmath22 and @xmath48 are masses of the orbiting particle and the nucleus . * relativistic corrections ( one can find them from the dirac equation ) ; * hyperfine structure ( due to the nuclear magnetic moment ) ; * recoil corrections ; * radiative ( qed ) corrections ; * nuclear - structure corrections . a structure of levels with the same value of the principal quantum number @xmath1 is a kind of signature of any atomic system . for most of the precision applications the substructure of interest is related to @xmath49 . the corrections decrease with a value of the principal quantum number as @xmath50 or faster . the only exception is the uehling correction for muonic and exotic atoms which scales as @xmath51 for medium and high @xmath15 . .various contributions to the energy levels . the results are in units of @xmath52 , where @xmath22 is the mass of the orbiting particle . here : @xmath48 is the nuclear mass and @xmath53 is the proton mass which enters equations if one measure the nuclear magnetic moment in units of the nuclear magneton . a contribution of the nuclear magnetic moment , i.e. , the hyperfine structure , appears if the nuclear spin is not zero . @xmath54 stands for the nuclear ( charge ) radius.[tmainqed ] [ cols="<,^,^,^ " , ] a. gubmeridze , th . stlker , d. bana , k. beckert , p. beller , h. f. beyer , f. bosch , s. hagmann , c. kozhuharov , d. liesen , f. nolden , x. ma , p. h. mokler , m. steck , d. sierpowski , and s. tashenov , phys . lett . * 94 * , 223001 ( 2005 ) . g. f. bassani , m. inguscio and t. w. hnsch ( eds . ) , _ the hydrogen atom _ , proceedings of the simposium , held in pisa , italy june , 30july , 2 , 1988 . 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quantum electrodynamics is the first successful and still the most successful quantum field theory . simple atoms , being essentially qed systems , allow highly accurate theoretical predictions . because of their simple spectra , such atoms have been also efficiently studied experimentally frequently offering the most precisely measured quantities . our review is devoted to comparison of theory and experiment in the field of precision physics of light simple atoms . in particular , we consider the lamb shift in the hydrogen atom , the hyperfine structure in hydrogen , deuterium , helium-3 ion , muonium and positronium , as well as a number of other transitions in positronium . additionally to a spectrum of unperturbed atoms , we consider annihilation decay of positronium and the @xmath0 factor of bound particles in various two - body atoms . special attention is paid to the uncertainty of the qed calculations due to the uncalculated higher - order corrections and effects of the nuclear structure . we also discuss applications of simple atoms to determination of several fundamental constants . simple atoms , precision measurements , bound states , quantum electrodynamics ( qed ) , hydrogen - like atoms , fundamental constants 12.20.fv , 12.20.ds , 31.30.jv , 06.02.jr , 31.30.gs , 36.10.dr , 13.40.em , 13.40.gp , 27.10.+h

introduction spectrum of simple atoms and main qed phenomena

arxiv

as a consequence of the interaction of galactic cosmic ray particles with air nuclei in the upper atmosphere a wide variety of secondary particles are produced . most of these particles are pions . they initiate an electromagnetic shower via @xmath2 production followed by the decay @xmath3 ( called the soft component ) . the shower has a muon component mainly from @xmath4 production followed by the decay @xmath5 ( called the hard component ) . the count rates of these particles by detectors at ground level is characterized mostly by the geomagnetic value of the site where the detector is placed . the solar activity , including the transient solar events such as solar flares , has a strong influence on the count rates of particles in the sub - gev to gev energy region and under certain conditions they give origin to the so called ground level enhancements ( gles ) . there is also a small fraction of muons that have their origin in photo - nuclear reactions induced by primary gamma rays . the ability to distinguish these `` photo - nuclear '' muons from background muons depends upon statistics , the strength and energy spectrum of the emitting source and some characteristics of the experiment such as the angular resolution . the tupi telescope can detect muons at sea level with energies greater than the @xmath6 gev required to penetrate the two flagstone or walls surrounding the telescope ( see appendix a ) . the tupi muon telescope is sensitive to primary particles ( including photons ) with energies above the pion production energy . in the case of charged particles , the minimal primary energy must be compatible with the ( niteroi - brazil ) geomagnetic cut - off (= 9.8 gv or 9.8 gev for proton ) . due to its limited aperture ( 9.5 degrees of opening angle ) , the tupi telescope is on the boundary between telescopes with a very small field of view , like the air cherenkov telescopes , and the small air shower arrays , characterized by a large field of view . sunspot groups can trigger solar flares and coronal mass ejection ( cme ) , and exhibit a complex magnetic field that harbors energy for powerful eruptions . solar flares and cmes occurs whenever there is a rapid large - scale change in the sun s magnetic field . in general the solar active region that produced the eruption has a complicated magnetic configuration . so far , we have found reports of the detection at ground level of gles associated to solar powerful flares of large scale , those with a x - ray prompt emission classified as x - class ( above @xmath7 ) @xcite , as well as the enhancement of muons at ground level from powerful solar flares have been reported @xcite . observations in satellites and at ground of solar flares @xcite have led to the identification of two classes of acceleration events : impulsive ( prompt ) and gradual ( post - eruptive or delayed ) . the impulsive events require selective acceleration such as the gyroresonant interaction with plasma waves . the energetic particles from these events arrive very quickly , around 15 - 25 minutes after a flare . in contrast , the gradual events have a strong association with coronal mass ejection ( cme ) and suggest that the particles in these events are accelerated by cme driven shocks . the energetic particles from these events are observed up to several hours after a flare . the effect on the interplanetary medium occurs preferentially during this post - eruptive phase @xcite . in order to see a possible photo - muon excess in the direction of the galactic center , since june 2003 a set of observations have been conducted using the tupi telescope and several gles have been found . here , we present a summary of the results obtained from these observations and a search for a possible correlation between these gles and satellite transient events due to solar x - ray activity . we present several correlations between flares whose x - ray prompt emission are of small scale and muon enhancements at ground level . this paper is organized as follows : in section 2 the tupi muon telescope is presented , the experimental setup and its sensitivity to small scale flares is briefly discussed . in section 3 the methods of observation including the raster search technique and pressure corrected relative intensity are presented . the section 4 contains the justification for the criteria of association between gles and solar flares used in the present study . the results of the search for the origin of the gles are shown in section 5 , including some comments , and section 6 contains conclusions and remarks . figure 1 is a scheme of the tupi muon telescope , it has an equatorial assembly and a servo - mechanism which allows the axis of the telescope to be pointed so as to accompany a given source @xcite . the telescope computes mainly the muon intensity in the atmosphere initiated by cosmic rays ( mainly protons ) , giving the coincidence counting rate of two - elements a and b ( plastic scintillator detectors with @xmath8 ) placed perpendicularly to the axis of the telescope . the main task of the first - level trigger is a coincidence between the a and b detectors . the second - level trigger is a veto for air showers coming from other directions , far off the telescope s axis direction and contains other two detectors ( c and d plastic scintillator ) off the telescope axis . each plastic scintillator is viewed by a 7 cm diameter photomultiplier and the unit has a nominal energy threshold of @xmath9 for charged particles . with these characteristics the telescope has a effective angular aperture or geometrical factor of @xmath10 , projecting to the space a cone with an open angle of @xmath11 in relation to the telescope s axis . the tupi muon telescope is installed on the campus of the universidade federal fluminense , niteri , rio de janeiro - brazil . the position is : latitude : @xmath12 s , longitude : @xmath13 w , at sea level . at sea level and in the sub - gev to gev energy region , the muon s flux is @xmath14 times higher than the nucleons flux , and @xmath15 times higher than the electrons flux @xcite . consequently the telescope detect mainly muons with energies greater than the @xmath6 gev required to penetrate the two flagstones or walls surrounding the telescope . the concrete reduces the noise due to other non - muon particles , for example it is opaque to electrons ( see appendix a for details ) . outside of the building where the tupi telescope is deployed , there is an automatic meteorological station and we have free access to the data . in addition , we have also our own barometer . together with the muon counting rate , the pressure is registered every 10 second . the station is very useful because its data is used as reference and calibration and other parameter as the outside temperature , atmospheric humidity , and wind speed , among others are available . the data acquisition is made on the basis of an advantech pci-1711/73 card with a counting rate up to 100 khz . all the steps from signal discrimination to the coincidence and anti - coincidence are made via software , using the virtual instrument technique . the application programs were written using the lab - view tools . it is commonly accept that a solar flare must be powerful to be a candidate for association with a gle . initially we do not believe that small - scale flares such as of the c class could cause muon enhancements . however , now we have other opinion and in this section , we will try to respond to the question , which are the main factors that increase the tupi telescope s sensitivity to small scale flares ? ( a)the gles as observed by the tupi telescope at sea level are characterized by an impulsive peak with a fast rise time ( see section 5 ) . this is a signature indicating that they are constituted by a bundle of muons produced in the atmosphere by the arrival of a coherent particle pulse , probably from the sun ( see section 4 ) . this mean that the particles from the pulse front arrive at the earth almost simultaneously . in order to discriminate ( to count ) a small coherent particle pulse , a detector working with a rather fast response time , better than milliseconds , is necessary in order to reduce the dead time , during which the detector may not respond to the incident radiation . the tupi telescope has a counting rate up to @xmath1 khz and at this frequency it is possible to obtain a response time as short as @xmath16 . table i shows the characteristics of several `` solar particles '' detectors at ground level @xcite , including the tupi telescope . from table i it is possible to see that typical neutron monitors ( nm ) , nm-64 , nm - scintillator and igy types , have operated with a counting rate of @xmath17 khz . \(b ) due to the tupi telescope s tracking system , the pitch angle defined as the angle between the sun - ward direction and the telescope axis direction , is always the same ( constant ) during a raster scan . this characteristic probably also helps to increases the sensitivity because the energetic solar charged particle propagation follows the interplanetary magnetic field lines ( imf ) , the rotation of the sun gives the magnetic field a spiral form ( garden hose effect ) the pitch angle of the imf at 1 au is @xmath18 ( this value does nt include the magnetic field of the earth ) . if the telescope axis is oriented near or close to the direction of these imf lines , as is schematized in fig.2 ( see also the section 4 ) , the solar particle sources will be magnetically well connected to the direction of the telescope axis and the detection efficiency will be close to maximum . this favorable situation increases the telescope s sensitivity . the atmospheric muon flux originating from the decay of charged pions and kaons produced by galactic primary cosmic rays in the atmosphere constitute the main source of background at sea level and in the energy region of sub - gev to gev . however , at these energies the muon flux is influenced by the magnetic field of the earth @xcite . consequently the muon angular distributions are quite different for different sites on the globe . there are two main geomagnetic effects on the muon flux observed at ground level : a ) the east - west effect , the muon flux is highest ( lowest ) for a direction coming from the west ( east ) . b)the azimuth dependence on the positive - to - negative ratio of muons , a considerable amount of the negative excess is observed for muons coming from the east , because a positive muon coming from the east has a longer path length than a negative one . these geomagnetic effects distort the zenith angle distribution of sub - gev to gev muons during a raster scan , because the measures ( galactic center ) begin around the south - east direction and they finish around the south - west to southerly directions . the observation of the galactic center ( @xmath19 , @xmath20 ) began on june 26 , 2003 and consists of on - source and off - source tracking runs or raster scans , where each tracking run represents a sidereal half - day ( 12 hours ) . this is approximately the time that the galactic center is above our horizon in every 24 hours . basically each tracking run is a time series which is generated registering the coincident counting rate every 20 seconds and initially in order to avoid the background contamination a very high ( pulse - height ) discrimination level has been used . however , starting from 2003 december , after a up - grade , the counter rate every 10 seconds and a low pulse - height discrimination have been used to observe a possible muon excess in the direction of the galactic center . until february 28 , 2005 we have completed 2160 hour of observation which constituted 180 raster scans ( 12 hour each ) . in this paper we are presenting the first six gles , all of them have the potential to be considered as solar flare associated , because there is some correlation with solar flares , whose prompt x - ray emission have been reported by the goes spacecraft ( noaa space environment center website)@xcite . in all cases except one , the gles were detected with a statistical significance of more than @xmath21 and they are described in section 5 . in order to taken into account the barometric pressure effect in the muon intensity , the muon counting rate is transformed to the `` relative intensity '' by using the natural logarithmic representation ( nlr ) . the algorithm comes from the exponential dependence of the counting rate on the pressure @xmath22,\ ] ] where @xmath23 is the barometric coefficient . it has a value of @xmath24 per mbar ( in neutron monitors experiments ) . however , for the muon component , @xmath23 has a value below @xmath25 , varying from station to station . @xmath26 is the reference s pressure ( average pressure ) . fig.3 shows the hourly average distribution of the atmospheric pressure , obtained in 2003 at niteroi city with @xmath27 . @xmath28 is the average counting rate . in the nrl representation the relative intensity is defined as @xmath29 in this way , the pressure correction on the relative intensity is given by @xmath30 following the last relation , it is possible to see that the relative muon intensity is in @xmath31 with the pressure , or in other words , if the atmospheric pressure @xmath32 is larger than the average value @xmath26 the relative muon intensity decreases and vise versa . in this paper , every raster scan ( raw data ) is divided in bins of 15 minutes and then we apply the pressure correction algorithm . from fig.3 it is possible to see that the bandwidth of the pressure variation is around @xmath33 mbar ( at 95% confidence level ) and correspond to a variation in the muon relative intensity of up to @xmath34 the occurrence of gles ( muons excess ) as observed by the tupi telescope can be interpreted as evidence of the arrival in the upper layers of the atmosphere of a bundle of protons and/or ions with energies exceeding the pion production and above the local geomagnetic cut - off ( 9.8 gv ) . however , the central question is : are they associated with small - scale flares ? in the affirmative case , what are the acceleration mechanisms to gev energies ? in order to give a convincing answer to these questions an analysis on the basis of spacecraft data was made . the statistical studies on the association between the soho / lasco cmes and the goes x - ray flares provides us with useful tools to the analysis on the nature of the gles apparently associated with small x - ray flares . in the case of gradual events , whose origin results from the acceleration of ambient coronal ions ( i.e. protons , alpha etc ) by shocks driven by cmes , a plausible explanation ( among others ) for the low rate of gles associated to small scale solar flares is the so called `` filtering effect '' @xcite . this effect has been obtained from recent solar observations on the basis of the soho / lasco cme catalog and the solar geophysical data ( sgd ) for the same period . the results have reveled that mass ejection are ubiquitous in solar flares . however , if the flare is less energetic , the hot plasma is surrounded by a strong magnetic field and can not escape from the corona . in this case there is not cme ejected to the interplanetary space , there is only the x - ray emission . this effect can explain the shape of the distribution of the x - ray peak fluxes of cme - associated flares and that it is of the type log - normal distribution . on the other hands , the distribution of the number of flares per unit energy and per unit time follows a power law @xmath35 , with @xmath36 and this index practically does not vary with the solar cycle @xcite . consequently the number of small flares is high , while the cmes associated with these small flares is low . they are in the left tail of the log - normal distribution of the x - ray peak flux of cme - associated flares . the mean value ( peak of the log - normal distribution ) is around @xmath37 . the above results are compatible with other statistical studies of solar flares and cmes . particularly with the waiting - time distribution , the distribution of times @xmath38 between events @xcite . both the solar flare and cme waiting - time distributions are closely related by a poisson distribution . the comparison between the waiting - time distribution for lasco cmes and for goes x - ray flares of greater than c1 class for the years 1996 - 2001 has shown an excess of flares in relation to the cmes in the waiting - time region up to @xmath39 hours . this is exactly the region of small scale flares . the nature of solar flares associated with gles via coronal mass ejection is poorly understood . solar flares and cmes often occur together , but not necessarily because the flare triggers the cme or vice versa . especially during solar maximum , there are cmes without an associated flare . however , the effect can be due to high solar activity , because the x - ray background is hight and the flares with a small scale x - ray emission can be masked , while small scale flares without an associated cmes are more frequents due to the `` filtering effect '' . the conditions under which a cme is associated with a flare are found by addressing the relationships between their positions and timing the flare start and the appearance of the cme . the association of events between soho / lasco cmes and goes-8 x - ray flares ( band - width @xmath40 ) @xcite have shown that the cmes not associated with metric radio burst ( cmes no metric type ii ) have a delay in relation to the flare start of @xmath41 ) . the x - ray peak flux of these flares have a mean value of @xmath42 , consequently most of them are similar to c class . the cmes such as metric type ii are associated mostly with flares of m class . the association between a x - ray flare and a gle requires taking into account the delay between the flare start and appearance of cme plus the time of flight between the sun and the earth of highly energetic particles . the time of flight is estimated from results of simulations under the assumption that the propagation of energetic particles released ( for instance ) by a cme driven shocks are injected into the interplanetary medium in coherent pulses of energetic particles and a realistic archimedean spiral field lines around the sun @xcite . here , we summarized simulation results for a typical arc length along the magnetic field ( garden hose direction ) , @xmath43 , as a function of the distance traveled , @xmath44 , which can be expressed as @xmath45 the constant of proportionality , @xmath46 , depend on the scattering mean free path , @xmath47 , and satisfies the constrain condition @xmath48 for @xmath49 , sometimes called `` scatter - free '' condition and implies that particles freely stream along the field at their maximum speed like a coherent particle pulse . there are two extreme cases for high energy solar particle propagation to @xmath50 au , depending on how the particles are injected into the interplanetary medium : when the scattering mean free path is very small compared to the scale length of the imf(i.e.@xmath51 ) . the particles propagation follows helical trajectories around the field of the imf . fluctuations of small scale of the imf act as scattering centers of the particles and the propagation is dominantly diffusive . in contrast , when the scattering mean free path is compatible with the distance from particles source ( i.e. @xmath52 the focusing effect of the interplanetary magnetic field lines became dominant and the propagation of the energetic particles is like coherent pulses , following trajectories around the field line of the imf . gles constituted by muon bundles , detected at sea level by a directional telescope , are a signature of primary particles arriving at the top of the atmosphere with a strong anisotropic pitch angle distribution `` coherent pulses '' , as well as , the very short rise time in the profile time of the gles suggest a non - diffusive coherent particle pulse transport . under the non - diffusive transport assumption and for a typical average value of @xmath53 , the time of flight of coherent pulses of energetic particles with a mean rigidity of @xmath54 is estimated as @xmath55 minutes . consequently , we look for the more energetic ( larger peak flux ) and nearest x - ray flare from the gles with a delay of @xmath56 hours . the criteria comes from the `` filtering effect '' , because the probability of occur of a cme associated flare increase with the energy . on the other hand , the @xmath57 hours window take into account the delay between flare start and appearance of cme plus the time of flight between the sun and the earth of highly energetic particles . on the other hand , even in gles associated with large solar flares , the acceleration mechanism producing particles of up to several tens of gev is not well understood . the situation becomes still more critic in the case of gles associated with solar flares of small scale . here , we argue the possibility of a `` scale - free '' energy distribution of particles accelerated by shock linked with cmes . this possibility comes from the shapes of the energy spectrum of the particles being close to a power - law distribution , because power law distributions are characterized as scale free distributions . the spectral index of these power - law distributions depend on time t in one and the same event and of event to event due to stochastic processes of acceleration and modulation @xcite . the energy spectrum of energetic particles ( protons and ions ) during the early stage of a gle is less steep than the energy spectrum at declining stage and probably the tail of the distributions extend to energies above 20 gev / nucleon . this scale - free hypothesis need further confirmation . we begin with a list ( table ii ) of all associated gles and solar flares established in the present study , according to the criteria established in the last section . in the table ii , a zero degree pitch angle represents the sun - ward direction and the coordinates right ascension and declination and pitch angle are with reference to the telescope axis . therefore the angular aperture of the telescope is a cone of opening angle 9.5 degrees . the plots and data files of the gles reported here are accessible to the public through our web - site @xcite ( see results ) . in second place we show the correlation between the x - ray solar emission and the tupi counting rate when there are no solar flares ( x - ray prompt emission ) . in this case , the solar x - ray emission ( background ) as reported by the goes spacecraft on @xmath58 is as shown in fig.4 ( upper panel ) , the x - ray emission ( band - width @xmath59 a ) is lower than @xmath0 ( the horizontal dash - dotted line shows this limit ) . in fig.4 ( central and lower panels ) , the time profiles of the telescope output ( raw data ) and the relative intensity ( pressure corrected ) are also presented . in spite of the telescope s relatively low angular resolution ( @xmath60 and the small statistics during a raster scan ( 12 hours ) , it is possible to see the characteristics indicated above , such as a muon flux excess from the west direction . the west - east asymmetry is defined as the ( normalized ) difference between the fluxes @xmath61 and is free of calibrations and experimental bias . a quantitative result of the west - east effect during a raster scan can be obtained from two symmetrical points , those with the same zenith angle , with one pointing to the west and the other to the east ( see the two arrows in fig.4 ) ) . in the present case we obtain a west - east asymmetry around 16% . this value is in agreement with similar observations @xcite . in figures 5 , 6 , 7 , 8 , 9 and 10 the time profiles of the goes12 x - ray in the band - width @xmath59 a and @xmath62 a ( upper panel ) , the tupi raw data ( central panel ) and the tupi relative intensity ( in the nlr representation ) pressure corrected ( lower panel ) are shown for six raster scans respectively . when a gle has been correlated with a solar flare , the x - ray peak ( in the @xmath59 a band - width ) of the flare is marked by a flare index . for example , c1.3 means @xmath63 . we would like to make some comments on the experimental data : ( a ) the gle found on @xmath64 is classified as marginal due to the low statistical significance level ( @xmath65 ) in the raw data . however , a clear muon excess in the relative intensity ( after pressure corrections ) can be seen it is practically in coincidence with a powerful x - ray ( @xmath66 class ) prompt emission . the beginning of the gle coincides approximately with the decline of the sun . due to the small difference between the x - ray prompt emission and the gle s peak , the event can be classified as impulsive . the impulsive emission of powerful flares ( such as those of x class ) in most of the cases is observed at ground level by the network neutron monitors . in all the other cases the delay of the gles in relation to the flares suggest , emission via cmes . cmes produce fast shocks which accelerate charged particles ( electrons , protons and ions ) in the magnetically open corona , over a region at least comparable to the size of the cme itself . ( b)in the occurrence of the gle on 2003/12/02 especially the second peak @xmath67 ut , the sun was below the horizon . even so under this condition there is a magnetically well connected field line . in the opposite case , and for low altitude of the sun ( early morning ) , the imf are not well connected with the point of observation on the earth . fig.2 summarized the situation . ( c)due to virtual instrument technique used in the data acquisition system , it is possible to examine the time profile of a gle for other off - line ( higher ) pulse - height amplitude discrimination , chosen via software @xcite . using this `` noise filter '' we have verified that the six gles here presented are not mere background fluctuations . ( d)if the earth s magnetic field is ignored , the best value in the pitch angle to observe solar energetic particles is @xmath68 ( see fig.2 ) . however , due to the geomagnetic effects , the pitch angle should be a little bigger ( around 15%-20% ) . due to the complexity of the phenomenon a full monte carlo is necessary in order to taking into account the effects of the joining of the imf with the earth s magnetic field on the propagation of the solar particles and their secondary particles produced in the atmosphere . the six gles analyzed here were obtained during an search of a possible excess of `` phot - muons '' in the direction of the galactic center , with a pitch angle bigger than that @xmath68 . we have described and analysed ground level enhancements observed during a search for enhancements from the galactic center with `` photo - muons '' at sea level , detected using the tupi telescope . the paper present the first results of an ongoing study of the associations between these gles and solar flares . the main conclusions are summarized as follows : ( a)the tupi muon telescope is sensitive to primary particles ( including photons ) with energies above the pion production threshold . the tupi telescope can detect muons at sea level with energies greater than the @xmath69 gev required to penetrate the two flagstones or walls surrounding the telescope . the concrete reduces the noise due to other non - muonic particles . for example , it is opaque to electrons . b)the muon flux is subject to several sources of modulation , such as the atmospheric pressure variation , and geomagnetic effect , among others . however , the temporal scale of these modulation phenomena are much larger than the gles duration . in addition , no anomalous changes in the atmospheric pressure and temperature were observed during the raster scan where the gles were found . \c ) the impulsive emission of powerful flares ( such as those of x class ) is almost always accompanied by observations at ground level by the network neutron monitors , the delay between the x - ray prompt emission and the beginning of the gle being around ( @xmath70 minutes ) . the gle @xmath64 has this characteristic . however , the other ground level enhancements have a delay of @xmath71 hours in relation to the x - ray prompt emission and suggest an association with gradual or post - eruptive acceleration processes . consequently , the shock driven by cme is an essential requirement for the particle acceleration efficiency . \d ) the tupi telescope has shown the ability to detect variations in the muon fluxes . for instance , it is possible to see the west - east effect embedded in the time series ( raw data ) . the efficiency of the tupi telescope for detecting the enhancement of muons at sea level from a coherent cluster of solar energetic particles , associated to solar flares of small scale is a consequence of several factors such as : ( 1 ) its high counting rate , @xmath1 khz . this value is around 100 times higher than the other telescopes at ground ( see table i ) . the requirement of a high counting rate is essential to discrimination of particles inside a small coherent particle pulse . ( 2)due to its raster scan system , the telescope axis can be oriented close to the directions of the imf lines , favoring the detection of the solar particles . ( 3)the shape of the lateral distribution function of muons at sea level , which is close to a flat distribution . for primary energies below 100 gev @xcite , the fraction of muons that hit the telescope when the core of the shower is at a distance @xmath72 from the telescope center is the same for @xmath72 up to @xmath71 km . in short , we have shown strong experimental evidences of the association between small scale flares and gles , but important questions as acceleration mechanism of the solar particles up to gev energies remain open and requires continued observations and further investigation . the ongoing experiment will deliver much better statistics in the next years . a careful study of the origin of the gles will be continued . this work was partially supported by faperj ( research foundation of the state of rio de janeiro ) in brazil . the authors wish to express their thanks to dr . a. ohsawa from tokyo university for help in the first stage of the experiment , to dr . m. olsen for reading the manuscript and to dr . j.l.fernandes de oliveira from uff university for the free access to the meteorological station data . we are also grateful to the various catalogs available on the internet and to their open data police , especially to the noaa s space environment center ( sec ) . the average muon energy loss rate @xcite in its propagation through the matter is given by @xmath73 where x is the material thickness ( in units of @xmath74 ) . the first term of the last equation ( with @xmath75 ) take into account the ionization energy loss and the second term ( with @xmath76 ) represent the energy loss due to radiative processes , and it is the dominant process only in the energy region above the critical energy ( several hundred gev ) . while for sub - gev to gev energies the ionization became the dominant process to the energy loss of muons . under this assumption the last equation can be written as @xmath77 the tupi telescope is inside a building , under two flagstones of concrete as is shown in fig.1 . the flagstones thickness have a dependence on the zenithal angle , with a average value of @xmath78 . consequently , the flagstones increases the detection muon energy threshold , because the telescope will can detect muons with energies greater than the 0.2 - 0.3 gev required to penetrate the two flagstones . in the case of electrons the energy loss by bremsstrahlung become dominant above a few tens of mev and is nearly independent on energy . for electrons of 100 to 1000 mev , the energy loss can be approximately expressed as @xmath79 with the continuous and fast decreasing of the electrons energy via bremsstrahlung , and for energies below the critical value ( @xmath80 ) the ionization process become more important and they are `` absorbed '' by the flagstones whose thickness is @xmath81 . this mean that the two flagstones are opaque to the electrons . j. bieber et al . , astrophys . journal , 622 ( 2002 ) . a. falcone and j. ryan , astr . , 283 , ( 1999 ) . d. b. swinson and m. a. shea , geophys . research lett . 8 , 1073 , ( 1990 ) . j. poirier and c. dandrea , preprint(astro - ph/0211490 ) . k. munakata et al . , in 27th icrc , hamburg , 3494 ( 2001 ) . d. f. smart , il nuovo ciment , 19c , 765 , ( 1996 ) . l. g. kocharov et al . , in 24th icrc , rome , 4 , 163 ( 1995 ) . c. r. augusto , c. e. navia and m. robba , nucl . methods phys . 503 , 554 ( 2003 ) . t. k. gaisser and t. stanev , phys . lett . , 592 , 228 ( 2004 ) . net of cosmic ray stations , http://cr0.izmiran.rssi.ru/common/links.htm s. hayakawa , cosmic ray physics , nuclear and astrophysical aspects ( wiley - interscience a division of john wiley @xmath82 sons , new york ) ( 1969 ) . noaa space environment center website , world wide web : http://www.sec.noaa.gov . i. aoki et al . preprint astro - ph/0401352 vl . n. b. crosby , n. j. aschwanden and b. r. dennis , solar phys . 143 , 271 ( 1993 ) . m. s. wheatland , solar phys . 214 , 361 ( 2003 ) . a. shanmugaraju , y. j. moon , m. dryer and s. umapathy , solar phys . 217 , 3001 ( 2003 ) . d. ruffolo and t. khumlumlert , in 24th icrc , rome , 4 , 277 ( 1995 ) . v. m. dvoraikov and v. e. sdobnov , in 24th icrc , rome , 4 , 235 ( 1995 ) . the tupi telescope website , http://www.if.uff.br/@xmath83navia m. tokiwa et al . , in 28th icrc , tsukuba , 2709 ( 2003 ) . c. e. navia , c. r. a. augusto , m. b. robba , m. malheiro and h. shigueoka , apj . 621 , 1137 ( 2005 ) . j. poirier , s. roesler , a. fasso , astropart . 17 , 441 ( 2202 ) . h. bichsel and s. r. klein , phys . lett . , 592 , 242 ( 2004 ) .

this paper presents first results of an ongoing study of a possible association between muon enhancements at ground observed by the tupi telescope and transient events such as the sun s x - ray activity . the analysis of the observed phenomenon by using the goes satellite archive data seems to indicate that on most cases the ground level enhancements ( gles ) could potentially be associated with solar flares . we show that small scale solar flares , those with prompt x - ray emission classified as c class ( power above @xmath0 at 1 au ) may give rise to gles , probably associated with solar protons and ions arriving to the earth as a coherent particle pulse . the tupi telescope s high performance with these energetic solar particles arises mainly from : ( 1 ) its high counting rate ( up to @xmath1 khz ) . this value in most cases is around 100 times higher than other detectors at ground and ( 2 ) due to its tracking system . the telescope is always looking near the direction of the imf lines . the gle s delay in relation of the x - ray prompt emission suggest that shock driven by corona mass ejection ( cme ) is an essential requirement for the particle acceleration efficiency .

introduction the tupi muon telescope method of observation gle associated flare results conclusions continuous charged lepton energy loss

arxiv

for an integrable function @xmath0 on the complex unit circle @xmath1 the _ toeplitz matrix _ @xmath2 of size @xmath3 is defined by @xmath4 where @xmath5 is the @xmath6th fourier coefficient of @xmath0 @xmath7 the function @xmath0 is called the _ symbol _ of @xmath2 . in this paper we will be interested in symbols @xmath0 that are rational . that is , we assume that there exist polynomials @xmath8 , @xmath9 and @xmath10 such that @xmath11 where the roots of @xmath9 ( or @xmath10 ) lie inside ( or outside ) the unit circle . thus we do not allow @xmath0 to have poles on the unit circle . we take @xmath8 so that it has no common roots with @xmath9 and @xmath10 . note that if @xmath12 , @xmath13 , and @xmath14 , then reduces to the laurent polynomial @xmath15 where @xmath16 thus we have @xmath17 for all @xmath18 and for all @xmath19 . the matrix @xmath2 is then a _ banded toeplitz matrix_. the integers @xmath20 and @xmath21 in correspond to the outermost non - zero diagonals in the lower and upper triangular part of this matrix , respectively . for a detailed discussion of banded toeplitz matrices see @xcite . we are interested in the asymptotic behavior of the eigenvalues of @xmath2 as @xmath22 . it is known that the eigenvalues accumulate on a particular curve in the complex plane that we will introduce shortly . moreover , there exists a measure on this curve describing the limiting distribution of the eigenvalues . it was shown in @xcite that for banded toeplitz matrices this limiting distribution is subject to an equilibrium problem that is naturally constructed out of the symbol . the purpose of the present paper is to extend this result to the case of rational symbols . let us first review some results on the asymptotic behavior of eigenvalues of rationally generated toeplitz matrices . let @xmath0 be as in and @xmath2 the associated toeplitz matrix . denote the spectrum of @xmath2 as @xmath23 to describe the asymptotic behavior of the spectrum we introduce , as in @xcite , two different limiting sets @xmath24 consisting of all @xmath25 for which there exists a sequence @xmath26 , with @xmath27 converging to @xmath28 , and the set @xmath29 consisting of all @xmath28 for which there exists a sequence @xmath26 , with @xmath27 having a subsequence converging to @xmath28 . it turns out that these limiting sets can be described in terms of solutions to the equation @xmath30 where @xmath31 following the analogy with , we define @xmath32 to avoid trivial cases , in what follows we always assume that @xmath33 , see e.g. @xcite . we also assume without loss of generality that @xmath34 see ( * ? ? ? * page 263 ) . note that @xmath35 in is a polynomial of degree @xmath36 in @xmath37 , with each of its coefficients depending linearly on @xmath28 . there can be at most one value of @xmath25 for which the leading coefficient vanishes . for all other @xmath25 , the polynomial @xmath35 has precisely @xmath36 roots @xmath38 ( counting multiplicities ) and we label them by absolute value as @xmath39 for which the polynomial @xmath40 has less than @xmath36 roots , say @xmath41 , we again order these roots @xmath42 as in and then we set @xmath43 . define the curve @xmath44 the fact of the matter is that @xmath45 this result was first established by p. schmidt and f. spitzer @xcite in the banded toeplitz case , using a determinant identity by h. widom @xcite . the generalization to the case of rational symbols is due to k.m . day @xcite , based on an extension @xcite of widom s determinant identity . let @xmath46 be the counting measure on the eigenvalues of @xmath2 @xmath47 where @xmath48 is the dirac measure at @xmath49 and each eigenvalue is counted according to its multiplicity . it turns out that the measures @xmath46 converge weakly to a measure @xmath50 on @xmath51 . in the banded case the measure @xmath50 is known to be absolutely continuous , and an explicit expression for this measure was given by i.i . hirschman @xcite . an alternative representation of @xmath50 can be obtained by setting @xmath52 in below , cf.@xcite . further results about @xmath50 in the banded case can be found in @xcite . for toeplitz matrices with rational symbol the limiting eigenvalue measure does not need to be absolutely continuous . indeed , it was shown by day @xcite that this measure has an absolutely continuous part together with at most two point masses . finally , we turn to the results of @xcite . consider the general system of curves @xmath53 for @xmath54 . each curve @xmath55 consists of finitely many analytic arcs . we equip every analytic arc of @xmath55 with an orientation and we define the @xmath56-side ( or @xmath57-side ) as the side on the left ( or right ) of the arc when traversing the arc according to its orientation . for @xmath54 we define the measure @xmath58 on the curve @xmath55 . here @xmath59 denotes the complex line element on each analytic arc of @xmath55 , according to the chosen orientation of @xmath55 . moreover , @xmath60 and @xmath61 are the boundary values of @xmath62 obtained from the @xmath56-side and @xmath57-side respectively of @xmath55 . these boundary values exist except for a finite number of points . note that is actually independent of the choice of the orientation . for the banded case it is shown in @xcite that each @xmath63 is a finite positive measure . moreover , @xmath50 is the measure of hirschman , that is , the limit of the normalized eigenvalue counting measures @xmath46 as given in . the main observation in @xcite is that the system of measures @xmath64 together uniquely minimizes an energy functional defined on the system of curves @xmath65 . the purpose of this paper is to prove that also for rational symbols the measures @xmath64 minimize an energy functional , thus generalizing the results in @xcite . the general definition of the energy functional involves point sources that do not occur in the banded toeplitz case this is related to the phenomenon that the limiting eigenvalue distribution possibly has point masses for rationally generated toeplitz matrices , as mentioned before . we also emphasize that the @xmath63 are absolutely continuous . it is to be understood that @xmath50 is the _ absolutely continuous part _ of the limiting eigenvalue distribution , with the possible point masses removed . our results will be stated in detail in the next section . first we introduce some definitions that will be used in the statement of our main theorems . for @xmath52 these definitions will be essentially the ones of day @xcite , but we will state the definitions for general values of @xmath66 . [ def : lambda12 ] recall the notations . define the coefficients @xmath67 , @xmath68 , by @xmath69 ( note the index shift . ) define @xmath70 such that @xmath71 respectively . define @xmath72 to be the maximal indices for which @xmath73 and @xmath74 respectively . here we make the convention that @xmath75 if @xmath76 and @xmath77 , @xmath78 . the numbers @xmath79 in definition [ def : lambda12 ] are the unique @xmath28-values for which the polynomial @xmath80 has some of its roots equal to @xmath81 ( for @xmath82 ) or to @xmath83 ( for @xmath84 ) . in fact , the numbers @xmath85 and @xmath86 are chosen such that @xmath87 has @xmath85 roots equal to zero , and @xmath88 has @xmath86 roots at @xmath83 . for all other values of @xmath28 , @xmath89 has precisely @xmath36 roots ( counting multiplicities ) which are all non - zero and finite . [ rem1 ] by definition we have that @xmath90 . the case where @xmath91 or @xmath92 can not occur since it would imply that all coefficients of @xmath35 are equal up to multiplication with a scalar . this would then imply that the numerator and denominator in are equal up to a scalar factor , contrary to our assumptions . note also that it is possible to have either @xmath93 or @xmath94 , but not simultaneously . indeed , in the latter case we would have that @xmath95 and hence @xmath96 . the latter implies that either @xmath97 or @xmath98 which contradicts the assumption @xmath33 made in the introduction . [ def : mk ] for each @xmath54 define @xmath99 @xmath100 and @xmath101 the numbers @xmath102 and @xmath103 will be the weights of certain point masses , see further . the quantity @xmath104 will be the total mass of the measure @xmath63 in . occasionally we will also consider @xmath102 , @xmath103 and @xmath105 for the indices @xmath106 or @xmath107 . note that the @xmath104 are strictly positive for all @xmath6 . indeed , from the definition of @xmath108 and the fact that @xmath109 and @xmath110 it is easy to check that @xmath111 for @xmath54 . moreover , if @xmath112 for some @xmath6 then @xmath113 . however , in remark [ rem1 ] we observed that this is not possible . [ example : bandedcase ] consider the banded case , i.e. @xmath114 and @xmath76 for all other @xmath6 . in that case we have @xmath115 , @xmath116 and @xmath117 . then the numbers @xmath102 , @xmath103 and @xmath105 in definition [ def : mk ] are given in the following table @xmath118 the last row of contains the masses of the measures @xmath63 appearing in @xcite . here are two examples of possible behavior when @xmath119 and @xmath120 : @xmath121 for the case where @xmath122 or @xmath123 , respectively . this occurs e.g. for the rational symbols @xmath124 below we will consider measures @xmath125 supported on contours in @xmath126 . if the support is unbounded then we will assume that @xmath127 for such a measure @xmath125 define its _ logarithmic energy _ as @xmath128 similarly , for measures @xmath129 define their _ mutual energy _ as @xmath130 ( compare with @xcite ) we call a vector of measures @xmath131 _ admissible _ if @xmath132 has finite logarithmic energy , @xmath132 is supported on @xmath55 , and @xmath132 has total mass @xmath133 for every @xmath54 , recall . the _ energy functional _ @xmath134 is defined by @xmath135 here we define @xmath136 to be @xmath137 when @xmath138 and @xmath81 otherwise . the quantities @xmath139 and @xmath140 are similarly defined . the _ equilibrium problem _ is to minimize the energy functional over all admissible vectors of positive measures @xmath141 . the equilibrium problem may be understood intuitively as follows . on each of the curves @xmath55 one puts charged particles with total charge @xmath104 . particles that lie on the same curve repel each other . the particles on two consecutive curves attract each other , but with a strength that is only half as strong as the repulsion on a single curve . particles on different curves that are non - consecutive do not interact with each other in a direct way . in addition , if @xmath142 and @xmath143 then we have an external field acting on the particles on the curve @xmath144 . similarly , if @xmath145 and @xmath146 we have an external field acting on the particles on @xmath147 . the external fields come from point charges at @xmath148 and @xmath149 , respectively . the minus signs in imply that these point charges are _ attractive_. such external fields are sometimes referred to as sinks . note that there are no external fields acting on the other measures @xmath132 , @xmath150 . note that the external fields acting on the measures @xmath151 and @xmath152 do not occur in @xcite . indeed , in that case we have @xmath115 and hence @xmath136 and @xmath153 in vanish , see example [ example : bandedcase ] . for more information on equilibrium problems with external fields , see @xcite . in order for the above equilibrium problem to make sense , we need the energy functional @xmath134 in to be bounded from below . a proof of this boundedness will be given in lemma [ lemma : boundedbelow ] . for the boundedness it is important to note that @xmath154 which follows immediately from the definitions of @xmath85 and @xmath86 . hence the sinks are not on the contours on which they are acting . the following is our main theorem . [ theorem : main ] recall the notations in , , and and assume that @xmath33 . then * the vector of measures @xmath155 defined in is admissible . * for each @xmath156 there exists a constant @xmath157 such that @xmath158 for @xmath159 . here we let @xmath160 and @xmath161 be the zero measures . * @xmath162 is the unique solution to the equilibrium problem described above . theorem [ theorem : main ] will be proved in section [ section : proofmaintheorem ] . note that the equalities in part ( b ) are precisely the euler - lagrange variational conditions of the equilibrium problem . part ( c ) will then be a consequence of the convexity of the energy functional @xmath134 and the fact that @xmath134 is bounded from below . it was proved in @xcite that in case of banded toeplitz matrices , the measures @xmath163 for @xmath164 also have an interpretation of being the limiting measures for certain generalized eigenvalues . for the rationally generated toeplitz matrices such a result remains valid . [ def : countingmeasuresrational ] for @xmath165 and @xmath166 we define the polynomial @xmath167 by @xmath168 and we define the _ @xmath6th generalized spectrum _ of @xmath2 by @xmath169 finally , we define @xmath170 as the normalized zero counting measure of @xmath171 @xmath172 where in the sum each @xmath28 is counted according to its multiplicity as a zero of @xmath167 . the toeplitz matrix @xmath173 in definition [ def : countingmeasuresrational ] may be interpreted as a shifted version of @xmath174 . for example if @xmath175 then by we have @xmath176 we will show that for each @xmath6 the sequence @xmath177 has a limit . moreover , the limiting measure will have point masses at @xmath148 ( if @xmath138 ) and @xmath149 ( if @xmath178 ) with weights at least @xmath102 and @xmath103 , respectively . on the other hand , if @xmath179 or @xmath180 then the total mass of the limiting measure is reduced with at least @xmath102 or @xmath103 respectively . these facts can already be seen at the level of the finite-@xmath181 measures @xmath170 as the following proposition shows . [ prop : combinatorial2 ] let @xmath66 . then the polynomial @xmath182 satisfies the following properties . * @xmath183 * @xmath184 where @xmath185 is a constant depending only on the symbol @xmath0 . denote by @xmath186 the quotient polynomial obtained from @xmath182 by removing all its factors @xmath187 ( if @xmath188 ) and @xmath189 ( if @xmath190 ) . then we have that * proposition [ prop : combinatorial2 ] will be proved in section [ subsection : proofpropcomb ] . since the measure is absolutely continuous , the best one can hope for is @xmath63 being the _ absolutely continuous part _ of the limiting @xmath6th generalized eigenvalues distribution . this means that the possible point masses at @xmath191 and @xmath192 should be stripped out in @xmath63 . this turns out to be indeed the case . [ theorem : geneigrat ] let @xmath66 . then @xmath193 and @xmath194 holds for every bounded continuous function @xmath195 on @xmath126 . from we see that @xmath102 and @xmath103 are the weights of the point masses at @xmath82 and @xmath84 in the limiting @xmath6th generalized eigenvalues distribution , if present . the rest of this paper is organized as follows . section [ section : proofmaintheorem ] contains the proof of theorem [ theorem : main ] . in section [ section : prooftheoremgeneig ] we prove proposition [ prop : combinatorial2 ] and theorem [ theorem : geneigrat ] . most of the proofs are inspired by the proofs given in @xcite for the corresponding statements in the banded case , hence we will often refer to that paper . finally , some illustrations of our results are given in section [ section : examples ] . in this section we will prove theorem [ theorem : main](a)(b ) . first we recall some elementary definitions and properties involving the algebraic equation @xmath196 . ( ( * ? ? ? * section 11.2),@xcite ) a point @xmath25 is called a _ branch point _ if @xmath196 has a multiple root . a point @xmath197 is an _ exceptional point _ of @xmath55 if @xmath28 is a branch point , or if there is no open neighborhood @xmath198 of @xmath49 such that @xmath199 is an analytic arc starting and terminating on @xmath200 . [ prop : structuregammak ] let @xmath66 . then the set @xmath55 in is the disjoint union of a finite number of open analytic arcs and a finite number of exceptional points . the set @xmath55 has no isolated points . the proof of this proposition is similar as in ( * ? ? ? * theorem 11.9),@xcite,@xcite . the condition is needed to ensure that the @xmath55 are proper curves , i.e. , they are 1-dimensional subsets of @xmath126 . a major role is played by the functions @xmath201 which , for @xmath54 are defined by @xmath202 the function @xmath201 is analytic in @xmath203 . occasionally we will also consider @xmath201 for the indices @xmath106 or @xmath107 . note that may be written alternatively as @xmath204 to discuss the integrability of this measure , we will need the asymptotic behavior of @xmath205 . the relevant facts are listed in the following proposition . [ prop : threeproperties ] let @xmath66 and recall the notations in section [ subsection : auxdef ] . then the following statements hold 1 . for any @xmath206 , there exists an @xmath207 such that @xmath208 as @xmath209 with @xmath210 . we have @xmath211 unless @xmath212 is a branch point . 2 . assume @xmath213 . then near the point @xmath191 there exists an @xmath214 such that @xmath215 as @xmath216 with @xmath210 . 3 . assume @xmath213 . then near the point @xmath192 there exists an @xmath214 such that @xmath217 as @xmath218 with @xmath210 . 4 . if @xmath219 , then @xmath220 as @xmath221 with @xmath210 . first we make some general observations . for any @xmath222 the polynomial @xmath223 has roots @xmath224 , @xmath225 , all of which are finite and non - zero ( although some of the roots might occur with higher multiplicity ) . zero roots or infinite roots can only occur if @xmath226 . as @xmath216 , then the @xmath85 smallest roots @xmath224 tend to zero like a power @xmath227 , while the other roots converge to non - zero constants . similarly , as @xmath218 , then the @xmath86 largest roots @xmath224 tend to infinity like a power @xmath228 , while the other roots converge to non - zero constants . these facts all follow from definition [ def : lambda12 ] . to prove the first equality of part ( a ) , fix @xmath229 and denote with @xmath230 , @xmath225 , the roots of @xmath231 ; by the discussion in the previous paragraph we have @xmath232 for each @xmath233 . pick one of the roots @xmath234 which has the highest multiplicity @xmath235 . writing @xmath236 where @xmath237 and @xmath238 are polynomials with @xmath239 and @xmath240 , it follows that @xmath241 for some constant @xmath242 . by taking the logarithmic derivative we obtain @xmath243 the first equality in part ( a ) follows from this and the fact that @xmath244 the second equality in part ( a ) ( for the case @xmath245 ) is proved in a similar way , this time using a decomposition @xmath246 where again @xmath239 and @xmath240 . to prove part ( b ) , first assume that @xmath138 . from the discussion in the first paragraph of this proof we obtain @xmath247 for all @xmath248 , while @xmath249 for @xmath250 and a suitable @xmath214 . hence @xmath251 by virtue of , with @xmath252 . similarly one can prove the case where @xmath179 . the proofs of parts ( c ) and ( d ) are similar as well . [ prop : mupositivemass ] for each @xmath6 we have that @xmath63 in is a positive measure on @xmath55 with total mass @xmath253 as defined in . first we prove that the density is locally integrable around the points @xmath191 , @xmath192 , @xmath83 ( at least those of them which lie on the curve @xmath55 ) . for @xmath83 this follows from the second equality in proposition [ prop : threeproperties](a ) . for @xmath191 this follows from proposition [ prop : threeproperties](b ) and taking into account that the @xmath254 terms at the @xmath56-side and @xmath57-side in cancel ; a similar argument holds for the point @xmath192 . the fact that the measure @xmath63 is positive follows from a cauchy - riemann argument as in ( * ? ? ? * proof of proposition 4.1 ) . finally , the statement that @xmath253 follows from a contour deformation argument as in ( * ? ? ? * proof of proposition 4.1 ) . more precisely , we have @xmath255 where @xmath256 is a clockwise oriented contour surrounding @xmath55 and those points @xmath191 and @xmath192 which are finite , and where @xmath257 denotes the residue of @xmath258 at @xmath28 . note that is valid even when some of the @xmath259 lie on the curve @xmath55 , @xmath78 , thanks to the local integrability of @xmath63 around these points . applying the residue theorem once again , this time for the exterior domain of @xmath256 , we then find for the first term in that @xmath260 the fact that @xmath253 then follows from and the residue expressions in proposition [ prop : threeproperties ] ; recall also . [ prop : cauchytransformcontour ] for each @xmath6 we have that @xmath261 and @xmath262 for a suitable constant @xmath263 . the quantities in the above proposition are only well - defined if @xmath264 , @xmath78 . however , one easily checks that @xmath191 and @xmath192 are removable singularities for the right hand sides of both and , due to the continuity of the corresponding left hand sides . the proof of follows by contour deformation in a similar way as in the proof of proposition [ prop : mupositivemass ] . the relevant expression is now @xmath265 where @xmath256 is a clockwise oriented contour surrounding @xmath55 , the point @xmath28 , and those points @xmath191 and @xmath192 which are finite . now the integrand in the integral over @xmath256 has zero residue at infinity and therefore this integral vanishes . using the residue expressions in proposition [ prop : threeproperties ] one then arrives at the right hand side of . finally , the proof of then follows by integrating , see also @xcite . now we are ready to prove theorem [ theorem : main](a)(b ) . taking into account proposition [ prop : mupositivemass ] , it suffices to show that the logarithmic energy @xmath266 is bounded for each @xmath66 . the latter follows by integrating over @xmath267 . then the left hand side becomes @xmath268 , so it suffices to show that each of the four terms in the right hand side is bounded . for the fourth term this is evident since @xmath63 has finite mass . for the two middle terms this follows from our earlier observation that @xmath63 is integrable around @xmath191 and @xmath192 ( assuming they are on the curve @xmath55 ) , which is still true when multiplying with the logarithmic singularities @xmath269 and @xmath270 . a similar argument holds for the first term . the proof of part ( b ) follows from and the auxiliary results @xmath271 for @xmath159 , and @xmath272 @xmath273 here the boundary terms @xmath102 , @xmath103 for @xmath106 or @xmath107 are defined by the usual formulae . these considerations imply the desired result for @xmath159 ; the cases @xmath148 and @xmath149 then follow by continuity . to prove theorem [ theorem : main](c ) we rewrite in the following way , compare with ( * ? * eq . ( 2.12 ) ) : @xmath274 we leave it to the reader to check the correctness of this identity ; note that the calculation makes use of the auxiliary result @xmath275 for @xmath66 , which follows from and . here we recall the boundary values @xmath276 . we also invoke the fact that @xmath277 whenever @xmath278 and @xmath279 are positive measures with @xmath280 . this is a well - known result if @xmath278 and @xmath279 have bounded support @xcite . if the support is unbounded this is a recent result of simeonov @xcite . [ lemma : boundedbelow ] the energy functional is bounded from below on the set of admissible vectors of measures @xmath141 . from we see that in order to show that the energy functional @xmath281 is bounded from below , it is sufficient to show that @xmath282 and @xmath283 are both bounded from below on the set of admissible vectors of measures @xmath141 . let us check this for the first term . we will use that @xmath284 a fact already observed in , which follows immediately from the definition of @xmath85 . now we distinguish between two cases . the first case is when @xmath179 . then the second term in drops out while on the other hand @xmath285 , so the contour @xmath144 is bounded and therefore the first term in is bounded from below as well . the second case is when @xmath138 . then standard arguments from potential theory show that the expression is minimized precisely when @xmath151 is the _ balayage _ of the dirac point mass at @xmath191 onto the curve @xmath144 , and in particular this expression is bounded from below ( * ? ? ? * chapter 2 ) . the above proof goes through because the constant factor in front of the first term in is precisely @xmath286 . if this constant factor is different from @xmath286 then the connection with balayage measures breaks down , and in fact if the constant is larger than @xmath286 then the energy functional is not bounded from below anymore . assume that @xmath287 is a vector of admissible measures satisfying the equalities in theorem [ theorem : main](b ) , and let @xmath141 be any admissible vector of measures . we need to prove that @xmath288 with equality if and only if @xmath289 . note that the equalities in theorem [ theorem : main](b ) are precisely the euler - lagrange variational conditions of the equilibrium problem . the result then follows from the fact that the energy functional @xmath134 is convex and bounded from below . more precisely , one can use exactly the same argument as in ( * ? ? ? * proofs of lemma 2.3 and theorem 2.3(c ) ) , taking into account . there are some modifications induced by the external fields , but this does not lead to problems since the latter act in a linear way on the measures . the proof of proposition [ prop : combinatorial2 ] is based on the reduction of a rationally generated toeplitz matrix into banded form , which will then allow us to follow the proof in ( * ? ? ? * proof of prop . let us recall from that @xmath290 where the numerator @xmath35 is a polynomial in @xmath37 . then we claim that for any @xmath66 and for any @xmath181 sufficiently large , the rationally generated toeplitz matrix with symbol @xmath291 can be reduced into banded form by the factorization @xmath292 where @xmath293 are non - singular lower and upper triangular toeplitz matrices respectively . the middle factor in the left hand side of is our rationally generated toeplitz matrix of interest , and shows that it can be reduced to the banded matrix pencil in the right hand side . here @xmath294 is a matrix whose size and entries are independent of @xmath181 but depend only on the symbol @xmath295 . for more information on factorizations of the type see e.g. ( * ? ? ? * prop . 2.12 ) and also @xcite . from we immediately deduce that @xmath296 where @xmath297 is a numerical constant , given by the product of the diagonal entries of the two triangular factors @xmath298 and @xmath299 in . we are now ready for the proof of proposition [ prop : combinatorial2 ] . the proof will follow by expanding the determinant in by a basic combinatorial argument , see also ( * ? ? ? * proof of prop . the proposition is obvious if @xmath300 . so we will assume below that @xmath301 , or equivalently @xmath302 first we consider the case where @xmath138 . by expanding the determinant in we find @xmath303 denotes the set of all permutations of @xmath304 , and we denote with @xmath305 the maximum of the row and column sizes of the matrix @xmath294 ; note that this number is independent of @xmath181 . by the band structure it follows that we only have non - zero contributions for the permutations @xmath306 that satisfy @xmath307 denote , for @xmath308 , @xmath309 the set @xmath310 contains all indices @xmath233 for which the @xmath311 entry lies in the union of the @xmath85 topmost bands of the banded matrix in . by assumption these bands include the main diagonal @xmath312 and by definition of @xmath85 we have that the entries in these bands are all divisible by @xmath187 . denote the number of elements of @xmath310 in by @xmath313 . then obviously @xmath314 where we minimize over all permutations @xmath308 satisfying . let @xmath308 satisfy . we give a lower bound for @xmath313 . since @xmath315 we obtain @xmath316 where @xmath317 is defined as @xmath318 for @xmath319 . from the above definitions we also have that @xmath320 by combining and we find that @xmath321 here @xmath322 is a correction term which is due to the presence of the matrix @xmath294 in the top left matrix corner in ; the number @xmath323 is clearly bounded from above . we then obtain @xmath324 where we used and , and where we put @xmath325 . the first statement in proposition [ prop : combinatorial2](a ) now follows from and . the proof of the second statement in proposition [ prop : combinatorial2](a ) ( for @xmath179 ) is similar to the one above . now one uses that all the entries @xmath326 in the bands indexed by @xmath327 have their @xmath28-coefficient @xmath328 , which then yields in a similar way to and that @xmath329 as desired . similar to part ( a ) . part ( c ) follows immediately from parts ( a ) and ( b ) , together with , in case where @xmath213 . the case where @xmath219 can be obtained as well , by observing that at least one of the numbers @xmath102 and @xmath103 must be zero in that case . the latter follows since otherwise the numerator and denominator in are equal up to multiplication with a scalar , contrary to our assumptions . to prove theorem [ theorem : geneigrat ] we need to manipulate the polynomial @xmath182 . to this end we will use a determinant identity by k.m . day which we state next . to state the identity , we need some notations . denote with @xmath330 and @xmath331 the zeros of @xmath332 and @xmath333 , respectively . recall the notation @xmath334 for the roots of @xmath35 . thus @xmath335 where @xmath242 , @xmath336 , @xmath337 are non - zero constants . the following theorem was proved under some additional hypotheses by k.m . day @xcite . other proofs are in @xcite , the former of them stated under the weakest assumptions . we state the theorem in the form that is most convenient for our purposes . [ theorem : day ] ( day s determinant identity ) . let @xmath66 and let @xmath338 be such that all roots of @xmath35 are distinct . then @xmath339 where the sum is over all subsets @xmath340 of cardinality @xmath341 and for each such @xmath342 we have @xmath343 and ( with @xmath344 ) @xmath345 with @xmath346 and @xmath347 . incidentally , observe that can be written alternatively as @xmath348 we note that in case where @xmath349 , our formulation of theorem [ theorem : day ] follows directly from the one of @xcite ; for the case @xmath350 it can be obtained from the result of @xcite by working with the transposed matrix . from we see that for large @xmath181 , the main contribution in comes from those subsets @xmath342 for which @xmath351 is the largest possible . for @xmath210 there is a unique such @xmath342 , namely @xmath352 now we are ready to show that the asymptotic distribution of the @xmath6th generalized eigenvalues of @xmath2 is described by the measure @xmath63 , together with possible point masses at @xmath191 and @xmath192 . first we prove this at the level of the cauchy transforms . [ prop : geneigrat ] let @xmath66 . then @xmath353 uniformly on compact subsets of @xmath203 . the above proposition implicitly assumes that @xmath264 , @xmath78 . however one checks that if @xmath354 then @xmath259 is a removable singularity for the right hand side of , due to the continuity of the left hand side , and then the uniform convergence still applies . as already mentioned , for large @xmath181 the dominant term in day s determinant identity theorem [ theorem : day ] is obtained by taking @xmath355 . then we find in the same way as in ( * ? ? ? * proof of corollary 5.3 ) that @xmath356 uniformly on compact subsets of @xmath203 , where the last transition of follows from and . now from proposition [ prop : cauchytransformcontour ] we see that the right hand side of equals the right hand side of . the proposition is proved . now we are ready for the from the convergence of the cauchy transforms in proposition [ prop : geneigrat ] we deduce that @xmath357 in the weak - star sense , which means that holds for every continuous @xmath195 that vanishes at infinity . now a priori , it is not immediate that holds for all bounded continuous functions since it is possible that @xmath170 has mass leaking to infinity as @xmath358 . however , from proposition [ prop : combinatorial2 ] it follows that this can not happen , i.e. , the measures @xmath359 are _ tight_. thus holds indeed for all bounded continuous functions . for more details see ( * ? ? ? * proof of theorem 2.6 ) . consider the rationally generated toeplitz matrix with symbol @xmath360 defined on the complex unit circle . we may compute the fourier series of this symbol explicitly and find @xmath361 so the rationally generated toeplitz matrix @xmath2 looks like @xmath362 equation now becomes @xmath363 and leads to @xmath364 . the roots of @xmath35 are given by @xmath365 and they should be labeled in such a way that @xmath366 for all @xmath28 . the roots @xmath367 and @xmath368 are coalescing precisely when @xmath369 , so the branch points are @xmath370 and @xmath371 . since @xmath364 , there is only one relevant index @xmath6 in , namely @xmath52 . the corresponding set @xmath51 is simply the line segment connecting the branch points @xmath370 and @xmath371 : @xmath372.\ ] ] this may be checked from a straightforward calculation . definitions [ def : lambda12 ] and [ def : mk ] now specialize as follows : @xmath373 , @xmath374 , @xmath375 , @xmath376 , and @xmath377 , @xmath378 and @xmath379 . thus the limiting eigenvalue distribution of the matrix @xmath2 for @xmath358 consists of an absolutely continuous part @xmath50 with total mass @xmath286 , supported on @xmath380 $ ] , and a singular part which is a point mass of mass @xmath286 at @xmath370 . the energy functional now specializes to @xmath381 so @xmath50 is the minimizer of over all measures @xmath382 on @xmath380 $ ] with total mass @xmath286 . the second term in can be interpreted as an attraction of @xmath50 towards the point @xmath383 . the measure @xmath50 is absolutely continuous with density given by ( with @xmath52 and @xmath364 ) . the density can be explicitly computed , but we will omit the result since it does not lead to considerable insight . we only mention that the density blows up like an inverse square root near both endpoints @xmath370 and @xmath371 . more precisely , it behaves approximately like @xmath384 near @xmath370 and like @xmath385 near @xmath371 . figure [ fig : density1 ] contains a plot of the limiting density . the figure shows that there is more mass near @xmath81 than near @xmath386 , which is due to the attraction towards @xmath383 in . figure [ fig : density2 ] shows the result of a numerical computation of the eigenvalues of @xmath2 with @xmath387 . note that approximately half of the eigenvalues is located at zero , according to proposition [ prop : combinatorial2 ] ; in fact we have @xmath388 in this case . on @xmath380 $ ] for the symbol . the density blows up like a square root near both endpoints @xmath386 and @xmath81 . there is more mass near @xmath81 due to the attraction towards @xmath383 . ] for the symbol , computed numerically in maple for @xmath387 using high precision arithmetic . all the eigenvalues are real . there are @xmath389 of them in the open interval @xmath390 , together with a @xmath389-fold eigenvalue at @xmath370 . ] we may consider the following modification of , @xmath391 where @xmath392 is some small number . it is still true that @xmath373 and @xmath374 for any @xmath393 , but for @xmath393 non - zero we now have @xmath394 , @xmath395 and @xmath396 . thus the limiting eigenvalue distribution of @xmath2 is absolutely continuous ( without point mass ) , it has total mass @xmath137 , and it is supported on the interval @xmath51 joining the two branch points @xmath397 from the above discussions , we see that the limiting eigenvalue distribution of @xmath2 is absolutely continuous if @xmath392 and has a point mass at the origin if @xmath398 . to understand this , note that for @xmath392 the energy functional contains attracting point charges at both @xmath373 and @xmath374 ( since @xmath394 ) . in the limit @xmath399 , the rightmost endpoint of @xmath51 in moves towards the point source at @xmath374 . this causes an increasing accumulation of mass near this endpoint which in the limit for @xmath398 gives birth to the point mass . the authors thank professor arno kuijlaars for stimulating discussions . g. baxter and p. schmidt , determinants of a certain class of non - hermitian toeplitz matrices , math . scand . 9 ( 1961 ) , 122128 . a. bttcher and s.m . grudsky , spectral properties of banded toeplitz matrices , siam , philadelphia , pa , 2005 . a. bttcher and b. silbermann , invertibility and asymptotics of toeplitz matrices , akademie - verlag , berlin , 1983 . a. bttcher and b. silbermann , introduction to large truncated toeplitz matrices , universitext , springer - verlag , new york 1998 . k.m . day , toeplitz matrices generated by the laurent series expansion of an arbitrary rational function , trans . ( 1975 ) , 224245 . day , measures associated with toeplitz matrices generated by the laurent expansion of rational functions , trans . ( 1975 ) , 175183 . m. duits and a.b.j . kuijlaars , an equilibrium problem for the limiting eigenvalue distribution of banded toeplitz matrices , siam j. matrix anal . 30 ( 2008 ) , 173196 . hirschman , jr . , the spectra of certain toeplitz matrices , illinois j. math . 11 ( 1967 ) , 145149 . t. hholdt and j. justesen , determinants of a class of toeplitz matrices , math . 43 ( 1978 ) , 250258 . nikishin and v.n . sorokin , rational approximations and orthogonality , amer . providence , ri , 1991 . t. ransford , potential theory in the complex plane , london math . soc . texts 28 , cambridge university press , cambridge , uk , 1995 . saff and v. totik , logarithmic potentials with external field , springer - verlag , berlin , 1997 . p. schmidt and f. spitzer , the toeplitz matrices of an arbitrary laurent polynomial , math . ( 1960 ) , 1538 . p. simeonov , a weighted energy problem for a class of admissible weights , houston j. math . 31 ( 2005 ) , 12451260 . ullman , a problem of schmidt and spitzer , bull . 73 ( 1967 ) , 883885 . h. widom , on the eigenvalues of certain hermitian operators , trans . 88 ( 1958 ) , 491522 .

we consider the asymptotic behavior of the eigenvalues of toeplitz matrices with rational symbol as the size of the matrix goes to infinity . our main result is that the weak limit of the normalized eigenvalue counting measure is a particular component of the unique solution to a vector equilibrium problem . moreover , we show that the other components describe the limiting behavior of certain generalized eigenvalues . in this way , we generalize the recent results of duits and kuijlaars @xcite for banded toeplitz matrices . * keywords * : toeplitz matrix , rational function , generalized eigenvalues , ( vector ) potential theory .

introduction statement of results proof of theorem proofs of proposition example acknowledgment

arxiv

it has been known for many decades that an accelerated detector , moving in a quantum field prepared in the ground state of the field modes for an inertial frame , will become excited@xcite . in the case of constant acceleration , @xmath2 , the frequency response of the detector is completely equivalent to the response of an inertial detector in a thermally excited field with a temperature @xmath3 given by @xmath4 . we call this unruh - davies radiation . quite clearly , such an effect would be very difficult to see given current technologies . in this paper we suggest an analogous system , based on detecting phonons of the vibrational modes of cold trapped ions@xcite . in many ways this parallels a theme , pioneered by unruh , of sonic equivalents for quantum fields in curved space time@xcite . our analogy is based on an alternative view of unruh - davies radiation in terms of the time dependent red shift seen by an accelerated observer@xcite . by controlling the trapping potentials of trapped ions it is possible to modulate the normal mode frequencies so that they have the same time dependent phase as red - shifted frequencies seen by a constantly accelerated observer . suppose now that the ions are prepared ( using laser cooling ) in the ground state of the normal modes of vibration of the time independent trap . if a suitable detector of the vibrational quanta for the trapped ions was available , they could be used to detect excitations out of the ground states of vibrational motion due to the frequency modulation . this would be analogous to the response of an accelerated detector to a scalar field prepared in a minkowski ground state . fortunately the ions themselves can be made to respond as phonon detectors . in ion trap implementations of quantum information processing , a laser is used to couple the vibrational motion of a trapped ion to an electronic transition between states which we denote @xmath5 @xcite . furthermore it is possible to readout one or the other of these internal electronic states , say @xmath6 , using a laser ( the readout laser ) to drive a cycling transition between @xmath6 and another electronic state , thereby producing fluorescence conditional on whether the ion was in state @xmath6 . such measurements are highly efficient and closely approximated by a perfect projective measurement of the internal electronic state . in effect this scenario defines a _ phonon detector _ that may be turned on and off at will . to be more specific , we can implement various kinds of phonon detectors by carefully tuning the laser frequency @xmath7 to one of the vibrational sidebands of the ion . this enables one to realize rather unconventional phonon detectors that respond to antinormally ordered moments ( blue sideband ) as well as the more conventional normally ordered moments ( red sideband ) as we explain in more detail below . the interaction hamiltonian describing the coupling of the internal and vibrational degrees of freedom of the @xmath8th ion in a linear array of ions in a trap ( in the interaction picture , and lamb - dicke limit ) can be written as @xcite ^(m)_i = _ 0 k _ m(t)_x(t ) where _ x(t)=e^-it _ - + , and @xmath9 is the ( scaled ) rabi frequency , @xmath10 is the detuning between atomic resonance and the laser , @xmath11 is the wavevector , @xmath12 is the angle the laser beam makes with the longitudinal axis for the linear ion chain , @xmath13 is the quantized local displacement of the @xmath8th ion about its equilibrium position and @xmath14 is the atomic lowering operator between the upper and lower atomic states @xmath15 and @xmath16 separated by frequency @xmath17 . we assume the equilibrium position of the @xmath8th ion is located at a node of the standing wave laser beam , we have used the rotating wave approximation to describe the interaction between the laser and the ion . in the lamb - dicke limit , terms of order @xmath18 have been neglected since the displacement of the ion is much less than the wavelength of light . in an experiment the @xmath19 is a quadrupole transition and is driven by a raman process . thus @xmath9 is an effective rabi frequency for this process . equation ( [ 1 ] ) could also be regarded as a discretised representation for the interaction of a scalar field , @xmath20 , and a local detector , with transition frequency @xmath21 , where @xmath22 for some discretisation length @xmath23 . in such an interpretation the interaction eq.([1 ] is equivalent to the unruh model of a particle detector@xcite , with only two internal energy levels . note that the detector can be turned on and off through the dependance on the external laser field in @xmath9 , a somewhat unusual feature for field quanta detectors . another unusual feature of this detector is that the transition frequency of the detector , @xmath24 can be varied by tuning the external laser . conventional detectors would have a fixed transition frequency . this latter feature will enable us to define different kinds of phonon detectors . the local displacement of the @xmath8th ion , @xmath13 , can be expanded in terms of creation and annihilation operators for _ global _ normal modes ( phonons ) of the @xmath25-ion system @xmath26 by @xcite _ m(t ) & = & _ p=1^n b^(p)_m _ p(t ) & = & _ p=1^n s^(p)_m ( _ p e^-i_p t - _ p^e^i_p t ) where the coupling constant is defined by s^(p)_m = . in the above @xmath27 are the normal mode trap frequencies given by @xmath28 in terms of the bare trap frequency @xmath29 and the eigenvalues @xmath30 . for the center of mass mode @xmath31 , @xmath32 and for the breathing mode @xmath33 , @xmath34{3}$ ] , where in the later @xmath35 are the components of the breathing mode eigenvector @xmath36 normalized to unity . further normal mode parameters for up to @xmath37 ions are computed numerically in james @xcite . defining the lamb - dicke parameter as @xmath38 we can write our hamiltonian in its final form ^(m)_i = _ m(t)_x(t ) where we have defined the operator @xmath39 and @xmath40 . this hamiltonian describes the coupling between the two level electronic transition ( the detector ) and the vibrational degrees of freedom whenever the external laser is turned on ( @xmath41 ) . note that we do not make any assumptions at this stage about the relative size of @xmath21 and @xmath27 . we wish to keep @xmath21 as a free parameter which may be varied to define different kinds of phonon detectors . in analogy with the unruh detector model , the field operator @xmath13 represents the scalar field at the position @xmath42 and time @xmath43 , @xmath44 . in physical terms this is the displacement of the @xmath8th ion as a function of time . it is worth noting an important difference between this model and the usual treatment of a particle detector . in the case of a usual detector the frequency term , @xmath21 , would be strictly positive thus defining the positive and negative frequency components of the dipole @xmath45 . in , the parameter @xmath21 can be positive or negative so we can not simply refer to positive or negative frequency components in absolute terms . however the operators @xmath45 will retain their usual definition as raising and lowering operators . in the case that @xmath46 , the laser is detuned below the atomic transition , which we refer to as red detuning . we can resonantly excite so called red sidband transitions when @xmath47 . near such a resonance ( @xmath48 ) we can make the rotating wave approximation and describe the interaction by the hamiltonian @xmath49 this describes the usual jaynes - cummings model of a two level system interacting with a bosonic degree of freedom . in physical terms it describes a raman process in which one laser photon and one trap phonon are absorbed to excite the atom ( see figure [ fig1 ] ) . a phonon detector defined this way would respond to the normally ordered moments of the phonon field amplitude . in the case that @xmath50 , the laser is detuned above the atomic transition , which we will refer to as blue detuning . the resonant term for the fisrt blue sideband is then given by @xmath51 again this is a raman process in which the atom is excited by the absorption of one laser photon and the _ emission _ of one phonon ( see figure [ fig1 ] ) . considered as a phonon counter this would correspond to a detector that responded to anti - normally ordered moments of the phonon field amplitude . using laser cooling techniques it is possible to cool the system very nearly to the ground state of the vibrational degrees of freedom . in reality this becomes more difficult as the number of ions , and thus normal modes increases . however for our purposes even one ion would suffice . in current experiments the cooling is sufficiently efficient to reach the ground state with probability 0.999@xcite . we thus assume an initial state of the form , ( 0)|= |g|0 _ where the initial vibrational state is a tensor product of the ground states of each of the normal modes , @xmath52 we will exponentially chirp the trap frequency up or down such that ( t ) = e^t with @xmath0 the chirp rate and focus our laser on the first ion ( @xmath53 ) . for a constant trap frequency the phonon annihilation operator satisfies the usual uncoupled mode equation @xmath54 with solution @xmath55 , @xmath56 which was used in . for an chirped trap frequency given by between an initial time @xmath57 and final time @xmath43 , the phonon mode now satisfies the following equation of motion , with solution _ p(t ) = -_p e^t _ p(t)_p(t ) = ( e^t_0 ) ( - e^t ) _ p(t_0 ) . where we consider the chirp - up case first ( with the chirp down case discussed subsequently ) . we now suppose that the coupling to the detector is turned on at the same time as the frequency modulation is turned on , and turned off at the same time as the frequency modulation is turned off . this is a rather different scenario to the usual discussion of the unruh - davies effect in which the detector is always coupled to the field and continuously accelerated , ie the red shift frequency modulation is always on . we are interested in the probability @xmath58 for the excitation of the @xmath8th ion from the ground state to all excited states of detector and field for a detector turned on at @xmath59 and turned off at @xmath60 . the detector in our case has only one excited state . we let the excited states of the vibrational degree of freedom be represented by a complete orthonormal basis @xmath61 . the total excitation probability is then @xcite p_m(t , t_0)=^2_t_0^tdt_t_0^t dt e^i(t-t)g(t,t ) where the field correlation function is defined as @xmath62 with the field now given by @xmath63 and we have neglected the phase factor arising from the initial time as it does not contribute to the correlation function . substituting this result into eq.([cor ] ) , p_m(t , t_0)=^2_p=1^n|i_p(t , t_0)|^2 where the integral is i_p(t , t_0 ) _ t_0^t dt e^t e^ ( e^t ) . with no loss of generality we may now set @xmath64 . we first consider the case @xmath46 , which is the red - sideband case . our objective is to calculate the excitation probability for the two level system near the red sideband transition which we label @xmath65 . to evaluate this integral we first change to dimensionless time , @xmath66 , so that i_p(t , t_0 ) = _ 0^t d e^a e^b e^ . where @xmath67 and @xmath68 . next we define the constant @xmath69 @xmath70 it then seems sensible to make the new change of variable @xmath71 so that the integral becomes @xmath72 in an experiment one needs to vary @xmath21 near the red or blue sideband , so we expect that @xmath21 and @xmath27 are of the same order . we now consider the limit in which @xmath73 for all the normal modes . in this limit @xmath74 , the integral does not depend much on @xmath69 , so we extend the lower limit of the integral to infinity . furthermore we suppose the time over which the detector and modulation are on is such that @xmath75 , in which case the integral does not vary much with @xmath3 . in that case we can extend the upper limit of the integral to infinity , so that @xmath76 changing the variable of integration to @xmath77 we obtain an expression for @xmath78 with arbitrary limits as i_p & = & _ 0^dy y^ia-1e^iy + & = & ( ia)e^-a/2[ip2 ] if we now use the identity @xmath79 $ ] for real @xmath80 @xcite we see that |i_p|^2= inserting this into we can write @xmath81 in the suggestive form p_m^r= _ p=1^n where we have defined the unruh temperature as k_bt . has the analogous form to a thermal spectrum at temperature @xmath3 as seen by a uniformly accelerated observer moving through a particle - free inertial vacuum . the `` thermal '' form of the probability is independent of the phonon frequencies @xmath27 since we have taken the upper limit @xmath43 to infinity . the last summation expression is just a numerical factor , which can be computed @xcite , or dropped in the case of a single ion in the trap . the analogy with the unruh effect @xcite for a uniformly accelerated observer in minkowski space can be seen as follows . as discussed in @xcite the chirping of the trap frequencies can be considered as arising from the the modification of the usual minkowski plane wave @xmath82 due to the motion of the accelerated observer . for an observer moving at constant velocity @xmath83 in minkowski space , a lorentz transformation ( lt ) of the phase @xmath84 viz @xmath85 and @xmath86 with the rapidity defined by @xmath87 , simply transforms it to @xmath88 which merely produces the constant doppler shifted frequencies in the new frame @xmath89 . for an accelerated observer , one has to perform a time dependent lt at each instant to the comoving frame that is instantaneously at rest with respect to the accelerated observer . the orbit of the accelerated ( rindler ) observer as described by an inertial minkowski observer is given by the rindler transformation @xmath90 and @xmath91 where @xmath2 is the constant uniform acceleration . under this transformation , the rindler observer experiences phase of the minkowski plane wave @xmath92 as transformed to @xmath93 . these are the chirped frequencies that appear in in exponentially expanding or contracting the trap . the fourier transform of this modified plane wave with respect to _ the rindler proper time _ @xmath94 can be considered as a measure of the noise spectrum seen by the accelerated observer viz s ( ) = | _ -^ dt e^ e^ ( i c / a e^a / c ) as in . here the perceived thermal temperature is defined from the only other frequency that can be formed from the rindler observers accelerated motion , @xmath95 . thus we get the usual unruh temperature defined by @xmath96 . in our ion trap analogy , the role of the acceleration frequency @xmath95 is played by the trap expansion rate @xmath0 . , ( b ) blue side band @xmath97 . the unruh effect takes place on the red sideband . we obtain an _ anti - normally ordered _ unruh effect if we tune to the blue side band . , width=384,height=288 ] it is important to note that in order to obtain the planck factor @xmath98 in , indicative of a bose - einstein ( be ) thermal distribution and the signature of the unruh effect , we made crucial use of a positive detuning @xmath99 , corresponding to a red sideband detuning in and . this resulted in the factor @xmath100 that appears in @xmath78 in . dividing the square of this factor into the @xmath101 function appearing in the denominator of @xmath102 , resulting from the term @xmath103 , produces the signature be thermal distribution function . if on the other hand , we had instead chosen @xmath104 corresponding to a negative detuning to the blue side band , the previous factor would become @xmath105 . dividing the square of this term into the @xmath101 function appearing in the denominator then produces alternatively the probability for excitation on the blue sideband p_m^b= _ p=1^n , with the same definition of the unruh temperature as in . we might label such a distribution an _ anti - normally ordered _ unruh effect since the vibrational excitation from the ground to the excited state takes place by an absorption of a photon and an emission of a phonon to the electronic state @xmath106 as depicted in , i.e. by a term such as @xmath107 , see eq.([blue ] ) . as discussed earlier , such a detector responds to the anti - normally ordered moments of the phonon field amplitude . note that as @xmath108 ( @xmath109 ) in we get a _ finite _ contribution to the probability for excitation @xmath110 . in the case of red sideband detuning , the usual unruh effect analogy in , @xmath111 as @xmath108 . this limit corresponds to a fixed trap frequency @xmath29 for which we get no excitation as described above . the above @xmath108 limiting cases can also be understood as follows . for a constant trap frequency @xmath112 the relevant integral to compute is i_p(,- ) = _ -^ dt e^i t e^i_p t = 2(+ _ p ) which arises from the @xmath113 term in @xmath114 when acting on the motionally cooled ground state . for blue sideband detuning @xmath104 we obtain a non - zero contribution from the delta function . in the usual ion trap excitation schemes for quantum computing @xcite , the resonant ( rotating wave approximation ) portion of the hamiltonian gives rise to the anti - jaynes - cummings type interaction , which permits transitions of the form @xmath115 . with our motionally cooled ground state with @xmath116 such transitions are possible from @xmath117 . thus @xmath113 represents a resonant contribution to the hamiltonian . for red sideband detuning , @xmath118 we do not obtain a contribution from the delta function . here the resonant portion of the hamiltonian gives rise to the usual jaynes - cummings type interaction eq.([red ] ) , which permits transitions of the form @xmath119 . with an initial motional ground state with @xmath116 such transitions are not possible . however , in writing down we have dropped the non - resonant anti - normally ordered terms . it is the @xmath113 non - rwa term in @xmath114 that can produce transitions from the motionally cooled ground state and is responsible for the unruh - like behavior of the probability @xmath110 for excitation out of this ground state . we can also give an unruh analogy interpretation of the zero contribution to the delta function @xmath120 in . the case of red sideband detuning @xmath118 , a constant trap frequency is analogous to an inertial observer moving with constant velocity in minkowski space , giving rise to a constant lorentz transformation as discussed above . for a constant velocity observer , an inertial detector would have a response function proportional to @xmath121 where @xmath122 is the energy of the detected particle and @xmath123 represents the atomic transition frequency producing the particle to be detected @xcite . since the detector only responds to positive energies @xmath124 , the contribution from the delta function @xmath121 is zero , which simply states that an inertial detector moving through the minkowski vacuum would detect no particle production . for a finite chirp between times @xmath57 and @xmath3 we can develop a general expression for @xmath58 in terms of the incomplete gamma functions @xmath125 and @xmath126 such that @xmath127 . let us write @xmath128 for general finite duration limits in with the definitions @xmath129 and @xmath130 . by scaling the integration variable to @xmath131 in the second integral on the right hand side and @xmath132 in the third integral on the right hand side one can easily show that i_p(t , t_0 ) = ( 1 -(i a ,- i b y_0 ) - (i a , - i b y_t ) ) , with the same definitions of @xmath2 and @xmath133 used in , and where we have defined the normalized gamma functions @xmath134 and @xmath135 such that @xmath136 . thus we obtain p_m(t , t_0 ) = _ p=1^n | 1 - (i ,- i e^t_0 ) - (i , - i e^t)|^2 . the previous expression for the total excitation probability in corresponds to @xmath137 using above which formally corresponds to sweeping the trap frequency from an initial zero value to an infinite final value . considering a more realistic situation applicable to experiments , let us consider @xmath138 and a finite trap expansion time @xmath3 , which corresponds to sweeping the trap frequency from @xmath139 . considering as a function of the detuning @xmath21 with parameters @xmath140 and @xmath141 we can recover under the following conditions 1 , e^t 1 p = ( 1,2 , , n ) which makes the incomplete gamma functions small compared to unity . as an example , taking @xmath142 and @xmath143 requires that @xmath144 note that approaches @xmath137 of as @xmath145 and @xmath146 in which the incomplete gamma functions approach zero . the main point for experimental purposes is that we only need to take finite limits of the order @xmath147 and @xmath148 to approximate the full unruh case , @xmath137 as shown in . and @xmath149 , @xmath150 with @xmath151 as the abscissa and @xmath152 as the ordinate for various values of @xmath153 ; red curve @xmath154 , green curve @xmath155 , blue curve @xmath156 . the smooth black curve is the pure unruh case @xmath137 given by . , width=384,height=288 ] as we have shown , the experimental signature of the exponential modulation of the trap frequency is the planck - like form for the excitation probability for the two level electronic system in each ion . in such experiments it is the ratio of the excitation probability on the red ( @xmath157 ) and blue ( @xmath158)sidebands that is determined : @xmath159 as this number is independent of the rabi frequency , the lamb - dicke parameter , and the time of interaction between the vibrational and electronic degrees of freedom@xcite . let us consider the case of a single ion with trap frequency @xmath29 . using eqs.([14],[14.5 ] ) we see that @xmath160 in a typical experiment one can detect @xmath161 values as low as @xmath162 with about @xmath163% error . this implies that @xmath164 . at secular frequencies of @xmath165 mhz , we need a modulation frequency of the order of a few hundred khz to mhz ; a not particularly difficult requirement for fast electronics . the key issue however is the absolute size of the excitation probabilities at the red and blue sideband . this is determined by the prefactor @xmath166 . defining @xmath167 the equations for the excitation probability at the red and blue sidebands are @xmath168 as we expect @xmath169 to be of the order of unity , we require that the secular frequency is within one order of magnitude of the effective rabi frequency . this corresponds to a rather weakly bound ion , but should be achievable if stimulated raman transitions are used to couple the two level system . for example the relevant transition in @xmath170be@xmath171 can have an effective rabi frequency of the order of @xmath172khz@xcite . if we use the centre of mass mode with secular frequency of @xmath173khz , and a lamb - dicke parameter of @xmath174 , the prefactor is @xmath175 . at a more conservative estimate of @xmath176 , well within the lamb - dicke regime , the prefactor drops to a value of @xmath177 . these numbers are encouraging enough to suggest the plausibility of observing an analogous unruh - like effect in today s linear ion traps . a related proposed unruh - analogy experiment that involves the acceleration of atoms through microcavities can be found in m.a . et al _ , phys . lett . * 91 * , 243004 ( 2003 ) , which was brought to our attention after this work was completed . james , appl . b * 66 * , 181 ( 1998 ) . note : eq.(40 ) of this referecne is for a dipole transition , but can also be used in our context as a quadrapole transition driven by a raman process , which modifies the definition of @xmath9 . for the total excitation probability stems from the expression @xmath178 where the amplitude for a transition from the vacuum to any arbitrary exicted state @xmath179 is given by the first order perturbation theory @xmath180 , and use has been made of the completeness relation @xmath181 . c. monroe , d.m . meekhof , b.e . king , s.r . jefferts , w.m . itano , d.j . wineland and p. gould , phys . * 75 * , 4011 ( 1995 ) . turchette , d. kielpinski , b.e . king , d. leibfried , d.m meekhof , c.j . myatt , m.a . rowe , c.a . sackett , c.s . wood , w.m . itano , c. monroe and d.j . wineland , phys . a , * 61 * , 063418 ( 2000 ) .

we propose an experiment in which the phonon excitation of ion(s ) in a trap , with a trap frequency exponentially modulated at rate @xmath0 , exhibits a thermal spectrum with an `` unruh '' temperature given by @xmath1 . we discuss the similarities of this experiment to the usual unruh effect for quantum fields and uniformly accelerated detectors . we demonstrate a new unruh effect for detectors that respond to anti - normally ordered moments using the ion s first blue sideband transition .

introduction trapped ion model. finite chirp, general expression discussion and conclusion.

arxiv

measurement of the anomalous magnetic moment of the muon is very important because it can in principle bring a discovery of new physics . experimental data dominated by the bnl e821 experiment , @xmath0 ppm ) , are not consistent with the theoretical result , @xmath1 , where @xmath2 . the discrepancy is 3.2@xmath3 : @xmath4 @xcite . in this situation , the existence of the inconsistency should be confirmed by new experiments . the past bnl e821 experiment @xcite was based on the use of electrostatic focusing at the `` magic '' beam momentum @xmath5 . an upgraded ( but not started up ) experiment , e969 @xcite , with goals of @xmath6 ppm and @xmath7 ppm is based on the same principle . since the muon _ g_-2 experiment is very important , a search for new methods of its performing is necessary . one of new methods has been proposed by farley @xcite . its main distinctions from the usual _ g_-2 experiments are _ i _ ) noncontinuous magnetic field which is uniform into circular sectors , _ ii _ ) edge focusing , and _ iii _ ) measurement of an average magnetic field with polarized proton beams instead of protons at rest . a chosen energy of muons can be different from the `` magic '' energy . its increasing prolongs the lab lifetime of muons . as a result , a measurement of muon _ g_-2 at the level of 0.03 ppm appears feasible @xcite . in the present work , we develop the ideas by farley . we adopt his propositions to measure the average magnetic field with polarized proton beams and to use a ring with a noncontinuous field for keeping the independence of the spin rotation frequency from the particle momentum . we also investigate the most interesting case when the beam energy can be arbitrary . however , we propose to perform the high - precision muon _ g_-2 experiment on an ordinary storage ring with a nonuniform field created by superconducting magnets . we prove that the independence of the spin rotation frequency from the particle momentum can be reached not only in a continuous uniform magnetic field @xcite and a noncontinuous and locally uniform one @xcite but also in a usual storage ring with a noncontinuous and nonuniform magnetic field . in the last case , the total length of straight sections of the ring should be appropriate . we also analyze possibilities to avoid the betatron resonance @xmath8 ( @xmath9 is the horizontal tune ) and consider corrections to the _ g_-2 frequency . the system of units @xmath10 is used . eq . ( [ eqst ] ) is useful for analytical calculations of spin dynamics with allowance for field misalignments and beam oscillations . this equation does not contain small terms which can be neglected . @xmath13 is an analogue of the @xmath14 factor for the electric dipole moment , @xmath15 . the sign @xmath16 denotes a horizontal projection for any vector . thereinafter , the electric dipole moment will be disregarded . the vertical magnetic field , @xmath17 , is the main field in the muon _ g_-2 experiment . the spin precession caused by this field is defined by @xmath18 . \label{eqstz}\end{aligned}\ ] ] let @xmath19 denotes the average value of @xmath20 . the spin coherence is kept when @xmath21 for a storage ring with a noncontinuous field , the quantity @xmath17 should be averaged . this condition defines a spin - isochronous ring , i.e. , the spin precession frequency is independent of the momentum at the first order . condition ( [ main ] ) can be satisfied for ordinary storage rings with magnets creating nonuniform field ( fig . beam direction is normal to the magnet faces and there is not edge focusing . the number of bending sections can be different . if the field created by the magnets is given by @xmath22 , the field index and betatron tunes into bending sections are equal to @xmath23 where @xmath24 , and @xmath25 is the ring radius . average angular frequency of spin precession is given by @xmath26 where @xmath27 is a half of the total length of the straight sections ( fig . the muon anomaly is equal to @xmath28 where the fundamental constants @xmath29 and @xmath30 are measured with a high precision . the magnetic field is the same for muons and protons when they move on the same trajectory . in this case , their momenta coincide . when the momentum increases ( @xmath31 ) , the magnetic field becomes weaker , but the time of flight in the magnetic field becomes longer . the spin precession is defined by @xmath32 condition ( [ main ] ) leads to @xmath33 and is satisfied when @xmath34 where @xmath25 corresponds to @xmath35 and @xmath36 . in this case @xmath37 and the following relation takes place : @xmath38 where @xmath39 is the orbit circumference . as a result , the momentum compaction factor is @xmath40 since @xmath41 where @xmath42 is the revolution period , the definition of @xmath43 can be brought to the usual form : @xmath44 evidently , the spin - isochronous ring ( @xmath45 ) is not isochronous in the usual sense , i.e. , the beam revolution frequency depends on the momentum . ( [ eql ] ) is not exact because it does not include a correction for the fringe field . this field also contributes to the average field , but it is independent of @xmath46 . the fringe field is important only near the magnet edges and causes the correction to @xmath47 of order of the ratio of the magnet gap to the ring radius ( @xmath48 ) . this correction depends on the number of the straight sections and can be analytically and numerically calculated because the magnet field is known with a needed accuracy . evidently , the correction to the local value of @xmath49 is given by @xmath50 the corrected values of @xmath47 also coincide for muons and protons because particles with equal momenta move in the same field . two other corrections to the angular velocity of spin precession caused by the longitudinal magnetic field and the vertical betatron oscillations are considered in section iv . while these corrections are different for the muons and protons , they are rather small ( @xmath51 ppm ) . the real value of the length of the straight section , @xmath27 , can slightly differ from @xmath47 . in the general case , @xmath52 the difference between the real and nominal values of the average angular frequency of spin rotation is given by @xmath53 it is important that eq . ( [ dof ] ) does not depend explicitly on @xmath54 . the first term in the r.h.s . of this equation disappears if we _ define _ @xmath55 . in this case , @xmath35 is the vertex of a parabola in the momentum space . to find @xmath35 and adjust the ring lattice , one can make measurements with proton beams . three measurements with different values of @xmath56 are sufficient . the average proton momentum can be kept with radio frequency ( rf ) cavities put into straight sections of the ring . the longitudinal electric field in the cavities does not influence the spin dynamics . condition ( [ main ] ) leading to eq . ( [ mcf ] ) should not be exactly satisfied . it can be shown that the relation @xmath45 leads to the betatron resonance @xmath8 which results in zeroth frequency of horizontal coherent betatron oscillation ( cbo ) of the beam as a whole and a loss of the beam @xcite . therefore , the total length of the straight sections should slightly differ from @xmath47 so that the cbo tune would be small but nonzero : @xmath57 typically , in a weak focusing ring @xmath58 . ( [ mcfg ] ) results in @xmath59 . we expect that the cbo tune about @xmath60 is sufficient to keep the beam . in this case , the appropriate choice of the total length of straight sections @xmath61 reduces the dependence of the spin rotation frequency on the beam momentum by two orders of magnitude . as a result , the use of proton beams for measuring the average magnetic field becomes quite possible . experimental details depend on the beam momentum . if it is higher than in the completed experiment ( see ref . @xcite ) , the muon lifetime in the laboratory frame increases and the rf cavities may be helpful not only for protons but also for muons to keep the spin coherence . otherwise , the use of low muon momentum ( @xmath62 gev/@xmath63 ) and much higher statistics ( see ref . @xcite ) may even be more preferable . in this case , the rf cavities are unnecessary for muons . the problem of taking into account corrections to the _ g_-2 frequency is very important . one of the main problems is an influence of the radial and vertical betatron oscillations on the average vertical magnetic field . we can consider the case when the velocity of unperturbed motion , @xmath64 , coincides with the absolute value of the velocity of perturbed motion . for the latter motion , the average longitudinal component of the velocity is approximately equal to @xmath65 it can be shown that the average magnetic field for the perturbed motion , @xmath66 , slightly differs from that for the unperturbed motion , @xmath67 : @xmath68 where @xmath69 is given by eq . ( [ cbot ] ) . approximately , @xmath70 when @xmath71 , the correction to the average vertical magnetic field for the betatron oscillations is rather small and may be even negligible . a noncontinuous vertical magnetic field leads to a longitudinal magnetic field on the edges of the magnets . possibly , the latter field is a reason of the main correction to the _ g_-2 frequency . it was asserted in ref . @xcite that this field causes `` the need to know @xmath72 for the muons to a precision of 10 ppb '' . however , we should take into account that the longitudinal magnetic field can not be neglected only on small segments of the beam trajectory near edges of magnets . as a result , the above estimate of precision should be decreased by several orders of magnitude . the correction for the longitudinal magnetic field can be carefully examined . as @xmath73 and @xmath74 , the longitudinal magnetic field acting on a particle oscillates . when the vertical velocity oscillation ( pitch ) is given by @xmath75 , @xmath76 where @xmath77 is the trajectory length and @xmath78 is the cyclotron frequency into bending sections . evidently , @xmath79 . to estimate the correction , we can suppose that @xmath80 . the length of the considered trajectory segment at the magnet edge is @xmath81 . calculations can be simplified if we present the angular velocity of the spin precession in the form @xmath82 and suppose that @xmath83 . this is nothing but an estimate because the vertical magnetic field strongly varies within the considered trajectory segment . to calculate the correction , we can use the results presented in ref . the corrected local angular frequency is given by @xmath84.\label{claf}\end{aligned}\ ] ] the correction to the average angular velocity of the spin precession caused by the longitudinal magnetic field is equal to @xmath85}\cdot\frac{b}{2\pi\rho } \nonumber\\ \approx -\omega^{(a)}\frac{(a+1)^2\psi_0 ^ 2r_0 } { 4\pi n(4n - a^2\gamma^2)b } , \label{eqstc}\end{aligned}\ ] ] where @xmath49 and @xmath86 are given by eqs . ( [ eqstz ] ) and ( [ eqsto ] ) , respectively . ( [ eqstc ] ) defines only the correction for one segment of the beam trajectory . to obtain the total correction , we should take into account the maxwell equation @xmath87 . if the _ g_-2 precession did not take place , the total effect of the longitudinal magnetic field would vanish . however , the total correction is nonzero owing to a non - commutativity of rotations and is provided by the spin component orthogonal to the beam polarization at the beginning of a beam turn . therefore , the total correction , @xmath88 , can be obtained with the multiplication of @xmath89 by the additional factor : @xmath90 the quantity @xmath81 is usually of the order of the magnet gap . if we substitute the parameters of the bnl e821 experiment into eqs . ( [ eqstc]),([eqstf ] ) , we obtain @xmath91 ppm for both muons and protons . to measure the total correction with an absolute accuracy of 0.01 ppm , one should determine the magnetic field parameters with a relative accuracy of @xmath92 . since the field of magnets is well known , extra measurements may be unnecessary . when the muon beam momentum is significantly decreased as compared with the bnl e821 experiment ( see ref . @xcite ) , the correction for the muons becomes an order of magnitude less . for low - momentum beams , one can suppress the vertical betatron oscillations and additionally reduce the corrections for both the muons and protons . in the proposed experiment , the correction for the vertical betatron oscillations ( pitch correction ) @xcite ( see also ref . @xcite ) should also be taken into account . known formulas @xcite give the order of magnitude of this correction ( @xmath93 ppm ) . the pitch correction can also be reduced with a suppression of the vertical betatron oscillations for low - momentum beams . specific calculations should allow for a noncontinuity and a nonuniformity of the magnetic field . in any case , all the corrections can be determined with an accuracy of 0.01 ppm or even better . the stabilization and monitoring the magnetic field is an important and rather difficult problem . to stabilize the magnetic field in a few minutes needed for measuring the proton spin precession frequency , superconducting magnets can be used . it is more difficult to avoid a change of the magnetic field when switching from muon to proton storage . however , such a change can be properly determined . the average magnetic field can be calculated if the beam momentum and the average radius or frequency of the beam orbit are known . a change of the average magnetic field brings a corresponding change of the average radius and frequency of the beam orbit . therefore , measuring the frequencies @xcite or positions of the muon and proton beam orbits allows to determine the shift of the average magnetic field . the average proton momentum is defined by the rf cavities . in addition to the muon measurements , proton beams before and/or after muon runs can be used . the use of these methods should provide a determination of the shift of the average magnetic field with a relative accuracy of 0.1 ppm or even better . as a result , the muon and proton measurements can be related with a high precision . the methods of measurement of the _ g_-2 precession in the proposed experiment and the farley s one are very similar . the important advantage of a noncontinuous nonuniform ring versus a noncontinuous uniform one is a possibility to avoid much shimming needed for creating the uniform magnetic field . shimming is even more difficult for the noncontinuous uniform ring than for a continuous uniform one because of the fringe field . we expect that the proposed experiment can be carried out with one of existing rings . the systematical errors considered above do not prevent to measure the muon _ g_-2 factor with a high precision . the sum of all systematical errors considered in the manuscript causes less systematic uncertainty than that in the planned e969 experiment @xcite . while there are many other systematical errors , we expect that the precision of the proposed experiment may be approximately the same or better than that of the planned e969 experiment . a more detailed theoretical analysis should be based on the matrix method . the use of the matrix method is necessary for further theoretical investigations . however , any theoretical analysis is not sufficient to calculate the spin dynamics in specific _ g_-2 rings with a needed accuracy . nevertheless , necessary calculations can be carried out with spin tracking . since the theoretical predictions and the experimental data do not agree , performing new experiments based on different ring lattices is necessary . such experiments will be very important even if they will not provide better precision as compared with the usual _ g_-2 experiments @xcite . in this work , we propose the new experiment to measure the muon _ g_-2 factor . the developed experiment does not require much shimming . this experiment could provide an independent experimental result with different systematics and the advantages mentioned in the farley s paper @xcite . the author is very much obliged to f.j.m . farley for valuable remarks and discussions . the author is also grateful to i.n . meshkov and y.k . semertzidis for helpful discussions and the referees for valuable comments and remarks . the work was supported by the belarusian republican foundation for fundamental research ( grant no . @xmath9410d-001 ) . s. granger and g.w . ford , phys . ( 1972 ) 1479 ; f.j.m . farley , phys . b 42 ( 1972 ) 66 ; f.j.m . farley and e. picasso , in _ quantum electrodynamics _ , edited by t. kinoshita ( world scientific , singapore , 1990 ) ; j.h . field and g. fiorentini , nuovo cimento soc . fis . a 21 ( 1974 ) 297 .

a new high - precision experiment to measure the muon _ g_-2 factor is proposed . the developed experiment can be performed on an ordinary storage ring with a noncontinuous and nonuniform field . when the total length of straight sections of the ring is appropriate , the spin rotation frequency becomes almost independent of the particle momentum . in this case , a high - precision measurement of an average magnetic field can be carried out with polarized proton beams . a muon beam energy can be arbitrary . possibilities to avoid a betatron resonance are analyzed and corrections to the _ g_-2 frequency are considered .

introduction @xmath11-2 ring with a noncontinuous magnetic field and magnetic focusing avoiding a betatron resonance corrections to the @xmath11-2 frequency discussion and summary acknowledgments

arxiv

when illuminated by the galactic interstellar radiation field , dust in the interstellar medium gives rise to scattered radiation known as diffuse galactic light ( dgl ) . efforts to observe the dgl at different wavelengths and attempts to derive scattering properties of the dust grains from such data have extended over the past half - century , and corresponding work directed at the far - ultraviolet spectral region has been ongoing during the past quarter - century . a detailed review of dgl studies was given by witt ( @xcite ) , while bowyer ( @xcite ) and henry ( @xcite ) reviewed in - depth the particular complications involved in the observation and interpretation of the far - ultraviolet background radiation due to several sources , including the dgl . major challenges facing dgl studies in the far - ultraviolet include the need for observations with high diffuse - source sensitivity which provide extensive sky coverage while simultaneously offering the means to separate the flux from discrete sources , such as stars and galaxies , from that of the diffuse background . for the interpretation of dgl data , there has been a demand ( @xcite ) for multiple - scattering models which take into account the inhomogeneity of the scattering interstellar medium and the anisotropic distribution of the interstellar radiation field , which is extreme for the far - ultraviolet sky ( @xcite ) . the successful flight of the far - ultraviolet space telescope ( faust ) ( @xcite ) , conducted as part of nasa s 1992 atlas-1 shuttle mission ( @xcite ) , has been a response to the first of these two challenges . the photon - counting imaging detector of faust ( @xcite ) allowed a separation of the diffuse background from that due to stars in the field to a level where unresolved stars contribute less than one percent of the total diffuse intensity in the field ( @xcite ) . at the same time , this instrument provided sensitive measurements of the diffuse background in fields covering over 1000 square degrees in representative directions ranging from high to intermediate latitudes ( @xcite ) . sasseen et al . ( 1995 ) showed that the dominant component in the measured far - ultraviolet background is due to starlight scattered by galactic dust by demonstrating a clear relationship between spatial power spectra of the iras 100@xmath2 m cirrus and the faust far - ultraviolet diffuse background images . an initial attempt to model the faust ultraviolet background and to constrain the scattering properties of the diffusely distributed dust was undertaken by sasseen & deharveng ( @xcite ) . this attempt , which used the simple model of jura ( @xcite ) for high - latitude dust illuminated by a constant plane source , met with only limited success however . the model was unable to correctly reflect the anisotropy and the longitudinal variation of the intensity of the illuminating radiation field . in this paper , we are using a substantially more sophisticated model to treat the radiative transfer for the diffuse background regions observed by faust . this model was first introduced and described briefly by witt & petersohn ( @xcite , hereafter wp ) in the context of an analysis of ultraviolet background observations made by the dynamics explorer 1 spacecraft ( @xcite ) . the organization of this paper is as follows . in 2 we will summarize the data reduction and present an initial analysis . this will establish that the measured intensities are a combination of dgl , which is dependent on the column density of the dust in the line of sight , and of roughly constant contributions from airglow and possible extragalactic diffuse background radiation . in 3 we will present the wp model in greater detail then was done in wp ( @xcite ) . following this , we carry out the radiative transfer analysis of the faust data in 4 , followed by a discussion of the implications of the derived dust properties for the likely processing of interstellar dust in the diffuse medium away from the galactic plane in 5 . in 6 we will summarize our results and formulate our conclusions . the data used for this analysis were obtained with the imaging telescope , faust , during a march 1992 shuttle flight . faust has a bandpass covering the range 140 - 180 nm and a 7.6@xmath3 circular field of view . the detector was a photon - counting microchannel plate with a wedge - and - strip position sensitive anode ( @xcite ) . the in - flight angular resolution of the all - reflecting camera ( d=161 mm , f/1.12 ) was approximately 3.8@xmath4 , which permitted the identification and removal of point sources down to a detection limit of 1 x 10@xmath5 erg s@xmath6 @xmath7 @xmath6 . each pixel found to contain excess flux due to a point source was replaced with a weighted average based on distance , of those surrounding pixels containing negligible flux from the source ( sasseen et al . 1995 ) . a summary of the data used for this analysis is given in table 1 . lccccrrr dorado & 2 & 49 & 266.2 & -44.4 & 1705@xmath890 & [email protected] & [email protected] ngpkm & 8 & 64 & 250.1 & 76.3 & 900@xmath856 & [email protected] & [email protected] m87 & 9 & 64 & 285.5 & 75.0 & 1431@xmath875 & [email protected] & [email protected] virgo p2 & 10 & 58 & 284.3 & 76.4 & 1023@xmath851 & [email protected] & [email protected] virgo p1 & 11 & 62 & 283.8 & 71.6 & 1052@xmath8137 & [email protected] & [email protected] centaurus & 14 & 59 & 302.8 & 21.7 & 3169@xmath8174 & [email protected] & [email protected] m83 & 16 & 64 & 314.7 & 32.8 & 1777@xmath8147 & [email protected] & [email protected] ngc 6752 & 18 & 62 & 336.8 & -25.2 & 2190@xmath8289 & [email protected] & [email protected] hydra 20a & 20 & 62 & 261.4 & 37.6 & 1376@xmath8221 & [email protected] & [email protected] hydra 20b & 20 & 49 & 267.5 & 38.1 & 1211@xmath8184 & [email protected] & [email protected] hydra 20c & 20 & 64 & 273.5 & 38.5 & 1361@xmath8228 & [email protected] & [email protected] hydra 21a & 21 & 63 & 314.5 & 35.3 & 2200@xmath8227 & [email protected] & [email protected] hydra 21b & 21 & 50 & 309.5 & 38.3 & 2379@xmath8241 & [email protected] & [email protected] hydra 21c & 21 & 62 & 304.3 & 40.7 & 2179@xmath8210 & [email protected] & [email protected] the list contains the same faust fields as studied by sasseen & deharveng ( @xcite ) , to which we have added data from the faust image # 2 in dorado . the columns in table 1 give , respectively , ( 1 ) the name of the faust field ( sasseen & deharveng 1996 ) ; ( 2 ) the faust image number ( @xcite ) ; ( 3 ) the number of 0.5@xmath3 x 0.5@xmath3 pixels in each image for which the diffuse uv intensity was determined ; ( 4 ) & ( 5 ) the galactic coordinates for the image center , ( 6 ) the average intensity of the diffuse radiation in the 140 - 180 nm faust band in units of photons @xmath7 s@xmath6 @xmath6 sr@xmath6 , ( hereafter referred to as `` units '' ) ; ( 7 ) the average intensity of the diffuse iras background radiation at 100@xmath2 m ( @xcite ) ; and ( 8) the average hi column density taken from the bell lab hi survey ( @xcite ) , supplemented with data from the parkes hi survey ( @xcite ; @xcite ) for declination @xmath9 . the averages listed in columns ( 7 ) and ( 8) were performed over the same 0.5@xmath3 x 0.5@xmath3 pixels for which the average uv intensities were determined . the data reduction , point source removal , and the treatment of foreground light sources for the faust images was described in detail by sasseen et al . an additional discussion of the impact of the space shuttle environment on the astronomical observations and the removal of such effects is provided by lampton et al . ( @xcite ) and chakrabarti et al . ( @xcite ) . with contributions due to direct starlight effectively removed ( @xcite ) , the residual uv intensities listed in column ( 6 ) of table 1 should therefore contain mainly a sum of dgl , residual airglow , and an essentially isotropic cosmic background . small , additional contributions to the diffuse emissions from the galaxy are likely in the form of h@xmath10 fluorescence ( @xcite ) and of line emission from c iv at 155.0 nm , si iv at 139.7 nm and o [ iii ] at 166.3 nm ( @xcite ) . martin et al . ( 1990 ) detected about 75 - 150 units at low and intermediate latitudes for h@xmath10 fluorescence , while martin & bowyer found @xmath11 20 units for the listed line emissions . these intensities are small compared to the variations in the airglow component , and lacking the facility to separate them spectroscopically in the faust experiment , we will include these emissions in the airglow component in subsequent sections and use dgl to refer only to dust - scattered radiation . we confirm the presence of a strong dgl component in the data by plotting the observed average uv intensities for our 14 target fields against the corresponding hi column densities in figure 1 . the dgl component is expected to depend upon the column density of dust and upon the illumination conditions of this dust set by the galactic radiation field . the column density of dust is correlated with the hi column density ( @xcite ) . hence , we expect a correlation between the dgl component and hi which figure 1 indeed reveals . we also note the interesting fact that the variation in the uv intensity in individual fields ( table 1 ) compared to the variation of the hi column densities for the same fields is larger on average ( ratio = 3.62 @xmath8 1.37 [ photon units/10@xmath12 @xmath7 ] ) than the same ratio ( fig.1 ) for the total data set ( 2.50 @xmath8 0.37 [ photon units/10@xmath12 @xmath7 ] ) . this is easily understood if one considers the difference in spatial resolution represented by the two data sets , 30@xmath4 x 30@xmath4 pixels for the uv intensity measurements vs. the 2@xmath3 fwhm width of the horn beam of the bell lab hi survey . we infer that there must be real intensity variations in the uv background radiation due to structure in the ism too small to be resolved by the hi survey . the positive intercept at n(hi ) = 0 in figure 1 suggests that the sum of residual airglow and extragalactic background is approximately 600 units . it is likely that this is an underestimate because the coupling of a forward - throwing phase function with the concentration of galactic light sources near the galactic plane produces a higher scattered light intensity per unit dust column at lower latitudes , i.e. , higher n(hi ) values , compared to higher latitudes , or lower n(hi ) values , thus steepening the slope of the correlation and lowering the corresponding intercept ( wp @xcite ) . we can test this by plotting the uv intensities against corresponding 100@xmath2 m iras background intensities , as shown in figure 2 . the iras background is correlated both with the dust column density and the density of the interstellar radiation field responsible for heating the grains along the line of sight . the steepening of the correlation suspected in figure 1 should therefore be reduced ; a residual effect could still result from the decrease of radiation density with z - distance from the galactic plane . in figure 2 , the intercept at zero iras intensity suggests a background of about 835 units , as suggested above . however , this is rendered somewhat uncertain by three factors : ( 1 ) the zero - point uncertainties ( @xcite ) of the iras 100@xmath2 m detector ; ( 2 ) the likely presence of an as yet undetermined extragalactic background component in the 100@xmath2 m intensity ; and ( 3 ) the exceptional weight of the dorado measurement on the slope of the correlation . since the high uv intensity at dorado is consistent with expectations from our model ( see figure 5 ) , we suspect that the 100@xmath2 m intensity of dorado is in error . eliminating dorado from the correlation in figure 2 ( dashed line ) , we arrive at an intercept of 550 units , in essential agreement with the intercept in figure 1 . the closeness of the correlation between dgl and far - ir background in fig . 2 is marginally better ( r = 0.93 , dorado omitted ) than the correlation between dgl and hi in figure 1 ( r = 0.79 ) ; this may be understood through the fact that both dgl and iras 100@xmath2 m intensities depend directly on the dust column density , while the dgl - hi correlation involves a possibly variable gas - to - dust ratio . we must attribute the real width in the dgl distribution for a given iras intensity as resulting from variations due to phase function effects in the far - ultraviolet , which would leave the iras background unaffected , and from deviations of the line - of - sight cloud spectrum from the average ( see 3.3.4 ) . this initial examination of the faust data revealed that the measured diffuse background intensities consist of a constant component of at least 600 units and a dgl component which is correlated with other measures of the galactic dust column density and with the galactic radiation density . provided the first component is , on average , the same for any random subset of the data , the latter component can be analyzed with an appropriate radiative transfer model . a successful model for the dgl must have the following properties : ( 1 ) it must contain the actual observed illuminating radiation field , ( 2 ) the scattering medium should have cloud structure , and ( 3 ) multiple scattering must be included . these requirements become clear when one considers the following points . one of the most important characteristics of the dgl at far - uv wavelengths is apparent from figure 1 : the dgl intensity for a given n(hi ) varies by factors of two - to - three with direction in the sky after the approximately constant background has been subtracted . while part of this variation may be due to a spatially variable gas - to - dust ratio or to dust associated with hii or h@xmath10 rather than hi , most of the observed variation is a result of the spatially , severely anisotropic , interstellar radiation field ( @xcite ; @xcite ) . a first requirement of any suitable dgl model for the uv is therefore that it must employ the known distribution of the far - uv sources of radiation . a radiation field based on the expected uv fluxes from known hot stars listed in the skymap star catalog ( @xcite ) has recently been presented by murthy & henry ( 1995 ) . another useful source is the measured uv fluxes of some 58,000 stars detected by the td-1 satellite ( @xcite ) , which we have used for this work . the interstellar medium responsible for the opacity of interstellar space is distributed in the form of structures commonly referred to as clouds . clouds containing grains with single - scattering albedo @xmath14 and exposed to an external radiation field will have an albedo less than the single - scattering albedo ( @xcite ) , and this difference in albedo will become larger with increasing optical depth of a cloud and with increasing phase function asymmetry ( @xcite ) . in the far - uv the optical depth per cloud is typically three times that at visible wavelengths , and previous studies suggest an increase of the phase function asymmetry toward shorter wavelengths ( @xcite ) . thus , a dgl model suitable for the analysis of far - uv data should incorporate a cloud structure for the scattering medium if it is to avoid a systematic underestimate of the far - uv albedo . van de hulst and de jong ( @xcite ) demonstrated that multiple scattering is an indispensable feature involved in the transfer of dgl . this applies even more to the uv because of the greater optical depths of the galactic dust layer . in the case of the present observations , which are restricted to intermediate and high galactic latitudes with n(hi ) @xmath15 8 x 10@xmath16 @xmath7 ( see table 1 ) , multiple scattering is marginally important . with corresponding uv optical depths of order 1.2 or less , the resultant albedo bias can be as high as 25% if the entire line - of - sight material is contained in a single cloud and a single grain albedo of 0.5 is assumed . we will , therefore , include multiple scattering in our model . the principle of the wp radiative transfer model follows the pioneering approach of mattila ( 1971 ) by taking into account the discrete cloud structure of the interstellar medium . mattila s treatment was limited to directions in the galactic plane which placed his calculations into the limit of very large optical depths . this meant that the dgl intensity was determined mainly by the adopted incident radiation field which was derived from photometric measurements of the integrated sky brightness as a function of galactic latitude and longitude . in the wp model , the approach was generalized to include all galactic latitudes . as a source for the ( mostly small ) optical depths at higher latitudes we adopted the bell laboratories hi survey ( @xcite ) , supplemented by the parkes hi survey ( @xcite ; @xcite ) . we used the conversion from hi column density to optical depth at the faust effective wavelength of 165 nm as given by sasseen & deharveng ( @xcite ) , @xmath17,\ ] ] which is based on the n(h ) - e(b - v ) relationship observed by bohlin et al . ( 1978 ) and diplas & savage ( @xcite ) , and the average galactic extinction curve of savage & mathis ( 1979 ) . in order to extend the radiative transfer model to directions with @xmath18 , we need to express the variation of the radiation field with z - distance from the plane . we start by constructing the radiation field in the galactic plane ( z = 0 ) utilizing the fluxes of about 58,000 stars measured by td-1 at @xmath19 = 156.5 nm ( fwhm = 330 ) which were kindly provided to us by j. murthy . maps of this radiation field may be found in the recent work of murthy & henry ( 1995 ) and in the atlas of the ultraviolet sky by henry et al . ( 1988 ) . the td-1 catalog is thought to be complete down to fluxes of 1.0 x 10@xmath20 erg @xmath7 s@xmath6 @xmath6 at 156 nm . the contribution by stars fainter than this limit to the integrated stellar fluxes appears to be negligible at this wavelength , certainly much less than 10% of the total ( gondhalekar 1990 ) . however , the dgl observed by td-1 ( @xcite ) is part of the illuminating radiation field seen by each cloud and needs to be added to the integrated stellar flux . we therefore computed the expected dgl radiation field for a dust albedo @xmath21 and a strongly forward directed phase function and added it to the integrated stellar fluxes . this increased the total background intensities by typically 25% . for the purpose of our calculation , we represented this radiation field ( at z = 0 ) as an 800-element , two - dimensional array , and expressed the average intensity in each array element in photon units . the spacing of the array elements and their sizes are non - uniform . at low galactic latitudes , and other directions where large gradients in the radiation field intensity occur , elements are spaced more closely , while at high galactic latitudes large elements are chosen . the transformation of the central - plane radiation field into one seen by a cloud located at distance z@xmath22 0 from the galactic plane was approximated following a method described by mattila ( @xcite,@xcite ) . for the purpose of this transformation only , we assumed that the principal source of the @xmath19 = 156.5 nm radiation field is galactic ob stars whose volume emissivity is distributed in a plane - parallel disk with an exponential z - distribution characterized by a vertical scale height @xmath23 = 60 pc . we assume this because about 90% of the total integrated stellar radiation at 156 nm arises near the galactic plane defined by @xmath24 ( gondhalekar 1990 ) . also , this assumption appears justified because the number of suspected subdwarfs in the td-1 catalog , though greater than anticipated , was only of order 10@xmath25 ( @xcite ) . the galactic distribution of these stars does not differ drastically from that of the overall stellar radiation field and their contribution to the radiation field was included in any case . by comparing the expected @xmath26-distribution of the integrated intensity seen by an observer at z = z@xmath27 with the symmetric distribution seen at z = 0 , we derived transformation factors as a function of z@xmath27 and @xmath26 . this was done by dividing the skewed radiation field distribution at z = z@xmath27 by the symmetric distribution at z = 0 . to obtain an actual radiation field applicable to z = z@xmath27 , we then multiplied the td-1 radiation field with the appropriate @xmath26-dependent transformation factors . as a result , radiation fields at increasing @xmath28z@xmath29 exhibit a growing asymmetry in the radiation emanating from the two galactic hemispheres with @xmath30 and @xmath31 , and the near - equatorial intensity peak shifts toward latitudes in the opposite hemisphere while maintaining the basic non - isotropy and longitudinal variability of the original td-1 field . we constructed a total of 17 radiation fields , for values of z@xmath27 = 0 , @xmath825pc , @xmath850pc , @xmath875pc , @xmath8100pc , @xmath8150pc , @xmath8200pc , @xmath8250pc , and @xmath8300pc . interstellar clouds exist with a wide range of optical depths ( e.g. crovisier 1981 ) . in order to reflect these conditions we adopted a cloud spectrum of three types of spherical clouds : uniform clouds of optical depth diameter ( at 156.5 nm ) of 0.5 , uniform clouds of optical depth diameter of 2.5 , and centrally condensed clouds ( @xmath32 r@xmath33 for r / r @xmath34 0.05 ; @xmath35 = constant for r / r @xmath11 0.05 ) of optical depth diameter of 10 . the probability ratio of encountering such clouds along any line of sight was taken to be 12:3:1 . assuming arbitrary impact parameters for a penetrating line - of - sight , such a cloud spectrum provides , on average , a column density distribution very similar to that derived by crovisier ( @xcite ) from the statistical properties of interstellar hi which were based on 21-cm line absorption surveys . the average optical depth per cloud for our adopted spectrum was found to be 0.593 by subjecting our spectrum of three clouds , in a frequency ratio of 12:3:1 , to random penetration by a line of sight . the crovisier study seemed to be particularly relevant in the current context : ( 1 ) its database consists mainly of observations out of the galactic plane , as is the case with the faust data ; ( 2 ) our source for the column density of dust is the correlation between e(b - v ) and hi ( @xcite ) ; ( 3 ) the range of total hi column densities covered by the crovisier study coincides with the range encountered in the faust fields in this study . the probability of encountering a cloud of any type at a distance z from the galactic plane was assumed to be given by an exponential distribution with a scale height of 110 pc , i.e. was assumed to be proportional to exp ( -z/110 pc ) ( crovisier 1981 ; @xcite ) . the computation of the expected dgl surface brightness in a given direction with our model involves several monte carlo processes : the multiple scattering radiative transfer within each cloud of our cloud spectrum , the determination of the number of clouds in a given direction , the determination of the type and location of the cloud along the line of sight , the determination of the individual surface brightness profile of each cloud along the line of sight , the determination of the penetration point of each cloud , the determination of the optical depth along the chosen line of sight through each cloud , and finally the integration of intensities of the clouds along the line of sight . the transfer solution for a spherical cloud of given optical depth diameter , dust scattering parameters given by albedo @xmath36 and phase function asymmetry @xmath37 , and density profile was found with the method outlined in witt & stephens ( 1974 ) and in fitzgerald et al . ( @xcite ) . the method traces the transfer of individual photons in reverse and produces weight matrices which express the probability that photons hitting the cloud from any direction will exit the cloud at a given projected radius in the direction of the observer . for the non - isotropic scattering phase function , the henyey - greenstein ( @xcite ) function was chosen . our approach has the advantage that the radiative transfer through individual clouds needs to be done only once for a given set of parameters . the specific surface brightness profile exhibited by a cloud at a given z@xmath27 can be obtained by multiplying the cloud s probability matrix with the appropriate radiation field matrix corresponding to the value of z@xmath27 . the average number of clouds along a given line of sight with total column density n(hi ) is found by dividing the corresponding optical depth derived from eq . ( 1 ) by the average optical depth per cloud @xmath38 = 0.593 ( 3.2.2 ) . for a typical line of sight in the galactic plane ( a@xmath39 = 1.9 mag / kpc ; @xcite ) , with an average ratio of n(hi)/e(b - v ) = 4.8 x 10@xmath40 atom @xmath7 mag@xmath6 ( @xcite ) , we find 7.1 clouds / kpc , in close agreement with estimates discussed by spitzer ( 1978 ) . for typical high-@xmath41 lines of sight , we generally find 2 , 1 , or no clouds . the actual number of clouds , n , for a given monte carlo simulation is then selected by a monte carlo process from a poisson distribution whose average is the average number of clouds just determined . the types of clouds ( optical thickness diameter , density distribution ) are found by random - number choices from within our cloud spectrum based upon their relative frequencies . the z - distances of these clouds are then derived from yet another series of monte carlo processes applied to the assumed exponential distribution of cloud distances from the galactic plane . this sampling of cloud distribution by number , types , and location along the line of sight was performed 300 times for each of the approximately 60 pixel positions within each faust field for a total of 1.8 x 10@xmath42 integrations per field . thus , the statistical properties of the interstellar medium are interwoven into the prediction of the dgl intensity . we therefore predict an actual distribution of expected dgl intensities for each pixel , whose fwhm provides a measure of the dgl uncertainty resulting from possible deviations from average conditions . examples of such distributions are shown in figure 3 and will be discussed below in 3.3.4 . the probability weight matrix of each of the randomly selected and positioned clouds ( 3.3.2 ) must be multiplied by the appropriate radiation field in order to predict their radial surface brightness profiles . the most abundant variety of clouds , the optically thin ones , exhibit a brightness profile with a maximum at the projected center ; the moderately optically thick clouds present a fairly flat surface brightness profile with limb darkening , while the highly optically thick clouds display a dark central region surrounded by a bright rim ( @xcite ) . the surface brightness value added by each cloud along the line of sight is determined by the selection of a random penetration point on the geometric cross section of a cloud . in order to correct the surface brightness of still more distant clouds seen in this direction , we also determine the optical depth through the entire cloud at the penetration point . if the intensities of the clouds along a given direction are denoted by i@xmath43 , i@xmath10 , i@xmath44 the total integrated intensity for this line of sight ( @xcite ) is @xmath45 in its attempt to represent the dgl radiative transfer in as realistic a manner as possible , the model described above involves numerous parameters . while most of these , such as the radiation field distribution and the optical depth distribution , are well constrained by available observations and may therefore be considered fixed , it is the nature of the adopted cloud spectrum that affects the model predictions most profoundly . as indicated in 3.1 , the greater the optical thickness of an individual cloud , the lower its resultant effective albedo for a given radiation field and single grain albedo . in addition , in directions with low column densities , a cloud spectrum incorporating clouds of higher optical depth naturally leads to instances of `` holes '' , i.e. directions with no clouds and thus , zero dgl intensity , because the beam - averaged optical depth must still be that constrained by the hi column density . if , on the other hand , the cloud spectrum consists entirely of very - low-@xmath46 clouds , there will always by numerous clouds along lines of sight with small overall column density , assuring a more uniform sky coverage with more efficiently scattering clouds . we illustrate these effects in figure 3 by showing the results of 100 samplings with 300 photons each along a single direction ( @xmath47 , n(hi ) = 5.15 x 10@xmath16@xmath7 ) for three different cloud spectra . the first consists of optically thin clouds of type 1 only , with an optical depth diameter of 0.5 at 156.5 nm wavelength . the average optical depth for cloud penetration is 0.33 . the predicted dgl intensity is 1871 @xmath8 59 units . the second is our adopted case of a cloud spectrum with ratios 12:3:1 divided among clouds of type 1 , 2 , and 3 , respectively . the average optical depth per cloud is now 0.59 and the predicted dgl intensity is 1329 @xmath8 52 units , some 30% lower than case 1 . the third case displayed is for a cloud spectrum 1:1:1 , which now emphasizes optically thick clouds . the average optical depth per cloud is 0.83 and the predicted dgl , as expected , is still lower at 1101 @xmath8 52 units . all calculations assumed the same scattering characteristics , namely @xmath48 and @xmath49 . for the analysis of the faust data we have adopted a cloud spectrum which is dominated by low-@xmath46 diffuse clouds , but also contains a small number of clouds of greater optical thickness in a ratio of 12:3:1 in terms of their average relative frequency along a given line of sight . this spectrum reflects the demonstrated presence of numerous small molecular clouds at high galactic latitudes ( @xcite ; @xcite ; @xcite ; see @xcite for a recent review ) . our adopted spectrum is also consistent with the observed distribution of hi column densities based on the 21-cm line absorption survey by crovisier ( @xcite ) . it should be clear from fig . 3 that the albedo derived from the faust data using our adopted cloud spectrum will be higher by about 30% than it would be , had we assumed low-@xmath46 clouds entirely . the dependence of the intensity of the predicted dgl upon the chosen cloud spectrum must also be kept in mind when we compare predicted dgl intensities with those actually observed ( see fig . our aim in this section is two - fold : ( 1 ) we want to separate the galactic component of the diffuse uv intensities measured by faust ( table 1 ) from the non - galactic components . contributions to the latter are terrestrial airglow and extragalactic background radiation . ( 2 ) we want to analyze the galactic component using the dgl model described in the previous section . the aim is to derive the average dust albedo @xmath36 and the average phase function asymmetry parameter @xmath37 which characterize the galactic dust in the diffuse ism at intermediate and high galactic latitudes . toward this end we calculated the expected average uv intensity for each of the 14 faust fields , each based on approximately 1.8 x 10@xmath42 integrations , for a wide grid of closely spaced values of @xmath36 and @xmath37 . which set of scattering parameters for dust grains will lead to the best representation of our data ? to decide this question , we assumed the following . ( a ) all faust lines of sight contain dust with identical average properties . since for our observations @xmath50 , most of the clouds studied are well outside the galactic plane and are representative of the low - density diffuse interstellar medium . ( b ) the measured surface brightness of each faust field contains an unknown component with a surface brightness @xmath34 0 units due to residual airglow and a component of extragalactic origin . we assume the latter to have an intensity of 300 units ( wp @xcite ) outside the galaxy . this component is assumed to be isotropic but to be attenuated for each line of sight by the galactic dust column derived from the n(hi ) values . we included this extragalactic component both into the illuminating radiation field ( essentially negligible ) and into the predicted intensity for each faust field . ( c ) on average , the faust fields at lower galactic latitudes are subject to the same airglow contamination as the faust fields at higher galactic latitudes . this does not require that each field have an identical airglow component , only that any random group of six or seven fields have similar airglow averages . thus , the galactic latitude dependence seen in our measured intensities is due entirely to dgl and the extinction - modulated extragalactic background with the former generally increasing with decreasing latitude and the latter decreasing with the increasing extinction at lower latitudes . given these assumptions , the best - fit model must satisfy these requirements : 1 . slope criterion : the model predictions of the dgl values for the faust fields , including the partially attenuated extragalactic radiation , must exhibit the same slopes as the observations when plotted against the corresponding values of n(hi ) , cosec@xmath41 , and the iras 100@xmath2 m intensities for the faust fields . positive airglow criterion : the average predicted intensities of dgl and extragalactic light for individual faust fields should in no case fall above the measured intensities since this would imply negative airglow . equal airglow criterion : a considerable range of dust scattering parameters ( @xmath51 ) will be able to satisfy the first two criteria . however , for most sets of values of ( @xmath51 ) , the value of the positive airglow derived from application of criteria ( 1 ) and ( 2 ) is different when observations and model predictions are plotted against n(hi ) and when plotted against cosec@xmath41 . the equal airglow criterion demands that the value of the airglow deduced from the comparison of intercepts is the same when observations and model predictions are plotted against n(hi ) and cosec@xmath41 and evaluated at n(hi ) = 0 and cosec@xmath41 = 0 . a much more limited set of values ( @xmath51 ) will satisfy this third criterion in addition to satisfying the first two . we do not include the intercept in the plot against the iras 100@xmath2 m diffuse intensity in this criterion because calibration uncertainties in the zero point of the iras intensity scale and the presence of some extragalactic component in the iras data would render the outcome uncertain . in figure 4 , we show all combinations of @xmath36 and @xmath37 for which complete models were calculated . different symbols indicate the slope found when predicted intensities are plotted against the n(hi ) values for the corresponding faust fields . the observed slope ( see fig . 1 ) is 2.5 [ photon units/10@xmath12@xmath7 ] . it is clear from fig . 4 that the fitting criterion ( 1 ) is met within a very narrow range of @xmath36-values ( 0.4 @xmath52 0.5 ) but for a large range of @xmath37-values . the positive airglow criterion eliminates all models identified by encircled symbols , as well as all those with still larger @xmath36-values which were left uncomputed . as an example , a model with @xmath36 = 0.45 and @xmath37 = 0.675 is shown in fig . 5 in comparison with the faust data plotted against the n(hi ) values of the corresponding faust fields . the model points , denoted by open circles with vertical error bars , follow the observed intensities closely , except they are displaced by a constant vertical offset attributed to the airglow component . for example , the model successfully reproduces the significant differences in intensity observed for groups of faust fields for which the hi column density is almost constant . specific cases are the groups consisting of hydra 21c , centaurus and hydra 21b , and virgo p1 , dorado , and virgo p2 . this gives us confidence both in the model s predictive power and in our assumption that the residual airglow contribution is roughly constant . note that the positive airglow criterion ( 2 ) is also met in the model shown in fig . 5 . to further illustrate the close match between the model predictions with the individual observations , we added an average airglow intensity of 530 units to the model values and overplotted these on the observations in figure 6 . in almost all cases , the observations are reproduced within better than one standard deviation . thus far we have found combinations of @xmath36 and @xmath37 which reproduce the slope of the observed data in figure 5 . as mentioned , we interpret the vertical offset between the least - squares slopes fitted to the observed points and the predicted points in fig . 5 as resulting from airglow . the airglow intensity derived from fig . 5 is about 530 units , on average . when the same data and predictions are plotted against cosec@xmath41 of the positions of the faust fields , we arrive at fig . again , the observed and the model slopes are nearly identical and the vertical offset between the two slopes is approximately 530 units , i.e. , the inferred airglow intensity is the same as derived from fig . 5 . if the @xmath37-values of the models are increased beyond 0.675 , the cosec@xmath41 plot yields higher airglow values than the n(hi ) plot as well as divergent slope values ; if @xmath37 is reduced below 0.675 , the reverse occurs . this is shown quantitatively in fig . 8 where lines of constant airglow as derived from the two types of plots are shown in the relevant area of the ( @xmath51 ) plane . allowable values for @xmath36 and @xmath37 must be located in the part of the plane where the two comparisons yield the same airglow values . the straight forward application of the three fitting criteria constrains the best solution to a narrow region surrounding the point @xmath36 = 0.45 and @xmath37 = 0.675 in the ( @xmath51 ) plane . using this solution , we have derived the likely airglow intensities for our 14 faust fields and have listed them in table 2 . lc dorado & 582 @xmath8 233 ngpkm & 400 @xmath8 70 m 87 & 883 @xmath8 93 virgo p2 & 470 @xmath8 79 virgo p1 & 559 @xmath8 142 centaurus & 657 @xmath8 368 m 83 & 449 @xmath8 368 ngc 6752 & 457 @xmath8 344 hydra 20a & 307 @xmath8 298 hydra 20b & 307 @xmath8 220 hydra 20c & 390 @xmath8 239 hydra 21a & 863 @xmath8 263 hydra 21b & 717 @xmath8 338 hydra 21c & 494 @xmath8 391 it was shown in fig . 4 that the albedo value of the `` best fit '' solution is sensitively dependent upon the slope of the measured faust diffuse intensity plotted against n(hi ) . because of the close correlation between n(hi ) and cosec@xmath41 and between n(hi ) and the iras 100@xmath2 m diffuse galactic background ( @xcite ; @xcite ) , the same would be the case , if plots are made against either of these two other quantities . there are two major sources of uncertainty in these observed slopes : ( 1 ) the residual airglow is variable with time and direction , and thus , a randomly variable component is added to the true dgl values . the values in table 2 are representative of the range of variability to be expected for the airglow component . only if our dgl model were perfectly accurate would table 2 give the actual airglow values , and we do not assume this . the presence of a random airglow component having the same distribution as table 2 among our data renders the slope of the linear regression in the plot against n(hi ) uncertain by @xmath8 0.34 [ photon units/10@xmath12@xmath7 ] . ( 2 ) the absolute calibration of the faust instrument is uncertain by about 15% ( @xcite ) . while this does not affect the uncertainty of the linear regression , it affects each data point by the same percentage , and thus both the intercept and the slope are uncertain . we find the impact of the systematic calibration uncertainty upon the slope to be @xmath8 0.37 [ photon units/10@xmath12@xmath7 ] . adding these uncertainties in quadrature , we find for the observed slope 2.5 @xmath8 0.5 [ photon units/10@xmath12@xmath7 ] . an additional uncertainly arises from the fact that two independent model calculations with identical parameters but different seeds for the random number generator produce model slopes which typically vary by @xmath8 0.10 [ photon units/10@xmath12@xmath7 ] , which must be added to the uncertainty above when observations and models are compared . we again plot in fig . 9 the ( @xmath51 ) plane as in fig . 4 in which the region of possible ( @xmath51 ) values is now delineated by the application of the three fitting criteria and the uncertainties in the slope discussed above . the outer boundary is set by the slope uncertainty in the plot of faust intensities against n(hi ) . the cross - hatched area is considered unlikely for possible solutions since the model , with no airglow , overpredicts the observed intensity . the equal airglow criterion and the slope criterion lead to a most probable value for the albedo , @xmath36 = 0.45 @xmath8 0.05 , and for the phase function asymmetry , @xmath37 = 0.68 @xmath8 0.10 . ultimately , we would like to know whether the population of dust grains is different in different astrophysical environments within the galaxy . standard dust grain models are based mainly on the observation of the average galactic extinction curve ( savage & mathis 1979 ) and observed depletions of heavy elements from the gas phase ( @xcite ; @xcite , @xcite ) , and are therefore representing mostly the diffuse interstellar medium near the galactic plane . however , differences in extinction characteristics between different lines of sight ( @xcite ; @xcite ; @xcite ; @xcite ) as well as differences in depletion patterns ( @xcite ) are well documented , and these are reflected in differences in the size distributions of grains derived for different environments ( e.g. @xcite ) . scattering characteristics , like extinction properties , are also dependent on grain composition and size . there is little reason to suspect that dust grains in different environments , e.g. reflection nebulae , hii regions , or the low - density diffuse ism , are similar with respect to their scattering properties . this is especially so in light of the considerable observational evidence that the extinction properties of the dust exhibit substantial variations in the uv , both in magnitude and in wavelength dependence . in table 3 we have compiled a number of values of ( @xmath51 ) which have been derived cccccc ngc 7023 & 144,152 & @xmath53 & 0.75 @xmath8 0.05 & @xmath54 4 & 1,5 ngc 7023 & 130 & 0.62 @xmath8 0.09 & 0.75 @xmath8 0.05 & @xmath54 4 & 2,5 sco ob2 assn & 177 & 0.64 @xmath8 0.09 & undet . & 3.5 & 3 ic 435 & 156 & 0.75 @xmath8 0.05 & 0.7 @xmath8 0.1 & 5.3 & 4,6 diffuse ism & 160 & 0.45 @xmath8 0.05 & 0.68 @xmath8 0.1 & 3.1 & this study recently from the observation and analysis of scattered uv light in reflection nebulae and star forming regions , all regions characterized by densities in the range 10@xmath55@xmath56 n(h ) @xmath57 10@xmath42 @xmath58 . the denser reflection nebulae appear to share albedo values in excess of 0.6 in the 130 - 170 nm spectral range , while the present study yields a value definitely lower than that . this difference is most easily understood if we assume that the average grain size is larger in reflection nebulae , or , what would amount to the same , that the number of very small grains ( size @xmath59 ) is reduced compared to the number of larger ( size @xmath60 ) grains . this is indeed supported by the fact that in reflection nebulae r@xmath39 = a@xmath39/e(b - v ) is usually found to be larger than the `` normal '' galactic value for the diffuse ism of r@xmath39 = 3.1 , which is true also for the regions compared in table 3 . in addition , most of the dust observed in this study is seen at relatively high galactic latitudes , placing it potentially at large z - distances . with our assumed z - distribution , 50% of the dust is located at @xmath28z@xmath61 pc . sembach and savage ( 1996 ) have shown that dust grains in such environments are subject to erosive processes which return a portion of their solid substance to the gas phase . kim and martin ( @xcite ) have shown that size distributions of interstellar grains characterized by r@xmath62 3.1 have higher albedos than the standard r@xmath39 = 3.1 grains throughout the entire near - ir to uv spectral range . there are still discrepancies , however , between the observationally derived albedo values for reflection nebulae and the values predicted by kim & martin . this may simply reflect the fact that their assumed graphite / silicate composition for the grains does not correspond entirely to reality . nevertheless , we have for the first time a relatively clear indication that the dust albedo in the uv in the diffuse ism of the galaxy is lower than the albedo in denser regions , about 0.45 vs. 0.65 , and that this difference is related to the typical grain sizes found in these environments , with smaller grains in the diffuse ism yielding a lower albedo . additional support for these conclusions is found in the observations and analysis of scattered far - uv radiation seen at high galactic latitudes by onaka and kodaira ( @xcite ) . while more limited in spatial coverage , their data set allowed them to conclude @xmath63 and @xmath64 , in agreement with our results . the suggestion of smaller dust grains , on average , at higher @xmath28z@xmath28- distances also finds support in the finding by kiszkurno & lequeux ( @xcite ) of a general steepening of the far - uv extinction curve with increasing @xmath28z@xmath28-distance from the galactic plane . this characteristic is attributed to a systematic shift toward smaller grain sizes by these authors . when comparing the phase function asymmetry values in table 3 , no clear difference can be established , and , in fact , may not exist . within existing uncertainties , we conclude that the phase function asymmetry of scattering by dust in the diffuse ism and in dense nebulae is identical . the phase function asymmetry is determined solely by those grains which contribute to the scattering . if , for example , the diffuse ism contains a separate population of very small grains ( @xmath59 ) which would provide absorption , the albedo would be lowered while the phase function would still entirely be determined by the larger scattering grains and would remain unchanged . in addition , a very small change in @xmath37 represents a significant change in the phase function when @xmath65 0.7 , and as long as any determination of @xmath37 is uncertain to @xmath8 0.1 , any real differences equal to small @xmath66 s will not likely be detectable . the only situation which allows more precise differences in @xmath37 to be derived is the determination of relative @xmath37-values as a function of wavelength in a given object ( e.g. @xcite ) . the positive correlation between the far - uv background intensity and the column density of neutral hydrogen is generally viewed as evidence for the galactic origin of a major part of the uv background . ( @xcite ; bowyer 1991 ) . however , a number of separate investigations ( bowyer 1991 , table 1 ) have produced a wide range of values for the slope of the dgl - n(hi ) correlation , often mutually exclusive . bowyer ( @xcite ) tabulates values ranging from 0.3 to 1.8 [ photon units/10@xmath12 hi @xmath7 ] from eleven different studies . to this , we now add our present result of 2.5 @xmath8 0.5 [ photon units/10@xmath12 hi @xmath7 ] , larger than any found previously . the significance of these apparent discrepancies is easily understood . the scattered light intensity per [ 10@xmath12 hi @xmath7 ] is determined by the scattering cross section per h - atom , the incident radiation field , and the phase function asymmetry @xmath37 . a relatively large value of @xmath37 , as we have deduced in this investigation , provides a strong coupling between the anisotropy of the illuminating radiation field and the anisotropy of the scattered radiation field . the faust observations were conducted in a region of the uv sky where the gradient in illuminating intensity between near - equatorial galactic latitudes and polar latitudes is one of the strongest in the entire sky . the fact that this strong gradient is indeed reflected in a larger value for the slope in the dgl - n(hi ) correlation is by itself an independent indication that the scattering phase function must be relatively large . correlations between measured uv background intensities with n(hi ) are therefore comparable only , if the data have been obtained from identical regions of the sky . we can demonstrate that our value for the slope of the dgl - n(hi ) correlation is well within the range expected for different regions of the sky with the following experiment . for this test , we adopted scattering properties @xmath36 = 0.45 , @xmath37 = 0.6 and the average n(hi ) values of the faust fields . we predicted the observable dgl , including 300 units of extragalactic background , for the actual faust fields and found a slope for the dgl - n(hi ) correlation of 2.58 [ photon units/10@xmath12 hi @xmath7 ] ( fig . 10 ) . retaining the galactic latitudes of the faust fields as well as their n(hi ) values , we calculated the dgl ( plus 300 units extragalactic background ) under the assumption that their galactic longitude was @xmath67 . this is a region where the illuminating radiation field has a much weaker gradient with latitude . the resulting slope in the dgl - n(hi ) correlation was now 0.72 [ photon units/10@xmath12 hi @xmath7 ] , near the lower end of the range tabulated by bowyer ( @xcite ) . this clearly identifies the anisotropy of the illuminating radiation field as the dominant cause of the anisotropy of the dgl in the presence of a strongly forward - directed scattering phase function . the high value of the slope of the dgl / iras 100@xmath2 m correlation found by sasseen & deharveng ( @xcite ) for these faust data is also consistent with the picture of anisotropic illumination coupled with a high value of @xmath37 . the present study did not permit a separate determination of the airglow and the partially attenuated extragalactic background components , except that their sum should be about 700 photon units averaged over the 14 faust fields . the reason for this is two - fold : we lacked a spectroscopic capability which would have been able to separate the known line emissions of the airglow from the quasi - continuous extragalactic background spectrum ; and the variations in airglow intensity among different faust fields were large in comparison with the weak galactic latitude dependence in the extragalactic background imposed by galactic foreground extinction . we assumed an extragalactic component of 300 units in agreement with the determination by wp ( @xcite ) and the summary of earlier results by paresce ( @xcite ) . this produced the deduced airglow intensities listed in table 2 . after correction for galactic foreground extinction , the extragalactic component is reduced to about 230 units for the high - latitude fields and to about 100 units for the faust fields with the highest n(hi ) values leading to an average for our sample of 162 @xmath8 46 photon units . a discussion of the origin of the extragalactic component and the potential contributors is beyond the scope of the present paper . however , we must examine whether the assumption of an extragalactic component , anticorrelated with n(hi ) through extinction , has a significant effect upon the final scattering characteristics derived from the dgl analysis . repeating our analysis with the assumption that an extragalactic component in the uv is completely absent , we find that the excess intensity must then be mostly attributed to a higher airglow . the resultant albedo and phase function asymmetry are essentially unchanged , 0.43 instead of 0.45 , and 0.65 instead of 0.675 , respectively . thus , should future studies reveal our assumption of 300 units for the extragalactic uv background as inaccurate , this would not affect the validity of the present results . on the other hand , an extragalactic component of _ more _ than 300 units is effectively excluded by the lower limit to the total far - uv background , including extragalactic , measured by onaka and kodaira ( @xcite ) who directly recorded intensities as low as 300 units at high galactic latitudes , effectively using the same bandpass . it is important to state , however , that it is not necessary to know the intensities of the airglow and the extragalactic background separately in order to determine the dust scattering properties of galactic dust , provided two conditions are met : ( 1 ) the observations must cover a sufficiently large range of galactic latitudes to allow the dependence of the measured intensity upon the hi column density to be revealed , and ( 2 ) the airglow component must be relatively constant during the observation . we have analyzed the 140 - 180 nm uv background intensity in 14 faust fields at intermediate and high galactic latitudes after contributions from discrete sources such as stars and galaxies had been removed . we conclude the following : \(1 ) we confirm that a major component of the measured uv background must be of galactic origin based on its strong correlation with n(hi ) , with cosec@xmath41 , and with the iras 100@xmath2 m background intensity . we interpret this component as dgl , produced by scattering of stellar photons by galactic dust . \(2 ) the slope in the faust dgl - n(hi ) correlation of 2.5 @xmath8 0.5 [ photon units/10@xmath12 hi @xmath7 ] is found to be steeper than that seen in earlier investigations in other galactic regions . we explain this discrepancy as owing to the existence of a very strong galactic latitude gradient in the illuminating radiation field in the faust region . \(3 ) we found our radiative transfer model capable of matching the detailed variation of the dgl intensity from one line - of - sight to another . the most important characteristic of the model responsible for this is the incorporation of the detailed anisotropy of the interstellar radiation field in the uv and its variation with z - distance from the galactic plane . \(4 ) the radiative transfer analysis of the dgl component with our model resulted in a set of well - constrained scattering parameters for dust in the diffuse ism at intermediate and high galactic latitudes : albedo @xmath68 and phase function asymmetry @xmath1 . \(5 ) the dust albedo is some 50% lower than that commonly derived for dense reflection nebulae and star forming regions . we interpret this difference as arising from a difference in size distributions with the grains in the diffuse ism being smaller on average . \(6 ) a contribution of about 700 @xmath8 200 photon units appears to be uncorrelated with galactic lines - of - sight . we interpret this component as due to a sum of residual airglow ( 530 @xmath8 190 photon units ) and isotropic extragalactic background radiation ( 160 @xmath8 50 photon units , after correction for galactic extinction ) . we thank jens petersohn for his invaluable assistance with the wp radiative transfer model during the early stages of this work . the referee , dr . ken sembach , provided a number of thoughtful and constructive comments which helped to improve the presentation of this paper , and for which we are grateful . anw and bcf acknowledge material support from nasa ltsa grant nagw-3168 to the university of toledo . lampton , m. , bowyer , s. , & deharveng , j. m. 1990 , in the galactic and extragalactic background radiation , proceedings of iau symposium no . 139 , s. bowyer and c. leinert , ed . , ( dordrecht : kluwer acad . publ . ) , 449

in 1992 the far - ultraviolet space telescope ( faust ) provided measurements of the ultraviolet ( 140 - 180 nm ) diffuse sky background at high , medium , and low galactic latitudes . a significant fraction of the detected radiation was found to be of galactic origin , resulting from scattering by dust in the diffuse interstellar medium . to simulate the radiative transfer in the galaxy , we employed a monte carlo model which utilized a realistic , non - isotropic radiation field based on the measured fluxes ( at 156 nm ) and positions of 58,000 td-1 stars , and a cloud structure for the interstellar medium . the comparison of the model predictions with the observations led to a separation of the galactic scattered radiation from an approximately constant background , attributed to airglow and extragalactic radiation , and to a well constrained determination of the dust scattering properties . the derived dust albedo @xmath0 is substantially lower than albedos derived for dust in dense reflection nebulae and star - forming regions , while the phase function asymmetry @xmath1 is indicative of a strongly forward directed phase function . we show the highly non - isotropic phase function to be responsible , in conjunction with the non - isotropic uv radiation field , for the wide range of observed correlations between the diffusely scattered galactic radiation and the column densities of neutral atomic hydrogen . the low dust albedo is attributed to a size distribution of grains in the diffuse medium with average sizes smaller than those in dense reflection nebulae .

introduction the data the radiative transfer model derivation of scattering properties discussion conclusions

arxiv

the recent surge in the investigation of non - invasive techniques for cancer detection through fluorescence spectroscopy@xcite , raman spectroscopy@xcite , elastic scattering spectroscopy@xcite , optical gated imaging , optical coherence tomography , diffuse optical tomography , polarization gated imaging @xcite , turbid medium polarimetry @xcite and phase contrast microscopy@xcite are some areas being actively pursued to understand the micro - structure variations through disease progression . it is well known that the elastic scattering spectrum contains rich morphological information about the biological tissue samples due to the inhomogeneity of the constituent organelles sizes@xcite . the angular and wavelength dependence of the elastic scattering spectra have been used to analyze such subtle variations in the morphological changes @xcite . in this work , we perform a monochromatic transmission imaging ( mti ) of the tissue samples which provides us with small angle scattering information of the biological samples . et al._@xcite have analyzed the angular dependence of the elastic light scattering spectra in order to analyze the changing multi - fractality of the morphological structures in the tissues . et al._@xcite , analyzed the multi - fractality of the refractive index variation captured through phase contrast microscopy where the visible range of the electro - magnetic spectrum was used . the mti presented in this work is a proof - of - concept study of the multi - fractality of elastic scattering in small forward angles through imaging where the contribution of the larger sized scatters is more pronounced@xcite . it is well known that the elastic light scattering spectra is the power spectrum of the refractive index variation@xcite . here , we restrict ourselves to a single wavelength in order to observe the refractive index fluctuation at a particular wavelength . the use of the wavelet based multi - fractal de - trended fluctuation analysis ( wbmfdfa ) allows the use of the multi resolution analysis ( mra ) capability of the wavelet transforms to isolate trends of different polynomial orders . this is particularly helpful in the context of inhomogeneous size distribution of scatterers in a biological sample . this method has been used in various contexts like determining the multi - fractality in light scattering spectra for pre - cancer detection@xcite and studying tissue multi - fractality through phase contrast microscopy@xcite . here we apply this method to explore the possibility of differentiating between various stages of cancer . this article is organized as follows : in the next section ( [ sec : emm ] ) , we give a brief description of the sample preparation and the experimental setup . in the subsequent section ( [ sec : theory ] ) , we briefly review the fourier analysis and the wbmfdfa . in section ( [ sec : results_d ] ) , we present our observations and discuss the results of the analysis . we conclude with future directions in section ( [ sec : conclusion ] ) . hematoxylin and eosin ( he ) stained healthy and histo - pathologically graded neoplastic biopsy samples of human cervical tissues sliced into 4 mm @xmath0 6 mm ( lateral ) , @xmath1 5 @xmath2 m thick sections were prepared on glass slides for the experiment . the 12 healthy and 22 dysplastic tissue samples were obtained from g. s. v. m. medical college and hospital , kanpur , india . he staining involves nuclear staining by the application of hemalum ( a complex of aluminium ions and haematoxylin ) followed by the staining of eosinophilic structures by eosin y@xcite . the slide preparation involved standard tissue dehydration , wax embedding and sectioning under a rotary microtome@xcite . a schematic representation of the experimental setup is shown in fig . ( [ fig : expsetup ] ) . ( -5,-1.5)(5,1 ) ( -3,0)(3,0 ) ( -5,-0.5)(-3,0.5 ) ( -2,-0.4)(-2,0.4 ) ( -1,-0.4)(-1,-0.05)(-1,0.4)(-1,0.05)(0,-0.4)(0,0.4 ) ( 0.3,-0.25)(0.3,0.25)(0.6,-0.17)(0.6,0.17)(0.3,-0.25)(0.6,-0.17)(0.3,0.25)(0.6,0.17)(1,-0.4)(1,0.4)(2,-0.4)(2,0.4)(5,-0.5)(3,0.5 ) ( -4,0)laser ( 4,0)emccd ( -2,-1)ndf ( -1,-1)a ( 0,-1)s ( 0.4,-1)o ( 1,-1)l@xmath3 ( 2,-1)l@xmath4 a 632.8 nm ( output power 5mw ) he - ne laser ( research electro - optics inc . , lhrr-0200,usa ) masked by a reflective neutral density filter ( ndf ) ( special optics inc . , 9 - 1051 , usa ; two ndfs were used : a ) with optical density 0.5 , 31.62% transmittance and optical density 0.9 with 12.58% transmittance ) was used to illuminate the sample s. an aperture was used to control the beam size to @xmath1 1 mm . the transmitted light was collected through a 20x objective ( labomed lp20x semi plan achro ) and collimating lenses on an electron - multiplying charge coupled device ( andor ixon3 897 , with a pixel size 16@xmath2 m @xmath0 16@xmath2 m and image size 512 @xmath0 512 ) . this recorded raw image was then subtracted by the image of a blank glass slide to remove the artifacts arising due to glass . the exposure time was kept at 0.4s . the resulting image was then cropped for isolating the epithelium and the stroma . false color images of the epithelium for healthy and dysplastic samples are shown in fig . ( [ fig : sample_image ] ) . these images were then subjected to fluctuation analysis . the image unfolding is a method to convert two dimensional data into one dimension by horizontally ( vertically ) concatenating the rows ( columns ) of the image . for example , a matrix of the form @xmath5 can be unfolded horizontally as @xmath6 and vertically as @xmath7 where @xmath8 is the matrix transpose . this method has been used to study the multi - fractal behavior of human cervical tissues earlier @xcite where the multi - fractality of such tissues were analyzed through phase contrast microscopy . the fourier power spectrum of a signal @xmath9 is given by @xmath10 where @xmath11 and @xmath12 are dual spaces of each other and for self - similar processes , the power spectrum is well known to follow a power law @xmath13 where @xmath14 is called the power law exponent . this is related to the hurst exponent by @xmath15 , @xmath16 $ ] which is a parameter used to describe the self - similarity of mono - fractal processes and is related to the fractal dimension through @xmath17@xcite . however , the ubiquitousness of multi - fractal processes in nature have been well studied and characterized@xcite . the multi - fractality is characterized by a described by a spectrum of exponents instead of a single exponent . in the next subsection , we will briefly review the multi - fractal analysis through wavelet based fluctuation analysis . the question of multi - fractal signals has been studied by stanley and his co workers extensively though the multi - fractal de - trended fluctuation analysis ( mfdfa)@xcite . the wavelet based multi - fractal de - trended fluctuation analysis ( wbmfdfa ) proposed by manimaran _ @xcite used the multi - resolution analysis capability of the wavelet transforms to perform the de - trending of the signals . in this method , we initially make the signal @xmath18 stationary by calculating the log - return series @xmath19 and normalize it to get the normalized log - return series : @xmath20 where @xmath21 is the time average of the log - return series and @xmath22 is the standard deviation of @xmath23 . subsequently , we calculate the profile of the series through @xmath24 we use this profile series to extract the fluctuations through discrete wavelet transform . the fluctuation extraction involves a wavelet decomposition using the db4 wavelet which has a support width of 7 and 8 filters@xcite . the profile series can be decomposed as @xmath25 where , @xmath26 is the mother wavelet db4 and @xmath27 is the father wavelet such that it is orthogonal to the mother wavelet . the coefficients @xmath28(@xmath29 ) are called the low pass ( high pass ) coefficients and capture the trend ( fluctuation ) . the profile is reconstructed at a particular level @xmath30 by taking only the low pass coefficients to extract the trend at level @xmath30 . this trend is subtract from the profile to obtain the fluctuations at each scale . however , due to the convolution errors , these obtained fluctuations could have edge artifacts which are removed by performing this fluctuation extraction on the reversed profile and taking the average@xcite . then these fluctuations are subdivided in to @xmath31 non - overlapping segments such that @xmath32 where the segment length @xmath33 is related to the wavelet scale @xmath30 by the number of filter co - efficients for the wavelet used and @xmath34 is the length of the fluctuations . we obtain the @xmath35 order fluctuation function @xmath36 for @xmath37 as @xmath38^{q/2}\right]^{\frac{1}{q}},\ ] ] and for @xmath39 @xmath40^{q/2}\right]^{\frac{1}{q}},\ ] ] the fluctuation function @xmath36 and the window size @xmath33 are related by @xmath41 where @xmath42 is called the generalized hurst exponent@xcite . however , the dependence of @xmath42 on @xmath43 does not make it the ideal parameter for the characterization of multi - fractality @xcite and hence a more sophisticated function called the singularity spectrum is required . the singularity spectrum @xmath44 is related to the generalized hurst exponent by the relations @xmath45 here , @xmath46 is the multi - fractal scaling exponent and is defined by the standard partition function based formalism @xcite and @xmath44 and @xmath47 are related by a legendre transform . a quantity @xmath48 or the width of the singularity spectrum can be an important parameter for the characterization and differentiation of the multi - fractality of a signal . this parameter has recently been used to characterize the network properties of financial markets@xcite . we shall use this width of the singularity spectrum to analyze and differentiate between healthy and dysplastic tissue images in the following section . we have shown the false color blank subtracted images of histo - pathologically characterized samples of healthy and dysplastic human cervical tissues in fig . ( [ fig : sample_image ] ) . the inhomogeneity of the tissue micro - structure due to the presence of various organelles with various refractive indices can be easily observed in the figures . we use the fourier analysis and the wbmfdfa in order to probe the refractive index variations associated with the structural changes occurring in the course of disease progression . the fourier power law analysis is based on the mono - fractal hypothesis and proffers the power law exponent @xmath14 which is related to the hurst exponent @xmath49 by @xmath50 . the fourier power in this case is a function of the spatial frequency . the values of the @xmath14 for the healthy and dysplastic epithelium and stroma are shown in fig . ( [ fig : ane ] ) and ( [ fig : a2e ] ) . we observe that the mean absolute @xmath14 for the epithelium ( stroma ) is @xmath51 ( @xmath52 ) for the healthy case while it is @xmath53 ( @xmath54 ) for the dysplatic case . this power law behavior in the spatial frequency domain can be attributed to the inhomogeneous size distribution of the scatterers in the tissue micro - structure . however , this power law exponent is not independent of the scale as has been observed in the case of the phase contrast microscopy@xcite and implies the multi - fractal nature of the scatterer composition of the tissues . in terms of distinguishing different stages of cancer , as compared to the results obtained in the analysis of the phase contrast microscopic images ( where the whole visible spectral region ( @xmath55nm-@xmath56 nm ) is used as opposed to this method where a monochromatic image at @xmath57 nm is used ) , the mean value of @xmath14 is lower . nevertheless , the trend of @xmath14 for dysplastic samples being lower than that of the healthy samples is corroborated at the @xmath57 nm wavelength . as expected , the @xmath14 for the epithelium @xmath58 is higher than that of the stroma @xmath59 . the densely packed structure of the connective fibers in the stroma as compared to the epithelium causes @xmath60 . a comparison of the mean values of @xmath59 and @xmath58 for health and dysplastic tissues is given in table ( [ tab : tab_param ] ) . to further verify this and to analyze the multi - fractal nature of the tissues , we proceed to study the images through the wbmfdfa . figure [ fig : hursttiss ] shows the calculated values of the hurst exponent for healthy and dysplastic tissues in fig . ( [ fig : hne ] ) and ( [ fig : h2e ] ) respectively . the hurst exponent for the epithelium @xmath61 and that for the stroma @xmath62 are also compared . we can see that while the healthy tissues show a mean @xmath61 of @xmath63 , the dysplastic tissues display a mean value of @xmath64 . similarly , the values of mean @xmath62 are @xmath65 and @xmath66 for healthy and dysplastic stroma respectively . this is in agreement with the results of the fourier analysis based power law calculations . as observed earlier , the @xmath67 . the comparative values of @xmath62 and @xmath61 are given in table ( [ tab : tab_param ] ) . we mentioned in the earlier section that multi - fractal signals require the singularity spectrum for characterization and that the width of the singularity spectrum @xmath68 is a parameter used to describe multi - fractality of the data under consideration . in fig . ( [ fig : gammaepi ] ) , the @xmath69 and @xmath70 for healthy [ fig : gne ] and dysplastic [ fig : g2e ] tissues is shown . the mean @xmath70 and @xmath69 are found to be @xmath71 and @xmath72 respectively . as compared to the differences between @xmath58 and @xmath59 and @xmath61 and @xmath62 ; the difference between @xmath70 and @xmath69 is more pronounced . still , the trend of @xmath73 is followed , also implying the higher multi - fractality of the epithelium as compared to the stroma . .[tab : tab_param ] comparison of the mean parameters calculated through the fourier and the wbmfdfa . $ ] represents the ensemble average over the different samples of healthy and dysplastic tissues . the parameters have been averaged over @xmath75 and @xmath76 samples of healthy and dysplastic cervical tissues respectively . [ cols="^,^,^,^,^ " , ] in conclusion , we have compared the normal and dysplastic human cervical epithelium and stroma in an effort to probe the differences between their multi - fractality towards developing a possible non - invasive optical technique for cancer detection . the use of monochromatic imagery for this purpose provides us information about the micro - structural changes in the tissues associated with disease progression . we observe that the multi - fractality of tissues decreases with progression of cancer . though the @xmath14 or @xmath68 are not drastically different , the trend of decreasing multi - fractality through the progression is sufficient to quantify the stage of cancer . for example , the decreasing @xmath14 shows that the as the cancer progresses , the tissue micro - structure goes from @xmath77 towards a flatter power spectrum . similarly , the hurst exponents show that with the increase in the cancer grades , the structural organization goes from a seemingly more random behavior to a spatially long term correlated behavior . the decreasing @xmath68 implies the decrease in multi - fractality with the disease progression . this work is a proof of concept about the exploration of monochromatic laser imagery to understand the variations of the tissue micro - structure when observed at a particular wavelength as compared to phase contrast microscopy where the whole visible range is used . however , we believe that due to the presence of various fluorescing enzymes in the cancerous tissues like nad@xmath78 , the response of the tissues to different wavelengths would be different and a more extensive study of the same is being undertaken which will be reported soon . a notable feature of this study is its candidature for _ in - vivo _ examination and characterization of the mti which is also under investigation currently . ghosh , n. , majumder , s. k. , patel , h. s. , and gupta , p. k. , `` depth - resolved fluorescence measurement in a layered turbid mediumby polarized fluorescence spectroscopy , '' _ opt . lett . _ * 30 * , 162164 ( jan 2005 ) . haka , a. s. , shafer - peltier , k. e. , fitzmaurice , m. , crowe , j. , dasari , r. r. , and feld , m. s. , `` diagnosing breast cancer by using raman spectroscopy , '' _ proc . ( usa ) _ * 102*(35 ) , 1237112376 ( 2005 ) . ghosh , s. , soni , j. , purwar , h. , jagtap , j. , pradhan , a. , ghosh , n. , and panigrahi , p. k. , `` differing self - similarity in light scattering spectra : a potential tool for pre - cancer detection , '' _ opt . express _ * 19 * , 1971719730 ( sep 2011 ) . soni , j. , jose , g. p. , ghosh , s. , pradhan , a. , sengupta , t. k. , panigrahi , p. k. , and ghosh , n. , `` probing tissue multifractality using wavelet based multifractal detrended fluctuation analysis : applications in precancer detection , '' in [ _ biomedical engineering and informatics ( bmei ) , 2011 4th international conference on _ ] , * 1 * , 448452 ( december 2011 ) . perelman , l. t. , backman , v. , wallace , m. , zonios , g. , manoharan , r. , nusrat , a. , shields , s. , seiler , m. , lima , c. , hamano , t. , itzkan , i. , van dam , j. , crawford , j. m. , and feld , m. s. , `` observation of periodic fine structure in reflectance from biological tissue : a new technique for measuring nuclear size distribution , '' _ phys . lett . _ * 80 * , 627630 ( jan 1998 ) . gurjar , r. s. , backman , v. , perelman , l. t. , georgakoudi , i. , badizadegan , k. , itzkan , i. , dasari , r. r. , and feld , m. s. , `` imaging human epithelial properties with polarized light - scattering spectroscopy , '' _ nature medicine _ * 7 * , 12451248 ( 2001 ) . drezek , r. , guillaud , m. , collier , t. , boiko , i. , malpica , a. , macaulay , c. , follen , m. , and richards - kortum , r. , `` light scattering from cervical cells throughout neoplastic progression : influence of nuclear morphology , dna content , and chromatin texture , '' _ j. biomed _ * 8 * , 7 ( 2003 ) . ghosh , n. , buddhiwant , p. , uppal , a. , majumder , s. k. , patel , h. s. , and gupta , p. k. , `` simultaneous determination of size and refractive index of red blood cells by light scattering measurements , '' _ appl . lett . _ * 88*(8 ) , 084101 ( 2006 ) . yu , c .- c . , lau , c. , odonoghue , g. , mirkovic , j. , mcgee , s. , galindo , l. , elackattu , a. , stier , e. , grillone , g. , badizadegan , k. , dasari , r. r. , and feld , m. s. , `` quantitative spectroscopic imaging for non - invasive early cancer detection , '' _ opt . express _ * 16 * , 1622716239 ( sep 2008 ) . kalashnikov , m. , choi , w. , yu , c .- c . , sung , y. , dasari , r. r. , badizadegan , k. , and feld , m. s. , `` assessing light scattering of intracellular organelles in single intact living cells , '' _ opt . express _ * 17 * , 1967419681 ( oct 2009 ) . hunter , m. , backman , v. , popescu , g. , kalashnikov , m. , boone , c. w. , wax , a. , gopal , v. , badizadegan , k. , stoner , g. d. , and feld , m. s. , `` tissue self - affinity and polarized light scattering in the born approximation : a new model for precancer detection , '' _ phys . _ * 97 * , 138102 ( sep 2006 ) . ilker r. apolu , rogers , j. d. , taflove , a. , and backman , v. , `` accuracy of the born approximation in calculating the scattering coefficient of biological continuous random media , '' _ opt . _ * 34 * , 26792681 ( sep 2009 ) . ghosh , s. , manimaran , p. , and panigrahi , p. k. , `` characterizing multi - scale self - similar behavior and non - statistical properties of fluctuations in financial time series , '' _ physica a _ * 390*(23 - 24 ) , 4304 4316 ( 2011 ) .

in this work , we report a wavelet based multi - fractal study of images of dysplastic and neoplastic he- stained human cervical tissues captured in the transmission mode when illuminated by a laser light ( he - ne 632.8 nm laser ) . it is well known that the morphological changes occurring during the progression of diseases like cancer manifest in their optical properties which can be probed for differentiating the various stages of cancer . here , we use the multi - resolution properties of the wavelet transform to analyze the optical changes . for this , we have used a novel laser imagery technique which provides us with a composite image of the absorption by the different cellular organelles . as the disease progresses , due to the growth of new cells , the ratio of the organelle to cellular volume changes manifesting in the laser imagery of such tissues . in order to develop a metric that can quantify the changes in such systems , we make use of the wavelet - based fluctuation analysis . the changing self- similarity during disease progression can be well characterized by the hurst exponent and the scaling exponent . due to the use of the daubechies family of wavelet kernels , we can extract polynomial trends of different orders , which help us characterize the underlying processes effectively . in this study , we observe that the hurst exponent decreases as the cancer progresses . this measure could be relatively used to differentiate between different stages of cancer which could lead to the development of a novel non - invasive method for cancer detection and characterization .

introduction experimental materials and methods theory results and discussions conclusion

arxiv

after the sax positioning of grb970228 ( costa et al . 1997 ) , and the discovery of an optical transient in the refined error box ( van paradijs et al , 1997 ) , the optical counterpart of grb970228 has been observed many times both with ground based instruments and with the hubble space telescope . several days after the event , an extended optical emission was detected where the optical transient ( ot ) had been seen in the discovery image , taken 21 h after the event ( van paradijs et al , 1997 ) . since then , the magnitude of such an extended emission has been measured many times , by several observers , using different instrumental set - ups . in this paper we review and compare the measurements gathered so far to investigate if the flux values recently measured by stis on hst ( fruchter et al , 1997 ) and by the 5 m palomar telescope ( djorgovski et al . , 1997 ) are consistent with the ground based ones obtained at early epochs . table 1 summarizes the data collected so far , both for the ot integrated magnitude ( ground measurements ) and for the contribution of the two components : point source and extended emission ( hst data ) . the first claim for an extended object , using an 1 hour ntt exposure taken on march @xmath0 , gave @xmath1 ( van paradijs et al 1997 ) . to this , one should add the keck measurement ( @xmath2 , metzger et al , 1997a ) obtained on march @xmath3 but announced after the discovery of the optical transient ( groot et al . 1997a ) . [ sym_tab ] indeed , all the march ground measurements agree in describing the extended object as elongated in the north - south direction with @xmath4 . hst observations were carried out in late march and early april using the wfpc and a broad v filter ( f606 ) . the extended source was resolved into a point source superimposed to a `` fuzz '' , which , according to sahu et al . ( 1997 ) , were detected at @xmath5 and @xmath6 , respectively . comparison of the march and april images showed that the point source was most probably fading , while nothing definite could be said on the diffuse emission ( sahu et al . 1997 ) . using the same data , galama et al ( 1997 ) estimate an r magnitude of @xmath7 for the point source and @xmath8 for the fuzz . in the same paper , the value of the magnitude measured by the ntt on march @xmath0 was also revised , bringing it to @xmath9 . more keck observations taken on april @xmath10 and @xmath3 gave , for the total emission , a @xmath11 ( metzger et al . 1997b ) i.e. significantly lower than both the hst one and the keck march @xmath3 data . in april the source became unobservable from the ground and from hst . the observability window opened again in late august when it was pointed both from the keck / palomar ( iau circ 6732 ) and from hst using , this time , the newly installed stis . both observations show the overall flux to be lower than that measured previsiously . on sept @xmath12 , djorgovski et al ( 1997 ) used the palomar 5 m telescope to obtain an r image of the field where the extended source was detected at @xmath13 . the stis instrument on board hst also observed the source on sept @xmath12 with the clear filter . fruchter et al , ( 1997 ) are barely able to detect the point source , now at @xmath14 , over a diffuse emission of @xmath15 , i.e. 0.8 magnitude fainter than in the hst march observation . this prompted a reanalysis of the march / april wfpc data which resulted in a reassessment of their magnitude value now estimated at @xmath16 , i.e. half of the flux published by sahu et al ( 1997 ) for two independent wfpc observations . although not stated , a similar downward revision should apply also to the r magnitude values published by galama et al ( 1997 ) for the hst observations . + however , even accepting that the hst data , after re - analysis , can be rendered consistent , it seems very difficult to reconcile the september stis / palomar data with the ntt / keck ones of early march . table 1 shows that a suggestion for fading of the extended component of the ot seems to be present . however , magnitude values do not always render easy the comparison of data taken through different filters . the suggestion for a significant fading of the extended source , implicit when comparing march to september data , becomes stronger when one computes the actual energy fluxes . this is done in table 1 , where we have transformed the magnitude values in @xmath17 . in the following , we shall compare the ntt data ( kindly provided to us by jan van paradijs ) with the hst ones . in order to do so we have to assume that the extended emission seen from the ground is indeed the superposition of the fuzziness seen by hst plus a point source . if we assume that the september stis / palomar flux values for the extended source are correct , and if we further assume no fading , we have to explain the extended total emission seen both by ntt and by keck with a combination of the stis / palomar fluxes plus a point source of suitable magnitude . even considering the revised ntt mag value given by galama et al ( 1997 ) , we have to account for a total flux of @xmath18 . since the extended source observed in september provides @xmath19 , the unseen point source should have been @xmath20 , i.e. definitely brighter that the extended one . however , in order to simulate the appearance of such a combination one should be able to locate the hst point source into the ntt nebulosity . this calls for an accurate superposition of the hst march data onto the ntt / susi frame . to take care of geometric distorsion , we have used the task `` mosaic '' which re - scales the hst / pc image , rebinning it to pixel size of 0.1 `` . the resulting image has been rebinned and rotated onto the susi one ( 0.13''/pixel ) using a standard technique which is certainly accurate to better than 1/2 pixel ( actually 1/10 would be a more realistic estimate ) . figure 1 shows the superposition of the hst march frame onto the susi one . only the central portion of the actual images is given . zooming on the ot , one sees clearly that the hst point source falls in the central part of the ntt nebulosity , where the emission is less intense and no hint of a point - like object is seen . however , the central region of the nebulosity is just where one should put a hypothetical point source of @xmath21 , corresponding to @xmath22 . this value is similar to the flux measured by the ntt for the faint source just south of the grb970228 counterpart . such a source is in interesting test case , since it is point - like in the hst / pc image ( star # 3 in figure 1 ) but it looks extended in the ntt image . however , inspection of figure 2 , where we have compared the right ascension and declination tracings of the two sources , shows unambigously their difference in shape for a comparable flux . while star # 3 is dominated by a clear peak superimposed to a region of higher background , the grb nebulosity does not show any obvious point - like contribution . this is somewhat surprising , since a point source of @xmath23 should have been far easier to detect than a @xmath13 extended one . moreover , we note that such a faint extended source would be hardly within reach of an 1 hour ntt exposure . + thus , the truly extended nature of the ntt source , coupled with the lack of point source at the hst location , leads to the conclusion that the nebulosity itself has faded away from march @xmath0 and sept @xmath12 . although comparing fluxes obtained with different instruments , different filters and different observing conditions is not straightforward , the compilation of the magnitudes values measured so far for the optical counterpart of grb970228 points toward a fading both of the point source and of the diffuse emission . while the fading of the point source is expected in all theoretical scenarios , the fading of the diffuse emission has far reaching consequences and , as such , is in need of a dedicated observing campaign . the data available are numerous , but too diverse to provide the constraints needed to assess with certainty if , and how much , the nebulosity has faded . indeed , for grb 970228 , it looks as if every new observation results in a downward revision of the values previously published . only more observations , directly comparable with those already in hand ( i.e. obtained with identical instrumental - ups ) can provide a definite aswer to this all important point . of particular importance could be new hst / pc data since the unfiltered stis image is not directly comparable to the pc ones , obtained with a broad v filter . caraveo p.a . et al,_a & a. _ * 326 * , l13 ( 1997 ) . costa e. et al._nature _ * 387 * , 783 ( 1997 ) . djorgovski s. et al _ iau circ 6732 _ ( 1997 ) fruchter , a. et al . _ iau circ . 6747_(1997 ) galama t. et al . _ nature _ * 387 * 479 ( 1997 ) groot j.p . . 6584 _ ( 1997a ) groot j.p . ( 1997b ) metzger m.r . . 6588 _ ( 1997a ) metzger m.r et al . . 6631 _ ( 1997b ) sahu k. et al . _ nature _ * 387 * 476 ( 1997 ) van paradijs et al._nature _ * 386 * 686 ( 1997 )

in view of the data gathered in september 1997 , we review the flux values collected so far for the `` fuzziness '' seen in the optical counterpart of grb970228 . comparison between the ground based data collected in march and the data of september 1997 suggests a fading of the fuzz . given the diversity of the data in hand , the magnitude of the effect and its significance are not easy to quantify . only new images , both from the ground and with the space telescope , directly comparable to the old ones could settle this problem .

introduction the data hst vs ntt conclusions

arxiv

i was very happy to accept an invitation from h.machner to give the summary talk for meson2000 . however i was quickly surprised and somewhat dismayed by many of the plenary talks , which discussed meson physics from viewpoints with which i was only vaguely familiar . a second theme of this meeting , beginning with prof . jarczek s welcoming talk , was the 600th anniversary of the refounding of the jagellonian university , and its place in polish and european history . late one night , while puzzling over how to reduce this rather broad meeting to a few remarks , the random thought occurred to me that it would be easier to summarize the last 600 years of european history than to summarize this conference . and , like many wild ideas , this one would not go away . so , please bear with me through fig.1 . [ fig_1 ] = 5truein one can learn several interesting things from fig.1 . the first is that , in physics at least , we have made considerable progress . at the beginning of this timeline scholars were mainly concerned with theological considerations that were not amenable to experimental confirmation . with the reestablishment of european universities and recovery of classical texts , physicists again turned to the study of astronomy , which led to a precise formulation of classical mechanics in the 17th century . the copy of _ de revolutionibus orbium coelestium _ ( copernicus ) which many of us have seen here in cracow is a stirring reminder of the work of our antecedents in this field . to my mind european culture peaked in the 18th century age of enlightenment , both in the development of scientific attitudes and socially , in fields as disparate as government ( jefferson ) , history ( gibbon ) , literature ( goethe ) and music ( mozart ) . with the emergence of the romantic movement in the 19th century , a general decline was evident ; theories of government became more radical , nationalism became fashionable in europe , and music and literature increasingly reflected the social problems of the age . to quote wittgenstein regarding music , as early as brahms ` i can begin to hear the sound of machinery . ' physics nonetheless continued dramatic advances , due in part to developments in this machinery and in mathematics , and the 19th century saw major achievements in the establishment of the theories of electromagnetism and classical thermodynamics . finally , in the 20th century europe entered very troubled times indeed , from which we are only now emerging . one should note that good things can arise even in times of adversity , such as the reappearance of an independent poland in 1919 . physics also went through crises at the beginning of the 20th century , but we must all agree that the resulting quantum physics and relativity are two of the most exciting and fascinating developments in the field . against this 600 year historical background the development of hadron physics has been strikingly rapid . the identification of the compact atomic nucleus , the home of most terrestrial hadrons , was due to rutherford in 1911 . the identification of the positively charged proton , the first known hadron , can also be dated to about 1911 . the first meson to be identified was the pion , found by lattes _ et al . _ in 1947 ( in cosmic rays ) , and it had been anticipated by yukawa as the carrier of the strong nuclear force . the familiar light mesons @xmath1 , @xmath2 , @xmath3 etc were found in the late 1950s to early 1960s , and the identification of these and the light baryons suggested the quark model to zweig , gell - mann and neemann in about 1963 . the identification of qcd as the theory of the strong interaction , in 1973 , was due mainly to its property of asymptotic freedom , which had been observed at slac in the late 1960s . the crucial confining property of qcd was at the time regarded as an unproven conjecture , and is still poorly understood . the mid to late 1970s saw the experimental establishment of the charm and beauty families of hadrons , the first searches for glueballs , and the development of new theoretical techniques such as lgt . the remarkable qcd predictions of glueballs and exotic mesons have taken longer to test experimentally , and the more widely accepted experimental candidates for these states were identified in the middle 1990s . this short time scale is most reassuring ; in fig.1 we can see that almost all the progress in strong interaction physics has been made in the last 10% of the timeline , and the study of qcd itself occupies only the final 5% . and finally , as if to close the circle , at the end of the millennium many theoreticians have again turned to theological speculations which are not amenable to experimental confirmation . after the first few plenary talks i was confirmed in my suspicion that hadron physics , even meson physics , is a very broad field with clearly identifiable communities that have little overlap . since much of the work in contributing to a new field involves learning the field s terminology or `` jargon '' , one can identify the different communities by the rate of recurrence of characteristic words or expressions in research papers or presentations at conferences . two obvious communities in nonperturbative qcd are the effective lagrangian / chiral symmetry specialists ( to whom the pion is the most interesting meson ) and the hadron spectroscopists ( to whom it is not ) . as a test of the idea of separating these groups by their use of language , i invented an order parameter which i call `` physicist chirality '' @xmath4 to distinguish them . @xmath0 is defined by the number of times a plenary speaker used the word `` chiral '' compared to the exotica words `` glueball '' , `` hybrid '' or `` exotic '' , @xmath5 this quantity is @xmath6 for a purely exotic physicist and @xmath7 for a purely chiral physicist . i had expected to find an interesting distribution of @xmath0 in this meeting , so i applied it to the first 18 plenary speakers . the result , shown in fig.2 , is rather disturbing . one sees clear evidence of `` phase separation '' in the presence of two almost completely disjoint communities in meson physics ! [ fig_2 ] 1.5 cm = 3truein for those who are keeping score , the two extreme cases on an absolute scale were w.weise ( 39 uses of chiral ) and a.szczepaniak ( over 29 uses of exotic ) . there were also 0/0 scores ( recorded at 0 ) , which suggested additional phases ; eventually i identified five subconferences at meson2000 , which are summarized in the following section . this subdiscipline of meson physics considers the pion to be the most interesting of mesons because it is the lowest - lying excitation , in condensed matter jargon the `` gap mode '' . in this field one discusses hadrons in terms of the @xmath8 `` order parameter '' , writes effective lagrangians for pions and nucleons , and derives the resulting low - energy scattering amplitudes and in - medium properties . apparently part of the game is to borrow as much of the condensed matter viewpoint and jargon as possible , and use it in a hadronic context . ( i am not being entirely frivolous about what appears to me to be an exercise in borrowing jargon , since i actually work in condensed matter physics @xcite . ) our first plenary speakers weise , thomas and ( rather surprisingly for an impartial experimentalist ) nefkens identified themselves as belonging to this community through their frequent references to chiral symmetry and order parameters . mosel , senger , grioni , cassing and oset also participated in this subconference , which was largely concerned with prospects for seeing mass shifts e.g. of kaons , nucleons and vector mesons in medium , and in - medium corrections to other processes , such as @xmath9 scattering . in these talks one could often hear loan words from condensed matter , extending in extreme cases to `` spectral functions '' and even `` good quasiparticle '' , which is a rather imprecise notion even in quantum magnetism . throughout i confess to having a feeling that we need one very clear , unambiguous experimental observation of a hadronic in - medium mass shift in a relatively narrow state , such as the @xmath3 , before this work on mass shifts can be regarded as supported by experiment ; the inclusive distribution of dilepton mass pairs , composed of many hypothetical broad contributions , seems less than convincing . cassing noted in particular that there are several interesting prospects for observing @xmath3 mass shifts . experiment has not contributed much to this field in - vacuum " recently , largely because the favored topics of low energy @xmath9 and @xmath10 scattering were studied long ago , _ albeit _ not with especially good statistics in @xmath9 . at this meeting however we heard plans for two new relevant measurements . one is a determination of the @xmath11 @xmath9 scattering length difference using @xmath12 atoms , discussed by gianotti . there are dispersion relations known as the roy equations that purportedly constrain these quantities accurately , so this will be a useful measurement . the second ( at daphne , discussed here by lauss ) is a measurement of @xmath13 @xmath14 and @xmath15 scattering lengths , similarly using radiative transitions in the hadronic atom . this second subconference was concerned with `` good auld hadrons '' , by which i mean low energy reactions and the spectroscopy of reasonably well established quark model states . there was a large experimental component from the various few - gev facilities that are studying these processes . meson production near threshold , especially as accessible at cosy , was discussed by speth . the interesting topics here are the various possible mechanisms for @xmath16 and @xmath17 production and the recurring question of the physical size of the @xmath18 `` quark core '' in a baryon . haberzettl gave a clear review of the complications of electroproduction amplitudes , in particular in the production of associated strangeness at cebaf . filippi told us about ozi violation in @xmath19 annihilation , especially in the final state @xmath20 , and discussed ways of distinguishing between intrinsic strangeness and rescattering effects such as @xmath21 . moskal discussed @xmath22 at cosy-11 , where @xmath23 and @xmath24 , and noted the importance of isi and the possibility of measuring the @xmath25 scattering length . salaburn discussed @xmath22 at disto , where one can also study @xmath26 . he noted that one may test hidden-@xmath27 components using polarized @xmath28 , and also that the @xmath29 data supports a simple @xmath1-exchange picture . stryer discussed baryon resonance production and decays , and noted that such a programme at cosy using @xmath30 beams would be a useful complement to the various photon facilities now planned or in operation . niskanen discussed the origin of isospin violation effects in nucleon - nucleon scattering and suggested channels in which these effects may be largest . klimala considered meson production in @xmath31 collisions and discussed how one might test models such as intermediate @xmath32 production . eyrich discussed prospects for strangeness production at cosy using tof , including @xmath33 ( current ) , @xmath34s ( possible in future ) , and the very interesting but long neglected @xmath35 exotic flavor channels ( in particular the sector @xmath36 ) . kupse summarized future plans for celsius / wasa , which include rare @xmath37 and @xmath17 decays , including the processes @xmath38 ( i violating , of interest in @xmath39pt ) and @xmath40 ( for which vmd and @xmath39pt give rather different predictions ) . finally , braccini reviewed the very interesting results on @xmath41 couplings of @xmath42 , @xmath27 and @xmath43 resonances which have come from lep and cornell recently . these results include observations of the possible radial excitations @xmath44 and @xmath45 , and in @xmath46 evidence for the light axials @xmath47 and @xmath48 and the @xmath49 , which is apparently produced through @xmath50 vector dominance . the subject of non-@xmath51 mesons , the so - called `` exotica '' , has seen exciting developments of late , with the announcement of glueball and spin - parity exotic candidates . this , i admit , was the conference i attended . barnes first gave a review of exotic mesons ( exotic meaning having quantum numbers forbidden to conventional @xmath51 states ) . we now have two light exotic candidates , the @xmath52 ( bnl , crystal barrel , ves ) and @xmath53 ( bnl , ves ) . unfortunately it may be too soon to celebrate , since lgt predicts that the @xmath54 exotic level should lie at about 2.0 gev , much higher than reported . klempt discussed glueballs , in particular the various states in the @xmath55 sector . these include the crystal barrel candidate @xmath56 , the possibility of a single very broad scalar `` der rote drache '' , and various models of scalar mixing and decays . szczepaniak summarized the cebaf halld project , which is a proposed high - statistics meson photoproduction experiment for the study of light exotic and @xmath27 meson spectroscopy . willutski reviewed results from bnl e852 , including evidence for the @xmath57 , and in @xmath58 for a @xmath59 @xmath60 and a @xmath61 @xmath62 . note that the @xmath60 is considerably lighter than expected by godfrey and isgur for a 2p state . close reviewed his very interesting work on the exchanged angular quantum numbers and @xmath63 dependences in diffractive meson production , which can be used as a `` discriminator '' between @xmath51 and @xmath64 candidates . clearly something very important about diffraction has been discovered here , although just what is as yet unclear . peters restricted himself to the `` past , present and future '' of meson spectroscopy , including the 10th anniversary of the crystal ball @xmath65 dalitz plot , developments in exotics , the importance of high statistics , complications in analyses , scalars , d decays , ... and future facilities . finally , stefanski reviewed results from the charm photoproduction experiment e791 at fermilab , and noted that the @xmath66 dalitz plots from @xmath67 and @xmath68 show evidence for strong isobar contributions , including @xmath69 , @xmath70 @xmath71 and @xmath72 . the interesting evidence of fsi effects in the complex relative phases of these states was also noted . subconference 4 ( on hep ) was the shortest , with just two plenary contributions . these contributions could be identified by the fact that the hadrons were clearly considered non - essential complications to the interesting physics . the first hep contribution was by sciaba , who summarized the status of the search for @xmath73 mixing . this has not yet been observed , but the limits are now rather close to theoretical expectations , `` watch this spot '' . next , fleischer reviewed the general subject of cp violation in b decays in impressive detail , and suggested several final states which may be of interest in future experimental studies . the final subconference i identified , with six plenary contributions , was on photon - hadron interactions . ( braccini s two - photon talk might also be listed here as a seventh contribution . ) the first contribution was by levi sandri , who reviewed the baryon resonance program at graal , and noted some interesting results , such as the fact that the @xmath74 @xmath75 ratio does not agree well with capstick and isgur s quark model predictions . bruncko reviewed desy results on vector photoproduction of @xmath2 , @xmath3 , @xmath76 , @xmath77 , @xmath78 and @xmath79 . a remarkable `` universal curve '' of electroproduction cross sections versus @xmath80 was shown for these states . steffens reviewed polarized deep inelastic scattering at hermes , especially the `` semi - inclusive '' processes @xmath81 , tests of schc , production mechanisms and parton distributions . arends reviewed the status of the dhg sum rule and concluded that there is no indication of disagreement with experiment . nikolaev discussed diffractive electroproduction of vector mesons and the interesting possibility of distinguishing 2s from d states through their different @xmath80 dependences . and finally , muccifora discussed charged and neutral @xmath16 electroproduction at hermes , which can be used to test the @xmath80 evolution of fragmentation functions . these results included surprising evidence of possible isospin symmetry violation above @xmath82 . although there were many interesting results presented at the meeting , i would like to take advantage of my rle as summary speaker to cite what seemed to me personally to be the single most remarkable new experimental and theoretical results . in experiment : the `` universal curve '' for the @xmath80 dependence of vector meson electroproduction appears to be a very suggestive observation , and presumably tells us something very general about hadron electromagnetic couplings . does this establish a vector dominance picture over direct photon - quark pqcd amplitudes ? if so , most quark model calculations of resonance photoproduction and electroproduction amplitudes may be inaccurate ! the question of just what this result teaches us should clearly be pursued . in theory : close has found remarkably simple and accurate results for diffractive scattering amplitudes , using an almost conserved vector coupling model ; this is telling us _ something _ profound about the long standing issue of just what the `` pomeron '' is at the quark - gluon level . as with many interesting discoveries , it is not yet clear what these results mean , but they suggest that progress in this long - standing question may now be possible . at this conference , the question of the limits of usefulness of various models must have occurred to many of the attendees . this story gives some indication as to feynman s attitude to the use of models in hadron physics . in about 1974 as a new caltech graduate student i was looking for an interesting thesis topic . this was an exciting period with many new developments in physics , such as supersymmetry , string theory , the parton model , non - abelian gauge theories , compact objects in astrophysics and so forth . although i was initially interested in rather formal problems in quantum gravity , the very practical and skeptical research atmosphere at caltech strongly encouraged graduate students to study topics that led to direct comparison with experiment . since qcd had just been proposed , and a group at mit had just published their first paper on the `` bag model '' which showed that one could derive many experimentally observable properties of light hadrons quite simply using quark and gluon `` cavity resonator '' modes , i began work on this model and suggested it as a thesis topic to my advisor jon mathews . since mathews was a rather pure mathematical physicist , he was unenthusiastic . he suggested however that i talk to feynman about this work , since feynman had heard about the model at a meeting and had since worked out many of its predictions for hadrons himself . i found that feynman , unlike mathews , was very interested in and excited by what could be derived in this simple model ; so much so that i rather courageously asked him if he would tell mathews that the study of this model was a suitable thesis topic . the resulting transition from initial to final states is shown in fig.3 below ( this is a feynman diagram in which feynman actually appears ) . [ fig_3 ] 1.5 cm = 3.2truein as usual , precisely what took place within the circle is unknown , but much can be inferred from the initial and final states . in this case , in the initial state @xmath83 the mit bag model was _ not _ a suitable ph.d . thesis topic , and in the final state @xmath84 it _ was _ a suitable topic . although the details of the interaction were not observed , @xmath84 made statements to the effect that _ the study of models is useful in hadron physics because one can abstract model - independent features_. i presume that this is the justification feynman gave to mathews . this has indeed proven to be the case for the bag model , since it was the first to predict a light @xmath85 exotic meson ; this exotic has been found by all subsequent approaches , including lgt . we now have two experimental exotic meson candidates with these quantum numbers , which were discussed in detail at this meeting , and the general topic of `` exotica '' is now widely considered the most interesting subject in light hadron spectroscopy . in summary , each model is wrong in detail , but they may nonetheless contain some common physical truth . it is a great pleasure to thank the organisers for their kind invitation to summarize the many conferences of meson 2000 . this research was supported in part by the doe division of nuclear physics , at ornl , managed by ut - battelle , llc , for the us department of energy under contract no . de - ac05 - 00or22725 , and by the deutsche forschungsgemeinschaft dfg at the university of bonn and the forschungszentrum jlich under contract bo 56/153 - 1 .

this short contribution is a _ lite _ meson2000 conference summary . as appropriate for the 600th anniversary of the jagellonian university , it begins with a brief summary of the last 600 years of european history and its place in hadron physics . next a `` physicist chirality '' order parameter @xmath0 is introduced . when applied to meson2000 plenary speakers this order parameter illustrates the separation of hadron physicists into disjoint communities . the individual plenary talks in meson2000 are next sorted according to the subconference associated with each of the 36 plenary speakers . finally , i conclude with a previously unreported feynman story regarding the use of models in hadron physics .

hadrons and the last 600 years of european history the many phases of the meson community the index of plenary speakers and their conferences personal favorites feynman story acknowledgements

arxiv

the universe at large appears to be remarkably homogeneous and isotropic and its dynamics governed by the gravitational force created by its material content . in such a flrw geometry mach s principle , that is the idea that local inertial frames are determined by an average motion of the matter in the universe , can hardly be put to test because of the too high symmetry of the model . however the universe is not perfectly homogeneous and isotropic : perturbations are present , in particular on very large scales , as testified by the discovery by the cobe satellite of cosmic microwave background anisotropies as well as the observations of voids with size not negligible with respect to the hubble radius [ 1 ] . such perturbations , which ought to be treated within a general - relativistic framework , could be used to test , at least in principle , mach s ideas - in a similar spirit as they were studied in the case of the flrw ( and of lematre - tolman - bondi ) universes in [ 12 ] . in particular , if we happened to live in a slowly rotating void , that is an underdense region separated from the almost flrw universe outside by a slowly rotating sheet - like structure or wall , then the apparent motion of distant stars could be used to measure the angular velocity of the wall , that is the transverse component of its peculiar velocity , which is inaccessible to the usual redshift measurements . as a first approach to this idea of using mach s principle in observational cosmology we shall here address the question of how to measure a global rotational velocity field in the universe . to do so we shall study a very simplified model which however encaptures the physics of the problem : we shall consider an empty void separated from the almost flrw ( dust ) universe outside by a spherically symmetric , slowly rotating shell comoving with the outer universe . if the assumption of spherical symmetry is reasonable , the assumption of an empty void , on the other hand , is drastic and will force us to consider very heavy shells which can not be treated perturbatively . however , since we are only interested here in the observable effects of the slow _ rotation _ of the shell , its weight is not so important . finally the assumption that the shell is comoving with the cosmic dust means that we consider the void long after its possibly explosive creation , when damping forces have slowed it down . we shall also ignore all perturbations which are not due to the rotation of the shell as they can be linearly added . such spacetimes have already been studied in the past . we shall not review here the enormous literature on _ non_-rotating voids in cosmology ( see e.g. [ 2 ] and references therein ) . rotating voids in cosmology , on the other hand , do not seem to have been much treated . indeed , slowly rotating voids in asymptotically _ flat _ spacetimes have mainly been used to study machian effects , how and to what extent they are embodied in general relativity . the classical papers of thirring [ 3 ] , treating a slowly rotating shell of small mass , were generalized by brill and cohen [ 4 ] who considered a slowly rotating shell separating an empty void from the schwarzschild solution at first order in the rotational perturbation . lindblom and brill [ 5 ] further extended their work by treating a freely falling , slowly rotating shell ; in addition , in order to interpret in physical terms the rotational perturbation inside the shell , they studied the apparent motion of a searchlight at the centre of the shell as seen from infinity . more recently , katz , lynden - bell and bik [ 6 ] solved the opposite problem ( which is more interesting from the astrophysical point of view ) of how fixed stars at infinity appear to rotate to an inertial observer at the origin of the void . an evident drawback of all these schwarzschild - shell models is that the whole universe is described by the shell in an asymptotically flat spacetime so that all rotations are defined with respect to the inertial frames at infinity . thus the rotation of all inertial frames within the void is determined , at least in part , by these asymptotic frames that play the role of something given absolutely , and the shell then can not be considered as the only source of inertia in the universe . lewis [ 8 ] was the first to discuss the dragging in a truly cosmological context . he considered a slowly rotating shell immersed at a given comoving position into a closed flrw spacetime . he gave the solution for the rotational perturbation at first order and discussed some machian features of the model . later , klein [ 7 ] , analysed more in detail a slowly rotating shell immersed at a given comoving position into a flrw spacetime with open flat spatial sections @xmath0 . he discussed the dragging of inertial frames inside the shell under the condition that the rotational perturbation is caused by the shell only . he found that perfect dragging ( that is the fact that an inertial observer inside the void measures a zero angular velocity for the shell ) is reached only as the radius of the shell goes to infinity , where the whole matter of the universe is distributed on the shell . in the next paper [ 14 ] , klein also considered some effects caused by slowly rotating shells in @xmath1 universes to the second order in the angular velocity . in section ii of the present paper we analyze , in contrast to [ 7,8 ] , the rotating voids in _ all _ types of the flrw universes in a _ unified _ manner , pointing out the differences between the open and closed cases . we give the spacetime metric as well as all angular velocities in terms of two parameters - of a constant characterizing the angular momentum of the shell , and of the comoving radius of the shell . using bardeen s formalism [ 10 ] to treat the rotational perturbation at first order we calculate the dragging of inertial frames inside the void and determine the conditions for perfect dragging to occur . for einstein - de sitter universes @xmath0 we recover , in an alternative way , the results of klein [ 7 ] . we extend his work to all open models @xmath2 and closed models , where new features arise , as , for example , the condition that the total angular momentum of the universe must vanish . we illustrate that perfect dragging can not be reached in a closed universe . in section iii we analyze , in the spirit of katz , lynden - bell and bik [ 6 ] , the apparent motion of the distant stars as seen by observers inside a void . as we are aware of , this problem has not been tackled before in a cosmological context . we thus study the propagation of light in the spacetimes obtained and describe how the two parameters of the problem , the angular momentum of the shell and its comoving radius , could in principle be measured . in the last section iv we argue that ( slow ) rotations in the open universes are absolute in the sense that there is no freedom to choose rotating axes . one is thus left with a preferred coordinate system and hence with the possibility to define all rotations with respect to this system . on the other hand , there is no such system in closed universes . we finally discuss the machian features of the model . aware of the controversies connected with the formulation and interpretation of mach s principle [ 11 ] , , [ 16 ] and ref . a short history and meaning of mach s principle is reviewed in [ 12 ] . ] we express all rotations in terms of , in principle , measurable quantities . the observations of the shell and the cosmic dust outside the shell , made by inertial observers inside , are in full conformity with mach s ideas ( for both open and closed universes ) : the matter of the models considered is not observed to be globally ( slowly ) rotating in a preferred direction . in this sense the inertial frames inside the void are determined by the average motion of the matter in the universe in the spirit of mach s principle . details of calculations of the matching across the shell , of the rotational metric perturbation outside the shell , and of the angular velocity of a star measured inside the void are given in appendices a , b , c . the void is empty so that spacetime inside the void can be represented by a portion of minkowski space . consider a family of standard clocks at rest with respect to each other - they define an inertial frame @xmath3 with the line element ds^2|_in =- d|t^2+d|r^2+|r^2(d^2+^2d|^2),where @xmath4 is the time measured by the clocks and @xmath5 their spherical coordinates . now one can use another time coordinate @xmath6 such that @xmath7 where the lapse function @xmath8 is to be specified later . one can also use coordinates which rotate with respect to @xmath5 , in such a way that @xmath9 where the angular velocity @xmath10 is a function of @xmath6 . in this new ( rotating ) frame @xmath11 with coordinates @xmath12 the line element ( [ 1 ] ) to first order in @xmath10 reads ds^2|_in=-^2(t)dt^2+d|r^2+|r^2(d^2+^2d^2)-2|r^2 _ 0(t)^2ddt . the shell is supposed to be spherically symmetric . its equation of motion can thus be written as @xmath13 ; @xmath14 is for the moment an arbitrary function of time . the metric induced on the shell by ( [ 2 ] ) is then ds^2|_in^=-[^2(t)-^2(t)]dt^2+l^2(t ) ( d^2+^2d^2 - 2_0(t)^2ddt),where @xmath15 and @xmath16 denotes the shell . outside the void , the universe is supposed to be almost perfectly homogeneous and isotropic , up to a rotational perturbation . there exists therefore a coordinate system in which the line element reads ds^2|_out =- dt^2+a^2(t).as usual , the curvature index @xmath17 for respectively spherical , flat or hyperbolic spatial sections , @xmath18 is the scale factor and function @xmath19 is the rotational metric perturbation . for the perturbation theory at first order to apply it must be small , that is @xmath20 . we will also suppose that the shell is comoving with the cosmic dust , thus placed at a constant comoving radius @xmath21 . the metric induced on such a shell by ( [ 5 ] ) then reads ds^2|_out^=-dt^2+a^2(t)l_0 ^ 2 , where @xmath22 is the value of the rotational perturbation on the shell : @xmath23 . let us now complete the matching of the two coordinate grids on the shell . the induced metrics ( [ 4 ] ) and ( [ 9 ] ) must be continuous [ 9 ] , [ 15 ] . this first gives the radial position of the shell @xmath14 in terms of its comoving coordinate @xmath24 and the scale factor @xmath18 as l(t)=l_0 a(t ) . second it implies that the angular velocity @xmath10 and the time shift @xmath8 of the coordinate system @xmath11 with respect to the system @xmath3 are _ 0(t)=_0(t),(t)=. israel s junction conditions [ 9 ] , [ 15 ] give the stress - energy tensor of the shell in terms of the jump in its extrinsic curvature ( see appendix a ) . if we consider the shell to be a rotating 2-dimensional perfect fluid , we can write _ ab=(_+p_)w_aw_b+p__ab , w^a=(1,0,_),where @xmath25 , @xmath26 are respectively the shell energy density and surface pressure and @xmath27 ( @xmath28 being the azimuthal coordinate of a fixed point of the shell in the outside coordinates ) , @xmath29 being the proper time on the shell . ] is the angular velocity of the shell with respect to the outside coordinate grid or , also , with respect to the rotating frame @xmath11 inside the void . the indices @xmath30 stand for @xmath31 and are raised and lowered by the induced metric on the shell ( [ 9 ] ) . the matching conditions ( app.a , eq.([14 ] ) ) result in _ ( t)&=&14g1al_0(- ) , + p_(t)&=&-12_-18gl_0a , + _ ( t)&=&_0(t)-116ga(_+p_)r|_0 , where we define @xmath32 . the energy density and pressure of the shell are determined in terms of @xmath24 and the scale factor , and its angular velocity will be known once the metric perturbation @xmath19 is determined . we note that @xmath33 is always positive and that the equation of state of the shell differs in general from that of the flrw universe outside . a number of global quantities can be associated with the shell . for example we can define its total ( proper ) mass as m_(t)=4a^2l_0 ^ 2_= al_0g(-).note that this mass is _ not _ equal to the ( proper ) mass @xmath34 of the universe which has to be removed in order to create the void . indeed m_(t)&=&a2ga^2l_0 ^ 3k=0 + m_(t)&=&3a4g(1+a^2)(_0-l_0)k=1 ( and a similar expression for @xmath35 ) . only in the the case of voids small compared to the hubble and curvature radii ( @xmath36 , @xmath37 ) do @xmath38 and @xmath34 coincide . otherwise , in general , @xmath39 . in any case , unless the void is much larger than the hubble radius , which is unlikely for causality reasons , @xmath38 represents a sizeable fraction of @xmath34 and therefore is large , which is not surprising since we assumed the void to be totally empty . more interesting is the angular momentum of the shell j_=_t^t _ dd,@xmath40 being the determinant of the induced metric ( [ 9 ] ) , which is conserved due to the axial symmetry of the problem at hand . using the decomposition ( [ 15]-[16 ] ) it can be rewritten as j_=83a^4l_0 ^ 4(_-_0)(_+p_)= -16gl_0 ^ 4a^3r|_0 . the source which governs the outside metric ( [ 5 ] ) is the stress - energy tensor of the cosmic dust : t_=u_u_,u^=(1,0,0,),where @xmath41 is the metric ( [ 5 ] ) , @xmath42 is the energy density of the dust , and @xmath43 is its ( small ) angular velocity . einstein s equations for the rotational metric perturbation @xmath19 are solved in appendix b. they give ( t , r)=-1a^3((r)+g(t ) ) , where ( r)=-2gr^3(1 + 2kr^2)j_+6g_l_0^rj(r^)r^4dr^. the radially dependent function @xmath44 , j(r)_l_0^rt^t _ dddr^=83a^5_l_0^r(-)r^4dr^ , is the conserved total angular momentum of the dust layer @xmath45 $ ] outside the shell . hence , the first term on the right hand side of ( [ 27 ] ) , proportional to @xmath46 , comes from the rotating shell while the second term is the contribution of cosmic matter outside the shell . as for the function @xmath47 , it is clear that it can be absorbed in the azimuthal coordinate @xmath48 and thus expresses the possibility to choose any global ( slowly ) rotating axes , i.e. the frame @xmath11 , but we keep this freedom for later purposes . einstein s equations thus yield us the metric perturbation @xmath19 in terms of the function of time @xmath47 and the radial dependent function @xmath44 . these functions will be determined by boundary conditions . @xmath49 _ the case of open universes _ in open universes the function @xmath47 is determined uniquely by the condition @xmath50 for all radii @xmath51 so that the perturbation @xmath19 must die away at infinity at least as @xmath52 . when @xmath53 vanishes , as will be the case , this amounts , by ( [ 25 ] ) and ( [ 27 ] ) , to choosing the coordinate system in such a way that g(t)=0k=0;g(t)=-4gj_k=-1.the metric ( [ 2 ] ) inside the void is then completely known : @xmath8 and @xmath10 are given by ( [ 11 ] ) and _ 0(t)&=&2gj_a^3l_0 ^ 3k=0 + _ 0(t)&=&2gj_a^3l_0 ^ 3k=-1 . let us finally turn to the metric outside the shell which still depends on the ( so far unknown ) function @xmath44 . the conservation laws for the dust universe imply @xmath54 so that from ( [ 22 ] ) we deduce that the combination @xmath55 does not depend on time . we thus define a function @xmath56 which describes the perturbation of the stress - energy tensor of cosmic dust outside the shell , @xmath57 , or , equivalently , its angular momentum distribution . in general it is not determined uniquely just by requiring @xmath58 at infinity and can in principle be fixed by initial conditions or deduced from observations . however , if we demand , in accord with klein [ 7 ] , that the only source for the rotational perturbation @xmath59 at any @xmath51 is the shell itself ( brought in some way into a slow rotation ) , we eliminate the contribution of the cosmic dust in ( [ 27 ] ) by choosing @xmath60 . then ( t , r)=(t , r)j(r)=0 . with the condition ( [ 31 ] ) the metric outside the shell is known in terms of the single parameter @xmath61 : ( t , r)&=&2gj_a^3r^3k=0 + ( t , r)&=&2gj_a^3r^3k=-1 . @xmath49 _ the case of closed universes _ in this case the condition @xmath50 holds for any small @xmath47 ( with @xmath62 small - cf . ( [ 25 ] ) ) and we are left with a freedom to choose ( slowly ) rotating axes . we can thus choose the gauge simply as g(t)=0.the metric inside the void is then known . in particular _ 0(t)=2gj_a^3l_0 ^ 3(1 + 2l_0 ^ 2 ) . in order now to determine the metric outside the void we first take into account the fact that the total angular momentum of the _ closed _ universe must vanish [ 7 ] , [ 12 ] . the choice @xmath63 , used in the open cases is not compatible with this constraint . this means that the integral in ( [ 27 ] ) can not be eliminated , i.e. the matter outside the shell _ does _ necessarily influence the rotational perturbation : the cosmic dust has to have some intrinsic angular momentum to counteract the rotation of the shell . nevertheless , even the zero total angular momentum condition does not determine the function @xmath64 uniquely . we shall , for definiteness , treat the example @xmath65 , where the constant is fixed by @xmath66 where we put as usual @xmath67 ( @xmath68 ) . then @xmath69 and the metric outside the shell is known in terms of the angular momentum of the shell @xmath61 . we obtain @xmath70^{-1},\ ] ] where ( ) = 132(12 - 82 + 4 ) , and the analogue of condition ( [ 31 ] ) reads @xmath71^{-1},\ ] ] i.e. j()=-j_. equation ( [ 27a ] ) can now be integrated to obtain the rotational perturbation outside the shell in the form ( t,)= 38 gj _ 1a^3 \{1 ^ 3- c(_0)},where @xmath72 is given by ( _ 0)= 1 ^ 3_0\{2_0(12 + 13 ^ 2_0)(3_0- 3_0)+2_0},the comoving radius of the shell , @xmath73 , being a parameter of the solution . the value of the rotational perturbation at the shell is obtained by the limit @xmath74 in ( [ 37 ] ) _ 0(t)=-gj_a^3 ^ 3_0[3_0 - 3_0 ] , which is identical to ( [ 34 ] ) . the function @xmath75 decreases monotonously as the function of the comoving radius of the shell @xmath76 ; it vanishes at @xmath77 . fig.[fig1 ] shows the rotational perturbation @xmath78 as a function of the radial coordinate @xmath79 for several positions @xmath76 of the shell . for @xmath80 the rotational perturbation vanishes at some point , say @xmath81 $ ] . for @xmath82 the rotational perturbation is negative in the whole spacetime outside the shell . however , we can always choose the function @xmath47 in such a way that @xmath78 vanishes at some given @xmath83 whatever is the value of @xmath76 - see section iv for a discussion of this point . let us now turn to the rotation of the shell . in ( [ 15 ] ) we defined the shell angular velocity with respect to the outside coordinate grid : @xmath27 . we can also define its proper angular velocity , @xmath84 , @xmath85 being the trajectory of a point of the shell in the inertial frame @xmath3 and @xmath29 being its proper time . as the shell is slowly rotating , we see at first order , i.e. for @xmath86 and @xmath87 , that |_=_-_0 . putting together equations ( [ 16 ] ) , ( [ 25 ] ) and ( [ 27a ] ) , and recalling that @xmath88 , we find for all @xmath89 : |_=3gj_a^3l_0 ^ 3 1 - -l_0 ^ 2a^2(a / a).note that in order to keep @xmath61 constant , the agular velocity of the shell in general depends on time through the scale factor @xmath18 . a _ fixed star _ , as defined in [ 6 ] in the case of the schwarzschild shell , denotes a source of light placed at infinity at rest with respect to the asymptotic inertial frame on the centre of the shell . we shall not in general consider such fixed stars as they are of little use in a cosmological context for the following reasons : 1 . the radius of the visible universe is limited by the horizon ; 2 . there is no infinity in closed cosmological models ; 3 . stars comove with the cosmic matter and in general they are not at rest with respect to the @xmath90 coordinate grid because of the presence of the ( time - dependent ) rotational perturbation . we shall thus talk about the apparent motion of _ distant stars _ rather than fixed stars , the 4-velocity of these stars being given by the 4-velocity of the cosmic dust , i.e. by ( [ 21 ] ) . the _ angular velocity of a distant star as measured by an observer in the inertial frame _ @xmath3 in the void is given by @xmath91 is observer s proper time interval between the arrival times of two photons emitted by the star and @xmath92 is the angular separation travelled by the star in the interval @xmath93 , as measured in @xmath3 . the results of appendix c show that this angular velocity of a star comoving with the cosmic dust at a given radius @xmath94 ( where @xmath95 for @xmath96 , respectively ) , emitting photons radially inwards in the equatorial plane , is given by the expression 0 mean that the quantities are evaluated at the time of emission and at the time when the first photon reaches the shell , respectively . we recall that the comoving radius of the shell is @xmath97 or @xmath98 . @xmath49 _ the case of open universes _ the scale factor of a @xmath1 universe is given in terms of the conformal time @xmath99 and a constant @xmath100 as @xmath101 . with the condition ( [ 31 ] ) that the rotational perturbation arises only due to the rotation of the shell , @xmath102 , we obtain from eq.([a18 ] ) staightforwardly |_star^m=-384gj _ a_m^3_0 ^ 21 + 2_0/_0 _ _ 0^_*d^4 [ _ 0+_0-]^4,where @xmath103 . since @xmath104 can never be positive , the distant stars are seen rotating _ backwards_. note that the stars placed just behind the shell , @xmath105 , do not rotate with respect to the inertial observers inside the void . interesting from the observational point of view is today s dependence of @xmath106 on the cosmological redshift @xmath107 of the star . @xmath108 is given as a function of the conformal time @xmath109 elapsed by the time @xmath110 when the first ray reaches the observer at the centre of the void . between @xmath111 and @xmath110 , when the light travels inside the void , the conformal time changes from @xmath108 to @xmath109 so that @xmath112 resulting in ( _ rec^2 + 4_0 ^ 2)^3/2=3_0_0 ^ 2+(_0 ^ 2 + 4_0 ^ 2)^3/2 , and @xmath109 is given as a function of today s estimated age of the universe , @xmath113 years , as @xmath114 equation ( [ a20 ] ) gives the expression for @xmath108 explicitly as @xmath115^{1/3}+ \left[x - y\right]^{1/3}-\chi_0\right\}^2 - 4\chi_0 ^ 2,\ ] ] in which @xmath116 ^ 2-\chi_0 ^ 6}.\ ] ] the equality ( [ a19 ] ) can now be evaluated as a dependence @xmath117 , @xmath118 being the measured redshift , @xmath119 ^ 2}-1.\ ] ] several curves illustrating today s angular velocity of a star parametrized by the comoving radius of the shell , as a function of @xmath118 , are plotted in fig.[fig2 ] . as the measured redshift depends on @xmath120 , when the light ray leaves outer universe , and not on @xmath121 at the time of observation , the cosmological redshift does not bring any information about the radius of the void . eq.([a19 ] ) also shows that in a nonstationary universe the distant stars do not rotate uniformly : for @xmath76 , @xmath94 fixed , @xmath104 depends on the time of observation through @xmath108 and ( [ a20 ] ) . @xmath104 decreases monotonously down to zero as @xmath122 ( then @xmath123 so that the rotational perturbation vanishes ) . the @xmath35 models give qualitatively the same results . = 3.1 in = 3.1 in : one can check in eq.([a20 ] ) that @xmath124 when @xmath125 , and @xmath126 for @xmath127 . since @xmath128 corresponds to the coordinate value of the particle horizon of an observer at the centre of the void at the time @xmath129 , while @xmath109 would correspond to today s particle horizon of a pure flrw observer , the presence of a void increases the particle horizon of the observers inside . @xmath49 _ the case of closed universes _ in the closed case also the cosmic dust contributes to the rotational perturbation : @xmath130 . equations ( [ 27a ] ) and ( [ 36 ] ) imply a(_*)[(_*,_*)-(_*,_*)]&= & -j_g1a_m a ( _ * ) , + d [ a^3]()d&=&-6gj_^4()-()()-(_0),so that eq.([a18 ] ) becomes |_star^m= -4gj_a^3_m , where @xmath131)^2}d\chi\quad\hbox{and}\quad \alpha(\eta_0)=\sqrt{1+{\sin^2\chi_0\sin^2\eta_0\over(1-\cos\eta_0)^2}}.\ ] ] @xmath104 in the above expression is negative for all @xmath94 , @xmath76 , @xmath108 . as in the case of open universes , the distant stars are seen rotating _ backwards_. in contrast with the open case , the cosmic dust induces the first term at the right hand side of ( [ a21 ] ) , which does not vanish , unlike the integral @xmath132 , in the limit @xmath105 . hence , if the dust outside the shell contributes to the rotational perturbation , the stars just behind the shell in general rotate with respect to the inertial observers inside the void ( see fig.[fig2 ] ) . the dependence @xmath108 on @xmath109 ( obtained similarly to the case @xmath1 ) , is given by @xmath133 which can be solved numerically ( only when @xmath134 one can obtain a simple analytic formula for @xmath135 ) . inserting @xmath135 into ( [ a21 ] ) , together with @xmath136}-1,\ ] ] we get today s dependence @xmath106 on the redshift ; fig.[fig2 ] shows several typical curves parametrized by @xmath76 . equation ( [ a18 ] ) allows to determine the angular velocity of the shell as measured by the inertial observer at the centre of the void . if one replaces @xmath137 by @xmath138 , and with @xmath139 , one gets the apparent angular velocity of the shell as |_^m=1_0[_-_0]_0=1_0 [ |_]_0 , @xmath140 is the proper angular velocity defined in subsection iiia . @xmath49 _ the case of open universes _ combining the above expression with the expression for the proper angular velocity of the shell ( [ a3 ] ) we find |_^m=192gj_a^3_m^3_0 ^ 6_0 . this expression can never be negative . thus , for any given finite @xmath76 the shell is seen rotating _ it decreases as @xmath76 increases and vanishes as @xmath141 . a similar behaviour is obtained for @xmath76 fixed in the limit @xmath122 ( fig.[fig4 ] ) . the case of @xmath35 open universes yields qualitatively the same results . @xmath49 _ the case of closed universes _ repeating the same steps as in the previous case we now get |_^m=24gj _ a_m^3 ^3_0[1-_0]^3 ( _ 0 ) 1 . it is easy to see that , just as in the case of open universes , @xmath142 can never be negative . the shell is thus always seen rotating _ forwards_. however , in contrast to the open case , @xmath142 , as a function of @xmath108 , shows a minimum which never reaches zero . one gets a similar behaviour for @xmath108 fixed and varying @xmath76 ( fig.[fig4 ] ) . we will return to this point in the next section . the discontinuity in the angular velocities at the shell , @xmath143 in the limit @xmath144 , shown in fig.[fig2 ] , allows one to determine observationally , at least in principle , the two parameters of the problem : @xmath46 and @xmath24 . the light emitted from sources placed on the shell , and just behind it , propagates in minkowski spacetime and is observed with zero cosmological redshift . the source rotating forwards must be on the shell ; the measurement of @xmath142 in eq.([a23 ] ) or ( [ a24 ] ) gives @xmath46 as a function of @xmath76 . in the case @xmath145 , an additional measurement of a counter - rotating @xmath146 source @xmath147 determines , due to ( [ a21 ] ) , both @xmath46 and @xmath76 . for @xmath1 eq.([a19 ] ) implies that @xmath148 , and the observer inside the void must therefore measure a source at a known redshift instead of a source just behind the shell . @xmath149 , where the redshift corresponds to the distance between @xmath76 and @xmath94 , determines , together with @xmath142 , both @xmath46 and @xmath76 . one can alternatively measure the angular velocities of distant stars at known redshifts and read off the parameters @xmath46 and @xmath76 from the curves at fig.[fig2 ] . the angular velocity of the shell then follows from ( [ a23 ] ) or ( [ a24 ] ) . for given values of @xmath46 and @xmath76 , the angular velocities of distant stars at known redshifts depend on the type of the flrw model so that their measurement can also in principle determine the curvature index @xmath150 . it is clear that above results depend quantitatively on our choice of the function @xmath64 expressing the initial condition on the motion of the cosmic matter outside the shell - another choice in ( [ 31 ] ) and ( [ 35 ] ) would have led to a different redshift dependence of @xmath151 . the actual measurement of the redshift dependence of @xmath151 would then determine @xmath64 . note however that the angular velocity of the shell , @xmath142 , does not depend on the choice of @xmath64 , see eq.([a22 ] ) . we note also that since @xmath152 depends on time , we can not require @xmath153 . hence , a discontinuity in the angular velocities @xmath142 , @xmath104 at the shell occurs for any choice of @xmath64 . the determination of the void radius , allowed only thanks to the presence of the rotational perturbation , is crucial for the observer inside the void . only if @xmath76 is known , one can properly adjust the redshift method to measure distances - this would not be possible inside a void in a non - perturbed flrw universe . the inertial frame @xmath3 attached to the centre of the void , as well as other inertial frames in the void all rotate with the same angular velocity with respect to the rotating frame @xmath11 or , equivalently , with respect to the observers inside the shell fixed at constant @xmath154 . the two coordinate grids are related by @xmath155 . from @xmath156 it follows that = _ 0(t ) , where the explicit expression for @xmath22 is ( [ 29 ] ) or ( [ 34 ] ) . the rotation of the spatial axes of the frame @xmath3 with respect to those of @xmath11 demonstrates _ the dragging of inertial frames _ inside the void . it will be useful to define the _ preferred _ observers outside the shell as those which are at rest in the outside coordinate grid in which the line element is given by ( [ 5 ] ) , i.e. at constant @xmath157 . the preferred observers are not in general freely falling ; in expanding and collapsing stages when @xmath158 they fall freely at the radii where the rotational perturbation vanishes . the choice of @xmath47 in ( [ 28 ] ) and ( [ 33 ] ) implies that the preferred observers fall freely at infinity in open cases , and at a certain @xmath83 in closed cases ( see discussion below eq.([39 ] ) ) . in the open universes the rotating axes were fixed uniquely by the condition @xmath50 . in this frame physically realizable preferred observers cover the whole outer universe and can be defined as _ globally non - rotating_. the frame dragging inside a void in the open universes can thus be regarded as _ absolute _ with respect to these observers . in a closed universe , on the other hand , the value of the rotational perturbation at the shell , @xmath22 , is a gauge dependent quantity . the function @xmath47 can be used to choose any arbitrary path @xmath159 as a locus of freely falling preferred observers , or to choose any value for @xmath22 ( cf . fig.[fig1 ] ) . in particular , by choosing @xmath160 @xmath161 , we obtain @xmath22 vanishing at all times . in this case the preferred observers fall freely at the shell and the spatial axes of the two frames @xmath3 and @xmath11 coincide . hence , the dragging of inertial frames inside a void in closed universes is _ relative _ , i.e. depending on the choice of gauge . now , since we have expressed all rotations in terms of mesurable quantities @xmath104 and @xmath142 ( their gauge invariance being evident since @xmath47 cancels out in both ( @xmath162 ) and ( @xmath163 ) we shall confront the observation of the angular velocity distribution of matter in the universe , made by inertial observers inside the void , directly with mach s ideas . by _ perfect dragging _ we shall mean that : an inertial observer inside the void is _ dragged perfectly by the shell _ if the angular velocity of the shell @xmath142 , measured in the frame @xmath3 , vanishes . @xmath49 _ the case of open universes _ in the case @xmath1 ( for @xmath35 we get similar results ) @xmath142 decreases as the radius of the shell @xmath76 or the expansion factor @xmath164 increase , and dies away as 1 ) @xmath165 for @xmath164 fixed ( fig.[fig4 ] ) or 2 ) @xmath166 for @xmath76 fixed . this behaviour is in agreement with klein s result [ 7 ] and can be interpreted from a machian point of view as follows : \1 ) when the radius of the shell is getting large , almost all matter of the universe can be regarded as redistributed on the shell . in this case , the angular velocity of the shell fully determines the angular velocity distribution of the matter in the universe and the frame dragging becomes perfect . this is in full conformity with mach s ideas since the universe does not rotate relative to local inertial frames inside the void . \2 ) in an expanding universe , the matter density as well as the rotational perturbation decrease as a function of time ; hence the influence of the matter on the inertial frames diminishes so that in this limiting case the dragging inside the void becomes perfect . @xmath49 _ the case of closed universes _ when the universe is closed , @xmath142 is positive for all times @xmath108 and for all comoving positions of the shell @xmath76 ; it never vanishes ( fig.[fig4 ] ) - there is no perfect dragging in closed universes . since @xmath142 does not depend on the choice of the function @xmath64 , this result holds for any initial angular momentum distribution of the cosmic dust outside the shell . thus , the influence of the shell on the inertial frames inside the void is always balanced by the cosmic dust moving to conserve the zero total angular momentum of the universe . in both cases , the fact that the whole matter in the universe ( the shell and the cosmic dust ) is not observed inside the void at any finite time to ( slowly ) rotate in a preferred direction is in full agreement with mach s observation . we thus conclude that inertial frames inside the void are just in this sense determined by the average motion of distant galaxies. t.d . would like to thank guillaume faye for his help with the numeric part of the paper . we thank joseph katz for his critical as well as friendly reading of the manuscript and dedicate him this work for his 70th birthday . j.b . and t.d . acknowledge the support from the grants j13/98:113200004 , gacr 202/99/0261 and gauk 114/2000 . j.b . also acknowledges the kind hospitality of the darc in meudon and the raymond and beverly sackler distinguished visiting fellowship at the institute of astronomy in cambridge . in the regions inside and outside the void , the independent tangent vectors at the shell are & e^ in_t=(1,a l_0,0,0),e^ out_t=(1,0,0,0 ) , + & e^ in , out_=(0,0,1,0),e^ in , out_=(0,0,0,1),where @xmath167 stands for @xmath168 . the normal vectors to the shell @xmath169 and @xmath170 , both pointing inside the shell , are determined by @xmath171 , and are normalized to unity : @xmath172 the extrinsic curvature is defined by @xmath173 where the indices @xmath30 stand for @xmath31 and are raised and lowered by the induced metric on the shell ( [ 9 ] ) . @xmath174 is the covariant derivative associated with the metric ( [ 2 ] ) or ( [ 5 ] ) . when the metric is given by ( [ 2 ] ) , the non - zero components of the extrinsic curvature of @xmath16 , calculated at first order in @xmath22 , are equal to k^in_tt&=&-l_0a , k^in_=l_0 a , k^in_=k^in_^2 , + k^in_t&=&-l_0 a_0 ^ 2 . similarly the non - zero components of the extrinsic curvature of @xmath16 , when computed with the metric ( [ 5 ] ) , are k^out_tt&=&0,k^out_=l_0 a , k^out_=k^out_^2 , + k^out_t&=&-l_0 a(12l_0r|_0+_0)^2 , where @xmath175 the integration of einstein s equations across the shell gives the stress - energy tensor @xmath176 of the shell as [ 9 ] -8gt_ab = k_ab-_abk^c_c,@xmath177 and @xmath178 are the coefficients of the induced metric ( [ 9 ] ) . einstein s equations relate the rotational perturbation @xmath19 and the dust angular velocity @xmath43 . they can be easily found in the the literature ( see e.g. [ 13 ] ) , recalling that @xmath19 is bardeen s vector metric perturbation in the poisson gauge . in the notations of [ 13 ] they read ad_(k_l)+2ad_(k|b_l)=0,d_id^i|b_k+k|b_k= 8ga^2(+p)(|b_k+|v_k ) , where @xmath179 stand for @xmath180 , @xmath181 is the covariant derivative associated with the spatial part of the metric ( [ 5 ] ) , @xmath182 , and @xmath183 . the first equation ( [ 23 ] ) implies 3ar+ar=0,the general solution of which is ( t , r)=-1a^3((r)+g(t ) ) . the second equation ( [ 23 ] ) translates as : ( 1-kr^2)^2r^2+(4 - 5kr^2)1rr= -16ga^2(+p)(-).this can be integrated and , using ( [ 20 ] ) and ( [ bb ] ) , written in terms of the constant of motion @xmath61 as = 6gr^4(j_+j(r ) ) , where @xmath44 is the angular momentum of the cosmic dust defined by ( [ 22 ] ) . further integration of the prevoius equation implies finally ( r)=-2gr^3(1 + 2kr^2)j_+6g_l_0^rj(r^)r^4dr^. in order to find @xmath184 , we need to compute the trajectories of two photons emitted by the star , and propagating first in the perturbed flrw universe outside the shell and then in the void inside the shell , and arriving at the centre of the void . consider thus a star comoving with the cosmic dust at some radius @xmath94 ( where @xmath79 is as usualy introduced by @xmath185 for @xmath96 , respectively ) , emitting photons radially inwards and for the sake of simplicity in the equatorial plane . the trajectory of a photon inside the shell , as described in the inertial frame @xmath3 , is given by the null geodesic and @xmath186 are constants and where @xmath187 stands for the photon @xmath188 or @xmath189 . these photons started from the shell at times @xmath190 defined by @xmath191 . recalling that the coordinate @xmath192 is related to the coordinate @xmath48 outside the void by @xmath155 and that the proper time in @xmath3 is related to the time in the outside coordinate grid by @xmath193 , introducing also the conformal time @xmath99 ( @xmath194 ) , we have @xmath93 to @xmath195 ( the interval , in the conformal time outside the shell , between the arrival times of the two photons on the shell ) and @xmath196 to @xmath197 ( the angular deviation of the two photons on the shell in the outside coordinate grid ) , and where the subscript 0 means that the quantities are evaluated at the time the photons reach the shell . outside the shell the ingoing photons propagate , at zeroth order , along the radial null geodesics : @xmath198 , @xmath199 , @xmath200 , where @xmath201 is an affine parameter and the bar denotes the zeroth order null geodesics in the flrw unperturbed universe . the non - zero components of the null wave vector are @xmath202 and @xmath203 . the coordinates @xmath204 and @xmath205 along the photons are thus given by -_0 = _ 0^p-,=/2 , @xmath206 is the conformal time at which the photon @xmath187 reaches the shell at @xmath76 , @xmath207 . the photons are emitted at @xmath208 and @xmath209 by the star placed at @xmath210 , so that we see from ( [ a11 ] ) that simply _ 0= _ * , which relates @xmath195 to @xmath211 . now at first order , the wave vector of the light propagating in the _ perturbed _ spacetime outside the shell is @xmath212 , @xmath213 solves the null geodesic equation up to first order in the perturbations : @xmath214 + 2r|k^r k^+1\{a a } ( |k^r)^2=0 , where @xmath19 is given by ( [ 32 ] ) for @xmath215 and by ( [ 37 ] ) for @xmath216 . equation ( [ a13 ] ) has a simple solution @xmath217 , where @xmath59 can now be expressed only as a function of conformal time since the trajectory of the photon is given at leading order by ( [ a11 ] ) : @xmath218 . hence , using @xmath219 , the @xmath220 coordinates of the trajectories of the photons can be expressed as -_0^p=__0^p^a(^)(^,_0+_0^p -^ ) d^ , where @xmath221 are their angles at which they reach the shell , @xmath222 . from ( [ a14 ] ) , taking into account that @xmath223 ( eq.([25 ] ) ) , we obtain _ * -_0=_0__0 ^ 1^ _ * d , which relates @xmath197 to @xmath211 and @xmath224 ( the angular deviation of the two photons at the time of emission in the outside coordinate grid ) - again the subscript * means that the quantities are evaluated at the time of emission . integrating by parts and changing the variable of integration @xmath99 to @xmath79 , ( [ a15 ] ) can be written as _ * -_0=_*([a]_*-[a]_0-__0^_*d(a^3)d1a^2d),where @xmath164 is now a function of @xmath225 , @xmath108 stands for @xmath226 ; the index 1 denoting the first ray is dropped out here and in the following . in order to relate @xmath224 to @xmath211 , we use the fact that the star is comoving with the cosmic dust so that _ * = [ a ] _ * _ * , where @xmath227 is the angular velocity of cosmic matter introduced in ( [ 21 ] ) and related to the rotational perturbation @xmath59 by ( [ 31 ] ) for @xmath228 and by ( [ 36 ] ) for @xmath216 .

we consider a spacetime consisting of an empty void separated from an almost friedmann - lematre - robertson - walker ( flrw ) dust universe by a spherically symmetric , slowly rotating shell which is comoving with the cosmic dust . we treat in a unified manner all types of the flrw universes . the metric is expressed in terms of a constant characterizing the angular momentum of the shell , and parametrized by the comoving radius of the shell . treating the rotation as a first order perturbation , we compute the dragging of inertial frames as well as the apparent motion of distant stars within the void . finally , we discuss , in terms of in principle measurable quantities , machian features of the model . 6.70 in

i introduction ii the spacetime metric iii angular velocity of the shell and apparent motion of distant stars iv dragging inside the void and machs principle acknowledgements extrinsic curvature of the shell the metric perturbation @xmath59 angular velocity of a distant star measured by inertial observers inside the void

arxiv

spin - exchange interactions are of fundamental importance for the magnetic properties of strongly correlated media @xcite . of particular interest are superexchange interactions that are induced by higher - order tunneling processes via virtually occupied intermediate states @xcite . in solid state systems such as cuo and mno , theses processes are of long - ranged nature , as they do not depend on a direct wave - function overlap of the electrons in contrast to direct exchange interactions . in the context of high - temperature superconductivity they are in the focus of current investigations @xcite . experiments with ultracold atoms in optical lattices present a promising tool for the study of fundamental properties of higher - order tunneling processes @xcite . here , typically second - order superexchange tunneling is considered . however , also models with third - order tunneling processes as the leading order are discussed in the literature , e.g. , strongly correlated fermions in frustrated lattices @xcite . recently , the direct time - resolved observation of superexchange processes has been demonstrated in experiments with ultracold atoms in deep double - well potentials @xcite . these experiments exploit the interaction blockade inhibiting first - order tunneling processes . however , for the interaction blockade the spin fluctuations arising from superexchange are diminished due to the small amplitude . there is great current interest in accomplishing long - ranged anti - ferromagnetic ordering with ultracold fermions in optical lattices . however , the small energy associated with superexchange can not compete with thermal fluctuations in present experiments with isotropic lattices @xcite . the realization of fermionic high - spin systems with ultracold atoms promises a deeper understanding of fundamental spin - spin interactions @xcite . these systems are of broad interest , since they offer the possibility to realize , e.g. , a multitude of new quantum phases @xcite , the simulation of su(n ) magnetism @xcite as well as spin changing collisions @xcite . recent experimental advances demonstrate long - lived coherent spin dynamics of fermionic @xmath1 atoms in optical lattices with tunable effective spin and reliable initial - state preparation @xcite . the interplay of spin - changing collisions and tunneling processes leads to new phenomena such as the instability of an initially band insulating state @xcite . this allows for the detailed study of many - body non - equilibrium dynamics in a highly controllable environment . here , we investigate the non - equilibrium superexchange dynamics of fermionic high - spin systems with spin - changing collisions in optical double - well and plaquette lattices . we find a regime dominated by superexchange with large amplitude and strong spin fluctuations in _ shallow _ potentials . usually , the observation of superexchange with ultracold quantum gases requires deep lattices with a large interaction gap @xmath2 that suppresses the first - order tunneling @xmath3 . in the latter regime , the superexchange has small amplitudes @xmath4 and is therefore extremely sensitive to thermal fluctuations . in a high - spin system , spin - changing collisions lift the pauli blocking of an initially prepared band - insulator state with @xmath5 . this causes the initial state to be unstable which in general allows strong particle - number fluctuations to arise @xcite . surprisingly , we find that the first - order tunneling is strongly suppressed in shallow few - well potentials . in this regime , an emerging tunneling- energy gap leads to an interaction blockade at large @xmath6 . we show how this phenomenon can be used to investigate superexchange processes with large amplitude corresponding to short time scales . this opens new possibilities for the dynamical study of superexchange in a non - equilibrium system , previously restricted to deep optical lattices with small superexchange amplitude @xmath4 . we present a comprehensive study of the exact time evolution of a band insulator in double - well potentials and four - well plaquettes . a possible experimental realization with @xmath1 and a measurement scheme for particle - number fluctuations is discussed . the required techniques , i.e. , the preparation of optical few - well potentials @xcite and the reliable initial state preparation @xcite have already been successfully demonstrated . system , a spin - changing collision transfers particles on a lattice site from @xmath7 to @xmath8 and vice versa . ( b ) a superexchange process exchanges two particles on neighboring lattice sites . here the orange @xmath9 particle on the left site is exchanged with the green @xmath10 particle on the right site . ( c ) a first - order tunneling processes inducing particle - number fluctuations . [ fig_schemes],scaledwidth=50.0% ] and the blue line the spin fluctuations @xmath11 averaged over a long period of time ( @xmath12 ) as a function of @xmath6 . the dashed lines shows the fluctuations @xmath13 of the respective ground - state of the system for comparison . the shading illustrates the crossover between the two regimes @xmath14 and @xmath15 . ( b ) the same data evaluated for the time evolution in a four - well plaquette . additional to the two limiting regimes , sharp resonances with high fluctuations occur for specific values of @xmath6 . [ fig_fluctuations],scaledwidth=90.0% ] in this paper we discuss the realization of a pure superexchange model with high - spin fermions . first , we introduce the corresponding superexchange as well as the full microscopic hamiltonian . in the subsequent sections , we demonstrate that the superexchange hamiltonian @xmath16 can be realized with particles of spin @xmath17 in an optical lattice in a surprisingly wide range of parameters , including shallow lattices with large amplitudes @xmath4 . here , @xmath18 ( @xmath19 ) creates ( annihilates ) a particle on lattice site @xmath20 in the spin - state @xmath21 and obeys the fermionic commutation relations . the first part of this hamiltonian describes spin - changing collisions with an amplitude @xmath22 as depicted in figure [ fig_schemes](a ) . two particles on a lattice site exchange their spins while preserving the total magnetization , i.e. , @xmath23 . the second part of the hamiltonian represents superexchange processes ( figure [ fig_schemes](b ) ) . these second - order tunneling processes exchange particles on neighboring lattices sites , which preserves the density distribution but can lead to strong spin - fluctuations already in a double - well system . the corresponding amplitude @xmath4 accounts for the increased interaction energy @xmath2 in the virtually occupied intermediate state . the interaction energy itself is omitted in as a constant offset , since neither spin - changing collisions nor the superexchange affects the particle - number distribution . the first - order tunneling ( figure [ fig_schemes](c ) ) is not present in the hamiltonian and any intersite dynamics is mediated by superexchange processes . a system of fermionic particles with spin @xmath17 in an optical lattice can be used to realize the hamiltonian . the full microscopic description is given by the hamiltonian @xmath24 the number operator @xmath25 counts particles on a lattice site @xmath20 . in contrast to the superexchange hamiltonian , the full model allows for first - order tunneling @xmath26 , e.g. , @xmath27 ( figure [ fig_schemes](c ) ) and a corresponding change in onsite - interaction energy @xmath2 . the spin - changing collisions represent a small perturbation of the standard hubbard hamiltonian @xmath28 . we assume an amplitude of @xmath29 , which is a realistic value for @xmath1 ( see section [ sec_spin - dependent ] ) . in this work , we investigate the dynamics of the band insulating ground state @xmath30 of the unperturbed hamiltonian @xmath31 . the robust experimental preparation of this two - component band insulator as well as the control over the spin - changing collisions @xmath32 via a magnetic field has been demonstrated in reference @xcite . in the following we discuss the dynamics of in a double - well and a four - well plaquette potential . both setups can be realized experimentally @xcite and can be solved numerically exact . we show that the time evolution is governed by the superexchange model even in the absence of a direct interaction blockade . this effect depends on the existence of the fluctuationless eigenstate of the unperturbed hamiltonian in shallow lattices . note that there is no such eigenstate in bosonic systems where the realization of a pure superexchange model is only possible in deep lattices . by superimposing two optical lattices of a short wavelength @xmath33 and a longer wavelength @xmath34 an array of double - well potentials can be realized @xcite . the preparation of the initial state can be achieved by loading atoms with @xmath5 into the optical lattice of the short wavelength before ramping up the second lattice to separate the individual double - wells . the time evolution can be determined by means of exact diagonalization of the microscopic hamiltonian . for a wide range of parameters @xmath6 we investigate the dynamics and evaluate the time - dependent particle - number fluctuations @xmath35 . in figure [ fig_fluctuations](a ) the time - averaged value of @xmath36 is shown as a function of @xmath6 ( solid black line ) and compared with the corresponding value of the ground state ( dashed line ) . for deep lattices , fluctuations are exponentially suppressed for decreasing @xmath6 due to an interaction blockade @xmath2 . surprisingly and in contrast to the ground state , we also find an exponential suppression of @xmath36 in shallow lattices for increasing @xmath6 . therefore , the time evolution of the band insulator is governed by the superexchange hamiltonian for almost all lattice depths . this phenomenon allows the simulation of the superexchange hamiltonian with ultracold atoms without an interaction blockade , i.e. , with large amplitudes @xmath4 . in the case of vanishing tunneling @xmath37 , i.e. , in a very deep lattice , both sites undergo in - phase rabi oscillations induced by the spin - changing collisions ( figure [ fig_double - well](a ) ) . the state remains fluctuationless due to a large interaction blockade @xmath2 . after an evolution time of @xmath38 , the state will eventually be the band - insulator @xmath39 . in both states , @xmath40 and @xmath41 , no tunneling is possible due to the pauli principle . for all other times , however , the state is in a superposition of @xmath42 and @xmath43 on both wells . for a finite amplitude @xmath44 first - order tunneling is in principle possible ( figure [ fig_schemes](c ) ) . surprisingly , the numerical results show very low occurrence of particle - number fluctuations @xmath36 even and especially for large values of @xmath6 . thus , the dynamics of is described by the superexchange operator . fig_double - well ] shows the time evolution for various parameters @xmath6 calculated with the full hamiltonian ( solid lines ) and with the superexchange hamiltonian ( dotted lines ) . only at values of @xmath45 , where @xmath36 has a maximum , finite deviations between the two models are present ( figure [ fig_double - well](b ) ) . in shallow lattices , periodic oscillations with full amplitude are reestablished ( figure [ fig_double - well](c ) ) . superexchange processes take place on a short time - scale , leading to the population of onsite states with finite magnetization @xmath46 such as @xmath47 ( black lines in figure [ fig_double - well](b , c ) ) . this directly corresponds to a non - zero value of spin fluctuations @xmath48 , whereas particle - number fluctuations @xmath36 are negligible . in contrast , in deep lattices spin fluctuations remain small . here , @xmath11 is on the order of @xmath49 and thus the black line in figure [ fig_double - well](a ) is not visible . the time - averaged value of the spin fluctuations in the evolution of is shown as a blue line in figure [ fig_fluctuations](a ) . we can identify two distinct regimes : one with almost no spin fluctuations @xmath50 for small @xmath6 and one with large spin fluctuations @xmath51 for large @xmath6 . the crossover between the two distinct regimes discussed above is not a unique feature of the double - well but occurs also in other few - well systems . in the following we identify the underlying mechanism of the fluctuationless time - evolution in shallow potentials for the particular case of a four - well plaquette . this mechanism is , however , general for all few - well systems that show the discussed behavior . a four - well plaquette is the natural extension of a double - well potential in two dimensions . it is equivalent to a one - dimensional four - well lattice with periodic boundary conditions and has been recently realized experimentally @xcite . despite the increased complexity compared with the double - well , the hamiltonian can still be diagonalized for a four - well lattice allowing to investigate the time - evolution of the initially prepared band - insulator exactly . the hamiltonian has a number of symmetries that can be exploited to reduce the numerical effort . these symmetries are the spatial translation , spin - flip , quadrupole and particle - hole symmetry . furthermore , the initial state also obeys several symmetries and we can restrict the basis to the corresponding subspace . we investigate a wide range of parameters and find the two regimes with @xmath52 , known from the double - well . the spin fluctuations @xmath11 that arise from the initial state and the ground state properties of the system ( figure [ fig_fluctuations](b ) ) are very similar to the double - well system . in contrast to the results of the double - well , however , strong fluctuations appear at certain values of @xmath6 . the eigenspectra allow for a qualitative understanding of the two limits @xmath37 and @xmath53 and also explain the emerging resonances with high fluctuations @xmath36 . in deep potentials @xmath37 , the onsite interaction @xmath2 is the dominating energy scale of the hamiltonian and the eigenstates separate in groups of fock states with integer @xmath2-excitations that are perturbed by @xmath26 and @xmath32 . the spectrum is shown in figure [ fig_plaquette](b ) , where each marker corresponds to an eigenstate @xmath54 with an energy @xmath55 and particle - number fluctuations @xmath36 . the latter directly correspond to the interaction energy @xmath56 @xmath57 red markers have finite overlap with the initial state , which is clearly confined to the group of eigenstates without fluctuations . here , the interaction blockade @xmath58 causes the dynamics to be determined by the superexchange hamiltonian , which is used in conventional realizations with ultracold atoms @xcite . the time - evolution of the initial state is shown in figure [ fig_plaquette](a ) , where the solid lines correspond to the full model and the dotted lines to the superexchange model . note that due to the very small amplitude of the superexchange @xmath4 spin fluctuations @xmath11 are suppressed and merely cause a slow dephasing of the onsite rabi oscillations . in very shallow potentials , the tunneling operator @xmath26 is the dominating energy scale of the hamiltonian . the spectrum for @xmath59 is shown in figure [ fig_plaquette](f ) . in a finite lattice , the spectrum is gaped due to the discrete energies of the bloch functions . in the case of four lattice sites and taking into account the conservation of momentum , the gap is @xmath60 . when this is large compared with @xmath2 and @xmath22 , the spectrum separates in groups with @xmath61 where @xmath62 is an integer number . the initial state obeys @xmath63 and @xmath64 and consequently lies in the group with @xmath65 . for @xmath66 , in this group the eigenenergies are given by the interaction energy @xmath67 , similar to the regime of the interaction blockade , as indicated by the dashed lines in the spectra . since the spin - changing collisions present only a small perturbation @xmath68 , the band isolator state is only perturbed by states with @xmath69 . for these states , the particle - number fluctuations are small and the band isolator state can only develop small particle - number fluctuations . interestingly , a subordinate energy gap @xmath70 arises between fluctuationless states with @xmath69 ( including @xmath40 ) and states with particle - number fluctuations and an energy @xmath71 . due to the tunneling - energy gap , the interaction blockade is reestablished in sufficiently shallow potentials , where now @xmath22 is the competing energy scale instead of @xmath3 . the gap @xmath58 amplifies the suppression of particle - number fluctuations for the initial state , but is not a necessary requirement for @xmath72 . the surprisingly low fluctuations in very shallow potentials thus arise from a tunneling - energy gap @xmath73 and a subordinate interaction blockade . in figure [ fig_plaquette](e ) the time evolution for @xmath59 is shown , where no deviations between the full model and the superexchange model can be seen . the time evolution of figure [ fig_plaquette](e ) in units of @xmath74 is identical for all lattice depths with @xmath75 and sufficiently small fluctuations @xmath36 , i.e. , away from resonances in figure [ fig_fluctuations](b ) . in the parameter regime @xmath76 , neither an interaction blockade @xmath58 nor a tunneling - energy blockade @xmath73 inhibits particle - number fluctuations ( see figure [ fig_plaquette](d ) ) . the eigenenergies are not determined by the interaction energy only and thus do not directly correspond to the particle - number fluctuations . however , few eigenstates have finite overlap with the initial state and thus , only at specific parameters @xmath6 fluctuations arise . at these resonances , the time evolution of the full model and the superexchange model differ significantly , as can be seen in figure [ fig_plaquette](c ) . the superexchange model ( dotted line ) predicts the same behavior as for very shallow potentials , whereas the correct calculations are strongly affected by the occurrence of triply occupied sites , which is plotted as a dashed blue line . , where the interaction - energy shifts have to be taken into account . ( b ) the bias can be tuned to fulfill the resonance condition for certain particle - number distributions @xmath77 , allowing for the measurement of triply occupied wells ( blue line ) . [ fig_measure_triplons],scaledwidth=65.0% ] an experimental proof of superexchange processes and the absence of first - order tunneling is the demonstration of strong spin fluctuations and vanishing particle - number fluctuations . the former can be accessed in an experiment with @xmath1 via the population of new spin components , as discussed in section [ sec_spin - dependent ] . the latter can be measured in an optical double - well setup with the technique illustrated in figure [ fig_measure_triplons](a ) . first , the lattice with the short wavelength @xmath33 is ramped up to freeze the populations . afterwards , the relative phase between the two lattice beams is tuned to introduce a bias @xmath78 between the wells . that way the third vibrational mode of the deeper well can be brought into resonance with the first one of the shallower well . the required bias @xmath79 between the left ( l ) and the right ( r ) well depends on the single - particle energies @xmath80 of the vibrational modes @xmath20 and the particle - number distribution @xmath77 that introduces interaction energy shifts . when a tunneling process occurs , the interaction energy on the left well is reduced by @xmath81 , while it is increased on the right well by @xmath82 , where @xmath83 is given by the wave - function overlap of the lowest mode with the third one . this technique has already been successfully demonstrated in reference @xcite . for every given particle - number distribution we schematically plot the resonance of a transfer to the third band as a function of the applied bias @xmath78 in figure [ fig_measure_triplons](b ) . the blue line corresponds to the case of three particles in the left and one in the right well . we assume a width of @xmath84 for the resonances , which is a pessimistic assumption compared with the measured width in the experiment of reference @xcite . figure [ fig_measure_triplons](b ) shows that it is clearly possible to tune the bias to a value such that particles will be transferred to the third band only if there are exactly three particles on the shallower site . the population of the third mode can be measured by means of the band - mapping technique afterwards . as a concrete example , we now consider @xmath1 prepared in the @xmath85 manifold as a possible species for an experimental realization of high - amplitude superexchange . as the initial state we suggest a mixture of @xmath86 and @xmath87 , which allows for rich spin - dynamics and is the most similar to a band insulator in a @xmath88 system . without tunneling , i.e. , in very deep lattices , rabi - oscillations between @xmath89 and @xmath90 occur in phase on every lattice site in analogy to the @xmath88 system . when tunneling is allowed , however , more complicated dynamics arise . consecutive to tunneling processes , new spin - components can be populated by other spin - changing collisions , e.g. @xmath91 . states with @xmath92 are only occupied by third - order processes and we neglect them in our simulations as their influence on the qualitative behavior is assumed to be small . this reduces the system effectively to 6 spin components . the spin fluctuations @xmath11 are no directly measurable quantity , however the population of the new spin components is proof for the occurrence of tunneling or superexchange processes . another difference compared with the spin-3/2 system is that the interaction energies @xmath2 and @xmath22 are no longer spin - independent but slightly differ from each other @xcite . the generalization of the microscopic hamiltonian to 6 spin components is straight forward , when the spin - dependent interaction energies are known . for our simulations we use parameters from @xcite . as a characteristic quantity for the potential depth we use the ratio @xmath93 , where the interaction energy @xmath94 corresponds to a spin - preserving collision in the initial state . the energy @xmath95 of the spin - changing collision @xmath96 is significantly smaller with @xmath97 . the quadratic zeeman - energy must be used to tune the spin - changing collisions into resonance , because the initial and final state of a spin - changing collision have different interaction energies . of course , this can not be done for all spin states at the same time and the exact time - behavior depends on the applied magnetic field . our simulations are performed with @xmath98 , but any other value near the resonance is suitable . in figure [ fig_6_components](a , b ) the time - behavior of the @xmath99-components is plotted for the double - well and the plaquette . they differ from the behavior of a @xmath88 system with spin - independent interaction energies in several ways . in the case of spin - dependent interactions the superexchange operator @xmath100 can be approximated by the matrix elements @xmath101 between an initial state @xmath102 and a final state @xmath103 with exactly two particles per lattice site , respectively . the sum is over all possible intermediate states @xmath104 and @xmath105 . for flat lattices , the time - evolution can be described by this operator as can be seen by comparing the dotted lines with the solid lines in figure [ fig_6_components](a , b ) . the new spin - components @xmath106 and @xmath107 occur after a few milliseconds , which corresponds to an increase in spin fluctuations @xmath11 . in the plaquette the coherent rabi - oscillations are damped out rapidly in strong contrast to the spin-3/2 system , where they last for arbitrarily long times . the reason for this is the larger number of different frequencies due to the spin - dependency of the interaction strengths . concerning the appearance of particle - number fluctuations @xmath36 , however , the qualitative behavior is identical to the spin-3/2 system , as shown in figure [ fig_6_components](c ) . the population of the components @xmath10 and @xmath108 ( dashed lines ) is proportional to the average spin fluctuations @xmath11 ( dotted lines ) . this is an intriguing advantage of the high spin @xmath109 as opposed to a pure spin-3/2 system . only for certain values of @xmath6 resonances with triply occupied sites occur , while for a wide regime of parameters , the state remains fluctuationless . due to the additional components , spin fluctuations increase to @xmath51 even in deep lattices for very long evolution times . the particle - number fluctuations , as discussed in section [ sec_measuren ] , as well as the population of the new spin - components are quantities that are accessible experimentally . in combination they are direct evidence for superexchange processes in the absence of direct tunneling . in conclusion , @xmath1 is a suitable atomic species for the proposed experiments to simulate superexchange hamiltonians in shallow lattices . we have found a mechanism for fermionic high - spin systems that allows for the investigation of superexchange processes in shallow lattices , i.e. , at large amplitudes @xmath4 . this regime corresponds to much faster time scales than in deep lattices and was not accessible yet . the non - equilibrium dynamics of an initially band - insulating state with two components has been calculated , and we found the intersite dynamics to be fully dominated by the superexchange . although , in general , spin - changing collisions circumvent pauli blocking and therefore allow for strong particle - number fluctuations in shallow lattices , we have shown that the first - order tunneling is strongly suppressed in few - well potentials while strong spin fluctuations arise via superexchange processes . the underlying mechanism is driven by an emerging tunneling - energy gap that reestablishes an interaction gap , which is usually only expected in deep lattices . furthermore , we have investigated the crossover of superexchange regimes with weak and strong spin fluctuations in deep and shallow lattices , respectively . we have discussed the experimentally relevant case of optical double - well and plaquette potentials . we have simulated the time evolution for a pure spin-3/2 system as well as for realistic parameters of @xmath1 and have shown that in both cases it is governed by the superexchange hamiltonian . the latter allows for a measurement of spin fluctuations via the population of certain spin components . the double - well potential and the four - well plaquette are both suited for an experimental realization . even larger lattices of up to six sites in one spatial direction show the existence of a sufficiently large tunneling - energy gap @xmath110 and a subordinate interaction - energy blockade . the necessary experimental tools , i.e. , the optical potentials for a double - well in one and two dimensions @xcite as well as the preparation of the initial band insulator with @xmath1 atoms have both been demonstrated successfully @xcite . we would like to thank k. sengstock , l. mathey and j.s . krauser for very fruitful discussions . we thank the deutsche forschungsgesellschaft ( dpg ) for financial support within grant for 801 and grant grk 1355 .

we show that fermionic high - spin systems with spin - changing collisions allow to monitor superexchange processes in optical superlattices with large amplitudes and strong spin fluctuations . by investigating the non - equilibrium dynamics , we find a superexchange dominated regime at weak interactions . the underlying mechanism is driven by an emerging tunneling - energy gap in shallow few - well potentials . as a consequence , the interaction - energy gap that is expected to occur only for strong interactions in deep lattices is reestablished . by tuning the optical lattice depth , a crossover between two regimes with negligible particle number fluctuations is found : first , the common regime with vanishing spin - fluctuations in deep lattices and , second , a novel regime with strong spin fluctuations in shallow lattices . we discuss the possible experimental realization with ultracold @xmath0k atoms and observable quantities in double wells and two - dimensional plaquettes .

introduction superexchange in spin-3/2 systems double-well plaquette measuring particle-number fluctuations spin-dependent interaction energies conclusions acknowledgments

arxiv

galaxy redshifts are important for a large number of studies related with the extragalactic universe , due to their direct correlation with the distance of the sources . photometric galaxy redshifts ( hereafter photo - z s ) are crucial in the current era of large surveys , based on massive datasets . they are used in a wide plethora of tasks , such as , for example , to constrain the dark matter and dark energy contents of the universe through weak gravitational lensing , to understand the cosmic large scale structure , by identifying galaxies clusters and groups , to map the galaxy color - redshift relationships , as well as to classify astronomical sources . more recently , the attention in this field has been focused on the techniques able to compute a probability density function ( pdf ) of the photo - z s for each individual astronomical source , with the goal to improve the knowledge about statistical reliability of photo - z estimations . in the machine learning context several methods have been proposed to approach this task , see for instance : ( ( * ? ? ? * bonnet 2013 ) , ( * ? ? ? * rau et al . 2015 ) , ( * ? ? ? * sadeh et al . 2015 ) , ( * ? ? ? * carrasco & brunner 2014 ) ) . here we present a new method , named metaphor ( machine - learning estimation tool for accurate photometric redshifts ) , a modular workflow including a machine learning engine to derive photo - z s and a method to produce their pdf s , based on the evaluation of photometric data uncertainties to derive a perturbation law of the photometry . with this law we perform the perturbation of the features , in a controlled , not biased by systematics , way . a proper error fitting , accounting for the attribute errors , allows to constrain the perturbation of photometry on the biases of the measurements . the conceptual flow of the metaphor pipeline is based on the following sequence of tasks : given a multi - band data sample containing the spectroscopic galaxy redshifts , _ ( i ) _ for each band involved , a photometry perturbation function is derived ; _ ( ii ) _ the data sample is randomly shuffled and split into a training and a test set ; _ ( iii ) _ the photometry of the test set is perturbed , thus obtaining an arbitrary number @xmath0 of test set replica ; _ ( iv ) _ finally , the machine learning engine is trained and the @xmath1 test sets ( @xmath0 perturbed plus the unperturbed one ) are submitted to the training model to derive the pdf of photo - z estimations . in the last step , the @xmath1 values , output of the trained network , are used to calculate , for each bin of redshift , the probability that a given photo - z value belongs to each bin . the binning step @xmath2 , as well as the number @xmath0 of perturbations , are user defined parameters , to be chosen accordingly to the specific requirements of the experiment . for a given photo - z binning step @xmath2 , we calculate the number of photo - z s for each bin ( @xmath3 ) and the probability that the redshift belongs to the bin is @xmath4 . the resulting pdf is thus formed by all these probabilities . at the end of the procedure , a post - processing module calculates the final photo - z estimation and pdf statistics . for instance , we evaluate the photo - z s in terms of a standard set of statistical estimators for the quantity @xmath5 on the objects in the blind test set : _ ( a ) _ bias : defined as the mean value of the residuals @xmath6 ; _ ( b ) _ @xmath7 : the standard deviation of the residuals ; _ ( c ) _ @xmath8 : the radius of the region that includes @xmath9 of the residuals close to 0 ; _ ( d ) _ @xmath10 : the normalized median absolute deviation of the residuals , defined as @xmath11 ; _ ( e ) _ fraction of outliers with @xmath12 ; _ ( f ) _ skewness : asymmetry of the probability distribution of a real - valued random variable around the mean . furthermore , in order to evaluate the cumulative performance of the pdf we compute the following three estimators on the _ stacked _ residuals of the pdf s : _ ( 1 ) _ @xmath13 : the percentage of residuals within @xmath14 ; _ ( 2 ) _ @xmath15 : the percentage of residuals within @xmath16 ; _ ( 3 ) _ @xmath17 : the weighted average of all the residuals of the _ stacked _ pdf s . the photometry perturbation is based on the following expression , applied on the given _ j _ magnitudes of each band _ i _ as many times as the number of perturbations of the test set : @xmath18 the term @xmath19 is a multiplicative constant , used to customize the photometric error trend on the base of the specific band photometric quality . this could result particularly useful in case of photometry obtained by merging different surveys ; the quantity @xmath20 is the weighting coefficient associated to each specific band used to weight the gaussian noise contribution to magnitude values ; finally , the term @xmath21 is a random value extracted from a normal distribution . we investigated four different types of the weighting coefficient @xmath20 . first one is a heuristically chosen real number between @xmath22 and @xmath23 , implying a same width of the gaussian noise for each point . the second choice is based on weighting the gaussian noise contribution using the individual magnitude error provided for each source . the third one is a polynomial fitting : a binning of photometric bands is performed , in which a polynomial fitting of the mean magnitude errors is used to reproduce the intrinsic trend of the distribution . the last option is a slightly more sophisticated version of the polynomial fitting , coupled with a minimum value chosen heuristically , thus resulting in a bi - modal perturbation function . as introduced , one of the most suitable features of metaphor is the invariance to the specific empirical model used as engine to estimate photo - z s . in order to demonstrate this capability , we tested the metaphor workflow using three different machine learning methods : mlpqna neural network ( * ? ? ? * ( byrd et al . 1994 ) ) , already successfully used in several astrophysical contexts ( ( * ? ? ? * brescia et al . 2013 ) , ( * ? ? ? * brescia et al . 2014a ) , ( * ? ? * cavuoti et al . 2012 ) , ( * ? ? ? * cavuoti et al . 2014a ) , ( * ? ? * cavuoti et al . 2014b ) , ( * ? ? ? * cavuoti et al . 2015b ) ) , the standard knn ( * ? ? ? * ( cover & hart 1967 ) ) , and random forest ( * ? ? ? * ( breiman 2001 ) ) . in particular , the experiment with a very basic machine learning model like knn method , would demonstrate the most general applicability of any empirical model engine within metaphor . furthermore , by considering that the methods mostly based on sed template fitting intrinsically provide the pdf of the estimated photo - z s , we compared metaphor with the _ le phare _ model ( * ? ? ? * ( ilbert et al . 2016 ) ) . the real data used for the tests were a galaxy spectroscopic catalogue sample extracted from the data release 9 ( dr9 ) of the sloan digital sky survey , sdss , ( * ? ? ? * ( york et al 2000 ) ) . by using mlpqna as internal engine for photo - z estimation , we reached values of @xmath24 , @xmath25 and @xmath26 of outliers . these statistical results are slightly worse than what we showed in a previous article ( * ? ? ? * ( brescia et al . 2014c ) ) , where we already used the mlpqna method to derive photo - z s for the galaxies of sdss - dr9 . however , this discrepancy is only apparent , by considering that the spectroscopic kb used in the cited work was much larger than the one used here ( @xmath27 training objects against only @xmath28 objects used for the training in this case ) . the decision of such limited sample used in the present experiment was induced by the different goal of the experiment . we performed a large number of experiments with mlpqna using @xmath29 photometric perturbations in order to find the best perturbation law . the most performing experiment turns out to be the one based on a bi - modal perturbation law with threshold @xmath30 and a multiplicative constant @xmath31 . this experiment leads to a stacked pdf with @xmath32 within @xmath33 $ ] , @xmath34 , @xmath35 of the objects falling within the peak of the pdf , @xmath36 falling within @xmath23 bin from the peak and @xmath37 falling within the pdf . after having found the best perturbation law , we executed @xmath38 perturbations of the test set . this experiment led to an increase in the statistical performances , obtaining @xmath39 and @xmath40 within the peak of the pdf , @xmath41 within @xmath23 bin from the peak and @xmath42 inside the pdf . the same configuration and perturbed data have been used to estimate photo - z s by replacing mlpqna with , respectively , the knn and random forest models within the metaphor workflow . in parallel , we derived also the photo - z pdf s with the _ le phare _ method . the statistical results for all these methods are summarized in table [ tab : stackedstat ] . .statistical results of photo - z s and related pdf estimation on the blind test set extracted from sdss - dr9 , obtained by three machine learning models ( mlpqna , knn and random forest ) , alternately used as internal engine of metaphor and by the sed template fitting method _ le phare_. the last three estimators are related to the cumulative pdf of the estimated photo - z s . see text for the explanation of the statistical estimators . [ cols="^,^,^,^,^",options="header " , ] although there is a great difference in terms of photo - z estimation statistics between _ le phare _ and mlpqna ( see table [ tab : stackedstat ] ) , the results of the pdf in terms of @xmath15 are comparable . but the greater efficiency of mlpqna induces an improvement in the range within @xmath13 , where we find @xmath43 of the objects against the @xmath44 for _ le phare_. both individual and _ stacked _ pdf s are more symmetric in the case of empirical methods . this is particularly evident by observing the skewness ( see table [ tab : stackedstat ] ) , which is @xmath45 times greater in the case of _ le phare_. the presented photo - z estimation results and the statistical performance of the cumulative pdf s , achieved by mlpqna , rf and knn through the proposed workflow , demonstrate the validity and reliability of the metaphor strategy , despite its simplicity , as well as its general applicability to any other empirical method . mb and sc acknowledge financial contribution from the agreement asi / inaf i/023/12/1 . mb acknowledges the prin - inaf 2014 _ glittering kaleidoscopes in the sky : the multifaceted nature and role of galaxy clusters_. ct is supported through an nwo - vici grant ( project number @xmath46 ) . bonnet , c. , 2013 , mnras , 449 , 1 , 1043 - 1056 breiman , l. , 2001 , machine learning , springer eds . , 45 , 1 , 25 - 32 brescia m. , cavuoti s. , longo g. , et al . , 2014a , pasp , 126 , 942 , 783 - 797 brescia , m. , cavuoti , s. , longo , g. , de stefano , v. , 2014c , vizier on - line data catalog : j / a+a/568/a126 brescia m. , cavuoti s. , dabrusco r. , mercurio a. , longo g. , 2013 , apj , 772 , 140 byrd , r.h , nocedal , j. , and schnabel , r.b . , mathematical programming , 63 , 129 ( 1994 ) carrasco , k. , brunner , r. j. , 2014 , mnras , 442 , 4 , 3380 - 3399 cavuoti , s. , brescia , m. , longo , g. , mercurio , a. , 2012 , a&a , 546 , 13 cavuoti s. ; brescia m. ; dabrusco r. ; longo g. & paolillo m. , 2014a , mnras 437 , 968 cavuoti , s. , brescia , m. , longo , g. , 2014b , proceedings of the iau symposium , vol . 306 , cambridge university press cavuoti , s. , brescia , m. , de stefano , v. , longo , g. , 2015b , experimental astronomy , springer , vol . 39 , issue 1 , 45 - 71 cover , t. m. , hart , p. e. , 1967 , ieee transactions on information theory 13 ( 1 ) ilbert , o. , arnouts , s. , mccracken , h. j. , et al . , 2006 , a&a , 457 , 841 rau , m. m. , seitz , s. , brimioulle , f. , et al . , 2015 , mnras , 452 , 4 , 3710 - 3725 sadeh , i. , abdalla , f. b. , lahav , o. , 2015 , eprint arxiv:1507.00490 york , d. g. , adelman , j. , anderson , j. e. , et al . , 2000 , aj , 120 , 1579

we present metaphor ( machine - learning estimation tool for accurate photometric redshifts ) , a method able to provide a reliable pdf for photometric galaxy redshifts estimated through empirical techniques . metaphor is a modular workflow , mainly based on the mlpqna neural network as internal engine to derive photometric galaxy redshifts , but giving the possibility to easily replace mlpqna with any other method to predict photo - z s and their pdf . we present here the results about a validation test of the workflow on the galaxies from sdss - dr9 , showing also the universality of the method by replacing mlpqna with knn and random forest models . the validation test include also a comparison with the pdf s derived from a traditional sed template fitting method ( le phare ) .

introduction the metaphor workflow the experiments acknowledgments

arxiv

the class of decision problems that can be solved in nondeterministic polynomial time , known as the np class , is central to the theory of computational complexity . informally , a decision problem is in np if its solution can be checked to be correct in polynomial time . obtaining that solution , however , is altogether a different matter : for some problems it can be done in polynomial time as well , while for others it is as yet unknown whether a polynomial - time procedure exists . in order to characterize such seemingly harder problems , it has proven useful to look at the problems that are `` complete '' for np ( the so - called np - complete problems ) , that is , problems in np for which the discovery of a polynomial - time procedure to find a solution would immediately warrant the existence of such a procedure for all the other problems in np as well @xcite . the np - complete problems are thus the hardest problems in np , and are in a sense essentially equivalent to one another in terms of how hard they are to be solved @xcite . but it has been known already for many years , both from practical experience ( e.g. , @xcite ) and from looking at the minutiae of the structure of np @xcite , that some np - complete problems are in fact harder than others , and that a given np - complete problem may have instances that are significantly harder than other instances of the same problem . following some initial results of about one decade ago @xcite , it is now known that , for np - complete problems like satisfiability and its derivations @xcite and graph coloring @xcite , sharp phase transitions with respect to some order parameter exist and are frequently correlated with the hardness of finding a solution . we are concerned in this paper with combinatorial optimization problems . hard optimization problems have been characterized in much the same way as their decision - form counterparts . by a minor technicality , however , they are best termed np - hard problems ( as opposed to np - complete problems ) to indicate only that they are at least as hard as the problems in np ( but not necessarily one of them ) @xcite . invariably , an optimization problem whose decision - form variant is np - complete , is np - hard . unlike decision problems , optimization problems are only now beginning to be looked at in order to explain the relative hardness of their instances , but we already have some empirical evidence of the presence of similar phase transitions @xcite . in the case of graph coloring , for example , the optimization problem asks for the graph s chromatic number the least number of colors needed to assign one color to each node without ever assigning the same color to neighbors in the graph ( i.e. , nodes that are connected by an edge ) . what is known for this problem comes from considering a sequence of graphs of increasing density ( number of edges per node ) and what happens along this sequence at the points in which the chromatic number increases . it has been discovered that finding the chromatic number just before these points is distinctly harder than just after them . also , sharp peaks in the size of the so - called backbone of each graph in the sequence are detected just before those points as well , thus indicating a strong positive correlation between problem hardness and backbone size . a graph s backbone in this case is the set of node pairs that are assigned the same color by every coloring that employs a number of colors given by the graph s chromatic number . so a graph with a large backbone presents many opportunities for an algorithm that seeks the optimum to waste time trying to assign two different colors to a node pair that belongs to the backbone . coloring a graph optimally is one of the problems that we treat in this paper . while the backbone size relates clearly , in an intuitive way , to why larger backbones tend to imply harder instances of the problem , we feel that it lends little combinatorial insight into the hardness of those instances , specifically into how the structure of the graph affects the hardness of coloring its nodes optimally . our contribution in the context of this problem , presented in section [ coloring ] , is to demonstrate that other indicators exist that relate just as clearly to the appearance of phase transitions related to the hardness of graph coloring while at the same time carrying what we think is better combinatorial intuition . the indicators we use come from considering a graph s chromatic polynomial and its set of acyclic orientations . the latter brings us to the second problem of interest in this paper , treated in section [ mis ] , which is the problem of finding an independent set of maximum cardinality in a graph . an independent set is a set of nodes that includes no neighbors . the cardinality of a maximum independent set of a graph is the graph s independence number . finding this number is also an np - hard problem , one that shares with optimal graph coloring a clean combinatorial interpretation in terms of the graph s acyclic orientations . this problem does not appear to have already been examined for phase transitions related to the hardness of its instances . throughout the paper , we use @xmath0 ( or simply @xmath1 , if @xmath2 and @xmath3 can be inferred from the context ) to denote an undirected graph with @xmath2 nodes and @xmath3 edges . all our empirical results are based on fixed sequences of graphs , each graph with @xmath2 nodes but increasingly more edges ( up to @xmath4 ) . for @xmath5 , the @xmath6th graph in this sequence , denoted by @xmath7 or simply by @xmath8 , has @xmath2 nodes , @xmath6 edges , and is generated according to the random - graph model that samples uniformly from the set of all graphs having the same number of nodes and edges @xcite ( equivalently , for @xmath9 , @xmath10 may be regarded as being obtained from @xmath8 by the random addition of a new edge ) . using one single sequence as the basis of each experiment precludes the smoothing effect of taking averages over larger ensembles , known to mask the appearance of very sharp phase transitions for problems like graph coloring @xcite . we are always careful , however , to make sure that the observed phenomena are also present , qualitatively , in several other sequences . for small values of @xmath6 , a graph @xmath8 in the sequence @xmath11 is likely to have isolated nodes ( nodes without neighbors ) . such nodes do not affect the graph s chromatic number , and affect its independence number only trivially ( every isolated node is a member of all maximum independent sets ) . so the sequence that is actually used in all our experiments is the sequence @xmath12 , where , for @xmath5 , @xmath13 is obtained by stripping @xmath8 of its isolated nodes . before proceeding , we pause momentarily to consider this issue of isolated nodes more carefully . let @xmath14 denote the number of isolated nodes of @xmath8 . we have @xmath15 , while for @xmath9 it is easy to see that @xmath16 where @xmath17 , for @xmath18 , is the probability that the addition of the @xmath19st edge to @xmath8 incorporates @xmath20 new nodes into @xmath13 . these three probabilities can be assessed easily as fractions of @xmath21 and lead to a simplification of ( [ diffeq ] ) as @xmath22 which is clearly solved by @xmath23 furthermore , for @xmath24 we obtain @xmath25 so @xmath14 obviously decreases rapidly with @xmath6 . we now turn to our two main sections . at the end , concluding remarks are given in section [ concl ] . let @xmath26 denote the chromatic number of @xmath1 . we show in figure [ trick ] two plots for @xmath27 , one indicating the time needed to find @xmath28 by a public - domain code @xcite that is based on the heuristic of @xcite , the other indicating the evolution of @xmath28 as @xmath6 is increased . we only show data for a certain range of @xmath6 values , since coloring instances outside this interval tend to be relatively trivial , thus requiring little time for solution . as expected , the chromatic number increases steadily as the graph gets denser ( acquires more edges ) , and does so increasingly rapidly . also , it is often the case that the time needed to find the chromatic number goes down significantly immediately after each increase in the chromatic number . in this section , we develop new combinatorial arguments that show that such sudden transitions are indeed to be expected . [ trick ] our study of the hardness of finding @xmath26 starts with an investigation of how abundant optimal colorings of @xmath1 are , that is , colorings that require exactly @xmath26 colors . in order to carry out this investigation , we resort to the chromatic polynomial of @xmath1 , denoted by @xmath29 , which gives the number of distinct ways in which @xmath1 can be colored by at most @xmath30 colors . this polynomial has several interesting properties . for example , it is a degree-@xmath2 polynomial in @xmath31 , the coefficient of @xmath32 is @xmath33 , and the coefficient of @xmath34 is @xmath35 @xcite . also , by definition @xmath26 is the least value of @xmath31 for which @xmath29 is positive , giving the number of distinct ways in which @xmath1 can be colored optimally . so finding @xmath29 is expected to be no easier than finding @xmath26 , although the following simple method can be used for relatively small graphs . let @xmath36 be any edge of @xmath1 , denote by @xmath37 the graph obtained from @xmath1 by removing @xmath36 , and by @xmath38 the graph obtained from @xmath1 by contracting the end nodes of @xmath36 into one single node . we see that @xmath1 can be colored in as many distinct ways as @xmath37 can , except for those colorings of @xmath37 that assign to the end nodes of @xmath36 the same color . but these are precisely the colorings of @xmath38 , so we get @xmath39 clearly , ( [ recursion1 ] ) defines a simple recursion for calculating @xmath29 that stops either at graphs that only contain isolated nodes or at graphs that are completely connected . if @xmath20 is the number of nodes in either case , then the former graphs admit @xmath40 distinct colorings and the latter @xmath41 distinct colorings , so the polynomial can be calculated easily at the bases of the recursion and upward from them . in figure [ poly10 ] , three plots are given for @xmath42 : one depicts the continual increase of the chromatic number as the number of edges is increased , while another shows the number of distinct optimal colorings for each graph @xmath13 , that is , @xmath43 ( the third plot is discussed shortly in what follows ) . for each graph , the chromatic polynomial has been computed using public - domain code @xcite based on the recursion of ( [ recursion1 ] ) . this computation quickly exhausts processing and memory resources as the numbers of nodes and edges increase thence the reason why we present data for @xmath42 only . so we must always bear in mind that our ability to draw general conclusions may be impaired . [ poly10 ] as the plots of figure [ poly10 ] indicate , the number of distinct optimal colorings increases dramatically at each step in the chromatic number , and after that decreases more or less steadily until immediately before the next step . in absolute terms , then , at each increase in the chromatic number optimal colorings become strictly more abundant . if the same could be shown to hold in relative terms as well ( i.e. , if the fraction of optimal colorings relative to some larger universe could be shown to undergo the same transition ) , then we might be able to continue and investigate in more detail how this relates to the time it takes to color the graphs optimally . however , we find that it is as yet unclear how to characterize such a larger universe . even so , it is worth examining the matter further , because at least in part the sudden jumps in the number of optimal colorings are really the product of well understood combinatorial growth at work . for example , let us examine what happens when the chromatic number increases from @xmath20 to @xmath44 along the sequence of graphs . suppose nodes @xmath45 and @xmath46 are the nodes that , when connected to each other by an edge , cause the graph s chromatic number to increase . clearly , each optimal coloring before the increase ( i.e. , with @xmath20 colors ) yields at least @xmath44 distinct optimal colorings after the increase ( i.e. , with @xmath44 colors ) : viewing colors as positive integers , first choose one of @xmath45 or @xmath46 and assign to it color @xmath44 , while the remaining nodes all retain their previous colors ; then assign color @xmath44 to all nodes that had color @xmath20 and this color to the one of @xmath45 or @xmath46 that was selected previously ; then proceed likewise until this same node has been assigned all colors ( @xmath44 down through @xmath33 ) . while this may all seem like obvious combinatorial growth at the points where the chromatic number changes , we remark that such an effect may also be present in the behavior of the backbones commonly used to characterize the phase transitions of graph coloring . in the setting that we just examined , nodes @xmath45 and @xmath46 clearly constitute one of the node pairs of the backbone or else the addition of an edge between them would not cause the chromatic number to increase , because there would be at least one optimal coloring with @xmath20 colors that would assign different colors to them . so let us consider the probability that randomly chosen nodes @xmath45 and @xmath46 constitute a backbone pair . we do so by first conceding , just for the sake of the argument , that nodes are uniformly distributed among the colors in all optimal colorings . in this case , the probability that @xmath45 and @xmath46 have the same color in all optimal colorings ( that is , that @xmath45 and @xmath46 form a backbone pair ) when the chromatic number is @xmath20 is @xmath47 , where @xmath48 is the number of optimal colorings with @xmath20 colors . but @xmath48 increases to at least @xmath49 when the chromatic number increases to @xmath44 , so the probability we just computed gets divided by at least @xmath50 . we then see that the sudden increases observed in the number of optimal colorings are probably also inherently related to what happens to backbones at the same points in the sequence of graphs . this provides a new perspective on the collapse of backbones at the coloring transitions while at the same time providing a better understanding of why it happens . another well understood combinatorial - growth effect is that , when the chromatic number is @xmath20 , every optimal coloring is essentially equivalent to @xmath51 others , each corresponding to a permutation of the colors among the nodes . this brings us to the third plot of figure [ poly10 ] , where the number of optimal colorings is shown normalized by the factorial of the current chromatic number . evidently , all sudden jumps are still there , but they now possess a stronger significance , because only one optimal coloring is counted out of all colorings that are equivalent to one another by straightforward permutation of colors ( that is , without implying a different partition of the node set ) . but let us return to the chromatic polynomial of @xmath1 . the usefulness of this polynomial goes beyond the counting of distinct colorings , as it provides the first link to yet another characterization of the hardness of optimal graph coloring , now based on the acyclic orientations of @xmath1 . an orientation of @xmath1 is an assignment of directions to the edges of @xmath1 ; it is acyclic if no directed cycles are formed , that is , if it is impossible to reach the same node twice by following edges according to their directions exclusively . if we let the set of the acyclic orientations of @xmath1 be denoted by @xmath52 and consider the number of acyclic orientations of @xmath1 in terms of what happens to the graphs @xmath37 and @xmath38 introduced earlier , then we have the following . every acyclic orientation of @xmath37 yields either one or two acyclic orientations for @xmath1 when @xmath36 is assigned a direction . the former case happens when one direction assignment for @xmath36 forms a directed cycle in @xmath1 but not the other , the latter when both assignments preserve acyclicity in @xmath1 ( it can not happen that both assignments form directed cycles because the orientation of @xmath37 is acyclic ) . similarly , every acyclic orientation of @xmath38 is necessarily one of those acyclic orientations of @xmath37 from which two orientations of @xmath1 are obtained . thus , @xmath53 for any edge @xmath36 of @xmath1 . the recursion in ( [ recursion2 ] ) is strikingly similar to the one in ( [ recursion1 ] ) , and this has been shown to give rise to the remarkable identity @xmath54 that is , the number of acyclic orientations of @xmath1 can be obtained from applying the chromatic polynomial of @xmath1 to the negative unit @xcite . obtaining ( [ nacyclic ] ) and some of its refinements @xcite from the relation between ( [ recursion1 ] ) and ( [ recursion2 ] ) comes as a consequence of the so - called theory of p - partitions and its order polynomials @xcite . of interest to us is that such polynomials quantify the following very useful relationship between the acyclic orientations of @xmath1 and its colorings . suppose that @xmath1 can be colored by @xmath20 colors . still viewing colors as positive integers , suppose also that we assign to the edges of @xmath1 an orientation that makes every edge point from the node with the higher color to the one with the lower . this orientation is clearly acyclic and induces no directed path in @xmath1 containing more than @xmath20 nodes . conversely , suppose we start from an acyclic orientation of @xmath1 . if @xmath20 is the number of nodes on the longest directed path in @xmath1 according to this orientation , then @xmath1 can be colored by at most @xmath20 colors , as follows . we assign color @xmath33 to the sinks ( nodes whose adjacent edges are all oriented inward ) , then color @xmath55 to the sinks that would be formed if the original sinks were to be removed from @xmath1 , then the lowest available color to the set of sinks that would appear next , and so on . note , in this process , that starting with a coloring yields a unique acyclic orientation . the converse is not necessarily true , however : starting with an acyclic orientation may yield more than one coloring of the nodes of @xmath1 . consider , for example , the acyclic orientation shown in figure [ 1orient2colorings ] and the two corresponding colorings shown in parentheses next to the nodes . what the process indicates , however , is that it is possible to seek optimal colorings for @xmath1 by looking for acyclic orientations of @xmath1 that are shortest in terms of how many nodes there are in a longest directed path . letting @xmath56 be the set of such orientations , what we have seen is that @xmath57 so we may have a tighter characterization of how abundant optimal colorings are by looking at shortest acyclic orientations instead of the chromatic polynomial applied to @xmath26 . [ 1orient2colorings ] the formal relationship between @xmath26 and the acyclic orientations of @xmath1 can be stated as follows @xcite , and strengthens important earlier results @xcite . for @xmath58 , let @xmath59 be the set of all directed paths in @xmath1 according to @xmath60 . for @xmath61 , let @xmath62 indicate the number of nodes in @xmath63 . then @xmath64 an illustration is given in figure [ length ] , where two acyclic orientations are shown for the same graph , together with the corresponding partition into sinks alluded to earlier . this partition is known as the sink decomposition of the graph according to the acyclic orientation @xcite . in the figure , each such decomposition is shown with a rightmost box containing the sinks , then another box to its left containing the sinks that appear if the former sinks are eliminated , and so on . the set of nodes in each box is normally referred to as a layer of the sink decomposition . notice that the number of layers in the sink decomposition for @xmath60 is precisely @xmath65 . in the case of figure [ length ] , the bottommost acyclic orientation has the smaller sink decomposition and thus corresponds to a better ( in this case , optimal ) assignment of colors ( a different color to the nodes in each layer ) . + [ length ] for @xmath42 , figure [ shortest10 ] depicts the relationship between the chromatic number and the number of acyclic orientations of each graph in @xmath12 . the thinner dashed plot in the figure gives the number of acyclic orientations of each graph . this number has been computed using the algorithm of @xcite , which although efficient in several aspects relevant to the analysis of enumerative algorithms , becomes prohibitive very quickly as the graph gets larger . the same enumeration process has been used to record the number of acyclic orientations that are shortest , that is , those whose sink decompositions have as many layers as the graph s chromatic number . this small addition to the algorithm employs straightforward depth - first search @xcite , and the results are shown as the thicker solid plot of the figure . [ shortest10 ] remarkably , an effect very similar to the one observed in figure [ poly10 ] is seen to occur now as well : the number of shortest acyclic orientations increases sharply whenever the chromatic number increases , and subsequently goes down until immediately before the next increase . these are also absolute data , but now it is obvious how to make them relative : we simply observe the percentage of all acyclic orientations that are shortest . this is shown in figure [ compcolor10 ] , which confirms that , also in relative terms , shortest acyclic orientations become significantly more abundant right after an increase in the chromatic number , becoming increasingly rarer from there onward until the next increase . [ compcolor10 ] this indicates that , in a sense , coloring a graph optimally becomes easier immediately after an increase in the chromatic number , and incrementally harder until the next transition is reached . behind this statement is the intuitive feeling that coloring algorithms should in general fare better when optimal solutions are more abundant , be such abundance assessed as some indicator of how many optimal colorings there are or as the fraction @xmath66 of shortest acyclic orientations . of course , this intuition calls for experimental support , but notice that at least the most nave of all random approaches a series of bernoulli trials inside @xmath52 until the first member of @xmath67 is found is certain to benefit from such abundance of shortest acyclic orientations , as clearly the expected time for its convergence is @xmath68 @xcite . more serious methods with the potential to benefit from the relative abundance of optimal acyclic orientations exist @xcite , though , and a systematic effort to assess their capabilities on sequences of random graphs is under way . the number ( or fraction ) of optimal acyclic orientations is , in principle , also subject to the same concerns with the disguising action of obvious combinatorial growth that we expressed earlier . to see what happens when we apply the same normalization by the factorial of the current chromatic number , we have in each of figures [ shortest10 ] and [ compcolor10 ] a plot with the results ( the one in thick dashes ) . the pronounced jumps are still present , but this normalization is only meaningful as inherited from the relationship between optimal colorings and shortest acyclic orientations . a better normalization may exist and we would like to digress on this possibility briefly , although several problems related to it are still open . suppose we take an acyclic orientation and turn all its sinks into sources ( nodes whose adjacent edges are all oriented outward ) . this necessarily yields another acyclic orientation , and the continual repetition of the process must eventually lead to a period of orientations . this attractor dynamics has several interesting properties @xcite ; in our context , the most crucial property is that the number of layers in the graph s sink decomposition is continually nonincreasing along the process of obtaining new acyclic orientations by turning sinks into sources . so the period at the core of each attractor comprises orientations that yield sink decompositions all with the same number of layers , this number being also no larger than that resulting from any other orientation in the same attractor . this means that , in addition to the chromatic indicator originally observed when this attractor dynamics was first analyzed ( the graph s interleaved multichromatic number , cf . @xcite ) , each attractor is related to finding the chromatic number as well , since finding a period whose orientations are shortest over all attractors immediately yields the chromatic number . a period , in summary , provides a means of expressing the equivalence of several acyclic orientations and may become suitable for the normalization we need if only more knowledge can be obtained on it . unfortunately , we thus far lack this necessary additional knowledge . to finalize , we comment on yet another interesting insight that can be gained from considering the relationship between a graph s colorings and its acyclic orientations . suppose that @xmath60 is an acyclic orientation of @xmath1 , and consider the random addition of an edge to @xmath1 between two nodes not currently connected . if we look at @xmath60 from the perspective of its sink decomposition , then let its layers be numbered @xmath33 through @xmath69 , starting at the layer that contains the sinks and onward . as illustrated in figure [ morelayers ] , in some cases the addition of the new edge will preserve the sink decomposition , while in others it will not . these two possibilities are shown in the leftmost sink decomposition in the figure as two dashed directed edges . the addition of one of them preserves the sink decomposition ( shown in the middle sink decomposition of the figure ) ; the addition of the other , which connects two nodes in the same layer , forces the sink decomposition to acquire another layer , as shown in the rightmost sink decomposition of the figure . [ morelayers ] let us then assume that , if the new edge is added between layers , then it is oriented in the direction of the lower - numbered layer . let also the @xmath70th layer have @xmath71 nodes , so that @xmath72 . the number of node pairs not yet connected by an edge is @xmath73 ; of these , the ones that have nodes in different layers amount to @xmath74 . if we let @xmath75 be the probability that adding the new edge does not disturb the sink decomposition , then @xmath75 is the probability that the edge is added between layers . that is , @xmath76 by ( [ prob ] ) , the addition of an edge between layers ( this happens with probability @xmath75 ) causes @xmath75 to decrease , as the only change in the formula is the concomitant subtraction of @xmath33 off both the numerator and the denominator . when the edge is added between nodes of the same layer ( with probability @xmath77 ) , the double summation in the numerator of ( [ prob ] ) may either remain the same or vary . if it remains the same or decreases , then @xmath75 , as before , decreases . in order to verify what happens otherwise , let @xmath78 denote the double summation . then @xmath75 is seen to vary by at least @xmath79 so @xmath75 increases unless @xmath80 , in which case it was zero to begin with and remains zero . if we now consider the number of edges that need to be randomly added to @xmath1 so that @xmath75 once again assumes the value it currently has , say @xmath81 , it is easy to see from ( [ prob ] ) that this number is given by @xmath82 , where @xmath83 is the increase that @xmath75 incurs along the way . the value of @xmath83 is very hard to quantify , but it seems reasonable to assume , at least for the sake of the argument , that it gets smaller as the graph gets denser ( i.e. , acquires more edges ) . in this case , the number of random edge additions needed for @xmath75 to return to the value @xmath81 is ever smaller as the process unfolds . overall , what we witness is a process that resembles the increase in @xmath26 as @xmath1 becomes denser , slow at first but increasingly rapid as the graph s density gets higher . we hope to obtain a better characterization of this resemblance as further research adds detail to the picture . if we succeed , it may be possible to obtain a generic prescription for determining all the points at which the chromatic number increases along the sequence , thus adding to what is already known @xcite . let @xmath84 denote the independence number of @xmath1 . we have used public - domain code @xcite based on @xcite to find @xmath85 for each graph in @xmath12 with @xmath86 . what this code finds is not a maximum independent set directly , but rather a maximum clique in the graph that is complementary to the graph of interest ( i.e. , has an edge joining two distinct nodes if and only if the graph of interest does not ) . a clique is a subgraph whose nodes are all connected to one another , so the correspondence to independent sets should be clear . this is illustrated in figure [ maxclique ] . [ maxclique ] figure [ dfmax ] has two plots for @xmath86 , one to indicate the time to find the graph s independence number , the other to indicate how this number evolves as the graph becomes denser . we show data for a certain density interval only . to the left of what is shown the graphs have isolated nodes and the behavior is uncharacteristic , while to the right times become too small to be indicative of any particularly interesting behavior . but inside the density interval used in the figure the independence number goes down nearly steadily , from @xmath87 for @xmath88 and @xmath89 can only be explained by the existence of two last isolated nodes in @xmath90 , one of which gets incorporated into @xmath91 , the other into @xmath92 . that a node should remain isolated through @xmath93 for @xmath86 is unlikely but entirely conceivable . in fact , the probability of such an event , given approximately by @xmath94 ( cf . ( [ isolated ] ) ) , is @xmath95 . ] and at some of these downward transitions there appears to be a considerable decrease in the time for optimal solution right after the decrease in the independence number . although this evidence is less ubiquitous than in the case of graph coloring , and for this reason certainly less compelling , this study seems to be the first one to address the issue of phase transitions related to finding a graph s independence number and for this reason we investigate the matter further . in this section , we introduce some combinatorial arguments that appear to relate to these phenomena . [ dfmax ] as for graph coloring , our initial step is to assess the abundance of maximum independent sets for a given graph @xmath1 , that is , independent sets of size @xmath84 . unlike the case of graph coloring , however , it is as yet unknown how to count this number exactly . this remains true even if we settle for maximal ( as opposed to maximum ) independent sets , that is , independent sets that can not be enlarged without losing the independence property , although in this case increasingly better upper bounds on the desired number have been discovered recently ( cf . @xcite and the references therein ) . but at this point it helps to recall that the sequence of graphs @xmath11 is randomly generated , each graph being drawn uniformly from the set of graphs having the same number of nodes and edges . we may therefore attempt to assess the number of maximal independent sets of a given size in each graph by computing the expected value of this number , which is in fact a random quantity . in order to accomplish this more easily , it is helpful to resort to the model of random graphs in which an edge exists between two nodes with constant probability , say @xmath63 , independently of the nodes . although this is not the model under which our graphs were generated , away from limiting situations it is safe to assume that the two models are equivalent to each other with @xmath96 for @xmath0 @xcite . let @xmath97 denote the expected number of maximal independent sets of size @xmath20 in @xmath1 . the number of candidate sets is @xmath98 , and the probability that each one is an independent set is @xmath99 . if a candidate is an independent set , then the probability that it is maximal is the probability that each of the remaining @xmath100 nodes is connected to at least one of its @xmath20 nodes , that is , @xmath101^{n - k}$ ] . we then get @xmath102^{n - k}.\ ] ] now let @xmath1 be a random graph generated in compliance with the edge density given by @xmath63 , and let the value of @xmath84 be known . letting @xmath103 in ( [ expectedmis ] ) yields the expected number of maximal independent sets in similar random graphs , whose size is , at least for one of them , the size of its maximum independent set . [ expected75 ] for @xmath86 , we show the evolution of this number for the sequence @xmath12 in figure [ expected75 ] , along with the evolution of the independence number . interestingly , every transition of the independence number to a smaller value causes the expected number of maximal independent sets of size @xmath85 to increase markedly . from there onward , this number decreases until the next similar transition occurs . this is one first indication , albeit imprecise , that the abundance of such maximal sets may be related to the increased ease with which the graph s independence number can sometimes be found immediately after a decrease in the independence number . however , as in our analysis of the chromatic polynomial in section [ coloring ] , it is not clear how to proceed and characterize the relationship more effectively . not only this , but it is still possible , as in the case of graph coloring , that some underlying inherent equivalence among maximal independent sets exists that would indicate a way of normalizing the data shown in the figure . we still do not know how that can be achieved . once again , though , a deep relationship exists between a graph s independent sets and its acyclic orientations . it is in this case both more complex and more subtle than in the case of graph coloring , so we describe it with the aid of an illustration right from the start . first consider figure [ width ] , where two acyclic orientations of the same graph are shown , each one alongside what is known as a chain decomposition of the graph according to it . a chain decomposition of a graph according to an acyclic orientation is a partition of the graph s node set such that the nodes in each set of the partition are arranged by the acyclic orientation as a single chain of nodes ( a directed path ) . in the case of figure [ width ] , each partition is displayed with the aid of boxes to enclose the nodes that go in each set . furthermore , each of the chain decompositions shown in the figure employs the minimum number of chains for the corresponding acyclic orientation . the topmost acyclic orientation admits a chain decomposition comprising one single chain , while for the other no chain decomposition exists with fewer than two chains . + [ width ] understanding how chain decompositions and independent sets are related involves several technicalities that we will not discuss formally but illustrate through examples instead . figure [ flow ] contains two flow networks , that is , two directed graphs with two distinguished nodes , @xmath6 and @xmath104 , on which we consider the maximum flow from @xmath6 to @xmath104 subject to certain edge capacities ( unit for all edges that emanate from @xmath6 or arrive at @xmath104 , infinite for all others ) . each flow network corresponds to one of the acyclic orientations of figure [ width ] and is constructed from that orientation as follows . for each node @xmath45 in @xmath1 , two nodes , @xmath105 and @xmath106 , are added to the flow network , along with a directed edge from @xmath6 to @xmath105 and one from @xmath106 to @xmath104 . if an edge exists between nodes @xmath45 and @xmath46 and the acyclic orientation directs that edge from @xmath45 to @xmath46 , then the flow network contains an edge directed from @xmath105 to @xmath107 . + [ flow ] the maximum flow from @xmath6 to @xmath104 in each of the networks of figure [ flow ] is indicated by solid edges ( edges that carry unit flow ) and dashed edges ( edges that carry zero flow ) . in general , the following interesting property holds for maximum flows in such networks @xcite . of the solid edges , those whose removal disconnects @xmath6 from @xmath104 minimally ( the minimum cut ) induce a node cover in @xmath1 , that is , a set of nodes that touches every edge . in figure [ flow ] , an edge in the minimum cut either leads from @xmath6 to a node enclosed in a box or from such a node to @xmath104 . nodes thus represented correspond to the node sets @xmath108 and @xmath109 in @xmath1 , each set a node cover . note that , by definition , the complements of these sets with respect to the node set of @xmath1 are necessarily independent sets of @xmath1in the figure , these are the sets @xmath110 and @xmath111 , each of whose nodes corresponds to one of the chains in figure [ width ] . formally , the relationship between @xmath84 and the acyclic orientations of @xmath1 is as given next @xcite , and brings sharper focus to the classic result established by dilworth s theorem @xcite . for @xmath58 , let @xmath112 be the set of all chain decompositions of @xmath1 according to @xmath60 . for @xmath113 , let @xmath114 be the number of chains in @xmath115 . then @xmath116 as one readily notices , ( [ alphafromomega ] ) is a sort of dual of ( [ chifromomega ] ) : it expresses @xmath84 as the number of chains in the minimum chain decomposition that has the most chains over @xmath52 . henceforth , we let @xmath117 denote the set of such widest acyclic orientations , understood as those acyclic orientations that achieve the minimum chain decomposition that is maximum over @xmath52 . it is now instructive to return briefly to figure [ width ] . for each acyclic orientation @xmath60 displayed in the figure , the number of chains in the corresponding chain decomposition is @xmath118 . of the two acyclic orientations , the bottommost achieves the maximum of this quantity over @xmath52 . we now investigate the use of @xmath119 as an indicator of how abundant the widest acyclic orientations of @xmath1 are and how this relates to the hardness of finding @xmath84 . this time , our procedure to enumerate all the acyclic orientations of a graph has been coupled with a public - domain code @xcite that implements the algorithm of @xcite to find the maximum flow in a network ( and also the minimum cut , as a by - product ) . [ widest10 ] the results are shown in figure [ widest10 ] , in which the number of all acyclic orientations of each graph is plotted alongside its independence number and the number of acyclic orientations that are widest ( those whose chain decomposition into the fewest possible chains requires @xmath85 chains ) . the figure indicates that the same sharp transitions that appear in figure [ expected75 ] are also observed for the acyclic orientations : every downward transition that the independence number undergoes is accompanied by a sharp increase in the number of widest acyclic orientations . when we assess such increases with respect to the set of all the acyclic orientations of each graph , we obtain what is shown in figure [ compmis10 ] . clearly , every decrease in the independence number corresponds to a marked increase in the fraction of acyclic orientations that are widest . we hope to eventually be able to conclude that such increases are correlated with the time it takes to find a graph s maximum independent set . as we mentioned in section [ coloring ] , obviously the unreasonable approach of random trials does benefit from an elevated @xmath120 ratio , but we believe this may also be the case for heuristics that exploit the role of acyclic orientations directly @xcite . [ compmis10 ] we have investigated the np - hard problems of coloring the nodes of a graph optimally and of finding a maximum independent set in a graph . for each of these problems , we first displayed empirical evidence that , as the number of edges in the graph is increased by the random addition of one edge at a time , significant variations take place in the time to solve the problem optimally . often these variations coincide with the transition of the value of the optimal solution to a new level , a higher one for graph coloring , a lower one for independent sets . for graph coloring , we have shown that the upward transitions in the chromatic number along the sequence of increasingly denser graphs coincide with sharp increases in the abundance of distinct colorings employing the optimal number of colors . more importantly , we have shown that the same phenomenon takes place when we consider the shortest acyclic orientations of the graphs , that is , those orientations whose sink decompositions have as many layers as the graphs chromatic numbers . in this case , sharp increases are observed in the ratio @xmath121 . our conclusions for the maximum independent set problem are similar . the downward transitions in the independence number that occur along the sequence of graphs coincide with sharp increases in the expected number of maximal independent sets of certain sizes . as for the acyclic orientations of the graphs , we have given evidence that the same type of phenomenon takes place regarding the widest acyclic orientations of the graphs , that is , those whose minimum chain decompositions have as many chains as the graphs independence numbers . in this case , sharp increases take place in the ratio @xmath122 . we find that the two special subsets of @xmath52 , @xmath67 and @xmath123 , underlie an interpretation of the phase transitions observed in optimal graph coloring and optimal independent sets that is full of combinatorial meaning . this meaning is directly related to the structure of the graphs involved and that of its acyclic orientations . in addition , it provides a direct interpretation of the hardness of the problems as given by the relative abundance of the acyclic orientations that give the optima . except for those shown in figures [ trick ] , [ dfmax ] , and [ expected75 ] , all our empirical results have been given for @xmath42 only . as we indicated earlier , the reason for this has been the severe combinatorial explosion that occurs when finding a graph s chromatic polynomial or its set of acyclic orientations . for @xmath124 , so far these computations could only be carried out through about @xmath125 edges , yielding results that fully support the conclusions we have drawn along the paper . but , instead of presenting such partial results , we opted for giving the reader data on the full evolution through the densest graphs for @xmath42 . we think the empirical evidence we have provided is only the beginning of a deeper investigation of how the abundance of certain acyclic orientations affects the hardness of optimal graph coloring and of finding maximum independent sets . given the aforementioned computational difficulties , the most promising avenue for continuation seems to be the search for additional analytic properties that can explain the findings we have described . perhaps the evolution of the probability in ( [ prob ] ) superficially described at the end of section [ coloring ] , and of an analogous indicator in the case of independent sets , should be investigated more deeply as a first step .

we study combinatorial indicators related to the characteristic phase transitions associated with coloring a graph optimally and finding a maximum independent set . in particular , we investigate the role of the acyclic orientations of the graph in the hardness of finding the graph s chromatic number and independence number . we provide empirical evidence that , along a sequence of increasingly denser random graphs , the fraction of acyclic orientations that are `` shortest '' peaks when the chromatic number increases , and that such maxima tend to coincide with locally easiest instances of the problem . similar evidence is provided concerning the `` widest '' acyclic orientations and the independence number . + + * keywords : * graph coloring , maximum independent sets , phase transitions , acyclic orientations .

introduction graph coloring independent sets conclusions

arxiv

the next generation gravitational - wave detectors , such as advanced ligo @xcite , advanced virgo @xcite and kagra @xcite interferometers , are expected to detect gw signals from mergers of two compact objects . these gravitational wave bursts ( gwbs ) have well defined `` chirp '' signal , which can be unambiguously identified . once detected , the gw signals would open a brand new channel for us to study the universe , especially the physics in the strong field regime . due to the faint nature of gws , an associated electromagnetic ( em ) emission signal in coincidence with a gwb in both trigger time and direction would increase the signal - to - noise ratio of the gw signal , and therefore would be essential for its identification . one of the top candidates of gwbs is merger of two neutron stars ( i.e. ns - ns mergers ) @xcite . the em signals associated with such an event include a short gamma - ray burst ( sgrb ) @xcite , an optical `` macronova '' @xcite , and a long lasting radio afterglow @xcite . numerical simulations show that binary neutron star mergers could eject a fraction of the materials , forming a mildly anisotropic outflow with a typical velocity about @xmath1 ( where @xmath2 is the speed of light ) , and a typical mass about @xmath3 ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? the radioactivity of this ejecta powers the macronova and the interaction between the ejecta and the ambient medium is the source of radio afterglow . usually , the merger product is assumed to be a black hole or a temporal hyper - massive neutron star which survives 10 - 100 ms before collapsing into the black hole ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? nonetheless , recent observations of galactic neutron stars and ns - ns binaries suggest that the maximum ns mass can be high , which is close to the total mass of the ns - ns systems ( * ? ? ? * ; * ? ? ? * and references therein ) . indeed , for the measured parameters of 6 known galactic ns binaries and a range of equations of state , the majority of mergers of the known binaries will form a massive millisecond pulsar and survive for an extended period of time @xcite . when the equation of state of nuclear matter is stiff ( see arguments in @xcite and @xcite and references therein ) , a stable massive neutron star would form after the merger . this newborn massive neutron star would be differentially rotating . the dynamo mechanism may operate and generate an ultra - strong magnetic field @xcite , so that the product is very likely a millisecond magnetar . evidence of a magnetar following some sgrbs has been collected in the swift data @xcite , and magnetic activities of such a post - merger massive neutron star have been suggested to interpret several x - ray flares and plateau phase in sgrbs @xcite . since both the gravitational wave signal and the millisecond magnetar wind both nearly isotropic , a bright electromagnetic signal can be associated with a ns - ns merger gwb regardless of whether there is a short gamma - ray burst ( sgrb ) - gwb association @xcite . even if there is an association , most gwbs would not be associated with the sgrb since sgrbs are collimated . @xcite proposed that the near - isotropic magnetar wind of a post - merger millisecond magnetar would undergo magnetic dissipation @xcite and power a bright x - ray afterglow emission . here we suggest that after partially dissipating the magnetic energy , a significant fraction @xmath4 of the magnetar spin energy would be used to push the ejecta , which drives a strong forward shock into the ambient medium . the continuous injection of the poynting flux into the blast wave modifies the blast wave dynamics and leads to rich radiation signatures @xcite . figure 1 presents a physical picture for several em emission components appearing after the merger . we here study the dynamics of such an interaction in detail , and calculate broadband afterglow emission from this forward shock . the postmerger hyper - massive neutron star may be near the break up limit , so that the total spin energy @xmath5 ( with @xmath6 for a massive neutron star ) may be universal . here @xmath7 ms is the initial spin period of the proto - magnetar . throughout the paper , the convention @xmath8 is used for cgs units , except for the ejecta mass @xmath9 , which is in unit of solar mass @xmath10 . given nearly the same total energy , the spin - down luminosity and the characteristic spin down time scale critically depend on the polar - cap dipole magnetic field strength @xmath11 @xcite , i.e. @xmath12 , where @xmath13 , and the spin down time scale @xmath14 , where @xmath15 cm is the stellar radius . after the internal dissipation of the magnetar wind that powers the early x - ray afterglow @xcite , the remaining spin energy would be added to the blastwave . the dynamics of the blastwave depends on the magnetization parameter @xmath16 of the magnetar wind after the internal dissipation . since for the confined wind , magnetic dissipation occurs upon interaction between the wind and the ejecta , in this paper , we assume that the wind is still magnetized ( moderately high @xmath16 ) , so that there is no strong reverse shock into the magnetar wind @xcite . as a result , the remaining spin energy is continuously injected into the blastwave with a luminosity @xmath17 , where @xmath18 denotes the fraction of the spin down luminosity that is added to the blastwave . the evolution of the blastwave can be described by a system with continuous energy injection @xcite . the newly formed massive magnetar is initially hot . a poynting flux dominated outflow is launched @xmath19 s later , when the neutrino - driven wind is clean enough @xcite . at this time , the front of the ejecta traveled a distance @xmath20 cm ( for @xmath21 ) , with a width @xmath22 cm . the ultra - relativistic magnetar wind takes @xmath23 s to catch up the ejecta , and drives a forward shock into the ejecta . balancing the magnetic pressure and the ram pressure of shocked fluid in the ejecta , one can estimate the shocked fluid speed as @xmath24 , which is in the same order of forward shock speed . so the forward shock would cross the ejecta in around @xmath25 . note that when calculating magnetic pressure , we have assumed a toroidal magnetic field configuration in the poynting flux , but adopting a different magnetic configuration would not significantly affect the estimate of @xmath26 . after the forward shock crosses the ejecta , the forward shock ploughs into the ambient medium . the dynamics of the blastwave during this stage is defined by energy conservation should be @xmath27 , where @xmath28 is the initial lorenz factor for the ejecta , which we take as unity for convenience . ] @xmath29 where @xmath30 is the swept mass from the interstellar medium . initially , @xmath31 , so the kinetic energy of the ejecta would increase linearly with time until @xmath32 , where the deceleration timescale @xmath33 is defined by the condition @xmath34 . by setting @xmath35 , we can derive a critical ejecta mass @xmath36 which separate regimes with different blastwave dynamics . for a millisecond massive magnetar , the parameters @xmath37 , @xmath38 , @xmath39 are all essentially fixed values . the dependence on @xmath40 is very weak ( 1/8 power ) , so the key parameters that determine the blastwave parameters are the ejecta mass @xmath9 and the magnetar injection luminosity @xmath41 ( or the magnetic field strength @xmath42 ) . if @xmath43 ( or @xmath44 ) , the ejecta can be accelerated linearly until the deceleration radius , after which the blastwave decelerates , but still with continuous energy injection until @xmath45 . conversely , in the opposite regime ( @xmath46 or @xmath47 ) , the blastwave is only accelerated to @xmath45 , after which it coasts before decelerating at @xmath48 . in the intermediate regime of @xmath49 ( or @xmath50 ) , the blastwave shows a decay after being linearly accelerated . there is another critical ejecta mass , which defines whether the blastwave can reach a relativistic speed . this is defined by @xmath51 . defining a relativistic ejecta as @xmath52 , this second critical ejecta mass is @xmath53 an ejecta heavier than this would not be accelerated to a relativistic speed . below we discuss four dynamical regimes . _ case i : @xmath43 or @xmath54 . _ this requires both a small @xmath41 ( or low @xmath42 ) and a small @xmath9 . we take an example with @xmath55 ( @xmath56 g ) and @xmath57 . to describe the dynamics in such a case , besides the spin down timescale @xmath58 , we need three more characteristic time scales and the lorentz factor value at the deceleration time @xmath59 where @xmath60 are the two time scales when the blastwave passes the non - relativistic to relativistic transition line @xmath61 during the acceleration and deceleration phases . with these parameters , one can characterize the dynamical evolution of the blastwave ( fig.[i]a ) , as shown in table 1 . based on the dynamics , we can quantify the temporal evolution of synchrotron radiation characteristic frequencies @xmath62 , and the peak flux , @xmath63 . the evolutions of the characteristic frequencies are presented in fig.[i]b and collected in table 2 . following the standard procedure in @xcite , we derive the synchrotron radiation characteristic frequencies and the peak flux density at @xmath33 , @xmath64 where @xmath65 . with the temporal evolution power law indices of these parameters ( table 2 ) , one can calculate the x - ray , optical and radio afterglow lightcurves . notice that there are two more temporal segments listed in table 2 , since @xmath66 crosses @xmath67 twice at @xmath68 respectively . we present the light curves in x - ray ( fig.[i]d ) , optical and radio ( 10ghz ) band ( fig.[i]c ) . the distance is taken as 300 mpc , the detection horizon of advanced ligo . _ case ii : @xmath49 or @xmath69 . _ the dynamics and the expressions of the characteristic parameters become simpler : @xmath70 the temporal indices of the evolutions of @xmath71 are listed in table 2 , and the expressions of @xmath72 and @xmath73 are shown in table 1 . as examples , we consider @xmath74 ( @xmath75 g ) vs. @xmath76 , which satisfies @xmath77 . similarly to case i , we have @xmath78 the expressions of @xmath72 and @xmath73 as well as the power - law indices for this case are also presented in table 1 and table 2 , respectively . the dynamics typical frequency evolution , and the light curves are presented in fig.[ii ] . we note that in this case ( and case iii ) , the synchrotron radiation properties are very sensitive to @xmath9 and @xmath4 . _ case iii : @xmath79 _ ( @xmath80 ) . as an example , we take @xmath81 g , and @xmath82 . for this example , the dynamics and the expressions of the characteristic parameters become @xmath83 and @xmath84 the power - law indices of various parameters for this case are also collected in table 2 , and the dynamics , frequency evolutions , and light curves are presented in fig . [ iii ] . _ case iv : @xmath85 . _ in this case , the blast wave never reaches a relativistic speed . the dynamics is similar to case iii , with the coasting regime in the non - relativistic phase . the dynamics for a non - relativistic ejecta and its radio afterglow emission have been discussed in @xcite . our case iv resembles what is discussed in @xcite , but the afterglow flux is much enhanced because of a larger total energy involved . for all the cases , bright broadband em afterglow emission signals are predicted . the light curves typically show a sharp rise around @xmath58 , which coincides the ending time of the x - ray afterglow signal discussed by @xcite due to internal dissipation of the magnetar wind . the x - ray afterglow luminosity predicted in our model is generally lower than that of the internal dissipation signal , but the optical and radio signals are much brighter . in some cases , the r - band magnitude can reach 11th at the 300 mpc , if @xmath9 is small enough ( so that the blastwave has a high lorentz factor ) and the medium density is not too low . the duration of detectable optical emission ranges from @xmath86 seconds to year time scale . the radio afterglow can reach the jy level for an extended period of time , with peak reached in the year time scale . these signals can be readily picked up by all - sky optical monitors , and radio surveys . the x - ray afterglow can be also picked up by large field - of - view imaging telescopes such as iss - lobster . since these signals are originated from interaction between the magnetar wind and the ejecta in the equatorial directions , they are not supposed to be accompanied with short grbs , and some internal - dissipation x - ray afterglows @xcite in the free wind zone . due to a larger solid angle , the event rate for this geometry ( orange observer in fig.1 ) should be higher than the other two geometries ( red and yellow observers in fig.1 ) . however , the brightness of the afterglow critically depends on the unknown parameters such as @xmath9 , @xmath42 ( and hence @xmath41 ) , and @xmath40 . the event rate also crucially depends on the event rate of ns - ns mergers and the fraction of mergers that leave behind a massive magnetar rather than a black hole . this afterglow signal is much stronger than the afterglow signal due to ejecta - medium interaction with a black hole as the post - merger product @xcite . the main reason is the much larger energy budget involved in the magnetar case . since the relativistic phase can be achieved , both x - ray and optical afterglows are detectable , which peak around the magnetar spindown time scale ( @xmath87 s ) . the radio peak is later similar to the black hole case @xcite , but the radio afterglow flux is also much brighter ( reaching jy level ) due to a much larger energy budget involved . the current event rate limit of @xmath88 mjy radio transients in the minutes - to - days time scale at 1.4 ghz is @xmath89 @xcite , or @xmath90 all sky . in view of the large uncertainties in the ns - ns merger rate and the fraction of millisecond magnetar as the post - merger product , our prediction is entired consistent with this upper limit . because of their brightness , these radio transients can be detected outside the advanced ligo horizon , which may account for some sub - mjy radio transients discovered by vla @xcite . recently , @xcite proposed another possible em counterpart of gwb with a wide solid angle . they did not invoke a long - lasting millisecond magnetar as the merger product , but speculated that during the merger process , a breakout shock from the merging neutron matter would accelerate a small fraction of surface material , which reaches a relativistic speed . such an outflow would also emit broad - band synchrotron emission by shocking the surrounding medium . within that scenario , the predicted peak flux is lower and the duration is shorter than the electro - magnetic signals predicted in @xcite and this work , due to a much lower energy carried by the outflow . detecting the gwb - associated bright signals as discussed in this paper would unambiguously confirm the astrophysical origin of gwbs . equally importantly , it would suggest that ns - ns mergers leave behind a hyper - massive neutron star , which gives an important constraint on the neutron star equation of state . with the gwb data , one can infer the information of the two nss involved in the merger . modeling afterglow emission can give useful constraints on the ejected mass @xmath9 and the properties of the postmerger compact objects . therefore , a combination of gwb and afterglow information would shed light into the detailed merger physics , and in particular , provide a probe of massive millisecond magnetars and stiff equations of state for neutron matter . we thank stimulative discussion with yi - zhong fan and jian - yan wei . we acknowledge the national basic research program ( 973 " program ) of china under grant no . 2009cb824800 and 2013cb834900 . this work is also supported by the national natural science foundation of china ( grant no . 11033002 & 10921063 ) and by nsf ast-0908362 . xfw acknowledges support by the one - hundred - talents program and the youth innovation promotion association of chinese academy of sciences .

if double neutron star mergers leave behind a massive magnetar rather than a black hole , a bright early afterglow can follow the gravitational wave burst ( gwb ) even if there is no short gamma - ray burst ( sgrb ) - gwb association or there is an association but the sgrb does not beam towards earth . besides directly dissipating the proto - magnetar wind as suggested by zhang , we here suggest that the magnetar wind could push the ejecta launched during the merger process , and under certain conditions , would reach a relativistic speed . such a magnetar - powered ejecta , when interacting with the ambient medium , would develop a bright broad - band afterglow due to synchrotron radiation . we study this physical scenario in detail , and present the predicted x - ray , optical and radio light curves for a range of magnetar and ejecta parameters . we show that the x - ray and optical lightcurves usually peak around the magnetar spindown time scale ( @xmath0 s ) , reaching brightness readily detectable by wide - field x - ray and optical telescopes , and remain detectable for an extended period . the radio afterglow peaks later , but is much brighter than the case without a magnetar energy injection . therefore , such bright broad - band afterglows , if detected and combined with gwbs in the future , would be a probe of massive millisecond magnetars and stiff equation - of - state for nuclear matter .

introduction the model detectability and implications

arxiv

the study of the fluid phases in mixtures of colloids and nonadsorbing neutral polymers has become increasingly important in recent years ; see refs . @xcite for recent reviews . these systems show a very interesting phenomenology , which only depends to a large extent on the nature of the polymer - solvent system and on the ratio @xmath8 , where @xmath9 is the zero - density radius of gyration of the polymer and @xmath10 is the radius of the colloid . experiments and numerical simulations indicate that polymer - colloid mixtures have a fluid - solid coexistence line and , for @xmath11 , where @xcite @xmath12 - 0.4 , also a fluid - fluid coexistence line between a colloid - rich , polymer - poor phase ( colloid liquid ) and a colloid - poor , polymer - rich phase ( colloid gas ) . on the theoretical side , research has mostly concentrated on mixtures of neutral spherical colloids and polymers in solutions under good - solvent or @xmath0 conditions . in the former case , predictions for the colloid - polymer interactions have been obtained by using full - monomer representations of polymers ( for instance , the self - avoiding walk model was used in refs . @xcite ) , field - theoretical methods @xcite , or fluid integral equations @xcite . moreover , some general properties have been derived by using general scaling arguments @xcite . at the @xmath0 point , the analysis is simpler , since polymers behave approximately as ideal chains . these theoretical results have then been used as starting points to develop a variety of coarse - grained models and approximate methods , see refs . @xcite and references therein , which have been employed to predict colloid - polymer phase diagrams . in this paper , we consider polymer solutions that either show good - solvent behavior or are in the thermal crossover region between good - solvent and @xmath0 conditions . we study the solvation of a single colloid in the solution , assuming that the monomer - colloid potential is purely repulsive . we determine the distribution of the polymer chains around a single colloidal particle , which is the simplest property that characterizes polymer - colloid interactions . we investigate numerically , by means of monte carlo simulations , how it depends on the quality of the solution , which is parametrized @xcite in terms of the second - virial combination @xmath13 , where @xmath14 is the second virial coefficient and @xmath9 is the zero - density radius of gyration . this adimensional combination varies between 5.50 in the good - solvent case @xcite and zero ( @xmath0 point ) . beside good - solvent solutions , we consider two intermediate cases : solutions such that @xmath15 is approximately one half of the good - solvent value , which show intermediate properties betweeen good - solvent and @xmath0 behavior , and solutions such that @xmath15 is 20% of the good - solvent value , which are close to the @xmath0 point . in each case we compute the polymer density profile around the colloid . these results are used to determine thermodynamic properties , like the surface tension and the adsorption @xcite , which are then compared with the available theoretical predictions . note that an analysis of the polymer depletion around a colloid in the thermal crossover region was already performed in refs . @xcite , but without a proper identification of the universal crossover limit @xcite . here , we wish to perform a much more careful analysis of the crossover behavior , following refs . @xcite . we focus on the dilute and semidilute regimes , in which the monomer density is small and a universal behavior , i.e. , independent of chemical details , is obtained in the limit of large degree of polymerization . in the dilute regime , in which polymer - polymer overlaps are rare , solvation properties are determined for a wide range of values of @xmath1 , from 0 up to 30 - 50 . in the semidilute regime , simulations of systems with large @xmath1 require considering a large number of colloids , which makes monte carlo simulations very expensive . hence , we only present results for @xmath16 and 2 . the paper is organized as follows . in sec . [ sec2 ] we define the basic quantities we wish to determine . first , in sec . [ sec2.1 ] and [ sec2.2 ] we introduce the surface tension , the adsorption , and the depletion thickness , and discuss their relation to the density profile of the polymers around the colloid . in sec . [ sec2.3 ] we discuss the low - density behavior and the relation between solvation properties and colloid - polymer virial coefficients that parametrize the ( osmotic ) pressure of a polymer - colloid binary solution in the low - density limit . [ sec2.4 ] discusses the different behavior that are expected as a function of @xmath1 and density and gives an overview of theoretical and numerical predictions . [ sec3 ] summarizes our polymer model and gives a brief discussion on how one can parametrize in a universal fashion the crossover between the good - solvent and the @xmath0 behavior ( for more details , see ref . @xcite ) . in sec . [ sec4 ] we present our results for the dilute regime , while finite - density results are presented in sec . [ sec6 ] discusses a simple coarse - grained model in which each polymer is represented by a monoatomic molecule , which represents a more rigorous version of the well - known asakura - oosawa - vrij model . in sec . [ sec7 ] we present our conclusions . two appendices are included , one explaining how to compute the virial coefficients of a binary mixture of flexible molecules , and one discussing the small-@xmath1 behavior of the virial coefficients . tables of results are reported in the supplementary material . let us consider a solution of nonadsorbing polymers in the grand canonical ensemble at fixed volume @xmath17 and chemical potential @xmath18 . temperature is also present , but , since it does not play any role in our discussion , we will omit writing it explicitly in the following . let us indicate with @xmath19 the corresponding grand potential . let us now add a spherical colloidal particle of radius @xmath10 to the solution and let @xmath20 be the corresponding grand potential . the insertion free energy can be written as the sum of two terms , one proportional to the volume @xmath21 of the colloid and one proportional to its surface area @xmath22 @xcite : @xmath23 where @xmath24 is the bulk pressure and @xmath25 is the surface tension . the latter quantity can be related to the adsorption @xmath26 defined in terms of the change in the mean number of polymers due to the presence of the colloid : @xmath27 where @xmath28 indicates the numbers of polymers present in the solution , @xmath29 and @xmath30 are averages in the presence and in the absence of the colloidal particle , respectively . differentiating eq . ( [ gamma - def ] ) with respect to @xmath18 , we obtain @xmath31 we also define the average bulk polymer density @xmath5 as @xmath32 the surface tension can also be defined in the canonical ensemble , as a function of the bulk polymer density @xmath5 . then , we have @xmath33 where @xmath34 for the bulk system ( @xmath35 as usual ) . the adsorption coefficient can be easily related to the polymer density profile around the colloid . assume that each polymer consists of @xmath36 monomers and define the average bulk monomer density @xmath37 . then , we write @xmath38,\end{aligned}\ ] ] where the average is performed at chemical potential @xmath18 and volume @xmath17 , @xmath39 is the colloid position , and @xmath40 , @xmath41 , @xmath42 , are the monomer positions . if we now define the monomer - colloid pair correlation function @xmath43 and the integral @xmath44 , \label{gmoncp}\ ] ] we obtain @xmath45.\ ] ] since @xmath46 for @xmath47 , a more transparent relation is obtained by defining @xmath48 , \label{gmoncp2}\ ] ] in which one only integrates the density profile outside the colloid . since @xmath49 , we have @xmath50 in the previous discussion we have considered the monomer - colloid correlation function , but it is obvious that any other polymer - colloid distribution function could be used . in order to compare our results with those obtained in coarse - grained models ( we will discuss them in sec . [ sec6 ] ) , we will also use the pair distribution function between the colloid and the polymer centers of mass . if @xmath40 , @xmath51 , are the positions of the monomers belonging to polymer @xmath52 , we first define the polymer center of mass @xmath53 then , the pair distribution function between a colloid and a polymer center of mass is defined by @xmath54 where the average is taken at a given value @xmath18 . in terms of this quantity @xmath55 , \label{gamma - gcmcp } \\ g_{cm , cp}(\mu_p ) & = & 4\pi \int_0^\infty r^2 dr\ , [ g_{{cm},cp}(r;\mu_p ) -1 ] . \nonumber \end{aligned}\ ] ] if we define for @xmath56 , @xmath57 can not be obtained directly by performing the integration from @xmath58 to @xmath59 . ] @xmath60 , we can write a relation analogous to eq . ( [ gamma - gmoncp ] ) . comparison of eqs . ( [ gamma - gmoncp ] ) and ( [ gamma - gcmcp ] ) implies @xmath61 and @xmath62 , hence in the following we will simply refer to these quantities as @xmath63 and @xmath64 . it is interesting to relate the pair correlation functions @xmath65 to the analogous correlation functions @xmath66 that are appropriate for a binary system consisting of polymers and colloids at polymer and colloid chemical potentials @xmath18 and @xmath67 , respectively . indeed , one can show that , in the limit @xmath68 , i.e. , when the colloid density goes to zero , one has @xmath69 eq . ( [ equality - g ] ) allows us to relate @xmath63 to thermodynamic properties of the binary mixture in the limit of vanishing colloid density . for this purpose we use the kirkwood - buff relations between structural and thermodynamic properties of fluid mixtures @xcite . the integral @xmath63 , which is relevant to determine adsorption properties , corresponds to one of the kirkwood - buff integrals @xcite defined as @xmath70 where @xmath71 and @xmath72 label the different species of the mixture . the integrals @xmath73 can be related to derivatives of the pressure with respect to the polymer and colloid densities . for @xmath74 we have @xcite @xmath75 which imply @xmath76 eqs . ( [ gamma - gmoncp ] ) and ( [ gamma - gamma ] ) can then be rewritten as @xmath77 , \label{gamma - kb } \\ \beta \gamma & = & { 1\over a_c } \int_0^{\rho_p } { k_c - 1\over \rho_p ' } d\rho_p ' - \beta p { v_c\over a_c } . \label{gamma - kb}\end{aligned}\ ] ] depletion effects can be equivalently parametrized by introducing the depletion thickness @xmath78 @xcite , which is an average width of the depleted layer around the colloid . it is defined in terms of the integral @xmath63 as @xmath79 so that @xmath80 since @xmath78 is only determined by @xmath63 , knowledge of @xmath78 is completely equivalent to that of the adsorption . the two quantites are related by @xmath81 . \label{gamma - ds}\ ] ] as we shall discuss below , @xmath82 for large polymer densities , hence in this limit @xmath83 it is interesting to discuss the limit @xmath84 , in which the colloid degenerates into an impenetrable plane . setting @xmath85 in eq . ( [ gmoncp2 ] ) , we obtain @xmath86.\ ] ] for @xmath87 , we have @xmath88 , where @xmath89 is the pair distribution function between an impenetrable plane at @xmath90 and a polymer . then , we obtain for @xmath87 @xmath91 with @xmath92.\end{aligned}\ ] ] taking the limit @xmath87 in eqs . ( [ dsoverrcvsgcp ] ) and ( [ gamma - ds ] ) , we obtain @xmath93 for @xmath94 the depletion thickness @xmath78 and the surface quantities @xmath95 and @xmath25 can be related to the virial coefficients that parametrize the expansion of the pressure of a binary colloid - polymer system in powers of the concentrations . these relations have already been discussed in the literature @xcite . they can be easily derived by using eqs . ( [ gamma - kb ] ) and ( [ gamma - kb ] ) . we start by expanding the pressure as @xmath96 where @xmath97 and @xmath5 are the colloid and polymer concentrations and we have neglected fourth - order terms . then , eqs . ( [ gamma - kb ] ) and ( [ gamma - kb ] ) give @xmath98 , \nonumber \\ \\ \beta\gamma & = & - { \rho_p\over a_c } \left[v_c - b_{2,cp } + { 1\over2 } ( 2 b_{2,pp } v_c - b_{3,cpp})\rho_p + \ldots \right ] . \label{gamma - gamma - smallrhop}\end{aligned}\ ] ] in the limit @xmath99 one should recover the results for an infinite impenetrable plane . this requires the coefficients appearing in the previous two expressions to be of order @xmath100 as @xmath87 . this is explicitly checked in app . [ appb ] and allows us to write @xmath101 explicit expressions for @xmath102 and @xmath103 are reported in appendix [ appb ] . for the depletion thickness we obtain @xmath104 + \ldots , \label{deltas - smallrhop}\ ] ] where we have defined the polymer volume fraction @xmath105 and the adimensional combinations @xmath106 and @xmath107 , where @xmath9 is the zero - density polymer radius of gyration . in the limit @xmath87 , we should obtain the density expansion of the depletion thickness for an impenetrable plane . using eq . ( [ deltas - piano ] ) we obtain @xmath108 depletion properties have been extensively studied in the past . here we present scaling arguments and literature results , that will be checked in the following sections by using our accurate monte carlo estimates . for an ideal ( noninteracting ) polymer solution the insertion free energy is exactly known @xcite : @xmath109 where @xmath9 is the zero - density radius of gyration . the depletion thickness follows immediately @xcite : @xmath110 for good - solvent polymers there are several predictions obtained by using the field - theoretical renormalization group . in the dilute limit @xmath111 , the surface tension has been determined @xcite both in the colloid limit in which @xmath84 and in the so - called protein limit @xmath112 . setting @xmath113 and @xmath114 in the results of ref . @xcite , we obtain for @xmath84 and @xmath111 @xmath115 note that the dilute behavior in the colloidal regime @xmath116 is similar to that observed in the ideal case . the coefficients corresponding to the planar term and to the leading curvature correction are close , while the second - curvature correction is absent in the ideal case and quite small for good - solvent chains . in the opposite limit @xmath117 general arguments predict @xcite @xmath118 the constant @xmath119 has been estimated by hanke _ et al_. @xcite : @xmath120 eq . ( [ gamma - gamma ] ) gives then @xmath121 . for the depletion thickness we obtain @xmath122 . finite - density corrections have been computed by maassen _ _ @xcite in the renormalized tree approximation . for @xmath111 they obtain @xmath123 . \label{ft - tree - phi0}\ ] ] in this approximation one does not recover the correct large-@xmath1 behavior ( [ gamma - largeq ] ) , hence we expect it to be valid only in the colloid regime . the zero - density behavior can be compared with that given in eq . ( [ ft - phi0 ] ) , which includes the leading ( one - loop ) @xmath124 correction . differences are small , of order 5% . we expect an error of the same order for the coefficients of the density correction . the behavior of @xmath25 in the semidilute regime is expected to have universal features . if the polymer volume fraction @xmath125 is large , we expect , on general grounds , the behavior @xcite @xmath126 where @xmath71 is an exponent to be determined and @xmath127 is a coefficient , which _ a priori _ can depend on @xmath1 . however , deep in the semidilute regime , the coil radius of gyration is no longer the relevant length scale . one should rather consider the density - dependent correlation length @xmath128 @xcite , which measures the polymer mesh size . the scaling behavior ( [ gamma - largephi ] ) should be valid for @xmath129 , and in this regime @xmath9 plays no role . therefore , @xmath1 is not the relevant parameter and @xmath127 is independent of @xmath1 . to determine the exponent @xmath71 we can use the same argument which allows one to determine the scaling behavior of the osmotic pressure in the semidilute regime . for large @xmath125 we expect thermodynamic properties to depend only on the monomer density @xmath130 and not on the number @xmath36 of monomers per chain . this requirement gives @xcite @xmath131 where @xcite @xmath132 . predictions ( [ gamma - largephi ] ) and ( [ gamma - largephi - alpha ] ) can also be obtained @xcite by noting that @xmath133 can only depend on the correlation length @xmath128 deep in the semidilute regime , i.e. , when @xmath134 . then , dimensional analysis gives @xmath135 using @xmath136 @xcite , we obtain again eq . ( [ gamma - largephi ] ) with @xmath71 given by eq . ( [ gamma - largephi - alpha ] ) . eq . ( [ gamma - largephi ] ) allows us to obtain the large-@xmath125 behavior of the adsorption and of the depletion thickness . using eq . ( [ gamma - gamma ] ) and the general scaling of the osmotic pressure @xcite @xmath137 , we obtain @xmath138 equivalently , one could have observed that @xmath139 , since @xmath128 is the only relevant length scale . using @xmath136 , we obtain @xmath140 . ( [ gamma - deltas - largephi ] ) implies then eq . ( [ gamma - largephi ] ) . the large-@xmath125 behavior was determined in the renormalized tree - level approximation obtaining @xcite @xmath141 . \label{gamma - largephi - meb}\ ] ] this result is fully consistent with eq . ( [ gamma - largephi ] ) , since the @xmath1 correction appearing in eq . ( [ gamma - largephi - meb ] ) vanishes for @xmath142 . the exponent of the @xmath1-dependent correction in eq . ( [ gamma - largephi - meb ] ) can be easily interpreted . consider the ratio @xmath143 . this quantity is adimensional , hence it is a universal function of adimensional ratios of the relevant length scales in the system . deep in the semidilute regime the relevant polymer scale is the correlation length @xmath128 , hence we expect @xmath144 now we take @xmath125 large so that @xmath145 . then , we can expand @xmath146 since @xmath136 , we obtain @xmath147 which reproduces the behavior ( [ gamma - largephi - meb ] ) . ( [ helfrich - xi ] ) is the semidilute analogue of the helfrich expansion in powers of @xmath1 that holds for @xmath111 . the only difference is the expansion variable : in the semidilute region , polymer size is characterized by @xmath128 , hence one should consider @xmath148 instead of @xmath149 . quantitative predictions for the large-@xmath125 behavior of @xmath95 and @xmath78 can be derived from eq . ( [ gamma - largephi - meb ] ) , by using eq . ( [ gamma - gamma ] ) and the large-@xmath125 behavior of @xmath150 . the latter can be derived from the results of ref . @xcite , which give @xmath151 for @xmath142 . thus , we obtain @xmath152 in the protein limit , in which @xmath1 is large , beside the regime @xmath153 in which eqs . ( [ gamma - largephi ] ) , ( [ helfrich - xi ] ) and ( [ deltas - largephi ] ) hold , there is a second interesting regime in which one has both @xmath154 and @xmath155 . for @xmath1 large , these conditions are satisfied both in the dilute limit and in the semidilute region , as long as @xmath125 is not too large . under these conditions , eq . ( [ gamma - largeq ] ) holds irrespective of the polymer density . therefore , eq . ( [ gamma - kb ] ) can be rewritten as @xmath156.\ ] ] for @xmath117 , the pressure term can be neglected compared with the term proportional to @xmath157 , hence the right - hand side is density independent . this implies that the integrand that appears in left - hand side is also density independent in the density region where @xmath155 and is equal to @xmath158 . for @xmath111 , using the virial expansion ( [ virial - expansion ] ) we can write @xmath159.\ ] ] therefore , we can identify @xmath160 . moreover , @xmath161 vanishes for @xmath112 ( a similar result holds for the higher - order virial coefficients ) . by using eq . ( [ gamma - gamma ] ) and eq . ( [ gamma - largeq ] ) we also predict for the adsorption @xmath162 since @xmath163 for large @xmath125 @xcite , this relation predicts @xmath164 . note that eq . ( [ deltas - largephi - largeq ] ) holds only for @xmath165 . as @xmath125 further increases , @xmath128 decreases and one finds eventually @xmath166 . then , eq . ( [ deltas - largephi - largeq ] ) no longer holds and a crossover occurs . for @xmath153 the asymptotic behavior @xmath167 sets in . ( [ deltas - largephi - largeq ] ) can be written in a more suggestive form , by noting that @xmath168 @xcite . hence @xmath169 we recover the same scaling that occurs in the dilute regime , with @xmath128 replacing @xmath9 as relevant polymer scale . c + to conclude , let us summarize the different types of behavior of the depletion thickness in the @xmath128-@xmath1 diagram for the good - solvent case . they depend on the relative size of the three different scales that appear in the problem : the radius of gyration of the polymer , the radius of the colloid and the correlation length @xmath128 . in the colloid regime in which @xmath170 , i.e. @xmath171 , depletion shows two different behaviors , depending on the ratio @xmath172 . in the dilute regime in which the relevant scale is the radius of gyration ( domain i in fig . [ phase ] ) , @xmath78 is of order @xmath9 with a proportionality constant that can be expanded in powers of @xmath1 ( helfrich expansion ) . if instead @xmath173 ( semidilute regime , domain iii in fig . [ phase ] ) , the relevant scale is the correlation length @xmath128 . the depletion thickness is proportional to @xmath174 with a proportionality constant that admits an expansion in powers of @xmath148 . since @xmath175 for @xmath142 , the limiting behavior is independent of the colloid radius . in the protein regime in which @xmath176 , i.e. , @xmath177 , depletion shows three different behaviors . in the dilute regime ( domain ii in fig . [ phase ] ) , @xmath178 , i.e. , @xmath78 is much larger than the colloid radius but much smaller than @xmath9 . in the semidilute regime , two different behaviors occur . if @xmath179 ( domain iv ) , the role of the radius of gyration is now assumed by the correlation length and we have @xmath180 . finally , as @xmath125 increases further , one finally finds @xmath181 and one observes again @xmath182 ( domain iii ) . the surface tension @xmath25 was also computed in the prism approach @xcite , obtaining @xmath183 . \label{prism}\ ] ] such an expression does not have the correct behavior for @xmath112 or @xmath142 . in the dilute regime and for small @xmath1 , comparison with the field - theoretical results ( we shall show that they are quite accurate ) shows that it only provides a very rough approximation , differences being of order 20 - 30% . the adsorption @xmath95 was computed numerically for the planar case ( @xmath184 ) in ref . @xcite , obtaining @xmath185 this expression allows us to compute @xmath25 for @xmath184 using the expression of the compressibility factor given in ref . @xcite . in the small - density limit we obtain @xmath186 , \label{lbmh - smallphi}\ ] ] while for @xmath142 we obtain @xmath187 we can compare these expressions with the field - theory results . the leading density correction in eq . ( [ lbmh - smallphi ] ) is approximately one half of that predicted by field theory , see eq . ( [ ft - tree - phi0 ] ) , while the large-@xmath125 expression ( [ lbmh - largephi ] ) predicts a surface tension that is 17% smaller than eq . ( [ gamma - largephi - meb ] ) . finally , we mention the phenomenological expression for the depletion thickness of fleer _ et al . _ @xcite @xmath188 which should be only valid in an intermediate range of values of @xmath1 @xcite , since it does not have the correct behavior in the limits @xmath84 and @xmath112 . there are no predictions for polymers in the thermal crossover region . in this case , a new scale comes in , the dimension @xmath189 of the so - called thermal blob @xcite . on scales @xmath190 , the polymer behaves as an ideal chain , hence for @xmath191 the surface tension should coincide with that appropriate for an ideal chain . this implies that for any finite value of @xmath192 we should recover the ideal result for the surface tension , provided that @xmath1 is large enough . in particular , we predict @xmath193 for all finite values of @xmath192 and @xmath117 . in practice , ( [ gamma - largeq ] ) holds also for finite @xmath192 , with the values appropriate for the ideal chain , @xmath194 and @xmath195 . if instead @xmath196 , we expect to observe a nontrivial crossover behavior . its determination is one of the purposes of the present paper . in order to determine full - monomer properties , we consider the three - dimensional lattice domb - joyce model @xcite . we consider @xmath197 chains of @xmath36 monomers each on a finite cubic lattice of linear size @xmath198 with periodic boundary conditions . each polymer chain is modeled by a random walk @xmath199 with @xmath200 ( we take the lattice spacing as unit of length ) and @xmath201 . the hamiltonian is given by @xmath202 where @xmath203 is the kronecker delta . each configuration is weighted by @xmath204 , where @xmath205 is a free parameter that plays the role of inverse temperature . this model is similar to the standard lattice self - avoiding walk ( saw ) model , which is obtained in the limit @xmath206 . for any positive @xmath207 , this model has the same scaling limit as the saw model @xcite and thus allows us to compute the universal scaling functions that are relevant for polymer solutions under good - solvent conditions . in the absence of colloids , there is a significant advantage in using domb - joyce chains instead of saws . for saws scaling corrections that decay as @xmath208 ( @xmath209 , ref . @xcite ) are particularly strong , hence the universal , large degree - of - polymerization limit is only observed for quite large values of @xmath36 . finite - density properties are those that are mostly affected by scaling corrections , and indeed it is very difficult to determine universal thermodynamic properties of polymer solutions for @xmath210 by using lattice saws @xcite . these difficulties are overcome by using the domb - joyce model for a particular value of @xmath207 @xcite , @xmath211 . for this value of the repulsion parameter , the leading scaling corrections have a negligible amplitude @xcite , so that scaling corrections decay faster , approximately as @xmath212 . as a consequence , scaling results are obtained by using significantly shorter chains . unfortunately , in the presence of a repulsive surface , new renormalization - group operators arise , which are associated with the surface @xcite . the leading one gives rise to corrections that scale as @xmath213 @xcite , where @xmath214 is the flory exponent ( an explicit test of this prediction is presented in the supplementary material ) , hence it spoils somewhat the nice scaling behavior observed in the absence of colloids . nonetheless , the domb - joyce model is still very convenient from a computational point of view . since interactions are soft , the monte carlo dynamics for domb - joyce chains is much faster than for saws . we shall use the algorithm described in ref . @xcite , which allows one to obtain precise results for quite long chains ( @xmath215 ) deep in the semidilute regime . the domb - joyce model can also be used to derive the crossover functions that parametrize the crossover between the good - solvent and @xmath0-point regimes , at least not too close to the @xmath0 point , see refs . @xcite for a discussion . indeed , if one neglects tricritical effects , which are only relevant close to the @xmath0 point @xcite , this crossover can be parametrized by using the two - parameter model @xcite . two - parameter - model results are obtained @xcite by taking the limit @xmath216 , @xmath217 at fixed @xmath218 . the variable @xmath219 interpolates between the ideal - chain limit ( @xmath220 ) and the good - solvent limit ( @xmath221 ) . indeed , for @xmath222 the domb - joyce model is simply the random - walk model , while for any @xmath223 and @xmath217 one always obtains the good - solvent scaling behavior . the variable @xmath219 is proportional to the variable @xmath192 that is used in the context of the two - parameter model . we normalize @xmath192 as in refs . @xcite , setting @xmath224 note that the crossover can be equivalently parametrized @xcite by using the second - virial combination @xmath225 ( @xmath9 is the zero - density radius of gyration ) , which varies between the good - solvent value @xcite @xmath226 and @xmath227 at the @xmath0 point . with normalization ( [ zdef - dj ] ) we have @xmath228 for small @xmath192 @xcite . the correspondence between @xmath15 and @xmath192 in the whole crossover region is given in ref . @xcite . as discussed in ref . @xcite , the two - parameter - model results can be obtained from monte carlo simulations of the domb - joyce model by properly extrapolating the numerical results to @xmath229 . for each @xmath192 we consider several chain lengths @xmath230 . for each of them we determine the interaction parameter @xmath231 by using eq . ( [ zdef - dj ] ) , that is we set @xmath232 . simulations of chains of @xmath230 monomers are then performed setting @xmath233 . simulation results are then extrapolated to @xmath217 , taking into account that corrections are of order @xmath234 @xcite . in this paper we have performed a detailed study of the depletion for two values of @xmath192 : @xmath235 and @xmath236 , which correspond to @xcite @xmath237 and @xmath238 . they correspond to polymer solutions of intermediate quality . since @xmath239 @xcite under good - solvent conditions , we have @xmath240 and 0.54 for @xmath241 and @xmath242 , respectively . hence , for @xmath241 we are quite close to the @xmath0 point , while @xmath243 is intermediate between the good - solvent and @xmath0 regimes . in this paper we discuss depletion effects close to neutral colloids , which are modelled as hard spheres that can move everywhere in space : their centers are not constrained to belong to a lattice point . this choice is particularly convenient since it drastically reduces lattice oscillations in colloid - polymer correlation functions . such oscillations are instead present if colloids are required to sit on lattice points , as was done in ref . colloids and monomers interact by means of a simple exclusion potential . if @xmath244 and @xmath245 are the coordinates of a monomer and of a colloid , we take as interaction potential @xmath246 as we have seen in sec . [ sec2.3 ] , the low - density behavior of the surface tension or , equivalently , of the depletion thickness can be obtained by computing the virial coefficients @xmath247 and @xmath248 . we will thus report the computation of these two quantities and also of @xmath249 , which would be relevant to characterize the effective interaction between two colloids in a dilute solution of polymers . then , we shall discuss the depletion thickness @xmath78 for @xmath250 and its first density correction .

the surface tension , the adsorption , and the depletion thickness of polymers close to a single nonadsorbing colloidal sphere are computed by means of monte carlo simulations . we consider polymers under good - solvent conditions and in the thermal crossover region between good - solvent and @xmath0 behavior . in the dilute regime we consider a wide range of values of @xmath1 , from @xmath2 ( planar surface ) up to @xmath3 - 50 , while in the semidilute regime , for @xmath4 ( @xmath5 is the polymer concentration and @xmath6 is its value at overlap ) , we only consider @xmath7 and 2 . the results are compared with the available theoretical predictions , verifying the existing scaling arguments . field - theoretical results , both in the dilute and in the semidilute regime , are in good agreement with the numerical estimates for polymers under good - solvent conditions .

introduction adsorption and depletion thickness polymer model and crossover behavior dilute behavior

arxiv

inductive coupling @xcite is a conventional method to realize the near - field wireless power transfer ( wpt ) for short - range applications up to a couple of centimeters . recently , magnetic resonant coupling ( mrc ) @xcite has drawn significant interests for implementing the near - field wpt due to its high power transfer efficiency for applications requiring longer distances , say , tens of centimeters to several meters . the transmitter and the receiver in an mrc - wpt system are designed to have the same natural frequency as the system s operating frequency , thereby greatly reducing the total reactive power consumption in the system and achieving high power transfer efficiency over long distances . the mrc - wpt system with a single pair of transmitter and receiver has been extensively studied in the literature for e.g. maximizing the end - to - end power transfer efficiency or the power delivered to the receiver with a given input power constraint @xcite . however , there is limited work on analyzing the mrc - wpt system under the general setup with multiple transmitters and/or receivers . the system with two transmitters and a single receiver or a single transmitter and two receivers has been studied in @xcite , while their analytical results can not be applied for a system with more than two transmitters / receivers . furthermore , to our best knowledge , there has been no work on rigorously establishing a mathematical framework to jointly design parameters in the multi - transmitter / receiver mrc - wpt system for its performance optimization . in this paper , as shown in fig . [ fig : eleccirtuit ] , we consider a point - to - multipoint mrc - wpt system , where one transmitter connected to a stable energy source sends wireless power simultaneously to a set of distributed receivers , each of which is connected to a given load . we extend the results in @xcite to derive closed - form expressions of the transmit power drawn from the energy source and the power delivered to each load , in terms of various parameters in the system . our results reveal a near - far fairness issue in the case of multiuser wireless power transmission , similar to its counterpart in wireless communication . particularly , a receiver that is far away from the transmitter and thus has a small mutual inductance with the transmitter generally receives lower power as compared to a receiver that is close to the transmitter . we then show that the near - far issue can be optimally solved by jointly designing the receivers load resistances to control their received power levels , in contrast to the method of adjusting the transmit beamforming weights to control the received power in the far - field microwave transmission based wpt @xcite . specifically , we first study the centralized optimization problem , where a central controller at the transmitter which has the full knowledge of all receivers , including their circuit parameters and load requirements , jointly designs the adjustable load resistances to minimize the total power consumed at the transmitter subject to the given minimum harvested power requirement of each load . although the formulated problem is non - convex , we develop an efficient algorithm to solve it optimally . then , for ease of practical implementation , we consider the scenario without any central controller and devise a distributed algorithm for adjusting the load resistances by individual receivers in an iterative manner . in the distributed algorithm , each receiver sets its load resistance independently based on its local information and a one - bit feedback shared by each of the other receivers , where the feedback of each receiver indicates whether the harvested power of its load exceeds the required level or not . finally , through simulation results , it is shown that the distributed algorithm can achieve close - to - optimal performance as compared to the solution of the centralized optimization . we consider an mrc - wpt system with one transmitter and @xmath0 receivers , indexed by @xmath1 , @xmath2 , as shown in fig . [ fig : eleccirtuit ] . the transmitter and receivers are equipped with electromagnetic ( em ) coils for wireless power transfer . an embedded communication system is also assumed to enable information sharing among the transmitter and/or receivers . the transmitter is connected to a stable energy source supplying sinusoidal voltage over time given by @xmath3 , with @xmath4 denoting a complex voltage which is assumed to be constant , and @xmath5 denoting the operating angular frequency of the system . each receiver @xmath1 is also connected to a given load ( e.g. a battery charger ) , named load @xmath1 , with resistance @xmath6 . it is assumed that the transmitter and each receiver @xmath1 are compensated by series capacitors with capacities @xmath7 and @xmath8 , respectively . let @xmath9 , with complex @xmath10 , denote the steady state current flowing through the transmitter . this current produces a time - varying magnetic flux in the transmitter s em coil , which passes through the receivers em coils and induces time - varying currents in them . we thus denote @xmath11 , with complex @xmath12 , as the steady state current at receiver @xmath1 . we denote @xmath13 ( @xmath14 ) and @xmath15 ( @xmath16 ) as the internal resistance and the self - inductance of the em coil of the transmitter ( receiver @xmath1 ) , respectively . we also denote the mutual inductance between em coils of the transmitter and each receiver @xmath1 by @xmath17 , with @xmath18 , where its actual value depends on the physical characteristics of the two em coils , their locations , alignment or misalignment of their oriented axes with respect to each other , the environment magnetic permeability , etc . for example , the mutual inductance of two coaxial circular loops that lie in the parallel planes with separating distance of @xmath19 meter is approximately proportional to @xmath20 @xcite . moreover , since the receivers usually employ smaller em coils than that of the transmitter due to size limitations and they are also physically separated , we can safely ignore the mutual inductance between any pair of them . the equivalent electric circuit model of the considered mrc - wpt system is shown in fig . [ fig : eleccirtuit ] , [ t ] in which the natural angular frequencies of the transmitter and each receiver @xmath1 are given by @xmath21 and @xmath22 , respectively . we set @xmath23 and @xmath24 , @xmath25 , to ensure that the transmitter and all receivers have the same natural frequency as the system s operating frequency @xmath26 , named _ resonant angular frequency _ , i.e. , @xmath27 . we assume that the transmitter and all receivers are at fixed positions and the physical characteristics of their em coils are known ; thus , @xmath28 , @xmath29 , are modeled as given constants . we treat the receivers load resistances @xmath30 , @xmath25 , as design parameters , which can be adjusted in real - time @xcite to control the performance of the mrc - wpt system based on the information shared among different nodes in the system via wireless communication . in this section , we first present our analytical results . a numerical example is then provided to draw useful insights from the analysis . define @xmath31^t$ ] and @xmath32 $ ] , where @xmath33 is the voltage vector and @xmath34 is the current vector that can be obtained as a function of @xmath33 . let @xmath35 , @xmath36 , and @xmath37^t$ ] . by applying kirchhoff s circuit laws to the electric circuit model in fig . [ fig : eleccirtuit ] , we obtain @xmath38^{-1 } \hspace{-1.5 mm } { \mbox{\boldmath{$ v $ } } } = { \mbox{\boldmath{$ a $ } } } ^{-1 } { \mbox{\boldmath{$ v $ } } } , \end{aligned}\ ] ] where @xmath39 is called the _ impedance matrix_. the determinant of @xmath40 is given by @xmath41 where it can be easily verified that @xmath42 . then , we define @xmath43 , which is called the _ admittance matrix_. let @xmath44 denote the element in row @xmath45 and column @xmath46 of @xmath47 . we simplify ( [ eq : v = zi ] ) as @xmath48^{t}v_{\text{tx}}.\end{aligned}\ ] ] it can also be shown that @xmath49 , @xmath50 , is given by @xmath51 by substituting ( [ eq : b_mn ] ) into ( [ eq : simplified_v = zi ] ) , it follows that @xmath52 the power drawn from the energy source , denoted by @xmath53 , and that delivered to each load @xmath1 , denoted by @xmath54 , are then obtained as @xmath55 where @xmath56 is the conjugate of @xmath57 . from ( [ eq : pl ] ) , it follows that the power delivered to each load @xmath1 increases with the mutual inductance between em coils of its receiver and the transmitter , i.e. , @xmath58 . this can potentially cause a near - far fairness issue since a receiver that is far away from the transmitter in general has a small mutual inductance with the transmitter ; thus , its received power is lower than a receiver that is close to the transmitter ( with a larger mutual inductance ) . we accordingly define @xmath59 as the sum ( aggregate ) power delivered to all loads , where we always have @xmath60 . in the following , we study impacts of changing the load resistance of one particular receiver @xmath1 , i.e. , @xmath61 , on the transmitter power @xmath53 , its received power @xmath62 and that delivered to each of the other loads @xmath63 , @xmath64 , i.e. , @xmath65 , as well as the sum power delivered to all loads @xmath66 , assuming that all other load resistances are fixed . [ prop:3 ] @xmath67 strictly increases over @xmath68 . this result can be explained as follows . from ( [ eq : it1 m ] ) , it is observed that the transmitter current @xmath57 strictly increases over @xmath68 . hence , due to the fact that the energy source voltage @xmath69 is fixed , it follows that @xmath53 given in ( [ eq : pt ] ) strictly increases over @xmath68 . [ prop:4 ] @xmath65 , @xmath70 , strictly increases over @xmath71 . however , @xmath62 first increases over @xmath72 , and then decreases over @xmath73 , where @xmath74 with @xmath75 . the above result can be justified as follows . from ( [ eq : ilm ] ) , it follows that for each receiver @xmath76 , @xmath77 , its current @xmath78 strictly increases over @xmath68 . this is because @xmath57 increases with @xmath61 , and as a result , @xmath78 increases due to the mutual coupling between em coils of receiver @xmath76 and the transmitter . hence , the received power @xmath65 defined in ( [ eq : pl ] ) also strictly increases over @xmath68 . on the other hand , it follows from ( [ eq : ilm ] ) that for receiver @xmath1 , its current @xmath79 strictly decreases over @xmath68 . moreover , from ( [ eq : pl ] ) , it follows that the decrement in @xmath80 is smaller than the increment of @xmath61 when @xmath81 ; thus , @xmath62 increases with @xmath61 in this region , while the opposite is true when @xmath82 . [ prop:5 ] if @xmath83 , @xmath84 strictly increases over @xmath68 , where @xmath85 ; otherwise , @xmath84 first increases over @xmath86 , and then decreases over @xmath87 , where @xmath88 this property is a direct consequence of property 2 . we consider an mrc - wpt system with @xmath89 receivers , where @xmath90v , @xmath91 , @xmath92h , @xmath93 , @xmath94h , @xmath25 , @xmath95 \mu$]h , and @xmath96rad / s . in this example , receiver @xmath97 is closest to the transmitter and thus it has the largest mutual inductance , while receiver @xmath98 is farthest . for the purpose of exposition , we fix @xmath99 . we plot @xmath53 , @xmath54 , @xmath25 , and @xmath66 , versus the resistance of load @xmath97 , @xmath100 , in fig . [ fig : eff - pl - versus - rl1 ] . .,width=472 ] it is observed that @xmath53 , @xmath101 , @xmath102 and @xmath66 all increase over @xmath103 . note that in this example , the condition @xmath83 holds in property 3 . however , @xmath104 first increases over @xmath105 , and then declines over @xmath106 . these results are consistent with our above analysis . finally , we point out that changing @xmath100 not only affects @xmath107 , but also the power delivered to other loads . for instance , receiver @xmath97 can help receivers @xmath108 and @xmath98 , which are farther away from the transmitter , to receive higher power by increasing @xmath109 . this is a useful mechanism that will be utilized to solve the near - far issue . in this section , we optimize the receivers load resistances @xmath30 , @xmath25 , to minimize the power drawn from the energy source at the transmitter subject to the given load constraints . we assume a central controller at the transmitter , which has full knowledge of the receivers , including their circuit parameters and load requirements , to implement the proposed centralized optimization . we assume that the resistance of each load @xmath1 can be adjusted over a given range @xmath110 , where @xmath111 and @xmath112 are lower and upper limits of @xmath61 due to practical considerations . we also assume that the power delivered to each load @xmath1 should be higher than a certain power threshold @xmath113 . hence , we formulate the following optimization problem to minimize the power drawn from the energy source at the transmitter . @xmath114 ( p1 ) is a non - convex optimization problem . however , in the next we propose an efficient algorithm to solve ( p1 ) optimally . we define an auxiliary variable @xmath115 . since @xmath116 , @xmath25 , we have @xmath117 , where @xmath118 and @xmath119 . then , we rewrite ( p1 ) as @xmath120 although ( p2 ) is still non - convex , we can solve it in an iterative manner by searching for the smallest @xmath121 , @xmath117 , under which ( p2 ) is feasible . staring from @xmath122 , we test the feasibility of ( p2 ) given @xmath121 by considering the following problem . @xmath123 if ( p3 ) is feasible , then we set the optimal objective value of ( p2 ) as @xmath121 , which can be attained by any feasible solutions to ( p3 ) . otherwise , we set @xmath124 , where @xmath125 is a small step size . we repeat the above procedure until ( p3 ) becomes feasible or @xmath126 . the following proposition summarizes the feasibility conditions for ( p3 ) . [ prop:4 - 1 ] given @xmath121 , with @xmath117 , ( p3 ) is feasible if and only if all conditions listed below hold at the same time : * @xmath127 , @xmath25 , where @xmath128 . * @xmath129 and/or @xmath130 , @xmath25 , where @xmath131 and @xmath132 . * @xmath133 , where @xmath134 , and @xmath135 . given any @xmath136 , where @xmath137 is given in c3 of proposition [ prop:4 - 1 ] , the corresponding feasible solution to ( p3 ) is obtained by a change of variable as @xmath138 , @xmath25 . note that the obtained @xmath139 solves ( p1 ) optimally . to summarize , the algorithm to solve ( p1 ) is given in table 1 , denoted by algorithm 1 . in this section , we present a distributed algorithm for ( p1 ) , where it is suitable for practical implementation when a central controller is not available in the system . in this algorithm , each receiver adjusts its load resistance independently according to its local information and a one - bit feedback from each of the other receivers indicating whether the corresponding load constraint is satisfied or not . we denote the feedback from each receiver @xmath1 which is broadcast to all other receivers as @xmath140 , where @xmath141 ( @xmath142 ) indicates that its load constraint is ( not ) satisfied . in section [ sec : performance ] , we show that the power delivered to each load @xmath1 , i.e. , @xmath62 , has two properties that can be exploited to adjust @xmath61 . first , @xmath62 strictly increases over @xmath143 , @xmath70 , which means that other receivers can help boost @xmath62 by increasing their load resistances . second , @xmath62 has a single peak at @xmath144 , assuming that other load resistances are all fixed . thus , over @xmath145 , receiver @xmath1 can increase @xmath62 by increasing @xmath61 ; similarly , for @xmath146 , it can increase @xmath62 by reducing @xmath61 . although receiver @xmath1 can not compute @xmath147 from ( [ eq : x_star ] ) directly due to its incomplete information on other receivers , it can test whether @xmath148 , @xmath144 , or @xmath146 as follows . let @xmath149 , @xmath150 , and @xmath151 be the power received by load @xmath1 when its resistance is set as @xmath152 , @xmath61 , and @xmath153 , respectively , where @xmath154 is a small step size . assuming all the other load resistances are fixed , receiver @xmath1 can make the following decision : + @xmath155 if @xmath156 and @xmath157 , then @xmath145 ; + @xmath155 if @xmath158 and @xmath157 , then @xmath159 ; and @xmath157 , then @xmath160 . ] + @xmath155 if @xmath158 and @xmath161 , then @xmath146 . next , we present the distributed algorithm in detail . the algorithm is implemented in an iterative manner , say , starting from receiver 1 , where in each iteration , only one receiver @xmath1 adjusts its load resistance , while all the other receivers just broadcast their individual one - bit feedback @xmath162 , @xmath64 , at the beginning of each iteration . initialize by randomized @xmath163 $ ] , @xmath25 . at each iteration for receiver @xmath1 , if @xmath164 , then it will adjust @xmath61 to increase @xmath62 . to find the correct direction for the update , it needs to check for its current @xmath61 whether @xmath148 , @xmath144 , or @xmath165 holds , using the method mentioned in the above . on the other hand , if @xmath166 , receiver @xmath1 can increase @xmath61 to help increase the power delivered to other loads when there exists any @xmath64 such that @xmath167 is received ; or it can decrease @xmath61 to help reduce the transmitter power when @xmath168 , @xmath70 . in summary , we design the following protocol ( with five cases ) for receiver @xmath1 to update @xmath61 . + c1 : if @xmath169 and @xmath148 , set @xmath170 . + c2 : if @xmath169 and @xmath165 , set @xmath171 . + c3 : if @xmath172 , @xmath173 , and @xmath174 , @xmath167 , set @xmath175 . + c4 : if @xmath172 , @xmath173 , and @xmath168 , @xmath70 , set @xmath176 . + c5 : otherwise , no update occurs . in addition , we assume that there is a maximum number of iterations , denoted by @xmath177 , after which the algorithm will terminate , regardless of whether it converges to a stable point @xmath139 or not . however , when the algorithm converges / terminates , the power constraints given in ( [ problem8:const1 ] ) may or may not hold for all loads , depending on the initial values of @xmath61 s . if constraint ( [ problem8:const1 ] ) holds for all loads , then the obtained @xmath139 is a suboptimal solution to ( p1 ) ; otherwise , it is infeasible for ( p1 ) . the distributed algorithm is summarized in table 2 , as algorithm 2 . we consider the same system setup as that in section [ subsec : numericalexample - performnce ] . we set @xmath178 and @xmath179 , @xmath29 . we also set @xmath180w , @xmath181w , and @xmath182 varying as @xmath183w . note that ( p1 ) is feasible under the above setting . for algorithm 1 , we use @xmath184 . for algorithm 2 , we use @xmath185 and @xmath186 , which is sufficiently large such that the algorithm converges to a stable point , while there is no guarantee that the power constraints given in ( [ problem8:const1 ] ) hold for all loads at this point . therefore , to evaluate the performance of algorithm 2 , we averaged its result over 200 randomly generated initial points for each of which the algorithm converged to a feasible solution to ( p1 ) . in fig . [ fig : simul - alg ] , we plot @xmath53 versus @xmath102 . it is observed that @xmath53 obtained by algorithm @xmath97 is lower than that by algorithm @xmath108 , while the gap is quite small , for all values of @xmath182 . this is expected since algorithm @xmath97 solves ( p1 ) optimally , while algorithm 2 in general only returns a suboptimal solution . in this paper , we study a point - to - multipoint mrc - wpt system with distributed receivers . we derive closed - form expressions for the input and output power in terms of the system parameters . similar to other multiuser wireless applications such as those in wireless communication and far - field microwave based wpt , a near - far fairness issue is revealed in our considered system . to tackle this problem , we propose a centralized algorithm for jointly optimizing the receivers load resistances to minimize the transmitter power subject to the given load constraints . for ease of practical implementation , we also devise a distributed algorithm for receivers to iteratively adjust their load resistances based on local information and one - bit feedback from each of the other receivers . we show by simulation that the distributed algorithm performs sufficiently close to the centralized algorithm with a finite number of iterations . as a concluding remark , mrc - wpt is a promising research area for which many tools from signal processing and optimization can be applied to devise new solutions , and we hope that this paper will open up an avenue for future work along this direction . j. murakami , f. sato , t. watanabe , h. matsuki , s. kikuchi , k. harakawa , and t. satoh , `` consideration on cordless power station - contactless power transmission system , '' _ ieee trans . 5037 - 5039 , sep . 1996 . v. j. brusamarello , y. b. blauth , r. azambuja , i. muller , and f. sousa , `` power transfer with an inductive link and wireless tuning , '' _ ieee trans . 924 - 931 , may 2013 . a. kurs , a. karalis , r. moffatt , j. d. joannopoulos , p. fisher , and m. soljacic , `` wireless power transfer via strongly coupled magnetic resonances , '' _ science _ , 83 - 86 , july 2007 . j. shin , s. shin , y. kim , s. ahn , s. lee , g. jung , s. jeon , and d. cho , `` design and implementation of shaped magnetic - resonance - based wireless power transfer system for roadway - powered moving electric vehicles , '' _ ieee trans . 1179 - 1192 , mar . 2014 . b. l. cannon , j. f. hoburg , d. stancil , and s. goldstein , `` magnetic resonant coupling as a potential means for wireless power transfer to multiple small receivers , '' _ ieee trans . power electron . 24 , no . 7 , pp . 1819 - 1825 , july 2009 . o. jonah , s. georgkopoulos , and m. tentzeris , `` optimal design parameters for wireless power transfer by resonance magnetic , '' _ ieee antennas wireless propagat . 1390 - 1393 , nov . 2012 . r. johari , j. krogmeier , and d. love , `` analysis and practical considerations in implementing multiple transmitters for wireless power transfer via coupled magnetic resonance , '' _ ieee trans . 1174 - 1783 , apr . 2014

magnetic resonant coupling ( mrc ) is a practically appealing method for realizing the near - field wireless power transfer ( wpt ) . the mrc - wpt system with a single pair of transmitter and receiver has been extensively studied in the literature , while there is limited work on the general setup with multiple transmitters and/or receivers . in this paper , we consider a point - to - multipoint mrc - wpt system with one transmitter sending power wirelessly to a set of distributed receivers simultaneously . we derive the power delivered to the load of each receiver in closed - form expression , and reveal a `` near - far '' fairness issue in multiuser power transmission due to users distance - dependent mutual inductances with the transmitter . we also show that by designing the receivers load resistances , the near - far issue can be optimally solved . specifically , we propose a centralized algorithm to jointly optimize the load resistances to minimize the power drawn from the energy source at the transmitter under given power requirements for the loads . we also devise a distributed algorithm for the receivers to adjust their load resistances iteratively , for ease of practical implementation . wireless power transfer , magnetic resonant coupling , multiuser charging control , optimization , iterative algorithm . [ section ] [ section ] [ section ] [ section ] [ section ] [ section ] [ section ]

introduction system model performance analysis centralized optimization distributed algorithm simulation results conclusion

arxiv

ultracold atoms offer the possibility to study few- and many - body quantum systems with exquisite control over microscopic interactions . this has led to spectacular experiments @xcite , bridging different areas of physics . while the research directions stimulated by experiments with ultracold atoms are very diverse , one can identify several overarching trends that gained momentum in recent years . the most prominent of these trends is the effort aimed at understanding quantum phase transitions @xcite , leading to the experimental studies of bose einstein condensation ( bec ) @xcite , bosonic superfluidity @xcite , quantum magnetism @xcite , many - body spin dynamics @xcite , efimov states @xcite , bardeen schrieffer ( bcs ) superfluidity @xcite and the bec bcs crossover @xcite . with the development of techniques for trapping atoms in periodic potentials of optical lattices @xcite and single atom detection @xcite , ultracold atoms became an ideal platform for quantum simulation of lattice models used in solid state physics @xcite . as demonstrated by the realization of the mott insulator superfluid transition with atoms trapped in a three - dimensional optical lattice @xcite , the experiments with ultracold atoms hold the promise of insight into the details of the hubbard model , which may help unravel the mechanism of high-@xmath0 superconductivity @xcite . the development of experimental methods for the production of ultracold polar molecules @xcite has widened the possibilities for quantum simulation of condensed matter models to a great extent . a combination of the rotational , spin and hyperfine degrees of freedom with the long - range dipolar interactions enabled by the dipole moment of molecules allows for engineering a great variety of lattice models that can not be realized with atoms @xcite . of particular interest is the possibility of creating quantum phases with topological order @xcite , which are resilient to perturbations preserving the topology and are , therefore , ideal for quantum computation . to this end , a major effort of current experiments is focused on preparing a dense ensemble of ultracold polar molecules trapped in optical lattices @xcite . another focus of research with quantum degenerate gases is on emergent phenomena , such as solitons @xcite , rotons @xcite , vortices @xcite , spin waves @xcite and polarons @xcite . these experiments aim to elucidate emergence in natural systems , and may potentially lead to the development of novel ultra - sensitive sensors of gravity and electromagnetic fields . while these studies cover a wide range of collective dynamics of ultracold atoms and molecules in a single quantum state , much less is known about the effect of internal degrees of freedom of ultracold particles . for example , the role of rotational transitions in the excitation spectrum of a molecular bose einstein condensate had not been addressed until very recently @xcite . yet , the internal degrees of freedom can be used to explore new regimes of collective phenomena , especially in molecular systems that provide a dense spectrum of internal excitations @xcite . in this article , we consider rotational excitations of polar molecules trapped on an optical lattice . these excitations give rise to rotational excitons analogous to collective electronic excitations in molecular crystals . in contrast to excitons in natural solids , the properties of rotational excitons can be dynamically controlled by tuning the energy level structure of the trapped molecules , which can be used to study new regimes of frenkel exciton physics not accessible in natural solid state crystals . for example , we show that non - linear interactions of excitons can be tuned to examine the competition between the dynamical and kinematic effects and demonstrate that rotational excitons can be used to study quantum localization in a dynamically tunable disordered potential . the rotational excitons can also be used as a basis for quantum simulation of condensed - matter models that can not be realized with ultracold atoms . as an example , we discuss the possibility of engineering the holstein model with polar molecules on an optical lattice . in order to present these results in the context of current research , we briefly describe the related work on quantum simulation of many - body hamiltonians with ultracold atoms and molecules and the basics of frenkel exciton physics . the difficulty of simulating a quantum many - body system on a classical computer increases exponentially with the number of quantum states . an alternative , currently at the focus of detailed research , is quantum simulation @xcite . quantum simulation involves the design of a controllable quantum system in order to simulate the properties of another , more complicated system ( analog quantum simulator ) . this idea is generally attributed to feynman @xcite , although quantum simulation is already mentioned in an earlier publication by manin @xcite . the key ingredient of quantum simulation is the mapping of the hamiltonian of the simulated system onto the hamiltonian of the simulator with controllable parameters . tuning the parameters of the controlled system can then be used to map out the phase diagram of the simulated system . the recent literature offers many ideas about how to build quantum simulators based on ultracold quantum gases @xcite , ultracold trapped ions @xcite , single photon sources and detectors @xcite , ultracold rydberg atoms @xcite , circuit qed @xcite , quantum dots and n v centers in diamond @xcite . ultracold atoms trapped in a periodic potential of overlapped laser beams ( optical lattices ) is considered as one of the most promising systems for quantum simulation of lattice models @xcite . the geometry and strength of the optical lattice potentials can be controlled by adjusting the laser intensity and the beam overlap angle , which enables control over the translational motion of atoms in the lattice @xcite . the separation of the lattice sites in an optical lattice is equal to half the wavelength of the trapping laser field . recent technological breakthroughs permit the detection of ultracold atoms with sub - wavelength resolution , allowing for single - site addressing @xcite . the most prominent example of quantum simulation with ultracold atoms is the realization of the hubbard model @xcite @xmath1 where @xmath2 or @xmath3 , @xmath4 and @xmath5 are the creation and annihilation operators for fermions , satisfying the anti - commutation rule @xmath6 , @xmath7 is the particle number operator , and @xmath8 is the energy of particles in state @xmath9 . the angle brackets @xmath10 indicate that the summation is over nearest neighbors only . in this simple form , the hubbard hamiltonian accounts for nearest neighbor tunnelling ( @xmath11 ) and on - site interactions for fermions in different spin states ( @xmath12 ) . the fermions with the same spin experience hard - core repulsion . this model is widely used for the studies of high-@xmath0 superconductivity @xcite and quantum magnetism @xcite . with bosonic atoms , it is possible to realize the bose - hubbard model : @xmath13 where @xmath14 and @xmath15 are the creation and annihilation operators for bosons , satisfying the commutation rule @xmath16 = \delta_{nm}$ ] , and @xmath17 is the particle number operator . this model has been used in many studies of the properties of bosonic gases such as superfluidity @xcite . if @xmath18 , the particles are delocalized around the lattice and the ground state of the system is a superfluid . this hamiltonian also allows for modelling the properties of fermionic systems when quantum statistics plays no role and the properties of the system are entirely determined by the relative efficiency of hopping ( governed by @xmath11 ) and the interaction ( @xmath12 ) . in particular , when @xmath19 ( the on - site repulsion dominates over the inter - site hopping ) , this bosonic hamiltonian reproduces transition to the mott insulator phase @xcite , where the number of atoms per lattice site is fixed . this transition is typical for electrons in a metal . it is thus possible to realize the same physical phenomenon with particles satisfying different quantum statistics . we shall discuss this in detail in section [ s - s - mappings ] . recent experiments demonstrated the possibility of creating the mott insulator phase with ultracold atoms filling up to 99% of the lattice sites @xcite . a major thrust of current research is to extend these experiments to ultracold molecules . the experiments with polar molecules in optical lattices offer the possibility of realizing lattice models with long - range interactions @xcite . in the bose - hubbard model the inter - site interactions are accounted for by adding to the hamiltonian ( [ h_bh ] ) a term @xmath20 describing density density correlations . in the limit of the strong on - site repulsion , when @xmath21 or 1 , the system reduces to the @xmath11-@xmath22 model , described by the hamiltonian @xmath23 with the constraint @xmath24 , which accounts for the infinite on - site repulsion . with @xmath25 this hamiltonian corresponds to the lattice analog of the tonks - girardeau gas @xcite . this model is equivalent to an anisotropic spin-@xmath26 _ xxz _ model @xcite . in addition , molecules allow for the possibility of realizing a variety of lattice spin models @xcite , such as @xmath27 where @xmath28 are the pauli matrices for lattice site @xmath29 . this is the so - called heisenberg model . if the interactions are isotropic in the plane perpendicular to the @xmath30 axis , i.e. for @xmath31 , this hamiltonian reduces to the _ xxz_-model @xcite . in the isotropic heisenberg model , @xmath32 . the _ xy_-model is for the case when @xmath33 . the scalar version of the heisenberg model ( @xmath34 ) is the ising model . in the ising model , spins are treated as scalars , which can take one of two values : @xmath35 . the hamiltonian becomes @xmath36 the ising model can be realized with ultracold molecules in the spin - less @xmath37 electronic state trapped on an optical lattice in a mott insulator phase with one molecule per lattice site . the rotational levels of @xmath37 molecules in an external electric field form an isolated two - level system , illustrated in figure [ f - exciton dispersion]a . the ground @xmath38 and excited @xmath39 states can be used as the spin states @xmath40 . the coupling constant @xmath41 is determined by the dipole - dipole interaction between molecules in different lattice sites . the rotational excitation @xmath42 thus leads to spin waves of the ising model . in order to simulate more complicated spin models , such as eq . ( [ h heisenberg ] ) , it is necessary to use molecules with more complex structure @xcite . the hamiltonians presented in section [ s - s - lattice models ] describe very different physical systems , consisting of particles with all possible statistics ( bosons , fermions , spin matrices and pseudospins ) . they are , however , mutually related . some of these hamiltonians can be mapped onto each other , even when the quantum statistics of mapping and mapped particles are different . consider an ensemble of arbitrary two - level systems on a lattice . the state of a two - level system in lattice site @xmath29 can be characterized by the operators @xmath43 and @xmath44 describing , respectively , the creation and destruction of the excited state . operators for different lattice sites necessarily commute , as they act on different variables . on the other hand , the same site can not accommodate more than one excitation , which is a feature of fermi statistics . the commutation relations for the operators @xmath44 are @xmath45 the operators with such `` mixed '' statistics are called paulions ( in condensed matter physics @xcite ) or hard - core bosons ( in atomic physics @xcite ) . using the @xmath46-operators , we can write the following general hamiltonian @xmath47 which is identical to the hard - core hamiltonian ( [ h t - v ] ) with the constraint ( [ h t - v constraint ] ) absorbed into the statistical properties of paulions ( [ paulion commutations ] ) . the prime over the sum symbols indicates that @xmath48 . for @xmath25 , i.e. when the density - density correlations are absent , the hamiltonian ( [ paulionic hamiltonian ] ) is identical to the ising hamiltonian ( [ h ising ] ) with @xmath49 . this hamiltonian is also used to model frenkel excitons in solid state molecular crystals @xcite . any unitary transformation preserves the commutation properties of the bosonic and fermionic operators . for example , the fourier transform of bosons must produce bosons . therefore , many articles have been devoted to mapping paulions onto `` effective '' particles with bosonic or fermionic statistics , @xmath50 and @xmath51 . they can be mapped onto fermionic operators by the jordan wigner transformation @xcite , and onto bosonic operators by the holstein primakoff transformation @xcite . ] . girardeau showed @xcite that in 1d the many - body wave function of hard - core bosons corresponds ( up to a sign ) to a many - body wave function of a gas of fictitious non - interacting spin - less fermions . the exact mapping was proposed later by chestnut and suna for a 1d system in the nearest neighbor approximation @xcite . this is a variant of the jordan wigner transformation @xcite , and it works very well in 1d . however , in higher dimensions it may cause problems : the effective fermionic operators are non - local , i.e. the expression for the fermionic operators in site @xmath29 is dependent on the occupation numbers at other sites . that is why this transformation is not as effective in dimensions higher than one . another possibility is to use the agranovich toshich transformation @xcite , which expresses @xmath52-operators through an infinite series of bosonic operators @xmath53 and @xmath54 , in a way ensuring that for any number of bosons per lattice site @xmath55 the eigenvalues of the number operator for paulions @xmath56 are only 1 or 0 , so that the unphysical states with @xmath57 do not occur . then the paulionic hamiltonian ( [ paulionic hamiltonian ] ) reduces to a sum of terms consisting of the same number of creation and annihilation bosonic operators , which we schematically denote by @xmath58 ; @xmath59 . keeping the pairwise interactions only and assuming that @xmath60 , one obtains : @xmath61 where @xmath62 thus , the agranovich toshich transformation reduces the pauilonic ( hard - core boson ) hamiltonian to a hamiltonian describing a gas of bosons interacting via delta - like pairwise interaction with the strength @xmath63 . higher - order corrections to the interaction energy can be obtained by including the omitted terms . the hamiltonian @xmath64 describes the kinematic interaction , which we discuss in section [ s - s - s - kinematic ] . in contrast to the wigner jordan transformation , this approach is very effective in 3d , where @xmath65 can be treated as a perturbation . for 2d systems , the delta - like scattering with the magnitude @xmath63 , which is the largest energy scale of the problem , is very strong and does not allow for a perturbative treatment @xcite . in 1d , the effect of scattering is even stronger and the transformation to bosons makes no sense at all . to summarize , in 1d paulions are well described by a gas of fermions , in 3d by bosons , in 2d they are something in between . finally , we note that in principle it is not necessary to use any of these transformations ; one can work directly with paulions taking into account their commutation relations ( [ paulion commutations ] ) . while the majority of research with atoms and molecules on optical lattices has so far focused on quantum transport of particles in a lattice potential or quantum simulation of lattice models , such as the ones described above , the ability to control the structure of molecules trapped on an optical lattice can also be exploited to study quantum energy transport and collective excitations reminiscent of excitons in solid state crystals . in this section , we discuss the formation of rotational excitons . these excitons have unique properties and can , in turn , be used as a basis for quantum simulation of new physical phenomena and new lattice models that can not be realized with atoms on an optical lattice . this is discussed in sections [ s - s - nonlinear ] [ s - s - s - holstein polaron ] . rotational excitation of polar molecules trapped on an optical lattice in the mott insulator phase with one molecule per site gives rise to the formation of rotational frenkel excitons @xcite . the frenkel exciton is a charge - less quasiparticle , which describes the excitation transfer in molecular crystals @xcite . here , we consider the transition between the absolute ground state @xmath38 of trapped polar molecules and their first rotational excited state . we assume that the molecules reside in the ground vibrational state of the ground electronic state @xmath66 . in the presence of a dc electric field @xmath67 , the three - fold degeneracy of the rotationally excited state @xmath68 is lifted and the state with the projection @xmath69 of the total angular momentum is detuned from the states with @xmath70 ( figure [ f - exciton dispersion]a ) . we assume that the detuning is large enough so that the latter can be disregarded and consider the isolated two - level system of the state @xmath71 and the rotational excited state with @xmath72 , denoted by @xmath73 . the molecular states are the eigenstates of the hamiltonian @xmath74 where @xmath75 is the rotational constant , @xmath76 is the operator of the angular momentum , and @xmath77 is the dipole moment of the molecule in site @xmath29 . the field - dressed states are linear combinations of the field - free rotational states @xmath78 the coefficients @xmath79 and @xmath80 are determined by the electric field strength @xmath67 . in the limit @xmath81 , the states @xmath38 and @xmath73 become , respectively , @xmath82 and @xmath83 . it is convenient to introduce the _ transition _ operators @xmath84 defined by the equations @xmath85 and @xmath86 . as discussed in section [ s - s - mappings ] , these operators describe hard - core bosons , or paulions , and the excitonic hamiltonian describes a hard - core boson gas with long - range interactions , which is equivalent to the ising model . the rotational excitons can thus be mapped onto other systems described by the _ t v _ model , or the ising model . using the @xmath87-operators , the total hamiltonian for @xmath88 molecules on an optical lattice can be written as @xcite @xmath89 where the first term accounts for the excitation transfer between lattice sites @xmath90 with the constant @xmath91 determined by the matrix elements of the inter - molecular excitation transfer due to dipole - dipole interaction @xmath92 the second term describes non - linear interactions between excitons : @xmath93 with the interaction constant @xmath94 determined by @xmath95 finally , @xmath96 contains terms that do not conserve the number of molecular excitations in the system : @xmath97 \bigl ( \hat p_n^\dag \hat p_{n } \hat p_{m } + \hat p_{n}^\dag \hat p_{m}^\dag \hat p_n \bigr ) . \hspace{2.cm } \label{h - n - c}\end{aligned}\ ] ] if @xmath38 and @xmath73 are states of well - defined parity ( such as , for example , the rotational states at zero electric field ) , the matrix elements @xmath98 as well as the linear and cubic terms in eq . ( [ h - n - c ] ) must vanish . this is the case for molecular solids , such as anthracene and naphtalene , which are often considered as prototype systems for ( electronic ) frenkel excitons . and @xmath99 , which do not conserve the number of particles , do not vanish in natural molecular crystals with the inversion symmetry . the usual model neglecting these terms is known as the heitler london approximation @xcite . these terms can be accounted for as corrections , whose magnitude depends on the ratio between the dipole - dipole interaction matrix element @xmath100 and the excitation energy of the molecules @xmath101 , as well as on the structure of the molecules @xcite . ] for an ensemble of molecules in an optical lattice , the magnitude of these terms can be tuned by applying an external electric field , which breaks the inversion symmetry ( parity ) of the molecules . to our knowledge , the role of the odd-@xmath102 terms in eq . ( [ h - n - c ] ) has not been explicitly studied and remains an interesting open problem . the possibility to tune these terms by varying the field dressing of the states @xmath103 and @xmath39 may lead to new interesting phenomena . in what follows , we assume that the effect of @xmath104 is small and neglect this term . due to the translational invariance of the optical lattice , the hamiltonian @xmath105 can be diagonalized by the transformation : @xmath106 to yield @xmath107 where @xmath108 and @xmath109 describe the bloch plane waves , @xmath110 is the ( linear ) momentum , and @xmath111 is the exciton energy forming a quasi - continuous band . for @xmath112 , we can consider the wave vector @xmath110 and the energy @xmath113 to be continuous variables . in the nearest neighbor approximation , @xmath114 and @xmath115 where @xmath116 is the lattice constant . the energies are shown in figure [ f - exciton dispersion ] for different angles @xmath117 between the one - dimensional molecular ensemble and the electric field @xmath67 ( which determines the sign of @xmath118 and , consequently , the shape of the dispersion curve ) . in the nearest neighbor approximation , @xmath119 is the exciton bandwidth . the magnitude of @xmath118 is a quantitative measure of the collective excitation effects ( in particular , @xmath120 is the timescale of the excitation transfer between molecules in adjacent lattice sites ) . note that @xmath121 at @xmath122 : at this angle the dipole - dipole interaction vanishes , and ( in the dipole approximation ) the molecules become decoupled . when this happens , no excitation transfer dynamics can occur . for polar molecules with permanent dipole moments of a few debye , the values of @xmath118 can be tens of khz ( for lics on an optical lattice with @xmath123 nm , @xmath124 khz ) . for small @xmath110 , where the dispersion is parabolic , we can introduce the effective mass @xmath125 . for @xmath126 ( @xmath127 ) the constant @xmath118 is positive ( negative ) , and , consequently , the effective mass is negative ( positive ) . quasiparticles with negative effective mass have counterintuitive propagation properties : they move in the direction opposite to their wave vector ( negative refraction @xcite ) . thus , by changing the angle @xmath117 it may be possible to tune the excitons from the regime of normal propagation to the regime of negative refraction . molecule in the presence of an electric field . the dashed lines show the positions of the field - free states . panels ( b ) and ( c ) : energy of rotational excitons in a one - dimensional array of lics molecules trapped on an optical lattice with the lattice constant @xmath128 nm in the presence of an electric field of 1 kv / cm . full curves exact numerical calculation ; dashed curves analytical result of eq . ( [ nna ] ) . the electric field is directed perpendicular to the array axis ( panel b ) and parallel to the array axis ( panel c ) . [ f - exciton dispersion ] ] the possibility of tuning the parameters of the exciton hamiltonians ( [ hexc ] ) , ( [ hdyn ] ) and ( [ h - n - c ] ) in an ensemble of polar molecules on an optical lattice opens the possibility to access new regimes of frenkel exciton physics that can not be observed in solid state crystals . here , we discuss a few examples that we presented in our recent papers @xcite , @xcite , @xcite . in this section we study non - linear interactions between rotational frenkel excitons and show that they can be dynamically tuned by an external electric field . this is important for applications of excitons in quantum information processing @xcite , where excitons can be used as qubits . in particular , we consider the interplay of the dynamical interactions arising from eq . ( [ hdyn ] ) and the kinematic interaction arising from the hard - core boson nature of the exciton operators and show that the relative importance of these two interactions can be tuned . there are two types of non - linear interactions for frenkel excitons . the first is the _ dynamical _ interaction given by eq . ( [ hdyn ] ) . this interaction is determined by the matrix elements @xmath129 of the dipole - dipole interaction ( see eq . ( [ d(n ) ] ) ) . in the wave vector representation , the dynamical interaction describes wave vector conserving scattering between two excitons with the exchange of momentum @xmath110 @xmath130 where @xmath131 . the second interaction mechanism originates from the hard - core repulsion of molecular excitations . since excitons are paulions , or hard - core bosons with long - range interactions , they effectively repel each other when placed in the same lattice site . as a result , the direct products of @xmath132-operators are not the eigenstates of the exciton hamiltonian @xmath133 ( [ hexc ] ) , as they would be for bosons or fermions . the commutation relations for paulions ( [ paulion commutations ] ) yield for a two - exciton state @xmath134 the following equation : @xmath135\ | \phi(k_1,k_2 ) \rangle + \\ \\ + \displaystyle \frac{1}{\cal n } \sum\limits_{{q_1 , q_2}\atop{q_1+q_2 = k_1 + k_2 } } [ e(q_1 ) + e(q_2)]\ | \phi(q_1,q_2 ) \rangle.\\ \end{array}\ ] ] the second term on the right - hand side may be interpreted as total wave vector conserving scattering between plane - wave - like one - exciton states @xmath136 @xcite . this type of scattering results from the kinematic interaction @xcite . this interaction can also be included as a perturbative term @xmath64 in the total hamiltonian for the effective bosons introduced via the agranovich toshich transformation ( section [ s - s - mappings ] ) . the non - linear properties of excitons ( in the two - body interaction approximation ) are determined by the balance between the dynamical and kinematic interactions . in the presence of an external electric field and with account of the dynamical interaction , the rotational frenkel excitons are described by the hamiltonian @xmath137 given in eqs . ( [ hexc ] ) and ( [ hdyn ] ) . in the nearest neighbor approximation @xmath105 and @xmath138 are parametrized by two constants : @xmath139 and @xmath140 . since @xmath118 determines the propagation properties of excitons , it is implicitly related to the strength of the exciton exciton kinematic interaction : roughly speaking , the strength of repulsive interaction ( scattering ) between two excitons at the same lattice site is proportional to @xmath141 . the constant @xmath142 describes the dynamical interaction . the paulionic corrections do not affect @xmath143 so one can use the bosonic commutation relations for the @xmath102-operators in this term , both in the site and wave vector representations . both @xmath142 and @xmath118 can be controlled by varying the magnitude and orientation of the electric field ( see figure [ f - d and j ] ) . panel ( a ) of figure [ f - d and j ] shows the dependence of @xmath142 and @xmath118 for a one - dimensional array of lics molecules on the field magnitude at a fixed angle between the array axis and the direction of the electric field . note that @xmath142 vanishes at zero electric field ( see eq . ( [ d(n ) ] ) ) . for @xmath144 , the field - dressed states @xmath71 and @xmath39 reduce adiabatically to the pure rotational states : @xmath145 and @xmath146 , and the matrix elements determining @xmath142 vanish according to the selection rules . panel ( b ) shows the dependence of @xmath142 and @xmath118 on the angle @xmath117 for a fixed field magnitude . note that both @xmath142 and @xmath118 vanish at @xmath147 . and @xmath118 as functions of electric field magnitude at a fixed angle @xmath148 . ( b ) @xmath142 and @xmath118 as functions of @xmath117 at a fixed electric field @xmath149 kv / cm . the calculations are for a 1d ensemble of lics molecules separated by @xmath116=400 nm ; @xmath150 = 72 khz . [ f - d and j ] ] figure [ f - d and j]b shows that the matrix elements @xmath142 and @xmath118 have the same sign , independent of @xmath117 . as discussed in section [ s - s - s - excitons ] , the sign of the effective mass is always opposite to the sign of @xmath118 , and , consequently , the sign of @xmath142 . the constant @xmath142 determines the dynamical interaction potential ( [ hdyn ] ) . due to the linearity of the schrdinger equation , a positive potential is attractive for a particle with negative mass and a negative potential is attractive for a particle with positive mass @xcite . therefore , the dynamical interaction in this system is always attractive @xcite . the kinematic interaction is always repulsive because it arises as a consequence of the hard - core boson nature of the excitation operators . figure [ f - d and j ] shows that the values of @xmath142 and @xmath118 can be tuned in a wide range of magnitudes . for the chosen example of lics molecules on an optical lattice with @xmath128 nm , @xmath142 can be varied in the interval from -80 khz to 40 khz and @xmath118 from -50 khz to 25 khz . their relative magnitudes determine the relative contributions of the kinematic and dynamical interactions . by changing the ratio between @xmath142 and @xmath118 , one can explore the interaction regimes dominated by the different interactions . in particular , prevailing attraction may result in the formation of bound two - exciton complexes known as biexcitons @xcite . they form in the low dimensional systems when @xmath151 @xcite . in 1d they appear as a single state band split form the continuum of the two - particle states . as a consequence of the correlation between the signs of @xmath142 and the effective mass , the biexciton state appears below ( above ) the two - particle continuum for negative ( positive ) @xmath142 , and can be continuously tuned between these two positions by varying the angle between the electric field and the intermolecular axis . the properties of biexcitons can be engineered by varying the ratio @xmath152 by tuning the magnitude of the electric field . in particular , for an ensemble of lics molecules , the biexciton begins to appear at @xmath153 kv / cm , when @xmath154 . with the increase of the electric field its binding energy ( and , correspondingly , its splitting from the continuum states ) increases , and the wave function shrinks , so that at @xmath155 a biexciton is a strongly correlated state of two molecular excitations separated by one or two lattice constants . the formation of a biexciton thus resembles the association of a molecule by combining two atoms . when @xmath156 , the dynamical interaction vanishes , and the kinematic interaction dominates . it should be mentioned that the effect of the kinematic interactions on frenkel exciton dynamics has not yet been observed in experiments . to this end , it would be useful to find a mechanism for tuning the kinematic interaction as well . we explore this in the following section . although the kinematic interaction is inherent to molecular crystals as an intrinsic consequence of the exciton operator statistics , it may be possible to generate excitations that do not experience kinematic interactions . to find the conditions for such excitations , we neglect the dynamical interactions and look for solutions of the two - particle schr " odinger equation @xmath157 in the form @xmath158 where @xmath159 is the relative wave vector of two interacting excitons , and @xmath160 . the expansion coefficients in eq . ( [ two - particle ] ) satisfy the following equation @xmath161 where @xmath162 is the total energy of two interacting excitons . the two - particle amplitude in the site representation , which depends on the relative distance @xmath163 between two molecular excitations , is @xmath164 . since @xmath165 , the right - hand side of eq . ( [ 2body disp eq ] ) vanishes . eq . ( [ 2body disp eq ] ) can be satisfied only if @xmath166 the corresponding wave function is @xmath167-time degenerate ( @xmath29-degeneracy ) : @xmath168 where the quantum number @xmath29 determines the fixed distance @xmath169 between the excited molecules : @xmath170 . these states describe two correlated excitations that do not experience the kinematic interaction . the key requirement ( [ kin suppress ] ) the @xmath171-independence of the two - particle energy @xmath172 is however not easy to satisfy . one possibility is to consider a specific choice of @xmath173 . in the nearest neighbor approximation , when @xmath174 , the two - particle energy written in terms of @xmath173 and @xmath171 is @xmath175 . it reduces to a constant if @xmath176 . thus , with the accuracy up to corrections coming from the interactions beyond the nearest neighbors and quadrupole interactions , such a pair immune to the kinematic interaction can be produced starting from a pair of excitons with @xmath177 . combined with the possibility to control the dynamical interaction strength by the electric field magnitude , this provides a system in which both attractive and repulsive interactions between quasiparticles can be independently tuned . another possibility to produce a two - particle state with the energy independent of @xmath171 can be realized using the three - level structure of the first rotationally excited state of @xmath37 molecules ( see figure [ f - exciton dispersion]a ) . at low electric fields , when the energy separation between the levels @xmath178 and @xmath179 is small , the different excitation branches may be mixed , which corresponds to configuration mixing in solid state molecular crystals @xcite . due to the symmetry properties of the molecular system considered here , the state with @xmath180 is decoupled from the states with @xmath181 if the electric field is parallel or perpendicular to the molecular ensemble . for other angles @xmath117 , the collective excitations give rise to three exciton modes which we denote @xmath182 , @xmath183 and @xmath184 , containing contributions from all three molecular transitions @xcite : @xmath185 where @xmath186 are the exciton operators corresponding to the molecular transition to the excited state with projection @xmath187 . operators ( [ pm(k ) ] ) diagonalize the four - level excitonic hamiltonian yielding @xmath188 at high electric field , the @xmath184-mode is split from the other two modes , leading to the isolated one - band exciton discussed throughout this article . however , the other two branches correspond to the excitations of the degenerate molecular states and remain mixed . their @xmath189 state can be accessed by microwave field with circular polarization . the non - zero @xmath171 states can be probed by raman transitions combining photons with linear and circular polarization . in the limit of high electric fields , the energies of these excitons can be written as @xmath190 where @xmath191 is the transition energy between the field - dressed ground state and the field - dressed excited state with @xmath181 ; @xmath192 , and @xmath193 are defined in eq . ( [ |g > and |e > ] ) . the kinematic interaction in the presence of several branches is more complicated , but it can be shown that the condition ( [ kin suppress ] ) also applies to the multi - branch problems . it can be seen that , at a particular angle @xmath194 , @xmath195 so the kinematic interaction must be absent . ( [ 33 ] ) is satisfied for arbitrary values of @xmath173 . we note that for the @xmath184-branch , the matrix elements of the dipole - dipole interaction vanish at @xmath196 so the @xmath184-state becomes dispersion - less . however , the matrix elements giving rise to the other two exciton states remain non - zero at this angle , see eqs . ( [ ealpha , ebeta ] ) . it is the cancellation of the dispersions of @xmath197 and @xmath198 that leads to the suppression of the kinematic interaction . dynamics of quantum particles in disordered potential has been extensively studied in relation to anderson localization @xcite , propagation of light @xcite and particles @xcite through disordered media , phase transitions between insulating and conducting states @xcite , to name a few examples . as a result of these studies , many properties of disordered systems , such as the role of the density of states or the correlation functions are well understood . at the same time , there are still many problems that remain open . a few examples include the role of disorder in the transition to the glassy state @xcite , transition from a superconductor to insulator with increasing disorder @xcite , and the counterintuitive behavior of conductivity in some disordered quasi - crystals @xcite . in this section we show that ultracold molecules trapped on an optical lattice offer the possibility to study quasiparticles in the presence of dynamically tunable disordered potential . we do not consider the effects of unavoidable natural disorder , such as unoccupied lattice sites , lattice potential inhomogeneity or fluctuations of the electric field . always present in experiments with atoms and molecules on optical lattices , the natural disorder induces localization and decoherence of excitons leading to homogeneous broadening of exciton dispersion curves . we assume that the effects of natural disorder can be reduced to a small fraction of the exciton bandwidth . here , we consider the possibility of applying an external disordered potential that could be varied to allow the observation of real - time dynamics of disorder - induced phenomena . there are several different models of disorder @xcite , and some of them can potentially be realized in an optical lattice with ultracold molecules . first , suppose that a small fraction of molecules trapped on the lattice ( host molecules ) is replaced with molecules of different kind ( impurities ) . this results in substitutional disorder . we assume that all impurity molecules are identical . the impurities break the translational symmetry of the system and therefore scatter excitons . this scattering is elastic and modifies the direction of exciton wave propagation , but not the absolute value of the exciton wave vectors . an impurity introduced into the molecular crystal in general differs from the host molecules by the molecular transition energy , @xmath199 , and by the dipole moment , @xmath200 , which modifies the dipole - dipole coupling strength . in the presence of a single impurity at the lattice site @xmath201 , we can write the total hamiltonian of the system as @xmath202 where @xmath203 and @xmath204 is the difference between the host - host and host - impurity excitation transfer constants . the exciton - impurity interaction can thus be described as a sum of a delta - function potential with strength @xmath205 and a perturbation due to the difference in the dipole moments of the host and impurity molecules . if the matrix of the operator ( [ h exc - imp ] ) is evaluated in the basis of exciton states in the site representation , one finds that @xmath205 perturbs the diagonal matrix elements and @xmath206 the off - diagonal matrix elements . thus , the constants @xmath205 and @xmath204 give rise to diagonal and off - diagonal disorder . for a properly chosen mixture of diatomic molecules , the magnitude and the sign of @xmath205 can be tuned by an external electric field . for example , figure [ disorder ] shows that this can be achieved in an array of lics molecules doped with lirb molecules . generally , it should be possible to tune @xmath205 from a negative value to a positive value in a mixture of @xmath37 diatomic molecules @xmath207 and @xmath208 , when the dipole moment of @xmath207 is greater and the rotational constant of @xmath207 is smaller . with @xmath209 ( upper curve ) and @xmath210 ( lower curve ) vs electric field for lics and lirb . [ disorder ] ] it may also be possible to realize a system with tunable diagonal disorder in a single - species ensemble of molecules by applying intense laser beams focused on a small part of the lattice . molecules at the focus of the beam must experience larger ac stark shifts than the molecules outside the beam focus . these molecules are equivalent to impurities because their energy level structure is different . the degree of detuning @xmath205 can be adjusted by varying the laser field strength . for small fields , it may be possible to neglect the off - diagonal disorder terms , which suggests a unique possibility to differentiate between the effects of diagonal and off - diagonal disorder . the properties of waves with arbitrary dispersion in the presence of local defects have been studied in ref . similar considerations were later applied to excitons in a crystal with substitutional disorder in ref . @xcite , where the exciton impurity scattering cross section , which characterizes their interaction , was derived . the exciton impurity interactions lead to the appearance of bound exciton states , which can capture excitons in solid state crystals @xcite . the constants @xmath205 and @xmath206 in eq . ( [ h exc - imp ] ) determine the character of the exciton impurity interactions in natural solids . we first consider an ensemble of molecules driven by several focused laser beams with the same strength , so that @xmath211 , and all impurities are characterized by the same value of @xmath205 . this system may allow for the possibility to explore the dependence of exciton impurity scattering cross sections not only on the exciton wave vector @xmath171 , but also on the impurity scattering strength @xmath205 , which is not possible in conventional solids . in particular , as we show below , the scattering cross section can be resonantly enhanced , if the potential produces a shallow bound state at small values of @xmath205 @xcite . for a particle with the parabolic dispersion @xmath212 in a @xmath213-dimensional delta - like potential @xmath214 the bound state is split from the continuum states by the energy @xmath215 , which depends on the dimensionality of the system . the binding energy can be obtained from the following equations : @xmath216^{-1 } { \rm \hspace{1.cm } for \hspace{0.3 cm } 2d } \\ \\ \displaystyle \sqrt{\frac{e_b^{\rm ( 1d)}}{{e}_{\rm loc } } } = -\frac{{\rm sgn } ( m _ * ) v_0}{{e}_{\rm loc } } \arctan \sqrt{\frac{{e}_{\rm loc}}{e_b^{(1d ) } } } { \rm \hspace{1.cm } for \hspace{0.3 cm } 1d}\\ \end{array}\ ] ] where @xmath217 is the localization energy of a particle with the mass @xmath218 in a region of dimension @xmath116 . note that these equations have solutions only for negative ( positive ) @xmath205 and positive ( negative ) mass @xmath219 . figure [ f - scattering cross section]a shows the behavior of the local states as functions of the dimensionless potential strength @xmath220 . in 1d and 2d , an attractive delta - like potential always produces a bound state , and in 3d only starting from a finite value : @xmath221 . resonant scattering may play an important role if the resonant enhancement of the scattering cross section at @xmath222 is reached at a finite value of @xmath205 . for instance , in 1d @xmath223 at vanishing @xmath205 , and the shallowing of the bound state is accompanied by the vanishing of the scattering potential itself . in turn , in 3d one can expect a resonant enhancement of the scattering cross section for the potential @xmath224 . near this potential strength , @xmath225 . in 2d , any potential @xmath226 produces a shallow bound state , whose energy tends exponentially to zero with vanishing impurity potential strength : @xmath227 $ ] ( we recall that @xmath205 and @xmath219 should have different signs for a bound state to appear ) . this exponential dependence leads to efficient resonant scattering for all @xmath228 : as the potential strength in 2d exponentially exceeds the kinetic energy of the scattered wave , then , in contrast to the 1d case , the resonant scattering can not be considered as weak , even at vanishingly small @xmath205 . at the same time , this means that for @xmath229 the local state is so close to zero , that the potential scatters resonantly only excitons with @xmath230 , which in fact do not propagate . on potential strength @xmath205 in different dimensions . ( b , c ) the exciton impurity scattering cross sections for 2d ( b ) and 3d ( c ) as functions of the potential strength for different values of @xmath231 ( shown near each line ) . ( d ) different propagation regimes in 2d ( top ) and 3d ( bottom ) for 1% of impurities ( thick solid line ) and 0.1% of impurities ( thin dashed line ) . [ f - scattering cross section ] ] for small wave vector excitons , @xmath232 , so that @xmath233 , and @xmath234 . borrowing the results from ref.@xcite , we write the scattering cross sections in 2d and 3d for @xmath235 as explicit functions of the potential strength @xmath205 , and express them as functions of the bound state energy @xmath215 and the kinetic energy @xmath236 ( these expressions correspond to the limit when both @xmath237 and @xmath215 are much smaller than @xmath238 ) : @xmath239 ^ 2 } = \frac{4 \pi^2 / ak}{\displaystyle \pi^2 + \left [ \ln \bigl(e_b^{(2d)}/t(k ) \bigr)\right]^2},\\ \\ \displaystyle \frac{\sigma_{\rm 3d}(k , v_0)}{\pi a^2 } = \frac{1}{\displaystyle \left(\frac{ak}{2 } \right)^2 + \left [ \left ( \frac{ak}{\pi}\right)^2 - \frac{2 e_{\rm loc}}{\pi v_0}{\rm sgn}(m ) -1 \right]^2 } = \frac{4 e_{\rm loc}/\pi^2}{e_b^{\rm ( 3d ) } + t(k)}. \end{array}\ ] ] figures [ f - scattering cross section]b and [ f - scattering cross section]c show the scattering cross sections for 2d and 3d ( note the divergence of @xmath240 at @xmath241 ) . these results demonstrate that by tuning the potential strength from zero to the critical value ( @xmath242 and @xmath243 ) , one can vary the value of the scattering cross section for small wave vector excitons by many orders of magnitude . the single - impurity scattering cross sections determine the regimes of exciton propagation in the lattice with a small admixture of impurities . according to the ioffe regel criterion @xcite , excitons with the wavelength @xmath244 are strongly localized when @xmath245 ( @xmath246 is the elastic mean free path ) . when @xmath247 excitons have plane - wave - like character , though , in some cases , weak localization of excitons can be achieved @xcite . excitons propagate without scattering when @xmath246 is on the order of the lattice size . for the ensemble of polar molecules trapped on an optical lattice , the elastic mean free path of excitons @xmath248 for a given concentration @xmath249 of impurities can be dynamically tuned by varying the strength of the electric field that modifies the disordered potential and , consequently , @xmath9 . this may allow a possibility to transfer excitons dynamically from the regime of ballistic propagation to the regimes of weak or strong localization . figure [ f - scattering cross section]d shows the @xmath250diagram of different propagation regimes for 2d ( top ) and 3d ( bottom ) lattices with the concentration of impurities 1% . the solid lines correspond to the condition @xmath251 . below these lines , the exciton wave length @xmath252 exceeds @xmath253 , which corresponds to the anderson localization regime . above the lines , the states are in general delocalized . the dashed line is plotted for the impurity concentration 0.1% . the diagrams are not complete without another important parameter , the phase - breaking length @xmath254 . this length accounts for inelastic scattering processes . the interference effects leading to weak and strong localization are only possible if @xmath255 @xcite . the phase - breaking length should be calculated for a given realization of the experimental system with the account of the major loss channels for rotational excitons . consider now a lattice with multiple impurities formed by molecules of a different type . this leads to significant values of @xmath256 . quantum particles in the presence of a random distribution of scattering centers undergo coherent localization , and in 1d all states are exponentially localized even in presence of disorder of arbitrarily small magnitude @xcite . however , when the disorder potential exhibits short - range correlations , particular states may become delocalized @xcite . delocalized states may even form a continuous band , if the correlations are long - range , so that a mobility edge exists between localized and delocalized states @xcite . studies of correlation - induced delocalization of quantum states are important for understanding quantum transport in disordered systems . the substitutional disorder introduces both the diagonal ( @xmath205 ) and off - diagonal ( @xmath206 ) disorder . as a consequence , @xmath205 and @xmath206 are correlated , and the spectrum of a 1d disordered lattice with a two - molecule mixture must always contain one delocalized state @xcite . the energy of this state is determined by the relation between @xmath205 and @xmath206 . the delocalization occurs because the diagonal and off - diagonal perturbations compensate one another . describing an exciton near the top of the energy spectrum for a 1d array of 1000 lics molecules with 10% homogeneously and randomly distributed lirb impurities . panels correspond to different values of @xmath205 : ( a ) @xmath257 , ( b ) @xmath258 khz , and ( c ) @xmath259 khz . the difference of the dipole moments of lics and lirb molecules leads to the value @xmath206 = -6.89 khz . figure is taken from ref.@xcite . [ f - loc deloc ] ] as demonstrated by figure [ disorder ] , @xmath205 can be tuned by shifting the rotational levels of host and impurity molecules simultaneously using a static electric field . in particular , at the field corresponding to the encircled region , @xmath205 vanishes , while @xmath206 remains finite . tuning the electric field around this value allows for the possibility to exploit both the positive and negative values of @xmath205 . choosing the appropriate value of @xmath205 can be used to induce the delocalization of any eigenstate of the system . for states near the origin of the brillouin zone , delocalization occurs at @xmath260 @xcite . figure [ f - loc deloc ] shows the evolution of a particular eigenstate with @xmath205 . at @xmath257 , the state is localized due to non - zero @xmath206 and exhibits the characteristic exponential profile . at @xmath261 khz , the diagonal disorder compensates the effects of the off - diagonal disorder , and the state becomes delocalized . tuning @xmath205 further results in the localization of this eigenstate . rotational frenkel excitons can be used for quantum simulation of model hamiltonians that can not be engineered with atoms , molecules or photons as probe particles . here , we discuss an important example illustrating the possibility of engineering the holstein polaron model with collective excitations of molecules on an optical lattice . polaron is a quasiparticle that describes an electron in a crystal lattice dressed by lattice phonons . the interaction properties of polarons are currently researched in an effort to understand the mechanism of high-@xmath262 superconductivity @xcite and quantum transport in open quantum systems @xcite . quasiparticles similar to polarons can be created by placing an impurity in a fermi degenerate gas of ultracold atoms , as demonstrated in several recent experiments @xcite . the impurity , produced by changing the internal state of one of the ultracold atoms , can be coupled to the fermi sea via a feshbach resonance . this gives rise to fermi polarons . however , lattice phonons are bosons . therefore , a better model of electrons in solid state crystals should be based on coupling a probe particle to bosons . while this may be achieved by placing an impurity in a bose einstein condensate of ultracold atoms , no such experiments have been reported to date . in a recent study @xcite , herrera and krems showed that the dipole - dipole interactions between polar molecules on an optical lattice can be exploited to engineer controllable couplings between excitons and lattice phonons . in this system , the phonons are associated with the oscillatory motion of molecules in the lattice potential @xmath263 where @xmath264 is a small deviation from the equilibrium position of the molecule in site @xmath29 , @xmath265 is the mass of the molecules , @xmath266 is the trapping frequency , and @xmath267 . the first term in eq . ( [ lattice ] ) describes uncoupled oscillations of molecules in their respective lattice sites and depends on the intensity of the trapping laser that determines the trapping frequency @xmath266 . the second term accounts for the collective motion by coupling molecules in different sites . by introducing the operators @xmath268 and @xmath269 for the phonon mode @xmath270 with wave vector @xmath171 , we can write the phonon hamiltonian as @xmath271 where @xmath272 the phonon spectrum is gapped , as @xmath273 , and resembles that of optical phonons in solid state crystals . the exciton phonon interaction is obtained by expanding the matrix elements of the dipole - dipole interaction in a taylor series to yield : @xmath274 where @xmath275 , @xmath276 , @xmath277 , and @xmath118 is the excitation transfer matrix element defined in eq . ( [ hexc ] ) . the first term in eq . ( [ h exc - phonon ] ) describes the phonon - modulated transition energies , the second the phonon - modulated excitation transfer . and @xmath118 as functions of the electric field magnitude at a fixed angle @xmath148 ( a ) and of the angle @xmath117 at @xmath278 ( b ) . calculations are for a 1d ensemble of lics molecules separated by @xmath116=400 nm ; @xmath150 = 72 khz . this is a modified version of the figure taken from ref.@xcite . [ f - polaron - fj ] ] the exciton phonon coupling constants @xmath279 and @xmath280 depend on the molecular states @xmath71 and @xmath39 , which can be tuned by an external electric field . the dependence of the constants @xmath281 and @xmath118 on the electric field magnitude @xmath67 and direction @xmath117 is shown in figure [ f - polaron - fj ] . in the limit @xmath282 the hamiltonian ( [ h exc - phonon ] ) describes the holstein polaron model @xcite ; this limit corresponds to large values of @xmath67 . in the opposite limit , when @xmath283 , it corresponds to the model of particle boson coupling by su , schrieffer and heeger @xcite ; this is the limit of small @xmath67 . varying in time as indicated in the inset . the field strength is 0.5 kv / cm . the figure is taken from ref.@xcite [ f - polaron - dyn ] ] the exciton phonon coupling for molecules in an optical lattice can also be controlled by varying the trapping frequency @xmath266 , which is proportional to the intensity of the trapping laser and enters both of the coupling constants as @xmath284 . increasing the exciton phonon coupling strength results in an increase of the effective mass of the polaron . in the limit of strong coupling , polarons are very massive and localized . the coherent propagation is impeded and polarons can only propagate by random hopping . by decreasing @xmath266 and therefore increasing the exciton phonon coupling , it may be possible to induce self - localization of excitons . the dynamics of an excitation initially produced on a single molecule is illustrated in figure [ f - polaron - dyn ] . these results demonstrate that polar molecules trapped on an optical lattice can be used as a dynamically controllable simulator of open quantum systems . the development of experimental techniques for cooling , trapping and controlling atoms and molecules has opened exciting opportunities for new studies of quantum many - body systems . ultracold atoms and molecules are extensively researched as paradigm systems for engineering novel states of quantum matter and for quantum simulation of model hamiltonians used in condensed matter physics . most of these studies use ultracold atoms or molecules as probe particles . here , we discuss the possibility of using collective excitations in an ensemble of molecules trapped on an optical lattice as a probe in a quantum simulation experiment . we have shown that rotational excitations of polar molecules on an optical lattice may lead to the formation of collective many - body excitations rotational frenkel excitons . these excitons have unique properties that allow tuning the linear and non - linear exciton interactions by modifying the rotational structure of ultracold molecules by an external electric field . we suggest that this can be exploited for the study of new regimes of frenkel exciton physics and the dynamics of quantum localization in disordered systems . we also suggest that rotational frenkel excitons can be used for quantum simulation of the holstein polaron model . this offers interesting possibilities to study quantum transport in open quantum systems with controllable interactions with the environment . in particular , the finite size and tunable properties of the phonon bath for molecules on an optical lattice suggests the possibility of exploring the transition from a non - markovian to markovian environment . to observe rotational excitons , one can measure the populations of the rotational states at different lattice sites . as described in ref . @xcite , this can be achieved by applying a gradient of an electric field and detecting resonant transitions from stark - shifted molecular levels . we thank our collaborators , felipe herrera , ping xiang and jess prez - ros , who contributed to the original publications @xcite forming the basis of this article . our work is supported by nserc of canada and the peter wall institute for advanced studies at the university of british columbia .

ultracold polar molecules trapped on an optical lattice is a many - body system that , under appropriate conditions , may support collective excitations reminiscent of excitons in solid state crystals . here , we discuss the rotational excitations of molecules on an optical lattice leading to rotational frenkel excitons . apart from solid hydrogen , there is no other natural system that exhibits rotational excitons . the rotational excitons have unique properties that can be exploited for tuning non - linear exciton interactions and exciton impurity scattering by applying an external electric field . we show that this can be used to explore the competing role of the dynamical and kinematic exciton exciton interactions in excitonic energy transfer and to study quantum localization in a dynamically tunable disordered potential . the rotational excitons can also be used as a basis for quantum simulation of condensed matter models that can not be realized with ultracold atoms . as an example , we discuss the possibility of engineering the holstein model with polar molecules on an optical lattice .

introduction quantum simulation of lattice models rotational frenkel excitons nonlinear interactions of frenkel excitons rotational excitons in a tunable disordered potential quantum simulation of holstein polaron model conclusions acknowledgements

arxiv

the first - order duffin - kemmer - petiau ( dkp ) formalism @xcite-@xcite describes spin-0 and spin-1 particles and has been used to analyze relativistic interactions of spin-0 and spin-1 hadrons with nuclei as an alternative to their conventional second - order klein - gordon and proca counterparts . the onus of equivalence between the formalisms represented an objection to the dkp theory for a long time and only recently it was shown that they yield the same results in the case of minimally coupled vector interactions , on the condition that one correctly interprets the components of the dkp spinor @xcite-@xcite . however , the equivalence between the dkp and the proca formalisms has already a precedent @xcite . the equivalence does not maintain if one considers partially conserved currents @xcite and the dkp formalism proved to be better than the klein - gordon formalism in the analysis of the @xmath0 decays , the decay - rate ratio @xmath1 , and level shifts and widths in pionic atoms @xcite . furthermore , the dkp formalism enjoys a richness of couplings not capable of being expressed in the klein - gordon and proca theories . a number of different couplings in the dkp formalism , with scalar and vector couplings in analogy with the dirac phenomenology for proton - nucleus scattering , has been employed in the phenomenological treatment of the elastic meson - nucleus scattering at medium energies with a better agreement to the experimental data when compared to the klein - gordon and proca based formalisms @xcite-@xcite . on the other hand , the dkp theory has also experienced a renewed interest due to the discovery of a new conserved vector current now , @xcite-@xcite , whose positive - definite time component would be a candidate to a probability current , and as a bonus a hope for avoiding klein s paradox for bosons @xcite . however , it has been shown that the proposed new current is a fiasco as a probability current @xcite . an effort to disembarrass the status of that new current was done @xcite but in @xcite it was shown to be indefensible . in ref . @xcite it also was shown that klein s s paradox may exist in the dkp theory with minimally coupled vector interactions . the dkp theory has also experienced a renewal of life in the context of applications to quantum chromodynamics @xcite , covariant hamiltonian dynamics @xcite , relativistic phase space @xcite , curved space - time @xcite , causal approach @xcite , @xcite , superluminal tunneling @xcite , bohm model @xcite , gho5 , @xcite , tunneling time @xcite , s - matrix @xcite , five - dimensional galilean invariance @xcite , pseudoclassical mechanics @xcite , bose - einstein condensation @xcite , homogeneous magnetic field @xcite , aharonov - casher phase @xcite , aharonov - bohm potential @xcite , position - dependent mass and vector step potential @xcite , time - dependent mass and time - dependent vector fields @xcite , tensor dkp oscillator ( tensor coupling with a linear potential ) @xcite-@xcite and its thermodynamics properties @xcite , vector dkp oscillator ( nonminimal vector coupling with a quadratic potential @xcite and minimal plus nonminimal vector couplings with a linear potential @xcite ) sextic oscillator ( tensor coupling with a linear plus a cubic potential ) @xcite , vector step potential @xcite , @xcite , @xcite , vector woods - saxon potential @xcite , vector deformed hulthen potential @xcite , vector square well @xcite , vector coulomb potentials boz10 , @xcite,@xcite-@xcite and nonminimal vector step potentials @xcite . the main purpose of the present article is to report on the properties of the dkp theory with the nonminimal vector coupling interaction . nonminimal vector potentials , added by other kinds of lorentz structures , have already been used successfully in a phenomenological context for describing the scattering of mesons by nuclei @xcite-@xcite , @xcite , cla2 . in this paper it is shown that charge - conjugation and time - reversal symmetries have some special features not displayed by minimal vector potentials , in particular nonminimal vector potentials do not couple to the charge . it is also shown that nonminimal vector couplings have been used improperly in the phenomenological description of elastic meson - nucleus scatterings @xcite-@xcite , @xcite , @xcite . furthermore , nonminimal vector potentials can be used as a model for confining bosons and that linear potentials lead to a sort of relativistic dkp oscillator . this article is organized as follows . in sec . ii we present the general dkp equation , discuss conditions on the interactions which lead to a conserved current and effects of parity , charge - conjugation and time - reversal transformations on the vector lorentz structures . adopting a specific representation for the dkp matrices , we set up the one - dimensional equations for the components of the dkp spinor ( iia for the spin-0 sector and iib for the spin-1 sector ) in the presence of minimal and nonminimal vector interactions . we point out that the space component of the nonminimal vector potential can not be absorbed into the spinor , as diffused in the literature . beyond that , we show that the space component of the nonminimal vector potential could be irrelevant for the formation of bound states for potentials vanishing at infinity but its presence is a sine qua non condition for confinement . in sec . iii we specialize to nonminimal vector linear potentials and discuss the solutions of the vector dkp oscillator in detail . the relevance of the nonminimal vector potential for the confinement of bosons is reinforced . an apparent paradox related to the localization of bosons in the presence of strong potentials is solved by introducing the concepts of effective mass and effective compton wavelength . finally , in sec . iv we draw conclusions . the dkp equation for a free boson is given by @xcite ( with units in which @xmath2)@xmath3where the matrices @xmath4 satisfy the algebra@xmath5and the metric tensor is @xmath6diag@xmath7 . the algebra expressed by ( [ beta ] ) generates a set of 126 independent matrices whose irreducible representations are a trivial representation , a five - dimensional representation describing the spin-0 particles and a ten - dimensional representation associated to spin-1 particles . the dkp spinor has an excess of components and the theory has to be supplemented by an equation which allows to eliminate the redundant components . that constraint equation is obtained by multiplying the dkp equation by @xmath8 , namely@xmath9this constraint equation expresses three ( four ) components of the spinor by the other two ( six ) components and their space derivatives in the scalar ( vector ) sector so that the superfluous components disappear and there only remain the physical components of the dkp theory . the second - order klein - gordon and proca equations are obtained when one selects the spin-0 and spin-1 sectors of the dkp theory . a well - known conserved four - current is given by @xmath10where the adjoint spinor @xmath11 is given by@xmath12with@xmath13 in such a way that @xmath14 ( the matrices @xmath4 are hermitian with respect to @xmath15 ) . the time component of this current is not positive definite but it may be interpreted as a charge density . the factor 1/2 multiplying @xmath16 , of no importance regarding the conservation law , is in order to hand over a charge density conformable to that one used in the klein - gordon theory and its nonrelativistic limit ( see e.g. @xcite ) . then the normalization condition @xmath17 can be expressed as@xmath18where the plus ( minus ) sign must be used for a positive ( negative ) charge , and the expectation value of any observable @xmath19 may be given by @xmath20where @xmath21 must be hermitian with respect to @xmath22 , namely @xmath23 , for insuring that @xmath24 is a real quantity . with the introduction of interactions , the dkp equation can be written as@xmath25where the more general potential matrix @xmath26 is written in terms of 25 ( 100 ) linearly independent matrices pertinent to the five(ten)-dimensional irreducible representation associated to the scalar ( vector ) sector . in the presence of interactions @xmath27 satisfies the equation@xmath28thus , if @xmath26 is hermitian with respect to @xmath15 then the four - current will be conserved . the potential matrix @xmath26 can be written in terms of well - defined lorentz structures . for the spin-0 sector there are two scalar , two vector and two tensor terms @xcite , whereas for the spin-1 sector there are two scalar , two vector , a pseudoscalar , two pseudovector and eight tensor terms @xcite . the tensor terms have been avoided in applications because they furnish noncausal effects @xcite-@xcite . considering only the vector terms , @xmath26 is in the form@xmath29a_{\mu } ^{\left ( 2\right ) } \label{pot}\]]where @xmath30 is a projection operator ( @xmath31 and @xmath32 ) in such a way that @xmath33 behaves as a scalar and @xmath34\psi $ ] behaves like a vector . notice that the vector potential @xmath35 is minimally coupled but not @xmath36 . one very important point to note is that this matrix potential leads to a conserved four - current but the same does not happen if instead of @xmath37 $ ] one uses either @xmath38 or @xmath39 , as in @xcite-@xcite , @xcite , @xcite , @xcite ) . as a matter of fact , in ref . @xcite is mentioned that @xmath38 and @xmath40 produce identical results . if the terms in the potential @xmath26 are time - independent one can write @xmath41 , where @xmath42 is the energy of the boson , in such a way that the time - independent dkp equation becomes@xmath43a_{\mu } ^{\left ( 2\right ) } \right ) \right ] \phi = 0 \label{dkp10}\]]in this case @xmath44 does not depend on time , so that the spinor @xmath45 describes a stationary state . note that the time - independent dkp equation is invariant under a simultaneous shift of @xmath42 and @xmath46 , such as in the schrdinger equation , but the invariance does not maintain regarding @xmath42 and @xmath47 . ( [ dkp10 ] ) for the characteristic pair @xmath48 can be written as@xmath49\phi _ { k}=0 \label{orto1}\]]and its adjoint form , by changing @xmath50 by @xmath51 , as@xmath52^{\dagger } = 0 \label{orto2}\]]by multiplying ( [ orto1 ] ) from the left by @xmath53 and ( [ orto2 ] ) from the right by @xmath54 leads to@xmath55\phi _ { k}=0 \label{orto3}\]]and@xmath56^{\dagger } \eta ^{0}\phi _ { k}=0 \label{orto4}\]]respectively . subtracting ( [ orto4 ] ) from ( [ orto3 ] ) and considering that the spinors fit boundary conditions such that@xmath57one gets@xmath58eq . ( [ orto6 ] ) is an orthogonality statement applying to the dkp equation . any two stationary states with distinct energies and subject to suitable boundary conditions are orthogonal in the sense that @xmath59 in addition , in view of ( [ norm ] ) the spinors @xmath60 and @xmath61 are said to be orthonormal if@xmath62 the dkp equation is invariant under the parity operation , i.e. when @xmath63 , if @xmath64 and @xmath65 change sign , whereas @xmath46 and @xmath66 remain the same . this is because the parity operator is @xmath67 , where @xmath68 is a constant phase and @xmath69 changes @xmath70 into @xmath71 . because this unitary operator anticommutes with @xmath72 and @xmath73 $ ] , they change sign under a parity transformation , whereas @xmath22 and @xmath74 $ ] , which commute with @xmath15 , remain the same . since @xmath75 or @xmath76 , the spinor components have definite parities . the charge - conjugation operation changes the sign of the minimal interaction potential , _ i.e. _ changes the sign of @xmath77 . this can be accomplished by the transformation @xmath78 , where @xmath79 denotes the complex conjugation and @xmath80 is a unitary matrix such that @xmath81 . the matrix that satisfies this relation is @xmath82 . the phase factor @xmath83 is equal to @xmath84 , thus @xmath85 . note also that @xmath86 , as should be expected for a charge current . meanwhile @xmath80 anticommutes with @xmath87 $ ] and the charge - conjugation operation entails no change on @xmath88 . by the same token it can be shown that @xmath89 and @xmath90 have opposite behavior under the time - reversal transformation in such a way that both sorts of vector potentials change sign under @xmath91 . the invariance of the nonminimal vector potential under charge conjugation means that it does not couple to the charge of the boson . in other words , @xmath90 does not distinguish particles from antiparticles . hence , whether one considers spin-0 or spin-1 bosons , this sort of interaction can not exhibit klein s paradox . for the case of spin 0 , we use the representation for the @xmath4matrices given by @xcite@xmath92where@xmath93@xmath94 , @xmath95 and @xmath96 are 2@xmath973 , 2@xmath972 and 3@xmath973 zero matrices , respectively , while the superscript t designates matrix transposition . here the projection operator can be written as @xcite @xmath98 in this case @xmath30 picks out the first component of the dkp spinor . the five - component spinor can be written as @xmath99 in such a way that the dkp equation for a boson constrained to move along the @xmath100-axis decomposes into @xmath101@xmath102@xmath103where@xmath104furthermore,@xmath105note that , in the absence of the nonminimal potential , the first line of ( [ dkp3 ] ) reduces to the klein - gordon equation , and that @xmath106 , @xmath107 and @xmath108 are the superfluous components of the dkp spinor ( the reason that @xmath109 is because of the one - dimensional movement ) . in the time - independent case one has@xmath110@xmath111@xmath112where@xmath113@xmath114meanwhile , @xmath115 \label{corrente4}\]]it is worthwhile to note that @xmath116 becomes negative in regions of space where @xmath117 ( a circumstance associated to klein s paradox ) and that @xmath90 does not intervene explicitly in the current . the orthonormalization formula ( [ orto8 ] ) becomes@xmath118regardless @xmath119 and @xmath90 . ( [ orto1 ] ) is in agreement with the orthonormalization formula for the klein - gordon theory in the presence of a minimally coupled potential puk . this is not surprising , because , after all , both dkp equation and klein - gordon equation are equivalent under minimal coupling . the form @xmath120 in eq . ( [ dkp3 ] ) suggests that the space component of the minimal vector potential can be gauged away by defining a new spinor @xmath121even if @xmath119 is time dependent . without any question@xmath122 in such a way that @xmath123 satisfies the dkp equation without @xmath124 . in refs . @xcite and @xcite the term involving @xmath125 was explicitly absorbed into the wave function . nevertheless , it seems that there is no chance to get rid from this term . as a matter of fact , we will show that the space component of the nonminimal vector potential plays a peremptory role for confining bosons . the possibility for ruling out @xmath119 but not @xmath126 is reinforced by the observation that the first derivative of a second - order differential equation , such as the term containing @xmath119 in the first line of eq . ( [ dkp4 ] ) , is a well - known trick in mathematics . it is noticeable that if @xmath127 as @xmath128 , confining solutions for a pure nonminimal vector potential will be possible on the condition that the space component of @xmath90 is stronger , or has a dominant asymptotic behavior , than its time component . otherwise , nothing but continuum states will be possible . in this last circumstance , a boson can tunnel into the classically forbidden region , an unexpected result in nonrelativistic mechanics and by no means related to klein s paradox . on the other hand , for a pure nonminimal vector potential going to zero at infinity , a necessary condition for the existence of bound - state solutions ( with @xmath129 ) is that @xmath130at any arbitrary point on the @xmath100-axis . in this case , it is the time component of the nonminimal vector potential that plays a leading role in establishing bound states . for the case of spin 1 , the @xmath4 matrices are @xcite@xmath131where @xmath132 are the 3@xmath973 spin-1 matrices @xmath133 , @xmath134 are the 1@xmath973 matrices @xmath135 and @xmath136 , while * * * * @xmath137 and @xmath96 * * * * designate the 3@xmath1383 unit and zero matrices , respectively . in this representation @xmath139__i.e . _ _ , @xmath30 projects out the four upper components of the dkp spinor . with the spinor written as @xmath140 , and partitioned as@xmath141@xmath142@xmath143the one - dimensional dkp equation can be expressed in the compact form @xmath144@xmath145@xmath146where @xmath147 is again given by ( [ dzao ] ) . in addition , expressed in terms of ( [ part ] ) the current can be written as @xmath148@xmath149@xmath150note that the third line plus the second equation in the middle line of ( dkp3 ) are the constraint equations which allow one to eliminate the superfluous components ( @xmath151 , @xmath152 , @xmath153 and @xmath154 ) of the dkp spinor . the component @xmath155 because the movement is restrict to the @xmath100-axis . meanwhile the time - independent dkp equation decomposes into@xmath156@xmath157@xmath158where@xmath159@xmath160now the components of the four - current are @xmath161@xmath162 \label{cur2}\]]and the orthonormalization expression ( [ orto8 ] ) takes the form @xmath163just as for scalar bosons , @xmath164 for @xmath117 and @xmath165 does not appear in the current . similarly , @xmath124 and @xmath90 do not manifest explicitly in the orthonormalization formula . from ( [ spin1-ti])-([k ] ) one sees that the solution for the spin-1 sector consists in searching solutions for two klein - gordon - like equations , owing to the term @xmath166 in ( [ k ] ) . it should not be forgotten , though , that the equations for @xmath167 and @xmath168 are not indeed independent because @xmath42 appears in both equations . evidently , matching a common value for the energy might compromise the existence of solutions for spin-1 bosons when compared to the solutions for spin-0 bosons with the very same potentials . this amounts to say that the solutions for the spin-1 sector of the dkp theory , if they really exist , can be obtained from a restrict class of solutions of the spin-0 sector . this limitation on the possible solutions for spin-1 bosons as compared for spin-0 bosons should not be a surprise if one remembers that , in the absence of any interaction , all the components of the free proca equation obey a free klein - gordon equation but with an additional constraint on the components of the proca field . having set up the spin-0 and spin-1 equations for vector interactions , we are now in a position to use the machinery developed above in order to solve the dkp equation with specific forms for nonminimal interactions . let us consider pure nonminimal vector linear potentials in the form@xmath169where @xmath170 and @xmath171 are dimensionless quantities . our problem is to solve ( [ dkp4 ] ) and ( [ spin1-ti ] ) for @xmath45 and to determine the allowed energies . although the absolute value of @xmath172 in @xmath66 is irrelevant in the effective equations for @xmath173 ( in the scalar sector ) and @xmath174 ( in the vector sector ) , it is there for ensuring the covariance of the dkp theory under parity . it follows that the dkp spinor will have a definite parity and @xmath175 and @xmath27 will be genuine four - vectors . for the spin-0 sector of the dkp theory one finds that @xmath176 obeys the second - order differential equation @xmath177where@xmath178the solution for ( [ ohs1 ] ) , with @xmath179 and @xmath180 , is precisely the well - known solution of the schrdinger equation for the nonrelativistic harmonic oscillator ( see , _ e.g. _ , @xcite):@xmath181@xmath182where @xmath183 , @xmath184 is a normalization constant , and @xmath185 is a @xmath186-th degree hermite polynomial in @xmath187 . notice that the condition @xmath180 requires that @xmath188 , meaning that the space component of the potential must be stronger than its time component in order to the effective potential be a true confining potential . nevertheless , there is no requirement on the signs of @xmath171 and @xmath170 . from ( [ eigen1 ] ) one obtains the discrete set of dkp energies ( symmetrical about @xmath189 as it should be since @xmath165 does not distinguish particles from antiparticles ) @xmath190 where @xmath191irrespective to the sign of @xmath170 . in general , @xmath192 is higher for @xmath193 than for @xmath194 . it increases with the quantum number and it is a monotonically decreasing function of @xmath170 . in order to insure the reality of the spectrum , the coupling constants @xmath195 and @xmath171 satisfy the additional constraint@xmath196if one squares ( [ const ] ) the resulting inequality is in general a quadratic algebraic inequality in @xmath171 ( or @xmath170 ) , which can be solved analytically . the price paid is that some spurious solutions can appear in this process , although , of course , these can be eliminated by checking whether they satisfy the original inequality . a more instructive procedure is to follow a graphical method , by which one seeks the regions of the functions of @xmath171 in ( [ const ] ) : a hyperbole on the left - hand side,@xmath197and a straight line on the right - hand side,@xmath198@xmath199 is a nonnegative function having two symmetric branches , and for @xmath200 it approximates the function @xmath201 . figure [ fig1 ] present results for the three first quantum numbers with @xmath202 . for @xmath203 , this figure shows clearly that @xmath204 only for some @xmath205 , although @xmath206 for all @xmath207 . the intersection points of @xmath208 and @xmath209 , for @xmath202 , correspond to @xmath210 figure [ fig1 ] also allows one to conclude that @xmath211 for @xmath212 . notice that there is a high density of states ( number of states in a fixed range of energy ) corresponding to an infinite set of quasi - degenerate solutions in the neighborhood of @xmath213 . in the weak - coupling limit , @xmath214 and @xmath215 , @xmath216 ) describe a genuine nonminimal vector dkp oscillator . nevertheless , the lorentz structure of the potentials plays no role in a nonrelativistic scheme , because one has to use the schrdinger equation with the potential @xmath217 . despite the effective harmonic oscillator potential appearing in ( [ ohs1 ] ) , the linear potentials given by ( [ pot2 ] ) do not furnish bound - state solutions in the schrdinger equation because the sum @xmath218 with @xmath219 is unbounded from below . on the other hand , for @xmath200 one has that@xmath220so that @xmath221 for @xmath193 . concerning @xmath194 , as far as @xmath171 increases , the spectrum moves towards @xmath189 , except for @xmath222 which maintains @xmath223 ( the spectrum acquiesces @xmath224 , [ fig3 ] and [ fig4 ] illustrate the spectrum in terms of @xmath225 for three different values of @xmath226 . for @xmath227 there is a spectral gap given by @xmath228and there are infinitely many energy levels above @xmath229 where , in the absence of interaction , there was the continuum . as far as @xmath171 increases , the spectrum moves towards @xmath189 , except for @xmath230 . the gap tends to vanish as @xmath171 becomes close to @xmath231 ( for @xmath232 ) , and so the positive- and negative - energy levels tend to be very close to each other . figures [ fig5 ] and [ fig6 ] illustrate the spectrum in terms of @xmath233 for two different values of @xmath170 . the charge density@xmath234dictates that @xmath176 must be normalized as@xmath235using the property @xcite@xmath236one finds that the normalization constant can be chosen to be@xmath237thus , for @xmath238 one has@xmath239then , using ( [ exp ] ) the quantity @xmath240 can be written as@xmath241now it is a simple matter to write down the uncertainty in the position:@xmath242if @xmath243 shrinks then @xmath244 ( uncertainty in the momentum ) will must swell , in consonance with the heisenberg uncertainty principle . nevertheless , the maximum uncertainty in the momentum is given by @xmath229 requiring that is impossible to localize a boson in a region of space less than half of its compton wavelength ( see , for example , @xcite-@xcite ) . nevertheless , if one defines an effective mass as @xmath245 and an effective compton wavelength as @xmath246 one will find that @xmath247 . it follows that the high localization of bosons , related to high values of @xmath248 ( @xmath200 ) never menaces the single - particle interpretation of the dkp theory . for @xmath249 and the quasi - degenerate solutions mentioned above are related to very delocalized states . as for the behavior in the neighborhood of @xmath250 one should note that , despite of @xmath251 and @xmath243 are independent of @xmath252 , the dkp spinor is not defined for @xmath253 . thus , @xmath254 must be ruled out of the theory . although positive- and negative - energy levels do not touch , they can be very close to each other for coupling constants moderately strong without any danger of reaching the conditions for klein s paradox . as for the spin-1 sector , proceeding as before , one finds that @xmath255 obeys the equation @xmath256where @xmath257 is defined as in ( [ ohs2 ] ) and@xmath258for bound states , to which we shall devote our attention , we must require @xmath259 and @xmath180 , as before . thus , the solution is expressed as@xmath260@xmath261where @xmath262 , @xmath263 is a normalization constant , and @xmath264 is a column matrix whose elements are normalization constants related to the solutions for @xmath265 and @xmath266 . hence , the necessary conditions for binding spin-1 bosons subject to linear potentials have been put forward . the formal analytical solutions have been obtained and it has been revealed that the solutions related to the spinor @xmath167 are formally the same as those ones for spin-0 bosons . now we move on to match a common energy to the spin-1 boson problem . the matching condition requires that the quantum numbers @xmath267 and @xmath268 must satisfy the relation@xmath269this constraint on the nodal structure of @xmath167 and @xmath168 dictates that acceptable solutions only occur for a countable number of possibilities for @xmath270 , _ viz . we showed that minimal and nonminimal vector interactions behave differently under charge - conjugation and time - reversal transformations . although klein s paradox can not be treated as unworthy of regard in the dkp theory with minimally coupled vector interactions , it never makes its appearance in the case of nonminimal vector interactions because they do not couple to the charge . in the case of a pure nonminimal vector coupling , both particle and particle energy levels are members of the spectrum , and the particle and antiparticle spectra are symmetrical about @xmath189 . if the interaction potential is attractive ( repulsive ) for bosons it will also be attractive ( repulsive ) for antibosons . however , there is no crossing of levels because possible states in the strong field regime with @xmath189 are in fact unnormalizable . these facts imply that there is no channel for spontaneous boson - antiboson creation and for that reason the single - particle interpretation of the dkp equation is ensured . the charge conjugation operation allows us to migrate from the spectrum of particles to the spectrum of antiparticles and vice versa just by changing the sign of @xmath42 . this change induces no change in the nodal structure of the components of the dkp spinor and so the nodal structure of the four - current is preserved . in view of recent developments on the construction of positive - definite inner product for the klein - gordon theory @xcite , we acknowledge that we took a very conservative stance when considering a current that can not be related to a probability current . the interesting possibility of a probability current in the dkp theory , constructed from the energy - momentum tensor , launched in @xcite-@xcite , though , received a severe criticism in @xcite and @xcite . it will then be challenging to construct a probability current in the dkp theory from a relativistically invariant positive - definite inner product . notwithstanding , the conserved charge current plus the charge conjugation operation are enough to infer about the absence of klein s paradox under nonminimal vector interactions , or its possible presence under minimal vector interactions . we showed that nonminimal vector couplings have been used improperly in the phenomenological description of elastic meson - nucleus scatterings potential by observing that the four - current is not conserved when one uses either the matrix @xmath38 or @xmath39 , even though the linear forms constructed from those matrices behave as true lorentz vectors . we also pointed out that the space component of the nonminimal vector potential can not be absorbed into the spinor . beyond that , we showed that the space component of the nonminimal vector potential could be irrelevant for the formation of bound states for potentials vanishing at infinity but its presence is an essential ingredient for confinement . for the one - dimensional problem , the dkp equation with nonminimal vector potentials was mapped into a sturm - liouville problem in such a way that the solution for linear potentials could be found by solving a schrdinger - like problem for the nonrelativistic harmonic oscillator . the behavior of the solutions for this sort of dkp oscillator was discussed in detail . that model reinforced the absence of klein s paradox . furthermore , due to the fact that there is no room for the boson - antiboson production , a boson embedded in this sort of background acquires an effective mass which permits it can be strictly localized . we also showed that the dkp oscillator for vector bosons is conditionally solvable . in addition to provide a better understanding of the dkp theory with a sort of coupling full of phenomenological relevance and not yet well explored in the literature , it was conceived an exactly solvable vector model relating to the confinement of bosons . this work was supported in part by means of funds provided by capes and cnpq . the authors would like to thank an anonymous referee for drawing attention to ref . @xcite , unavailable for us and partially cited in ref . @xcite . e. fischbach _ et al_. , phys . * 26 * , 1200 ( 1971 ) ; e. fischbach _ et al_. , phys . * 27 * , 1407 ( 1971 ) ; e. fischbach and m. m. nieto , phys . lett . * 29 * , 1046 ( 1972 ) ; n. g. deshpande and p. c. mcnamee , phys . d * 5 * , 1012 ( 1972 ) ; a. o. barut and z. z. aydin , phys . d * 6 * , 3340 ( 1972 ) ; e. fischbach , m. m. nieto , and c. k. scott , phys . d * 7 * , 207 ( 1973 ) ; z. z. aydin and a. o. barut , phys . d * 7 * , 3522 ( 1973 ) ; m. d. scadron and r. l. thews , phys d * 9 * , 2180 ( 1974 ) ; e. fischbach _ et al_. , phys . d * 9 * , 2183 ( 1974 ) ; e. fischbach , m. m. nieto , and c. k. scott , prog . phys . * 51 * , 1585 ( 1974 ) ; f. t. meiere _ et al_. , phys . d * 8 * , 4209 ( 1973 ) ; e. friedman , g. klbermann , and c. j. batty , phys . c * 34 * , 2244 ( 1986 ) . a. datta , _ high spin field theories and relativistic quantum mechanics of bosons _ , in _ bosons , ferromagnetism and crystal growth research _ , horizons in world physics , edited by e. seifer ( nova publishers , new york , 2007 ) vol . 257 , chap . 119 - 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vector couplings in the duffin - kemmer - petiau theory are revised . it is shown that minimal and nonminimal vector potentials behave differently under charge - conjugation and time - reversal transformations . in particular , it is shown that nonminimal vector potentials have been erroneously applied to the description of elastic meson - nucleus scatterings and that the space component of the nonminimal vector potential plays a crucial role for the confinement of bosons . the dkp equation with nonminimal vector linear potentials is mapped into the nonrelativistic harmonic oscillator problem and the behavior of the solutions for this sort of dkp oscillator is discussed in detail . furthermore , the absence of klein s paradox and the localization of bosons in the presence of nonminimal vector interactions are discussed .

introduction the dkp equation and the vector couplings the nonminimal vector linear potential conclusions

arxiv

_ graph analytics _ is rapidly establishing itself as a major discovery tool in such diverse application domains as road systems , social networks , natural language processing , biological pattern discovery , cybersecurity , and more . graph analytics tasks for big data networks are typically run on distributed architectures such as clusters of loosely - coupled commodity servers , where the challenge is to minimize communication overhead between processors , while each processor can store only a small fraction of the entire network . a number of computational models proposed in recent years @xcite provide excellent rigorous frameworks for studying algorithms for massive data , subject to these constraints . under this new computational paradigm , state - of - the - art graph algorithms often do not scale up efficiently to process massive instances , since they either require superlinear memory or exhibit long critical paths resulting in a large number of communication rounds . in this work we focus on _ graph decomposition _ , which is a fundamental primitive for graph analytics as well as for several other application contexts , especially in distributed settings , where decompositions are often at the base of efficient parallel solutions . we develop an efficient parallel decomposition algorithm for partitioning the nodes of an unweighted , undirected graph into disjoint , internally connected _ clusters _ , which is able to control the maximum radius of the clusters ( the maximum distance of a node in a cluster to the cluster s center ) . similarly to other known decomposition approaches , our algorithm grows clusters from several batches of centers which are progressively selected from the uncovered nodes . however , rather than fixing the radius of each grown cluster _ a priori _ , or randomly delaying the activation of the centers , as in previous works , we activate a new batch of centers every time that the number of uncovered nodes halves , while continuing growing the clusters of previously activated centers . the idea behind such a strategy is to force more clusters to grow in poorly connected regions of the graph while keeping both the total number of clusters and the maximum cluster radius under control . in the _ metric @xmath0-center _ clustering problem @xcite the goal is to partition an undirected graph into @xmath0 clusters so that the maximum radius of the clusters is minimized . the problem is np - hard and we are therefore interested in efficient approximations . given an @xmath1 node unweighted undirected graph , building on our parallel graph decomposition method , we obtain a randomized @xmath2-approximation algorithm to the @xmath0-center problem . the algorithm can be implemented on the mapreduce ( mr ) model of @xcite in a number of parallel rounds proportional to the maximum cluster radius using overall linear space , as long as each processing node is provided with @xmath3 local space , for any constant @xmath4 . in order to derive a more explicit bound on the parallel complexity , we analyze the maximum cluster radius , hence the number of rounds , as a function of the doubling dimension of the graph ( see definition [ doublingdim ] ) , showing that for a graph of diameter @xmath5 and doubling dimension @xmath6 the algorithm can provide a decomposition into @xmath0 clusters with maximum cluster radius @xmath7 . next , we apply our graph decomposition strategy to a challenging problem in the context of graph analytics , namely , the approximation of the graph diameter , a global property of a graph , in a number of parallel rounds which is substantially less than the diameter itself , and using linear global space and local memory at each processor sufficient to store only a small fraction of the graph . we remark that known parallel approaches to estimating the diameter of a graph either require up to quadratic space ( e.g. , using transitive closure ) or require a number of parallel rounds which is inherently linear in the diameter ( e.g. , using straightforward parallelization of breadth - first search or neighborhood function estimations ) . to estimate the diameter of a graph @xmath8 , we first compute a decomposition of @xmath8 of suitable granularity with our novel algorithm , and then we estimate the diameter through the diameter of the _ quotient graph _ , that is , the graph whose nodes correspond to the clusters and whose edges connect clusters containing nodes which are adjacent in @xmath8 . the granularity is chosen so that the size of the quotient graph is small enough so that its diameter can be computed using limited local memory and very few communication rounds . we show that on any unweighted , undirected connected graph @xmath8 with @xmath1 nodes and @xmath9 edges , our algorithm returns an upper bound to its diameter which is a factor @xmath2 away from the true value , with high probability . the algorithm can be implemented on the aforementioned mr model using overall linear space and @xmath3 local space , with @xmath10 , in a number of parallel rounds which is @xmath11 where @xmath12 is the doubling dimension of the graph and @xmath13 is any constant less than @xmath14 . observe that for graphs with small ( e.g. , constant ) doubling dimension , which arise in important practical realms @xcite , the number of rounds can be made asymptotically much smaller than the diameter if sufficient yet sublinear local memory is available . while a similar approach for diameter estimation has been used in the past in the external - memory setting ( see section [ previouswork ] ) , the algorithm presented here , to the best of our knowledge , is the first linear - space distributed algorithm for the problem requiring a number of parallel rounds which is sublinear in the diameter . a very desirable feature of our algorithms is that they lend themselves to efficient practical implementations . we report on an extensive set of experiments conducted on a number of large graphs . a first batch of experiments shows the effectiveness of our decomposition strategy in minimizing the maximum cluster radius , compared against the recent parallel decomposition strategy of @xcite , while a second batch shows that the approximation obtained by our diameter approximation algorithm is in fact much smaller than the asymptotic bound , even for graphs of unknown doubling dimension ( less than twice the diameter in all tested cases ) and that the algorithm s performance compares very favorably to the one exhibited by direct competitors such as breadth - first search and the ( almost exact ) diameter estimation algorithm hadi @xcite . for graphs of very large diameter , the speed - up of our algorithm can be of orders of magnitude . the rest of the paper is organized as follows . section [ previouswork ] summarizes relevant previous work on graph decomposition , @xmath0-center clustering , and diameter estimation . section [ clustering ] presents our novel decomposition and discusses how it can be employed to approximate the @xmath0-center problem . section [ diameter ] presents our decomposition - based algorithm for diameter approximation . section [ mrimplementation ] analyzes our strategies in the mr model of @xcite . section [ experiments ] reports on the experimental results , and , finally , section [ conclusions ] offers some concluding remarks . parallel clustering algorithms relevant to this work have been studied in @xcite . in @xcite the notion of @xmath15-cover for a weighted graph @xmath8 is introduced , which is essentially a decomposition of the graph into nondisjoint clusters , where each node is allowed to belong to @xmath16 distinct clusters and for any two nodes at weighted distance at most @xmath17 in the graph , there is a cluster containing both . a @xmath15-cover is obtained by growing clusters of decreasing radii from successive batches of centers . the algorithm presented in @xcite is similar but returns disjoint clusters and guarantees a bound on the average number of edges between clusters . in @xcite an alternative clustering algorithm is proposed which assigns to each node @xmath18 a random time shift @xmath19 , taken from an exponential distribution with parameter @xmath20 , and grows a cluster centered at @xmath21 starting at time @xmath22 , where @xmath23 is the maximum shift , unless by that time node @xmath21 has been already covered by some other cluster . the authors show that in this fashion the graph is partitioned into clusters of maximum radius @xmath24 , with high probability , while the average number of edges between clusters , hence the size of the quotient graph , is at most @xmath25 . none of the above clustering approaches guarantees that the maximum radius of the returned clusters is ( close to ) minimum with respect to all possible decompositions of the graph featuring the same number of clusters . the related _ metric @xmath0-center _ optimization problem requires that given a set @xmath26 of @xmath1 points in a metric space , a subset @xmath27 of @xmath0 points be found so to minimize the maximum distance between any @xmath28 and @xmath29 . the problem is np - hard even if distances satisfy the triangle inequality @xcite , but polynomial - time 2-approximation sequential algorithms are known @xcite . recently , a constant - approximation mapreduce algorithm was developed in @xcite under the assumption that the distances among all @xmath30 pairs of points are given in input . the problem remains np - hard even if @xmath26 represents the set of nodes of an undirected connected graph @xmath8 , and the distance between two nodes is the length of the shortest path between them . to the best of our knowledge , no low - depth linear - space parallel strategy that yields a provably good approximation for this important graph variant is known in the literature . in recent years , efficient sequential algorithms for estimating the diameter of very large graphs have been devised , which avoid the costly computation of all - pairs shortest paths or the memory - inefficient transitive closure by performing a limited number of breadth - first searches ( bfs ) from suitably selected source nodes @xcite . unfortunately , due to the inherent difficulty of parallelizing bfs @xcite these approaches do not appear amenable to efficient low - depth parallel implementations . external - memory algorithms for diameter estimation which employ a clustering - based strategy similar to ours have been recently proposed in @xcite . the algorithm by @xcite basically selects @xmath0 centers at random and grows disjoint clusters around the centers until the whole graph is covered . the author shows that the diameter of the original graph can be approximated within a multiplicative factor of @xmath31 , with high probability , by computing the diameter on the quotient graph associated with the clustering with suitable edge weights . in @xcite a recursive implementation of this strategy is evaluated experimentally . this approximation ratio is competitive with our result only for polylogarithmic values of @xmath0 . however , observe that for such small values of @xmath0 the radius of the @xmath0 clusters must be within a small ( polylogarithmic ) factor of the graph diameter , and thus the parallel number of rounds can not be substantially sublinear in the diameter itself . efficient pram algorithms for approximating shortest path distances between given pairs of nodes are given in @xcite . for sparse graphs with @xmath32 , these algorithms feature @xmath33 depth , for any fixed constant @xmath34 , but incur a polylogarithmic space blow - up due to the use of the @xmath15-covers mentioned above . the algorithms are rather involved and communication intensive , hence , while theoretically efficient , in practice they may run slowly when implemented on distributed - memory clusters of loosely - coupled servers , where communication overhead is typically high . moreover , their depth is not directly related to the graph diameter . in @xcite , an efficient algorithm , called anf , is devised to tightly approximate the _ neighborhood function _ of a graph @xmath8 , which , for every @xmath35 , gives the number of pairs of nodes at distance at most @xmath36 in @xmath8 , and , therefore , it can be used to estimate the diameter . on a connected graph @xmath8 of diameter @xmath5 , anf executes @xmath5 iterations and maintains at each node @xmath37 a suitable succinct data structure to approximate , at the end of each iteration @xmath36 , the number of nodes at distance at most @xmath36 from @xmath37 . a mapreduce implementation of anf , called hadi , has been devised in @xcite using apache hadoop . little experimental evidence of hadi s performance on large benchmark graphs is available . however , as confirmed by our experiments ( see section [ experiments ] ) , for large - diameter graphs hadi s strategy , even if implemented using faster engines than hadoop , runs very slowly because of the large number of rounds and the high communication volume . a very fast , multithreaded version of anf , called hyperanf , has been devised in @xcite for expensive tightly - coupled multiprocessors with large shared memories , which are not the architectures targeted by our work . let @xmath38 be an undirected connected graph with @xmath39 nodes and @xmath40 edges . for any two nodes @xmath41 let @xmath42 denote the number of edges in the shortest path between @xmath21 and @xmath37 in @xmath8 . also , for any @xmath18 and @xmath43 , let @xmath44 denote the minimum distance between @xmath21 and a node of @xmath45 . we now present an algorithm ( cluster ) that partitions @xmath26 into disjoint clusters around suitably selected nodes called _ centers _ , so that the radius of each cluster , defined as the maximum distance of a cluster node from the center , is small . as in @xcite , our algorithm activates batches of centers progressively , so to allow more clusters to cover sparser regions of the graph . however , unlike those previous works , we can show that our algorithm minimizes the maximum cluster radius , within a polylogarithmic factor , a property that later will turn out crucial for the efficiency of the diameter - approximation algorithm . a parameter @xmath46 is used to control the size of each batch of activated centers . when two or more clusters attempt to cover a node concurrently , only one of them , arbitrarily chosen , succeeds , so to maintain clusters disjoint . the algorithm s pseudocode is given below . we define a crucial benchmark for analyzing each iteration of the algorithm . let @xmath0 be an integer , with @xmath47 , and let @xmath48 be a subset of at most @xmath49 nodes . we define @xmath50 suppose that we have partially grown some clusters covering a subset @xmath51 . we know that by continuing to grow these clusters plus @xmath0 new clusters centered in uncovered nodes , @xmath52 growing steps are necessary to cover the whole graph . we have : [ thm1 ] for any integer @xmath53 , with high probability , cluster@xmath54 computes a partition of @xmath26 into @xmath55 disjoint clusters of maximum radius @xmath56 where @xmath57 , and @xmath58 for @xmath59 . the bound on the number of clusters follows by observing that the number of clusters added in each iteration is a binomial random variable with expectation @xmath60 , and that at most @xmath61 iterations are executed overall . as for the upper bound on @xmath62 , it is sufficient to show that in the @xmath63th iteration , with @xmath64 , the radius of each cluster ( new or old ) grows by @xmath65 , with high probability . let @xmath66 be the set of nodes that at the beginning of iteration @xmath63 are already covered by the existing clusters . by construction , we have that @xmath67 and , for @xmath68 , @xmath69 . hence , @xmath70 . let @xmath71 . it is easy to verify that @xmath72 by definition of @xmath73 we know that there must exist @xmath74 nodes , say @xmath75 , such that each node of @xmath76 is at distance at most @xmath77 from either @xmath66 or one of these nodes . let us consider the partition @xmath78 where @xmath79 is the set of nodes of @xmath76 which are closer to @xmath66 than to any of the @xmath80 s , while @xmath81 is the set of nodes of @xmath76 which are closer to @xmath80 than to @xmath66 or to any other @xmath82 . let @xmath83 and note that @xmath84 therefore , we have that @xmath85 . since @xmath86 , it is easy to see that for any @xmath87 and @xmath88 , in the @xmath63th iteration a new center will be chosen from @xmath81 with probability at least @xmath89 . hence , by the union bound , we conclude that a new center will fall in every @xmath81 with @xmath87 and @xmath88 , with probability at least @xmath90 . when this event occurs , @xmath91 cluster growing steps will be sufficient to reach half of the nodes of @xmath76 ( namely , the nodes belonging to @xmath92 . the theorem follows by applying the union bound over the @xmath93 iterations . an important issue , which is crucial to assess the efficiency of the diameter approximation algorithm discussed in the next section , is to establish how much smaller is the maximum radius @xmath62 of the clusters returned by cluster with respect to the graph diameter @xmath5 , which is an obvious upper bound to @xmath62 . our analysis will express the relation between @xmath62 and @xmath5 as a function of the _ doubling dimension _ of the graph , a concept that a number of recent works have shown to be useful in relating algorithms performance to graph properties @xcite . [ doublingdim ] consider an undirected graph @xmath38 . the _ ball of radius @xmath94 _ centered at node @xmath37 is the set of nodes at distance at most @xmath94 from @xmath37 . also , the _ doubling dimension _ of @xmath8 is the smallest integer @xmath95 such that for any @xmath96 , any ball of radius @xmath97 can be covered by at most @xmath98 balls of radius @xmath94 . the following lemma provides an upper bound on @xmath62 in terms of the doubling dimension and of the diameter of the graph @xmath8 . [ radius - bound ] let @xmath8 be a connected @xmath1-node graph with doubling dimension @xmath12 and diameter @xmath5 . for @xmath99 , with high probability , cluster@xmath54 computes a partition of @xmath26 into @xmath100 disjoint clusters of maximum radius @xmath101 let @xmath102 be the smallest maximum radius achievable by any decomposition into @xmath74 clusters . it is easy to see that each @xmath103 is a lower bound to @xmath102 , whence @xmath104 . by iterating the definition of doubling dimension starting from a single ball of radius @xmath5 containing the whole graph , one can easily argue that @xmath8 can be decomposed into ( at most ) @xmath74 disjoint clusters of radius @xmath105 . the bound on @xmath62 follows since @xmath106 . observe that for graphs with diameter @xmath107 and low ( e.g. , constant ) doubling dimension , @xmath62 becomes @xmath108 when @xmath74 is large enough . indeed , some experimental work @xcite reported that , in practice , big data networks of interest have low doubling dimension . also , for applications such as the diameter estimation discussed in the next section , it is conceivable that parameter @xmath74 be made as large as @xmath109 , for some constant @xmath4 . in fact , the gap between the graph diameter and @xmath62 can be even more substantial for irregular graphs where highly - connected regions and sparsely - connected ones coexist . for example , let @xmath8 consist of a constant - degree expander of @xmath110 nodes attached to a path of @xmath111 nodes , and set @xmath112 . it is easy to see that @xmath113 , for @xmath64 . hence , cluster@xmath54 returns @xmath114 clusters of maximum radius @xmath115 , which is exponentially smaller than the @xmath116 graph diameter . algorithm cluster can be employed to compute an approximate solution to the @xmath0-center problem , defined as follows . given an undirected connected graph @xmath117 with unit edge weights , a set @xmath118 of @xmath0 _ centers _ is sought which minimizes the maximum distance @xmath119 in @xmath8 of any node @xmath28 from @xmath29 . as mentioned in section [ previouswork ] this problem is np - hard . the theorem below states our approximation result . [ k - center ] for @xmath120 , algorithm cluster can be employed to yield a @xmath121-approximation to the @xmath0-center problem with unit edge weights , with high probability . fix @xmath122 so that our algorithm returns at most @xmath0 clusters with high probability , and let @xmath29 be the set of centers of the returned clusters . without loss of generality , we assume that @xmath29 contains exactly @xmath0 nodes ( in case @xmath123 , we can add @xmath124 arbitrary nodes to @xmath29 , which will not increase the value of objective function ) . let @xmath62 be the maximum radius of the clusters returned by our algorithm . as proved in lemma [ radius - bound ] , we have that , with high probability @xmath125 . we conclude the proof by arguing that @xmath126 . consider the optimal solution to the @xmath0-center problem on the graph , and the associated clustering of radius @xmath119 . let @xmath127 be a spanning tree of the quotient graph associated with this clustering . it is easy to see that @xmath127 can be decomposed into @xmath74 subtrees of height @xmath128 each . merge the clusters associated with the nodes of each such subtree into one cluster and pick any node as center of the merged cluster . it is easy to see that every node in the graph is at distance @xmath129 from one of the picked nodes . since @xmath130 , we conclude that @xmath131 , and the theorem follows . let @xmath8 be an @xmath1-node graph with @xmath132 connected components . it is easy to see that for any @xmath133 , algorithm cluster@xmath54 works correctly with the same guarantees stated in theorem [ thm1 ] . also , observe that for @xmath134 , the @xmath0-center problem still admits a solution with noninfinite radius . given @xmath135 , we can still get a @xmath2-approximation to @xmath0-center on @xmath8 as follows . if @xmath136 we simply run cluster@xmath54 with @xmath137 as before . if instead @xmath138 we run cluster@xmath139 and then reduce the number of clusters @xmath140 returned by the algorithm to @xmath0 by using the merging technique described in the proof of theorem [ k - center ] . it is easy to show that the approximation ratio is still @xmath2 . let @xmath8 be an @xmath1-node connected graph . as in @xcite , we approximate the diameter of @xmath8 through the diameter of the quotient graph associated with a suitable clustering of @xmath8 . for the distributed implementation discussed in the next section , the clustering will be made sufficiently coarse so that the diameter of the quotient graph can be computed on a single machine . in order to ensure a small approximation ratio , we need a refined clustering algorithm ( cluster2 ) , whose pseudocode is given in algorithm [ alg : cluster2 ] , which imposes a lower bound on the number of growing steps applied to each cluster , where such a number is precomputed using the clustering algorithm from section [ clustering ] . let @xmath46 be an integral parameter . we have : [ correctness ] for any integer @xmath53 , with high probability algorithm cluster2@xmath54 computes a partition of @xmath26 into @xmath141 clusters of radius @xmath142 , where @xmath62 is the maximum radius of a cluster returned by cluster@xmath54 . the bound on @xmath143 is immediate . let @xmath17 be the number of clusters returned by the execution of cluster@xmath54 within cluster2@xmath54 . by theorem [ thm1 ] , we have that @xmath144 , with high probability . in what follows , we condition on this event . for @xmath145 , define @xmath146 as the smallest integer such that @xmath147 , and let @xmath148 . for @xmath149 , define the event @xmath150 `` at the end of iteration @xmath151 of the * for * loop , at most @xmath152 nodes are still uncovered '' . we now prove that the event @xmath153 occurs with high probability . observe that @xmath154 since @xmath155 clearly holds with probability one . consider an arbitrary @xmath63 , with @xmath156 , and assume that @xmath157 holds . we prove that @xmath158 holds with high probability . let @xmath66 be the set of nodes already covered at the beginning of iteration @xmath151 . since @xmath159 holds , we have that @xmath160 . clearly , if @xmath161 then @xmath158 must hold with probability one . thus , we consider only the case @xmath162 let @xmath163 and observe that since @xmath62 is the maximum radius of a partition of @xmath8 into @xmath17 clusters , we have that @xmath164 . by the definition of @xmath77 , there exist @xmath17 nodes , say @xmath165 , such that each node of @xmath76 is at distance at most @xmath77 from either @xmath66 or one of these nodes . let us consider the partition @xmath166 where @xmath79 is the set of nodes of @xmath76 which are closer to @xmath66 than to any of the @xmath80 s , while @xmath81 is the set of nodes of @xmath76 which are closer to @xmath80 than to @xmath66 or to any other @xmath82 ( with ties broken arbitrarily ) . let @xmath167 it is easy to see that @xmath168 . since we assumed that @xmath169 , we have that for every @xmath81 with @xmath170 , @xmath171 as a consequence , since @xmath145 , a new center will be chosen from @xmath81 in iteration @xmath151 with probability at least @xmath89 . by applying the union bound we conclude that in iteration @xmath151 a new center will fall in every @xmath81 with @xmath170 , and thus the number of uncovered nodes will at least halve , with probability at least @xmath172 . by multiplying the probabilities of the @xmath173 conditioned events , we conclude that event @xmath153 occurs with high probability . finally , one can easily show that , with high probability , in the first @xmath146 iterations , @xmath174 clusters are added and , by conditioning on @xmath153 , at the beginning of each iteration @xmath151 , @xmath175 , @xmath176 new clusters are added to @xmath177 , for a total of @xmath178 clusters . suppose we run cluster2 on a graph @xmath8 , for some @xmath179 , to obtain a set @xmath177 of clusters of maximum radius @xmath143 . let @xmath180 denote the quotient graph associated with the clustering , where the nodes correspond to the clusters and there is an edge between two nodes if there is an edge of @xmath8 whose endpoints belong to the two corresponding clusters . let @xmath181 be the diameter of @xmath180 . we have : [ segment ] if @xmath5 is the true diameter of @xmath8 , then @xmath182 , with high probability . let us fix an arbitrary pair of distinct nodes and an arbitrary shortest path @xmath183 between them . we show that at most @xmath184 clusters intersect @xmath183 ( i.e. , contain nodes of @xmath183 ) , with high probability . divide @xmath183 into _ segments _ of length @xmath62 , and consider one such segment @xmath185 . clearly , all clusters containing nodes of @xmath185 must have their centers belong to nodes at distance at most @xmath143 from @xmath185 ( i.e. , distance at most @xmath143 from the closest node of @xmath185 ) . recall that @xmath186 . for @xmath187 , let @xmath188 be the set of nodes whose distance from @xmath185 is between @xmath189 and @xmath190 , and observe that any cluster intersecting @xmath185 must be centered at a node belonging to one of the @xmath188 s . we claim that , with high probability , for any @xmath191 , there are @xmath192 clusters centered at nodes of @xmath188 which may intersect @xmath185 . fix an index @xmath191 , with @xmath187 , and let @xmath193 be the first iteration of the for loop of algorithm cluster2 in which some center is selected from @xmath188 . it is easy to see that , due to the smooth growth of the center selection probabilities , the number of centers selected from @xmath188 in iteration @xmath193 and in iteration @xmath194 is @xmath173 , with high probability . consider now a center @xmath37 ( if any ) selected from @xmath188 in some iteration @xmath195 . in order to reach @xmath185 , the cluster centered at @xmath37 must grow for at least @xmath189 steps . however , since in each iteration active clusters grow by @xmath196 steps , by the time the cluster centered at @xmath37 reaches @xmath185 , the nodes of @xmath185 have already been reached and totally covered by clusters whose centers have been selected from @xmath188 in iterations @xmath193 and @xmath194 or , possibly , by some other clusters centered outside @xmath188 . in conclusion , we have that the nodes of segment @xmath185 will belong to @xmath197 clusters , with high probability . the theorem follows by applying the union bound over all segments of @xmath183 , and over all pairs of nodes in @xmath8 . let @xmath198 . it is easy to see that @xmath199 . moreover , since @xmath186 and @xmath200 , we have from theorem [ segment ] that @xmath201 . the following corollary is immediate . [ diam - thm ] let @xmath8 be an @xmath1-node connected graph with diameter @xmath5 . then , the clustering returned by cluster2 can be used to compute two values @xmath202 such that @xmath203 , with high probability . in order to get a tighter approximation , as in @xcite , after the clustering we can compute the diameter @xmath204 of the following weighted instance of the quotient graph @xmath205 . specifically , we assign to each edge @xmath206 a weight equal to the length of the shortest path in @xmath8 that connects the two clusters associated with @xmath21 and @xmath37 and comprises only nodes of these two clusters . it is easy to see that @xmath207 is an upper bound to the diameter @xmath5 of @xmath8 , and @xmath208 . it is important to remark that while in @xcite the approximation factor for the diameter is proportional to the square root of the number of clusters , with our improved clustering strategy the approximation factor becomes independent of this quantity , a fact that will also be confirmed by the experiments . as we will see in the next section , the number of clusters , hence the size of the quotient graph , can be suitably chosen to reduce the complexity of the algorithm , based on the memory resources . as a final remark , we observe that the proof of theorem [ segment ] shows that for any two nodes @xmath209 in @xmath8 their distance @xmath210 can be upper bounded by a value @xmath211 . as a consequence , by running cluster2@xmath54 with @xmath212 and computing the @xmath213-size all - pairs shortest - path matrix of the ( weighted ) quotient graph @xmath180 we can obtain a linear - space distance oracle for @xmath8 featuring the aforementioned approximation quality , which is polylogarithmic for farther away nodes ( i.e. , nodes at distance @xmath214 . we now describe and analyze a distributed implementation of the clustering and diameter - approximation algorithms devised in the previous sections , using the mapreduce ( mr ) model introduced in @xcite . the mr model provides a rigorous computational framework based on the popular mapreduce paradigm @xcite , which is suitable for large - scale data processing on clusters of loosely - coupled commodity servers . similar models have been recently proposed in @xcite . an mr algorithm executes as a sequence of _ rounds _ where , in a round , a multiset @xmath45 of key - value pairs is transformed into a new multiset @xmath215 of pairs by applying a given reducer function ( simply called _ reducer _ in the rest of the paper ) independently to each subset of pairs of @xmath45 having the same key . the model features two parameters @xmath216 and @xmath217 , where @xmath216 is the maximum amount of global memory available to the computation , and @xmath217 is the maximum amount of local memory available to each reducer . we use to denote a given instance of the model . the complexity of an algorithm is defined as the number of rounds executed in the worst case , and it is expressed as a function of the input size and of @xmath216 and @xmath217 . considering that for big input instances local and global space are premium resources , the main aim of algorithm design on the model is to provide strategies exhibiting good space - round tradeoffs for large ranges of the parameter values . the following facts are proved in @xcite . [ prefixsorting ] the sorting and ( segmented ) prefix - sum primitives for inputs of size @xmath1 can be performed in @xmath218 rounds in with @xmath219 . [ matrixmult ] two @xmath220-matrices can be multiplied in @xmath221 rounds in . we can implement the sequence of cluster - growing steps embodied in the main loops of cluster and cluster2 as a progressive shrinking of the original graph , by maintaining clusters coalesced into single nodes and updating the adjacencies accordingly . each cluster - growing step requires a constant number of sorting and ( segmented ) prefix operations on the collection of edges . moreover , the assignment of the original graph nodes to clusters can be easily maintained with constant extra overhead . by using fact [ prefixsorting ] , we can easily derive the following result . [ clusteringmr ] cluster ( resp . , cluster2 ) can be implemented in the model so that , when invoked on a graph @xmath8 with @xmath1 nodes and @xmath9 edges , it requires @xmath222 rounds , where @xmath94 is the total number of cluster - growing steps performed by the algorithm . in particular , if @xmath223 , for some constant @xmath4 , the number of rounds becomes @xmath224 . the diameter - approximation algorithm can be implemented in the model by running cluster2@xmath54 for a value of @xmath74 suitably chosen to allow the diameter of the quotient graph to be computed efficiently . the following theorem shows the space - round tradeoffs attainable when @xmath217 is large enough . [ mr - diameter ] let @xmath8 be a connected graph with @xmath1 nodes , @xmath9 edges , doubling dimension @xmath225 and diameter @xmath5 . also , let @xmath226 be two arbitrary constants . on the model , with @xmath227 and @xmath228 , an upper bound @xmath229 to the diameter of @xmath8 can be computed in @xmath230 rounds , with high probability . fix @xmath231 so that cluster2@xmath54 returns @xmath232 clusters with high probability . ( in case the number of returned clusters is larger , we repeat the execution of cluster2 . ) let @xmath205 be quotient graph associated with the returned clustering . if @xmath233 , we can compute the diameter of @xmath180 in one round using a single reducer . otherwise , by employing the sparsification technique presented in @xcite we transform @xmath180 into a new graph @xmath234 with @xmath235 , whose diameter is a factor at most @xmath236 larger than the diameter of @xmath180 . the sparsification technique requires a constant number of cluster growing steps similar in spirit to those described above , which can be realized through a constant number of prefix and sorting operations . hence , the transformation can be implemented in @xmath237 rounds in . once @xmath238 is obtained , its diameter and the resulting approximation @xmath239 to @xmath5 can be computed in one round with a single reducer . therefore , by combining the results of lemmas [ radius - bound ] , [ correctness ] , and [ clusteringmr ] , we have that cluster@xmath54 runs in @xmath240 rounds , and cluster2@xmath54 runs @xmath241 rounds . hence , we have that the total number of rounds for computing @xmath239 is @xmath242 . alternatively , we can set @xmath243 so to obtain a quotient graph @xmath180 with @xmath244 nodes . we can compute the diameter of the quotient graph by repeated squaring of the adjacency matrix . by applying the result of fact [ matrixmult ] with @xmath245 and observing that @xmath246 , we conclude that the computation of the quotient graph diameter requires only an extra logarithmic number of rounds . in this fashion , the total number of rounds for computing @xmath239 becomes @xmath247 . the theorem follows by noting that for both the above implementations , the quality of the approximation is ensured by corollary [ diam - thm ] . we remark that while the upper bound on the approximation factor is independent of the doubling dimension of the graph , the round complexity is expressed as a function of it . this does not restrict the generality of the algorithm but allows us to show that for graphs with small doubling dimension , typically graphs with low expansion , the number of rounds can be made substantially smaller than the graph diameter and , in fact , this number decreases as more local memory is available for the reducers , still using linear global space . this feature represents the key computational advantage of our algorithm with respect to other linear - space algorithms , that , while yielding tighter approximations , require @xmath248 rounds . we tested our algorithms on a cluster of 16 hosts , each equipped with a 12 gb ram and a 4-core i7 processor , connected by a 10 gigabit ethernet network . the algorithms have been implemented using apache spark @xcite , a popular engine for distributed large - scale data processing . we performed tests on several large graphs whose main characteristics are reported in table [ tab : datasets ] . the first graph is a symmetrization of a subgraph of the twitter network obtained from the law website @xcite . the next four graphs are from the stanford large network datasets collection @xcite and represent , respectively , the livejournal social network and three road networks . the last graph is a synthetic @xmath249 mesh , which has been included since its doubling dimension is known , unlike the other graphs , and constant ( @xmath250 ) , hence it is an example of a graph where our algorithms are provably effective . we compared the quality of the clustering returned by algorithm cluster ( see section [ clustering ] ) against that of the clustering returned by the algorithm presented in @xcite and reviewed in section [ previouswork ] , which , for brevity , we call mpx . recall that cluster uses a parameter @xmath74 to control the number of clusters , while mpx uses ( an exponential distribution of ) parameter @xmath20 to decide when nodes are possibly activated as cluster centers , hence indirectly controlling the number of clusters . both algorithms aim at computing a decomposition of the graph into clusters of small radius , so we focused the experiments on comparing the maximum radius of the returned clusterings . however , since the minimum maximum radius attainable by any clustering is a nonincreasing function of the number of clusters , but neither algorithm is able to precisely fix such a number a priori , we structured the experiments as follows . we aimed at decomposition granularities ( i.e. , number of clusters ) which are roughly three orders of magnitude smaller than the number of nodes for small - diameter graphs , and roughly two orders of magnitude smaller than the number of nodes for large - diameter graphs . we ran mpx and cluster setting their parameters @xmath20 and @xmath74 so to obtain a granularity close enough to the desired one , and compared the maximum cluster radius obtained by the two algorithms . in order to be conservative , we gave mpx a slight advantage setting @xmath20 so to always yield a comparable but larger number of clusters with respect to cluster . table [ tab : comparison - miller ] shows the results of the experiments for the benchmark graphs . each row reports the graph , and , for each algorithm , the number of nodes ( @xmath251 ) and edges ( @xmath252 ) of the quotient graph associated with the clustering , and the maximum cluster radius ( @xmath253 ) . the table provides a clear evidence that our algorithm is more effective in keeping the maximum cluster radius small , especially for graphs of large diameter . this is partly due to the fact that mpx starts growing only a few clusters , and before more cluster centers are activated the radius of the initial clusters is already grown large . on the other hand , mpx is often more effective in reducing the number of edges of the quotient graph , which is in fact the main objective of the mpx decomposition strategy . this is particularly evident for the first two graphs in the table , which represent social networks , hence feature low diameter and high expansion ( thus , probably , high doubling dimension ) . in these cases , the few clusters initially grown by mpx are able to absorb entirely highly expanding components , thus resulting in a more drastic reduction of the edges . for the diameter approximation , we implemented a simplified version of the algorithm presented in section [ diameter ] , where , for efficiency , we used cluster instead of cluster2 , thus avoiding repeating the clustering twice . also , in order to get a tighter approximation , we computed the diameter of the weighted variant of the quotient graph as discussed at the end of section [ diameter ] . we performed three sets of experiments , which are discussed below . the first set of experiments aimed at testing the quality of the diameter approximation provided by our algorithm . the results of the experiments are reported in table [ tab : diameter - approximation ] . for each graph of table [ tab : datasets ] we estimated the diameter by running our algorithm with two clusterings of different granularities ( dubbed _ coarser _ and _ finer _ clustering , respectively ) reporting , in each case , the number of nodes ( @xmath251 ) and edges ( @xmath252 ) of the quotient graph @xmath180 , the approximation @xmath239 and the true diameter @xmath5 . since the quotient graphs turned out to be sufficiently sparse , the use of sparsification techniques mentioned in section [ mrimplementation ] was not needed . we observe that in all cases @xmath254 and , in fact , the approximation factor appears to decrease for sparse , long - diameter graphs . also , we observe that , as implied by the theoretical results , the quality of the approximation does not seem to be influenced by the granularity of the clustering . therefore , for very large graphs , or distributed platforms where individual machines are provided with small local memory , one can resort to a very coarse clustering in order to fit the whole quotient graph in one machine , and still obtain a good approximation to the diameter , at the expense , however , of an increased number of rounds , which are needed to compute the clustering . with the second set of experiments , we assessed the time performance of our algorithm against two competitors : hadi @xcite , which was reviewed in section [ previouswork ] and provides a rather tight diameter ( under)estimation ; and breadth first search ( bfs ) , which , as well known , can be employed to obtain an upper bound to the diameter within a factor two . hadi s original code , available from @xcite , was written for the hadoop framework . because of hadoop s known large overhead , for fairness , we reimplemented hadi in spark , with a performance gain of at least one order of magnitude . as for bfs , we implemented a simple and efficient version in spark . table [ usvshadi ] reports the running times and the diameter estimates obtained with the three algorithms where , for our algorithm , we used the finer clustering granularity adopted in the experiments reported in table [ tab : diameter - approximation ] . the figures in the table clearly show that hadi , while yielding a very accurate estimate of the diameter , is much slower than our algorithm , by orders of magnitude for large - diameter graphs . this is due to the fact that hadi requires @xmath255 rounds and in each round the communication volume is linear in the number of edges of the input graph . on the other hand bfs , whose approximation guarantee is similar to ours in practice , outperforms hadi and , as expected , is considerably slower than our algorithm on large - diameter graphs . indeed , bfs still requires @xmath255 rounds as hadi , but its aggregate communication volume ( rather than the per round communication volume ) is linear in the number of edges of the input graph . as remarked in the discussion following lemma [ radius - bound ] , a desirable feature of our strategy is its capability to adapt to irregularities of the graph topology , which may have a larger impact on the performance of the other strategies . in order to provide experimental evidence of this phenomenon , our third set of experiments reports the running times of our algorithm and bfs on three variants of the two small - diameter graphs ( livejournal and twitter ) obtained by appending a chain of @xmath256 extra nodes to a randomly chosen node , with @xmath257 , thus increasing the diameter accordingly , without substantially altering the overall structure of the base graph . the plots in figure [ fig : tails ] clearly show that while the running time of our algorithm is basically unaltered by the modification , that of bfs grows linearly with @xmath258 , as expected due to the strict dependence of the bfs number of rounds from the diameter . a similar behaviour is to be expected with hadi because of the same reason . putting it all together , the experiments support the theoretical analysis since they provide evidence that the main competitive advantages of our algorithm , which are evident in large - diameter graphs , are the linear aggregate communication volume ( as in bfs ) coupled with its ability to run in a number of rounds which can be substantially smaller than @xmath5 . we developed a novel parallel decomposition strategy for unweighted , undirected graphs which ensures a tighter control on both the number of clusters and their maximum radius , with respect to similar previous decompositions . we employed our decomposition to devise parallel polylogarithmic approximation algorithms for the @xmath0-center problem and for computing the graph diameter . the algorithms use only linear overall space and , for a relevant class of graphs ( i.e. , those of low doubling dimension ) , their parallel depth can be made substantially sublinear in the diameter as long as local memories at the processing nodes are sufficiently large but still asymptotically smaller than the graph size . while the improvement of the approximation bounds and the parallel depth of our algorithms is a natural direction for further research , the extension of our findings to the realm of weighted graphs is a another challenging and relevant open problem . we are currently exploring this latter issue and have devised a preliminary decomposition strategy that , together with the number clusters and their weighted radius , also controls their hop radius , which governs the parallel depth of the computation . g. e. blelloch , a. gupta , i. koutis , g. l. miller , r. peng , and k. tangwongsan . near linear - work parallel sdd solvers , low - diameter decomposition , and low - stretch subgraphs . in _ spaa _ , pages 1322 , 2011 . m. zaharia , m. chowdhury , t. das , a. dave , j. ma , m. mccauly , m. j. franklin , s. shenker , and i. stoica . resilient distributed datasets : a fault - tolerant abstraction for in - memory cluster computing . in _ nsdi _ , pages 1528 , 2012 .

we develop a novel parallel decomposition strategy for unweighted , undirected graphs , based on growing disjoint connected clusters from batches of centers progressively selected from yet uncovered nodes . with respect to similar previous decompositions , our strategy exercises a tighter control on both the number of clusters and their maximum radius . we present two important applications of our parallel graph decomposition : ( 1 ) @xmath0-center clustering approximation ; and ( 2 ) diameter approximation . in both cases , we obtain algorithms which feature a polylogarithmic approximation factor and are amenable to a distributed implementation that is geared for massive ( long - diameter ) graphs . the total space needed for the computation is linear in the problem size , and the parallel depth is substantially sublinear in the diameter for graphs with low doubling dimension . to the best of our knowledge , ours are the first parallel approximations for these problems which achieve sub - diameter parallel time , for a relevant class of graphs , using only linear space . besides the theoretical guarantees , our algorithms allow for a very simple implementation on clustered architectures : we report on extensive experiments which demonstrate their effectiveness and efficiency on large graphs as compared to alternative known approaches .

introduction previous work clustering algorithm diameter estimation distributed implementation and performance analysis experimental results conclusions

arxiv

the phase locked - loop ( pll ) circuits were invented in the first half of the twentieth century and nowadays are widely used in modern telecommunications and computers . pll is essentially a nonlinear control system and its _ real model _ is described by a nonlinear nonautonomous system of differential equations ( _ mathematical model in the signal space _ ) . in practice , simulation and simplified mathematical models are widely used for the analysis of pll - based circuits @xcite . in the following it will be shown that 1 ) the use of simplified mathematical models and 2 ) the application of non rigorous methods of analysis ( e.g. , simulation ) may lead to wrong conclusions concerning the operability of _ real model _ of classical pll . consider the classical pll nonlinear models in the signal and signal s phase spaces @xcite . * _ real model of the classical pll in the signal space _ [ pll - signal - ics ] ) or its _ nonlinear mathematical model in the signal space _ ( corresponds to the spice - level simulation ) : + @xmath0 + * _ model of the classical pll in signal s phase space _ [ pll - phase - ics ] ) ; system with averaged @xmath1 gives the _ nonlinear mathematical model in signal s phase space _ : @xmath2 + let us construct matlab simulink model , which corresponds to the model in the signal space ( see fig . [ simulink_model_signal ] ) . here all elements are standard blocks from simulink library except for the vco . the vco subsystem is shown in fig . [ simulink_vco_signal ] . the vco subsystem consists of one input , which is amplified by l ( gain block ) . the integration of the sum of amplified input signal and the vco free - running frequency omega_free forms the phase of the vco output . the vco output corresponds to @xmath3 . now consider matlab simulink model , which corresponds to the model in signal s phase space ( see fig . [ simulink_model_phase ] ) . the pd subsystem is shown in fig . [ simulink_pd_phase ] . the vco subsystem in signal s phase space is shown in fig . [ simulink_vco_phase ] . the vco output in signal s phase space corresponds to @xmath4 . consider a passive lead - lag loop filter with the transfer function @xmath5 , @xmath6 , @xmath7 and the corresponding parameters @xmath8 , @xmath9 , @xmath10 , @xmath11 . the input signal frequency is @xmath12 , initial phase is zero : @xmath13 , and the vco input gain @xmath14 . this example shows the importance of initial state of filter ( see fig . [ lpf_ics ] ) : while the real model ( see fig . [ pll - signal - ics ] ) with nonzero initial state of loop filter @xmath15 does not acquire lock ( black color ) , the same real model with zero initial state of loop filter @xmath16 acquires lock ( red color ) . here the vco free - running frequency @xmath17 . for real model with nonzero initial state of loop filter ( red ) , real model with zero initial state of loop filter ( black ) . ] this example shows that the initial phase difference @xmath18 between the vco signal and input signal may affect stability of the classical pll . in fig . [ ex_phase ] the real model ( see fig . [ pll - signal - ics ] ) with zero initial phase difference acquire lock ( red color ) , the same real model with nonzero initial phase difference @xmath19 is out of lock ( black color ) . here the vco free - running frequency @xmath17 and the initial state of loop filter is @xmath20 . for real model with nonzero initial phase difference ( black ) , real model with zero initial phase difference ( red ) . ] examples 1 and 2 shows that while the term `` initial frequency '' ( without an explanation ) is sometimes used instead of the the term `` free - running frequency '' in engineering definitions of various stability ranges , it may lead to a misunderstanding ( see corresponding discussion in @xcite ) . this example shows that the pll model in signal s phase space may not be equivalent to the pll real model in the signal space . in fig . [ double_freq ] the real model ( see fig . [ pll - signal - ics ] ) does not acquire lock ( red color ) , the equivalent signal s phase space model acquires lock ( black color ) . here the vco free - running frequency @xmath17 , the initial state of loop filter is @xmath21 , and the initial phase difference @xmath22 . for signal s phase model ( black ) , real model ( red ) . ] these examples shows the importance of analytic methods for investigation of pll stability . more precisely , it is shown that the simulation may lead to wrong results . in fig . [ pll_hidden ] the pll model in signal s phase space simulated with relative tolerance `` 1e-3 '' does not acquire lock ( black color ) , but the pll model in signal s phase space simulated with standard parameters ( relative tolerance set to `` auto '' ) acquires lock ( red color ) . in @xcite the simetrics spice model of the two - phase pll with lead - lag filter gives two essentially different results with default sampling step and minimum sampling step set to @xmath23 . ] . here the input signal frequency is @xmath24 , the vco free - running frequency @xmath25 , the vco input gain is @xmath26 , the initial state of loop filter is @xmath27 , and the initial phase difference is @xmath28 . for signal s phase space model with standard integration parameters ( red ) , signal s phase space model with relative tolerance set to `` 1e-3''(black ) . ] consider now a phase portrait ( the loop filter state @xmath29 versus the phase difference @xmath30 ) corresponding to signal s phase model ( see fig . [ pll_hidden_phase_portret ] ) . the solid blue line in fig . [ pll_hidden_phase_portret ] corresponds to the trajectory with the loop filter initial state @xmath31 and the vco phase shift @xmath32 rad . this line tends to the periodic trajectory , therefore it will not acquire lock . the solid red line corresponds to the trajectory with the loop filter initial state @xmath33 and the vco initial phase @xmath34 . this trajectory lies just under the unstable periodic trajectory and tends to a stable equilibrium . in this case pll acquires lock . all trajectories between stable and unstable periodic trajectories tend to the stable one ( see , e.g. , a solid green line ) . therefore , if the gap between stable and unstable periodic trajectories is smaller than the discretization step , the numerical procedure may slip through the stable trajectory . in other words , the simulation will show that the pll acquires lock , but in reality it is not the case . the considered case corresponds to the coexisting attractors ( one of which is so - called hidden oscillation ) and the bifurcation of birth of semistable trajectory @xcite . an oscillation in a dynamical system can be easily localized numerically if the initial conditions from its open neighborhood lead to long - time behavior that approaches the oscillation . thus , from a computational point of view , it is natural to suggest the following classification of attractors , based on the simplicity of finding the basin of attraction in the phase space @xcite : _ an attractor is called a _ hidden attractor _ if its basin of attraction does not intersect with small neighborhoods of equilibria , otherwise it is called a _ self - excited attractor_. _ for a _ self - excited attractor _ its basin of attraction is connected with an unstable equilibrium and , therefore , self - excited attractors can be localized numerically by the _ standard computational procedure _ , in which after a transient process a trajectory , started from a point of an unstable manifold in a neighborhood of an unstable equilibrium , is attracted to the state of oscillation and traces it . thus self - excited attractors can be easily visualized . in contrast , for a hidden attractor its basin of attraction is not connected with unstable equilibria . for example , hidden attractors are attractors in the systems with no equilibria or with only one stable equilibrium ( a special case of multistable systems and coexistence of attractors ) . the derivation of mathematical model in signal s phase space and the use of the results of its analysis to draw conclusions about the behavior of real model in the signal space have need for a rigorous foundation . but the attempts to justify analytically the reliability of conclusions , based on such engineering approaches , and to study the nonlinear models of pll - based circuits are quite rare in the modern engendering literature @xcite . one of the reasons is that `` _ nonlinear analysis techniques are well beyond the scope of most undergraduate courses in communication theory _ '' @xcite . note once more that various simplifications and the analysis of linearized models of control systems may result in incorrect conclusions ( see , e.g. , the counterexamples to the filter hypothesis , aizerman s and kalman s conjectures on the absolute stability of nonlinear control systems @xcite , and the perron effects of the largest lyapunov exponent sign reversals @xcite , etc . ) . in the work it is shown that 1 ) the consideration of simplified models , constructed intuitively by engineers and 2 ) the application of non - rigorous methods of analysis ( e.g. , simulation and linearization ) can lead to wrong conclusions concerning the operability of the classical phase - locked loop . similar examples for nonlinear costas loop models can be found in @xcite . authors were supported by saint - petersburg state university and russian scientific foundation . the authors would like to thank roland e. best ( the founder of best engineering company , switzerland ; the author of the bestseller on pll - based circuits @xcite ) for valuable discussions . g. a. leonov , n. v. kuznetsov , m. v. yuldahsev , and r. v. yuldashev , `` analytical method for computation of phase - detector characteristic , '' _ ieee transactions on circuits and systems - ii : express briefs _ , vol . 59 , no . 10 , pp . 633647 , 2012 . g. a. leonov , n. v. kuznetsov , m. v. yuldashev , and r. v. yuldashev , `` nonlinear dynamical model of costas loop and an approach to the analysis of its stability in the large , '' _ signal processing _ , vol . 108 , pp . 124135 , 2015 . r. e. best , n. v. kuznetsov , g. a. leonov , m. v. yuldashev , and r. v. yuldashev , `` simulation of analog costas loop circuits , '' _ international journal of automation and computing _ , vol . 11 , no . 6 , pp . 571579 , 2014 , 10.1007/s11633 - 014 - 0846-x . n. kuznetsov , g. leonov , m. yuldashev , and r. yuldashev , `` rigorous mathematical definitions of the hold - in and pull - in ranges for phase - locked loops , '' in _ 1st ifac conference on modelling , identification and control of nonlinear systems_.1em plus 0.5em minus 0.4emifac proceedings volumes ( ifac - papersonline ) , 2015 , pp . 720723 . n. v. kuznetsov , g. a. leonov , m. v. yuldashev , and r. v. yuldashev , `` hold - in , pull - in , and lock - in ranges of pll - based circuits : rigorous mathematical definitions and limitations of classical theory , '' _ arxiv e - prints _ , 2015 . g. bianchi , n. kuznetsov , g. leonov , m. yuldashev , and r. yuldashev , `` limitations of pll simulation : hidden oscillations in spice analysis , '' _ arxiv:1506.02484 _ , 2015 , http://arxiv.org/pdf/1506.02484.pdf . g. a. leonov and n. v. kuznetsov , `` hidden attractors in dynamical systems . from hidden oscillations in hilbert - kolmogorov , aizerman , and kalman problems to hidden chaotic attractors in chua circuits , '' _ international journal of bifurcation and chaos _ , 23 , no . 1 , 2013 , art . 1330002 . n. kuznetsov , g. leonov , m. yuldashev , and r. yuldashev , `` nonlinear analysis of classical phase - locked loops in signal s phase space , '' _ ifac proceedings volumes ( ifac - papersonline ) _ , vol . 82538258 , 2014 . n. v. kuznetsov , g. a. leonov , and v. i. vagaitsev , `` analytical - numerical method for attractor localization of generalized chua s system , '' _ ifac proceedings volumes ( ifac - papersonline ) _ , vol . 4 , no . 1 , pp . 2933 , 2010 . v. o. bragin , v. i. vagaitsev , n. v. kuznetsov , and g. a. leonov , `` algorithms for finding hidden oscillations in nonlinear systems . the aizerman and kalman conjectures and chua s circuits , '' _ journal of computer and systems sciences international _ 50 , no . 4 , pp . 511543 , 2011 . n. v. kuznetsov and g. a. leonov , `` on stability by the first approximation for discrete systems , '' in _ 2005 international conference on physics and control , physcon 2005 _ , vol . proceedings volume 2005.1em plus 0.5em minus 0.4emieee , 2005 , pp . 596599 . n. kuznetsov , o. kuznetsova , g. leonov , p. neittaanmaki , m. yuldashev , and r. yuldashev , `` simulation of nonlinear models of qpsk costas loop in matlab simulink , '' in _ 2014 6th international congress on ultra modern telecommunications and control systems and workshops ( icumt ) _ , vol . 2015-january.1em plus 0.5em minus 0.4emieee , 2014 , pp . n. kuznetsov , o. kuznetsova , g. leonov , s. seledzhi , m. yuldashev , and r. yuldashev , `` bpsk costas loop : simulation of nonlinear models in matlab simulink , '' in _ 2014 6th international congress on ultra modern telecommunications and control systems and workshops ( icumt ) _ , vol . 2015-january.1em plus 0.5em minus 0.4emieee , 2014 , pp . n. kuznetsov and g. leonov , `` hidden attractors in dynamical systems : systems with no equilibria , multistability and coexisting attractors , '' _ ifac proceedings volumes ( ifac - papersonline ) _ , vol . 19 , pp . 54455454 , 2014 . e. v. kudryasoha , o. a. kuznetsova , n. v. kuznetsov , g. a. leonov , s. m. seledzhi , m. v. yuldashev , and r. v. yuldashev , `` nonlinear models of bpsk costas loop , '' _ icinco 2014 - proceedings of the 11th international conference on informatics in control , automation and robotics _ , vol . 1 , pp . 704710 , 2014 . r. best , n. kuznetsov , o. kuznetsova , g. leonov , m. yuldashev , and r. yuldashev , `` a short survey on nonlinear models of the classic costas loop : rigorous derivation and limitations of the classic analysis , '' in _ american control conference ( acc)_.1em plus 0.5em minus 0.4emieee , 2015 , pp . 12961302 , http://arxiv.org/pdf/1505.04288v1.pdf .

nonlinear analysis of the classical phase - locked loop ( pll ) is a challenging task . in classical engineering literature simplified mathematical models and simulation are widely used for its study . in this work the limitations of classical engineering phase - locked loop analysis are demonstrated , e.g. , hidden oscillations , which can not be found by simulation , are discussed . it is shown that the use of simplified dynamical models and the application of simulation may lead to wrong conclusions concerning the operability of pll - based circuits .

introduction simulation of the classical phase-locked loop in matlab simulink conclusion

arxiv

the recent discovery of a new boson at the large hadron collider ( lhc)@xcite will lead to a detailed study of its properties in the next years , and will require a careful comparison of the experimental data with the results of precise calculations for the higgs sector of the standard model ( sm ) . vector boson fusion ( vbf ) , @xmath0 , is the second largest channel for detecting the higgs boson . it is a pure electroweak process and its study is very important for determining the couplings and for a deeper knowledge of the higgs sector@xcite . however , at the 10 tev energy scale of lhc , even the electroweak radiative corrections become important @xcite and should be included in computer simulations together with qcd corrections @xcite . with a partonic center - of - mass energy @xmath1 of several tev - more than an order of magnitude larger than the masses @xmath2 of the gauge bosons - the radiative corrections contain large sudakov logarithmic terms @xmath3 where @xmath4 are the weak coupling constants and the logarithms @xmath5 emerge from the two different energy scales . these terms dominate the perturbative expansion and may even require resummation when the fixed order perturbation expansion breaks down . however , for vbf the scattering amplitude is proportional to the vacuum expectation value ( vev ) , and standard resummation methods do not apply because the effective operator is not a gauge singlet . that makes vbf a special interesting process to deal with . the sudakov logarithms can be regarded as infrared logarithms since they diverge as @xmath6 , and by using an effective theory they can be converted to ultraviolet logarithms and summed by standard renormalization group techniques . quite recently the soft - collinear effective theory ( scet ) @xcite has been shown to provide a simple way to obtain the sum of the series of leading - logs ( ll ) @xmath7 , next - to - leading - logs ( nll ) @xmath8 , next - to - next - to - leading - logs ( nnll ) @xmath9 , etc . @xcite . in the effective theory the single terms contributing to the scattering amplitude are multiplied by the general factor @xcite @xmath10 { \nonumber}\\ & \quad \times \exp\left\{-\int_{\mu_l}^{\mu_h } \frac { { \rm d}\mu}{\mu } \left [ a(\alpha(\mu))\log\frac{\mu^2}{\mu^2_h}+b ( \alpha(\mu ) ) \right ] \right\ } { \nonumber}\\ & \quad \times \exp f ( \alpha(\mu_h))\end{aligned}\ ] ] where @xmath11 is the low energy scale and @xmath12 is the high energy scale . the coefficient @xmath13 is the high scale matching coefficient , @xmath14 and @xmath15 are the low scale matching coefficients , @xmath16 is the non - cusp anomalous dimension and @xmath17 is the coefficient of the cusp anomalous dimension . the ll series is summed by the one - loop cusp anomalous dimension ; the sum of the nll series requires the two - loop cusp anomalous dimension , the one - loop non - cusp anomalous dimension , and the one - loop low scale matching coefficient @xmath15 ; the nnll series is given by the the three - loop cusp , two - loop @xmath16 and @xmath15 , one - loop @xmath14 and @xmath13 , etc . in fact the exponentiated form of eq.([amp ] ) only requires the inclusion of electroweak corrections at low orders , while the unexponentiated form of fixed - order calculations would require electroweak corrections of any order for achieving the same accuracy . in some recent papers @xcite the method as been used for calculating the electroweak corrections to higgs production via vbf . numerical results at nll order were obtained for the cross section , including the effect of parton distribution functions ( pdfs ) @xcite . most of the corrections were obtained in analytical form by scet and might be easily included in the other software packages that have been developed , without the need of tedious loop calculations . while pdfs imply a 3% error , many of the retained terms at nll order are very small , less than 1% , and could be neglected in order to speed up the numerical integration of the cross section . in this paper the simpler ll order resummation is compared with the full nll calculation for higgs production via vbf , and the accuracy of the two approximations is discussed . the ll calculation of the cross section is found to deviate from the nll result by less than @xmath18 up to 10 tev center - of - mass energy . while the present study gives a useful evaluation of the accuracy in the scet resummation , it provides a very simple and fast analytical way for including the electroweak radiative corrections in software packages . in fact at nll order the one - loop anomalous dimension turns out to be a @xmath19 matrix in the operator basis @xcite , and the running integral in eq.([amp ] ) requires a numerical computation . moreover two - loop beta functions are required for the running of the couplings at the same order , yielding coupled equations that again must be solved by a numerical routine . on the other hand , at ll order the only one - loop term that is required is the cusp anomalous dimension , and by use of the simple uncoupled one - loop beta functions the running integral in eq.([amp ] ) is analytical , yielding a diagonal correction factor for the differential cross section . moreover , an exact cancellation of terms yields the same correction factor that would be found for the quark scattering @xmath20 , plus higgs rescattering and wave function renormalization terms . nll terms are still shown to be relevant for a full description of the dependence on scattering angles of the differential cross section , but such small dependence is averaged in the integrated cross section . the paper is organized as follows : in sec . [ sec : kinematics ] the kinematics of the vbf process is described and some details on the integration of the cross section are reported ; the general calculation framework is described in sec . [ sec : operators ] where the operator basis set is defined for the effective theory and matched onto the low scale and high scale gauge theories ; in sec . [ sec : running ] the running of the wilson coefficients is discussed and the anomalous dimension is shown to take a simple diagonal form at ll order ; explicit analytical expressions are derived for the running integral in sec . [ sec : llorder ] and the numerical results at ll order and nll order ar compared with other fixed order perturbative calculations in sec . [ sec : numerical ] . [ fig : tree ] , @xmath21 and @xmath22-channel diagrams are reported.,title="fig:",scaledwidth=48.0% ] at tree - level , the feynman diagrams for the process @xmath23 are shown in fig.1 . the two outgoing jets have a large rapidity gap that characterizes the vbf channel , and the background is usually suppressed by cuts ( for vbf cuts see e.g. refs . @xcite ) . we denote by @xmath24 , @xmath25 the momenta of the incoming quarks , and by @xmath26 , @xmath27 the momenta of the outgoing quarks ( jets ) . the momentum of the higgs boson is denoted by @xmath28 , and it is assumed to be on - shell , @xmath29 . quark masses are neglected , @xmath30 , and all momenta are taken to be incoming , according to the conventions of ref . the kinematic can be expressed in terms of generalized mandelstam variables : @xmath31 in scet , a set of light - cone vectors is associated to each collinear direction . they are defined as @xmath32 , @xmath33 , with the plus ( minus ) sign for incoming ( outgoing ) particles . for the quarks @xmath34 while for the higgs @xmath35 . the colliding quarks ( @xmath36 ) carry a fraction @xmath37 of the total hadron momenta , @xmath38 where @xmath39 is the center - of - mass energy , and the fractions @xmath37 are integrated over in the cross section by use of pdfs that describe their distribution in the proton . the momenta of the outgoing jets can be written as @xmath40 where @xmath41 are the energies and @xmath42 the angles with the axis of the beam , while @xmath43 is the azimuthal angle between the outgoing quarks . the momentum of the higgs is fixed by momentum conservation . in the effective theory , in order to evaluate the cross section , we square the matrix elements of the effective operators , sum over helicities , flavors , channels and integrate over phase space : @xmath44 where @xmath45 are the effective operators and @xmath46 are the pdf for the flavor @xmath47 at momentum fraction @xmath48 , and a factor of 1/2 must be included for identical particles in symmetric phase - space integrations . the wilson coefficients @xmath49 are obtained in terms of tree - level high scale coefficients @xmath50 by matching at the high scale , running down to the low scale and matching at the low scale according to eq.([amp ] ) , @xmath51 which must be regarded as a matrix product in the operator basis set . the three - body phase space @xmath52 reads@xcite @xmath53 \ , { \mathrm{d}}\varphi\ , \theta\bigg(\sum_{j=1}^4 p_j^0\bigg ) { \nonumber}\\ & \quad \times { \delta}\bigg[\bigg(\sum_{j=1}^4 p_j\bigg)^2 - m_h^2\bigg ] { \nonumber}\\ & = \frac{1}{2 ^ 6 \pi^4 } \int\ ! { \mathrm{d}}(\cos \theta_3)\ , { \mathrm{d}}(\cos \theta_4)\ , { \mathrm{d}}\varphi\ , { \mathrm{d}}e_3\ , \frac{e_3 f_1}{f_2 ^ 2 } \,.\end{aligned}\ ] ] where @xmath54 { e_\mathrm{cm}}e_3 { \nonumber}\\ f_2 & = [ x_1+x_2 + ( x_2 - x_1)\cos \theta_4 ] { e_\mathrm{cm}}-2[1-\cos \theta_3 \cos \theta_4 { \nonumber}\\ & \quad - \cos \varphi \sin \theta_3 \sin \theta_4 ] e_3\,.\end{aligned}\ ] ] the remaining integrals in eq . must be carried out numerically with the boundary conditions @xmath55{e_\mathrm{cm } } } \,,\end{aligned}\ ] ] while we already included the bounds of the pdfs , the boundaries can be restricted by further cuts that we might impose on the phase space in the integration . electroweak corrections can be obtained by scet in the framework of ref . the extension to the vbf process was derived in ref . @xcite , and the explicit expressions are reported in ref . here we only consider the sm gauge group . extensions like the minimal left - right symmetric gauge group@xcite will be the subject of an other paper . the first step consists of matching onto scet at a high scale @xmath56 . here the effects of symmetry breaking are suppressed , and the matching can be done in the unbroken gauge theory . next we run the effective operators down to a low scale @xmath57 by renormalization group equations . at the low - scale , the @xmath58 and @xmath59 boson are integrated out : we match onto a @xmath60 effective theory , only containing gluons and photons . in this low - scale matching , the effects of @xmath61 symmetry breaking must be considered . they only enter in this low - energy matching . at the high scale , the operators for vbf are given by @xcite @xmath62 in the higgs sector we introduce the operators @xmath63 @xmath64 where @xmath65 is the collinear scalar doublet @xmath66 that gives rise to a higgs boson , with its collinear wilson line @xmath67 . the field @xmath68 is a soft scalar that gives rise to a vev when the symmetry is broken . for the quarks we introduce the operators @xmath69 with the subscript @xmath70 that labels the momentum @xmath71 of the particle . collinear wilson lines @xmath72 are included in all these fermion fields , according to collinear gauge invariance . in order to keep the notation as general as possible , the projectors @xmath73 have been suppressed , and both left- and right - handed quarks have been allowed . of course the field @xmath74 is supposed to be a fermion doublet ( singlet ) if it is left - handed ( right - handed ) . each helicity is considered separately and the contributions are combined at the end . it is quite obvious that the operators @xmath75 and @xmath76 can only make sense if the quarks are left - handed , while the operators @xmath77 and @xmath78 are allowed if at least one of the incoming particles is left - handed . as discussed in ref.@xcite this operator basis is not complete , because it only suffices for quarks and for the @xmath79-channel . however for incoming anti - quarks and for the @xmath80-channel the corresponding operators are obtained by interchanging the particle labels . thus we will only discuss the case of incoming quarks in the @xmath79-channel . at ll and nll order the tree - level high - scale matching suffices , and can be done in the unbroken phase of the electroweak gauge theory . the coupling of the scalar doublet to the gauge fields is described by the lagrangian terms @xmath81 in fig . 1 we only need to consider the @xmath79-channel , because the @xmath22-channel contribution is suppressed in vbf . matching onto the operators in eq . , yields @xmath82 while the other coefficients vanish at tree level . in order to keep the notation as general as possible , a new variable @xmath83 has been defined : @xmath84 if the particle with label @xmath47 is left - handed and @xmath85 if the particle is right - handed . the hypercharge is @xmath86 for left - doublets , while for right - handed particles @xmath87 for up - type quarks and @xmath88 for down - type quarks . the running of the wilson coefficients follows by the rg equation @xmath89 in terms of the anomalous dimension @xmath90 of the operators . that is a matrix equation , corresponding to the 10 operators in eq . . using the notation of eq.([amp ] ) the anomalous dimension can be written as the sum of cusp and non - cusp terms @xmath91 we only need the one - loop cusp anomalous dimension at ll order , while two - loop cusp and one - loop non - cusp terms were required at nll order and were reported in detail in ref . @xcite . finally , let us consider the low energy matching . the tree - level result suffices at ll order . the one - loop calculation was required at nll order and was also reported in ref . @xcite . at low energies we must take into account the effects of electroweak symmetry breaking . we match onto a basis of operators in the broken phase of the gauge group . for the higgs part of the operator , given in eq . , the soft scalar field simply attains a vev and the collinear scalar field produces a higgs . thus at tree - level the low energy matching yields @xmath92 where @xmath93 is the vev and @xmath94 is the higgs field . the quark part in eq . gets matched onto the set @xmath95 here @xmath80 and @xmath96 denote up and down - type quarks . for operators @xmath97 , the pairs of fields @xmath98 can be left - handed or right - handed , whereas in @xmath99 and @xmath100 all fields must be left handed . we match \{@xmath101 , @xmath102 } onto the operators @xmath103 in eq . , while we ignore @xmath104 and @xmath105 which vanish at tree level . the matching is described by the @xmath106 matrix @xmath107 which is defined according to @xmath108 so that @xmath109 the matrix @xmath107 follows @xmath110 at ll order , one - loop corrections enter through the cusp anomalous dimension . for vbf the one - loop terms contributing to the anomalous dimension were derived in ref . @xcite and reported in detail in ref . they can be written as the sum of three terms @xmath111 the first term @xmath112 is the collinear anomalous dimension , it is diagonal and contains the large logarithms that contribute to the cusp anomalous dimension , plus wave function renormalization terms . resummation at ll order requires the inclusion of these one - loop terms , and they are easily obtained by the sum of single - particle collinear terms @xcite for all the external particles @xmath113 as reported in ref . @xcite , with the shorthand @xmath114 the quark one - loop collinear terms @xmath115 read @xmath116 l_i - \frac{9 { \alpha}_2}{16\pi } \eta_i - \frac{3 { \alpha}_1}{4\pi } y_i^2\,,\ ] ] while the higgs one - loop collinear term @xmath117 is @xmath118 l_h - \frac{3 { \alpha}_2}{4\pi } - \frac{{\alpha}_1}{4\pi } + \frac{3y_t^2}{16\pi^2}\,,\ ] ] where we included the contribution from the top yukawa @xmath119 to the higgs wave function renormalization . the second term @xmath120 in eq.([eq : ga_total ] ) is the soft anomalous dimension that is obtained by summing over the soft functions @xcite , and is reported in ref . @xcite in some detail . this soft term does not contain any large logarithm and does not contribute to the cusp anomalous dimension . it only depends on the scattering angles of the external particles , and its effect largely cancels in the integration of the cross section . at ll order this term can be neglected . the third term @xmath121 in eq.([eq : ga_total ] ) is a specific new contribution occurring in the vbf process @xcite , and can be written as the sum of @xmath122 and @xmath123 parts plus a term @xmath124 arising from the rescattering of the higgs boson . the first two parts contain cusp diagonal terms proportional to @xmath125 , and smaller non - cusp off - diagonal terms that do not depend on the scale @xmath126 . at ll order we only need to retain the diagonal cusp terms and the rescattering term that is non - cusp but is large because of the large self - coupling @xmath127 of the higgs boson . we also include the diagonal wave function renormalization terms . according to ref . @xcite , the cusp parts of the @xmath122 contribution read @xmath128\ ] ] for the coefficients @xmath129 , @xmath130 , @xmath131\ ] ] for the coefficient @xmath132 and @xmath133\ ] ] for the coefficient @xmath134 . the cusp part of the @xmath123 contribution is @xmath135,\ ] ] and the higgs rescattering contribution is given by @xmath136 where @xmath137 for the coefficient @xmath138 . summing all the terms , we find that the dependence on @xmath139 cancels ( up to non - cusp terms like the difference @xmath140 that can be neglected at ll order ) , yielding the following simple result for the total anomalous dimension @xmath141 where the sum is over the four external quarks . here @xmath142 is the sum of the retained diagonal non - cusp terms , including constant wave function renormalization and higgs rescattering terms @xmath143 -\frac{3{\alpha}_1}{4\pi}\left [ \sum_i y_i^2 + \frac{1}{2}\right ] + \frac{3y_t^2}{16\pi^2}+\hat { \gamma}^\lambda\ ] ] where @xmath144 for the coefficient @xmath138 . some of these terms are small , but they are constant , do not depend on scattering angles , and their weight might sum up in the integration of the cross section . all other non - cusp terms have been neglected . the simple result of eq.([eq : ga_tot ] ) says that the one - loop cusp anomalous dimension is the same that we would obtain by the sum of the collinear terms for the external quarks , neglecting the higgs particle . however the higgs momentum would affect the kinematic of the quarks anyway . moreover the higgs rescattering and wave function renormalization terms are not small and have been included in the non - cusp part @xmath145 of the anomalous dimension . at ll order the running of the coupling constant can be evaluated by uncoupled one - loop beta functions . that , together with the simple diagonal form of the one - loop cusp anomalous dimension in eq.([eq : ga_tot ] ) , allows for a fully analytical evaluation of the electroweak radiative corrections . by insertion of eq.([eq : d0 ] ) in eq.([amp ] ) , the low scale wilson coefficients follow from eq.([uc ] ) that now reads @xmath146 where the product runs over the four couplings @xmath147 , having defined @xmath148 and @xmath149 , and the exponentiated running factors follow by integration of the corresponding anomalous dimension @xmath150 here @xmath151 is the term proportional to the coupling @xmath152 in the anomalous dimension for the wilson coefficient @xmath153 . by inspection of eq.([eq : ga_tot ] ) , we see that such terms can be written as functions of @xmath154 @xmath155\ ] ] where for each coupling @xmath152 there is a different coefficient @xmath156 , while @xmath157 is a function of the external momenta @xmath158 that in general also depends on the chosen coefficient @xmath153 . the explicit expressions of the functions are @xmath159 \ , , { \nonumber}\\ { \cal f}^{(2)}_x(p)=&\frac{3}{4\pi}\sum_i \eta_i \log({\bar{n}}_i \cdot p_i ) - \frac{9}{16\pi}\left [ \sum_i \eta_i + d^{{\mathcal{o}}}\right ] \ , , { \nonumber}\\ { \cal f}^{(y)}_x(p)=&\frac{3}{4\pi } \ , , { \nonumber}\\ { \cal f}^{(\lambda)}_x(p)=&\frac{c^{{\mathcal{o}}}}{\pi } \ , , \end{aligned}\ ] ] while the coefficients are @xmath160 in all these definitions the sums are over the external quarks . the couplings also have an implicit dependence on @xmath79 , dictated by the one - loop beta - functions @xmath161 these can be easily integrated , yielding @xmath162 where the one - loop coefficients are well known@xcite @xmath163 by a simple integration , the running factors in eq.([eq : a ] ) take the explicit general form @xmath164 } } \,\end{aligned}\ ] ] and their product in eq.([eq : wc ] ) gives an analytical expression for the electroweak radiative correction at ll order . the cross section follows by insertion of the low scale wilson coefficients from eq.([eq : wc ] ) in the phase space integral of eq.([eq : si_ew ] ) and integrating by pdfs . the closed analytical form of eq.([eq : aint ] ) greatly speeds up the numerical integration of the cross section at ll order compared with nll order . we discuss the accuracy of the two orders in the next section by a direct comparison . in this section the accuracy of the approximation is tested by a comparison between ll and nll orders . all the details of the numerical integration are kept exactly the same as reported in ref . @xcite where the cross section was evaluated at nll order . we summarize them briefly . the low energy matching scale is chosen to be @xmath165 . the couplings and parameters of the standard model have been set at the electroweak scale @xmath165 according to the data of ref.@xcite . the higgs mass is assumed to be @xmath166 gev . the high scale @xmath167 is set at the larger value between @xmath168 and the geometric average @xmath169 . at this scale the sum of logarithmic terms @xmath170 reaches its minimum in the one - loop matching . as discussed in ref . @xcite this choice has the merit of stopping the running whenever one of the mandelstam variables is too small , while keeping the neglected one - loop matching terms as small as possible . on the other hand the sensitivity to the choice of the high scale was shown to be small and comparable to the sensitivity of the standard tree - level cross section . should make the omitted terms small , as for the method of minimal variance@xcite . ] we use cteq6 pdfs@xcite and neglect the very small contribution of t and b quarks . moreover we neglect the s - channel contribution and interference terms that are known to be small with vbf cuts . the masses of the vector gauge bosons are restored in the denominators of the coefficients in eq . , as their effects become important when the total cross section is evaluated . hereafter , in order to compare with the nll calculation of ref . @xcite , we adopt the same cuts on angles and transverse moment : @xmath171 , @xmath172 , @xmath173 gev . no qcd corrections have been included in both calculations , and only the virtual electroweak corrections are considered by the method . the terms contributing to the cross section at tree - level are reported in fig.2 . the @xmath174 process dominates , followed by the other left - left terms . the left - right contributions are two order of magnitude smaller , while the right - right terms are very small and have been neglected . in fig.2 a sum over the generations of quarks is included through the pdfs . as a first comparison we study the single terms in the phase space , and for each of them we calculate the k - factor which is defined as the ratio between the cross section with electroweak corrections on , and the tree - level cross section without any radiative correction @xmath175 where the differential cross sections are evaluated for fixed values of @xmath176,@xmath177 , @xmath178 and @xmath39 , and for a given set of angles . here the simple ll calculation is compared with the nll result of ref . @xcite . in figs . 3 - 10 the k - factor is reported as a function of @xmath179 with @xmath180 and @xmath181 at a typical set of parameters : @xmath182 tev , @xmath183 and @xmath184 . the agreement of ll and nll results is very good at small angles . as expected , at ll order the dependence on angles is reduced in comparison with the nll result , because the neglected terms have a larger dependence on the scattering angles . however for @xmath185 , in the physically relevant phase - space region , the difference is small and the ll result ( solid line ) interpolate the nll data ( squares ) quite well . the dependence on the azimuthal angle @xmath186 is very small and negligible for @xmath185 , in perfect agreement with the nll order . we find a better agreement for the left - right processes ( fig . 7 - 10 ) where the radiative corrections are larger , than for left - left processes ( fig . 3 - 6 ) where the corrections are smaller . thus the very large suppression of the left - right processes is mainly due to the role played by the collinear anomalous dimension . it is remarkable that such large suppression has no relevant effects on the total cross section which is dominated by the left - left processes . the single terms are integrated by pdfs over the phase space and summed up , yielding the cross section @xmath187 . by comparison with the tree level cross section , the relative electroweak correction is defined as the ratio @xmath188 and is displayed in fig.11 as a function of the center - of - mass energy . for comparison the nll result of ref . @xcite is also reported in fig.11 together with the output of the code hawk@xcite that is based on a fixed order perturbative calculation . while differences are negligible below 2 tev , at large energies ll and nll corrections grow faster than predicted by fixed order one - loop calculations , reaching 9% at the full lhc energy @xmath182 tev , to be compared with 5% predicted by hawk . at ll order the correction is a bit smaller with respect to nll order , but the difference is less than 1% up to @xmath189 tev . thus the simple ll calculation provides an analytical electroweak correction that seems to be more reliable than fixed order calculations at high energies . while fixed - order perturbative calculations might miss part of the correction at the lhc energy scale , the simple ll resummation contains the main terms , and gives integrated corrections that are more accurate than pdfs . insertion of the simple analytical result of eqs.([eq : wc]),([eq : aint ] ) in software packages would be straightforward , and would increase the accuracy of simulations .

using soft - collinear effective theory , the leading - log radiative electroweak corrections are written in a closed and analytical form for the hadronic cross section of higgs production through vector boson fusion , @xmath0 , one of the most promising channels for studying the higgs boson at the lhc . the simple leading - log resummation is compared with a full next - to - leading - log calculation , and its accuracy is found to be of order 1% up to 10 tev , i.e. better than the accuracy of pdfs . corrections are found to be larger than predicted by one - loop fixed order approximations at lhc energies . the method provides a simple way of incorporating the electroweak corrections in software packages , improving the accuracy of simulations .

introduction kinematics of vbf and cross section effective theory and operator basis set anomalous dimension and running radiative corrections at ll order ll vs. nll order: numerical results

arxiv

we begin with some terminology . by a _ graph _ we mean a finite set of vertices and edges such that there is at most one edge between a pair of vertices and every edge has two distinct vertices . a graph is said to be _ 3-connected _ if it can not be disconnected or reduced to a single vertex by removing fewer than 3 vertices together with the edges containing them . for example , the complete graph @xmath37 is 3-connected if and only if @xmath46 . let @xmath7 be a 3-connected graph embedded in @xmath0 and let @xmath6 be an edge of @xmath7 . if there is a ball @xmath47 in @xmath0 such that @xmath48 is an arc in the interior of @xmath6 whose union with an arc in @xmath49 has non - trivial knot type @xmath50 , then we say that @xmath47 is a _ ball for the local knot @xmath50 of @xmath6 _ and we say that the pair @xmath51 has _ knot type @xmath50_. it is shown in @xcite that for any edge of an embedded 3-connected graph the local knots on that edge are well defined . if a graph is not 3-connected this is not necessarily the case . for example , see figure 3 . [ h ] with ball @xmath52 , and on the right we see the `` same knot '' is a local knot on @xmath53 with ball @xmath54.,title="fig : " ] suppose that some ball @xmath47 meets @xmath7 in an arc @xmath55 . if @xmath56 has non - trivial knot type then we say the arc @xmath57 is _ knotted _ , otherwise we say the arc @xmath57 is _ unknotted_. let @xmath6 be an edge of @xmath7 which contains a local knot with ball @xmath47 . let @xmath58 be an arc obtained from @xmath6 by replacing the knotted arc @xmath59 by an unknotted arc of @xmath47 with the same endpoints and let @xmath19 be obtained from @xmath7 by replacing @xmath6 by @xmath58 . if @xmath58 has no local knots in @xmath19 , then we say that @xmath47 is an _ unknotting ball _ for @xmath6 . we show below that every locally knotted edge has an unknotting ball and this ball is unique up to an isotopy setwise fixing @xmath7 . this is in contrast with the observation that balls for local knots are generally not unique up to isotopy as illustrated in figure 1 . we construct a neighborhood of @xmath7 out of balls and tubes around vertices and edges as follows . let @xmath60 and @xmath61 denote the vertices and edges respectively of @xmath7 . for each vertex @xmath62 , we let @xmath63 denote a closed ball around @xmath64 whose intersection with @xmath7 consists of the single vertex @xmath64 together with an arc of every edge containing @xmath64 such that if @xmath65 then @xmath66 . let @xmath67 denote the union of all of these balls . for each embedded edge @xmath68 , let @xmath69 denote a solid cylinder @xmath70 whose core is @xmath71 , such that @xmath72 , @xmath73 for @xmath74 , and @xmath75 meets @xmath67 in a pair of disks @xmath76 . let @xmath77 denote the union of all these solid cylinders , and let @xmath78 . suppose that @xmath6 is an edge in an embedded graph @xmath7 with vertices @xmath79 and @xmath80 , and @xmath47 is a ball such that @xmath48 is an arc of @xmath6 . observe that there is an isotopy of @xmath8 fixing every vertex of @xmath7 taking @xmath47 to a ball whose boundary meets @xmath81 in two disks contained in @xmath82 and @xmath83 . thus we shall assume , when needed , that a ball for a local knot of @xmath6 has this form . we will use the characteristic submanifold theorem for pared manifolds together with some definitions stated below . a * pared @xmath84-manifold * @xmath85 is an orientable @xmath84-manifold @xmath86 together with a family @xmath87 of disjoint incompressible annuli and tori in @xmath88 . a pared manifold @xmath85 is said to be * simple * if it satisfies the following three conditions : \1 ) @xmath86 is irreducible and @xmath89 is incompressible \2 ) every incompressible torus in @xmath86 is parallel to a torus component of @xmath87 \3 ) any annulus @xmath57 in @xmath86 with @xmath90 contained in @xmath89 is either compressible or parallel to an annulus @xmath91 in @xmath88 with @xmath92 and such that @xmath93 consists of zero or one annular component of @xmath87 . a pared manifold @xmath85 is said to be * seifert fibered * if there is a seifert fibration of @xmath86 for which @xmath87 is a union of fibers . a pared manifold @xmath85 is said to be * i - fibered * if there is an @xmath94-bundle map of @xmath86 over a surface @xmath95 such that @xmath87 is the preimage of @xmath96 . we use the following version of the characteristic submanifold theorem for pared manifolds proved independently by jaco and shalen @xcite and johannson @xcite . henceforth we shall refer to this result simply as jsj . @xcite let @xmath85 be a pared manifold with @xmath86 irreducible and @xmath89 incompressible . then , up to an isotopy of @xmath85 , @xmath86 contains a unique finite family @xmath97 of disjoint incompressible tori and annuli with boundaries in @xmath89 with the following two defining properties : \1 ) if @xmath98 is the closure of a component of @xmath99 , then the pared manifold @xmath100 is either simple , seifert fibered , or @xmath94-fibered . \2 ) no such family has fewer elements than @xmath97 . furthermore , if @xmath5 is an annulus in @xmath86 with boundaries in @xmath89 which is incompressible and boundary incompressible , then there is an isotopy of the pair @xmath101 taking @xmath5 to an annulus which is contained in either @xmath102 or in a component @xmath100 that is seifert fibered or @xmath94-fibered . we now state and prove theorem 1 . [ p : uniqueness ] let @xmath7 be a 3-connected graph embedded in @xmath0 . then any locally knotted edge @xmath6 has an unknotting ball which is unique up to an isotopy of @xmath8 fixing the vertices of @xmath7 . furthermore , if @xmath9 are pairwise disjoint balls for local knots of an edge @xmath6 , then @xmath6 has an unknotting ball which contains @xmath10 . let @xmath103 and @xmath104 , and let @xmath97 be a jsj family for @xmath85 . thus each annulus in @xmath105 has its boundary in @xmath106 . we begin by making two observations which will be used later . first suppose , for the sake of contradiction , that there is a component of @xmath107 whose closure @xmath108 is @xmath94-fibered . since @xmath86 is contained in @xmath0 , @xmath98 is a product of a surface cross an interval where the ends of the product are in one or two components of @xmath106 . since @xmath7 is 3-connected , removing the vertices corresponding to these components and the edges containing them does not separate @xmath7 . also , there is at most one edge between any pair of vertices and no edge from a vertex to itself . it follows that there are precisely two annuli in @xmath109 , and hence these annuli must be parallel in the product . since this contradicts the minimality of @xmath97 , no components of @xmath107 are @xmath94-fibered . next consider an annulus in @xmath97 whose boundary components are in a single @xmath110 , and suppose , for the sake of contradiction , that its boundaries are non - isotopic curves in @xmath111 . thus the surface between these two curves in @xmath110 intersects @xmath7 . also , since the annulus is incompressible in @xmath112 , the disjoint disks bounded by the curves in @xmath110 each intersect @xmath7 . it follows that the sphere obtained by capping off the annulus with these two disks separates @xmath7 and each component contains at least one vertex other than @xmath64 . since this contradicts the 3-connectedness of @xmath7 , the boundary components of such an annulus must be isotopic in @xmath111 . we now consider the locally knotted edge @xmath6 . let @xmath79 and @xmath80 be the vertices of @xmath6 and choose a ball for a local knot of @xmath6 whose boundary meets @xmath81 in the disks @xmath113 and @xmath114 . by removing these disks we obtain an annulus which is incompressible in @xmath86 . it now follows from jsj that there is an isotopy of the pair @xmath115 taking this annulus to an annulus @xmath57 which has boundaries in @xmath82 and @xmath83 and is contained either in @xmath97 or in a seifert fibered component of @xmath107 . consider the case where the annulus @xmath57 is contained in a seifert fibered component @xmath116 of @xmath107 . let @xmath117 be the disk in @xmath118 bounded by @xmath119 . it follows from the fibered structure of @xmath98 that every component of @xmath120 is an annulus . in particular , the component of @xmath120 containing @xmath121 has precisely one boundary which is in @xmath117 and is isotopic to @xmath121 in @xmath122 , and all of the other components of @xmath120 have either 0 or 2 boundary components which are in @xmath117 and are isotopic to @xmath121 in @xmath122 . since every boundary component of @xmath123 is a boundary of some component of @xmath120 , it follows that there are an odd number of boundary components of @xmath124 which are in @xmath117 and are isotopic to @xmath121 in @xmath122 . suppose , for the sake of contradiction , that every annulus of @xmath123 with at least one boundary component which is in @xmath117 and is isotopic to @xmath121 in @xmath122 has its other boundary component in @xmath82 . such an annulus has isotopic boundaries in @xmath122 , and since @xmath57 separates @xmath98 and one boundary of it is in @xmath117 the other boundary must be in @xmath117 as well . thus there must be an even number of boundary components of @xmath124 which are in @xmath117 and are isotopic to @xmath121 in @xmath122 . since we saw above that this number is actually odd , there must be some such annulus @xmath91 whose boundary component is in @xmath110 with @xmath125 . now suppose , for the sake of contradiction , that @xmath126 . consider the sphere obtained from @xmath127 by adding the annulus between @xmath57 and @xmath91 in @xmath122 together with disks in @xmath128 and @xmath63 . since @xmath57 and @xmath91 are incompressible in @xmath86 , this sphere separates @xmath7 into components each containing at least one vertex of @xmath7 other than @xmath80 and @xmath64 . since this contradicts the 3-connectedness of @xmath7 , we must have @xmath129 . furthermore , if we cap off @xmath91 in @xmath82 and @xmath83 , we obtain a sphere bounding a ball which intersects @xmath7 in an arc of @xmath6 and contains the annulus @xmath57 . since @xmath7 is 3-connected , we can cap off any annulus in @xmath97 with boundaries in @xmath82 and @xmath83 to obtain a sphere bounding a ball whose intersection with @xmath7 is an arc of @xmath6 . choose @xmath5 to be maximal among all such annuli . that is , by capping off @xmath5 in @xmath82 and @xmath83 we obtain a sphere bounding a ball @xmath47 which intersects @xmath7 in an arc of @xmath6 and contains all of the other such annuli . then it follows from our argument above that any incompressible annulus in a seifert fibered component of @xmath107 with boundaries in @xmath82 and @xmath83 will be contained in @xmath47 . hence any incompressible annulus in @xmath86 with boundaries in @xmath130 and @xmath83 is isotopic in @xmath115 to an annulus which is properly embedded in @xmath131 . furthermore , if such an annulus is contained in @xmath132 , then it is isotopic in @xmath133 to @xmath5 . it follows from our choice of @xmath5 that @xmath47 is an unknotting ball for @xmath6 . let @xmath134 be another unknotting ball for @xmath6 , and let @xmath58 be obtained from @xmath6 by replacing the knotted arc @xmath135 by an unknotted arc of @xmath134 . after an isotopy of @xmath8 fixing the vertices , we can assume that @xmath136 meets @xmath81 in disks in @xmath82 and @xmath83 . by removing these disks we obtain an annulus @xmath137 , which we prove as follows is isotopic to @xmath5 in @xmath115 . since @xmath134 is a ball for a local knot of @xmath6 , @xmath137 is incompressible in @xmath86 . hence , by our choice of @xmath5 , we can assume that @xmath137 is properly embedded in @xmath131 . let @xmath138 be the torus obtained from the annuli @xmath5 and @xmath137 by adding annuli in @xmath82 and @xmath83 . suppose that @xmath137 is not isotopic to @xmath5 . if @xmath138 bounds a solid torus in @xmath86 , then the meridian of the solid torus does not have intersection number @xmath139 with a component of @xmath140 . however , in this case , by adding a thickened disk to the solid torus along a component of @xmath140 we would get a punctured lens space . as this is impossible in @xmath0 , we can assume that @xmath138 bounds a knot complement in @xmath86 . however , since @xmath141 , this would imply that @xmath47 is a ball for a local knot in the edge @xmath58 which is contrary to our assumption that @xmath134 is an unknotting ball for @xmath6 . therefore @xmath137 is isotopic to @xmath5 in @xmath115 , and hence @xmath134 is isotopic to @xmath47 by an isotopy of @xmath14 pointwise fixing the vertices of @xmath7 . now let @xmath9 be pairwise disjoint balls for local knots of @xmath6 . since each @xmath142 meets @xmath7 in an arc in the interior of @xmath6 , without loss of generality , we can assume that each @xmath143 meets @xmath81 in two disks contained in @xmath144 . for each @xmath33 , let @xmath145 denote the annulus obtained from @xmath143 by removing these two disks . since @xmath146 , for each @xmath33 , @xmath147 . by a standard cut - and - paste argument , we can assume that no components of @xmath148 are disks . thus if some @xmath145 is not contained in @xmath47 , then there is an annulus component of @xmath148 in @xmath132 . we can extend this annulus parallel to @xmath5 to obtain an annulus in @xmath132 with boundaries in @xmath82 and @xmath83 which is incompressible in @xmath132 . hence , we can obtain an isotopy of @xmath133 taking this annulus to @xmath5 . thus we can remove any annuli from @xmath148 while pointwise fixing @xmath7 without introducing any new intersections of @xmath5 with some @xmath149 . this gives us an unknotting ball which contains @xmath150 as required . let @xmath7 be a 3-connected graph embedded in @xmath0 with locally knotted edges @xmath29 , , @xmath30 . then there is a collection of pairwise disjoint unknotting balls for these edges . by theorem 1 , we know that there is a collection @xmath52 , , @xmath151 of unknotting balls for @xmath29 , , @xmath30 respectively . we see as follows that we can isotop @xmath52 , , @xmath151 to be pairwise disjoint while setwise fixing @xmath7 . first we will isotop @xmath54 , , @xmath151 off of @xmath52 , then we will isotop @xmath152 , , @xmath151 off of @xmath54 , and so on . let @xmath153 be a circle of intersection of @xmath154 with @xmath155 that bounds an innermost disk @xmath156 on @xmath154 . since there are at least two innermost disks on @xmath154 bounded by circles of @xmath157 and @xmath29 intersects @xmath154 in precisely two points , we can choose the disk @xmath158 so that it intersects @xmath29 in at most one point . similarly , we can choose @xmath159 to be a disk bounded by @xmath153 in @xmath155 which intersects @xmath53 in at most one point . since @xmath160 if @xmath161 , the sphere @xmath162 meets @xmath7 in at most two points . by hypothesis , @xmath7 is 3-connected . thus a sphere can not meet @xmath7 in precisely one point , and if a sphere meets @xmath7 in two points then those two points must be on the same edge . since the sphere @xmath162 meets each @xmath163 in at most one point , it follows that @xmath162 can not intersect either @xmath163 . thus one component of @xmath164 is a ball which is disjoint from @xmath7 . using this ball , we can isotop @xmath159 to a disk parallel to @xmath158 by an isotopy of @xmath54 that fixes @xmath7 . this isotopy removes @xmath153 as a circle of intersection in @xmath165 . thus , by inducting on the number of circles of intersection , we can make @xmath54 disjoint from @xmath52 . since we have changed @xmath54 by an isotopy pointwise fixing @xmath7 , the new @xmath54 will still be an unknotting ball for @xmath53 . we continue this process to obtain pairwise disjoint unknotting balls for @xmath29 , , @xmath30 . in our study of topological symmetry groups , we would like to use local knots to prevent certain automorphisms from being induced by any homeomorphism of the embedding . we begin with some more terminology . by theorem 1 , we know that unknotting balls are unique up to an isotopy of @xmath8 fixing every vertex of @xmath7 . thus if @xmath47 is an unknotting ball for an edge @xmath6 , and the pair @xmath166 has knot type @xmath50 , then we can unambiguously say that @xmath6 has _ knot type @xmath50 in @xmath7 _ without making reference to a particular unknotting ball . now let @xmath47 be a ball such that @xmath48 is an unknotted arc @xmath57 in an edge @xmath6 . let @xmath19 be obtained from @xmath7 by replacing @xmath57 with an arc @xmath91 in @xmath47 such @xmath167 has knot type @xmath50 and @xmath168 . then we say that @xmath19 is obtained from @xmath7 by _ adding the local knot @xmath50 to @xmath6_. suppose that @xmath6 has knot type @xmath169 in @xmath7 . let @xmath47 be an unknotting ball for @xmath6 , and let @xmath170 such that @xmath135 is an unknotted arc @xmath57 . let @xmath19 be obtained from @xmath7 by adding the local knot @xmath171 to @xmath6 within @xmath134 , and let @xmath58 be the resulting edge . since @xmath172 , the ball @xmath47 is also an unknotting ball for @xmath58 in @xmath19 . now by definition of the connected sum of knots , the pair @xmath173 has knot type @xmath174 . hence the edge @xmath58 has knot type @xmath174 in @xmath19 . finally , we will use the following terminology in the statement of the knot addition lemma . we say that an edge @xmath6 is _ inverted _ by an automorphism @xmath175 if @xmath175 interchanges the vertices of @xmath6 . if @xmath176 is a set and @xmath28 is a group acting on @xmath177 , then we will use the notation @xmath178 to denote the orbit of @xmath179 under @xmath28 . [ l : add ] let @xmath7 be an embedding of a 3-connected graph in @xmath0 and let @xmath28 be a ( possibly trivial ) subgroup of @xmath15 . let @xmath29 , , @xmath30 be edges of @xmath7 with distinct orbits under @xmath28 . let @xmath169 , , @xmath37 be distinct prime knots , which are not local knots of @xmath7 , such that @xmath180 is invertible if and only if @xmath163 is inverted by some element of @xmath28 . then for each @xmath33 , the local knot @xmath181 can be added to the edges in @xmath182 to create an embedding @xmath19 such that @xmath183 . furthermore , for each @xmath33 , let @xmath184 be the embedding of @xmath163 in @xmath19 . then @xmath185 , and if @xmath184 is inverted by an element of @xmath18 then @xmath184 is also inverted by an element of @xmath28 . let @xmath179 denote a set consisting of one point in the interior of each edge of @xmath7 . for each @xmath186 , let @xmath187 denote a neighborhood of the point @xmath188 in the interior of @xmath77 such that @xmath189 is an unknotted arc . let @xmath95 denote the set of all the @xmath187 . thus the balls in @xmath95 are pairwise disjoint . it follows from the uniqueness of @xmath67 , @xmath77 , and @xmath179 up to an isotopy fixing the vertices of @xmath7 that there is a group @xmath190 of orientation preserving diffeomorphisms of @xmath8 inducing @xmath15 , such that for every @xmath191 , @xmath192 , @xmath193 , and @xmath194 . note that @xmath190 need not be isomorphic to @xmath15 . let @xmath195 denote a subgroup of @xmath190 which induces @xmath28 on @xmath7 . since @xmath95 is setwise invariant under @xmath190 , for any @xmath196 we can define the orbit @xmath197 . for @xmath198 , , @xmath39 , let @xmath199 denote the point of @xmath179 on @xmath163 , and let @xmath200 . let @xmath201 be the arc @xmath202 , let @xmath203 denote an arc in @xmath142 with the same endpoints as @xmath201 containing the local knot @xmath180 , and let @xmath184 denote @xmath163 after @xmath201 has been replaced by @xmath203 . by hypothesis , @xmath180 is prime and is not a local knot of @xmath7 . also , @xmath180 is invertible if and only if @xmath163 is inverted by some element of @xmath28 . if @xmath163 is not inverted by any element of @xmath28 , then we assign an orientation to @xmath163 , which in turn induces an orientation on both @xmath180 and all of the edges of @xmath204 . now let @xmath6 be an arbitrary edge of @xmath204 . then precisely one ball @xmath196 intersects @xmath6 . let @xmath58 be obtained from @xmath6 by replacing the unknotted arc @xmath205 by an arc in @xmath47 with the same endpoints containing the local knot @xmath180 such that if @xmath180 is non - invertible then the orientation of @xmath180 with respect to the oriented edge @xmath58 is the same as the orientation of @xmath180 with respect to the oriented edge @xmath184 . if @xmath163 is inverted by some element of @xmath28 , then @xmath180 was chosen to be invertible so the orientation of @xmath180 is not important . let @xmath19 be obtained from @xmath7 by adding local knots to the edges in each orbit @xmath182 in this way . in order to prove that @xmath206 , let @xmath207 . since @xmath208 , there is a diffeomorphism @xmath209 which induces @xmath210 on @xmath7 . we need to define an orientation preserving diffeomorphism of @xmath211 which induces @xmath210 on @xmath19 . for each @xmath33 , let @xmath212 . then @xmath213 . hence we can define @xmath214 . also , since the set of vertices @xmath215 , we know @xmath216 . we extend @xmath217 within each set of balls @xmath218 as follows . let @xmath219 , let @xmath57 denote the unknotted arc @xmath48 , and let @xmath91 denote the knotted arc @xmath220 . first suppose that @xmath221 . in this case , we let @xmath222 denote the arc @xmath223 . observe that since @xmath209 , the ball @xmath224 , and hence @xmath222 contains the local knot @xmath180 . thus the pairs @xmath225 and @xmath226 both have knot type @xmath180 , and if @xmath180 is non - invertible , then the knotted arcs @xmath91 and @xmath222 each has its orientation consistent with that of @xmath163 . thus we can extend @xmath217 within @xmath47 so that @xmath227 . now suppose that @xmath228 . if @xmath229 fixes the endpoints of @xmath57 , then we can extend @xmath217 to @xmath225 in such a way that @xmath217 pointwise fixes the knotted arc @xmath91 . if @xmath229 interchanges the endpoints of @xmath57 then @xmath229 inverts @xmath163 and hence the knot @xmath180 is invertible . thus we can extend @xmath217 to @xmath225 in such a way that @xmath217 inverts @xmath91 . in this way we have extended @xmath217 to every ball in @xmath230 such that @xmath231 . now @xmath232 induces @xmath175 on @xmath19 . it follows that @xmath206 . in order to prove that @xmath233 , let @xmath234 . then @xmath235 is induced on @xmath19 by an orientation preserving diffeomorphism @xmath217 of @xmath211 . since @xmath206 , the orbit @xmath236 is a set of edges in @xmath19 . suppose that each edge in @xmath204 has knot type @xmath237 in @xmath7 ( where @xmath237 might be the trivial knot ) . then each edge in @xmath238 has knot type @xmath239 . since @xmath7 is 3-connected , it follows from @xcite that adding the local knot @xmath180 to an edge of @xmath7 does not cause any local knot to be added to any other edge of @xmath7 . thus since @xmath180 is a prime knot that is not a local knot of @xmath7 , the edges in @xmath238 are the only edges in @xmath19 containing @xmath180 among their local knots . it follows that for each @xmath33 , @xmath240 . by our construction , for each @xmath241 , the neighborhood @xmath75 is a ball for the local knot @xmath180 in the corresponding edge @xmath58 of @xmath19 . thus by theorem 1 and lemma 1 , we can choose a collection of pairwise disjoint unknotting balls for the edges in @xmath242 such that for each @xmath243 the unknotting ball for @xmath58 contains the ball @xmath75 . for each @xmath33 , let @xmath244 denote the subset of these unknotting balls which are unknotting balls for the edges in @xmath238 . now @xmath244 and @xmath245 are each sets of unknotting balls for the edges of @xmath19 in @xmath238 . since unknotting balls are unique up to isotopy by theorem 1 , there is an isotopy of @xmath211 fixing the vertices of @xmath19 which takes @xmath245 to @xmath244 . hence there is an orientation preserving diffeomorphism @xmath246 of @xmath247 fixing the vertices of @xmath19 such that for each @xmath33 , @xmath248 . now @xmath249 is a diffeomorphism of @xmath211 which leaves each @xmath244 setwise invariant and induces @xmath235 on @xmath19 . for each @xmath33 , the collection of balls in @xmath244 contains both @xmath204 and @xmath236 . thus @xmath250 . so we can define @xmath251 . we extend @xmath229 to the balls within each @xmath244 as follows . let @xmath47 be one of the balls in @xmath244 , let @xmath252 denote the arc @xmath48 which is contained in some edge @xmath6 of @xmath204 , and let @xmath253 denote the arc @xmath254 which is contained in some edge @xmath58 of @xmath238 . since @xmath146 and @xmath47 is an unknotting ball for @xmath58 in @xmath19 , @xmath47 must be an unknotting ball for @xmath6 in @xmath7 as well . thus since @xmath6 and @xmath58 have knot types @xmath237 and @xmath239 respectively , the pairs @xmath255 and @xmath256 also have knot types @xmath237 and @xmath239 respectively . suppose that @xmath257 . let @xmath258 and @xmath259 . since @xmath260 , the pair @xmath261 must also have knot type @xmath239 . since the edges in @xmath238 are the only ones in @xmath19 which contain @xmath180 among their local knots , @xmath262 . hence @xmath263 is the embedding in @xmath19 of some edge @xmath264 in @xmath7 . thus @xmath265 has knot type @xmath237 . since @xmath266 , it follows that the ball @xmath267 and hence @xmath268 is an unknotting ball for @xmath265 in @xmath7 . thus the pair @xmath269 has knot type @xmath237 . now the pairs @xmath270 and @xmath271 both have knot type @xmath237 . recall , that @xmath180 is a prime knot which is not contained in @xmath7 . in particular , @xmath180 is not among the prime factors of @xmath237 . thus it follows from schubert @xcite that if @xmath237 is non - invertible , then @xmath239 is non - invertible as well . hence @xmath272 takes the oriented knot in @xmath256 to the oriented knot in @xmath273 . it follows that we can extend @xmath229 within @xmath47 so that @xmath274 . now suppose that @xmath275 . if @xmath272 fixes the endpoints of @xmath253 , then we can extend @xmath229 to @xmath270 in such a way that @xmath229 pointwise fixes the arc @xmath252 . suppose that @xmath272 interchanges the endpoints of @xmath253 . then @xmath272 inverts @xmath276 . thus @xmath239 must be invertible . since @xmath237 and @xmath180 have distinct knot types , it follows that @xmath237 is invertible . therefore , we can extend @xmath229 to @xmath270 in such a way that @xmath229 inverts @xmath252 . in this way we have extended @xmath229 to every ball in @xmath277 such that @xmath278 . since the vertices of @xmath7 are disjoint from @xmath277 , the diffeomorphism @xmath279 induces @xmath235 on @xmath7 . it follows that @xmath233 . finally , in order to show that @xmath280 , first observe that since @xmath206 , @xmath281 . now let @xmath282 . then for some @xmath283 , we have @xmath284 . since @xmath184 contains the local knot @xmath180 , the edge @xmath58 also contains the local knot @xmath180 . however , by our construction of @xmath19 , the only edges of @xmath19 containing @xmath180 are the edges in @xmath285 . thus @xmath286 . hence @xmath280 . furthermore , suppose that @xmath184 is inverted by some element of @xmath18 . then @xmath180 must be invertible . from our construction , it follows that @xmath163 must also be inverted by an element of @xmath28 . the finiteness theorem below allows us to focus on topological symmetry groups which are induced by finite subgroups of @xmath287 ( i.e. , the group of orientation preserving diffeomorphisms of @xmath0 ) . @xcite [ l : reembed ] let @xmath7 be a 3-connected graph embedded in @xmath0 . then there is an embedding @xmath288 of @xmath7 in @xmath0 such that @xmath289 and @xmath290 is induced by an isomorphic finite subgroup of @xmath287 . we use the knot addition lemma together with the finiteness theorem to prove the following theorem . note that we use @xmath182 to mean the orbit of the edge @xmath163 under the action of the group of automorphisms @xmath28 . let @xmath7 be an embedding of a 3-connected graph in @xmath0 and let @xmath28 be a ( possibly trivial ) subgroup of @xmath15 . let @xmath29 , @xmath30 be a set of edges in @xmath7 whose orbits under @xmath28 are distinct . suppose that any @xmath31 which pointwise fixes @xmath29 and has @xmath32 for each @xmath33 , also pointwise fixes a subgraph of @xmath7 that can not be embedded in @xmath34 . then there is an embedding @xmath19 of @xmath7 with @xmath35 . let @xmath169 , , @xmath37 be distinct prime knots which are not local knots of @xmath7 and are invertible if and only if @xmath163 is inverted by some element of @xmath28 . we use the knot addition lemma to add the local knot @xmath180 to every edge in @xmath182 for each @xmath33 . thus we obtain an embedding @xmath19 such that @xmath291 . furthermore , for each @xmath33 , let @xmath184 be the embedding of @xmath163 in @xmath19 . then @xmath185 , and if @xmath184 is inverted by an element of @xmath18 then @xmath184 is also inverted by an element of @xmath28 . in order to show that @xmath292 , let @xmath293 . for each @xmath33 , the edges in @xmath294 are the only edges of @xmath19 containing the knot @xmath180 . thus @xmath295 . hence for some @xmath296 , @xmath297 . since @xmath206 , @xmath298 is an element of @xmath18 which setwise fixes @xmath299 . if @xmath298 inverts @xmath299 , then @xmath29 is inverted by some @xmath300 . in this case , let @xmath301 . otherwise , let @xmath302 . in either case , @xmath175 is an element of @xmath233 which pointwise fixes @xmath299 . also , for each @xmath33 , @xmath303 . thus @xmath175 is an element of @xmath15 which pointwise fixes @xmath29 and has @xmath32 for each @xmath33 . so by hypothesis , @xmath175 pointwise fixes a subgraph of @xmath7 which can not be embedded in @xmath34 . now by the finiteness theorem there is an embedding @xmath288 of @xmath7 in @xmath0 such that @xmath289 and @xmath290 is induced by an isomorphic finite subgroup of @xmath287 . in particular , the automorphism @xmath304 and hence is induced on @xmath288 by a finite order @xmath305 . now @xmath229 pointwise fixes a subgraph of @xmath288 which can not be embedded in @xmath34 . since @xmath229 has finite order , by smith theory @xcite @xmath229 must actually be the identity . thus @xmath175 is the identity automorphism on @xmath19 . thus either @xmath306 or @xmath307 . in either case , @xmath308 . hence @xmath309 as required . observe that the result below follows immediately from the subgroup theorem . [ l : fundamentaledge ] let @xmath7 be a 3-connected graph embedded in @xmath0 which has an edge @xmath6 that is not pointwise fixed by any non - trivial element of @xmath15 . then for every ( possibly trivial ) subgroup @xmath28 of @xmath15 , there is an embedding @xmath19 of @xmath7 with @xmath36 . in this section we use the subgroup corollary to prove theorem 3 . we begin by stating some previous results that we will use . the following theorem tells us which individual automorphisms of a complete graph @xmath37 can be induced by an orientation preserving diffeomorphism of some embedding of @xmath37 in @xmath0 . this result follows from theorems 1 and 2 of @xcite . [ t : automorphism ] @xcite let @xmath45 , and let @xmath175 be an automorphism of @xmath37 of order @xmath41 . then there is an embedding of @xmath37 in @xmath0 such that @xmath210 is induced by an orientation - preserving diffeomorphism @xmath272 if and only if one of the following holds : 1 . @xmath41 is even and @xmath310 , all cycles of @xmath210 are of order @xmath41 , and @xmath210 fixes no vertices ; 2 . @xmath311 , all cycles of @xmath210 are of order two , and @xmath210 fixes at most two vertices ; 3 . @xmath41 is odd , all cycles of @xmath210 are of order @xmath41 , and @xmath175 fixes at most three vertices ; 4 . @xmath41 is an odd multiple of 3 , all cycles of @xmath210 are of order @xmath41 except one of order 3 , and @xmath210 fixes no vertices . we will also use the following technical lemma , which follows from results in section 2 of @xcite . [ z3]@xcite let @xmath288 be an embedding of @xmath37 in @xmath0 such that the group @xmath312 is induced on @xmath288 by a finite subgroup @xmath313 . then there are at most two sets of 3 vertices which are each setwise invariant under @xmath28 . furthermore , if a non - trivial @xmath296 fixes any vertices of @xmath288 , then the set of fixed vertices of @xmath272 is equal to one of these sets of 3 vertices . we will prove theorem [ t : realize ] by proving two propositions , one in which @xmath15 is cyclic or dihedral and the other in which @xmath15 is a subgroup of @xmath314 for some odd @xmath41 . by the subgroup corollary , it suffices to prove in each proposition that there is an edge which is not pointwise fixed by any non - trivial element of @xmath15 . [ p : rank1 ] let @xmath7 be an embedding of @xmath37 in @xmath0 with @xmath45 , such that @xmath15 is cyclic or dihedral . then for any @xmath208 , there is an embedding @xmath19 of @xmath37 in @xmath0 such that @xmath36 . suppose that @xmath15 is @xmath315 or @xmath316 . then @xmath15 contains an element @xmath317 of order @xmath41 which is induced by an orientation preserving diffeomorphism of @xmath14 . by the automorphism theorem , @xmath317 has at least one @xmath41-cycle . let @xmath64 be a vertex in this @xmath41-cycle . then the edge @xmath318 is not pointwise fixed by any @xmath319 , with @xmath320 . if @xmath321 , then @xmath322 . hence the result follows from the subgroup corollary . suppose that @xmath323 . then there is an order 2 automorphism @xmath324 which is induced by an orientation preserving diffeomorphism of @xmath14 such that @xmath325 . assume that for some @xmath33 , @xmath326 pointwise fixes the edge @xmath318 . then both @xmath327 and @xmath328 . thus @xmath329 since @xmath330 has order 2 . hence @xmath331 . but we know that @xmath332 . hence @xmath333 , and so @xmath334 . since @xmath64 is in an @xmath41-cycle , this is impossible if @xmath310 . hence if @xmath335 , then @xmath318 is not fixed by any element of @xmath316 , and thus again the result follows from the subgroup corollary . now , we consider the case where @xmath336 . in this case , @xmath337 and @xmath330 both have order 2 and together generate @xmath338 . by the automorphism theorem , @xmath337 , @xmath330 , and @xmath339 each fix at most two vertices , and each have at least three 2-cycles since @xmath340 . so there are at least four vertices which are fixed by neither @xmath337 nor @xmath330 . if @xmath337 and @xmath330 agree on these four vertices , then the automorphism @xmath341 would fix 4 vertices . as this is contrary to the automorphism theorem , there is at least one vertex @xmath64 which is fixed by neither @xmath337 nor @xmath330 such that @xmath342 . it follows that @xmath64 , @xmath343 ) , @xmath344 are all distinct vertices . we see that @xmath345 is also distinct from these three vertices as follows . if @xmath346 , then @xmath347 ; if @xmath348 , then @xmath349 ; and if @xmath350 then @xmath351 . all three cases are impossible since @xmath64 , @xmath343 ) , @xmath344 are distinct . hence @xmath64 , @xmath343 ) , @xmath344 , and @xmath345 are all distinct . it follows that none of @xmath337 , @xmath330 or @xmath339 pointwise fixes @xmath318 . now our result follows again from the subgroup corollary . the remaining cases in the proof of theorem [ t : realize ] are when @xmath15 is a subgroup of @xmath40 for some odd @xmath41 . these subgroups are described by the following lemma . this result is undoubtedly known , but we were unable to find a reference , so we include a proof here . [ t : subgroups ] let @xmath352 be odd , and let @xmath190 be a non - trivial subgroup of @xmath353 . then @xmath190 is isomorphic to one of the following groups where @xmath354 , @xmath355 are odd : @xmath356 , @xmath357 , @xmath358 , @xmath338 , @xmath359 , @xmath360 , @xmath361 , @xmath362 , @xmath363 , or @xmath364 such that for any nontrivial elements @xmath365 and @xmath366 we have @xmath367 . each element of @xmath316 can be thought of as either a rotation or a reflection of a circle . define a homomorphism @xmath368 by @xmath369 if @xmath188 is a reflection and @xmath370 otherwise . let @xmath371 . now , @xmath372 . every subgroup of @xmath373 has rank at most 2 , and hence @xmath374 . thus @xmath28 is either trivial or isomorphic to @xmath357 or @xmath361 for some odd @xmath375 . if @xmath28 is trivial , then @xmath376 is an isomorphism from @xmath190 to a subgroup of @xmath377 . thus @xmath190 is either the trivial group , @xmath356 , or @xmath378 , so we are done . hence we shall assume that @xmath379 or 2 . let @xmath380 , then @xmath381 . if @xmath382 , then @xmath383 , and we are done . thus we shall assume that @xmath384 or 2 . if @xmath385 then without loss of generality we can assume that the generator of @xmath386 is either @xmath387 or @xmath388 , and if @xmath389 then we can assume the generators of @xmath386 are @xmath388 and @xmath390 . we consider two cases , according to whether or not @xmath386 is generated by @xmath387 . * case 1 : * @xmath386 is generated by @xmath387 . in this case , @xmath391 where both @xmath188 and @xmath392 are reflections . now for every nontrivial @xmath296 , @xmath393 and @xmath394 . thus , @xmath190 is either isomorphic to @xmath359 or @xmath364 depending on whether @xmath28 is isomorphic to @xmath357 or @xmath361 , respectively . thus , in case 1 we are done . before we consider case 2 we make the following observation . note here we use @xmath6 to denote the identity element . * observation * : if @xmath395 where @xmath188 is a reflection and @xmath392 a rotation , then @xmath396 and @xmath397 . if @xmath395 where @xmath188 is a rotation and @xmath392 a reflection , then @xmath398 and @xmath399 . * proof of observation * : we prove the first assertion as follows . we know that @xmath400 , since @xmath41 is odd and @xmath188 is a reflection . also , @xmath401 since @xmath402 . thus @xmath403 . now since @xmath392 is a rotation , it follows that @xmath396 . also , since @xmath404 and @xmath405 are both in @xmath190 , we have @xmath397 . the second assertion is proved similarly . we will use this observation in the proof of case 2 . * case 2 : * @xmath386 is not generated by @xmath387 . without loss of generality we can assume that either @xmath406 or @xmath407 . in the first case , @xmath408 , where @xmath409 is a reflection and @xmath410 is a rotation ; and in the second case , @xmath411 , where @xmath409 and @xmath412 are reflections and @xmath410 and @xmath413 are rotations . now , by applying the observation we see that in the first case @xmath414 and @xmath415 , and in second case , @xmath416 and @xmath417 . thus either @xmath418 or @xmath419 . in either case , @xmath415 and @xmath409 is a reflection . thus for any @xmath420 , we know that @xmath421 , and @xmath422 is a reflection and @xmath423 is a rotation . so we can apply the observation to conclude that @xmath424 . now since @xmath425 and @xmath426 are both in @xmath28 , it follows that @xmath427 . * subcase ( a ) * @xmath428 . in this case , without loss of generality we can assume that @xmath429 , where @xmath430 and @xmath423 are rotations and @xmath426 is of odd order @xmath354 . then either @xmath431 or @xmath432 . as we saw above @xmath424 . hence for some @xmath433 , @xmath434 . this gives @xmath435 and @xmath436 . assume that neither @xmath430 nor @xmath423 is the identity . then @xmath433 and @xmath437 must each divide @xmath41 or equal 0 , since @xmath438 . since @xmath41 is odd , @xmath433 and @xmath437 can not both divide @xmath41 , unless @xmath439 . thus either @xmath440 or @xmath439 . if @xmath440 , then @xmath441 and if @xmath439 then @xmath442 . hence without loss of generality we assume that @xmath442 . now @xmath443 or @xmath444 . in the former case , @xmath445 , since @xmath354 is odd . in the latter case , @xmath446 , since @xmath425 and @xmath447 generate @xmath359 and they both commute with @xmath448 . now since @xmath354 is odd , @xmath449 . * subcase ( b ) * @xmath450 . in this case , without loss of generality we can assume that @xmath451 , where @xmath430 , @xmath423 , @xmath452 , @xmath453 , are rotations such that @xmath426 and @xmath454 are of odd orders @xmath354 and @xmath455 respectively . hence either @xmath456 or @xmath457 . in either case , by arguing as we did before subcase ( a ) , we conclude that @xmath458 , @xmath425 , @xmath459 , @xmath460 . thus @xmath461 . it follows that for some @xmath462 and @xmath64 of order @xmath463 and @xmath464 respectively , @xmath465 . also @xmath463 , @xmath466 and are odd . thus either @xmath467 or @xmath468 . in the former case , @xmath469 , and in the latter case @xmath470 . thus again we are done . [ p : product ] let @xmath7 be an embedding of @xmath37 in @xmath0 with @xmath45 , such that @xmath15 is a subgroup of @xmath314 for some odd @xmath41 . then for every ( possibly trivial ) subgroup @xmath28 of @xmath15 , there is an embedding @xmath19 of @xmath37 in @xmath0 such that @xmath36 . in proposition 1 we have dealt with the case where @xmath15 is cyclic or dihedral . so we shall assume that @xmath15 is neither . now it follows from lemma [ t : subgroups ] that @xmath15 is isomorphic to one of the following groups , where @xmath354 , @xmath355 are odd : 1 . @xmath471 2 . @xmath472 3 . @xmath473 4 . @xmath474 such that for any nontrivial elements @xmath365 and @xmath366 we have @xmath367 observe that in all 4 cases , @xmath475 . we choose @xmath317 , @xmath476 such that @xmath477 , the order of @xmath337 is @xmath354 , the order of @xmath330 is @xmath455 , and @xmath478 only contains the identity . recall that for any vertex @xmath64 , @xmath479 and @xmath480 denote the @xmath317-orbit and @xmath481-orbit respectively of @xmath64 . we prove in the following two cases that there is a vertex @xmath64 which is contained in both an @xmath354-cycle of @xmath317 and an @xmath455-cycle of @xmath481 such that the two cycles intersect only in @xmath64 . * case 1 : * not both @xmath482 and @xmath483 . by the automorphism theorem , there are at most 3 vertices which are not contained in an @xmath354-cycle of @xmath317 and at most 3 vertices which are not contained in an @xmath455-cycle of @xmath481 . since @xmath340 , we can pick a vertex @xmath64 which is contained in both an @xmath354-cycle of @xmath317 and an @xmath455-cycle of @xmath481 . now suppose that for some @xmath33 and @xmath433 such that @xmath484 and @xmath485 , @xmath486 . let @xmath487 , then @xmath488 for some @xmath489 . hence @xmath490 . thus @xmath491 fixes @xmath492 , and hence fixes every vertex in @xmath479 . similarly , @xmath491 fixes every vertex in @xmath480 . we know that @xmath491 is not the identity since @xmath493 only contains the identity . thus by the automorphism theorem , @xmath491 fixes at most 3 vertices . therefore @xmath494 contains at most 3 vertices . on the other hand , we chose @xmath64 so that @xmath495 and @xmath496 . thus @xmath497 and @xmath498 . this is a contradiction since @xmath354 , @xmath355 and they do not both equal 3 . thus we conclude that there is a vertex @xmath64 which is contained in both an @xmath354-cycle of @xmath317 and an @xmath455-cycle of @xmath481 such that @xmath499 . * case 2 : * @xmath500 . by the finiteness theorem , there is a re - embedding @xmath288 of @xmath7 such that @xmath289 and @xmath290 is induced by an isomorphic subgroup of @xmath287 . thus @xmath501 is induced on @xmath288 by a finite subgroup of @xmath287 . now it follows from the @xmath502 lemma that there are at most two sets of 3 vertices which are each setwise invariant under both @xmath317 and @xmath481 , and if either @xmath317 or @xmath481 has any fixed vertices then the set of its fixed vertices is equal to one of these sets of 3 vertices . since @xmath340 , we can pick @xmath64 to be a vertex which is not in one of the above sets of 3 vertices . thus @xmath479 contains 3 vertices and can not be setwise fixed by @xmath481 . now @xmath503 and @xmath504 . suppose that for some @xmath33 and @xmath433 , @xmath505 and @xmath506 , we have @xmath486 . since @xmath500 , @xmath507 and @xmath508 . thus @xmath509 . thus @xmath510 , which is contrary to our choice of @xmath64 . hence there is a vertex @xmath64 which is contained in both an @xmath354-cycle of @xmath317 and an @xmath455-cycle of @xmath481 such that @xmath499 . let @xmath64 be the vertex given by case 1 or 2 , and let @xmath6 be the edge @xmath511 . now @xmath512 for all @xmath513 and @xmath514 except when both @xmath515 and @xmath516 . thus the edge @xmath6 is not pointwise fixed by any non - trivial element of @xmath471 . if @xmath517 , then we can apply the subgroup corollary to conclude that for every subgroup @xmath208 , there is an embedding @xmath19 of @xmath37 in @xmath0 such that @xmath36 . suppose that @xmath518 . then there is an order 2 automorphism @xmath519 such that @xmath520 and @xmath521 . we saw above that the edge @xmath6 is not pointwise fixed by any non - trivial element of the subgroup @xmath522 of @xmath15 . assume for the sake of contradiction that some @xmath523 pointwise fixes @xmath6 . this means that both @xmath524 and @xmath525 . so @xmath526 , and hence : @xmath527 thus @xmath528 , and hence @xmath529 . but @xmath64 is contained in an @xmath354-cycle of @xmath337 and @xmath354 is odd . so this is impossible . therefore @xmath6 is not pointwise fixed by any non - trivial element of @xmath472 . hence again we can apply the subgroup corollary to conclude that for every subgroup @xmath208 , there is an embedding @xmath19 of @xmath37 in @xmath0 such that @xmath36 . next suppose that @xmath530 . then , in addition to @xmath337 , @xmath330 , and @xmath531 , there is an order 2 automorphism @xmath532 such that @xmath533 , @xmath534 , and @xmath535 . we saw above that the edge @xmath6 is not pointwise fixed by any non - trivial element of the form @xmath536 or @xmath537 . by an analogous argument we see that @xmath6 is also not pointwise fixed by any non - trivial element of the form @xmath538 . now for the sake of contradiction assume that for some @xmath33 and @xmath433 , @xmath539 pointwise fixes @xmath6 . thus both @xmath540 and @xmath541 . so @xmath542 , and hence : @xmath543 @xmath544 thus @xmath545 . however , since @xmath354 , @xmath355 , we have contradicted the fact that @xmath499 . so , once again , the edge @xmath6 is not fixed by any non - trivial element of @xmath15 , and thus we can again apply the subgroup corollary . finally , suppose that @xmath546 . then in addition to @xmath337 and @xmath330 , @xmath15 contains an order 2 automorphism @xmath547 such that @xmath548 and @xmath549 . if for some @xmath33 and @xmath433 , @xmath550 pointwise fixes @xmath6 , then by substituting @xmath547 for @xmath551 in the previous paragraph we would again obtain a contradiction . thus yet again we conclude that @xmath6 is not fixed by any non - trivial element of @xmath15 , so we can apply the subgroup corollary . theorem [ t : realize ] now follows immediately from propositions 1 and 2 . the authors would like to thank curtis bennett and michael aschbacher for several very helpful conversations which led to significant improvements in this paper . we also would like to thank the anonymous referee for helpful suggestions .

it is shown that for any locally knotted edge of a 3-connected graph in @xmath0 , there is a ball that contains all of the local knots of that edge which is unique up to an isotopy setwise fixing the graph . this result is applied to the study of topological symmetry groups of graphs embedded in @xmath0 . schubert s 1949 result @xcite that every non - trivial knot can be uniquely factored into prime knots is a fundamental result in knot theory . hashizume @xcite , extended schubert s result to links in 1958 . then in 1987 , suzuki @xcite generalized schubert s result to spatial graphs by proving that every connected graph embedded in @xmath0 can be split along spheres meeting the graph in 1 or 2 points to obtain a unique collection of prime embedded graphs together with some trivial graphs . [ h ] on the right is not isotopic ( setwise fixing the graph ) to either of the spheres @xmath1 or @xmath2 on the left.,title="fig : " ] although the set of prime factors of a knot or embedded graph is unique up to equivalence , the set of splitting spheres is generally not unique up to an isotopy setwise fixing the knot or graph . for example , consider the embedding of the complete graph @xmath3 which is illustrated on both the left and right sides of figure [ nonstandard ] . the edge @xmath4 contains two trefoil knots . the spheres @xmath1 and @xmath2 ( illustrated on the left ) are splitting spheres for these two knots . however , one of the balls bounded by @xmath5 ( illustrated on the right ) meets @xmath6 in an arc whose union with an arc in @xmath5 is a single trefoil knot . thus @xmath5 is also a splitting sphere for one of the two local knots in @xmath6 . however , @xmath5 is not isotopic ( fixing the embedded graph setwise ) to either of the spheres @xmath1 or @xmath2 . by contrast , in this paper we show that for any locally knotted edge of an embedded 3-connected graph , there is a ball meeting the graph in an arc containing all of the local knots of that edge which is unique up to an isotopy fixing the graph . we call such a ball an _ unknotting ball _ for that edge . our main theorem is the following . [ p : uniqueness]let @xmath7 be a 3-connected graph embedded in @xmath0 . then any locally knotted edge @xmath6 has an unknotting ball which is unique up to an isotopy of @xmath8 fixing the vertices of @xmath7 . furthermore , if @xmath9 are pairwise disjoint balls for local knots of an edge @xmath6 , then @xmath6 has an unknotting ball which contains @xmath10 . we shall apply this theorem to the study of topological symmetry groups of graphs embedded in @xmath0 . the concept of the _ topological symmetry group _ was first introduced by jon simon @xcite as a way of describing the symmetries of non - rigid molecules . let @xmath11 be an abstract graph . an _ automorphism _ of @xmath11 is a permutation of the vertices of @xmath11 which preserves adjacency . we use @xmath12 to denote the group of automorphisms of @xmath11 . given an embedding @xmath7 of an abstract graph @xmath11 in @xmath0 , the _ topological symmetry group _ , @xmath13 , is defined to be the subgroup of @xmath12 induced on the vertices of @xmath11 by diffeomorphisms of the pair @xmath14 . if we only allow orientation preserving diffeomorphisms , we obtain the orientation preserving topological symmetry group @xmath15 . in this paper we are only interested in the orientation preserving topological symmetry group . thus for simplicity , we abuse notation and refer to @xmath15 as the _ topological symmetry group _ rather than the _ orientation preserving topological symmetry group_. flapan , naimi , pommersheim , and tamvakis @xcite proved that not every finite group can occur as @xmath15 for some embedded graph @xmath7 in @xmath0 . for example , the alternating groups @xmath16 for @xmath17 can not occur as @xmath15 for any embedded graph . for most abstract graphs @xmath11 , it is not known what groups can occur as @xmath15 for some embedding of @xmath11 in @xmath0 . in this paper , we consider the simpler question of whether all of the subgroups of a given @xmath15 can occur as @xmath18 for some re - embedding @xmath19 of @xmath7 in @xmath0 . for example , let @xmath11 denote a circle with three vertices . if the embedded graph @xmath7 is an unknotted circle , then @xmath20 , the dihedral group with 6 elements . if we re - embed @xmath7 as a non - invertible knot @xmath19 , then @xmath18 is the subgroup @xmath21 ( in figure [ d3 ] , @xmath19 contains the non - invertible knot @xmath22 ) . on the other hand , for any embedding @xmath23 of @xmath11 in @xmath0 there will be an orientation preserving diffeomorphism of @xmath24 which cyclically permutes the three vertices ( obtained by slithering @xmath23 along itself ) inducing an order 3 automorphism of @xmath23 . hence there is no embedding @xmath25 of @xmath11 such that @xmath26 is either @xmath27 or the trivial group . thus not all of the subgroups of @xmath15 can occur as @xmath26 for some re - embedding @xmath25 of @xmath7 in @xmath0 . [ h ] in contrast with the above example , we use local knotting together with theorem 1 to prove that in many cases every subgroup of a topological symmetry can occur as the topological symmetry group of another embedding of the graph . in particular , we prove the following . let @xmath7 be an embedding of a 3-connected graph in @xmath0 and let @xmath28 be a ( possibly trivial ) subgroup of @xmath15 . let @xmath29 , @xmath30 be a set of edges in @xmath7 whose orbits under @xmath28 are distinct . suppose that any @xmath31 which pointwise fixes @xmath29 and satisfies @xmath32 for each @xmath33 , also pointwise fixes a subgraph of @xmath7 that can not be embedded in @xmath34 . then there is an embedding @xmath19 of @xmath7 with @xmath35 . observe that the result below follows immediately from the subgroup theorem . [ l : fundamentaledge ] let @xmath7 be a 3-connected graph embedded in @xmath0 which has an edge @xmath6 that is not pointwise fixed by any non - trivial element of @xmath15 . then for every ( possibly trivial ) subgroup @xmath28 of @xmath15 there is an embedding @xmath19 of @xmath7 with @xmath36 . we shall apply this corollary to the study of the topological symmetry groups of complete graphs . the complete graphs @xmath37 are an interesting family of graphs to focus on because the automorphism group of @xmath37 is the symmetric group @xmath38 , which is the largest automorphism group of any graph with @xmath39 vertices . on the other hand , flapan , naimi , and tamvakis @xcite have classified the groups that can occur as the topological symmetry group of some embedding of a complete graph in @xmath0 . more specifically , they proved the following . [ t : fnt ] @xcite a finite group @xmath28 is isomorphic to @xmath15 for some embedding @xmath7 of a complete graph in @xmath0 if and only if @xmath28 is a finite cyclic group , a dihedral group , a subgroup of @xmath40 for some odd @xmath41 , or @xmath42 , @xmath43 , or @xmath44 . although this result restricts the types of groups which can occur , for a given complete graph @xmath37 , it is still not known precisely which of the above groups occur . in this paper , we use the subgroup corollary to prove the following . [ t : realize ] let @xmath45 and let @xmath7 be an embedding of @xmath37 in @xmath0 such that @xmath15 is a finite cyclic group , a dihedral group , or a subgroup of @xmath40 for some odd @xmath41 . then for every ( possibly trivial ) subgroup @xmath28 of @xmath15 , there is an embedding @xmath19 of @xmath37 such that @xmath36 . note that in @xcite , we classify all values of @xmath39 such that there is an embedding @xmath7 of @xmath37 with @xmath15 equal to @xmath42 , @xmath43 , or @xmath44 .

unknotting balls adding local knots to spatial graphs topological symmetry groups of complete graphs acknowledgments

arxiv

the observation of diffraction and interference of electron waves would provide the crucial test of the existence of wave nature of electrons . this observation was first seen in 1927 by c. j. davisson and l. h. germer.@xcite they studied electron scattering from a target consisting of a single crystal of nickel ( ni ) and investigated this phenomenon extensively . electrons from an electron gun are directed at a crystal and detected at some angle that can be varied ( see fig.[fig01 ] ) . for a typical pattern observed , there is a strong scattering maximum at an angle of 50@xmath0 . the angle for maximum scattering of waves from a crystal depends on the wavelength of the waves and the spacing of the atoms in the crystal . using the known spacing of atoms in their crystal , they calculated the wavelength that could produce such a maximum and found that it agreed with the de broglie s hypothesis for the electron energy they were using . by varying the energy of the incident electrons , they could vary the electron wavelengths and produce maxima and minima at different locations in the diffraction patterns . in all cases , the measured wavelengths agreed with de broglie s hypothesis . the davisson - germer experiment itself is an established experiment.@xcite there is no controversy for them . how about the physical interpretation ? one can see the description of the experiments and its physical interpretation in any standard textbook of the modern physics , which is one of the required classes for the physics majors ( undergraduate ) in u.s.a . nevertheless , students as well as instructors in this course may have some difficulty in understanding the underlying physics , since the descriptions of the experiments are different depending on textbooks and are not always specific.@xcite as far as we know , proper understanding has not been achieved fully so far . in some textbooks,@xcite the ni layers are thought to act as a reflective diffraction grating . when electrons are scattered by the ni ( 111 ) surface ( single crystal ) , the electrons strongly interact with electrons inside the system . thus electrons are scattered by a ni single layer . the ni ( 111 ) surface is just the two - dimensional layer for electrons . in other textbooks,@xcite electrons are scattered by ni layers which act as a bulk system . the 3d character of the scattering of electrons appears in the form of bragg points in the reciprocal lattice space.@xcite the 3d bragg reflection can occur when the bragg points lie on the surface of ewald sphere , like the x - ray diffraction . here we will show that the ni layers act as a reflective diffraction grating . the 2d scattering of electrons on the ni ( 111 ) , ni(100 ) , and ni(110 ) surfaces will be discussed in terms of the concept of the bragg rod ( or bragg ridge ) which intersects the surcae of the ewald sphere.@xcite we will show that the experimental results@xcite obtained by davisson and germer can be well explained in terms of this model . @xmath1 is wave vector of incident electron beam and @xmath2 is wave vector of outgoing electron beam.,width=264 ] is drawn in the direction of the incident electron beam . @xmath3 , which is the inplane reciprocal lattice vector , parallel to the surface . ewald sphere ( radius ( @xmath4 ) is centered at the point o. the point o@xmath5 is the origin of the reciprocal lattice vectors . the bragg reflection occurs when the surface of the ewald sphere intersects the bragg rod originated from the nature of the 2d system : @xmath6 . @xmath7 ev . @xmath8 for the ni(111 ) plane . the lattice constant of conventional fcc ni is @xmath9.,width=264 ] . the red lines are denoted by the bragg rods arisen from the character of the 2d system . the bragg reflection occurs when the wave vector of the reflected wave is on the point [ denoted by the blue points , which are not the bragg points ] , where the ewald sphere intersects the bragg rod . @xmath3.,width=264 ] in 1925 , davisson and germer investigated the properties of ni metallic surfaces by scattering electrons . their experiments ( davisson - germer experiment ) demonstrates the validity of de broglie s postulate because it can only be explained as a constructive interference of waves scattered by the periodic arrangement of the atoms of the crystal . the bragg law for the diffraction had been applied to the x - ray diffraction , but this was first application to the electron waves . we now consider the bragg reflections in the 2d system . the bragg reflections appear along the reciprocal rod , which is described by @xmath10 , where @xmath10 ( @xmath11 ) is the in - plane reciprocal lattice vector parallel to the surface . the incident electron wave ( @xmath12 , @xmath13 ) is reflected by the surface of the 2d system . @xmath2 ( @xmath14 ) is the wavevector of the out - going electron wave ( @xmath15 ) . here we use the notation @xmath16 as the wavelength , instead of the conventional notation @xmath17 . the ewald sphere is formed of the sphere with the radius of @xmath18 @xmath19 . the scattering vector @xmath20 is defined by @xmath21 and o@xmath5 is the origin of the reciprocal lattice space . the 2d system is located at the origin of the real space o. the direction normal to the surface of the system is anti - parallel to the direction of the incident electron wave . since the system is two - dimensional , the reciprocal lattice space is formed of bragg rods . the bragg reflections occur when the bragg rods intersect the surface of the ewald sphere.@xcite because of the 2d system , the bragg points of the 3d system are changed into the bragg rods . then the bragg condition occurs under the condition ( see fig.[fig03 ] ) , @xmath22 where @xmath23 the scattering angle @xmath24 is related to the angle @xmath25 as @xmath26 in the electron diffraction experiment , we usually need to use the wavelength ( @xmath16 ) , which is taken into account of the special relativity,@xcite @xmath27 or @xmath28 where @xmath16 is the wavelength , @xmath29 where @xmath30 is the planck s constant and @xmath31 is the velocity of light , @xmath32 ( in the units of ev ) is the kinetic energy of electron . @xmath33 ( @xmath34 ) is the rest energy . @xmath35 is the rest mass of electron . in the non - relativistic limit , we have @xmath36 in the unit of @xmath37 . when @xmath7 ev , @xmath16 is calculated as @xmath38 . suppose that ni ( 111 ) plane behaves like a three - dimensional system . the 3d bragg reflection occurs only if the bragg condition @xmath39 is satisfied , where @xmath20 is the scattering vector and @xmath40 is the reciprocal lattice vectors for the 3d system . in the experimental configuration as shown in fig.[fig02 ] . @xmath40 is one of the reciprocal lattice vectors for the fcc ni , and appears in the form of bragg point . this bragg point should be located on the surface of the ewald sphere with radius ( @xmath41 ) centered at the point o ( see fig.[fig02 ] ) . no existence of such a bragg point on the ewald sphere indicates that the 3d bragg scattering does not occur in the present situation ( fig.[fig02 ] ) . the primitive cell by definition has only one lattice point . the primitive translation vectors of the fcc lattice are expressed by @xmath42 where there is one lattice point ( or atom ) per this primitive cell and @xmath43 is the lattice constant for the conventional cubic cell ( @xmath44 for fcc ni).@xcite the corresponding reciprocal lattice vectors for the primitive cell are given by [ eq11 ] @xmath45 the reciprocal lattice vector is described by @xmath46 where @xmath47 , @xmath48 , and @xmath49 are integers . the translation vectors of the conventional unit cell ( cubic ) are expressed by @xmath50 where there are two atoms per this conventional unit cell.@xcite the reciprocal lattice vectors are defined by [ eq13 ] @xmath51 in general , the reciprocal lattice vector is given by @xmath52 with @xmath53 there are relations between @xmath54 and @xmath55 . note that all indices of @xmath54 are odd or even . there is a selection rule for the indices @xmath54 . the structure factor @xmath56 for the 2d system@xcite is given by @xmath57 where @xmath58 then @xmath56 depends only on @xmath59 and @xmath60 , forming the bragg rod ( or bragg ridge ) in the reciprocal lattice space . the structure factor @xmath56 for the 3d system@xcite is given by @xmath61 where @xmath62 is the position vectorof each atom , @xmath63 then @xmath64 depends only on @xmath59 , @xmath60 , and @xmath65 , which leads to the bragg points . let @xmath66 be defined by the contribution of atom @xmath67 to the electron concentration . the electron concentration is expressed by @xmath68 over the @xmath69 atoms of the basis . then we have @xmath70 or @xmath71 ) for ni(111 ) plane . note that @xmath72 , @xmath73 and @xmath74 are the reciprocal lattice vectors for the primitive cell where one atom exists , and @xmath75 , @xmath76 , and @xmath77 are the reciprocal lattice vectors for the conventional cell . 2d reciprocal lattice plane , which is viewed from the direction of ( @xmath78 ) . the green lines form a bragg rod along the direction of ( @xmath79 ) , arising from the 2d character of the system . the red circle shows the 3d bragg point of fcc ni . the blue circle is not the 3d bragg point and lies on the 2d bragg rods . , width=264 ] ) [ ni(111 ) plane ] . bragg rod forming along the direction ( @xmath79 ) . the red circle denotes the 3d bragg point of fcc ni . , width=264 ] here we discuss the experimental results obtained by davisson and germer in terms of the model described in the section [ model ] . here we note that @xmath80 with @xmath81 the unit vector along the direction of the vector @xmath82 is given by @xmath83 the component of @xmath72 parallel to the unit vector @xmath84 is @xmath85 similarly , we have @xmath86 which is equal to @xmath87 the component of @xmath72 , @xmath73 , and @xmath74 , perpendicular to the unit vector @xmath84 are [ eq26 ] @xmath88 then we get [ eq28 ] @xmath89 the 2d reciprocal lattice vector formed by bragg rods ( @xmath90 , @xmath91 , @xmath92 , @xmath93 , @xmath94 , @xmath95 ) is shown by figs.[fig04 ] and [ fig05 ] , where the magnitude of the reciprocal lattice vector is given by @xmath96 where @xmath97 . note that @xmath98 can be also obtained as @xmath99 figure [ fig06 ] shows the 2d reciprocal lattice vectors formed by the bragg rods with the six - fold symmetry . this implies that the corresponding 2d triangular lattice is formed in the real space . the direction of the fundamental lattice vector @xmath100 is rotated by 30@xmath0 with respect to the direction of the fundamental reciprocal lattice vector @xmath98,@xcite where @xmath101 using the geometry as shown in fig.[fig03 ] , the bragg condition can be obtained as @xmath102 where @xmath103 = 1 , 2 , , 3 , ..... and @xmath104 is the fundamental reciprocal lattice ( see fig.[fig06 ] ) . note that @xmath105 and @xmath106 are also possible for @xmath107 and @xmath108 , respectively . here we only consider the case of integer @xmath103 . we introduce the length @xmath109 such that @xmath110 where @xmath111 equation ( [ eq34 ] ) with @xmath112 corresponds to the expression for the reflective diffraction grating , where @xmath113 for the ni(111 ) plane . this value of @xmath114 agrees well with that reported by davisson and germer.@xcite we note that the left side of eq.([eq36 ] ) is the path difference between two adjacent rays for the reflective diffraction grating ( see fig.[fig07 ] ) . when @xmath32 = 54 ev , the wavelength can be calculated as @xmath115 , using eq.([eq07 ] ) . from the result of the davisson - germer experiment,@xcite @xmath116 . we get @xmath117 . this wavelength is exactly the same as that calculated based on the de broglie hypothesis . is rotated by 30@xmath0 with respect to the axis of the reciprocal lattice @xmath118 . @xmath119 . @xmath120 . @xmath121 . @xmath114 ( @xmath123 ) is the distance such that ni ( 111 ) plane acts as a reflective diffraction grating,@xcite @xmath124.,width=264 ] for ni(111 ) plane.@xcite @xmath125 . @xmath9 for ni . the blue points denote ni atoms on the 2d layer.,width=302 ] vs the kinetic energy @xmath32 for the ni ( 111 ) plane . the data denoted by points ( green ) were reported by davisson and germer.@xcite the red solid line for @xmath98 ( @xmath112 ) . the blue dotted line for @xmath126 ( @xmath127 ) . the purple dashed line for @xmath128 ( @xmath129 ) , where @xmath130.,width=302 ] figure [ fig08 ] shows the plot of the angle @xmath25 as a function of the kinetic energy @xmath32 , which is expressed by eq.([eq33 ] ) , where @xmath103 = 1 , 2 , and 3 . in fig.[fig08 ] , we also plot the experimental data obtained by davisson and germer ( denoted by green points ) . we find that all the data lie well on the predicted relation between @xmath25 and @xmath32 for @xmath103 = 1 , 2 , and 3 . the six - fold symmetry of the 2d reciprocal lattice vectors was experimentally confirmed by davisson and germer for the ni(111 ) plane [ @xmath32 = 54 ev and @xmath6].@xcite the rotation of the ni sheet around the ( 111 ) direction leads to nealy six - fold symmetry of the intensity as a function of azimuthal for latitude . note that the intensities at @xmath132 , @xmath133 , and @xmath92 ( denoted as ( 111 ) plane by davissson and germer)@xcite are stronger than those from @xmath134 , @xmath135 , and @xmath136 ( denoted as ( 100 ) plane by davisson and germer).@xcite we also note that when @xmath32 = 65 ev , davisson and germer observed @xmath137 , where @xmath16 can be evaluated as @xmath138 , using eq.([eq36 ] ) with @xmath109 by eq.([eq35]).@xcite in this case , the intensities at @xmath132 , @xmath91 , and @xmath92 are much weaker than those from @xmath139 , @xmath135 , and @xmath140 . in other words , the intensity vs azimuthal pattern is strongly dependent of the kinetic energy of electrons . for the ideal case of scattering from a true 2d network of atoms , the intensity vs azimuthal should show the perfect six - fold symmetry . the intensity is the same for @xmath132 , @xmath91 , @xmath92 , @xmath141 , @xmath142 , and @xmath136 . in the davisson - germer experiment,@xcite it may be possible that the primary electrons penetrate several atomic layers into the system . the deeper they penetrate , the more scattering events in the direction perpendicular to the surface , enhancing the contribution of the 3d scattering to experimental results . this leads to the change of the intensity of the bragg reflections as a function of azimuthal , in comparison with the case of pure 2d scattering.@xcite -direction , where @xmath75 is the reciprocal lattice vector of the conventional cubic lattice . ni(100 ) plane . the red circle denotes the 3d bragg points for fcc ni.,width=264 ] , @xmath91 are the reciprocal lattice vectors , which is viewed from the @xmath75-direction , where @xmath75 is the reciprocal lattice vector of the conventional cell . @xmath143 . , width=264 ] the unit vector along the ( 1,0,0 ) direction is defined by @xmath144 the components of @xmath72 and @xmath73 , parallel to the unit vector @xmath145 are [ eq40 ] @xmath146 then the components of @xmath72 and @xmath73 perpendicular to the unit vector @xmath145 are @xmath147 then we get the 2d reciprocal lattice vectors formed by bragg rods , having the four - fold symmetry around the vector @xmath145 , @xmath148 using the geometry as shown in fig.[fig10 ] , the bragg condition can be expressed in terms of @xmath149 for the ni(100 ) plane , where @xmath150 is the length of spacing for the reflective diffraction grating for ni(100 ) plane @xmath151 . vs the kinetic energy @xmath32 for the ni ( 100 ) plane . the data denoted by points ( green ) were reported by davisson and germer.@xcite the red solid line for @xmath98 . the blue dotted line for @xmath126 . the purple dashed line for @xmath152 , where @xmath153.,width=302 ] figure [ fig11 ] shows the plot of the angle @xmath25 as a function of the kinetic energy @xmath32 , which is expressed by eq.([eq42 ] ) , where @xmath103 = 1 , 2 , and 3 . in fig.[fig11 ] , we also plot the experimental data obtained by davisson and germer ( denoted by green points).@xcite we find that all the data fall fairly well on the predicted relation between @xmath25 and @xmath32 for @xmath103 = 1 , 2 , and 3 , in particular for @xmath112 . when @xmath32 = 190 ev , the wavelength can be calculated as @xmath154 , using eq.([eq07 ] ) . from the result of the davisson - germer experiment , @xmath155,@xcite on the other hand , we get @xmath156 for the ni(100 ) plane . this wavelength is almost the same as that calculated based on the de broglie s hypothesis . + @xmath76 direction , where @xmath75 and @xmath76 are the reciprocal lattice vector of the conventional fcc lattice . ni(110 ) plane . the red circle denotes the 3d bragg point . the blue circle does not denote the 3d bragg point and lies on the 2d bragg rod . , width=264 ] the unit vector along the ( 110 ) direction is defined by @xmath157 the components of @xmath72 , @xmath72 , and @xmath72 , parallel to the unit vector @xmath158 are [ ad45 ] @xmath159 the components of @xmath72 , @xmath73 , and @xmath74 , perpendicular to the unit vector @xmath158 are @xmath160 then we get the magnitude of the 2d reciprocal lattice vector ( rectangular lattice ) @xmath161 , @xmath91 are the reciprocal lattice vectors , which is viewed from the @xmath162 direction.,width=302 ] using the geometry as shown in fig.[fig13 ] , the bragg conditions for @xmath163 and @xmath164 can be expressed by @xmath165 and @xmath166 respectively , where @xmath167 and @xmath168 . the lengths @xmath169 and @xmath170 are equivalent spacings of the 2d rectangular lattice ( real space ) . figure [ fig14 ] shows the plot of the angle @xmath25 as a function of the kinetic energy @xmath32 , which is expressed by eq.([eq47 ] ) , where @xmath103 = 1 , 2 , and 3 . in fig.[fig14 ] , we also plot the experimental data obtained by davisson and germer ( denoted by green points).@xcite we find that all the data lie fairly well on the predicted relation given by eq.([eq47 ] ) between @xmath25 and @xmath32 for @xmath171 with @xmath103 = 2 . vs the kinetic energy @xmath32 for the ni ( 110 ) plane . the data denoted by points ( green ) were reported by davisson and germer.@xcite @xmath172 [ eq.([eq47 ] ) ] with @xmath173 , red solid line ) . @xmath174 ( @xmath173 , blue dotted line ) . @xmath174 ( @xmath175 , purple dashed line).,width=302 ] when @xmath32 = 143 ev , the wavelength can be calculated as @xmath176 , using eq.([eq07 ] ) . from the result of the davisson - germer experiment , @xmath177 , on the other side , we get @xmath178 using @xmath179 . this wavelength is almost the same as that calculated based on the de broglie s hypothesis . we note that the @xmath180-spacing @xmath181 for the reflective diffraction grating is @xmath182 , for the ni(110 ) plane . this value of @xmath181 agrees well with that reported by davisson and germer.@xcite the essential feature of the davisson - germer experiment for the ni(111 ) , ni(100 ) , and ni(110 ) planes is that the 2d bragg scattering occurs . the bragg rods are formed in the reciprocal lattice space . the component of the scattering vector * _ q _ * parallel to the surface is equal to the 2d surface reciprocal lattice vector of the bragg rods . the electron beam is reflected from a single layer , leading to the eflective diffraction grating with the @xmath180-spacing @xmath114 . davisson and l.h . germer , phys . rev . * 30 * , 705 ( 1927 ) . davisson and l.h . germer , nature * 119 * , 558 ( 1927 ) . davisson and l.h . germer proc . science * 14 * , 317 ( 1928 ) . davisson and l.h . germer proc . science * 14 * , 619 ( 1928 ) . c. davisson , _ the discovery of electron waves _ , p.387 ( 1937 ) . nobel prize lecture . < http://www.nobelprize.org/nobel_prizes/physics/laureates/1937/davisson-lecture.pdf>[<http://www.nobelprize.org/nobel_prizes/physics/laureates/1937/davisson-lecture.pdf > ] r.k . gehrenbeck , phys . today , january , 34 ( 1978 ) . r. eisberg and r. resnick , _ quantum physics of atoms , molecules , solids , nuclei , and particles _ , second edition ( john wiley & sons , new york , 1985 ) . tipler and r.a . llewellyn , _ modern physics _ , fifth edition ( w.h . freeman and company , 2008 ) . serway , c.j . moses , and c.a . moyer , _ modern physics _ , third edition ( thomson , brooks / cole , 2005 ) . k.s . krane , _ modern physics _ , third edition ( john wiley & sons , 2012 ) . s.t . thornton and a. rex , _ modern physics for scientists and engineers _ , fouth edition ( cengage , learning , 2013 ) . wichmann , _ quantum physics _ ( education development center inc . 1971 ) . c. kittel , _ introduction to solid state physics _ , fourth edition ( john wiley & sons , new york , 1971 ) . clarke , _ surface crystallography an introduction to low energy electron diffraction _ ( john wiley & sons , new york , 1985 ) . j. als - nielsen and d. mcmorrow , _ elements of modern x - ray physics _ ( john wiley & sons , ltd . , new york , 2001 ) . h. lth solid surfaces , _ interfaces and thin films _ , fourth , revised and extended edition ( springer , berlin , 2001 ) . cowley , _ diffraction physics _ , third revised edition ( elsevier science b.v . , amsterdam , 1995 ) . a. guinier , _ x - ray diffraction in crystals , imperfect crystals , and amorphous bodies _ ( w.h . freeman and company ( san francisco , 1963 ) .

the physical interpretation for the davisson - germer experiments on nickel ( ni ) single crystals [ ( 111 ) , ( 100 ) , and ( 110 ) surfaces ] is presented in terms of two - dimensional ( 2d ) bragg scattering . the ni surface acts as a reflective diffraction grating when the incident electron beams hits the surface . the 2d bragg reflection occurs when the ewald sphere intersects the bragg rods arising from the two - dimension character of the system . such a concept is essential to proper understanding of the davisson - germer experiment for undergraduate modern physics course

introduction [model]model: ewald sphere and 2d bragg scattering fundamental structure factor for ideal 2d and 3d systems: bragg rods and bragg points discussion conclusion

arxiv

observers of interstellar sight lines are familiar with the strong @xmath10@xmath7@xmath11 permitted absorption bands of co below 1545 . less familiar to observers are the intersystem bands of co , which involve the triplet states @xmath12 , @xmath13 , and @xmath14 , whose electric dipole transitions to the ground state @xmath11 are spin - forbidden . these triplet states have potential energy curves that cross the curve of @xmath10see figure 1 of rostas et al . ( 2000)and via mutual perturbations acquire a small percentage of @xmath15 character at the expense of the singlet state . the resulting triplet - singlet @xmath6-values are relatively small , thus turning the intersystem bands into very useful astrophysical probes along interstellar ( is ) sight lines for which the _ a@xmath7x _ bands are optically thick , see ( * ? ? ? * hereafter mn ) , ( * ? ? ? * hereafter f94 ) . to exploit such probes , accurate @xmath6-values of intersystem transitions are required . * hereafter rostas et al . ) presented predictions from extended calculations of the triplet - singlet mixing coefficients , partly as an attempt to resolve discrepancies between previous theoretical ( mn ) and astrophysical @xmath6-values . rostas et al . also conducted low - resolution laboratory measurements at room temperature to improve upon the experimental results of @xcite . high - resolution laboratory measurements on two intersystem bands were also reported by @xcite . using the _ hubble space telescope _ ( @xmath16 ) , we obtained a high - resolution echelle spectrum of the o9.5v star x per ( hd 24534 ) with the space telescope imaging spectrograph ( stis ) . permitted _ a@xmath7x _ bands along this line of sight were first modeled by @xcite , who based his analysis of @xmath0co on archival @xmath16 goddard high - resolution spectrograph ( ghrs ) observations . for our stis observations , the weakest permitted band available for analysis is the _ a@xmath7x _ ( 8@xmath70 ) band , which still has a large optical depth ( @xmath17 ) in the cores of the strongest lines , @xmath17(@xmath18(0 ) ) @xmath19 70 . therefore , the column density ( @xmath20 ) of @xmath0co toward x per can be measured more reliably from the observed intersystem bands , which have a much smaller @xmath17 . in this letter , we determine @xmath20(@xmath0co ) through a comparison with our previous observations of intersystem bands toward @xmath1 oph ( f94 ) . then we derive is @xmath6-values toward x per for all seven `` old '' bands and for five intersystem bands seen in is spectra for the first time . finally , we compare our @xmath6-values with the predictions of rostas et al . regarding nomenclature , we make use in this paper of the short notation introduced by rostas et al . , where , e.g. , @xmath2112 stands for _ d@xmath7x _ ( 12@xmath70 ) . stis spectra of x per were acquired in 2001 february and march with grating e140h during 10,728 s over five orbits , yielding the data sets o64812010@xmath7030 and o64813010@xmath7020 . the star was observed through the smallest aperture ( 0.1 @xmath2 0.03 ) , which was designed to provide the highest resolving power , @xmath18 = 200,000 @xcite . elaborate reduction procedures had to be employed in order to correct for odd - even sub - pixel effects in the unbinned ( 2048-pixel ) exposures . normal reduction routines were then followed by processing the spectra in the stsdas environment of iraf . the five orbital exposures were co - added into a single spectrum of 42 orders , sporting a signal - to - noise ratio of 70 , or 100 for features detected and combined from two adjacent overlapping orders . since the continuum is modulated by shallow photospheric features from x per , short spectral segments centered on features ( bands ) of interest were rectified and normalized before being compared to a fitted synthetic model of the pertinent feature . flux and wavelength calibrations were performed by the space telescope science institute pipeline and supplied with the binned , 1024-pixel exposures . a previous analysis of @xmath0co by @xcite inferred that this line of sight involves two unresolved velocity components . however , the significantly higher resolution of our spectrum shows profile asymmetries due to underlying structure , which required four components for a good fit . this is the same number of components found in preliminary fits of high - resolution spectra of x per ( d. e. welty , private communication ) , although we derived a different set of relative velocities , fractions , and component widths . some 87% of the total @xmath20(@xmath0co ) resides in a main component with a doppler parameter ( @xmath22-value ) of 0.39 km s@xmath5 . the other three components have between 2% and 9% shares of @xmath20 , with @xmath22-values between 0.17 and 2.09 km s@xmath5 , as will be fully described in a future paper . the highly saturated _ a@xmath7x _ bands could be fitted only once the total @xmath20 was established via the intersystem bands . our method of analysis employed a multi - parametric spectrum synthesis code based on voigt profile line transfer equations given in @xcite . the computed absorption spectrum was convolved with a gaussian instrumental profile ( jenkins & tripp 2001 ) and then matched with the data via rms - minimization down to relative parameter steps of 10@xmath23 or less . in order to determine @xmath6-values for intersystem bands , @xmath20(@xmath0co ) toward x per must be known . for optically thin features , the equivalent width ( @xmath24 ) is proportional to @xmath17 , i.e. , @xmath24 @xmath25 @xmath26 . we can , therefore , utilize the published value of @xmath20 toward @xmath1 oph and the ratio of @xmath24s for optically thin bands toward the two stars in the determination of @xmath20 toward x per . the result should be independent of the intersystem @xmath6-values and of the modeled cloud structure , provided we exclude the @xmath24 ratio for the optically thick @xmath2714 . column 4 of table 1 lists the ratios of band @xmath24 toward x per over band @xmath24 toward @xmath1 oph . among the optically thin bands , the highest @xmath24 ratio belongs to @xmath284 , which was the faintest intersystem band analyzed by f94 . in addition , @xmath284 also sports the largest deviation of derived @xmath6-value in the low-@xmath17 sample of f94 . it is clear that f94 underestimated the low - resolution @xmath24 for @xmath284 due to placing the continuum too low ( see their figure 1 ) . dropping the @xmath284 band , and thus avoiding a 5% increase in the average , we obtain @xmath29(x per)/@xmath29(@xmath1 oph ) = 5.54 @xmath30 0.87 . under the necessary and fulfilled condition of optical thinness , this is also the ratio of @xmath20(x per ) over @xmath20(@xmath1 oph ) . here we shall adopt the @xcite column density toward @xmath1 oph , @xmath20(@xmath0co ) = ( 2.54 @xmath30 0.16 ) @xmath2 10@xmath31 @xmath4 , which was also employed by f94 . based on the measured @xmath20 ratio , we infer that @xmath20(@xmath0co ) toward x per is ( 1.41 @xmath30 0.22 ) @xmath2 10@xmath3 @xmath4 . kaczmarczyk ( 2000b ) found ( 1.0 @xmath30 0.2 ) @xmath2 10@xmath3 @xmath4 from an analysis of permitted _ a@xmath7x _ bands ; the agreement between the two determinations is at the 2-@xmath32 level . recall , however , that kaczmarczyk s model has fewer cloud components and larger @xmath22-values , readily accounting for its reduced column density . the crossings of vibrational levels in a triplet state and vibrational levels in the @xmath10 state occur near certain rotational @xmath33 levels . therefore , mixing coefficients depend on @xmath33 and thus intersystem @xmath34-values generated by interaction with @xmath10 have a dependence on @xmath35 . whereas all previous calculations of mixing coefficients focused on the vibrational level of the a state ( @xmath36 ) closest to the perturbing triplet level , i.e. , a single-@xmath36 treatment , rostas et al . were the first to use multi-@xmath36 computations that included all @xmath37 levels with @xmath38 = 0 to 12 . they derived @xmath34-values that for some intersystem bands were in better agreement with their own experimental results ( @xmath35 = 300 k ) , and/or with the is results of f94 ( @xmath35 = 4.2 k ) . still , the calculated @xmath6-values of rostas et al . are `` a few percent '' off , because the changes in @xmath39@xmath40 of the `` parent '' _ a@xmath7x _ bands induced by mixing with other triplet levels were ignored . in addition , differences of a few percent are expected due to the adopted sources for _ a@xmath7x _ @xmath6-values : mn and we used values from @xcite , while rostas et al.s preference lies with values published by @xcite . the mn line oscillator strength , @xmath41@xmath40 , which we use as input for the fits , includes @xmath39@xmath40 for the permitted _ a@xmath7x _ band and the usual h@xmath42nl - london rotational factor , as well as the single-@xmath36 strength of the perturbation . according to rostas et al . , for triplet levels that cross the @xmath37 levels at low values of @xmath33 , single-@xmath36 and multi-@xmath36 calculations should not differ by a large factor since the interaction with the nearest `` parent '' @xmath36 level is predominant . therefore , using @xmath41@xmath40-values from mn is a valid approach while searching for relatively small corrections due to multi-@xmath36 interactions . this condition is appropriate for all bands studied here , except for @xmath217 , @xmath2110 , and to a lesser extent , @xmath284 . on the other hand , intersystem bands that are farther from their nearest parents have `` @xmath43-independent '' @xmath34-values due to their high-@xmath33 crossing , especially under is conditions . again , using @xmath41@xmath40-values from mn poses no difficulties for scaling the integrated band @xmath6-values thanks to negligible high-@xmath33@xmath44 level populations . indeed , we made no attempt to fit rotational lines independently of each other ; our syntheses did not indicate a need to adjust individual @xmath41@xmath40-values . rather than varying the @xmath6-values , our variable parameters were @xmath20(@xmath0co ) , which scales the entire band , and @xmath35 , which determines its shape . eleven @xmath0co bands exhibit similar excitation temperatures with averages of @xmath45 = 6.1 @xmath30 1.0 k and @xmath46 = 5.9 @xmath30 0.4 k. lower @xmath35 were suggested by the fit to @xmath2714 , presumably due to effects of higher @xmath17 . although @xmath34 depends on @xmath35 , the difference in @xmath6-values between @xmath35 = 4.2 k and @xmath35 = 6.0 k is @xmath47 1% , which we ignore here . one additional variable fitting parameter was the heliocentric radial velocity . for the main component , the average value from nine @xmath0co intersystem bands is 14.7 @xmath30 1.0 km s@xmath5 , where the 1 @xmath32 error is dominated by stis wavelength calibration uncertainties . this radial velocity is in very good agreement with the fits mentioned in 2 . table 1 lists results of synthetic fits for 12 intersystem bands detected in our spectrum of x per . for the reference @xmath6-values we used the mn @xmath41@xmath40-values of the @xmath18(0 ) lines , which agree to @xmath30 1% of @xmath34-values calculated for @xmath35 = 4.2 k , except for @xmath2714 and @xmath2717 , which have low-@xmath33 crossings and for which the @xmath18(0 ) @xmath6-values are off by @xmath47 2% . each ratio of a fitted @xmath34-value over the reference mn value was computed from the ratio of each band s fitted @xmath20(@xmath0co ) over the actual @xmath20 determined in 3 . column 5 lists the resulting band @xmath6-values toward x per . for comparison , in columns 6@xmath78 we list ratios of @xmath34 from this paper , from f94 , and from theoretical predictions in rostas et al . over the values from mn . the rostas et al . values are based on multi-@xmath36 calculations while the mn values refer to single-@xmath36 results . the theoretical ratios in column 8 lack formal error bars , although deviations due to the presence of other triplet levels and to absolute scales should amount to a few percent . the last column directly compares @xmath6-values from x per to those from rostas et al . the first seven bands listed in table 1 were previously detected toward @xmath1 oph by f94 . as discovered by f94 , the measured @xmath6-value of @xmath217 toward x per is five times greater than the value listed by mn in their tables 5 and 14 . however , mn list another @xmath6-value for @xmath217 in their table 7 , based on measurements by eidelsberg et al . ( 1992 ) ; the latter result is more in line with both is @xmath6-values and was confirmed in room temperatures experiments by rostas et al . the more rigorous multi-@xmath36 calculation reproduces the astronomical @xmath6-values much better than a single-@xmath36 prediction because @xmath217 is relatively far from its nearest parent , @xmath372 . our new @xmath6-value is 6% ( 0.4 @xmath32 ) lower than the prediction by rostas et al . , while the f94 result differs by 1% only . consequently , we do not confirm the recent laboratory result by stark et al . ( 2002 ) that @xmath34(@xmath217 ) should be reduced by ( 29 @xmath30 15)% relative to rostas et al . our synthesis of the optically thick @xmath2714 toward x per returns an @xmath6-value 6% smaller than that listed by mn , @xmath48 the f94 result of @xmath724% . the agreement with the newer predictions is better : the f94 @xmath6-value is lower by 19% , whereas the x per result is identical to that of rostas et al . according to them , single-@xmath36 and multi-@xmath36 @xmath6-values are also identical : 8.3 @xmath2 10@xmath23 . thus , ironically , the strong @xmath2714 can not help to distinguish between the two methods because its ( single ) interaction with @xmath374 is extremely predominant . all other five bands , @xmath2711 , @xmath2717 , @xmath2112 , @xmath284 , and @xmath285 , are also found to be in very good agreement with the multi-@xmath36 predictions of rostas et al . , i.e. , to within 20% . of these , the f94 @xmath27 bands already showed an excellent agreement with mn ( @xmath9 6% ) , and are within 7% of rostas et al . the last three f94 bands showed large differences ( down to @xmath749% ) from mn predictions . all three show better agreement with rostas et al . : @xmath2112 now differs by @xmath716% , @xmath285 differs by @xmath723% , while @xmath284 is off by @xmath737% ( 3.1 @xmath32 ) . the f94 @xmath6-value for @xmath284 remains their most discrepant result ( after @xmath217 ) . this weakest band of f94 was affected by an improper continuum placement , forcing its exclusion from our derivation of @xmath20(@xmath0co ) in 3 . stark et al.s ( 2002 ) result for @xmath284 , that its @xmath6-value is marginally too small by ( 17 @xmath30 15)% relative to rostas et al . , is very similar to the x per measurement : ( @xmath711 @xmath30 15)% . out of the seven bands under discussion , the largest x per difference relative to rostas et al.s predictions belongs to @xmath2717 ( + 20% = + 1.0 @xmath32 ) , a band for which a few weaker @xmath33@xmath44 = 2,3 lines are blended with another intersystem band , @xmath2112 . in our spectrum there are first is detections of four @xmath0co intersystem bands : @xmath218 , @xmath219 , @xmath2110 , and @xmath2812 . of these , @xmath34 was predicted by rostas et al . for @xmath218 , while single-@xmath36 @xmath6-values were listed by mn for @xmath218 and @xmath2812 . we identified @xmath219 and @xmath2110 as intersystem co features thanks to their characteristic shapes . final confirmation was provided by a comparison with laboratory bandhead wavelengths in table 27 of @xcite , or through the use of molecular state constants from tilford & simmons to compute reasonably accurate line positions . the newly detected bands are shown in fig . our wavelength calculations with the constants for the @xmath49 state identify the new _ d@xmath7x _ bands as the @xmath50 ( @xmath51 = 1 ) branch of the @xmath13 state . according to the lists of mn , @xmath52 ( @xmath51 = 2 ) components are at least ten times weaker . whereas @xmath218 was already mentioned by mn , who adopted eidelsberg et al.s ( 1992 ) room temperature @xmath6-value , the newer experiments of rostas et al . led to a refined value . our determination clearly favors the new value of rostas et al . calculated for an is temperature , since the x per @xmath6-value is @xmath724% ( @xmath71.7 @xmath32 ) away from their value , but is @xmath761% ( @xmath77.6 @xmath32 ) away from the value listed by mn . the strongest of the new bands , @xmath219 , has a @xmath24 between those of @xmath217 and @xmath2112 . its strong detection contradicts its non - inclusion in the tables of mn , who imposed a limiting @xmath6-value smaller than inferred here . the third _ d@xmath7x _ band , @xmath2110 , has the smallest secure @xmath6-value found in this study , @xmath19 6 @xmath2 10@xmath53 . the @xmath2110 band also lacks a previously published @xmath6-value . toward x per , it is blended with the red wing of the much wider @xmath541402 , which was successfully removed by profile rectification . the new is detection of @xmath2812 close to @xmath378 confirms to within 7% the @xmath6-value given by mn , but it was not included in the study of rostas et al . however , in the last column of table 1 we compare @xmath219 , @xmath2110 , and @xmath2812 @xmath6-values with unpublished multi-@xmath36 calculations ( m. eidelsberg , private communication ) . two bands agree to @xmath30 12% with these predictions , while the @xmath6-value of our faintest band , @xmath2110 , is @xmath734% ( @xmath72.3 @xmath32 ) away from its predicted value . the high @xmath20(co ) toward x per also provides us with a clean detection of a fully resolved intersystem band from the @xmath8co isotopomer , namely , @xmath2112 . this band was suggested to exist by mn in the @xmath1 oph spectrum of @xmath376 analyzed by @xcite . for @xmath8co , the @xmath2112 band was predicted to have substantial mixing , borrowing 19% of the @xmath376 @xmath6-value according to mn , versus the mere 1% borrowed by @xmath2112 from @xmath376 in @xmath0co . since the @xmath378 band of @xmath8co has @xmath55 1 , we could find @xmath20(@xmath8co ) toward x per directly . using the four - component model from @xmath0co , we derived @xmath20(@xmath8co ) = ( 1.94 @xmath30 0.08 ) @xmath2 10@xmath56 @xmath4 , which yields a @xmath0co/@xmath8co ratio of 73 @xmath30 12 toward x per . as can be seen in table 1 , the measured @xmath6-value of @xmath8co @xmath2112 is 13% ( 2 @xmath32 ) larger than the single-@xmath36 prediction in mn . unfortunately , multi-@xmath36 calculations for @xmath8co have yet to be performed . the agreement between the band @xmath6-values determined from our measurements of absorption along the is line of sight to x per and the calculations of rostas et al . is very good . rostas et al . suggested that multi-@xmath36 computations are superior for intersystem @xmath6-values whenever the spin - forbidden band is not fully overlapped by the permitted band . we concur , because the 11-band average of column 9 in table 1 is 0.95 @xmath30 0.16 , i.e. , the @xmath75% difference can be accounted for by known @xmath35 approximations and _ a@xmath7x _ @xmath6-value uncertainties , the latter estimated to be @xmath57 10% by rostas et al . as a caveat , had we retained the @xmath284 band in the calculation of @xmath20(@xmath0co ) in 3 , the average would have changed to 0.90 @xmath30 0.15 due to a systematic adjustment of all @xmath6-values by @xmath75% . individually , nine intersystem bands show very good agreement with their predictions : @xmath2711 , @xmath2714 , @xmath217 , and @xmath2112 differ by @xmath58 ( 7% ) , while @xmath2717 , @xmath284 , @xmath285 , @xmath219 , and @xmath2812 agree to within 1 @xmath32 ( 20% ) . such small differences between empirical and predicted @xmath6-values may be primarily attributed to assigned observational uncertainties ( @xmath9 17% for the stis data ; 16% for @xmath20 toward x per ) rather than to the possibility that current quantum - mechanical treatments are inadequate . the bands @xmath219 , @xmath2110 , and @xmath2812 were not included in rostas et al . , but a comparison with unpublished multi-@xmath36 calculations shows that two @xmath6-values agree to within 12% , while @xmath2110 , our weakest and deblended band , is @xmath734% off . the agreement between empirical results and theoretical predictions assures that accurate column densities can be extracted from is spectra showing very strong absorption in the permitted bands of @xmath0co . finally , there is a need for theoretical multi-@xmath36 calculations of triplet - singlet mixing coefficients in @xmath8co . we thank dr . m. eidelsberg and dr . d. e. welty for providing results before publication , and an anonymous referee for helpful comments . the research presented here was supported in part by a nasa grant for @xmath16 program go-08622 and nasa grant nag5 - 11440 to u. toledo . black , j. , & van dishoeck , e. 1988 , , 331 , 986 chan , w. f. , cooper , g. , & brion , c. e. 1993 , chem . phys . , 170 , 123 eidelsberg , m. , rostas , f. , breton , j. , & thieblemont , b. 1992 , , 96 , 5585 eidelsberg , m. , jolly , a. , lemaire , j. l. , tchang - brillet , w .- . l. , breton , j. , & rostas , f. 1999 , , 346 , 705 federman , s. r. , cardelli , j. a. , sheffer , y. , lambert , d. l. , & morton , d. c. 1994 , , 432 , l139 jenkins , e. b. , & tripp , t. m. 2001 , , 137 , 297 kaczmarczyk , g. 2000a , , 312 , 794 2000b , , 316 , 875 lambert , d. l. , sheffer , y. , gilliland , r. l. , & federman , s. r. 1994 , , 420 , 756 morton , d. c. , & noreau , l. 1994 , , 95 , 301 rostas , f. , eidelsberg , m. , jolly , a. , lemaire , j. l. , le floch , a. , & rostas , j. 2000 , , 112 , 4591 sheffer , y. , federman , s. r. , lambert , d. l. , & cardelli , j. a. 1992 , , 397 , 482 stark , g. , smith , p. l. , yoshino , k. , esmond , j. r. , matsui , t. , & ito , k. , 2002 , , 138 , 305 tilford , s. g. , & simmons , j. d. 1972 , j. phys . chem . ref . data , 1 , 147 . llrcccccccc @xmath0co & @xmath2711 [ @xmath372 ] & [email protected] & [email protected] & [email protected](@xmath75 ) & [email protected] & [email protected] & 1.14 & [email protected] + & @xmath2714 [ @xmath374]&[email protected] & [email protected] & [email protected](@xmath74 ) & [email protected] & [email protected] & 0.94 & [email protected] + & @xmath2717 [ @xmath376 ] & [email protected] & [email protected] & [email protected](@xmath75 ) & [email protected] & [email protected] & 1.00 & [email protected] + & @xmath217 [ @xmath372 ] & [email protected] & [email protected] & [email protected](@xmath75 ) & [email protected] & [email protected] & 5.07 & [email protected] + & @xmath2112 [ @xmath376 ] & [email protected] & [email protected] & [email protected](@xmath75 ) & [email protected] & [email protected] & 0.69 & [email protected] + & @xmath284 [ @xmath372 ] & [email protected] & [email protected] & [email protected](@xmath75 ) & [email protected] & [email protected] & 0.81 & [email protected] + & @xmath285 [ @xmath373 ] & [email protected] & [email protected] & [email protected](@xmath75 ) & [email protected] & [email protected] & 0.69 & [email protected] + & @xmath218 [ @xmath373 ] & [email protected] & & [email protected](@xmath76 ) & [email protected] & & 0.52 & [email protected] + & @xmath219 [ @xmath374 ] & [email protected] & & [email protected](@xmath75 ) & & & & ( [email protected] ) + & @xmath2110 [ @xmath375 ] & [email protected] & & [email protected](@xmath76 ) & & & & ( [email protected] ) + & @xmath2812 [ @xmath378 ] & [email protected] & & [email protected](@xmath75 ) & [email protected] & & ( 0.96 ) & ( [email protected] ) + @xmath8co & @xmath2112 [ @xmath376 ] & [email protected] & & [email protected](@xmath73 ) & [email protected] & & & +

in an echelle spectrum of acquired with the space telescope imaging spectrograph we have identified individual rotational lines of 11 triplet - singlet ( intersystem ) absorption bands of @xmath0co . four bands provide first detections for interstellar clouds . from a comparison with the @xmath1 oph sight line we find that x per is obscured by a higher @xmath0co column density of 1.4 @xmath2 10@xmath3 @xmath4 . together with the high spectral resolution of 1.3 km s@xmath5 , this allows ( i ) an improved measurement of previously published @xmath6-values for seven bands , and ( ii ) an extraction of the first astrophysical oscillator strengths for _ d@xmath7x _ ( 8@xmath70 ) , ( 9@xmath70 ) , and ( 10@xmath70 ) , as well as for _ e@xmath7x _ ( 12@xmath70 ) . the @xmath8co _ d@xmath7x _ ( 12@xmath70 ) band , previously suspected to exist toward @xmath1 oph , is now readily resolved and modeled . our derived intersystem @xmath6-values for @xmath0co include a few mild ( @xmath9 34% ) disagreements with recent predictions from a perturbation analysis calculated for the interstellar excitation temperature . overall , the comparison confirms the superiority of employing multiple singlet levels in the calculations of mixing coefficients over previous single - level predictions .

introduction observations and modeling the column density of @xmath0co band oscillator strengths concluding remarks

arxiv

the study of short period variable stars in open clusters is fundamental in stellar evolution . since all the cluster members are assumed to have the same interstellar reddening , distance , age and chemical abundance , it is possible to put strong constraints on these physical parameters of the cluster variables in asteroseismic model calculations [ e.g. @xcite ] . besides the great success of multichannel photoelectric photometers to study short period variables in open clusters [ e.g. @xcite ; @xcite ; @xcite ] , ccd technique working in the time - series photometry mode has been preferred for open clusters observations ( e.g. @xcite ; @xcite ; @xcite ; @xcite ) . indeed a ccd camera allows to obtain high precision photometric data by measuring the program star and reference stars simultaneously in the ccd s field of view ( fov hereafter ) under the same weather and instrumental conditions . the number of stars observed simultaneously with a ccd camera may vary from a couple , in case of a small fov and sparse fields ( e.g @xcite ) , up to thousands for mosaic ccds pointing near the plane of the milky way . taking the advantage of ccd cameras we have carried out a search for new short period variable stars in the direction of the coma berenices and upgren 1 open clusters . coma berenices ( melotte 111 hereafter , ra@xmath023@xmath1 , dec@xmath200@xmath3 , j2000.0 ) is the second closest open cluster to the sun being more distant than the hyades ( @xmath4 45 pc ) but closer than the preasepe ( @xmath4 180 pc ) . the _ hipparcos _ intermediate astrometry data place it at @xmath5 pc @xcite . an almost solar metallicity has been derived in the cluster ( e.g. [ fe / h ] @xmath6 dex , @xcite , [ fe / h ] @xmath7 dex , @xcite ) . the age of the cluster is estimated at between 400 and 600 myr ( e.g. @xcite ) . several investigations which have addressed the stellar population in melotte 111 support the fact that the cluster has relatively few members and , particularly that it is poor in low - mass stars . ( e.g. , @xcite ; @xcite ; @xcite ; @xcite ) . several searches for new low - mass cluster members have been recently performed but without more success than early studies ( e.g. @xcite ; @xcite ; @xcite ; @xcite ) . in particular @xcite have addressed the membership of the stars in direction of coma berenices using sdss iii apogee radial velocity measurements , confirming just eight k / m dwarf new candidate members of the cluster . given that melotte 111 is relatively sparse and the verification of membership in the cluster has been challenging , the detection of new variable stars that are members of the cluster is very important . meolotte 111 covers about 100 @xmath8 on the sky , but its central part occupies only about 25 @xmath8 . the core of the cluster is estimated between 5 - 6 pc @xcite . the catalogue of variable stars in open cluster by @xcite lists 57 variable stars belonging to the melotte 111 open cluster . upgren 1 ( ra@xmath035@xmath1 , dec@xmath922@xmath3 , j2000.0 ) is an association of seven f - type stars located at a distance of @xmath4 117 pc and considered to be a remnant of an old galactic open cluster ( @xcite , @xcite ) . @xcite studied these stars with a multichannel astrometric photometer and proposed that the cluster is composed of two dynamically different groups . we have observed four stars in this association . the paper is organized as follows . in sect . [ sec : obs ] , the acquisition of the data and the description of the observations are presented . in sect . [ sec : lcur ] , the analysis of differential light curves and the fields observed are discussed . [ sec : av1224 ] is devoted to the analysis of the light curve and the physical parameters of melotte 111 av 1224 . in sect . [ sec : con ] we summarize our conclusions . the ccd observations have been made with the 0.84-m f/15 ritchey - chrtien telescope at observatorio astronmico nacional at sierra san pedro mrtir ( oan - spm ) , during ten consecutive nights , between april 11 and 20 , 2009 . the telescope hosted the filter - wheel ` mexman ' with the marconi ( e2v ) ccd camera , which has a 2048 @xmath10 2048 pixels array , with a pixel size of 15 @xmath10 15 @xmath11m@xmath12 . the gain and readout noise of the ccd camera are 1.8 e@xmath13/adu and 7.0 e@xmath13 , respectively . the typical field of view in this configuration is about 8@xmath3 @xmath10 8@xmath3 arcmin@xmath12 with the scale of 0 . 28@xmath14/pixel . to search for short - period variable stars in direction of melotte 111 we observed three fields centered at the following coordinates @xmath15 @xmath16 ; @xmath17 @xmath18 and @xmath19 , @xmath20 . the images of these fovs are shown in figures [ fig : field1 ] , [ fig : field2 ] , [ fig : field4 ] respectively . the field in direction of upgren 1 was centered at the coordinates @xmath21 , @xmath22 ( fig . [ fig : field3 ] ) . in each figure the target stars are labeled with numbers and their corresponding identifications are given in the caption of the figure . the log of observations is shown in table [ tab : log ] where the dates , fov , hjd start , hjd end , filters , exposure times and number of frames are listed . we note that the star designation melotte 111 av refers to the star running number of the astrometric catalogue for the area of coma berenices by @xcite . sky flats , dark and bias exposures were taken each night . all ccd images were preprocessed to correct overscan , trim unreliable and useless regions , subtract bias frames , correct flat fielding and reject cosmic rays using the iraf / ccdred package . then , instrumental magnitudes of the stars were computed using the point spread function fitting method of the iraf / daophot package @xcite . the photometry is gathered in tables where frame no . , hjd , airmass , ut , photometric instrumental magnitudes and photometric errors are included . typical internal errors of the single frame photometry for stars are of about 0.001 mag in the bands used . the differential light curves were derived on a star - to - star basis computing the difference in magnitude of one star with respect to the others . in that way each star was checked for variability relative to at least two reference stars . we proceeded to search for short - period variable stars in each observed field using the differential ccd time - series photometry obtained in the previous section . our calculations for a v=15 mag star indicated that with 8 hours of data we can achieve a detection threshold around 1 mmag ( millimagnitude ) level . assuming a 4@xmath23 definition for the threshold , the noise level should be 0.25 mmag . this detection threshold is very typical for most ground - based observations . the search for stellar pulsations was done in two steps . first , all differential light curves were visually inspected for the presence of obvious variability . in this way we searched the light curves for features like eclipsing binaries , planetary transits , flares and high amplitude pulsations . second , all light curves were subject to fourier analysis . this latter step is very convenient in analysis of periodicities . it may uncover a periodic change with either a very tiny amplitude not easily seen directly in the light curves or a short period like those present in pulsating white dwarfs or sdb type variables . we calculated fourier transform up to the nyquist frequency . some comments on the observed field can be made . the fov 1 ( fig . [ fig : field1 ] ) includes six stars . two stars , nsv 5613 ( bd+27 2129 ) and nsv 5615 ( bd+27 2130 ) , have been classified in the literature as variable stars of melotte 111 . nsv 5613 was reported as suspected variable of melotte 111 by @xcite while nsv 5615 has been classified as a rr lyrae type variable star by @xcite . since then no new observations of these targets has been reported to the best of our knowledge . these stars are also listed in the international variable star index ( vsx ) database of the american association of variable star observers ( aavso ) . moreover both stars are included in the catalogue of variable stars in open clusters by @xcite , but no period information is given for them . we have observed both targets and the adjacent field stars through a strmgren @xmath24 filter with 30 sec of exposure times which correspond to a nyquist frequency of 1980 c / d ( see table [ tab : log ] for details ) . the light curves and amplitude spectra of these two stars are shown in fig . [ fig : nsv ] . as can be seen there is no evidence of periodic variations either in the light curves or in the amplitude spectra . we conclude that these stars have been wrongly classified as variables , as they do not pulsate at all . the fov 2 ( fig . [ fig : field2 ] ) is the most crowded field we have observed . the light curves of the 19 targets numbered in the figure has been derived . the following stars are listed as members of melotte 111 in the simbad database : melotte 111 av 1176 , melotte 111 av 1192 , melotte 111 av 1196 , melotte 111 av 1204 and melotte 111 av 1207 . this fov was observed through a johnson @xmath25 filter with an exposure time between 60 sec corresponding to a nyquist frequency of 635 c / d . after carefully analyzing the light curves and amplitude spectra of all the stars in this fov we conclude that none of the stars present significant variations attributed to intrinsic pulsations . as was mentioned before the fov 3 ( fig . [ fig : field3 ] ) was set in direction of upgren 1 . we have observed four stars of this association namely hd 109509 , hd 109530 , hd 109542 and bd+37 2295 through a @xmath24-strmgren filter with a exposure time of 20 sec resulting in a nyquist frequency of 2274 c / d . after checking carefully the light curves and the amplitude spectra we conclude that none of the stars is variable . five stars in fov 4 ( fig . [ fig : field4 ] ) were checked for variability . three stars are presumable members of melotte 111 , namely melotte 111 av 1224 , melotte av 1236 , melotte 111 av 1248 . we found evident brightness changes on a time scale of few hours in melotte 111 av 1224 . the adjacent stars in the field were found not to be variable stars . an in depth analysis of the light curve is given in the next section . [ cols="<,^,^,^,^,^ , > " , ] a summary of a search for new short - period pulsating variables in direction of the open clusters melotte 111 and upgren 1 has been presented . 35 stars were checked for variability in four observed fields . we did not confirm the variability in the stars nsv 5612 and nsv 5615 considered as variable stars of the melotte 111 open cluster . on the contrary , the star melotte 111 av 1224 was found to be a new eclipsing binary star . follow - up ccd observations of melotte 111 av 1224 allowed us to estimate the orbital period and ephemeris of the system . based on strmgren standard photometry and low - resolution spectra we conclude that the primary component is most likely an early k - type dwarf . the analysis of the strmgren standard photometry place it to 392 pc much more farther that melotte 111 open cluster ( @xmath4 87 pc ) . therefore , melotte 111 av 1224 is not dynamically associated with the melotte 111 open cluster . this is consistent with the fact already pointed out in early investigations that the melotte 111 open cluster has relatively few members and particularly that it is poor in low - mass stars . although a classical beta lyrae ( eb ) binarity classification can not be ruled out , our analysis of the light curve of melotte 111 av 1224 revealed properties similar in many respects to those of the w uma systems , which are characterized by having short orbital periods ( 0.2 - 0.8 d ) and are composed of f - k type stars sharing a common envelope . however the evolutionary history of the system is not clear due to the missing of radial velocity data . we have found that both models , overcontact and semidetached , systems fit the observed light curves equally well . we think that the system is undergoing cyclic variations with alternating phases of true contact and semidetached , but almost contact , phases . during the contact phases the characteristic w - uma light curve should be observed , while during semidetached phases the surface temperature of the components should be different , thus producing beta lyrae ( eb ) type light curve . therefore we are probably seeing the semidetached phases of the system . to date our observations represent the most extensive work on melotte 111 av 1224 . overall we believe that our results are the best that we can achieved based solely on photometric observations made in the @xmath25 filter . for a better understanding of the properties , both spectroscopic observations and photometric data at multiple wavelengths are needed . this work has received financial support from the unam via grant in114309 . based on observations collected at the 0.84 m telescope at the observatorio astronmico nacional at san pedro mrtir , baja california , mexico . special thanks are given to the technical staff and night assistants of the san pedro mrtir observatory . we thank j. miller for a careful proofreading of this manuscript . this research has made use of the simbad database operated at the cds , strasbourg ( france ) .

we report the results of ccd photometric observations in the direction of the coma berenices and upgren 1 open clusters with the aim at searching for new short - period variable stars . a total of 35 stars were checked for variability . as a result of this search the star designated in the simbad database as melotte 111 av 1224 was found to be a new eclipsing binary star . follow - up strmgren photometric and spectroscopic observations allowed us to derive the spectral type , distance , reddening and effective temperature of the star . a preliminary analysis of the binary light curve was performed and the parameters of the orbital system were derived . from the derived physical parameters we conclude that melotte 111 av 1224 is most likely a w - uma eclipsing binary that is not a member of the coma berenices open cluster . on other hand , we did not find evidence of brightness variations in the stars nsv 5612 and nsv 5615 previously catalogued as variable stars in coma berenices open cluster . techniques : photometric , spectroscopic open clusters : individual : , , stars : variability stars : individual : , , . . 97.30.dg , 97.10.ri , 97.10.vm , 97.10.zr , 97.10.sj

introduction observations and data reduction analysis of differential light curves results and conclusions

arxiv

in the ionization region of conventional magnetrons the electron pressure is negligibly small compared to the confining magnetic pressure . under the extreme conditions of a hipims discharge , however , this may not be the case . we calculate the pressure of the highly energetic ( or _ hot _ ) electrons localized in the spoke and argue that they possess enough free energy to trigger an anomalous diffusion process . to that end , we construct an effective kinetic equation for the spoke - averaged hot electron distribution function @xmath4 . the spoke volume is @xmath5 , where @xmath6 is the area ( seen from the top ) , and @xmath7 is the height ( seen from the side ) . further dynamic quantities are the cold electron density @xmath8 , with temperature @xmath9 , and the ion density @xmath10.quasineutrality holds and reads @xmath11 we assume that the hot electrons enter the spoke region with energy @xmath12 at a rate @xmath13 . here , @xmath14 is the secondary electron emission coefficient and @xmath15 is the ion flux that reaches the target with the characteristic bohm velocity @xmath16 . moreover , we have introduced the ion lifetime @xmath17 . confined by the magnetic field and the target potential , the electrons undergo a bouncing motion until they have lost all their energy by interaction with the gas and metal neutrals ( and emerge as _ cold_),or until they have diffused out of the ionization region . with respect to the ionization channel , we assume that the newly generated electrons are cold , and introduce an effective collision frequency @xmath18 and an effective ionization energy @xmath19 , i.e. the frequency at which electron - ion pairs are generated , and the average dissipated energy respectively.for argon , @xmath20 at high electron energies @xcite.we set @xmath21 , where @xmath22 is the average lifetime of the hot electrons with respect to the volume losses , i.e. inelastic collision processes.for the diffusion losses , we define a loss frequency @xmath23.these considerations allow us to formulate the kinetic equation for the hot electrons as @xmath24 the boundary condition at @xmath25 expresses the assumption that the total influx @xmath26 of hot electrons is distributed over the spoke volume @xmath27 : @xmath28 the balance equation for the ions accounts for volume generation by ionization and surface losses by diffusion to the wall , @xmath29 where @xmath30 is the minimal hot electron energy , i.e. about 10@xmath31 . + thus , we obtain a set of coupled differential equations , governed by the three time constants @xmath32 , @xmath22 , and @xmath33.the system can be reduced by observing that the pde ( [ kineq ] ) is solved by any function of the type @xmath34 ; the boundary condition ( [ bbc ] ) then gives the distribution @xmath35 inserting this into the ion balance equation ( [ iondyn ] ) yields a single integro - differential equation for the ion density @xmath36 this equation is linear and homogeneous , and its most natural solution is an exponential . employing the ansatz @xmath37 and defining the effective multiplication factor @xmath38 , we obtain a nonlinearequation for the time constant @xmath39 : @xmath40 figure [ fig3 ] shows that the ion current exhibits an exponential growth before the occurrence of a pronounced electron transport , and a steady state as the anomalous transport occurs . we examine the case before the occurrence of the anomalous transport , setting the diffusion constant @xmath41 , and the steady state case for the ion density , setting @xmath42 . in the latter case , the diffusion term ( or hot electron transport ) is taken into account . setting @xmath43 cm , @xmath44 , @xmath45 ev and @xmath46 v , we estimate the volume loss constant to be @xmath47 , and the ion time constant to be @xmath48 . + during the early phase , when no hot electron diffusion takes place , we indeed obtain a fast exponential growth , with @xmath49 . + during the second phase we obtain a diffusion loss constant @xmath50 , representing the time scale for locally generated hot electrons to diffuse out of the closed magnetic field region , which corresponds the time scale of the `` jet '' reported by ni et al . @xcite . the calculated density of hot electrons is @xmath51 , and the average energy of hot electrons calculated as @xmath52 results @xmath53 ev . therefore , in order to estimate the hot electrons pressure , it is necessary to estimate the ion density . we set @xmath54 @xmath3 , @xmath55 a , and we assume that half of the discharge current is driven through the spoke . given the two - species nature of the sheath in this late phase , the bohm velocity is defined by employing the average energy of the hot electrons . under these assumptions , the ion density results @xmath56 m@xmath57 , which is a reasonable value at this gas pressure . + finally , the hot electron pressure can be estimated as @xmath58 kpa . on the other hand , the average magnetic pressure in the ionization region , with @xmath59 mt , results @xmath60 kpa . + these rough estimates lead us to the conclusion that in the high emissivity region , where most of the secondary electrons are produced and most of the inelastic collision processes take place , @xmath61 exceeds @xmath62 . therefore _ locally _ the high hot electron pressure represents a reservoir of free energy , that the magnetic pressure can not balance . if we postulate this free energy to give rise to local electron deconfinement and enhanced cross field diffusion , then the streak camera images reported by ni et al . @xcite can be explained : hot electrons generated at the target , diffusing across the magnetic field lines and leaving the magnetic confinement region ( still obeying quasineutrality ) excite the atoms and ions on their path causing the observed trail of emission . we postulate the sharp emission cutoff to be a consequence of gas rarefaction and suppressed secondary electrons production . localized sputtering processes will yield a large number of target atoms . these atoms are ionized and accelerated back to the target , sputtering the target material and generating additional secondary electrons . the transit time over a point on the racetrack of a spoke is about one @xmath63s , as shown in figure [ fig3 ] . the time needed to an ar or al ion to be accelerated from distance of 1 mm to the target by an electric field of @xmath64 104 v / m @xcite is around 250 ns . therefore , several sputtering cycles can be expected to take place within a single spoke . assuming half of the discharge current to be distributed only in the spoke , it is reasonable to expect that abundant sputtering would result in the depletion of ar neutrals in the target vicinity , i.e. ar gas rarefaction by sputtering wind @xcite , @xcite . the gas rarefaction will locally change the composition of the impinging flux from an ar dominated to an al dominated flux , which in turn will hinder the secondary electron production . energy conservation principles prevent singly charged al ions from generating secondary electrons : the ionization energy of an al atom ( 5.98 ev ) should be bigger than two times the electron work function of the sputtering target ( 4.16 ev ) , which is not the case @xcite . this means that only secondary electrons generated during the ar dominated impinging flux at the beginning of the spoke contribute to ionization and excitation of particles in that same spoke . excitation and resulting emission will diminish once the energetic electrons reach the open magnetic field lines . transition to the open magnetic field lines and lack of additional energetic electrons due to an al dominated impinging flux results in the sharp edge observed in the emission contour ( figure [ fig2]d ) . the already mentioned results with the streak camera reported in ref . @xcite agree well with the postulate that localized gas rarefaction , and inability of singly charged metal ions to generate secondary electrons , result in a sharp emission edge . figure [ fig2 ] shows different shapes of the spoke . we argue that this characteristics can be explained based on the different ionization potentials . even though singly charged metal ions can not generate secondary electrons , doubly charged ions have sufficiently high ionization energy to generate secondary electrons . if the doubly charged ionization potential is low enough , then there is a good probability to generate a sufficient amount of doubly charged ions that will contribute in production of secondary electrons . however , the secondary electron generation would continue during the transit of the ionization region with reduced efficiency : indeed the energy required to produce a double charged ion is the sum of the first and the second ionization potentials , and these double charged ions present a reduced secondary electron yield . for instance , @xmath14 is 0.08 for ti@xmath65 impinging on a ti target , compared to 0.12 for ar@xmath66 . these secondary electrons will excite and ionize the species resulting in a continuous emission ( i.e. without forming a sharp edge ) , but with diminishing intensity due to the already mentioned reduced secondary electron generation efficiency . to address this hypothesis , we measured the plasma emission in a single spoke condition , using a fast iccd camera for 6 elements with different second ionization potentials as shown in figure [ fig2 ] . the results show that the emission of elements having lower ionization potential then ar has a diffuse shape , while the emission of elements having higher ionization potential then ar have a triangular shape with a sharp edge . these results agree reasonably with the hypothesis that low second ionization potential would result in a diffusely shaped emission , due to the prolonged secondary electron production . + in this contribution we postulate a model to describe the observed light inhomogeneities in the hipims discharge , based on the localization of the discharge current . this results in a localized secondary electron generation , which in turn leads to an anomalous diffusion across the magnetic field lines . the anomalous cross field diffusion results from the electron deconfinement due to the pressure of energetic electrons exceeding the magnetic pressure . the localization of the ion production and the anomalous electron diffusion has been detected using a double flat probe correlated with a light emission signal . consequences of the current localization are gas rarefaction and anomalous electron diffusion . these in turn result in a triangular shaped light emission with sharp cut off , for elements with second ionization potential higher than the ar ionization potential ; and in a diffuse shaped light emission , for elements with second ionization potential lower than the ar ionization potential . + this work has been supported by the german science foundation ( dfg ) within the frame of the special research unit sfb - tr 87 . the authors gratefully acknowledge fruitful discussions with a von keudell , t de los arcos , and t mussenbrock .

a time resolved analysis of the emission of hipims plasmas reveals inhomogeneities in the form of rotating spokes . the shape of these spokes is very characteristic depending on the target material . the localized enhanced light emission has been correlated with the ion production . based on these data , the peculiar shape of the emission profiles can be explained by the localized generation of secondary electrons , resulting in an energetic electron pressure exceeding the magnetic pressure . this general picture is able to explain the observed emission profile for different target materials including gas rarefaction and second ionization potential of the sputtered elements . recently we have reported on hipims discharges exhibiting spokes as localized light emission rotating in the @xmath0 direction with angular frequencies in the range of a 100 khz with different mode numbers and with transition from stochastic to periodic behaviour depending on the discharge parameters @xcite , @xcite . kozyrev et al . @xcite and anders et al.@xcite have independently observed the same phenomena . kozyrev et al . explained the inhomogeneity of the discharge as an effect of a high density ion current driven along the @xmath1 drift direction generated by an azimuthal electric field @xmath2 @xcite . the azimuthal electric field was attributed to the azimuthal modulation in the plasma density . anders et al . presented a model where the spoke is an ionization zone where electrons are decelerated due to the interaction with charged particles @xcite , @xcite . the model predicts the existence of an azimuthal electric field due to a reduced electron azimuthal velocity . similarly to kozyrev s model , the charged particles are removed from the target due to the @xmath1 drift . brenning et al . @xcite , @xcite offered a physical model to describe the spoke based on the alfven s critical ionization velocity ( civ ) . the ionization within the structure creates an electric field as a result of a charge separation based on the civ theory . the model postulates a modified two stream instability accelerating the electrons towards the target and generating a drift opposite to the @xmath0 direction , therefore explaining the rotation speed being one order of magnitude lower then @xmath0 speed . these phenomenological models can explain the observed lower than @xmath0 speed of the spokes and the transport of charged particles in the @xmath1 direction . however , these models do not explain the different shapes of the spokes depending on the target material @xcite@xcite , and the appearance of the jets " reported by ni et al . @xcite . + here , we measure the locally emitted light and electron and ion saturation current indicating localization of the ionization processes and thereby also of the discharge current . a localized discharge current leads to a localized ar gas rarefaction , which is responsible for the observed emission shape , and a high density of energetic secondary electrons . the experimental setup consists of a double flat probe ( 2fp ) and a photomultiplier tube ( pmt ) with 2 apertures , as shown in figure [ fig1 ] . the double flat probe consists of two flat concentric surfaces , one rectangular with a surface of 0.5 @xmath3 and a surrounding one with surface of 1 @xmath3 . the inner surface was biased , either to + 30 v or to -30 v , to measure the electron or ion saturation current , respectively . the outer surface of the 2fp was not biased , and therefore the floating potential was measured . the probe was mounted at 10 mm from the racetrack , see figure [ fig1 ] . to correlate the light emitted from the racetrack and the electric signal measured at the 2fp , the light emitted from the racetrack , closest to the 2fp , was collected using two apertures and photomultiplier tube ( pmt ) . fast iccd camera measurements facing the target , with acquisition time of 100 ns was used , as described in @xcite . + figure [ fig2 ] shows fast iccd camera shots of discharge with ti ( 13.6 ev ) , nb ( 14.0 ev ) , cr ( 16.5 ev ) and al ( 18.8 ev ) targets . the values in the brackets correspond to the second ionization potential of the element . the results show that for elements having lower ionization potential than the first ar ionization potential ( 15.8 ev ) , such as ti and nb , the emission shape of the spoke is elongated and diffuse . while for elements having higher ionization potential than ar , such as cr and al , the emission shape of the spoke is triangular with a sharp edge . the emission profiles from discharges with mo ( 16.2 ev ) and cu ( 20.3 ev ) ( not shown here ) , exhibit also a triangular shape with a sharp edge . + figure [ fig3 ] and [ fig4 ] presents the comparison between the light signal ( top ) , ion saturation current ( middle ) and floating potential ( bottom ) oscillations for the discharge with al and ti target , respectively . + _ al target _ : the light signal of the discharge with al target exhibits a peak intensity followed by a sharp drop ( figure [ fig3 ] ) of intensity that corresponds to the triangular shape emission profile shown in figure [ fig2]d . results presented in figure [ fig3 ] show a simultaneous increase in the optical emission , in the ion saturation current , and a drop in the floating potential . when applying a positive bias it was observed that peaks of the electron current correspond in time and shape to the occurrence of the peaks in the floating potential . therefore , a drop in the floating potential can be interpreted as an increase in the electron flux . the signal of ions and electrons at the position of the 2fp provides twofold information . first , that the high plasma emission region can be identified as the ionization zone , being suggested by anders et al . @xcite ; second , that the charged particles diffuse across the magnetic field lines reaching the 2fp , indicating strong cross field transport of electrons and ions , sometimes referred to as anomalous diffusion @xcite . the oscillations presented in figure [ fig3 ] have typical profiles of elements with second ionization potential higher than ar , such as al . + _ ti target _ : the emission profile for a ti target shows a diffuse shape without any sharp drop in intensity , as shown in figure [ fig4 ] and figure [ fig2]a . even though the plasma emission differs , it is possible to recognize the localized production of energetic electrons . as in figure [ fig3 ] , the drop in floating potential is closely followed by an increase in ion saturation current and plasma emission . however , instead of a sharp drop in ion saturation current and plasma emission the ion current remains high for a certain time and the plasma emission exhibits a smooth decay . we postulate that this different shape is dominated by the different dynamics of the generation of secondary electrons . the oscillations presented in figure [ fig4 ] have typical profiles for elements with second ionization potential lower than ar , such as ti . + the results indicate that two predominant phenomena are taking place : anomalous transport of energetic electrons , and different shapes of the plasma emission for different materials .

anomalous electron transport explanation of sharp emission cutoff plasma emission shapes

arxiv

development of a reconfigurable network of one - dimensional p - wave superconductors is a prerequisite for the demonstration of topologically protected qubits @xcite . the majority of experimental and theoretical effort is devoted toward superconductor / semiconductor nanowire hybrids @xcite or topological insulator ( ti ) 2d edges @xcite , where combination of spin - orbit interaction and magnetic field lifts fermion doubling while retaining both spin polarizations . coupled to a conventional s - wave superconductor such one - dimensional systems mimics kitaev s chain with majorana fermions formed at the ends of the wire @xcite . there is another completely overlooked class of objects where fermion doubling is lifted but both spin polarizations are preserved , namely domain walls ( dw ) formed during quantum hall ferromagnetic ( qhfm ) transition , where dws form a quasi one - dimensional helical channels @xcite . unlike nanowires , where multi - channel transport can render induced superconductivity to be topologically trivial @xcite , single energy level transport in qhfm dws is topologically protected by a cyclotron gap . qhfm transitions , where polarization of the top - most energy level changes polarization , have been observed in gaas quantum wells with vanishing g - factor @xcite , in magnetic semiconductors @xcite , and for some fractional qhe states in gaas @xcite , cdmnte @xcite and graphene @xcite . dws formed in a fractional qhe regime are of particular interest because they provide a path to form higher - order non - abelian excitations @xcite , which are more suitable for topologically protected quantum computation @xcite . qhfm transition results from a competition between cyclotron , zeeman and exchange energies , and is conventionally investigated by varying in - plane ( zeeman ) magnetic field . during qhfm transition a uniform 2d gas spontaneously phase - separate into insulating magnetic domains @xcite and electrical current flows along a random network of dws . in cdmnte qw landau level ( ll ) spin - splitting energy has two contributions @xcite : a positive s - d exchange term @xmath2 and a negative zeeman term @xmath3 , where @xmath4 is an overlap between electron wavefunction in a well and mn , @xmath5 is mn concentration , @xmath6 mev , @xmath7 for cdte , @xmath8 is bohr magneton , and we neglected @xmath9 and @xmath10-dependence of exchange at low @xmath10 and moderate @xmath9 . competition between these two terms leads to an unusual energy spectrum , fig.[effz]a , where landau levels with opposite spins cross at some magnetic field @xmath11 . engineering placement of mn within the quantum well allows electrostatic control of the s - d overlap @xmath12 and , thus , a local control of qhfm transition @xcite @xmath13 . at domains boundaries two counter - propagating edges with opposite polarization form dws , as shown schematically in fig.[effz]b for the qhfm transition at a filling factor @xmath0 . coupled to superconducting contacts these dws should support majorana fermions in the integer qhe regime and parafermions in the fractional qhe regime . past studies of qhfm were focused on the physics of a 2d phase transition @xcite and did not investigate dws _ per se_. in this work we use electrostatic control of exchange interaction and qhfm transition to form conductive channels between domains with different polarization and to study electrical transport through channels of different length and width . and effective mn doping @xmath14 . ferromagnetic phase transition for filling factor @xmath0 are marked by red dots . ( b ) formation of helical domain walls in a quantum hall regime . spin polarization of the top energy level is switched under a a top gate ( yellow ) and a dw is formed along the gate boundary . with induced superconductivity from superconducting contacts ( green ) non - abelian excitations ( magenta ) are expected to be formed at the boundaries between topological superconductivity induced in the helical dw and s - wave superconducting contacts . ] devices were fabricated from cdmnte / cd@xmath15mg@xmath16te qw heterostructures grown by molecular beam epitaxy ( mbe ) , see refs . @xcite for details . qw is 30 nm wide and is modulation doped with iodine . mn is introduced into the qw as 7 @xmath17-doping layers spaced by 6 monolayers of cdte starting 13 nm from the bottom of the quantum well in the growth direction . effective mn concentration is @xmath18 as determined from the position of qhfm transition @xmath11 . we found @xmath11 to depend on the cooldown conditions and led illumination which most likely affects wavefunction profile within the quantum well and , thus , the overlap @xmath4 . electrostatic control of @xmath19 affords @xmath20 control of the exchange and @xmath11 with both front ( @xmath21 ) and back ( @xmath22 ) gates , see fig.[qhfm]b . low temperature density and mobility are @xmath23 @xmath24 and @xmath25 @xmath26/v@xmath27s respectively . samples are patterned in a number of alternating gated and ungated hall bar sections with typical sizes @xmath28 m , see fig.[qhfm]a , with thin ( @xmath29 nm ) semitransparent ti layer serving as a front gate . gated and ungated sections are separated by narrow constrictions of various widths with gate boundary aligned with the constriction ( inset in the fig.[qhfm]a ) . copper foil glued to the back of samples serves as a back gate . this sample design allows measurements and control of qhfm transition in ungated and gated regions separately and investigate transport across gate boundaries . we fabricated a number of samples with lithographical width of constrictions in the 1 - 15 @xmath30 m range . the electrical width is reduced by @xmath31 nm due to the depletion of a 2d electron gas near mesa edges caused by ar ion milling , and by another @xmath32 nm ( here bohr radius @xmath33 is for cdte dielectric constant @xmath34 and effective electron mass @xmath35 ) due to the formation of edge channels in the qhe regime @xcite , overall @xmath36 @xmath30 m reduction . ohmic contacts are produced by soldering freshly cut indium pallets similar to previous studies @xcite . electron transport is measured in a dilution refrigerator in a temperature range @xmath37 mk with a standard ac technique using excitation current @xmath38 . , @xmath39 and @xmath40 chiral edge channels at @xmath0 . @xmath39 and @xmath40 channels hybridise forming a quantum hall dw . ( b ) dependence of qhfm position at filling factor @xmath0 versus back gate ( black dataset and black axis ) and front gate ( red dataset and red axis ) voltages . ( c ) after cooldown densities under gate and outside the gate differs by a factor of 2 - 3 due to surface fermi level pinning by gate . in order to align densities one need to apply high voltage on a front gate . after aligning densities and placing @xmath0 in vicinity of qhfm transition one can observe qhfm transitions on both ungated and gated sides . ] ti front gate , evaporated directly on the cdte surface , is found to modify surface pinning potential and reduce electron density under the gate by a factor of 2 , see top panel in fig . sharp peaks near 3.5 t and 7 t are qhfm transitions due to landau levels crossings modeled in fig . adjusting front and back gate voltages we can position the 7 t qhfm transition between @xmath40 and @xmath39 states within the @xmath0 , as shown in the middle and bottom panels in fig . [ qhfm]c . for the same electron densities in gated and ungated regions a slight difference in wavefunction profiles results in a difference in overlaps @xmath4 and in the field of qhfm transitions @xmath11 . the difference @xmath41 is in the @xmath42 t range depending on cooldown parameters , on the wavelength of led used , led current and sample temperature during led illumination . qhfm separation @xmath43 defines the gradient of landau levels in the vicinity of a gate boundary , which controls the width of dws @xcite and of the conducting channel formed along the gate boundary . magnetoresistance in the vicinity of the qhfm transition is plotted in fig . [ tdep ] , where @xmath0 qhe state extends between @xmath44 t@xmath45 t. @xmath46 below 7.0 t corresponds to a well - defined @xmath0 state with @xmath40 topmost energy state , while @xmath46 above 7.4 t is a @xmath0 state with the topmost energy level @xmath39 . resistance of the qhfm transition peak for wide regions shows activation behavior with an energy gap @xmath47 k , see dashed lines in fig . [ tdep](b , c ) , the gap can be attributed to spin - orbit mixing of @xmath40 and @xmath39 states @xcite . the center of qhfm transitions in gated region is at @xmath48 t , while for ungated @xmath49 t. transitions separation @xmath50 t is large enough that resistance in the midpoint @xmath51 t vanishes at low @xmath52 mk . thus , qhfm transition at a gate boundary should occur in the range @xmath53 t@xmath54 t. indeed , as shown in the middle and the bottom panels in fig . [ tdep]a resistance measured across gate boundaries peaks within that field range . for short channels @xmath55 m resistance saturates at low temperatures to a non - zero value , see fig . [ tdep](b , c ) where temperature dependence of peak resistance for different channels in two different samples with @xmath56 t for sample a and @xmath57 t for sample b is plotted , with solid lines being fits to a constant + activation function . it is important to note that at @xmath52 mk contribution of wide regions to @xmath58 is negligible , and peak resistance reflects conduction through the channel formed along a gate boundary . m constrictions ( middle and lower panes respectively ) . dashed lines mark gated and ungated qhfm transitions which are separated by @xmath43 . note that for the lowest temperature resistance across 6 @xmath30 m constriction is almost vanishes , while resistance across 4 @xmath30 m constriction saturates at low temperatures . ( b , c ) temperature dependencies of resistances measured across constrictions of different widths in devices a and b. dashed lines are for temperature dependencies of qhfm transitions in corresponding device . ] a hallmark of the electron transport through helical dws is resistance invariance under magnetic field reversal because dws are formed from two counter - propagating edges , fig . [ effz]b . indeed , we found @xmath59 for qhfm transitions measured across gate boundaries , fig [ bdep]a . in contrast , presence of chiral edge states leads to @xmath60 , for example when @xmath61 changes into @xmath0 across the gate boundary @xmath62 while @xmath63 , fig [ bdep]b . and @xmath39 states at @xmath0 is the same for both field directions . dashed lines mark position of ungated and gated qhfm peaks . ( b ) resistance across a chiral state formed between @xmath61 and @xmath0 states is quantized at @xmath64 and between @xmath0 and @xmath65 states at @xmath66 for positive field direction and is zero for negative fields . ] now we analyze transport in short constriction at low temperatures , where saturation resistance is found to be @xmath67 k@xmath68 in all measured samples . here we refer to saturation resistances measured for @xmath11 being close to the middle of the @xmath0 plateaus in both gated and ungated regions ; saturated resistance increases when @xmath11 is shifted closer to the @xmath0 boundary as shown in the supplementary material @xcite . within landauer - bttiker formalism resistance in the ballistic regime , where two non - interacting counter - propagating edge channels are formed along a gate boundary with no inter - channel scattering , is @xmath69 @xmath70 k@xmath68 , fig . [ dwmod ] . parameterizing inter - channel scattering by a resistor @xmath71 reduces expected resistance down to @xmath72 @xmath73 k@xmath68 , still much larger than measured values . thus , the proper model for dws is a single channel with backscattering , parameterized by effective channel resistance @xmath74 k@xmath68 in fig . [ dwmod]c . across a quantum hall dw formed between @xmath40 and @xmath39 states . ( a ) in a ballistic regime there is no scattering between @xmath40 and @xmath39 channels and @xmath75 . in the presence of scattering parameterized by resistance @xmath76 resistance @xmath77 is plotted in ( d ) for model ( b ) and in ( e ) for model ( c ) . ] transport in mesoscopic devices in the presence of disorder is characterized by conductance fluctuations @xcite . indeed , quasi - periodic conductance fluctuations are clearly observed in small channels , as shown in fig . [ ucf ] for a @xmath78 m constrictions . quasi - period @xmath79 of these oscillations is @xmath80 mt . similar quasi - periodic resistance fluctuations were observed in mesoscopic devices for transitions between neighboring quantum hall states @xcite . one possible interpretation of resistance fluctuations is formation of a few domains with a small network of dws spanning across the constriction . some static disorder , such as mn doping fluctuations , potential fluctuations due to remote impurities , or surface roughness with characteristic size of 0.2 @xmath30 m ( see atomic force micrograph of the device surface in fig . [ ucf]e ) , which result in the fluctuation of perpendicular component of magnetic field , may act as pinning centers for domain formation . experimentally we found , though , that fluctuations pattern changes drastically every time magnetic field is ramped outside the @xmath0 state , fig . [ ucf]b , which means that dynamic fluctuations rather than static impurities define the conduction path within the channel . we also note that the width of the gate - defined potential gradient , which defines the width of the conductive channel , is of the order of the 2d gas - to - gate distance 100 nm , of the same order as the width of a dws @xcite , which makes formation of multiple domains within the channel quite unlikely . assuming the width of the conduction path to be 100 nm , period of quasi - periodic oscillations translates into @xmath81 m elongated area which spans most of the estimated electrical width of the 2 @xmath30 m constriction . from exponential decay of fluctuation s amplitude we estimate the phase coherence length @xmath82 m at the base temperature , see supplementary materials @xcite . m constriction measured at a base temperature @xmath83 mk while magnetic field was swept in a narrow rage 6.8 - 7.5 t. fluctuations have similar pattern with a quasi - period of @xmath84 mt . ( b ) fluctuation pattern changes drastically after wide @xmath9 sweeps 5 - 10 t. ( c ) energy levels in the vicinity of the qhfm transition for gated ( dotted lines ) and ungated ( solid lines ) regions . ( d ) energy diagram of a dw formed at the gate boundary , wiggling lines indicate schematically a role of disorder and shaded areas are localized states in the tails of landau levels . conduction occurs via extended localized states in the middle of the gap , creating a conductive channel of helical nature . ( e ) atomic force microscopy image of the wafer surface morphology . ( f ) schematic of a conducting channel which provides backscattering between @xmath0 edge states . magenta path formes a loop responsible for the interference patten which leads to the resistance fluctuations . ] theoretically , transport in the qhfm regime should differ markedly from the transport carried by edge states along sample boundary . at sample boundary low lying landau levels always cross fermi energy forming conducting channels . at a qhfm transition significant rashba spin - orbit interaction results in @xmath85ev anticrossing gap between @xmath39 and @xmath40 levels which form dws @xcite . thus , there should be no conduction in the qhfm transition regime at low temperatures , consistent with vanishing resistance in wide samples . however , disorder results in the broadening of landau levels and introduces extended states within the spin - orbit gap , as illustrated in fig . the @xmath86 gap is smaller than the cyclotron gap @xmath87 t away from the transition ( @xmath88 mev / t , or 50 - 150 @xmath30ev ) and in - gap states are extended enough to hybridize and provide a conduction path at low temperatures . these in - gap states seems to be primarily defined by weakly localized charges in the tails of the landau levels since large @xmath9 change alters the interference pattern , also pattern slowly changes over several hours even if the field is kept close to the qhfm transition . in order to calculate transport through such channels of in - gap states we model a single qhfm dw numerically , see supplemental material for details @xcite . in the model we assume that the primary source of localized states are potential fluctuations due to remote doping and used zero - field mobility to calculate the strength and the density of the fluctuations . in the model we also include surface roughness , which leads to the deviation of magnetic field orientation and orientation of mn spins from the @xmath89direction at high fields , effectively a magnetic disorder . thus calculated conductivity of a dw dominated by the conduction via in - gap states is @xmath90 @xmath91 , which is in good agreement with the measured resistance across the constriction @xmath92 @xmath93 if we use the model in fig.[dwmod]c to convert the channel conductance into measured resistance . most importantly , theoretical modeling shows that wavefunctions of the in - gap states retain the helical character of the dws , similarly to the helical properties of spin - orbit states in quantum dots @xcite . such feature provides a necessary ingredient for the formation of a spinless p - wave superconducting order in the presence of s - wave superconducting proximity effect . in this work we demonstrated gate control of qhfm transition at @xmath0 in high mobility 2d gases in cdte : mn heterostructures and investigated conducting channels formed along domain walls at the boundary between ferromagnetic domains . we show that for long channels conduction is suppressed at low temperatures , consistent with spin - orbit - induced anticrossing of landau levels , but for short ( @xmath1 m ) channels conduction saturates at low temperatures . conduction through these channels does not depend on the direction of magnetic field , unlike conduction through chiral channels formed at the boundary of different qhe states . we expect that the in - gap states , which dominate the conduction , retain the helical character of domain walls and , thus , should have large overlap with s - wave superconductors forming a building block for an electrostatically reconfigurable network of topological superconductors . authors acknowledge support by the department of defense office of naval research award n000141410339 ( a.k , t.w . , g.s . , y. l - g . and l.r . ) and by the national science centre ( poland ) grant dec-2012/06/a / st3/00247 ( v.k . , z.a . , g.k . , and t.w . ) and by the foundation for polish science through the ira programme financed by eu within sg op programme ( v.k . , z.a . , g.k . , and t.w . ) . 29ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1070/1063 - 7869/44/10s / s29 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.80.1083 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.105.077001 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.105.177002 [ * * , ( ) ] link:\doibase 10.1038/nphys2429 [ * * , ( ) ] link:\doibase 10.1126/science.1222360 [ * * , ( ) ] link:\doibase 10.1038/nphys2479 [ * * , ( ) ] link:\doibase 10.1038/nature17162 [ * * , ( ) ] link:\doibase 10.1126/science.aaf3961 [ * * , ( ) ] link:\doibase 10.1038/ncomms10303 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.82.402 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.84.127 [ * * , ( ) ] link:\doibase 10.1126/science.290.5496.1546 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.89.266802 [ * * , ( ) ] link:\doibase 10.1103/physrevb.41.7910 [ * * , ( ) ] link:\doibase 10.1103/physrevb.90.115302 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.111.076802 [ * * , ( ) ] link:\doibase 10.1038/ncomms2340 [ * * , ( ) ] link:\doibase 10.1103/physrevx.4.011036 [ * * , ( ) ] link:\doibase 10.1038/nphys588 [ * * , ( ) ] link:\doibase 10.1016/s0022 - 0248(00)00113 - 5 [ * * , ( ) ] link:\doibase 10.1103/physrevb.94.075309 [ * * , ( ) ] link:\doibase 10.1103/physrevb.82.245120 [ * * , ( ) ] link:\doibase 10.1103/physrevb.46.4026 [ * * , ( ) ] link:\doibase 10.1103/physrevb.35.1039 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.60.1542 [ * * , ( ) ] link:\doibase 10.1103/physrevb.44.12933 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.87.256801 [ * * ( ) , 10.1103/physrevlett.87.256801 ] * supplementary materials * + mesoscopic transport in electrostatically - defined spin - full channels in quantum hall ferromagnets + images of constrictions with the gate boundaries are shown in fig . numbers indicate lithographical width of constrictions along gate boundaries . we found that condition of led illumination has a great impact on the width and electrostatic control of the qhfm transition . illumination of a sample with red led at @xmath9410 k results in very wide ( @xmath95 t ) qhfm transition which position is sensitive to the gate voltage ( @xmath96 t per 100 v on a back gate voltage ) . similar results were obtained by illuminating sample with a green led at low temperatures @xmath97 mk . illumination with green led at high temperatures ( @xmath9410 k ) results in a 2d gas with @xmath98 higher carrier density and narrow ( @xmath87 t ) qhfm transition which position is almost insensitive to the applied gate voltage . the optimal qhfm transition width and control was achieved by illuminating samples with green led at low temperatures and subsequent heating to 1 k , where after 2 - 4 hours 2d gas the system relaxed into an intermediate state with @xmath99 t - wide qhfm transition and @xmath100 t/100 v transition control . thus prepared 2d gases vary slightly between cooldowns for the same sample and between different samples . dependence of resistance across gate boundaries for different constriction sizes measured at base @xmath101 mk is shown in fig.[sidep ] . we conclude that for excitation currents @xmath102 na joule heating is negligible . separation of qhfm transitions in gated and ungated regions @xmath43 reflects the value of the s - d exchange gradient near the gate boundary and , as a result , position of ferromagnetic domains formation . for @xmath103 there is no s - d exchange gradient and domains are randomly formed within the 2d plane . narrow field sweeps within @xmath104 range across the qhfm transition results in the formation of different domain configurations , different conduction paths and different patterns of conduction fluctuations , fig . [ shyst](a ) . often there is no domain wall formed in the vicinity of the constriction , in this case no conduction is observed as shown for the up - sweep in ( a ) . in contrast , for @xmath105 t the gate - induced s - d exchange gradient stabilizes the domain wall position and conducting channels are always formed . the conduction channel is well defined and the resistance fluctuations pattern is reproducible over multiple field sweeps , fig . [ shyst](b ) . m channels are shown for ( a ) @xmath103 and ( b ) @xmath105 t. for each temperature point consecutive @xmath9-scans in bot field directions were recoded . ] even for large qhfm transitions separation @xmath105 t and at the lowest @xmath101 mk there is a slow change in the pattern of resistance fluctuations with time , fig . [ stime_dep ] . a characteristic time scale for the pattern change is @xmath106 hours , as determined from a half width at the half height of the autocorrelation function @xmath107 . most likely the conduction path and the fluctuation pattern are affected by gate voltage - induced slow motion of localized charges in the vicinity of the conduction channel . m channel is plotted in the color plot for @xmath105 t and @xmath108 mk . data was recorded for @xmath109 sweeps . ( b ) resistance auto - correlation function as a function of time offset @xmath110 t . ] the value of maximum resistance of the conducting channel formed between states with opposite polarization depends not only on the length of the channel and qhfm separation @xmath43 , but also on the position of the qhfm transition within the @xmath0 plateau . in fig . [ nudep ] we simultaneously change density in gated and ungated regions and sweep qhfm in the channel @xmath111 across the @xmath0 plateau while keeping @xmath43 approximately constant . magnetoresistance in the gated and ungated regions is plotted in the left plot , and across the 2 @xmath30 m constriction in the right plot . in the inset resistance saturation value @xmath112 and activation energy @xmath113 are extracted from temperature dependence of peak resistance . it is clear that extrema of @xmath112 and @xmath113 depend on the position of @xmath114 within the @xmath0 plateau , with minimum @xmath112 and maximum @xmath113 occur at @xmath0 . data in the main text is taken for @xmath114 placed close to the center of the @xmath0 plateau in both gated and ungated regions . in the vicinity of @xmath0 qhe state is plotted for gated and ungated regions ( left ) and across a 2 @xmath30 m constriction for different front gate @xmath21 and back gate @xmath22 voltages at @xmath115 mk . here position of the qhfm transition @xmath114 is shifting relative to the center of the @xmath0 plateau in both gated and ungated regions . in the inset the value of saturation resistance @xmath112 and activation energy @xmath113 are plotted as a function of a filling factor of @xmath114 , where @xmath112 and @xmath113 are extracted from fits of temperature dependence of peak value of the resistance to a constant + activation function . ] m constriction at different temperatures . ( b ) corresponding fluctuations @xmath116 of resistance r along gate boundary . inset shows standard deviation of resistance across 2 @xmath30 m constriction . ] temperature dependence of resistance fluctuations across a @xmath78 m constriction is shown in fig . [ stdep]a , where magnetic field was swept in a narrow range near the the qhfm transition ( 6.8 - 7.5 t ) in order to preserve the fluctuation pattern . channel resistance @xmath76 can be calculated from the measured resistance @xmath77 using landauer - bttiker formalism discussed in the main text : @xmath117 where both @xmath76 and @xmath77 are expressed in units of @xmath118 . fluctuations of the measured resistance @xmath119 are obtained by subtracting a smooth background from the resistance @xmath77 , and fluctuations of the resistance of the conducting channel @xmath116 are calculated as @xmath120 thus extracted fluctuations of channel resistance are plotted in fig . [ stdep]b for a wide temperature range . in the inset rms amplitude of @xmath116 is plotted as a function of temperature . from exponential decay of rms(@xmath116 ) with temperature @xmath121mk , we estimate that phase coherence length @xmath122 exceeds @xmath123 nm below 100 mk for @xmath124 m . thus the phase coherence is preserved over the length of the channel . in order to model our system , we consider @xmath125 electrons confined to a @xmath126 rectangle , subjected to magnetic field @xmath127 . we take @xmath128 $ ] , where @xmath129 is the magnetic length and @xmath0 is the filling factor . @xmath130\nonumber\\ & + & \frac{e^2}{2\epsilon_r } \sum_{i , j}{\frac{1}{|{\bf{r}}_i-\bf{r}_j|}}\end{aligned}\ ] ] here @xmath131 is the effective electron mass in cdte and @xmath132 is the rashba constant . the spin dependent potential @xmath133 mimics variation of zeeman energy across the sample as a result of applied gate voltage . we consider remote impurity potential @xmath134 $ ] , where the number of impurities @xmath135 and @xmath136 s denote the position of the randomly placed impurities in the doping layer and @xmath137 $ ] . surface roughness ( see fig . 6e of the main text ) translates into curvy profile of the quantum well , and as a consequence , into the deviation of magnetic field orientation hence mn spin orientation from z - direction . in order to model this effect of surface roughness we introduce the spin dependent random potential @xmath138\sigma_z$ ] . we choose @xmath139 mev , @xmath140 nm , and @xmath141 @xmath30ev and @xmath142 nm . parameters for remote dopants are chosen to be consistent with the electron mobility that has been measured experimentally ( @xmath143 cm / vs at @xmath144 ) . the electron - electron interaction is taken into account using the hartree - fock approximation . the self - consistent procedure is done in the basis set of five orbital landau states , each with two spin projections . in our numerical procedure , the spin - dependent potential and random impurities are chosen to be symmetric with respect to the reflection about a line parallel to the @xmath145-axis that bisects @xmath146 ; @xmath147 and @xmath148 , @xmath149 . periodic boundary conditions are used in both @xmath150 and @xmath145 directions . the hartee - fock procedure reduces the hamiltonian to a non - diagonal and non - local effective single particle form @xcite . we compute the conductance of our finite system using green s function approach @xcite . knowing the single particle hartree - fock and impurity potential , we discretize the problem on a lattice of @xmath151 points . we place our leads in the channels separated by @xmath152 . the hamiltonian describing the system with leads is given by @xmath153 where @xmath154 describes the lead , @xmath155 the coupling between lead and the localized electron states in the domain wall area ( @xmath156 label the lead ) . the conductance is given by @xmath157 where @xmath158 denotes the retarded ( advanced ) green s function of the interacting electron gas , @xmath159^{-1}$ ] , @xmath160 is the energy ( we take @xmath161 ) , @xmath162 are the coupling matrices , and the contact retarded and advanced self - energies @xmath163 and @xmath164 are given by @xmath165^{-1}\hat v_{is}\\ \hat\sigma_i^{a}&=&v_{is}^{\dagger}\left [ \left(e - i\eta\right)\hat i-\hat h_{i}\right]^{-1}\hat v_{is}~.\end{aligned}\ ] ] we compute conductance using ( [ eqg ] ) and extract conductivity of the dw @xmath166 . when both magnetic and remote impurities are present , the averaged conductivity for five realizations of disorder is found to be @xmath167 . if magnetic fluctuations are ignored , we obtain @xmath168 . the calculated value @xmath169 corresponds to the channel resistance @xmath170 k@xmath68 or @xmath171 k@xmath68 . this value of @xmath77 is in a good agreement with the measured resistance @xmath172 @xmath93 , suggesting that the model captures the essential physics of conduction in the channels formed along domain walls . 2ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop _ _ ( , ) @noop _ _ ( , )

development of a two - dimensional systems with reconfigurable one - dimensional topological superconductor channels became primary direction in experimental branch of majorana physics . such system would allow to probe non - abelian properties of majorana quasiparticles and realize the ultimate goal of majorana research - topological qubit for topologically protected quantum computations . here we propose and develop a new platform to support and manipulate non - abelian excitations which is based on ferromagnetic transitions in a quantum hall effect regime . electrostatic control of ferromagnetic transition at @xmath0 in high mobility 2d gases in cdmnte allows formation of domain walls along gate boundary , and we show that short ( @xmath1 m ) channels are conducting at low temperatures . we show that channels resistance is invariant under field inversion , unlike chiral channels formed between different qhe states . coupled to a s - wave superconductor these channels should support majorana excitations .

introduction samples results and discussion summary acknowledgments afm images of samples 2deg preparation local heating by excitation current. dependence of conductance fluctuations on @xmath43. time evolution of mesoscopic fluctuations. dependence of helical channel conductance on the position of @xmath11 within the @xmath0 plateau. temperature dependence of resistance fluctuations and phase coherence length modeling of domain wall conduction in the qhfm regime.

arxiv

active galactic nuclei ( agns ) exhibit a lot of spectral components over a wide waveband ; optical / ultraviolet ( uv ) bump , power - law component in hard x - rays , excess component in soft x - ray band ( so called soft excess ) , warm absorber feature around 1 kev in some objects , and reflection / fe k@xmath11 fluorescence line around a few to a few tens of kev ( mushotzky , done & pounds 1993 ; koratkar & blaes 1999 for reviews ) . those spectral components are commonly thought to be powered by gas accretion onto a massive black hole . the most well - known disk - accretion model is called as the standard disk model ( shakura & sunyaev 1973 ) . according to the standard disk model , the spectrum at each radius of the disk is assumed to be a blackbody radiation with a local effective temperature , @xmath12 . this simple picture was supported by their luminosity and by rough agreement of optical / uv spectral energy distribution ( sed ) between observations and models ( e.g. , shields 1978 ; malkan & sargent 1982 ) . the standard model , however , has limitations ; ( i ) if uv turn - over around 2000 @xmath13 is indeed an indication of the temperature at the innermost radius ( e.g. , malkan 1983 ; sun & malkan 1989 ) such disks are too cool to produce enough soft x - ray photons . ( ii ) inversely , if the soft excess component at 0.1 to 1 kev is due to radiation from the innermost region , as is often interpreted , such a disk only produces by 1.52 orders of magnitude less optical / uv flux than what is observed . ( iii ) when optical and soft x - rays spectra are simultaneously fitted with the disk spectrum , the luminosity of the disks ( @xmath14 ) often exceeds the eddington luminosity ( @xmath15 ) . ( iv ) observed optical spectra of qsos are typically redder ( @xmath8 , where @xmath16 ; francis et al . 1991 ) than those of the simplest standard accretion disks ( @xmath17 ) . in other words , a successful model spectrum of agns should deviate from that of the standard accretion disk . ( v ) hard power - law x - ray can not be reproduced . then , a number of authors have tried to distort the disk spectrum toward the high energy regime so that the disk can emit substantial soft x - ray and optical / uv radiation simultaneously . one promising idea is the comptonization within the disk in the vertical direction ( czerny & elvis 1987 ; wandel & petrosian 1988 ; laor & netzer 1989 ; ross , fabian & mineshige 1992 ) . the most accurate treatment of comptonization in the framework of the standard model was made by shimura & takahara ( 1993 , hereafter st93 ) who solved radiative transfer and vertical structure simultaneously and presented emergent spectra integrated over radii ( see also shimura & takahara 1995 , hereafter st95 ) . the effect of comptonization is more prominent at higher accretion rates ( e.g. , ross et al . 1992 ; st93 ; st95 ) . however , there still remain discrepancies between models and observations . ( i ) although comptonization tends to increase @xmath0 , the far - uv ( fuv ) spectra of these accretion disks exhibits @xmath0 @xmath2 1 at best when the luminosity of the disk approaches the eddington limit ( ross et al . 1992 ; st95 ) . on the other hand , the observations of distant quasars showed steeper fuv spectra ( @xmath6 1.8 2.2 , zheng et al . ( ii ) the observed small spectral indices in soft x - ray ( @xmath18 @xmath19 ; e.g. , walter & fink 1993 ; laor et al . 1997 ) are not achieved by any disk models ( e.g. , nandra et al . 1995 for mrk 841 ; laor 1997 for a sample of low - redshift qsos ) , because the higher energy tail of those disk models is the superposition of wien laws , thus exhibiting exponential roll - over . ( iii ) these accretion disks still can not reproduce the hard x - rays . then , hard x - ray emission should be treated as additional components in these models . with these problems kept in mind , we , in this present study , aim to produce the overall sed simultaneously by disk - corona models . the spectrum to be reproduced with these models is a composite one obtained from several independent observations ( see zheng et al . 1997 ; laor et al . it is composed of ` typical ' spectral indices over broad - bands from near - ir to hard x - ray , optical to x - ray flux ratio , and the energy cut - off of the hard power - law component ( see 3.1 for more detailed description ) . we can , for the first time , reproduce such broad - band spectra . in 2 we review the assumptions and numerical method used in the calculation . numerical results and comparisons between the models and the composite spectrum are presented in 3 . the final section is devoted to discussion and summary . the numerical code used in this study is basically the same as that of st93 except for some modifications . we assume for the configuration of the system that the accretion disk main body is sandwiched between coronal layers in the vertical direction ( e.g. , haardt & maraschi 1991 ) , and that the whole system is geometrically thin ( i.e. , plane - parallel slab geometry ) . we treat the disk - corona system consisting of fully ionized hydrogen , thermal plasma around a schwarzschild black hole of mass @xmath20 . gas evaporation from the disk to corona and -condensation from corona to disk ( e.g. , meyer & meyer - hofmeister 1994 ) are not included here for simplicity ; the disk and coronal layers are interacting only via radiation and pressure ( e.g. , nakamura & osaki 1993 ; ycki , collin - souffrin & czerny 1995 ) . although magnetic fields may also affect the disk - corona structures , they are not included in this calculations . the equation of hydrostatic equilibrium in the vertical direction is given by @xmath21 where @xmath22 and @xmath23 are the radial and the vertical coordinates , respectively ; and @xmath24 and @xmath25 are the gas pressure and the number density of electrons , respectively . for the radiation field , we adopt the diffusion approximation . then , the radiative flux @xmath26 at some frequency @xmath27 is given by @xmath28 where @xmath29 and @xmath30 are the radiation - energy density per unit frequency per unit volume and the mean free path of a photon with a frequency @xmath27 , respectively . we set @xmath31 ; we do not include the bound - free nor free - bound transitions in the calculations . here , @xmath32 and @xmath33 are the cross sections of thomson scattering and bremsstrahlung absorption , respectively . the thomson optical depth of the disk - corona measured from the mid - plane @xmath34 is related to the height from the mid - plane @xmath23 , as @xmath35 the total thomson optical depth from the mid - plane ( through the boundary between the disk - corona ) to the surface of the corona , @xmath36 , is a free parameter . the equation of state is that @xmath37 , where @xmath38 and @xmath39 are temperature of protons and electrons , respectively . the dissipated energy per unit surface area of the disk - corona system @xmath40 in newtonian approximation is written as ( shakura & sunyaev 1973 ) @xmath41 where @xmath42 and @xmath43 [ @xmath44 are the total ( above and below the mid - plane ) accretion rate and the keplerian angular velocity , respectively . the innermost radius of the disk - corona is assumed to be @xmath45 , where @xmath46 [ @xmath47 lt - day @xmath48 is the schwarzschild radius . for the case of a non - rotating black hole under non - relativistic treatment , @xmath49 [ @xmath50 corresponds to the accretion disk shining at @xmath15 [ @xmath51 . a constant fraction @xmath3 of mass accretion is assumed to be dissipated in the corona with a thomson optical depth of @xmath4 , where advective energy transport of protons is also included in addition to radiative cooling of electrons ( see figure 1 ) . a remaining fraction , @xmath5 , dissipates within the disk layer . advection in the disk layer ( i.e. , slim disk model that will be discussed later ) is not included because of small radial velocity there . the advective cooling rate in the corona per unit surface area , @xmath52 , is taken from the expression of optically - thin advection dominated accretion flow ( adaf ) in one - temperature case ( i.e. , @xmath53 ; see , kato , fukue & mineshige 1998 , p272 ) : @xmath54 here , @xmath55 is the viscosity parameter which controls the efficiency of the advection in the corona . we perform numerical calculations under the condition that the advective cooling should be less than the dissipated energy in the coronal layer ; @xmath56 . the heating rate in the disk and corona , @xmath57 and @xmath58 , respectively , and advective cooling rate in the corona per unit volume , @xmath59 , are assumed to be proportional to matter density at each site . as a result , the fraction of advective cooling in the dissipated energy is @xmath60 the energy balance in each layer is as follows : @xmath61 with @xmath62 where @xmath63 is the energy exchange rate due to coulomb collisions taken from guilbert and stepney ( 1985 ) ; and @xmath64 is the radiative cooling rate . we consider bremsstrahlung and comptonization for emission mechanisms . then , radiative cooling rate @xmath64 is described as @xmath65 where @xmath66 and @xmath67 are the bremsstrahlung emissivity ( rybicki & lightman 1979 ) and the net rate of energy transfer from electrons to photons via comptonization per unit volume per unit frequency , respectively . the latter is described by the kompaneets equation ( rybicki & lightman 1979 ; see hua & titarchuk 1995 for comparisons of the analytical treatment of comptonization with monte carlo simulations ) . thus , the radiative transfer equation for @xmath26 is written as @xmath68 for the expression of the free - free absorption and emission ( @xmath33 and @xmath66 , respectively ) , we take the gaunt factor to be unity , for simplicity ( see rybicki & lightman 1979 ) . following st93 , we take @xmath69 [ @xmath70log(@xmath71 ) ] as an independent variable for the vertical coordinate . to calculate the spectra integrated over the whole disk ( 3.1 ) , we divide the disk - corona from @xmath45 to @xmath72 into 20 consentric rings so that each ring radiates approximately the same luminosity ( cf . ross et al . 1992 ; st95 ) . in total , input free parameters required for the calculations are @xmath20 , @xmath42 , @xmath3 , @xmath4 , @xmath36 , and @xmath55 . the number of these parameters is similar to that of relevant observed parameters which we aim to explain simultaneously ; e.g. , @xmath73 , @xmath74 , @xmath75 , @xmath76 , @xmath77 , @xmath78 , etc . then , outputs are @xmath79 , @xmath80 , @xmath81 , @xmath82 , @xmath83 , and @xmath84 . spectrum of the whole disk - corona system is obtained by summing up the emergent spectra of all the rings . the emergent spectrum at each ring does not so strongly depend on @xmath36 as long as @xmath85 , but is sensitive to @xmath4 . we , thus , take @xmath36 as a constant over all rings , for simplicity . currently , we do not have a good theory to predict the radial dependence of @xmath4 and @xmath3 that can , in principle , be determined by physics of evaporation / condensation . thus , @xmath4 and @xmath3 are also simply assumed to be constant over all rings . the effects of changing @xmath4 and @xmath3 will be discussed later . general relativistic effect is not included . since no doppler broadening is considered , the total spectrum represents the case of a face - on disk . convection / conductive energy transport is not included . shakura , sunyaev & zilitinkevich ( 1978 ) have shown that convection transports no more than @xmath2 30 % of the vertical energy flux . the spectrum to be reproduced with our models is a composite one ( dotted lines in figure 2a ) , which is a useful probe to see whether a model for the sed of agns can work or not . the vertical normalization is determined so as to give rise to a representative optical luminosity among low - redshift quasars from the bright quasar survey adopting @xmath86 of 50 km s@xmath87 mpc@xmath87 and @xmath88 of 0.5 with an assumption of isotropic emission ( see laor et al . there are , however , two major problems that one should keep in mind when dealing with the composite spectrum ( koratkar & blaes 1999 ; laor 1999 ) . ( i ) we observe objects with different redshifts in different wavebands ; spectral index in fuv is obtained from distant qsos ( zheng et al . 1997 ) , while that of soft x - ray is mainly from nearby objects ( e.g. , walter & fink 1993 ; laor et al . also , sample used in each wavebands contains objects that are not necessarily the same objects . ( ii ) soft excess might be an instrumental / calibration problem ; _ bepposax _ observations did not detect soft x - ray excess in some objects while excess was seen for the same objects with other telescopes ( e.g. , matt 1999 ) . although the results are still controversial , we assume that soft excess really exists in most agns throughout this paper . in what follows , we try to reproduce broad - band spectra similar to the composite one , as the first step . we first show the most successful case with @xmath89 and @xmath90 that gives rise to luminosity of about 5 % of the eddington limit for a non - rotating black hole under non - relativistic treatment . thick line in figure 2a shows an example of the resultant broad - band spectra . since the innermost ring from 3.0 @xmath46 to 4.0 @xmath46 does not satisfy the restriction that @xmath91 for the parameter sets described in fig . 2a , we , hereafter , plot integrated spectra from 4.0 @xmath46 to 300 @xmath46 . in other words , all dissipated energy is assumed to be carried out by advection in @xmath92 . to account for the radiation at the surfaces of coronae above and beyond the disk , the resultant spectrum is multiplied by two . a significant fraction @xmath93 of mass accretion occurs at the corona . in this case , advective cooling in the corona is comparable with the radiative cooling at @xmath94 ; @xmath95 [ eq . ( 6 ) ] . a spectrum of the standard disk with the same @xmath20 and @xmath42 is also depicted for comparison ( dashed line ) . presence of multiple spectral components is the most noteworthy feature of the present model . this is because different radiative mechanisms play roles in different wavebands in fig . 2a ; thermal radiation of the disk in optical / uv , unsaturated comptonization in fuv / soft x - ray , and a combination of the power - law component due to comptonization , bremsstrahlung , and a reflection in hard x - ray . note that the underlying radiative processes in soft hard x - rays are distinct from those of the traditional explanation , in which uv soft x - ray component is due to blackbody whereas hard power - law component is due to comptonization of the soft photons . in our model , hard x - ray spectrum looks a power - law with @xmath6 0.51.0 due to the combination of multiple radiative mechanisms , in contrast . figure 2b shows contributions to the total spectrum ( @xmath96 ; thick line ) due to individual rings . the outermost ring ( @xmath97 ) contribute as much as one third of the total spectrum at a few tens of kev . usual models for x - ray emission of agns assume that only inner region radiates x - ray ( i.e. , one - zone ) ; our model differs from the traditional model in terms of radial dependence of the x - ray spctrum , as well . like optically - thin adaf ( e.g. , narayan , yi & mahadevan 1995 ; manmoto , mineshige & kusunose , 1997 ) , x - ray emission arises from wide spatial range ( @xmath98 ) , in contrast with a usual belief . interestingly , contribution from the outer rings to the total x - ray spectrum decreases towards lower - energy x - ray band . provided that inner rings are more time - variable than outer rings , resultant x - ray spectrum will get softer when the luminosity increases , as is actually observed ( e.g. , done 1995 ; leighly 1996 ) . the most powerful test of our model will be gravitational microlensing ( yonehara et al . 1998 ) which provides information as to the size of emission region as a function of wavelength on au ( @xmath99 for @xmath100 ) scales . the effect of advective cooling in the coronal layer to the emergent spectrum is demonstrated in figure 2c . thick dashed line is the resultant spectrum without advective cooling , which corresponds to a two - temperature treatment of the study by shimura , mineshige , & takahara ( 1995 ) , but with a constant @xmath4 . lower dashed curves represents the spectra of each ring in the case without advective cooling . thick solid curve has the same meaning as in fig . 2a and 2b . in our models above ( fig . 2a2c ) , we assumed constant @xmath3 and @xmath4 over all rings . this will be a reasonable assumption since janiuk , ycki & czerny ( 2000 ) , for example , estimated @xmath3 as a function of radius under some assumptions , finding that @xmath3 is a slowly increasing variable with radius , and that 0.2 @xmath101 1.0 for various parameter sets at 3 to 300 @xmath46 . the coronal gas is probably originated in evaporated gas cumulated in the flow ( meyer & meyer - hofmeister 1994 ; liu , meyer & meyer - hofmeister 1997 ) . in other words , we have assumed that a disk corona has already formed until @xmath2 300 @xmath46 . if , however , evaporation of disk material is still active at @xmath98 , @xmath4 and @xmath3 are both likely to increase with a decreasing radius . to what extent the resultant spectrum changes in such a case is demonstarated in figure 2d . spectra for the outermost ring with @xmath102 ( solid curve ) and @xmath103 ( dashed curve ) are shown . in the latter case , hard x - ray emission is not strong as the former case , in which bremsstrahlung emission at the outermost ring is the main origin of hardening in the total x - ray spectrum ( thick solid curve ) . then , we are lead to the conclusion that the corona must have developed until @xmath72 so as to have large values of @xmath3 and @xmath4 at the outermost ring , thereby reproducing the observed spectrum . optical spectral index of qsos has been recognized as a discrepancy between disk models and observations ; francis et al . ( 1991 ) reported that the bright qsos , in which stellar contamination to the optical fluxes is relatively negligible , typically show spectral index in optical , @xmath0 at 15005000 @xmath104 , of 0.32 ( @xmath105 ) , while the classical standard disk model predict that of @xmath106 ( i.e. , @xmath107 ) . in order to emphasize the optical spectra , a part of fig 2a is enlarged and shown in figure 3 with additional spectra of disk models integrated over 3 @xmath108 . thick solid and dotted lines have the same meanings as in fig . 2a , and they both show similar spectral index , @xmath8 . the shaded area inndicates the relevant waveband , 15005000 @xmath104 . the reason why the current model has a better agreement with the observations in terms of the optical spectral index is as follows . the long - dashed line is a spectrum of the standard disk ( i.e. , eq . 4 ) , but without the inner boundary term ( the term within the parenthesis on the right - hand - side of eq . 4 , which is sometimes omitted when roughly estimating the spectral index of the disk model ) . it shows @xmath9 . spectra of the standard disks with the inner boundary term are drawn by the short - dashed lines . the upper one represents the disk with @xmath109 and @xmath110 , while the lower one is for @xmath111 and @xmath112 showing somewhat redder spectrum ( @xmath113 ) due to the inner - boundary term . the emergent local spectrum changes its shape in a sense that the spectrum is distorted and shifted towards shorter wavelength when the electron scattering in the disk is taken into account ( czerny & elvis 1987 ) . such a spectral shift is more efficient in inner region than in outer region . as the result , disk models with electron scattering finally show redder optical spectra , @xmath114 in lower solid curve and @xmath115 in upper solid curve . the difference of spectral indices in the two solid curves , which are obtained by radial integrations of st93 model ( i.e. , no corona ) , are due mainly to the difference of their black - hole masses . the disk with larger @xmath116 gets cooler at fixed @xmath117 and @xmath118 , as is expected for the standard accretion disk ( eq . 14 described later ) . the optical band becomes to measure the flux at inner part of the disk , where it gets close to the edge of the integral interval ( e.g. , 3 @xmath119 ) . then , the emergent spectrum is somewhat curved in a sense that it is redder than the spectrum from outer region ( e.g. , figure 16 in koratkar & blaes 1999 ) . note that the redder spectra are achieved even within the framework of original st93 and st95 models , as well as our disk - corona model . if one wants to discuss in more details about optical spectra , heavy metals , bound - bound , free - bound , bound - free transitions and precise estimation of the contribution from broad line region ( e.g. , balmer continuum ) must be included . figure 4 shows the black - hole mass dependence of the spectra with a fixed ratio of @xmath42 to @xmath15 ( i.e. , fixed @xmath120 ) . the peak frequency of the big blue bump varies with @xmath20 in a similar fashion to that of the standard accretion disk in which @xmath121 on the other hand , the cut - off frequency of the hard x - ray ( i.e. , coronal electron temperature ) and the spectral slopes in x - rays are rather insensitive to the black - hole mass . accretion - rate dependence of the emergent spectra with a fixed blackhole mass ( @xmath109 ) is shown in figure 5 . the spectral slope at 0.1 2.0 kev stays almost constant , @xmath122 = 1.46 , 1.58 , 1.62 for @xmath123 = 0.1 , 0.5 and 0.7 , respectively . we finally present the vertical structure of the disk / corona . number density of electrons ( dotted line ) and temperature of protons ( crosses ) and electrons ( solid line ) at @xmath94 are shown in figure 6 . the left side of the figure where @xmath69 [ @xmath70 log(@xmath71 ) ] = 3 corresponds to the mid - plane ( i.e. , @xmath124 ) , and the boundary between the disk and corona is located at @xmath125 ( i.e. , @xmath4 = 0.6 ) . parameters used here are the same as those of fig . it turns out that the height of the boundary measured from the mid - plane is 0.03 @xmath46 , and that of the surface of the corona ( at @xmath69 of @xmath126 ) is 0.3 @xmath46 . then , the disk - corona system is indeed geometrically thin ; the height of the corona from the mid - plane @xmath127 . the most widely accepted interpretation , to date , of soft and hard x - ray emission mechanisms is that soft excess is due to unscattered photons propagating from the disk - corona boundary to the coronal surface , while hard power - law emission is attributed to unsaturated comptonization , i.e. , photons that are compton up - scattered during the propagation within the corona . monte - carlo simulations of propagating photons in the framework of two zone treatment ( disk and coronal layers ) indeed reproduce bump - like feature in soft x - ray and power - law component due to comptonization in hard x - ray ( e.g. , haardt & maraschi 1991 ; nakamura & osaki 1993 ) . however , the bump - like feature due to unscattered photons still keeps its spectral shape as it was at the disk - corona boundary even after it goes away from the coronal surface : i.e. , it should look like blackbody radiation . then , higher energy tail of the soft excess is predicted to be as steep as the wien law , in contradiction with observed small spectral indices in soft x - ray ( @xmath18 @xmath19 ) . thermal bremsstrahlung emission was proposed for hard x - ray power - law component by schlosman , shaham & shaviv ( 1984 ) , in which the emission comes from the disk - corona transition layer with an temperature gradient , where thermal conduction from the overlying corona is balanced with radiative cooling . they show that such a layer emits hard x - rays efficiently in the case of an accretion disk model proposed by sakimoto & coroniti ( 1981 ) , in which viscous torque is proportional to gas pressure , but that it does not work in standard accretion disks where the torque is in proportion to the total pressure unless viscosity parameter in the disk is less than @xmath128 . we should note here that non - thermal , power - law electron energy distribution models , similarly to our results , produce two spectral power - laws for soft and hard x - ray emissions ( e.g. , zdziarski & lightman 1985 ) , although ours is a thermal model . shimura et al . ( 1995 ) proposed a model in which fuv hard x - ray is attributed to unsaturated comptonization in a corona above a disk . their claim is that their model , with @xmath4 being a strongly increasing function with increasing radius , reproduces a single power - law in x - ray bands , not a broken power - law ( i.e. , soft excess and hard power - law ) . then , the corona could be patchy so that the disk is covered over only in part with the corona . in our model , however , bremsstrahlung emission from the corona and reflection of the coronal emission at the disk - corona boundary also contribute to hard x - ray as well as the unsaturated comptonization . as a result , we have different spectral slopes at soft and hard x - ray bands . it largely depends on our assumption of constant @xmath3 and @xmath4 . in other words , we need @xmath3 and @xmath4 to be almost constant or weak functions of radius in order to expect sufficient bremsstrahlung emission from the corona at outer region and then in order to reproduce the observed spectrum within the current framework . theoretical works to check the assumption are needed as next steps . without advective cooling in the corona , we also have rather straight spectrum from fuv to hard x - ray ( see fig . the effect of advection seems to bent the total spectrum so as that the spectrum has two power - laws in x - ray bands . in the extremely high accretion rate , the disk spectra may be able to reproduce the observed steep spectral decline at @xmath129@xmath130 , since comptonization is more efficient at higher accretion rates and since an increase of accretion rate changes the disk dynamics from the standard disk to optically - thick adaf . optically - thick adaf , which is so called the slim disk model , radiates with higher temperature than the standard disk ( szuszkiewicz , malkan & abramowicz 1996 ; mineshige et al . narrow - line seyfert 1 galaxies ( nls1s ) have peculiar spectral and temporal features that are not seen in normal seyfert nuclei and qsos ( e.g. , boller , brandt , & fink 1996 ; brandt , mathur , & elvis 1997 ; leighly 1999a , 1999b ; grupe et al . 1998 , 1999 ) . these features are often attributed to small black - hole mass and to high accretion rate ( i.e. , @xmath131 ; e.g. , pounds , done , & osborne 1996 ; hayashida et al . 1998 ; mineshige et al . then , such extreme accretion rate is not relevant in this study , where we are going to construct a disk model for normal seyferts / qsos that perhaps have moderate accretion rate [ @xmath132 ( 0.01 0.1 ) of @xmath15 ; e.g. , wandel 1999 ] . to account for systematically large @xmath77 in nls1s , we would need to reduce compton-@xmath133 parameter in the corona . the configuration of cold ( @xmath2 10@xmath134 k ) and hot ( @xmath2 100 kev ) regions that radiate optical / uv bump and x - rays , respectively , is an unsolved problem . optically - thin adaf is a possible energy source of x - ray emissions . however , optically - thin adaf itself is basically faint ( e.g. , mahadevan 1997 ) compared with observed luminosity of agns . then , the luminosity of the power - law x - ray emission indicates that cold material is located adjacent to such hot flow as a source of seed photons : for example , corona - like flows above and below a disk ( e.g. , liang & price 1977 ; haardt & maraschi 1991 ) . separation in the vertical direction has been also suggested from the stability against thermal and secular instabilities in inner , radiation - pressure dominated region of the disk ( ionson & kuperus 1984 ; nakamura & osaki 1993 ; mineshige & kusunose 1993 ) , the correlated variabilities between the iron - line flux and the x - ray continuum ( e.g. , nandra 1999 ; see also , however , lee 1999 ) , and need for a cold disk as a ` mirror ' for reflection - component / broad - fluorescence line ( e.g. , tanaka et al . 1995 ; mushotzky 1995 ; nandra 1997 ) . another potential problem is that those layers could be inhomogeneous and highly time dependent ; the corona can be patchy ( shimura , mineshige , & takahara 1995 ; di matteo 1998 ; kawaguchi et al . 2000 ; machida et al . 2000 ) , or the cold gas may exist as numerous blobs within the corona instead of the slab geometry ( guilbert & rees 1988 ; lightman & white 1988 ; sivron & tsuruta 1993 ; collin - souffrin 1996 ; krolik 1998 ; raska 1999 ) . nevertheless , spatially and temporally averaged treatment of the cold and hot regions will still be useful for study of time - averaged spectrum . appropriate treatment of mass evaporation / condensation process is another future issue . in other words , @xmath3 and @xmath4 may be variable in terms of @xmath22 , @xmath20 , @xmath42 , etc . for instance , see janiuk ( 2000 ) for @xmath3 as a function of @xmath22 with various @xmath42 and viscosity parameter , and ycki ( 1995 ) for @xmath4 v.s . @xmath42 . time variability provides strong constraints on spectral models . then , it will be worthwhile to briefly comment on the temporal behaviour , although we are now working on a model for the steady accretion disk - corona with an aim to reproduce the observed , time - averaged spectrum . followings are qualitative arguments expected from our current model . more details are out of the scope of this paper , and will be remained as future works . for the optical - to - uv time - lag observed in ngc 7469 ( wanders et al . 1997 ; collier et al . 1998 ) and marginally detected in ngc 4151 ( peterson et al . 1998 ) , our model can explain the lag as collier et al . ( 1998 ) demonstrated using the standard accretion disk model . that is because optical / uv emitter in our model is , like other spectral models , an accretion disk that has a similar dependence of ( surface ) temperature upon radius to the standard accretion disk . it should be noted here that such a time - lag has not yet been established in all agns ( edelson et al . 2000 ) since the fuv , soft x - ray and a part of hard x - ray emission come from the same region in our model , they are expected to vary without serious time - lag ( larger than an order of days ) , except for difference of escaping ( diffusion ) time at different energies . chiang et al . ( 2000 ) reported the inter - band lags in ngc 5548 from @xmath2 240 ksec simultaneous observations ; 0.14 - 0.18 kev flux leads 0.78 kev flux by @xmath2 10 ksec , and 5.4 kev flux by @xmath2 30 ksec . as discussed in section 6 in their paper , these lags are consistent with a coronal size of 10 @xmath135 for a @xmath136 black - hole . incidentally , the blackhole mass of ngc 5548 is estimated to be about @xmath136 through the reverberation mapping method ( wandel , peterson & malkan 1999 ; kaspi et al . 2000 ) . however , the @xmath2 4-day x - ray lag behind uv near the peak flux levels and almost simultaneous variations near the minimum fluxes detected in ngc 7469 ( nandra et al . 1998 ) is not solved by our model easily , as well as other models , to date , fail to do . finally , our model qualitatively explains following two issues about spectral variability in x - rays on short timescales ( less than months ) that are reported by numerous x - ray observations ( 3.2 ) . i ) soft x - ray flux is more variable than hard x - ray flux . ii ) hard x - ray becomes softer when it gets brighter . we study emission spectrum emerging from the vertical disk - corona structure composed of two - temperature plasma by solving hydrostatic equilibrium and radiative transfer self - consistently . the key question is what physical condition exhibits a soft x - ray excess with a spectral index @xmath0 ( @xmath1 ) of 1.6 and a hard x - ray tail with @xmath0 of 0.7 at the same time . in our model , a fraction @xmath3 of viscous heating is assumed to be dissipated in a corona with a thomson optical depth of @xmath4 , where advective cooling is also included , and a remaining fraction , @xmath5 , dissipates within a main body of the disk . our model can nicely reproduce the observed composite spectrum of agns , which shows soft x - ray excess with @xmath0 of about 1.5 and hard tail extending to @xmath2 50 kev with a different slope ( @xmath137 ) . our model should be checked with individual objects in the future , though . the broken power laws ( @xmath6 1.5 below 2kev and @xmath2 0.5 above ) are the results of different emission mechanisms : unsaturated comptonization in soft x - rays and a combination of the comptonization , bremsstrahlung , and a reflection of the coronal radiation at the disk - corona boundary in hard x - rays . previous models , where soft x - ray excess is attributed to blackbody or saturated comptonization of the disk blackbody , tended to deal with limited wavebands separately , while we tried to fit the broad - band sed simultaneously . that is the reason why we propose here emission mechanisms , that are different from previous models , in soft and hard x - ray bands . also , our model differs from traditional models in terms of x - ray emitting region ; hard x - ray emission causes from spatially spread region up to 300 schwarzschild radii . the emergent optical spectrum is redder ( @xmath8 ) than that of the standard disk ( @xmath9 ) , being consistent with observations , due to the different efficiencies of spectral distortion and shift of disk emission at different radii . the cut - off frequency of the hard x - ray ( reflecting the coronal electron temperature ) and x - ray spectral slopes are insensitive to the black - hole mass , while the peak frequency of the big blue bump is sensitive to the mass . we wish to thank hitoshi negoro for helpful discussions and encouragement , makoto kishimoto , masaru matsuoka , juri poutanen , ken ebisawa and lev titarchuk for useful comments . we would also like to thank an anonymous referee for rewarding suggestions . t.k . appreciates the organizers of the guillermo haro advanced lectures on the starburst - agn connection held in mexico , 2000 , for providing him the excellent lectures and acknowledges useful discussions with the participants , especially dario trevese and agnieszka janiuk . this work was supported in part by the research fellowship of the japan society for the promotion of science for young scientists ( 4616 , tk ) , and by the grants - in aid of the ministry of education , science , sports and culture of japan ( 10640228 , sm ) . boller th . , brandt w.n . , fink h.h . 1996 , a&a 305 , 53 brandt w.n . , mathur s. , elvis m. 1997 , mnras 285 , l25 chiang j. et al . 2000 , apj , 528 , 292 collier , s. j. , et al . 1998 , apj , 500 , 162 collin - 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souffrin s. , and czerny b. 1995 , mnras , 277 , 70 zheng , w. kriss , g. a. , telfer , r. c. , grimes , j. p. , & davidsen , a. f. 1997 , apj , 475 , 469 yonehara , a. , mineshige , s. , manmoto , t. , fukue , j. , umemura , m. , & turner , e. l. 1998 , apj , 501 , l41 zdziarski , a. a. , & lightman a. p. 1985 , apj , 294 , l79 fig . 2a. resultant spectrum from the disk - corona structure integrated over 4.0 to 300@xmath46 ( thick line ) . parameters used in this model are listed in the figure . with those parameter sets , advective cooling in the corona is comparable with the radiative cooling at @xmath94 ; @xmath95 . dashed line indicates the integrated spectra of the standard disk with the same @xmath20 and @xmath42 . spectral indices of the observed composite spectra ( dotted lines ) are : @xmath0 of 1.4 at nir ( @xmath138 ) , 0.3 at optical ( @xmath139 ) , 1.0 at uv ( @xmath140 ) , 1.8 at fuv ( @xmath141 ) , 1.6 at soft x - ray ( 0.22.0 kev ) , 0.7 at hard x - ray ( @xmath142 2.0kev ) , 1.5 between optical and x - ray ( 2500 @xmath132kev ) . the energy cut - off for the hard x - ray power - law is assumed to be 100 kev . 2c. the effect of advective cooling in the corona layer to the emergent spectrum . thick dashed line is the resultant spectrum without advective cooling . lower dashed curves represents contributions to the total spectrum due to each ring . thick solid curve has the same meaning as in fig . 2a and 2b . 2d. spectra for the outermost ring with @xmath102 ( solid curve ) and @xmath103 ( dashed curve ) . it is shown that hard x - ray emission in the latter case is not strong as the former case , where bremsstrahlung emission at the outermost ring is the main origin of hardening in the total x - ray spectrum ( thick solid curve ; see fig . 3. spectra of various models with optical spectral indices @xmath0 ( @xmath105 ) . the shaded area inndicates the relevant waveband , 15005000 @xmath104 . thick solid and dotted lines have the same meanings as in fig . 2a . spectra of the standard disks [ shakura & sunyaev 1973 ( ss73 ) , i.e. , eq . 4 ] are shown as the short - dashed lines as in fig . a long - dashed one is calculated through eq . 4 without inner boundary term ( the last term in eq . 4 ) . finally , thin solid curves are obtained by radial integrations of st93 model ( i.e. , no corona ) . lower three calculations are made for @xmath144 and @xmath145 at 3 @xmath108 , while upper two curves are for @xmath146 and @xmath147 at the same radii . 4. black - hole mass dependence of the emergent spectra with a fixed ratio of @xmath42 to @xmath15 . parameters used here are the same as those of fig . 2a and 2b . the cut - off frequency of the hard x - ray ( reflecting the coronal electron temperature ) and x - ray spectral slopes are insensitive to the black - hole mass , while the peak frequency of the big blue bump is sensitive to the mass . 6. vertical structure of the matter density ( dotted line ) , proton temperature ( crosses ) , and electron temperature ( solid line ) . the mid - plane of the disk is located at @xmath69 of 3 , and the boundary between the disk and corona is at @xmath125 . it turns out that the height of the boundary measured from the mid - plane is 0.03 @xmath46 , and that of the surface of the corona ( at @xmath69 of @xmath126 ) is 0.3 @xmath46 . note that @xmath148/@xmath149 = 1 corresponds to @xmath150 k ( @xmath151 500 kev ) .

recent multi - waveband observations of seyfert nuclei and qsos established significant deviations in the spectral shape of the big blue bump from a blackbody one ; soft x - ray excess has a spectral index @xmath0 ( @xmath1 ) of 1.6 and hard x - ray tail with @xmath0 of @xmath2 0.7 . we construct a disk - corona model which accounts for such broad - band spectral properties . we study emission spectrum emerging from a vertical disk - corona structure composed of two - temperature plasma by solving hydrostatic equilibrium and radiative transfer self - consistently . a fraction @xmath3 of viscous heating due to mass accretion is assumed to be dissipated in a corona with a thomson optical depth of @xmath4 , where advective cooling is also included , and a remaining fraction , @xmath5 , dissipates within a main body of the disk . our model can nicely reproduce the soft x - ray excess with a power - law shape and the hard tail extending to @xmath2 50 kev . the different spectral slopes ( @xmath6 1.5 below 2kev and @xmath2 0.5 above ) are the results of different emission mechanisms and different sites ; the former slope is due to unsaturated comptonization from the innermost zone and the latter is due to a combination of the comptonization , bremsstrahlung and a reflection of the coronal radiation at the disk - corona boundary from the inner to surrounding zone ( @xmath7 300 schwarzschild radii ) . the emergent optical spectrum is redder ( @xmath8 ) than that of the standard disk ( @xmath9 ) , being consistent with observations , due to the different efficiencies of spectral distortion of disk emission at different radii . further , we find that the cut - off frequency of the hard x - ray ( @xmath2 coronal electron temperature ) and broad - band spectral shape are insensitive to the black - hole mass , while the peak frequency of the big blue bump is sensitive to the mass as the peak frequency @xmath10 . e - mail : [email protected]

introduction basic assumptions and equations spectral energy distribution of agns discussion and conclusion

arxiv

softly broken gauge symmetries are frequently an important property in quantum field theory ( qft ) . one example is supersymmetry , which must be ( most likely softly ) broken in order to address the phenomenological applications and eventually experimental tests @xcite . another interesting application of the softly symmetry breaking is the effective qft approach to the propagating torsion @xcite . the completely antisymmetric component of torsion can be described by the dual axial vector coupled to fermions through the axial vector current . the presence of the symmetry breaking mass of the axial vector is required for the consistency of the effective theory in the low - energy sector . indeed , the massive couterterm shows up at 1-loop level . in many cases , one is interested not only in the classical aspects of the theory , but also in the derivation of quantum corrections . the subject of the present paper is the calculation of 1-loop effective action for the softly broken gauge theory of propagating torsion in curved space - time . here , the kinetic term and the interactions terms in the classical action are gauge invariant while the massive terms are not . consequently , the standard methods for evaluating the effective action face serious technical difficulties . as a strategy , we shall apply the st@xmath0ckelberg procedure @xcite , that is , we are going to restore the gauge symmetry by introducing an extra field or a set of fields . more details and applications of the method to models with softly broken gauge symmetry in curved spacetime can be found in ref . @xcite . we are going to show that our approach means much simpler and more efficient calculation of quantum corrections . the difference is especially explicit for the massive torsion - fermion system which was originally elaborated in ref . the present method provides an independent verification of our previous result in ref . @xcite and also enables one to perform the calculations in an arbitrary curved space - time , something that was impossible in the framework used in ref . torsion @xmath1 is an independent ( along with the metric ) quantity describing the spacetime manifold . it is defined by the relation ( see , e.g. , refs . @xcite for introduction ) @xmath2 it proves useful to divide torsion into three irreducible components @xmath3 as already known in literature . the interaction with the dirac fermion is described in a quantum consistent way by the action for the theory of effective fermion - torsion system @xcite ( see also refs . @xcite ) , s_tf = d^4x \ { - 14s_^2 - 12m^2 s^2_+ i|^ ( _ + i ^5s _ ) + m|}. [ t5 ] here @xmath4 , @xmath5 is the torsion mass , we consider only one non - vanishing component of torsion , @xmath6 , and @xmath7 is the covariant derivative without torsion . to calculate the 1-loop effective action for this model , one has to apply the generalized method of schwinger - dewitt @xcite in the transverse vector space , as was done in ref . @xcite . following the approach discussed in ref . @xcite , one can apply the st@xmath0ckelberg procedure by introducing a new scalar field , @xmath8 , and restoring the gauge symmetry in the following way : s_tf^ & = & d^4x \ { - 14s_^2 + 12m^2(s_- ) ^2 + & + & i|^ ( _ + i _ 1 ^ 5s _ ) + m | ( ) } , [ t555 ] the gauge symmetry must be supplemented by @xmath9 . the original theory ( [ t5 ] ) is restored when we use the gauge fixing condition @xmath10 . in order to obtain the one - loop divergences for the original theory ( [ t5 ] ) , one has to put @xmath11 in the general expression for the divergences of theory ( [ t555 ] ) , which can be computed by the standard schwinger - dewitt method . then the final result reduces to ^(1)_div & = & - d^nx\ { 4 ^ 2 m^2 s^s_- 13 ^2 s_^2 + 4i^2|^_^+ . 2i^2|^_+ ( - 4 ^ 2 m ) |+ |r + ( |)^2 } , + [ r1 ] where @xmath12 and @xmath13 . it is worth mentioning that the above result is more general than the result of ref . indeed , it is valid in curved spacetime , where a new non - minimal coupling with curvature shows up . of course this term is relevant for dynamics of dirac particles in the curved background , but the theory contains the @xmath14-term which has non - trivial consequences . in fact , at 2-loop level , this term is responsible for appearance of the feynman diagrams drawn in fig . 1 . detailed calculation of these diagrams reveals the appearance of the @xmath16-type counterterm , which introduces longitudinal degrees of freedom breaking unitarity . this undesireble contribution can not be compensated by another 2-loop diagrams , unless some artificial fine - tunning between different coupling constants takes place . even if theory ( [ t5 ] ) is not consistent at the quantum level , it can pehaps mimic some fundamental theory , as an effective theory . in this sense , it would be interesting to study the phenomenological consequences of the coupling term , @xmath17 . for instance , this term seems to introduce some kind of modified fermion mass , giving rise to an interesting non - trivial effect on the mass renormalization . s. dimopoulos and h. georgi , nucl . b193 ( 1981 ) 150 ; see also m. drees , r.m . godbole and p. roy , _ theory and phenomenology of sparticles : an account of four - dimensional @xmath18 supersymmetry in high energy physics _ , ( world scientific , 2004 ) .

the soft breaking of gauge or other symmetries is the typical quantum field theory phenomenon . in many cases one can apply the st@xmath0ckelberg procedure , which means introducing some additional field ( or fields ) and restore the gauge symmetry . the original softly broken theory corresponds to a particular choice of the gauge fixing condition . in this paper we use this scheme for performing quantum calculations for fermion - torsion theory , softly broken by the torsion mass in arbitrary curved spacetime . + g. de berredo - peixoto departamento de fsica , ice , universidade federal de juiz de fora , campus universitrio + juiz de fora , mg 36036 - 330 brazil + [email protected] keywords : softly symmetry breaking , renormalization , propagating torsion , curved space . + pacs numbers : 04.62.+v , 11.10.gh , 11.15.-q , 11.30.-j .

introduction massive softly broken torsion field coupled to fermion one-loop effective action and quantum + (in)consistency

arxiv

in a great variety of non - equilibrium situations , critical behavior is observed as a system evolves from one of its possible states to another . some examples are charge density waves , fluctuating interfaces and lines , cracks and fractures , and the barkhausen effect in magnets . these systems evolve , respectively , from a state without current to a state with current , from a stationary to a moving state , from a connected to a ruptured state , from a downward to an upward magnetization . under specific conditions ( _ i.e. _ preparation of the system ) , the transition between the two states is critical ( or continuous ) , exhibiting diverging correlation lengths , and scaling laws . the qualitative descriptions of the dynamics of the different physical situations mentioned are very similar in the key parameters and mechanisms which govern criticality . the quantitative descriptions are also close , in that the dynamics of motion can be described by continuum field equations , and share many common features . in this article we focus on magnetic systems . more specifically , on a lattice of spins with ferromagnetic exchange ( coupling ) . while our qualitative and quantitative analyses will be in this framework , some aspects of the discussion may apply to other systems , such as depinning transition of flux lines or fractures in disordered media . what do we mean by `` the key parameters and mechanisms which govern criticality '' ? consider a ferromagnet in an external magnetic field which increases slowly from @xmath1 to @xmath2 . each spin feels a local field equal to the average of the surrounding spins multiplied by the coupling constant ( @xmath3 in the case of the @xmath4 spin ) , plus the external magnetic field @xmath5 . at @xmath6 , all the spins point `` downward '' ( @xmath7 for unit spins ) , thus @xmath8 initially . at zero temperature , each spin simply points in the direction of the local field @xmath9 , and so none of the spins change before @xmath5 reaches @xmath10 , at which point they all flip upwards . the magnetization @xmath11 thus jumps from @xmath12 to @xmath13 at @xmath14 . this scenario for a perfectly clean system is modified by introducing some disorder . at each lattice point occupied by a spin , add a random field , @xmath15 , to the local field , @xmath9 . ( specifically , let @xmath15 be an uncorrelated random variable , chosen from a gaussian distribution centered at zero . ) then , the spins flip in a much less coherent way : as soon as @xmath16becomes positive , the @xmath4 spin flips . the upward @xmath15 s enhance the increase in magnetization for low @xmath5 , whereas the downward @xmath15 s suppress it for high @xmath5 . this results in the reduction of the magnitude of the jump in magnetization . clearly , if we broaden the random field distribution , _ i.e. _ increase the amount of disorder , the discontinuity in @xmath11 is further suppressed , until the curve @xmath17 eventually becomes smooth for a high enough disorder ( fig . 1 ) . we can imagine a sequence of hysteresis curves , corresponding to a succession of increasing amounts of disorder , say for the variance of the random field , @xmath18 , going from zero to infinity . the curve displays a discontinuity for small @xmath18 , and is smooth for large @xmath18 . the transition between the two regimes occurs at a critical point reminiscent of continuous or second - order phase transitions @xcite : at the critical amount of disorder the discontinuity collapses to a point , at which the slope is infinite . this is referred to as a _ critical hysteresis _ , for which , at a given magnetic field , the susceptibility diverges . the amount of disorder and the magnetic field are the two parameters we have to tune to observe criticality . = 16truecm in the last few years , disorder - induced critical hysteresis in magnets has been the subject of much interest@xcite . dahmen , sethna , and others studied * * this problem via a mean - field approximation , one - loop momentum - space renormalization , and numerical simulations . also , they describe a mapping of this non - equilibrium problem onto the equilibrium random field ising model , which can in turn be mapped ( close to the upper critical dimension ) onto the pure ising model in two lower dimensions@xcite . throughout their work , they consider a scalar order parameter . they study the dynamics of an ising ( or discrete ) spin field driven by an increasing magnetic field and in the presence of a random field , at zero temperature . the question we ask here is the following : how are the phase diagram and critical behavior modified if the order parameter is vectorial instead of scalar ? more precisely stated : in the renormalization group ( rg ) framework , is the fixed point which controls the above - mentioned hysteretic criticality , one and the same for both the ising and the vectorial cases ? and if not , how do the exponents differ ? in equilibrium , the disordering of scalar and vector systems is described by distinct universality classes@xcite . also , in the closely related context of depinning transitions , the distinction between interfaces ( scalar ) and flux lines ( vector ) was noted in ref . @xcite . the answer can readily be guessed on symmetry grounds . indeed , symmetry considerations lead to two distinct cases . in the first , _ and generic one _ , the critical hysteresis curve is such that the susceptibility diverges at a non - vanishing value of either the magnetic field or the magnetization . then , although we consider continuous spins , the full rotational symmetry of the problem is broken at the critical point , a unique preferred direction is picked , and ising - like critical behavior results . it is similarly argued , in ref . @xcite , appendix e , and ref . @xcite , appendix l , that the universality class of the random field scalar model extends also to random bonds scalar models ( with a positive non - zero mean of the bonds values ) and to random anisotropies @xmath0 models . on the other hand , if both the magnetic field and the magnetization vanish when the susceptibility diverges , we may have a fully rotation invariant system at the critical point . in that case , we expect @xmath0-like criticality , with exponents that differ from those of the ising model . evidently , a vanishing magnetization at @xmath19 is not a sufficient condition for a full rotational invariance . in particular , due to its history , the system might well display higher order anisotropies , such as , _ e.g. _ @xmath20 , where @xmath21 is the component of the spin field parallel to @xmath22 and @xmath23 ia any parpendicular component . this issue is resolved by a renormalization group analysis , which confirms our various guesses . furthermore , it discloses the possibility of a `` transverse critical point '' , corresponding to an instability of the magnetization component _ perpendicular _ to the external magnetic field . the present paper is organized as follows . in sec . ii , we construct the equation of motion of a vector spin field . a path integral formalism is described in sec . iii , with which the problem of renormalizing the equation of motion is recast into that of renormalizing a partition function . the time dependences , as well as the subtleties associated with `` many energy minima '' , are also examined . in sec . iv , the renormalization group treatment of the problem is presented . first , we define the coordinates and fields rescalings , and calculate the free propagator . then , we discuss successively the scalar and vector models . for the latter , the different cases ( hysteretic or non - hysteretic , longitudinal or transverse criticality ) are analysed , and the corresponding recursion relations and exponents are obtained . the equations of motion for a scalar field are discussed in refs . @xcite , and their generalization to a multicomponent field is straightforward . nonetheless , for completeness and to emphasize our perspective , we present here a didactic introduction to the equations of motion for vectorial spins , at zero temperature@xcite . consider a @xmath24-dimensional lattice , with a spin @xmath25 at each site @xmath26 , subject to a magnetic field which changes slowly from @xmath1 to @xmath2 , say linearly in time , @xmath27 . the rate @xmath28 can be made arbitrarily small in magnitude and points along the first axis of our coordinates , _ i.e. _ @xmath29 , @xmath30 . the time dependent magnetic field implies a time dependent energy function @xmath31 . at zero temperature ( @xmath32 ) , the spins simply follow the local minimum of this energy function according to @xmath33 starting from a uniform downward pointing configuration at @xmath34 . the parameter @xmath35 controls the relaxation rate of spins towards the time - dependent local energy minimum of @xmath36 . the smaller @xmath35 , the faster spins relax , and the less they lag behind the energy minimum @xcite . the following `` key features '' guide us in constructing the hamiltonian @xmath36 . first , to describe a ferromagnet , we include in @xmath36 couplings @xmath37 which tend to align the spins . in addition to the external uniform field @xmath22 which drives the system , we include quenched random fields @xmath38 . the @xmath38 s are uncorrelated gaussian random variables , chosen from the distribution @xmath39 = n\exp \left ( -\sum_i \frac{{\bf h}_i^2}{2r}\right ) , \ ] ] where @xmath40 is a normalization factor . for calculational convenience we shall work with soft spins ( whose magnitude can take any real value ) , which can be thought of as a coarse - grained picture of a hard spin field . to avoid the unphysical instability of spins diverging in magnitude , we introduce an on - site potential @xmath41 , which constrains the magnitude to remain close to 1 ( or some finite number ) . the potential @xmath42 is spherically symmetric ( a `` double well '' in the scalar model and a `` mexican hat '' in the vector model ) and is expressed through its taylor expansion about the origin , as @xmath43 whether or not @xmath42 is analytic at the origin is unimportant , since @xmath44 is constrained to be close to 1 ( and not to 0 ) . the full hamiltonian is now given by @xmath45 .\ ] ] the gradient descent with this hamiltonian leads to the equation of motion @xmath46 assuming that @xmath37 is a function of the separation between spins results , in the continuum limit , in @xmath47 while rewriting the problem in the continuum limit , we must impose some limit on how fine - grained the spin field @xmath48 may be because of its lattice origin . in other words , @xmath48 is a superposition of fourier components whose wave numbers are restricted from zero to some cutoff @xmath49 . finally , if the exchange function decays fast enough ( as its argument increases ) for its fourier component to be non - singular at the origin of momentum space , _ i.e. _ if @xmath50 , then @xmath51 and the equation of motion can be written as @xmath52 where the coefficient @xmath53 in the the expansion of the potential has been modified in order to take the @xmath54 term of eq . ( [ short - range ] ) into account . in order to identify the critical properties of our model , we should ideally solve its equation of motion . in practice , we study the behavior of eq . ( [ motion ] ) under a coarse - graining transformation . this allows us to locate and characterize a scale - invariant point . as a first step , we recast the equation of motion into a path integral ( or generating functional ) , which incorporates the whole history of the system . the generating functional is written as the sum over all paths of the exponential of some action , which is then renormalized perturbatively . the advantage of reformulating the problem in this way is that we can express the perturbative treatment in a diagrammatic fashion similar to other field theories . we define the generating functional @xcite simply as the sum over all paths of a delta function which makes each spin follow the time evolution given by eq . ( [ motion ] ) , _ i.e. _ @xmath55 where @xmath56 stands for @xmath57 . the jacobian merely normalizes the value of @xmath58 to unity , and we shall henceforth ignore it @xcite . let us rewrite the delta function in its representation as the integral of an exponential [ @xmath59 . then , after absorbing a factor @xmath26 in a redefinition of @xmath60 , and dropping an infinite multiplicative constant ( along with the jacobian ) , we have @xmath61 the _ generating functional _ @xmath58 enables us to evaluate all correlation and response functions . for example , the solution of eq . ( [ motion ] ) is @xmath62 and its response to the magnetic field is given by @xmath63 also , we can change the origin of time by a trivial reparametrization of the magnetic field , as _ e.g. _ in @xmath64 the latter expression takes the form @xmath65 if @xmath5 is increased linearly in time at a rate @xmath66 , whence @xmath67 from which it follows that the dynamic susceptibility is calculated as @xmath68 since we are interested in the average of the correlations and responses over the random field , from now on we deal with the average of @xmath58 . this enables us to forget the stochastic variable @xmath69 , trading it for a new term in the `` averaged action '' . taking advantage of the gaussian nature of @xmath69 , @xmath70 with ( using eq . ( [ distribution ] ) ) @xmath71 we have reformulated the theory , originally described by a dynamical differential equation , in terms of an action @xmath72 $ ] , which depends on the entire history of all spins . thus @xmath73 is a functional of the path which the system follows ; the probability weight @xmath74 picks the physical path and averages it over disorder . we can then study the symmetries and renormalization of the theory , as for equilibrium field theories . the motion comprises two time scales : @xmath35 and @xmath75 . the behavior of the system depends of course on the ratio of the two , and not on their respective values . for the calculation of static exponents , we let @xmath76 . consider for example the exponent @xmath77 , with which the correlation length diverges . a diverging correlation length gives rise to an infinite susceptibility , detected by a non - vanishing response ( of the magnetization ) to an infinitesimal increase of the magnetic field . clearly , such a behavior is obtained in our problem only if the magnetic field increases infinitely slowly [ _ i.e. _ @xmath78 , or equivalently , if @xmath76 . as @xmath35 is a measure of how much the system lags behind its local energy minimum , for @xmath76 the system always stays at the time dependent minimum ( as seen by setting @xmath79 in the equation of motion ) . in other words , the spin avalanches resulting from a small change in @xmath5 spread instantaneously . in the @xmath76 limit , time evolution of the spin field is simply motion with the minimum in the energy landscape . there is nevertheless a subtlety involved as to the choice of the minimum . for example , because of the double - well shape of the potential , a scalar spin has the choice , during some time interval of its history , between two positions , both of which locally minimize the hamiltonian . if we solve the equation of motion , there is no ambiguity , since we specify one of the energy minima as the initial condition , and then follow its time evolution . in particular , the initial condition corresponding to the case of a magnetic field which increases from @xmath1 to @xmath2 is one in which all the spins occupy the left minimum of their double wells @xcite . in the path integral formalism , however , no initial condition is specified . the weight @xmath80 picks all possible solutions , corresponding to different initial conditions . let us illustrate this point with , as in ref . @xcite , the zero - dimensional non - random version of our model , defined by the equation of motion @xmath81 with @xmath76 . we can imagine @xmath82 as a bead sitting at the minimum of the quartic potential @xmath83 , and moving with it . with @xmath84 and @xmath85 , @xmath86 has a single minimum if @xmath87 is larger than some value @xmath88 , and two minima if @xmath87 is smaller than @xmath88 . for @xmath5 very negative , the bead sits at the single minimum of the curve , which becomes the left minimum when @xmath5 is between @xmath89 and @xmath88 . at @xmath90 , the left minimum disappears and the bead moves to the right minimum . thus , in particular , @xmath91 and @xmath92 : the motion of @xmath82 _ is hysteretic _ ( fig . 2 ) . = 10truecm the action for this model is @xmath93 let _ @xmath94 _ be the average of @xmath82 with respect to the weight @xmath80 . calculating _ @xmath94 _ perturbatively , it is easily seen ( by counting the possible occurrences of @xmath82 and @xmath95 in a diagram ) that each diagram comes with an odd number of @xmath5 s . therefore , _ @xmath96 _ , and in particular , _ @xmath97 _ , in apparent contradiction with the above solution . but as mentioned earlier , the partition function is merely the integral of a delta function which imposes the equation of motion . in the present case , it imposes @xmath98 thus , in calculating _ @xmath94 _ , we pick all minima of the quartic form . in other words , _ @xmath99 _ is the sum of two terms , _ @xmath100 _ , corresponding to the left and right minima . the physical , `` bead '' solution is equal to _ @xmath101 _ up to @xmath90 and then _ @xmath102 _ for @xmath5 larger than @xmath88 . similarly , in the case of our original problem , the quantity @xmath103 of eqs . ( [ sol ] ) and ( [ response ] ) does not coincide with the physical solution we are looking for . in addition to the latter , @xmath103 contains other unwanted solutions corresponding to additional energy minima . we shall come back to this difficulty and circumvent it in the next section , both for the scalar and the vector models . our renormalization group transformation consists of the usual three steps . first , we coarse - grain the system , _ i.e. _ , integrate out the modes with wave number between @xmath104 and @xmath49 ( @xmath105 ) . second , we rescale coordinates ( _ i.e. _ change our length and time units ) , by setting @xmath106 or , equivalently @xmath107 with this change of units , the coarse - grained fields vary on the same length scales as the original ones , and the lattice cutoff is preserved . third , we rescale the fields according to @xmath108 or , equivalently @xmath109 with the choice @xmath110 the time derivative and laplacian terms of the action , as well as the @xmath111 terms , become scale invariant . the recursion relations will be calculated to the lowest non - trivial order in the interaction . to this order , no corrections of the parameters @xmath35 , @xmath112 , or @xmath113 of the action in eq . ( [ action1 ] ) occur in the coarse - graining transformation , _ i.e. _ @xmath35 , @xmath112 , and @xmath113 are invariant under our perturbative renormalization . as in the momentum space rg treatment of the @xmath114 theory , we consider the quadratic part of the action as a gaussian ( free ) theory , and the rest as a perturbation ( interaction ) . following refs . @xcite , for calculational convenience we treat the disorder - induced @xmath115 term as an interaction , instead of including it in the gaussian part . the free theory thus consists only of the @xmath116 part of the action , and the corresponding bare propagators are @xmath117 where @xmath118 denotes an average with respect to the gaussian weight , and the indices run from 1 to @xmath119 . the parameter @xmath120 is the @xmath121-independent coefficient of the quadratic @xmath122 term in the action . fourier transforming back in time , we have @xmath123 /\eta \text { \quad if\quad } t^{\prime } > t \end{array } \right . \text{.}\ ] ] ( this is calculated for @xmath124 . as we shall see , @xmath120 is indeed negative at criticality . ) in the @xmath76 limit , the propagator becomes @xmath125 that is , the contraction of @xmath126 and @xmath127 is non - vanishing only if the two times are equal ( actually , only if @xmath128 is infinitesimally higher than @xmath129 ) . with this propagator , it is easily seen diagrammatically that , although the disorder - induced @xmath115 terms couple different times , a renormalized vertex at time @xmath129 is a function only of the other vertices _ at the same time _ @xmath129 . thus , slices of the action at different times renormalize independently from each other , and flow to their respective fixed points . this justifies the procedure of refs . @xcite of setting @xmath5 constant for the calculation of static exponents . ( in what follows , we shall need to correct the action of eq . ( [ action1 ] ) to take care of the problem of multiple solutions . we shall , for example , expand @xmath73 about a uniform but time - dependent value of the field , say some function @xmath130 . hence the vertices ( coefficients in @xmath73 ) , and in particular the `` masses '' @xmath120 , become functions of the parameter @xmath131 . note that the above analysis of the @xmath132 limit is then legitimate only if @xmath133 is continuous , so that @xmath134 . below , we shall define our @xmath135 s in terms of the magnetization or analogous quantities . thus , our analysis holds if we approach criticality from the high disorder side . ) the zero - dimensional model discussed in sec . iii is nothing but the single - spin equivalent to the scalar field . the apparent contradiction mentioned therein is also present in the full model . let us call @xmath136 and @xmath137 the solutions of eq . ( [ motion ] ) , for a magnetic field @xmath5 increasing from @xmath1 to @xmath2 , or decreasing from @xmath2 to @xmath1 , respectively . the magnetization measured experimentally is the average over space , or equivalently over the random field , of the above solutions , @xmath138 generically , as is inferred from the single - spin case and observed experimentally , the magnetization displays hysteresis . in particular , @xmath139 and @xmath140 . on the other hand , the action of eq . ( [ action1 ] ) is invariant under the transformation @xmath141 . this implies that the average of @xmath142 [ eq . ( [ sol ] ) ] satisfies @xmath143 , and @xmath144 . as explained above , @xmath142 contains unphysical contributions ( corresponding to the many minima in the energy landscape ) in addition to the physical solution ( @xmath145 in the case of an increasing @xmath5 ) , _ i.e. _ @xmath146 hence the action @xmath147 does not describe the magnetization , unless it is corrected in such a way as to remove the unphysical solutions . these are inopportunly introduced through eq . ( [ delta ] ) , which should in fact be written as @xmath148 therefore , if we substract the quantity @xmath149 from the functional @xmath80 , we obtain a well defined theory , which incorporates only the physical solution and yields all the correct correlation and response functions . although this method is impossible to implement ( since it explicitely involves the unphysical solutions ) , a restricted , weaker version is applicable , and fully serves our purposes . the investigation of the critical behavior and the calculation of the corresponding exponents relies primarily on correlation and response functions , namely the field density @xmath82 and its susceptibility @xmath150 , _ which are linear in _ @xmath82 . for such objects , @xmath151 can be replaced with @xmath152 . furthermore , only averaged quantities are of interest to the study of the critical point , and linearity allows to perform the average inside the argument of the delta function , leading to @xmath153 where @xmath154 a comparison with eqs . ( [ delta ] ) and ( [ averaged ] ) implies that a weight @xmath80 which properly describes the averaged theory is obtained by adding @xmath155 to the argument of the action . indeed , for @xmath156 } \text{,}\ ] ] we have @xmath157 ; therefore shifting the field by @xmath158@xmath159 yields @xmath160 } = \int { \cal d}\hat s\,{\cal d}s\,\left [ s{\bf \left ( x , t\right ) } -\sigma \left ( t\right ) \right ] e^{s\left [ \hat s , s\right ] } = \left\langle s\right\rangle -\sigma \left ( t\right ) = m_{-}\left ( t\right ) \text{.}\ ] ] the average spin dynamics is thus properly described by the corrected action @xmath161 \nonumber \\ & = & \int dtd^dx\,\hat s\left ( -\eta \partial _ ts+k\nabla ^2s+h+a_0+a_1s+a_2s^2+a_3s^3+\cdots \right ) + \int dtdt^{\prime } d^dx\,\frac r2\hat s\left ( { \bf x},t\right ) \hat s\left ( { \bf x},t^{\prime } \right ) \text{,}\end{aligned}\ ] ] where @xmath162 although we can not obtain the precise form of @xmath163 without solving for the many minima , its qualitative shape is easily found . consider a very large ( positive or negative ) magnetic field . the unphysical minima are due only to the spins in a very large random field @xmath164 , such that the potential they feel still has two minima . when we take the average , the minima @xmath165 contribute to @xmath158 only with a very small weight [ @xmath166 , implying @xmath167 furthermore , it follows from the definition of @xmath158 that @xmath168 at @xmath19 . thus , @xmath169 has a bell shape , shifted to the right since the sum of @xmath158 and @xmath170 is an odd function of @xmath5 ( fig . 3 ) . the parameter @xmath163 , which corrects the coefficients in @xmath73 , breaks the up - down symmetry present in the action of eq . ( [ action2 ] ) . this is physically required , since in an hysteretic system , the ( non - equilibrium ) magnetization breaks that symmetry , in particular at @xmath19 . = 12truecm to discuss the relevant terms in the action , we apply a renormalization transformation in @xmath171 dimensions . all terms of the action higher than cubic in @xmath82 are irrelevant , and after a large enough number of coarse - graining steps , we are left with the renormalized action @xmath172 where @xmath173 s are functions of @xmath5 through @xmath158 , and @xmath174 is negative to prevent the spins from diverging in magnitude . furthermore , it is easily shown that @xmath175 is an even ( odd ) function of @xmath158 for @xmath26 odd ( even ) . in particular , since @xmath176 as @xmath177 , we also have that @xmath178 as @xmath177 . now , in order to find the critical point , let us expand the field about some value @xmath179 , so as to cancel @xmath180 . with the choice @xmath181 , the action in terms of @xmath182 becomes @xmath183 where the coefficients have been corrected by @xmath184 . since @xmath185 as @xmath177 , @xmath186 crosses @xmath184 at a given @xmath187 , _ i.e. _ @xmath188 . hence @xmath189 , which implies in general that @xmath190 . the action thus reduces to that studied in refs . @xcite , and is critical for a specific amount of disorder @xmath113 . the first question in the case of vector spins is whether the action of eq . ( [ action1 ] ) correctly describes the system . the single - spin problem is identical to that of a bead moving in a mexican hat tilted by @xmath191 , the sum of the external @xmath192 and the quenched random field @xmath193 . the bead simply turns around the bump of the hat , _ i.e. _ the spin always points in the direction of @xmath194 , as in the equilibrium problem . the motion is governed by a single minimum , and there is no hysteresis . similarly , mean field theory reduces the time evolution of the spin field to that of a single degree of freedom , and yields no hysteresis for any value of the disorder . these observations may suggest that eq . ( [ action1 ] ) properly describes the problem , and that criticality occurs at @xmath195 , simultaneously for the components of the field parallel and perpendicular to @xmath22 . however , although a single spin has only one minimum in its energy landscape , a configuration of several spins may have many . consider a system composed of two spins of unit length , in an increasing magnetic field @xmath22 , and with random fields @xmath196 and @xmath197 perpendicular to @xmath22 , as illustrated on fig . = 9truecm the hamiltonian for this system is @xmath198 if @xmath199 and @xmath200 are the angles of the spins with respect to @xmath22 , the minimum energy path that the spins follow imposes @xmath201 , and in terms of @xmath202 , the energy is @xmath203 for @xmath204 and @xmath19 , the energy as a function of @xmath202 is a symmetric double well centered on @xmath205 ( fig . 5 ) , similar to the scalar single - spin energy landscape , with two minima at @xmath206 . a non - vanishing @xmath5 tilts the double well , and ultimately suppresses one of the two minima . the ferromagnetic interaction causes the two spins to pull each other back before jumping ahead , thereby investing the @xmath119-component field s motion with a hysteretic , uniaxial - like character . that is , the two - spin toy system as a whole goes over an energy barrier , reminiscent of the motion of a scalar spin and in contrast with that of a single multicomponent spin which turns around the energy barrier . interestingly , the ising ( hysteretic ) behavior of the _ longitudinal _ component results from the interaction , and the presence of a _ transverse _ random field . = 10truecm in the zero dimensional ( single spin ) or infinite dimensional ( mean field ) case , the system follows a single minimum and there is no hysteresis . when fluctuations are included , however , the above toy system shows that many minima appear , allowing for a hysteretic behavior . thus , as in the scalar case , the action of eq . ( [ action1 ] ) has to be corrected in order to remove the unphysical minima . following the procedure of sec . iv.b , we replace @xmath207 by @xmath208 , to obtain , if @xmath135 is parallel to @xmath22 , the corrected action @xmath209 \nonumber \\ & = & \int dtd^dx\,\hat s_{\vert } \left\ { -\eta \partial _ t\left ( s_{\vert } + \sigma \right ) + k\nabla ^2s_{\vert } + h+c_1\left ( s_{\vert } + \sigma \right ) + c_2\left [ \left ( s_{\vert } + \sigma \right ) ^2 + { \bf s}_{\bot } ^2\right ] \left ( s_{\vert } + \sigma \right ) + \cdots \right\ } \nonumber \\ & & + \int dtdt^{\prime } d^dx\,\frac r2\hat s_{\vert } \left ( { \bf x},t\right ) \hat s_{\vert } \left ( { \bf x},t^{\prime } \right ) \nonumber \\ & & + \int dtd^dx\ , \hat { \bf s}_{\bot } \cdot \left\ { -\eta \partial _ t{\bf s}_{\bot } + k\nabla ^2 { \bf s}_{\bot } + c_1{\bf s}_{\bot } + c_2\left [ \left ( s_{\vert } + \sigma \right ) ^2+{\bf s}_{\bot } ^2\right ] { \bf s}_{\bot } + \cdots \right\ } \nonumber \\ & & + \int dtdt^{\prime } d^dx\,\hat { \bf s}_{\bot } \left ( { \bf x},t\right ) \cdot \hat { \bf s}_{\bot } \left ( { \bf x},t^{\prime } \right ) , \label{compact}\end{aligned}\ ] ] where the subscripts @xmath210 and @xmath211 refer to the components parallel and perpendicular to the magnetic field . expanding the polynomials , and absorbing the parameter @xmath158 in thereby corrected coefficients , leads to the form @xmath212 clearly , only even powers of @xmath213 are present in the first line , while only odd powers occur in the third . a term of order @xmath214 in @xmath215 , trivially ( _ i.e. _ ignoring the coarse graining step which couples different order terms ) scales with @xmath216 . in dimensions @xmath217 , all terms are relevant , and a non - trivial fixed point at long length scales can not in general be reached by tuning merely two quantities , namely the external magnetic field and the amount of randomness . similarly for @xmath218 , in which terms up to @xmath219 are relevant . in @xmath171 dimensions , however , all terms with @xmath220 are irrelevant under the rg , leading to an effective action of the form of eq . ( [ expanded ] ) , with any terms not displayed set to zero . including the corrections due to coarse graining , to lowest non - trivial order ( in the interaction and in @xmath221 ) , the recursion relations for the remaining 10 vertices read @xmath222 \\ a_1^{\prime } & = & b^2\left [ a_1 + 3\left ( i+2a_1\right ) a_3+\left ( n-1\right ) \left ( i+2b_1\right ) \bar a_3\right ] \\ b_1^{\prime } & = & b^2\left [ b_1+\left ( n+1\right ) \left ( i+2a_1\right ) b_3+\left ( i+2b_1\right ) \bar b_3\right ] \\ a_2^{\prime } & = & b^{1+\epsilon /2}\left [ a_2 + 18a_2a_3 + 2\left ( n-1\right ) \bar a_2\bar b_3 + 2\left ( n-1\right ) b_2\bar a_3\right ] \\ \bar a_2^{\prime } & = & b^{1+\epsilon /2}\left [ \bar a_2 + 2a_2\bar a_3 + 2\left ( n+1\right ) \bar a_2b_3 + 2b_2\bar a_3 + 4\bar a_2\bar a_3\right ] \\ b_2^{\prime } & = & b^{1+\epsilon /2}\left [ b_2 + 4b_2\bar b_3 + 2\left ( n+1\right ) b_2b_3 + 2b_2\bar a_3 + 4a_2\bar b_3 + 4\bar a_2\bar b_3\right ] \\ a_3^{\prime } & = & b^\epsilon \left [ a_3 + 18a_3 ^ 2 + 2\left ( n-1\right ) \bar a_3\bar b_3\right ] \\ \bar a_3^{\prime } & = & b^\epsilon \left [ \bar a_3 + 4\bar a_3 ^ 2 + 6a_3\bar a_3 + 2\left ( n+1\right ) \bar a_3b_3 + 4\bar a_3\bar b_3\right ] \\ b_3^{\prime } & = & b^\epsilon \left [ b_3 + 2\left ( n+7\right ) b_3 ^ 2 + 2\bar a_3\bar b_3\right ] \\ \bar b_3^{\prime } & = & b^\epsilon \left [ \bar b_3 + 4\bar b_3 ^ 2 + 2\left ( n+1\right ) b_3\bar b_3 + 6a_3\bar b_3 + 4\bar a_3\bar b_3\right ] \end{array } \right . , \ ] ] where @xmath223 , and a factor of @xmath224 is absorbed in a redefinition of @xmath225 , @xmath226 , @xmath227 , @xmath228 , @xmath229 , @xmath230 , and @xmath231 . ( for the derivation of the corresponding relations for the scalar model , the reader is referred to refs . @xcite ; the extension to the effective action of eq . ( [ expanded ] ) is straightforward . ) clearly , a non - trivial fixed point for all 10 vertices can not be reached in general if only two quantities are to be tuned . however , if @xmath113 is appropriately tuned , @xmath232 flows to its fixed ( finite ) value , while @xmath233 grows indefinitely . then , under the rg , the interactions in the perpendicular components become less and less important with respect to the quadratic term . after sufficient rescalings , the theory becomes gaussian in the perpendicular fields which can be integrated out , resulting in an effective action for @xmath21 identical to the scalar action of eq . ( [ scalaction ] ) . thus , the critical point of our @xmath234 model is generically described by the same action as in refs . @xcite , yielding identical recursion relations and exponents . this can be physically understood in the following way : if a configuration of several spins gives rise to many minima , a coarse - grained vector - spin system roughly looks like an ising system , leading to scalar - like critical behavior . the latter occurs for a non - vanishing value of the magnetic field or the magnetization , at which the full rotational symmetry of the @xmath0 model is broken . in the case considered above , the longitudinal field is massless and the transverse components massive . alternatively , one should be able to reach another fixed point which incorporates the reverse situation . in eq . ( [ expanded ] ) , let us expand the longitudinal field about some parameter @xmath235 , so chosen as to cancel the linear term @xmath236 , _ i.e. _ we write @xmath237 with @xmath238 satisfying @xmath239 the theory is then gaussian in @xmath240 . integrating out the longitudinal field , the action _ for the transverse field _ reduces to eq . ( [ renoraction ] ) with @xmath241 instead of @xmath242 , and @xmath243 , _ i.e. _ @xmath244 , \ ] ] where @xmath245 denotes the corrected mass ( see below ) . similarly to the scalar model , this action is critical for an appropriate value of the disorder , or equivalently of the `` mass '' . the recursion relations for @xmath245 and @xmath246 b_3 $ ] can be read off from eq . ( [ recursion ] ) by setting @xmath247 , as @xmath248 \left ( i+2r_{\bot } \right ) u_{\bot } \right\ } \\ u_{\bot } ^{\prime } & = & b^\epsilon \left\ { u_{\bot } + 2\left [ \left ( n-1\right ) + 8\right ] u_{\bot } ^2\right\ } \end{array } \right . \text{.}\ ] ] from these recursion relations we can obtain the static exponent @xmath77 with which the correlation length @xmath249 diverges . by definition @xmath250 , which along with @xmath251 and @xmath252 , yields @xmath253 , _ i.e. _ @xmath254 , and @xmath255 note that the action of eq . ( [ transaction ] ) is identical to the critical action for @xmath256 ( weakly coupled ) scalar fields , with rotational symmetry . it follows immediately from this consideration that the recursion relations and exponents for the longitudinal ( ising - like ) fixed point are identical to those calculated here , with @xmath257 . in the reduced action , the transverse components bare mass becomes @xmath258 the critical line @xmath259 is in the lower half ( @xmath260 ) of the ( @xmath261 ) plane . since @xmath262 as @xmath263 , eq . ( [ lambdaeq ] ) implies that @xmath264 also goes from @xmath265 to @xmath266 as the field is increased ( similarly to @xmath267 [ eq . ( [ renoraction ] ) ] , we have @xmath268 ) . thus , @xmath245 crosses the critical line for a given value of @xmath5 , provided that @xmath269 , _ i.e. _ unless @xmath233 is too negative ( for a mexican hat potential , we expect @xmath270 ) . this may not be true if @xmath236 is a discontinuous function of @xmath5 . but @xmath271 is clearly continuous for critical and higher disorders . ( a very high disorder , though , may suppress this transverse criticality by renormalizing @xmath245 into large negative values . ) whether or not the fixed point controlling the transverse criticality is reachable experimentally depends on what region of the space of the theory s parameters is swept when the physical quantities at hand in the experiment are varied . unlike many equilibrium problems in which the symmetry is broken by an infinitesimal field , criticality can occur here at large values of @xmath5 , allowing for non - negligible higher order terms , such as @xmath272 or @xmath273 , in the hamiltonian . these terms appear in the theory as modified ( strong ) functional dependences of the coefficients @xmath274 , @xmath275 , @xmath276 , @xmath277 , on the magnetic field , which , depending on the trend , may favor a transverse instability . the transverse critical point corresponds to an infinite susceptibility at some @xmath278 , resulting in a spontaneous _ transverse magnetization_. a similar phenomenon was noted for the case of a pure ( thermal ) system in ref . there , however , the appearance of a transverse magnetization is due to a magnetic field which oscillates at high frequency . it is a purely dynamical effect , not observed in the @xmath76 limit . in our case , the effect is due to the presence of a quenched randomness , and its non - equilibrium nature lies in the metastability of the minima occupied by the system during its history . this transverse instability , though , differs from the longitudinal criticality in that it occurs for a range of disorders , rather than at a specifically tuned amount of randomness . its underlying physical mechanism may for example be illustrated by two beads , attached to each other by a spring , and flowing on the two sides of a mexican hat s bump , as it is progressively tilted . at some point , it might become favorable for one of the beads to jump on the opposite side of the rim , and to continue its motion next to the other bead , corresponding to a transverse ordering . at @xmath278 , the transverse components choose one of the many minima , which breaks the rotational symmetry in the @xmath279-dimensional transverse space . we therefore have to correct the action of eq . ( [ expanded ] ) for @xmath280 , as we did for the longitudinal field . this however does not alter the analysis of the longitudinal criticality . by shifting @xmath281 , we eliminate the transverse linear term and can then follow the same procedure as before . we used the fact that the correction to eq . ( [ expanded ] ) vanishes for @xmath177 , and this clearly still holds here . in the above , we examined two distinct fixed points , corresponding to an infinite longitudinal susceptibility and a transverse ordering , respectively . for a specific choice of the theory s parameters , the two should merge into a single , rotationally invariant ( in the @xmath119-dimensional space of the fields ) fixed point , at which all components become simultaneously critical . in addition to the magnetic field and disorder , how many quantities should we tune to reach such an @xmath0 _ fixed point _ ? while eq . ( [ expanded ] ) is a suitable formulation of the model when the symmetry is broken in the longitudinal direction , the form of eq . ( [ compact ] ) is more appropriate close to a fully rotationally invariant theory . an rg transformation can be carried out for it , as was done for the action of eq . ( [ expanded ] ) . since @xmath155 is spatially uniform , only its @xmath282 mode is non - vanishing . the parameter @xmath158 , therefore , does not participate in the coarse graining transformation and its normalization is given by the rescalings of the coordinates and the fields , as @xmath283 the recursion relations for @xmath5 , @xmath53 , and @xmath284 , on the other hand , are obviously given by eq . ( [ recursion ] ) , with @xmath285 , @xmath286 , @xmath287 , and @xmath288 . by tuning both the magnetic field @xmath5 , and @xmath187 , defined as the zero of @xmath158 ( @xmath289 ) , to zero , the action of eq . ( [ compact ] ) reduces to the form of eq . ( [ transaction ] ) , with @xmath119 fields @xmath215 rather than the @xmath256 components of @xmath281 , which is critical for a given value of the disorder . consequently , the recursion relations and exponents are identical to those characterizing the transverse critical point , but with @xmath119 replacing @xmath256 . furthermore , since @xmath19 and the action is fully rotationally invariant , hysteresis is suppressed at the @xmath0 critical point ; in particular , @xmath290 . finally , by identifying the relevant parameter @xmath158 , we have established that only a single additional physical quantity needs to be tuned to reach the @xmath0 fixed point ( provided that @xmath187 crosses zero as the physical quantity in question is varied , at @xmath19 ) . the symmetric fixed point is characterized by three relevant parameters , namely @xmath291 and @xmath5 , with the usual exponents @xmath292 and @xmath293 , and @xmath294 with @xmath295 , to @xmath296 . the parameter @xmath294 is a measure of the deviation from a symmetric ( non - hysteretic ) theory at @xmath19 , and , according to the above exponents , is much less relevant than the other symmetry breaking term @xmath5 , or of the mass @xmath291 , to @xmath296 . @xcite notes the fact that the `` critical region '' is unusually large in the scalar model , as signaled by power laws with surprisingly high cutoffs even a few percent to one hundred percent away from the critical disorder , and briefly discusses possible origins of this phenomenon . this observation , along with the weakness of @xmath158 s relevance , suggests in our case that even away from the @xmath297 fixed point , an @xmath0-like behavior might be displayed at short ranges by the system , before crossing over to an ising - like behavior at long enough time and length scales@xcite . as mentioned in the introduction , a vanishing magnetization does not a priori ensure a fully rotationally invariant system ; higher moments could in general display anisotropy . in that case , a complete theory would break the rotational symmetry , leading to anisotropic exponents different from @xmath298 , @xmath299 , and @xmath300 above , which are the outcome of a restricted action that properly describes only quantities linear in the spin field . in general we may ask the following question , which applies to the three cases , ising , transverse , and @xmath301 . would a complete theory yield the same exponents as our resticted theory ? the latter correctly describes the magnetization @xmath302 . that is , if one were able to calculate the path integral @xmath303 } \text{,}\ ] ] one would obtain @xmath304 as a function of @xmath305 , @xmath113 , and an additional tuning parameter ( corresponding to @xmath187 ) , and thus the associated exponents describing the critical singularity of the magnetization . it is easy to see that , as for the non - random equilibrium thermal landau - ginzburg model , the exponents @xmath298 , @xmath299 , and @xmath300 have a one - to - one correspondance with the exponents that characterize the singular behavior of the magnetization , as well as with those associated with a number of other quantities , such as the correlation length . this immediately implies an affirmative answer to the above question ; the requirement that the action should describe the physical magnetization is a strong enough constraint for the theory to correctly generate the singularity of , _ e.g. _ , the correlation length . evidently , this argument does not apply to the critical behavior of higher moments of the spin field distribution , and whether or not the latter are isotropic at criticality is an interesting open question . in fact , even the isotropy of @xmath298 at the @xmath306 critical point may seem surprising at first , since the history of the system apparently sets up an anisotropic context for the spins motion . it is consistent , however , with the fact noted in sec . iv.a , that different time slices of the action decouple under the rg in the @xmath76 limit , in which the static exponents are calculated . as the reader may have noticed , our recursion relations and exponents in @xmath307 are none other than the recursion relations and exponents for the pure ( equilibrium ) @xmath0 model in @xmath308 . indeed , dimensional reduction to two lower dimensions holds perturbatively @xcite . this allows us to obtain , with no further calculational effort , the expansion for @xmath77 to higher orders in @xmath309 @xcite . the dynamic exponent @xmath310 , on the other hand , can not be obtained from a time - independent model . how is the renormalization procedure modified when @xmath311 ? in the coarse - graining process , vertices which couple different times are generated . in particular , a non - local quadratic term of the type @xmath312 is generated in the action , where @xmath313 is some function ( containing free propagators , disorder , etc ) . expanding @xmath314 about @xmath128 , this is rewritten as @xmath315 the first term contributes to the coarse - grained `` mass '' , the second results in a correction to @xmath310 . in fact , the perturbative calculation we just briefly described is similar to that done by by krey @xcite for the time - dependent thermal random field model , where he calculates @xmath310 to @xmath316 , giving to lowest non - trivial order in @xmath309 , @xmath317 ( here , @xmath119 is the number of components that become massless at criticality . ) the main aim of the present paper was to extend earlier work on critical hysteresis to the case of an ordering of continuous symmetry . our central result is that , generically , criticality is ising - like , _ i.e. _ the critical exponents calculated for a scalar field model @xcite still hold for a vector field . by tuning a single additional quantity , however , a fully symmetric fixed point can be reached , to which @xmath318-like exponents are associated . furthermore , the possibility of a spontaneous non - equilibrium transverse magnetization is unveiled and examined using a perturbative renormalization scheme . in addition , we have clarified several issues pertaining to the path integral formalism , in particular to the structure of time dependences , and to the problem of multiple solutions . these analyses are useful for our treatment of the problem , as well as for a clearer understanding of earlier works . an interesting question which remains open is that of the lower critical dimension . in the fully rotational case , naive dimensional reduction suggests that the lower critical dimension is 4 . if this were the case , no vector criticality should be observed in three dimensions . several scenarios are possible . for example , the hysteresis curve may display a jump for low disorder and be continuous for high disorder , but without the intermediate limit case of a continuous curve with a diverging slope at a given point . a more probable scenario is one in which the hysteresis curve is already smooth for an infinitesimal amount of disorder . rds is grateful to prof . s. coleman for illuminating and interesting discussions , and has benefitted from conversations with dr . k. dahmen and dr . this work was supported by the nsf through grant no . dmr-93 - 03667 . in refs . @xcite , the effective action is written as an expansion about mean field theory ( mft ) . in such a formulation , the linear term in the action ( corresponding to @xmath236 in eq . ( [ expanded ] ) ) is vanishing . the `` masses '' , _ i.e. _ the coefficients @xmath319 and @xmath320 ( corresponding to @xmath232 and @xmath233 in eq . ( [ expanded ] ) ) of the parallel and perpendicular quadratic terms are calculated as the elements of a particular response tensor within mft @xcite . in this appendix , we calculate @xmath319 and @xmath320 , starting from the mean field equations of motion , and find that , generically , @xmath321 . this appendix thus reaffirms that criticality is ising - like in general , as found in sec . iv , and makes contact with the earlier methodology of refs . @xcite . mft is defined by an infinite - range coupling , leading to the equations of motion @xmath322 where @xmath323 is defined self - consistently by @xmath324 , and @xmath325 is a test field . the `` masses '' are the static components of the tensor @xmath326 , where @xmath327 is the identity matrix @xcite . we have @xmath328 ( we consider @xmath76 , so responses are non - vanishing only at equal times , and only the static part remains . also , in mft , the magnetization is always parallel to the magnetic field , leading to vanishing off - diagonal elements and to the relation @xmath329 @xcite . ) in order to calculate the relevant tensor , let us split the field @xmath330 between @xmath331 and @xmath22 , in such a way as to satisfy eq . ( [ susceptibility ] ) for effective magnetization and magnetic field . that is , we write the equations of motion in the following fashion , with corrected @xmath331 and @xmath22 , and with no additional field , @xmath332 for the longitudinal component , and @xmath333 for the transverse components . whence @xmath334 and @xmath335 thus , @xmath336 in general . sethna , k. dahmen , s. kartha , j. a. krumhansl , b. w. roberts , and j. d. shore , phys . lett . * 70 * , 3347 ( 1993 ) ; k. dahmen and j. p. sethna , phys . rev * 71 * , 3222 ( 1993 ) ; o. percovi , k. dahmen , and j. p. sethna , phys . rev . lett . * 75 * , 4528 ( 1995 ) . y. imry and s .- k . ma , phys . rev . lett . * 35 * , 1399 ( 1975 ) ; for a recent review of random field phenomena see _ e.g. _ t. nattermann , `` theory of the random field ising model '' , to appear in _ spin glasses and random fields _ , ed . young ( world scientific ) ( cond - mat/9705295 ) . as in the equilibrium case , the disorder induced criticality is not affected by thermal fluctuations . calculationally , this can be detailed as follows . a non - zero temperature is implemented by a stochastic term , analogous to the quenched randomness term , in the equation of motion . when we reformulate the model as a path integral , two similar terms are produced , corresponding to thermal and quenched disorder , respectively . unlike the latter , the former is uncorrelated both in space _ and in time_. as a consequence , the two @xmath95 fields occur at equal time in the temperature induced term , which is thereby less relevant than its ( marginal ) disorder induced counterpart . thus , thermal fluctuations do not modify the critical behavior , unless they change the symmetries of the problem . indeed , a non - vanishing temperature will in general displace the critial point in parameter space . if , in the @xmath132 limit , thermal fluctuations allow the system to equilibrate for each value of the exterior magnetic field , the magnetization history is non - hysteretic , regardless of the amount of disorder . ( if @xmath337 is non - vanishing , the lag in the response rounds off the crititcal point , leading to a finite dynamic susceptibility . ) the absence of hysteresis clearly does not modify the predictions of the exponents in the scalar case . in the case of a multicomponent field , however , the scalar and transverse criticalities would not be at experimental reach , enforcing @xmath0-like exponents . to be more accurate , the spins not only follow their local minima , but also precess . however , since @xmath5 can be increased arbitrarily slowly , the precession time scale can be separated as much as we want from the `` @xmath338 '' time scale . then , considering the motion on an intermediate time scale , we find that the precession averages out , and we are effectively left with the above langevin equation . the path integral is defined as the limit of a time - discretized formulation . in the simplest ( ito ) discretization scheme in which the response to a force at time @xmath129 occurs at time @xmath339 , the jacobian is clearly independent of the spin field @xmath82 . in this case , the contraction of @xmath95 and @xmath82 ( response function ) vanishes at equal time . other discretization procedures lead to more complicated jacobians , which , in all cases , cancel the naive equal time responses resulting from the discrete equations . ( see also @xcite and references therein . ) the trajectory is determined by eq . ( [ motion ] ) _ and an initial condition . _ for a non - vanishing @xmath35 , there exists a solution of eq . ( [ motion ] ) for any initial condition . as the @xmath132 limit is approached , all the solutions of the same energy `` basin '' collapse onto the corresponding minimum . the @xmath76 limit suppresses the lag in the response , but does not overthrow causality . the latter is ensured by the prescription that a solution should nt jump from one minimum to another before the first one disappears . then , a single minimum is visited if we impose that the solutions , incorporated by the generating functional , exist from @xmath34 to @xmath340 . however , if we allow solutions to start or end at finite times , and such solutions _ are picked _ by the delta function in eq . ( [ delta ] ) , many minima are simultaneously included in the path integral formulation of the theory .

earlier work on dynamical critical phenomena in the context of magnetic hysteresis for uniaxial ( scalar ) spins , is extended to the case of a multicomponent ( vector ) field . from symmetry arguments and a perturbative renormalization group approach ( in the path integral formalism ) , it is found that the generic behavior at long time and length scales is described by the scalar fixed point ( reached for a given value of the magnetic field and of the quenched disorder ) , with the corresponding ising - like exponents . by tuning an additional parameter , however , a fully rotationally invariant fixed point can be reached , at which all components become critical simultaneously , with @xmath0-like exponents . furthermore , the possibility of a spontaneous non - equilibrium transverse ordering , controlled by a distinct fixed point , is unveiled and the associated exponents calculated . in addition to these central results , a didactic `` derivation '' of the equations of motion for the spin field are given , the scalar model is revisited and treated in a more direct fashion , and some issues pertaining to time dependences and the problem of multiple solutions within the path integral formalism are clarified .

introduction equations of motion path integral formalism perturbative renormalization conclusion acknowledgments expansion about mean field theory

arxiv

let us start our story with the dirichlet problem . this problem of finding a harmonic function in a , say , smoothly bounded domain @xmath0 matching a given continuous function @xmath1 on @xmath2 gained huge attention in the second half of the nineteenth century due to its central role in riemann s proof of the existence of a conformal map of any simply connected domain onto the disk . later on riemann s proof was criticized by weierstrass , and , after a considerable turmoil , corrected and completed by hilbert and fredholm - cf . @xcite for a very nice historical account . here , we want to focus on algebraic properties of solutions to the dirichlet problem when @xmath3 is an ellipsoid and the data @xmath1 possess nice algebraic properties . thus , we first present the following proposition . [ prop : dp ] let @xmath4 be an ellipsoid . the solution @xmath5 to the dirichlet problem @xmath6 where @xmath7 is a polynomial of @xmath8 variables , is a harmonic polynomial . moreover , @xmath9 proposition [ prop : dp ] was widely known in the nineteenth century for @xmath10 ( perhaps due to lam ) and was proved with the use of ellipsoidal harmonics ( see ( * ? ? ? * section 58 ) for a discussion of laplace s equation in elliptic coordinates ) . it is still widely known nowadays for balls but often disbelieved for ellipsoids . the elder author has won a substantial number of bottles of cheap wine betting on its truthfulness at various math events and then producing the following proof that was related to him by harold s. shapiro . we do not know who thought of it first , but we hope the reader will agree that it deserves to be called , following p. erds , the `` proof from the book '' . denote by @xmath11 the finite - dimensional space of polynomials of degree @xmath12 in @xmath8 variables . let @xmath13 be the defining quadratic for @xmath2 . consider the linear operator @xmath14 defined by @xmath15 the maximum principle yields at once that @xmath16 , so @xmath17 is injective . since @xmath18 , this implies that @xmath17 is surjective . hence , given @xmath19 with @xmath20 , we can find a polynomial @xmath21 such that @xmath22 . the function @xmath23 is then the solution of ( [ eq : dp ] ) . proposition [ prop : dp ] was extended @xcite to the case of entire data . namely , entire data @xmath1 ( i.e. , an entire function of variables @xmath24 ) yields an entire solution to the dirichlet problem in ellipsoids . this result was sharpened by armitage in @xcite who showed that the solution s order and type are dominated by that of the data . one might get bold at this point and ask does the proposition [ prop : dp ] extend to say rational or algebraic data , i.e. , does a smooth data function in ( [ eq : dp ] ) that is a rational ( algebraic ) function of @xmath25 imply rational ( algebraic ) solution @xmath5 ? the answer is a resounding `` no '' but the proofs become technically more involved - see @xcite . it was conjectured in @xcite that prop . [ prop : dp ] ( without the degree condition ( [ eq : deg ] ) ) characterizes ellipsoids . recently , using `` real fischer spaces '' , h. render confirmed this conjecture for many algebraic surfaces @xcite . in two dimensions , the conjecture was confirmed under a degree - related condition on the solution in terms of the data @xcite . this utilized a suprising equivalence , established by n. stylianopoulos and m. putinar @xcite , of the conjecture to the existence of finite - term recurrence relations for bergman orthogonal polynomials . in order to state the degree conditions and the associated recurrence conditions , assume @xmath3 is a domain in @xmath26 with @xmath27-smooth boundary . let @xmath28 be the bergman orthogonal polynomials ( orthogonal w.r.t . area measure over @xmath3 ) , and consider the following properties for @xmath3 . 1 . there exists @xmath29 such that for a polynomial data of degree @xmath30 there always exists a polynomial solution of the dirichlet problem posed on @xmath3 of degree @xmath31 . there exists @xmath32 such that for all @xmath33 , the solution of the dirichlet problem with data @xmath34 is a harmonic polynomial of degree @xmath35 in @xmath36 and of degree @xmath37 in @xmath38 . there exists @xmath32 such that @xmath28 satisfy a finite @xmath39-recurrence relation , i.e. there are constants @xmath40 such that @xmath41 4 . the bergman orthogonal polynomials of @xmath3 satisfy a finite - term recurrence relation , i.e. , for every fixed @xmath42 , there exists an @xmath43 , such that @xmath44 , @xmath45 . 5 . for any polynomial data there exists a polynomial solution of the dirichlet problem posed on @xmath3 . properties @xmath46 and @xmath47 are essentially equivalent @xcite , and @xmath48 , @xmath49 , and @xmath50 . in @xcite the authors used ratio asymptotics of orthogonal polynomials to show that @xmath51 and equivalently @xmath52 each characterize ellipses . the weaker statement that @xmath53 characterizes ellipsoids was proved in arbitrary dimensions @xcite . for more about the khavinson - shapiro conjecture stated in @xcite , we refer the reader to @xcite and the references therein . the mean value property for harmonic functions can be rephrased as saying that _ the average of any harmonic function over concentric balls is a constant_. as we formulate precisely below , there is a mean value property for ellipsoids which says _ the average of any harmonic function over confocal ellipsoids is a constant_. consider a heterogeneous ellipsoid @xmath54 and let @xmath3 be its interior . a family of ellipsoids @xmath55 , @xmath56 where @xmath57 is called a confocal family for @xmath58 these are ellipses with the same foci . note that the shapes of confocal ellipsoids differ , and as @xmath59 , @xmath60 look like a spheres . observe that when @xmath61 , @xmath62 @xmath63 is called the _ focal ellipsoid_. the following classical theorem goes back to maclaurin who considered prolate spheroids in @xmath64 ( @xmath65 ) . general ellipsoids were treated later by laplace ( * ? ? ? 2 ) . let @xmath5 be , say , an entire harmonic function . then @xmath66 for all @xmath67 . from now on , for the sake of brevity , we shall only consider the case @xmath68 . maclaurin s theorem is a corollary ( via a simple change of variables , see ( * ? ? ? 16 ) or ( * ? ? ? * ch . 13 ) ) of the following result of sgeirsson @xcite . `` suppose @xmath69 , where @xmath70 , @xmath71 satisfy the ultrahyperbolic equation @xmath72 then if @xmath73 , @xmath74 denote respectively the mean values of @xmath5 over @xmath75-dimensional balls of radius @xmath76 centered at @xmath77 , we have @xmath78 . '' here , we offer a purely algebraic approach to maclaurin s theorem @xcite , ( * ? ? ? * ch . 13 ) . the following notions are due to e. fischer @xcite ( see also ( * ? ? ? iv ) ) . let @xmath79 be the space of homogeneous polynomials of degree @xmath80 . if @xmath81 , then @xmath82 introduce an inner product on @xmath79 ( called the fischer inner product ) , by letting @xmath83 if @xmath84 , @xmath85 then @xmath86 the main point of introducing such an inner product is that the operators @xmath87 and multiplication by @xmath88 are adjoint with respect to the fischer inner product . let @xmath89 denote the space of homogeneous harmonic polynomials of degree @xmath30 . it follows from the definition ( [ eq : fischer ] ) that @xmath90 is a reproducing kernel for @xmath89 , i.e. , for all @xmath91 , @xmath92 this fact , along with hilbert s nullstellensatz , easily yields the following lemma ( see ( * ? ? ? 13 ) for a detailed proof ) . [ lemma : span ] the linear span of harmonic polynomials @xmath93 for all @xmath94 the isotropic cone equals @xmath89 . it suffices to check ( [ eq : mac ] ) for harmonic homogeneous polynomials , and in view of lemma [ lemma : span ] , we just have to check it for polynomials @xmath95 fix @xmath96 . let @xmath97 be the semi - axes of @xmath98 . we have to show that @xmath99 changing variables in both integrals @xmath100 , @xmath101 we see that it suffices to show the following : @xmath102 where @xmath103 is the unit ball in @xmath104 . or , since for @xmath105 @xmath106 it suffices to check the following assertion . * assertion . * the polynomial @xmath107 depends only on @xmath108 , for @xmath109 . the assertion follows at once from the rotation invariance of @xmath110 ( * ? ? ? 13 ) , @xcite . the following application is noteworthy . let @xmath3 be an ellipsoid with semiaxes @xmath111 , and @xmath112 be the exterior potential of @xmath3 . recall that @xmath63 denotes the focal ellipsoid defined above . the following corollary of maclaurin s theorem describes a so - called _ mother body _ @xcite , i.e. , a measure supported inside the ellipsoid which generates the same gravitational potential as the uniform density ( outside the ellipsoid ) but is in some sense minimally supported ( in this case supported on @xmath63 , a set of codimension one with connected complement ) . [ cor : motherbody ] for @xmath113 @xmath114 where @xmath115 @xmath116 is lebesgue measure on @xmath117 . since the integrand is harmonic , we have by maclaurin s theorem @xmath118 where we set @xmath119 . after simplifying this integral using fubini s theorem , the corollary is established by applying the lebesgue dominated convergence theorem as @xmath120 ( * ? ? ? we note in passing that finding relevant mother bodies for oblate and prolate spheroids ( supported on a disk and segment respectively ) could be a satisfying exercise . since the density of the distribution @xmath121 is real analytic in the interior of @xmath63 ( viewed as a set in @xmath122 ) we note the following corollary : the potential @xmath123 extends as a multivalued harmonic function into @xmath124 . an extension of this fact and a `` high ground '' explanation based on holomorphic pde in @xmath125 is discussed in section [ sec : cp ] . considering that force is the gradient of potential , the following theorem , due to newton , can be paraphrased in a rather catchy way : `` there is no gravity in the cavity '' . [ thm : newton ] let @xmath126 , and consider the ellipsoidal shell @xmath127 between two homothetic ellipsoids . the potential @xmath128 of uniform density on @xmath129 is constant inside the cavity @xmath3 . in fact , ellipsoids are characterized by this property , i.e. , newton s theorem has a converse @xcite . a consequence of newton s theorem is that the gravitational potential @xmath130 of @xmath3 is a quadratic polynomial inside @xmath3 . namely , @xmath131 with @xmath132 , where @xmath133 . indeed , denoting by @xmath134 ( for @xmath126 ) the dilated ellipsoid , one computes that its gravitational potential is @xmath135 . since newton s theorem implies that ( @xmath5 is the potential of the original ellipsoid ) , @xmath136 inside @xmath3 , the smaller ellipsoid , then taking partial derivatives @xmath137 , w.r.t . @xmath138 , @xmath139 , yields that @xmath140= @xmath141 . thus all these partial derivatives are homogeneous of degree zero inside @xmath3 . they are also obviously continuous and , hence , are constants , thus yielding @xmath130 to be a quadratic as claimed . denoting @xmath142 , consider the single layer potential @xmath143 where @xmath144 is the mass density and @xmath145 on @xmath146 is the surface area measure . @xmath147 is called an _ equilibrium potential _ if @xmath148 on @xmath146 and hence inside @xmath3 . for the sake of brevity we focus on @xmath149 leaving the case @xmath58 as an exercise . the quantity @xmath150 is called capacity . on the way to proving ivory s theorem , we note an explicit formula for the equilibrium potential . again , @xmath149 @xcite . with @xmath103 as above , in @xmath151 @xmath152 where @xmath153 , @xmath154 , and @xmath155 is the maclaurin `` quadrature measure '' supported on the focal ellipsoid @xmath63 ( cf . cor . [ cor : motherbody ] ) . the rhs of ( [ eq : ivory ] ) is harmonic in @xmath151 ( in fact , in @xmath156 ) since @xmath157 is harmonic there and @xmath158 . on @xmath146 , by maclaurin s theorem and newton s theorem @xmath159 moreover since @xmath160 has continuous first derivatives throughout @xmath161 , we can differentiate ( [ eq : mu ] ) on @xmath146 and thus obtain @xmath162 thus , the rhs of ( [ eq : ivory ] ) equals @xmath147 on @xmath146 . both functions are harmonic in @xmath151 and vanish at infinity and the statement follows . the equipotential surfaces of the equilibrium potential @xmath147 are confocal with @xmath146 . for the proof , one simply notes that the rhs of ( [ eq : ivory ] ) changes only by a constant factor when @xmath3 is replaced by a confocal ellipsoid @xmath163 namely , @xmath164 while @xmath165 . for the classical proof of ivory s theorem , see @xcite , ( * ? ? ? * lecture 30 ) . let us pause for a moment and apply these properties of ellipsoids to two problems in fluid dynamics . in the first problem , involving a slowly moving interface , viscosity plays an important role . in the second problem , viscosity is completely neglected , while vorticity plays the dominant role . imagine a blob of incompressible viscous fluid within a porous medium surrounded by an inviscid fluid . suppose there is a sink at position @xmath166 in the region occupied by viscous fluid . averaging the navier - stokes equations over pores @xcite leads to darcy s law for the fluid velocity @xmath167 in terms of the pressure @xmath110 @xmath168 incompressibility implies that @xmath169 except at the sources / sinks . the pressure of the inviscid fluid is assumed constant . neglecting surface - tension ( by far , the most controversial of these assumptions @xcite ) the pressure matches at the interface , which gives a constant ( assume zero ) boundary condition for @xmath110 , so @xmath110 is nothing more than the harmonic green s function with a singularity at @xmath166 . the mathematical problem is then to track the evolution of a domain @xmath170 whose boundary velocity is determined by the gradient of its own green s function . see @xcite for an engaging exposition of the two - dimensional case of this problem . given a harmonic function @xmath171 , richardson s theorem @xcite describes the time dependence of the integration of @xmath5 over the domain occupied by the viscous fluid . in the language of integrable systems this represents `` infinitely many conservation laws '' . let @xmath171 be a function harmonic in @xmath170 for all @xmath172 . then @xmath173 where @xmath166 is the position of the sink with pumping rate @xmath174 . an alternative setup places the viscous fluid in an unbounded domain with a single sink at infinity @xcite ; a reformulation of richardson s theorem implies that the potential inside the cavity of the shell regions @xmath175 is constant . thus , it is a consequence of newton s thm . [ thm : newton ] that an increasing family of homothetic ellipsoids is an exact solution . in fact , this is the only solution starting from a bounded inviscid fluid domain that exists for all time and fills the entire space @xcite ( also , cf . @xcite ) . returning to the case when the viscous fluid is bounded , suppose the initial domain @xmath176 is an ellipsoid and consider the problem of determining sinks and pumping rates such that @xmath177 shrinks to zero volume as @xmath178 . as a consequence of the mean value property , one can solve this problem exactly thus removing all of the fluid provided we can stretch our imaginations to allow a continuum of sinks . starting from the given ellipsoid @xmath176 , the evolution @xmath170 is a family of ellipsoids confocal to @xmath176 shrinking down to the ( zero - volume ) focal set @xmath63 , and the pumping rate is given by the time - derivative of the quadrature measure appearing in corollary [ cor : motherbody ] . based on the observation that motion in the atmosphere is roughly stratified into horizontal layers , the quasigeostrophic approximation @xcite provides a simplified version of the euler equations ( governing inviscid incompressible flow ) . further assumptions reduce the entire dynamics to a scalar field , the potential vorticity , which in the high reynold s number limit , forms coherent regions of uniform density @xcite . even with these simplifications , the problem can still be quite complicated . for instance , approximating the regions of potential vorticity by clouds of point - vortices , one encounters the notoriously difficult @xmath8-body problem . the _ quasigeostrophic ellipsoidal vortex model _ developed by dritschel , reinaud , and mckiver @xcite , simulates the interaction of ellipsoidal regions of vorticity ( see fig . [ fig : vortex ] , included here with their kind permission ) . as these regions interact , the length and alignment of semiaxes can change , but non - ellipsoidal deformations are filtered out . ( note that a single ellipsoid is stable for a certain range of axis ratios @xcite . ) the effect that one ellipsoid has on another is determined by its exterior potential , and thus the mean value property can be used to replace the ellipsoid by a two - dimensional set of potential vorticity on its focal ellipse ( with density determined by corollary [ cor : motherbody ] ) which can be further approximated by point vortices . it is interesting to single out the two - dimensional case of the moving interface problem which serves as a model for viscous fingering in a hele - shaw cell @xcite . conformal mapping techniques lead to explicit exact solutions @xcite that can even exhibit the tip - splitting depicted in fig . [ fig : hs ] . the vortex dynamics problem also admits many sophisticated analytic solutions in the two - dimensional case @xcite . for a compelling survey discussing _ quadrature domains _ as a common thread linking these and several other fluid dynamic problems , see @xcite . [ fig : twophase ] in yet another physically distinct setting , ellipsoids appear as exact solutions to a certain two - phase problem in fluid dynamics @xcite . in this case there are no sources or sinks , but rather a linear straining flow at infinity ( see fig . [ fig : twophase ] ) . the ( fixed - volume ) ellipsoid changes shape but remains an ellipsoid ( see @xcite for details ) . the problem mentioned in section [ sec : mvp ] of analytically continuing the exterior potential @xmath179 inside the region @xmath3 occupied by mass was studied by herglotz @xcite , and can be reformulated as studying the singularities of the solution to the following cauchy problem posed on the initial surface @xmath180 . @xmath181 where the notation @xmath182 indicates that @xmath183 along with its gradient vanishes on @xmath146 . the fact that @xmath183 carries the same singularities in @xmath3 as the analytic continuation @xmath5 of @xmath130 is a consequence of the fact that @xmath5 itself is given by the piecewise function @xmath184 the reason is that @xmath5 is harmonic on both sides of @xmath146 and @xmath185-smooth across @xmath146 . an extension of morera s theorem ( attributed to s. kovalevskaya ) implies that @xmath5 is actually harmonic across @xmath146 , i.e. , @xmath146 is a removable singularity set for @xmath5 . thus , @xmath5 is the desired analytic continuation of @xmath130 across @xmath146 , and the singularities of @xmath5 in @xmath3 are carried by @xmath183 . further reformulating the problem , note that the so - called _ schwarz potential _ of @xmath146 , @xmath186 , has the same singularities as @xmath183 and solves a cauchy problem for laplace s equation : @xmath187 this is a rather delicate ( ill - posed according to hadamard ) problem , and our discussion of it will pass from @xmath188 to the complex domain @xmath125 . let us first consider the more intuitive cauchy problem for a hyperbolic equation where similar behavior can be observed while staying in the real domain . @xmath189 where @xmath190 is , say , a real analytic curve in @xmath26 . for hyperbolic equations the mantra is `` singularities propagate along characteristics '' . if the solution is singular at some point @xmath191 , then one can trace the source of this singularity back to @xmath190 by following the characteristic cone with vertex at @xmath191 . one expects to find a singularity in the data itself at a point where this cone intersects @xmath190 , but what if the data function has no singularities as in ( [ eq : hyperbolic ] ) ? it is still possible for a singularity to propagate to the point @xmath191 if the characteristic cone from @xmath191 is tangent to @xmath190 . the point of tangency is called a _ characteristic point _ of @xmath190 . ) is regular except on the tangent characteristic @xmath192 . ] for example , suppose @xmath193 . we can solve ( [ eq : hyperbolic ] ) exactly : @xmath194 the solution is singular on the characteristic @xmath195 which is tangent to the initial curve @xmath190 at the point @xmath196 . the singularities in the solution of ( [ eq : sp ] ) also propagate along tangent characteristics , but the characteristic points ( the `` birth places '' of singularities ) reside on the complexification of @xmath146 , the complex hypersurface given by the same defining equation . at four characteristic points intersect @xmath26 precisely at the foci . ] all solutions of the cauchy problem ( [ eq : sp ] ) with entire data @xmath1 on @xmath197 extend holomorphically along all paths in @xmath125 that avoid the characteristic surface @xmath198 ( consisting of all characteristic lines tangent to @xmath146 ) . the intersection @xmath199 is the focal ellipsoid . according to the properties of the schwarz potential discussed above , this provides a @xmath125-explanation of a rather physical fact that @xmath63 supports a measure solving an inverse potential problem . johnsson s proof of this theorem can be described as a globalization of leray s principle , a local theory governing propagation of singularities . as johnsson notes , there is an unexpected coincidence between potential - theoretic foci ( points where singularities of @xmath200 are located ) and algebraic foci in the classical sense of plcker @xcite . understanding this correspondence and extending it to higher - degree algebraic surfaces is part of a program advocated by the first author and h. s. shapiro . the case @xmath201 is more transparent @xcite , and for @xmath202 it is virtually unexplored except for some axially - symmetric fourth - degree examples @xcite . newton s theorem can be reformulated in terms of a single layer potential obtained by shrinking a constant - density ellipsoidal shell to zero thickness ( while rescaling the constant ) , leading to a non - constant density @xmath203 , where @xmath204 is the defining quadratic of the ellipsoid . this is sometimes called the _ standard single layer potential _ ( it is different from the equilibrium potential discussed in section [ sec : ivory ] ) . the modern approach due to v. i. arnold and , then , a. givental @xcite , views the force at @xmath166 induced by infinitesimal charges at two points @xmath205 on a line @xmath206 through @xmath166 as a sum of residues for a contour integral in the complex extension @xmath207 of @xmath206 . the vanishing of force follows from deforming the contour to infinity . the detailed proof can be found in ( * ? ? ? 14 ) . the same proof can be used to extend newton s theorem beyond ellipsoids to any _ domain of hyperbolicity _ of a smooth , irreducible real algebraic variety @xmath146 of degree @xmath80 . a domain @xmath3 is called a domain of hyperbolicity for @xmath146 if for any @xmath208 , each line @xmath206 passing through @xmath166 intersects @xmath146 at precisely @xmath80 points . for example , the interior of an ellipsoid is a domain of hyperbolicity , and if a hypersurface of degree @xmath209 consists of an increasing family of @xmath80 ovaloids then the smallest one is the domain of hyperbolicity . defining the standard single layer density on @xmath146 in exactly the same way as before , except that the sign @xmath210 or @xmath211 is assigned on each connected component of @xmath146 depending whether the number of obstructions for `` viewing '' this component from the domain of hyperbolicity of @xmath146 is even or odd , the arnold - givental generalization of newton s theorem implies , in particular , that the force due to the standard layer density vanishes inside the domain of hyperbolicity ( cf . @xcite for more general statements and proofs ) . as a final remark , returning to ellipsoids , and even taking @xmath201 , let us note an application to gravitational lensing of corollary [ cor : motherbody ] . the two - dimensional version of maclaurin s theorem plays a key role in formulating analytic descriptions for the gravitational lensing effect for certain elliptically symmetric lensing galaxies @xcite ( cf . @xcite for terminology ) . here the projected mass density that is constant on confocal ellipses produces at most 4 lensed images @xcite . the density that is constant on homothetic ellipses produces at most 6 images @xcite , also cf . @xcite . the same technique that applies maclaurin s theorem to density that is not constant but is constant on each scaled ellipse can also be applied to the case when the ellipses are allowed to rotate as they are scaled . this leads to a lensing equation involving a gauss hypergeometric function that describes the images lensed by a spiral galaxy @xcite ( an investigation initiated during an reu ) . in connection to the converse to newton s theorem , whenever the rare focusing effect in graviataional lensing produces a continuous `` halo '' ( aka einstein ring - cf . @xcite for some striking nasa pictures ) around the lensing galaxy ( of any shape ) , the `` halo '' necessarily turns out to be either a circle or an ellipse @xcite . but this alley leads to the beginning of another story . p. ebenfelt , d. khavinson , h. s. shapiro , _ algebraic aspects of the dirichlet problem with rational data _ , quadrature domains and their applications , oper . theory adv . appl . , 156 , birkhuser , basel , 2005 , 151 - 172 . c. d. fassnacht , c. r. keeton , d. khavinson , _ gravitational lensing by elliptical galaxies and the schwarz function _ , in b. gustafsson and a. vasilev ( eds . ) , trends in complex and harmonic analysis , birkhuser ( 2007 ) , 115 - 129 . d. khavinson , n. stylianopoulos , _ recurrence relations for orthogonal polynomials and the khavinson - shapiro conjecture _ , in `` around the research of vladimir mazya ii , partial differential equations '' , pp . 219 - 228 , international mathematical series , vol . 12 , ed . by a. laptev , springer , 2010

ellipsoids possess several beautiful properties associated with classical potential theory . some of them are well known , and some have been forgotten . in this article we hope to bring a few of the `` lost '' pieces of classical mathematics back to the limelight .

dirichlets problem the mean value property for harmonic functions the equilibrium potential of an ellipsoid. ivorys theorem ellipsoids in fluid dynamics the cauchy problem: a view from @xmath125 epilogue

arxiv

the formation of the present - day massive elliptical galaxies remains one of the most controversial issues of galaxy evolution and structure formation . in cdm hierarchical merging models ( e.g. @xcite ) , massive ellipticals form at relatively low redshift ( e.g. @xmath1 ) through the merging of spiral galaxies . in such scenarios , massive , old and passively evolving ellipticals are extremely rare objects at @xmath2 . in marked contrast , other scenarios suggested that massive ellipticals formed at higher redshifts ( e.g. @xmath3 ) through an intense initial starburst event followed by pure luminosity evolution ( ple ) of the stellar population to nowadays ( e.g. @xcite ) , thus implying a constant comoving number density of passively evolving ellipticals at @xmath4 and @xmath2 . in this scenario , a substantial number of extremely red objects ( eros ) with the colors of an old stellar population at @xmath2 ( e.g. @xmath5 ) and @xmath6 surface brightness profiles typical of dynamically relaxed spheroidals is expected to be found in near - ir selected ( i.e. stellar mass selected ) galaxy samples . imaging surveys with typical fields of @xmath71 - 50 arcmin@xmath8 provided very discrepant results . some found a clear deficit of old ellipticals at @xmath9 or @xmath2 compared to passive evolution models ( e.g. @xcite ) , whereas others found evidence for a constant comoving density ( @xcite ) . a large part of the above discrepancies is certainly due to the strong ero angular clustering ( i.e. field - to - field density variations ) that was discovered thanks to wider field surveys ( @xcite ) . the results of such surveys suggest that the observed angular clustering is the signal of the underlying 3d large scale structure of massive ellipticals@xcite , and showed that the surface density of @xmath2 passive elliptical _ candidates _ is consistent with that expected in ple models , thus suggesting that most field ellipticals were fully assembled at least by @xmath102.5 ( @xcite ) . however , follow - up observations are needed to confirm that most eros are passive ellipticals because it is known that some eros are dust - reddened starburst galaxies , thus representing a `` contamination '' in color selected samples of elliptical candidates at @xmath11 ( @xcite ) . fig.1 shows the spectrum of a @xmath12 passive elliptical as observed with the eso vlt equipped with the optical imager spectrograph fors2 ( @xcite ) . no emission lines are present , and the main features are the 4000 continuum break together with strong caii h&k absorptions and other weaker absorption lines . when such a spectrum is redshifted to @xmath13 , the 4000 and the caii lines exit from the accessible optical spectral range and the redshift identification relies only on weak absorptions . 2 shows an example of an elliptical candidate at @xmath14 that can be taken as a clear example of the difficulties in identifying the nature and the redshifts of high-@xmath0 passive ellipticals ( see also @xcite ) . elliptical with @xmath15 ( from @xcite),scaledwidth=80.0% ] when elliptical candidates are too faint for optical spectroscopy and/or expected to be at @xmath16 , the only possibility is to move to near - ir spectroscopy in order to search for the 4000 break and the caii h&k lines redshifted at @xmath17 m . however , the general faintness of the target continua makes such observations difficult and time consuming , and the results of both keck and vlt seeing limited near - ir spectroscopy provided so far rather ambiguous results ( @xcite ) . despite the difficulties of optical and near - ir spectroscopy , a substantial number of @xmath2 passively evolving ellipticals has been identified in recent surveys @xcite . the inferred ages of the stellar populations ( @xmath18 gyr ) are consistent with such galaxies being formed at remote cosmological epochs . in addition , morphological studies based on hst imaging further confirmed the existence of a population of dynamically relaxed high-@xmath0 spheroids through the analysis of their surface brightness profiles ( e.g. @xcite . elliptical candidate with @xmath19 and @xmath20 plotted together with a bruzual & charlot 2 gyr old ssp model ( from @xcite).,scaledwidth=80.0% ] despite the successful observations of vlt , keck and hst to uncover a substantial population of passively evolving ellipticals at @xmath2 , many questions need still to be answered before drawing final conclusions on the formation and evolution of massive galaxies : what is the fraction of ellipticals as a function of @xmath0 and @xmath21 ? what are their masses , ages and redshifts of formation ? what is the evolution of their luminosity function and clustering with respect to model predictions ? addressing the above questions requires the use of different observing techniques , and three ideal vlt instruments could play a crucial role : * _ ao aided near - ir imager . _ such an instrument , especially if aided by a laser guide star ( lgs ) system allowing a flexible pointing of the telescope ( i.e. not limited to targets close to bright natural stars ) , would allow to study the morphology of elliptical candidates not only to confirm their nature by analysing their surface brightness profiles , but also to derive radial color gradients and to study the evolution of the fundamental plane ( e.g. via the kormendy relation ) once the redshifts are known . + passive ellipticals as a function of their colors , magnitudes and redshifts . the x - axis shows the @xmath22 and @xmath21 magnitudes , assuming a color of @xmath23 which is characteristic of an old passively evolving system at @xmath11 . the y - axes show the @xmath24 color and the redshift ( the @xmath24 color being a function of @xmath0 for @xmath11 ) . the continuous and dashed diagonal lines indicate the location of @xmath25 and @xmath26 objects respectively . the vertical continuous line indicate the present limit of seeing limited near - ir _ continuum _ spectroscopy with vlt+isaac . the vertical dashed line indicates a conservative limit of ao aided near - ir continuum spectroscopy expected for the vlt . different observing regimes can be envisaged . continuum optical spectroscopy can be realistically performed down to @xmath27 . near - ir spectroscopy without ao can be done only for relatively bright targets ( @xmath28 , @xmath29 ) and a @xmath30 magnitude gain is expected with the aid of ao . for targets fainter than @xmath31 , only imaging can be performed . for targets with , say , @xmath32 , optical _ and _ near - ir imaging can be done with reasonable integration times and photometric accuracy , but for galaxies with @xmath33 optical imaging becomes very difficult and most of the photometric information comes from near - ir observations only . , scaledwidth=100.0% ] * _ ao aided near - ir spectrograph . _ the scientific aim of such an instrument would be to spectroscopically confirm the nature of the passive elliptical candidates and to measure their redshifts . a simultaneous coverage of the near - ir spectral range ( @xmath34 ) would be essential to reduce the observation time and to overcome the problem of matching and inter - calibrating independent @xmath35 spectra ( e.g. isaac ) . a high throughput in the @xmath36 and @xmath22 bands would be also crucial to work efficiently in the spectral range where the 4000 break falls for @xmath37 . a mos capability would be obviously important to increase the multiplex . such a spectrograph should also have a low spectral resolution mode ( e.g. r@xmath7200 - 500 ) in order to derive , together with optical photometry , accurate continuum spectrophotometry for the faintest targets in order to estimate their `` spectrophotometric '' redshifts whenever it is impossible to measure their spectroscopic redshifts ( e.g. cimatti et al . 1999 ; soifer et al . 1999 ) . a moderately high spectral resolution would be important to estimate the masses of the brightest targets through the velocity dispersion of the absorption lines . it should be noted here that for faint galaxy spectroscopy ( where the slit width can not be much narrower than the size of the galaxy ) , low - order ao corrections are sufficient to provide a significant improvement of the s / n ratio thanks to reduction of the sky background compared with seeing - limited spectroscopy made with typical slit widths of 1@xmath38 . * _ near - ir wide - field imager _ ( e.g. 20@xmath3920 arcmin@xmath8 ) . a vlt near - ir wfi with optimal seeing sampling ( e.g. 0.15@xmath40/pixel ) would be crucial to exploit the image quality of the vlt and to push the photometry to the limits of a 8m - class telescope . such an instrument would allow to perform ultradeep surveys and to select and to study high-@xmath0 ellipticals beyond the spectroscopic limits . in this respect , it would be essential to have such an imager equipped with a set of medium - band filters in order to derive spectral energy distributions and @xmath41 with a high level of accuracy . such an instrument would play a key role in statistical studies such as the evolution of the luminosity function and of the clustering of ellipticals . a significant population of @xmath2 passively evolving elliptical candidates was unveiled thanks to recent wide - field imaging surveys . the confirmation of their nature and the measurement of their redshifts are very challenging even with 8 - 10 m class telescopes and the results obtained so far are limited to the brightest objects ( @xmath42 ) . much observational work can still be done with the 1st generation vlt instruments such as the red - upgraded fors2 ( optical imaging and spectroscopy ) , conica and sinfoni ( ao - aided near - ir imaging and spectroscopy ) . an ao- or mcao - aided near - ir imager - spectrograph ( with mos capability ) and a near - ir wfi equipped with medium - band filters are the most desirable 2nd generation vlt instruments expected to play a crucial role in the understanding of the formation and evolution of the massive ellipticals . kauffmann g. 1996 , mnras , 281 , 487 baugh c.m . , cole s. , frenk c.s . 1996 , mnras , 283 , 1361 baugh c.m . , cole s. , frenk c.s . , lacey c.g . 1998,apj,498,504 eggen , o. j. ; lynden - bell , d. ; sandage , a. r. 1962 , apj , 136 , 748 larson r.b . 1974 , mnras , 166 , 686 arimoto n. & yoshii y. 1987 , a&a , 173 , 23 zepf s.e . 1997 , nature , 390 , 377 franceschini a. et al . 1998 , apj , 506 , 600 barger a.j . et al . 1999 , aj , 117 , 102 menanteau f. et al . 1999 mnras , 309 , 208 treu t. , stiavelli m. 1999 , apj , 524 , l27 stiavelli m. , treu t. 2000 , astro - ph/0010100 rodighiero g. et al . 2001 , mnras , in press ( astro - ph/0101262 ) totani t. , yoshii j. 1997 , apj , 501 , l177 benitez n. et al . 1999 , apj , 515 , l65 broadhurst t.j . , bouwens r.j . 1999 , apj , 530 , l53 schade d. et al . 1999 , apj , 525 , 31 scodeggio m. , silva d.r . 2000 , a&a , 359 , 953 i m m. et al . 2001 , apj , in press ( astro - ph/0011092 ) daddi e. et al . 2000 , a&a , 361 , 535 mccarthy p.j . 2000 , astro - ph/0011499 daddi e. et al . 2001 , a&a , in press ( astro - ph/0107340 ) daddi e. , cimatti a. , renzini a. 2000 , a&a , 362 , l45 cimatti a. , andreani p. , rttgering h. , tilanus r. 1998 , nature , 392 , 895 cimatti a. et al . 1999 , a&a , 352 , l45 dey a. et al . 1999 , apj , 519 , 610 cimatti et al . , in preparation ( see http://www . arcetri.astro.it/@xmath7k20/ ) liu m.c . 2000 , aj , 119 , 2556 stockton a. 2001 , astro - ph/0104191 soifer b.t . 1999 , aj , 118 , 2065 glassman t.m . , larkin j.e . 2000 , apj , 539 , 570 moriondo g. , cimatti a. , daddi e. 2000 , a&a , in press spinrad h. et al . 1997 , apj , 484 , 581 cohen j.g . et al . 1999 , apj , 512 , 30 dunlop j.s . 1999 , astro - ph/9912380

the results and the present limits of the observations of high-@xmath0 ellipticals are discussed in the framework of vlt imminent and future instruments .

when and how did massive ellipticals form ? the results of vlt 1st generation instruments future prospects for the vlt summary

arxiv

to properly represent the multichannel physics of the feshbach resonances between k and rb atoms in our three - body calculations , we allow the k atom to carry the spin degrees of freedom with a spin state @xmath80 . the potential between k and rb atoms @xmath94 , where @xmath95 is the distance between the k and rb atoms , is constructed to reproduce the long - range ( @xmath96 bohr ) behavior of the two - body wave function near the relevant feshbach resonances . we use lennard - jones 6 - 12 potentials with the same @xmath97 @xcite for the diagonal potentials @xmath98 . the open - channel potentials ( associated with the lowest zero - field energy @xmath89 ) have short - range cutoffs such that the single - channel scattering lengths reproduce the background scattering lengths near the feshbach resonances investigated in our current study . for the @xmath2k-@xmath1rb interaction , this open - channel scattering length is @xmath99 bohr , which corresponds to the background scattering length of the isolated @xmath3-wave feshbach resonance near @xmath100 g. for the @xmath0k-@xmath1rb interaction , the open - channel scattering length is @xmath101 bohr , which is the global background scattering length of the overlapping feshbach resonances near 39 and 79 g. the closed - channel potential(s ) is constructed in the way such that its bound - state wave function reproduces the main characters of the closed - channel wave function found in the coupled - channel two - body calculations . in both @xmath2k-@xmath1rb and @xmath0k-@xmath1rb systems , the closed - channel component of the low - energy two - body scattering wave function is found to have a strong character of the triplet @xmath102 vibrational level of the corresponding system . based on this observation , we adjust the short - range cutoffs of the closed - channel potentials of @xmath2k-@xmath1rb and @xmath0k-@xmath1rb systems such that the potentials have triplet scattering lengths of @xmath103 and @xmath104 bohr , respectively , and use the the @xmath102 vibrational levels of the closed channels to give rise to the feshbach resonances in our spinor models . the single - atom energies @xmath89 are chosen for each spin channel to be on the order of the hyperfine splittings between the relevant k - rb scattering thresholds , and more importantly , to accurately position the feshbach resonances in our spinor models . the magnetic moments @xmath88 are set to reproduce the magnetic moment differences between the open and closed channels found in coupled - channel two - body calculations . in figure [ fig : sm_3b ] , we show the scattering length from the coupled - channel calculations and those from our two- and three - spin models used in our three - body calculations . the coupled - channel calculations include only the @xmath3-wave basis . for the @xmath2k-@xmath1rb system , although there is a @xmath105-wave feshbach resonance located at about 0.8 g higher than the @xmath3-wave resonance , this @xmath105-wave resonance is very narrow @xcite and produces very small perturbation to the @xmath3-wave feshbach resonance . moreover , the atom - dimer resonance was observed on a branch of the @xmath3-wave feshbach resonance different from the branch where the @xmath105-wave resonance locates , therefore the effect of the @xmath105-wave resonance can be safely neglected in our study of the @xmath2k-@xmath1rb-@xmath1rb system . no significant higher partial - wave resonances are found in the @xmath0k-@xmath1rb system in the range of magnetic field in our current study . 3ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) link:\doibase 10.1103/physreva.76.020701 [ * * , ( ) ] link:\doibase 10.1103/physreva.87.032706 [ * * , ( ) ]

the multichannel efimov physics is investigated in ultracold heteronuclear admixtures of k and rb atoms . we observe a shift in the scattering length where the first atom - dimer resonance appears in the @xmath0k-@xmath1rb system relative to the position of the previously observed atom - dimer resonance in the @xmath2k-@xmath1rb system . this shift is well explained by our calculations with a three - body model including the van der waals interactions , and , more importantly , the multichannel spinor physics . with only minor difference in the atomic masses of the admixtures , the shift in the atom - dimer resonance positions can be cleanly ascribed to the isolated and overlapping feshbach resonances in the @xmath2k-@xmath1rb and @xmath0k-@xmath1rb systems , respectively . our study demonstrates the role of the multichannel feshbach physics in determining efimov resonances in heteronuclear three - body systems . if physical systems exhibit properties that are independent of details of interaction , they are called universal @xcite . universality has played a central role in the analysis of quantum degenerate gases , _ e.g. _ , the effects of binary collisions were successfully characterized by a single parameter , the @xmath3-wave scattering length @xmath4 , independent of the details of the two - body potential . for few - body phenomena , however , it has been well known that an additional parameter _ e.g. _ three - body parameter @xcite is necessary for a complete description of the system . efimov states , an infinite series of three - body bound states with discrete scale invariance when a two - body scattering length diverges @xcite , provided us a unique opportunity to investigate the properties of three - body parameter both theoretically and experimentally . combined experimental effort to observe efimov - related resonance provided us with unexpected constancy of three - body parameters@xcite , while detailed theoretical analysis showed the origin of this constancy in some limiting cases@xcite . recently , a newly developed three - body spinor model that included both van der waals interactions and multichannel feshbach physics has succeeded in reproducing many experimentally observed efimov features in homonuclear atomic systems @xcite . this model involves additional parameters that characterize feshbach resonances , such as the background scattering length of an open channel normalized by the van der waals length ( @xmath5 ) and the resonance strength of a closed channel ( @xmath6 ) . it was impressive to see that predictions from a three - body model constructed to reproduce only two - body feshbach physics match almost perfectly with the experimentally observed three - body features in homonuclear systems @xcite . this achievement suggests that the necessity of including precise few - body short - range chemical forces in studies of universal few - body phenomena a task far beyond our current capability may be removed . extending this universal theory to _ hetero_nuclear systems is the next big challenge . in addition to the mass ratio , heteronuclear systems have the extra complication of having both inter- and intra - species scattering lengths . near broad feshbach resonances ( @xmath7 ) , the single - channel universal van der waals theory @xcite predicted a universal dependence of the three - body parameter on the intra - species scattering length . the predictions have been confirmed to hold in @xmath8li-@xmath9cs @xcite and @xmath10li-@xmath1rb @xcite mixtures , where the feshbach resonances with intermediate widths have @xmath11 even down to unity @xcite . for homonuclear systems , the universal prediction @xcite appeared to hold for @xmath12k even near narrow feshbach resonance with @xmath13 @xcite . the effect of the finite feshbach resonance width , as well as of the structure of the complicated resonance structure more commonly seen in realistic atomic systems , has therefore not been well demonstrated and understood . moreover , the recent experimental confirmations of the universal van der waals theory are for the systems with large mass ratios ( two heavy and one light atoms ) the so - called `` efimov - favored '' systems , where the mechanism of the universality of the three - body parameter is considered to be different from the mechanism in the systems with small or moderate mass ratios @xcite the `` efimov - unfavored '' systems . an independent investigation is therefore still needed for the latter systems . in this study , we show the role of multichannel feshbach physics in determining the three - body parameter in the `` efimov - unfavored '' systems of k - rb admixtures . our three - body spinor theory successfully reproduces the difference in the positions of the efimov - like atom - dimer resonance observed in our @xmath0k-@xmath1rb experiment and the previous jila @xmath2k-@xmath1rb experiment @xcite . as the feshbach resonances involved in the isotopic systems include both isolated and overlapping resonances , our study also demonstrates that efimov resonance features in atomic systems can be fully described by three - body van der waals models that reproduce the relevant two - body feshbach spectra . in previous experiments , the three - body parameter was mostly determined by the positions of the efimov resonances in three - body recombination . this approach is feasible in studies of the `` efimov - favored '' heteronuclear systems , thanks to the increased universal scaling constant @xmath14 and therefore , the significantly decreased efimov scaling cycle @xmath15 @xcite for the ground efimov resonance to be observed . for an `` efimov - unfavored '' system such as rb - rb - k , the size of scattering length needed for seeing an efimov resonance in three - body recombination is too large to be realized experimentally . we therefore measure the positions of the efimov - like atom - dimer resonances instead , which could be observed at significantly lower scattering length @xcite . for a comprehensive analysis of the heteronuclear efimov resonance , we also investigate the three - body loss in @xmath0k-@xmath1rb admixtures on the negative side of the feshbach resonance . as will be shown in our later discussion , the absence of a resonance in three - body recombination in our measurements is consistent with the universal predictions @xcite . there are several experimental groups working on different isotopes for the krb mixture @xcite . we compare our results with those obtained by the jila group for @xmath2k-@xmath1rb mixture @xcite . two krb systems are nearly identical in the single channel theory : the van der waals length ( @xmath16 ) are 71.9 @xmath17 and 72.2 @xmath17 @xcite , and the scaling parameters @xmath14 for two rb atoms and one k atom are 0.6444 and 0.6536 , for @xmath0k-@xmath1rb mixture and @xmath2k-@xmath1rb mixture , respectively@xcite . however , they are different in the multi - channel theory : the mixture of @xmath0k atoms and @xmath1rb atoms both in the lowest hyperfine state have one broad resonance at 39.4 g and one intermediate resonance at 78.82 g. special treatment is needed for describing these overlapping feshbach resonances @xcite . the mixture of @xmath2k atoms and @xmath1rb atoms both in the lowest hyperfine state have one intermediate resonance at 546.9 g. this resonance is well isolated from other feshbach resonances and thus it can be simply described by the combination of @xmath6 ( @xmath18 ) and @xmath5 ( @xmath19)@xcite . the purpose of comparing these two krb systems is twofold . first , if these two systems show identical efimov structures , it is very likely that a single - channel theory can adequately explain the three - body physics in heteronuclear systems ; conversely , if the results do not match , it should be investigated whether the difference can be reproduced by using the multichannel van der waals theory introduced in the homonuclear case . the details of our experimental setup can be found in ref . @xcite . in summary , we prepared a dual - species bose - einstein condensate ( dual - bec ) comprising @xmath0k atoms and @xmath1rb atoms in a crossed optical dipole trap . both atoms are in the @xmath20 state , where @xmath21 corresponds to the atomic angular momentum and @xmath22 is its projection . the typical number of each of the @xmath1rb and @xmath0k atoms in a dual - bec is @xmath23 . when it comes to the study of inelastic atom - dimer collisions , it is necessary for dimers to be produced efficiently . this is especially true for a bose - bose mixture , in which a large contribution from atom - dimer and dimer - dimer inelastic collisions limits the efficiency with which the dimers in traps are produced . we resolve this problem by using a three - dimensional optical lattice . some preparatory steps are needed before the dual - bec can be loaded onto the optical lattice potential . first , we compensate for the differential gravitational sag between the @xmath0k and @xmath1rb atoms by introducing an additional laser beam whose wavelength is 809 nm @xcite . second , we decompress the bec by decreasing the trapping frequencies in the horizontal directions . this is necessary for increasing the number of lattice sites that have exactly one k and one rb atom when they become a dual mott insulator phase . typical trap frequencies for k and rb are @xmath24 hz and @xmath25 hz , respectively , where the @xmath26 axis is the axis of gravity . finaly , we set the magnetic field @xmath27 g , where interspecies scattering length @xmath28 . at this magnetic field , the dual bec is miscible because the interspecies scattering length is much smaller than the intra - species scattering length ( @xmath29 @xcite , @xmath30 @xcite ) . as we raised the optical lattice potential , the dual bec was transformed into a dual mott insulator . then magnetic field was swept across the feshbach resonance at 78.82 g and the atoms were adiabatically associated into molecules . for measuring the atom(rb)-dimer(krb ) loss coefficient , we selectively removed the k atoms by applying a radio - frequency spin - flip followed by a radiative force from a laser beam tuned to the closed transition for k atoms . before imaging , the atoms and molecules were spatially separated in the horizontal direction via application of a magnetic field gradient during a time - of - flight . furthermore , molecules were dissociated into atoms by sweeping the magnetic field across feshbach resonance . the typical numbers of rb atoms and krb molecules are @xmath31 and @xmath32 , respectively . k-@xmath1rb system . while the dimer - dimer ( krb - krb ) loss coefficient ( red triangles ) does not show any prominent feature , the atom - dimer ( rb - krb ) loss coefficient ( blue circles ) shows a resonant feature . lines of best fit are illustrated by solid lines . the fit for the dimer - dimer loss coefficient assumes a linear dependence on @xmath4 , while the fit for the atom - dimer loss assumes equation ( [ eq : beta_ad ] ) . the typical initial densities of the atoms and dimers are @xmath33@xmath34and @xmath35@xmath34 , respectively . the typical temperature is 60 nk . , width=302 ] the atom - dimer loss coefficient @xmath36 was determined by placing the atom - dimer mixture into a dipole trap and measuring the number of dimers and atoms after a variable holding time @xmath37 . the rate equation for the number of dimers @xmath38 can be expressed as follows : @xmath39 in this equation , @xmath40 and @xmath41 are the number of @xmath42 atoms and @xmath43 dimers , respectively ; @xmath44 and @xmath45 are the density of the @xmath42 atoms and @xmath43 dimers , respectively ; and @xmath46 and @xmath47 are the loss coefficient for the atom - dimer and dimer - dimer collisions , respectively . assuming a thermal distribution for the atoms and dimers in the dipole trap , the right - hand side of equation ( [ eq : dimer ] ) can be calculated using the number and temperature of atoms and dimers from the time - of - flight images . both @xmath36 and @xmath48 can be evaluated by comparing the experimental data from different initial conditions . can be measured exclusively . a molecular sample can be prepared by tuning the magnetic field to @xmath49 g , and then applying a magnetic field gradient . since the magnetic moment of the molecule is zero at 77 g , the k and rb atoms can be removed . from the data with finite @xmath40 , @xmath36 can be determined by subtracting the contribution from @xmath48 . ] . experimental results of the atom - dimer resonances for rb+@xmath0krb and rb+@xmath2krb collisions . these values are determined by fitting of equation ( [ eq : beta_ad ] ) . subscripts of `` sys '' and `` fit '' denote systematic error and fitting error , respectively . [ cols="^,^,^,^",options="header " , ] figure [ fig : atom_dimer ] shows the measured atom - dimer and dimer - dimer loss coefficient , @xmath50 . the magnetic field was converted into the scattering length by using the a(b ) from our multichannel two - body calculation . the calculation uses the atomic potentials in refs . @xcite and is calibrated carefully to give the correct positions of the feshbach resonances . the dimer - dimer loss coefficient @xmath51 does not show any prominent features . with a linear function , we obtained @xmath52 , where @xmath53 @xmath54/s and @xmath55 @xmath54/s . ] the resonant feature was clearly observed in the atom - dimer loss coefficient @xmath56 , and the overall shape of the resonance was quite similar to that of the @xmath1rb-@xmath2k@xmath1rb mixture . the peak position , however , was different . we can quantify the difference in the peak positions by fitting the curves with the results obtained from the effective field theory , which includes three fitting parameters ( @xmath57 , @xmath58 , @xmath59 ) . @xcite . ] @xmath60+\mathrm{sinh}^2(\eta_\ast ) } \frac{\hbar a}{m_1}\label{eq : beta_ad}\ ] ] in equation ( [ eq : beta_ad ] ) , @xmath61 represents the resonance position , @xmath62 is the resonance width , and @xmath63 is the overall magnitude of the loss . note that @xmath64 is the mass of the k atom in this case , and @xmath14 is the scaling parameter . we can fit @xmath36 using equation ( [ eq : beta_ad ] ) and compare the results with those obtained for the @xmath2k-@xmath1rb system . the results of both fits are summarized in table [ tab : fit_results ] . an isotopic comparison showed that @xmath58 matches within the error bars , while @xmath57 and @xmath59 are different between the two isotopes . the difference in @xmath63 can be attributed to the systematic uncertainty in density calibration , whereas the difference in @xmath57 comprises a fluctuation in the magnetic field and the uncertainty of @xmath4-to - b conversion . ] signifies the difference in the position of the peak . thus , it is worth asking whether the difference can be attributed to the difference of the properties of the feshbach resonances . to better understand why there was an isotopic difference , we also checked the three - body recombination rate . recent studies on the three - body recombination coefficient of the k - rb systems @xcite showed that there is no efimov - related resonance in the region of @xmath65 . furthermore , the single - channel universal theory on the heteronuclear efimov resonance for broad resonance predicts that there is no resonance in the region of @xmath66 @xcite . dependence , and the amplitude factor is determined by a fitting on the range of the positive scattering length @xmath67 . on the negative side , the amplitude factor is half of its value on the positive side @xcite . the @xmath68 dependence on the negative side was observed for @xmath69 , and it saturates because of a finite density . the typical density and temperature of the cloud are approximately @xmath70 @xmath71 and 400 nk , respectively . this corresponds to a thermal de broglie length of @xmath72 and a reciprocal wavenumber @xmath73 . , width=302 ] experimental details on how the three - body loss coefficient was measured will be presented elsewhere @xcite . measuring the three - body loss coefficient for a heteronuclear system of bosonic atoms is problematic because we have to distinguish between competing processes . in the case of the three - body loss for the @xmath0k-@xmath1rb mixture in the vicinity of the @xmath0k-@xmath1rb feshbach resonance , there are two major contributions : k - k - rb and k - rb - rb . @xmath74/s @xcite and fairly small compared with the contributions from the heteronuclear loss . ] therefore , increasing the signal - to - noise ratio of data is mandatory . in our experiment , the main source of noise in the data analysis originated from fluctuations in the initial number of atoms . we eliminated these fluctuations by taking multiple images of the same cloud using phase - contrast imaging . additionally , we enhanced the three - body loss by increasing the atomic density . hz and @xmath75 hz , respectively . the typical time scales for the loss measurement are of the order of several tens of milliseconds . ] the observed three - body loss coefficient ( fig.[fig : tbc ] ) did not show any efimov - like structures @xcite . krb and ( b ) rb+@xmath2krb ( experimental data obtained from ref . @xcite ) collisions . in both graphs , results from numerical calculations ( shown in solid lines ) show reasonable agreement with experimental results ( shown in circles and triangles ) . the result from the fit using the effective field theory ( dotted line ) is also shown in ( b ) . the temperature of each measurement was @xmath7660 nk ( solid circles ) , @xmath76150 nk ( open triangles ) , and @xmath76300 nk ( solid triangles ) , respectively . the theoretical results ( solid lines ) are multiplied by 2 and 5 , which are thermally averaged at 70 nk and 300 nk for rb+@xmath0krb and rb+@xmath2krb loss coefficients , respectively . the peak positions of theoretical curves are 395 @xmath17 and 222 @xmath17 for rb+@xmath0krb and rb+@xmath2krb loss coefficients , respectively . the resonance position in rb+@xmath0krb loss coefficients shown in ( a ) are insensitive to the temperature change in the range of temperature we study , whereas the calculated coefficients shown at 70 nk give the best agreement with the line shape of the measured loss coefficients at lower scattering lengths . , width=302 ] the significant shift in the positions of the atom - dimer resonances in the two isotopic k - rb admixtures clearly can not be explained by the small differences in the van der waals lengths or the universal scaling constants . in fact , the single - channel , universal van der waals three - body theory fails here by a large margin with an incorrectly predicted atom - dimer resonance near @xmath77100 @xmath17 . the failure of the universal theory raises the important question of whether the general universality , _ i.e. _ , the independence of efimov physics on short - range chemical forces , is still valid . our approach to address the question above is to perform three - body calculations with a spinor model that reproduces the relevant two - body feshbach spectra in each of the isotopic systems . such a model is relatively easy to build in the rb - rb - k system , because the multichannel physics is only important for the k - rb pairs , whereas a single - channel description is a good approximation for the rb - rb interaction in the whole range of magnetic field of our current interest . our theory therefore allows the k atom to carry the ( pseudo)spin degrees of freedom and treats rb atoms as spinless . specifically , the total three - body wave function @xmath78 is expanded as @xmath79 , where @xmath80 is the ( pseudo)spin state of the k atom . with the spinor model , we solve the three - body schrdinger equation in the form of @xmath81 where @xmath82 is the three - body kinetic energy operator , @xmath83 , @xmath84 , and @xmath85 the single- and multi - channel two - body potentials of the three pairs of the atoms , and @xmath86 the single - atom energy level of the k atom . the proper magnetic - field dependence of the three - body hamiltonian is built in the single - atom energy as @xmath87 , where the magnetic moment @xmath88 and the zero - field energy @xmath89 are chosen to mimic the realisitc magnectic moments and the hyperfine splittings @xcite . we solve eq . ( [ eq_3b ] ) and calculate the atom - dimer loss coefficients with essentially the same potential models and numerical techniques used in ref . @xcite . in figure [ fig : from_yujun ] we show our numerically calculated atom - dimer loss rates compared with the data from our and jila s experiments . to properly reproduce the isolated and overlapping characters of the feshbach resonances in @xmath2k - rb and @xmath0k - rb pairs , two - spin - state and three - spin - state model interactions are used for @xmath90 in the @xmath2k - rb - rb and @xmath0k - rb - rb calculations , respectively . without fitting parameters , the calculated atom - dimer resonance positions agree well with both of the experimentally observed positions , and consequently , reproduce the atom - dimer resonance shift in the isotopic systems . for the @xmath0k - rb - rb system , we point out that in order to correctly predict the atom - dimer resonance position , it is necessary for the three - body model to reproduce both the background ( 39 g ) and overlapping ( 79 g ) feshbach resonances . a model that reproduces only the local properties of the overlapping resonance ( _ i.e. _ , the local @xmath91 and @xmath11 ) does not give the atom - dimer resonance position correctly . another observation is that regardless of the number of spin states , the calculated loss rates in the two isotopic systems have similar magnitude when the scattering length is low . this suggests that the observed shift of the atom - dimer resonance position going beyond the single - channel van der waals theory is the manifestation of the difference in the underlying two - body feshbach physics . the short - range chemical forces are clearly not involved . in summary , we measured the heteronuclear atom - dimer loss coefficients of @xmath1rb atoms and @xmath0k@xmath1rb feshbach molecules at ultracold temperatures . the observed loss coefficient showed an efimov - related resonance at @xmath92@xmath93 , which shifted from previous measurements for different isotopes of potassium . to explain this shift , we modeled the system using a three - body spinor theory that reproduced the properties of feshbach resonances . this theory was successful in reproducing the experimental results of the atom - dimer resonance for both isotopes . these results show the important role of the multichannel feshbach physics in shifting the positions of the three - body efimov resonances , and demonstrates the independence of these three - body resonances on the short - range chemical forces in the heteronuclear atomic systems even near relatively narrow feshbach resonances . the authors would like to thank shimpei endo and pascal naidon for their valuable suggestions . this work was supported by jsps kakenhi grant - in - aid for scientific research(b ) , grant number jp23340117 . 42ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.117.163201 [ * * , ( ) ] @noop * * , ( ) @noop @noop * * , ( ) @noop * * , ( ) @noop _ _ , @noop ph.d . thesis , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physreva.76.020701 [ * * , ( ) ] @noop * * , ( ) * supplemental material for `` isotopic shift of atom - dimer efimov resonances in k - rb mixtures : critical effect of multichannel feshbach physics '' * k-@xmath1rb 39 g and 79 g overlapping feshbach resonances , and ( b ) near the @xmath2k-@xmath1rb 547 g @xmath3-wave feshbach resonnace . the arrows indicate the positions of the atom - dimer resonances observed in the experiments.,width=604 ]

atomic interaction models used in three-body calculations

arxiv

the author is grateful to henk taale and the _ ministry of transport , public works and water management _ for supplying the freeway data . k. nagel and m. schreckenberg , j. physique i france * 2 * , 2221 ( 1992 ) ; k. nagel and s. rasmussen , in _ artificial life iv _ , edited by r. a. brooks and p. maes ( mit press , cambridge , ma , 1994 ) ; m. schreckenberg , a. schadschneider , k. nagel , and n. ito , phys . e * 51 * , 2939 ( 1995 ) ; k. nagel , phys . e * 53 * , 4655 ( 1996 ) . m. bando , k. hasebe , a. nakayama , a. shibata , and y. sugiyama , phys . e * 51 * , 1035 ( 1995 ) . t. nagatani , j. phys . a * 28 * , l119 ( 1995 ) ; t. nagatani , physica a * 223 * , 137 ( 1996 ) . m. j. lighthill and g. b. whitham , proc . a * 229 * , 317 ( 1955 ) . p. i. richards , oper . res . * 4 * , 42 ( 1956 ) . j. payne , in _ mathematical models of public systems _ , edited by g. a. bekey ( simulation council , la jolla , ca , 1971 ) , vol r. d. khne , in _ proceedings of the 9th international symposium on transportation and traffic theory _ , edited by i. volmuller and r. hamerslag ( vnu science , utrecht , 1984 ) . b. s. kerner and p. konhuser , phys . e * 48 * , r2335 ( 1993 ) ; b. s. kerner and p. konhuser , phys . rev . e * 50 * , 54 ( 1994 ) . m. hilliges and w. weidlich , transportation research * 29 * , 407 ( 1995 ) . d. helbing , phys . e * 51 * , 3164 ( 1995 ) . w. f. phillips , transportation planning and technology * 5 * , 131 ( 1979 ) . d. helbing , phys . e * 53 * , 2366 ( 1996 ) . d. helbing , in _ traffic and granular flow _ , edited by d. e. wolf , m. schreckenberg , and a. bachem ( world scientific , singapore , 1996 ) ; d. helbing and a. greiner , modeling and simulation of multi - lane traffic flow , phys . e , submitted ( 1996 ) . d. helbing , derivation and empirical validation of a refined traffic flow model , physica a , in print ( 1996 ) . c. wagner _ _ , second order continuum traffic flow model , phys . rev . e , submitted ( 1996 ) . d. helbing , _ verkehrsdynamik . neue physikalische modellierungskonzepte _ ( springer , berlin , in preparation ) . note that the situation on european freeways is somewhat different from american ones due to other legal regulations : because of the higher speed limit ( if there is any ) , overtaking is only allowed on the left - hand lane . therefore , trucks mainly use the right lane , on which the average velocity is lower ( cf . [ f2a ] ) . d. c. gazis , r. herman , and g. h. weiss , oper . 10 * , 658 . note that a synchronized state of traffic in the sense of b. s. kerner and h. rehborn [ phys . e * 53 * , r4275 ] ( i.e. with the same velocities @xmath28 on all lanes @xmath1 ) only occurs above an average density @xmath21 of about 35 vehicles per kilometer and lane ( cf . [ f2a ] ) . d. c. gazis and r. herman , transpn . * 26 * , 223 ( 1992 ) . similar results have been obtained by j. treiterer _ _ for american freeways [ in _ proceedings of the 6th international symposium on transportation and traffic theory _ , edited by d. buckley ( sydney , 1974 ) ] . ( 9.42,2.2 ) ( 1.53,1.8)(0,0)rottepolderplein ( 2.83,1.8)(0,0)s17 ( 7.56,1.8)(0,0)badhoevedorp ( 4.51,1.3)(1,0)0.4 ( 4.51,1.1)(1,0)0.4 ( 0.3,1.3)(0,0)@xmath53 ( 0.3,1.1)(0,0)@xmath54 ( 0,1.4)(1,0)9.42 ( 0,1.2)(9.42,1.2 ) ( 1,0.5)(0,1)0.9 ( 2.06,0.5)(0,1)0.9 ( 2.56,0.5)(0,1)0.9 ( 3.01,0.5)(3.01,1.4 ) ( 3.51,0.5)(0,1)0.9 ( 4.71,0.5)(0,1)0.9 ( 6.71,0.5)(0,1)0.9 ( 7.41,0.5)(0,1)0.9 ( 7.71,0.5)(0,1)0.9 ( 8.42,0.5)(0,1)0.9 ( 1,0.2)(0,0)43.31 km ( 1.96,0.2)(0,0)42.25 ( 2.66,0.2)(0,0)41.75 ( 3.51,0.2)(0,0)40.80 ( 4.71,0.2)(0,0)39.60 ( 6.51,0.2)(0,0)37.60 ( 7.21,0.2)(0,0)36.90 ( 7.91,0.2)(0,0)36.60 ( 8.82,0.2)(0,0)35.89 km ( 0,1)(1,0)0.75 ( 0.75,1)(1,-1)0.15 ( 1,1)(1,-1)0.15 ( 1,1)(1,0)1.06 ( 2.06,1)(-1,-1)0.15 ( 2.31,1)(-1,-1)0.15 ( 2.31,1)(1,-1)0.15 ( 2.56,1)(1,-1)0.15 ( 2.56,1)(1,0)0.45 ( 3.01,1)(-1,-1)0.15 ( 3.26,1)(-1,-1)0.15 ( 3.26,1)(1,0)3.9 ( 7.16,1)(1,-1)0.15 ( 7.41,1)(1,-1)0.15 ( 7.41,1)(1,0)0.3 ( 7.71,1)(-1,-1)0.15 ( 7.96,1)(-1,-1)0.15 ( 7.96,1)(1,0)1.46

from single vehicle data a number of new empirical results about the temporal evolution , correlation , and density - dependence of macroscopic traffic quantities have been determined . these have relevant implications for traffic modeling and allow to test existing traffic models . + with the aim of optimizing traffic flow and improving today s traffic situation , several models for freeway traffic have been proposed , microscopic @xcite and macroscopic ones @xcite . only some of them have been systematically derived from the underlying laws of individual vehicle dynamics @xcite . most models are phenomenological in nature @xcite . these base on various assumptions , the correctness of which has not been carefully discussed up to now , mainly due to a lack of empirical data or difficulties to obtain them . therefore , this paper presents some fundamental empirical observations which allow to test some of the models . the empirical relations have been evaluated from single vehicle data of both lanes of the dutch freeway a9 between haarlem and amsterdam ( cf . fig . [ f1 ] ) @xcite . these data were detected by induction loops at discrete places @xmath0 below the lanes @xmath1 of the roadway and include the passage times @xmath2 , velocities @xmath3 and lengths @xmath4 of the single vehicles @xmath5 . consequently , it was possible to calculate the number @xmath6 of vehicles on lane @xmath1 which passed the cross section at place @xmath0 during a time interval @xmath7 $ ] , the _ traffic flow _ @xmath8 and the macroscopic _ velocity moments _ @xmath9^k \ , .\ ] ] if nothing else is mentioned , the interval length was chosen @xmath10min , since this allowed to separate the systematic temporal evolution of the macroscopic traffic quantities from their statistical fluctuations @xcite . the _ vehicle densities _ @xmath11 were calculated via the flow formula @xmath12 a detailled comparison with other available methods for the determination of the average velocities @xmath13 and the vehicle densities @xmath14 from single vehicle data will be given in @xcite . finally , the _ lane averages _ of the above quantities were defined according to @xmath15 @xmath16^k \ , , \ ] ] and @xmath17 where @xmath18 denotes the number of lanes and @xmath19 . therefore , we have the following relation : @xmath20 for reasons of simplicity , most macroscopic traffic models describe the dynamics of the total cross section of the road in an overall manner by equations for the density @xmath21 and the average velocity @xmath22 . however , one would expect that a realistic description requires a model of the traffic dynamics on the single lanes and their mutual coupling due to overtaking and lane - chaning maneuvers @xcite . this could cause a more complex dynamics like density oscillations among the lanes @xcite . in order to check this , we will investigate the correlation between neighboring lanes . figure [ f2 ] shows that the temporal course of the densities @xmath23 and @xmath24 is almost parallel . a similar thing holds for the average velocities @xmath25 and @xmath26 @xcite . the difference between the curves is mainly a function of density ( cf . figure [ f2a ] ) : at small densities , the vehicles can move faster on the left lane than on the right one , whereas at high densities the left lane is more crowded than the right one . in addition , figure [ f3 ] shows that the variances @xmath27 behave almost identically on the neighboring lanes ( although the order of magnitude of the average velocities @xmath28 changes considerably ) . this strong correlation between neighboring lanes probably arises from overtaking and lane - changing maneuvers . it justifies the common practice to describe the dynamics of the total cross section of the road in an overall manner . now we face the question , how a realistic traffic model must look like . due to the conservation of the number of vehicles , the dynamics of the vehicle density is given by the _ continuity equation _ @xcite @xmath29 where @xmath30 and @xmath31 are the rates of vehicles which enter or leave the freeway at on- and off - ramps , respectively . lighthill , whitham and richards have suggested to specify the flow @xmath32 in accordance with an empirical flow - density relation @xmath33 @xcite : @xmath34 this relation has been called into question , since the resulting model can not describe the emergence of phantom traffic jams or stop - and - go traffic @xcite . therefore , some researchers have introduced an additional dynamical equation for the average velocity @xmath35 which allows to describe instabilities of traffic flow @xcite . however , others have interpreted these phenomena as effects of fluctuations or of phantom bottlenecks caused by slow , overtaking vehicles like trucks @xcite . hence we check relation ( [ equi ] ) in figure [ f4 ] . it is found that ( [ equi ] ) becomes invalid above a density of about 12 vehicles per kilometer and lane , where a hysteresis effect occurs @xcite . this indicates a transition from stable to unstable traffic flow . an empirical proof of emerging stop - and - go traffic is presented in figure [ f5 ] . during the rush hours between 7:30am and 9:30am , average velocity breaks down at place @xmath36 km because of the on - ramp at @xmath37 km . nevertheless , the traffic situation recovers at the successive cross sections , i.e. average velocity increases again . in spite of this , the initially small velocity oscillations at @xmath38 km grow considerably in the course of the road . this corresponds to emerging stop - and - go traffic ( i.e. alternating periods of acceleration and deceleration ) . at the same time , the wavelength of the oscillation increases . this is in good agreement with computer simulations which show a merging of density clusters leading to larger wave lengths @xcite . after we have found that we need a dynamic velocity equation for an adequate description of the spatio - temporal evolution of traffic flow , we have to clear up the question , whether we also need a dynamic equation for the variance @xmath39 \rangle = \theta + \langle ( v_i - v)^2 \rangle\ ] ] or not . theoretical considerations on the basis of gas - kinetic approaches have shown that the velocity equation depends on the variance , for which a separate equation can be derived @xcite . nevertheless we will try out the equilibrium approximation @xmath40 where @xmath41 is the empirical variance - density relation ( cf . figure [ f6a ] ) . figure [ f6 ] shows that this approximation fits the temporal evolution of the variance in a satisfactory way as long as the average velocity @xmath22 does not rapidly change . however , when the velocity breaks down or increases , the variance shows mysterious peaks . these are a consequence of having built the temporal averages @xmath42 over finite time intervals @xmath43 , where @xmath44 with the actual velocity distribution @xmath45 . in linear taylor approximation we find @xmath46 \nonumber \\ & = & v_a(x , t ) + \frac{t}{2 } \frac{\partial v_a(x , t)}{\partial t } \ , .\end{aligned}\ ] ] since @xmath47 is varying around zero , the measured value @xmath22 fluctuates around the actual value @xmath48 . for the variance we find @xmath49 ^ 2 } \nonumber \\ & = & \overline{\theta_a } + \frac{t^2}{4 } \overline{\left ( \frac{\partial v_a}{\partial t } \right)^2 } \ , .\end{aligned}\ ] ] therefore , time averaging leads to a positive correction term which becomes particulary large , where the average velocity changes rapidly , but vanishes in the limit @xmath50 . this correction term describes the variance peaks in figure [ f6 ] quite well . consequently , the dynamics of the variance can be reconstructed from the dynamics of the vehicle density @xmath51 and the average velocity @xmath35 . summarizing our results , we were able to demonstrate the following by empirical data : 1 . the dynamics of neighboring lanes is strongly correlated so that the total freeway cross section can be described in an overall way . 2 . there is a transition from stable to unstable traffic flow at a critical density @xmath52 of about 12 vehicles per kilometer and lane . 3 . emergent stop - and - go traffic exists , so that a realistic traffic model must contain a dynamic velocity equation . 4 . the variance can be well approximated by an equilibrium relation , if corrections due to time averaging are taken into account . these conclusions seem to be also valid for other stretches of freeway systems , at least european ones . the empirical findings question the fluid - dynamic model by lighthill , whitham and richards @xcite . they are in favour of the phenomenological models by payne @xcite , phillips @xcite , khne @xcite , kerner and konhuser @xcite , hilliges @xcite as well as a recent model by helbing @xcite which has been systematically derived from the microscopic vehicle dynamics via a gas - kinetic level of description . the last of these models fits the instability region best , in particular the surprisingly low critical density @xmath52 @xcite .

acknowledgments

arxiv

in order to calculate the derivative @xmath135 one should calculate the corresponding derivatives of the function @xmath136 . so one obtains @xmath137 and @xmath138 @xmath139 we took into account that for the equilibrium value of @xmath32 , the condition @xmath140 holds , from which follows that @xmath141 . similarly @xmath142 is also equal to zero . so , one obtains : @xmath143 for integration over @xmath28 in eq . ( 6 ) , the following relations are also used : @xmath144 where we used the right hand side of eq . ( 3 ) for @xmath145 at @xmath33 . widest - label y. ishibashi , _ incommensurate phases in dielectrics _ , edited by r. blinc and a.p . levanyuk , modern problems in condensed matter sciences , vol.*14.2 * , north - holland ( 1986 ) . a.m. arutyunyan , b. brezina , s.kh . esayan , v.v . lemanov , sov . , solid state , * 24 * , 814 , ( 1982 ) . g. dolino , j.p . bachheimer , phys.stat.sol . ( a ) , * 41 * , 673 , ( 1977 ) . alexandrov , a.n . vtyurin , v.f . shabanov , sov . phys . , pisma jetp , * 28 * , 153 , ( 1978 ) . golovko , a.p . levanyuk , sov . phys . , jetp , * 50 * , 780 , ( 1979 ) . m. iizumi , j.d . axe , g. shirane and k. shimaoka , phys . rev . * b 15 * , 4392 ( 1977 ) . sannikov , a.p . levanyuk , sov . phys . , solid state , * 4 * , 1005 , ( 1978 ) . i.e. dzyaloshinski , sov . jetp , * 47 * , 336 , 992 , ( 1964 ) . golovko , sov . , solid state , * 22 * , 2960 , ( 1980 ) .

we show that a ferroelectric phase transition takes place in the incommensurate phase of the @xmath0 crystal . the ferroelectric character of the ic phase explains the second - harmonic generation observed in the corresponding temperature range . the crystals of the @xmath1 family ( about 20 crystals ) @xcite undergo two successive phase transitions on cooling : an incommensurate ( ic ) transition at temperature @xmath2 with a modulation vector near the @xmath3 point in the brillouin zone , and upon further cooling at temperature @xmath4 a commensurate lock - in transition to the triple - period ferroelectric phase with the modulation vector @xmath3 . the most striking peculiarity of the @xmath0 crystal , which belongs to the @xmath1 family , is the second - harmonic generation ( shg ) in the ic phase,@xcite of the same intensity magnitude as that in the @xmath3- phase . low temperature @xmath3- phases of the @xmath1 type crystals are improper ferroelectrics , and therefore the shg is a normal manifestation of this feature . however , so far there was neither pointed out a plausible reason for the shg in the ic phase of @xmath0 , nor explained why it takes place not in all the crystals of the @xmath1 family . it is worth noting that the shg was observed also in the ic phase of quartz by dolino and bachheimer ( 1977)@xcite and in the ic phase of the ammonium fluoroberyllate @xmath5 by alexandrov et al . ( 1978)@xcite , and both observations also can be explained using the theory developed below . the attempt to explain the shg in the ic phase of @xmath5 was carried out by golovko and levanyuk@xcite based on the spatial dispersion of the dielectric constant . however , the expected effect appeared to be small , and besides , it is not explained why in such a case the shg is not observed in all the crystals with the ic phases . in the present paper we show that within the ic phase of the @xmath0 crystal , necessarily a transition to the ferroelectric ic phase takes place . we analyze the exact solution in frame of the landau theory . the physical reason for such a ferroelectric transition is the coupling between the crystal s polar symmetry vibrational mode @xmath6 and some displacements of the ic domain walls . we demonstrate that relative displacements of the domain walls in the domainlike ic structure induce polarization along the @xmath7 direction , and can be viewed as an additional phonon branch of a polar symmetry . this mode is not explicitly introduced in the equations below . it is a component of the ic phase @xmath8 function , and is taken into account implicitly , in the integral form . the coupling of the two modes renormalizes the frequency of the lower mode down to zero within the ic range , which induces a ferroelectric transition . structure of the ic modulation of @xmath0 was defined in the neutron diffraction by iizumi _ et al_.@xcite the thermodynamic potential describing the ic phase of @xmath9 was detailed studied by sannikov and levanyuk.@xcite for the two - component order parameter with components @xmath10 and @xmath11 it can be written as : @xmath12 where @xmath6 is polarization vector in the @xmath7 direction , @xmath13 is electric field in the @xmath7 direction . in case of @xmath14 this potential would describe a phase transition from the symmetric high - temperature phase to the triple - period commensurate phase at the temperature @xmath15 . for @xmath16 the equilibrium values of the modulation phase @xmath17 are @xmath18 ( otherwise @xmath19 ) . however , in virtue of the lifshitz invariant ( @xmath20 ) the phase transition takes place from the high - temperature phase to the ic phase at @xmath21 with @xmath22 and @xmath23 . substituting @xmath24 in potential ( 1 ) one obtains for the @xmath25- and @xmath6- dependent part of potential ( 1 ) the following expression : @xmath26 minimization of this expression with respect to @xmath25 gives the equation : @xmath27 after single integration over @xmath28 and taking for the integration constant the form @xmath29 one obtains : @xmath30 where @xmath31 . the integration constant @xmath32 , as well as the polarization vector @xmath6 , can be defined via minimization of the potential with respect to these parameters after substituting the solution of eq . ( 3 ) in the potential ( 2 ) . prior to considering the ferroelectric transition in this system let us discuss the solutions of eq . ( 3 ) for the case of @xmath33 . the solution to this kind of equations for the ic structures was analyzed in the pioneer work by dzyaloshinski @xcite ( it was also detailed studied by golovko@xcite ) , and for eq . ( 3 ) , it has the form : @xmath34 where @xmath35 is the _ elliptic amplitude function _ and @xmath36 is the ic vector at the temperature of the ic transition . ( in the present paper the notations of the wolfram research , `` mathematica '' program for the elliptic functions are used . ) substituting solution ( 4 ) in the potential ( 2 ) and minimizing over the parameter @xmath32 one obtains : @xmath37 is the _ complete elliptic integral of the second kind _ , which has the asymptotic + @xmath38 eq . ( 5a ) can be numerically solved with respect to @xmath32 for given coefficients @xmath39 , and one obtains that the parameter @xmath32 monotonously decreases from @xmath40 at @xmath41 ( i.e. , at the ic transition @xmath2 ) down to @xmath42 at the continuous lock - in transition @xmath43 . as it follows from eq . ( 5a ) , at the lock - in transition point the ic amplitude is tending to @xmath44 which gives that @xmath45 , if the remaining coefficients are of the same order of magnitude . in fig . 1(a ) the solutions given by eq . ( 4 ) for @xmath46 , @xmath47 and @xmath33 for the three values of the ic amplitude are plotted . at the ic transition @xmath2 ( @xmath48 ) the corresponding graph must look like a straight line with slope @xmath36 ( @xmath49 , i.e. , sinusoidal modulation ) , and on cooling it acquires the domain - like features . due to the symmetry , each point @xmath28 on the curves of fig . 1(a ) is associated with a nonzero polarization , which is spatially distributed as @xmath50 with zero - integral effect . meanwhile , each domain there has a nonzero - integral polarization . in the areas of any of two neighboring domains @xmath51 and @xmath52 respectively , giving rise to the @xmath6-s of opposite signs . in fig . 1(b ) the transformation of the curves of fig . 1(a ) under the applied electric field @xmath53 is shown . the field @xmath53 is taken into account via replacing in eq . ( 3 ) of the polarization vector @xmath6 by @xmath54 . an identical transformation of the domains takes place in the presence of nonzero polarization @xmath6 in crystal ( when @xmath55 ) . corresponding solutions were found both analytically and numerically ( using the `` mathematica '' software ) with the same result . note that the integral polarization vector contributed by the nonequal domains in fig . 1(b ) is not zero . now we show , that the above described transformation of the domain structure from that shown in fig . 1(a ) to fig . 1(b ) takes place spontaneously , i.e. , without applying an external electric field . in other words , there takes place a ferroelectric phase transition inside the ic phase . let us calculate the dielectric susceptibility of the crystal in the ic phase : @xmath56 differentiation of potential ( 2 ) gives : @xmath57 @xmath58 is the ic period , given by eq . ( 10 ) below . in order to calculate the derivative @xmath59 , we introduce notation @xmath60 integration of eq . ( 3 ) gives in this notation , @xmath61 , from which follows that @xmath62 . the latter allows one to calculate the derivative @xmath63 via calculation of the corresponding derivatives of the function @xmath64 in the integral form ( see the appendix ) . after substituting of the calculated derivative in eq . ( 6 ) , one obtains : @xmath65 where @xmath66 is a _ hypergeometric function_. this hypergeometric function has the asymptotic @xmath67 which well fits the function in all the range @xmath68 , i.e. , not only at @xmath69 . as one can see from eq . ( 10 ) , the ic period has an asymptotic : @xmath69 , @xmath70 ( as shown below , on approaching the lock - in temperature @xmath43 the coefficient @xmath32 is tending to zero as @xmath71 ) . so , for the susceptibility @xmath72 one obtains : @xmath73 and the stability coefficient @xmath74 necessarily vanishes ( since the @xmath32 -dependent term in eq . ( 8) is negative and diverging with @xmath69 ) , inducing ferroelectric phase transition at the temperature @xmath75 . a loss of stability necessarily takes place even for the case of large @xmath76 ( i.e. , even for a not soft polar mode of the high temperature phase ) , and it is expected in the medium temperature range of the ic phase ( @xmath77 ) . however , some alternative situations are possible near the loss of stability @xmath75 . as it follows from eq . ( 3 ) , the polarization @xmath6 which appears below the transition should be sufficiently small : @xmath78 otherwise , the expression under the root in eq . ( 3 ) takes negative values for some @xmath17 , what means that the crystal does not fall within the ic phase , but within the commensurate lock - in phase instead , skipping the ferroelectric ic phase . in terms of fig . 1(b ) , it is equivalent to the infinite increasing of the one domain s size , which covers all of the crystal area . the right hand side of eq . ( 9 ) is very small , since @xmath79 in eq . ( 9 ) is a small parameter , and @xmath32 tends to zero in the low - temperature range of the ic phase ( see also below ) . in the case of a first - order ferroelectric phase transition giving rise to a large @xmath6 ( in landau theory , it means a negative fourth - order term in the potential s @xmath6- expansion ) , inequality ( 9 ) does not hold , and the transition transfers the crystal directly to the lock - in phase , skipping the ferroelectric ic phase . even for some order - disorder transitions , which are very close to the second - order type ( the polarization vector @xmath6 below the order - disorder transitions is larger , due to the small curie constant compared to that for the displacement type ) the expected ferroelectric transition is more likely to turn into the transition to the lock - in phase , skipping the ferroelectric ic phase . note that among the crystals of the @xmath1 family only @xmath0 is of a displacement type , and therefore for this crystal the ferroelectric phase transition ( from the ic to the ferroelectric ic phase ) can be considered as a most likely . in other words , the ic structure of @xmath0 becomes ferroelectric with the modulation , spatially distributed like that in fig . 1(b ) . one should expect that the ferroelectric transition takes place in the beginning of the domainlike modulation formation , i.e. , at temperatures near the curve i in fig . 1(a ) , when the @xmath32 coefficient falls below unity . as it follows from eqs.(5a , 5b ) , for estimates of inequality ( 9 ) , the asymptotic of the parameter @xmath69 can be used : @xmath80 near the lock - in point we presented @xmath81 , where @xmath82 is some constant . so , eq . ( 9 ) can be presented as @xmath83 , when @xmath84 ( we neglected a logarithmically diverging term @xmath85 , which does not change order of magnitude of the estimated values , since , as discussed below , @xmath86 can not be sufficiently small for it ) . for obtaining of the inharmonic fourth order term in the @xmath6 expansion of the potential one should calculate the derivative @xmath87 we shall introduce here only the result of this calculation . in addition to the @xmath88 term in eq . ( 2 ) , one obtains a negative , strongly diverging as @xmath89 with @xmath69 term . if the sum of the inharmonic @xmath90 terms becomes negative earlier than @xmath75 , then takes place a first - order transition . the latter makes more probable the first order transition , even if the coefficient @xmath91 is not small . now we shall discuss some features of the ic phase for @xmath33 case in order to compare it with behavior of the ferroelectric ic phase . to study the temperature evolving of the ferroelectric ic structure upon cooling one should minimize potential ( 2 ) also over the ic amplitude @xmath92 . however , we imply in the present paper only increasing of the ic amplitude @xmath92 upon cooling , and do not discuss the character of the lock - in transition . first we discuss some structure peculiarities of the non - ferroelectric ic phase of @xmath0 . the length of each plateau in fig . 1(a ) is six times shorter than the ic wavelength @xmath93 , where @xmath94 is the ic wavevector [ @xmath94 is equal to the average slope of the given curve in fig . 1(a ) ] . from this geometrical fact it follows , that the ic modulation argument @xmath31 can be presented as @xmath95 , where @xmath96 is a periodic function with period @xmath97 . as it follows from eq . ( 4 ) , roughly @xmath96 can be approximated by @xmath98 . the ic satellite reflections observed in the x - ray diffraction are induced by the fourier - components of the modulation function @xmath99 $ ] . the ic satellites observed for such a modulation should appear in the points @xmath100 ; @xmath101 near the main bragg- reflections @xmath102 . in other words , the manifestation of the domain - like ic structure in @xmath0 in diffraction should be only the simultaneous observation of the ic satellites of about the same intensity ( asymmetrically disposed with respect to the @xmath103 point ) in the points @xmath104 and @xmath105 near @xmath106 . intensive ic satellites in the points @xmath107 , which are observed in diffraction , are not contributed by the domain features of the modulation , and can not be viewed as a manifestation of a domainlike ic structure . the satellites @xmath104 and @xmath105 were not observed for @xmath0 , which proves that the domainlike structure is not well developed in @xmath0 , and the ferroelectric phase exists only in the medium range of the ic phase . so , as it follows from the exact solution of eq . ( 3 ) , any crystal of the @xmath1 type with well developed ic domain structure ( i.e. , sufficiently small @xmath32 ) should be ferroelectric . the ferroelectric transition does not change the translation symmetry of the ic phase , and therefore the satellites disposition for the ferroelectric ic phase should not essentially differ from that for the @xmath33 case . we also discuss the character of increasing of the ic modulation period on cooling , which is logarithmically diverging , as it follows from eq . ( 4 ) . the ic period for the modulation given by eq . ( 4 ) is @xmath108 where @xmath109 is the _ complete elliptic integral of the first kind_. with @xmath69 the function @xmath110 has the asymptotic @xmath111 and @xmath112 which is logarithmically diverging . the temperature @xmath113 can approach the lock - in temperature @xmath43 not closer than the temperature fluctuation @xmath114 , i.e. , @xmath115 where @xmath116 is the crystal volume and @xmath117 is the heat capacity . so , for any crystal of a finite size ( @xmath118 ) , the period @xmath119 can increase only about 10 times compared to @xmath120 , in virtue of the logarithmical character of the divergence ( it is also confirmed by our numerical simulations ) . besides , in the real experiment the temperature @xmath113 can not approach the lock - in temperature @xmath43 even closer than the instrumental temperature resolution @xmath121 , which does not allow to observe any diverging ic period , and the lock - in transition always should be observed as a discontinuous phase transition with jump in the wavevector @xmath94 down to @xmath122 . for calculation of the ferroelectric ic modulation s period [ the @xmath28-length of the six neighboring domains in fig . 1(b ) ] , one should integrate eq . ( 3 ) from @xmath123 through @xmath124 , which gives : @xmath125 @xmath126\;\,\end{aligned}\ ] ] where @xmath127 is the _ elliptic integral _ , and @xmath128 , i.e. , the condition @xmath129 coincides with eq . ( 9 ) , and therefore it always holds in the ic phase . the parameter @xmath130 decreases down to zero with increasing of the polarization vector @xmath6 on cooling . integral ( 11 ) was calculated exactly , and after being simplified to the form of eq . ( 12 ) for @xmath131 . the first term in eq . ( 12 ) , which gives the size of the larger domains in fig . 1(b ) , is diverging with @xmath132 as @xmath133 . meanwhile , the size of the smaller domains is given by the second term in eq . ( 12 ) , and it is always finite , though it also increases with cooling . summarizing , we showed that in the ic phase of the @xmath9 crystal takes place ferroelectric phase transition , which explains the shg in the ic phase of this crystal . formation of the ferroelectric ic phase is most likely near the displacement - type phase transitions , though it can occur also in those of the order - disorder type . the ic period increases on cooling towards the lock - in temperature @xmath43 as @xmath134 , though the size of the domains , polarized opposite to the prevailing polarization , remains finite .

calculation of @xmath135

arxiv

the migdal and liashberg theory of `` strong - coupling superconductivity''@xcite has been very successful in qualitative description of superconductivity in real materials@xcite . for example , it explains the deviation from the universal value @xmath0 in bardeen - cooper - schrieffer ( bcs ) theory , or the dependence of critical magnetic field on the transition temperature . moreover , in some situation , it is known that the strong - coupling effect is not only a quantitative but also a qualitative effect ( e.g. , refs . the strong - coupling effect modifies the spectrum of the quasiparticles . therefore , one can expect that it may change the structure of low - energy states within the vortices in type - ii superconductors . as far as we know , however , there have been only a few studies of the strong - coupling effect on a vortex on the basis of microscopic theories . two - dimensional chiral _ p_-wave superconductivity is considered to be realized in sr@xmath1ruo@xmath2 @xcite . this state is topologically non - trivial and attracts much attention in these days . within the vortices of this superconductor , reflected in the topology of this system , there is a zero - energy bound state , which is expected to be very robust against not - so - strong impurities@xcite . recently , the relationship between this robustness and the odd - frequency pairing also has been discussed@xcite . in the present paper , we calculated the self - consistent liashberg equation to study how the strong - coupling feature affects the vortex of a chiral _ p_-wave superconductor microscopically . we also calculated the free - energies of the vortices and discuss its stability . in this study , we consider an isolated vortex in the two - dimensional spinless chiral _ p_-wave superconductor with isotropic fermi surface . we use quasiclassical theory@xcite ; we assume that the product of the coherence length of the superconductor @xmath3 and the fermi wavevector @xmath4 is much larger than unity . the quasiclassical green s function @xmath5 is a @xmath6 matrix and obeys the eilenberger equation @xmath7&= \check 0 \label{eq : eilenberger } , \end{aligned}\ ] ] where @xmath8 are the matsubara frequencies , @xmath9 is a fermi velocity , @xmath10 denotes a direction of momentum on the fermi surface such that @xmath11 , @xmath12 ( @xmath13 ) are the pauli matrices , @xmath14 is the elementary charge , @xmath15 is the speed of light , @xmath16 is the vector potential , and @xmath17 is the self - energy . the quasiclassical green s function satisfies the normalization condition @xmath18 and its bulk value is @xmath19 to incorporate strong - coupling effect , we use liashberg equation to calculate the self - energy @xmath20 from the quasiclassical green s function ; @xmath21 where @xmath22 is the cutoff of the matsubara frequencies , @xmath23 is the density of states on the fermi level , and @xmath24 denotes the average over the fermi surface and is defined @xmath25 . we took 47 equally spaced points in the momentum space . we assume that the interaction between electrons @xmath26 has the following form : @xmath27 where @xmath28 is a characteristic frequency of a mediated boson , and @xmath29 is a constant parameter . we set these parameters so that @xmath30 , where @xmath31 is the critical temperature of the superconductivity . we choose this value so that the strong - coupling effect is very large but not unrealistic ) is about @xmath32 at @xmath33 . for example , cecoin@xmath34 exhibits such a large value ( @xmath35 ) @xcite . ] . we set the cutoff of the matsubara frequencies @xmath36 , and confirm that this cutoff is considered to be sufficiently large by comparison of the magnitude of the bulk pair - potential with those for @xmath37 and @xmath38 . we also define @xmath39 and use it as a characteristic length of the spatial modulation of the self - energy . the vector potential @xmath16 is obtained from the quasiclassical green s function as @xmath40 where @xmath41 is a trace over the nambu space . we define @xmath42 as a characteristic length of the electromagnetic entities . we set @xmath43 in this paper . to discuss the stability of the isolated vortices , we calculated the free - energy deviation from the normal state @xmath44 with the following equation : @xmath45 where @xmath46 is the magnetic field , and @xmath47 is a solution of @xmath48&= \check 0 , & \check g_s^2=-{\pi}^2\check\tau_0.\end{aligned}\ ] ] the above expression of @xmath44 is a simple extension of the weak - coupling bcs one@xcite . we used the 15-points gauss - kronrod quadrature formula to integrate respect to @xmath49 . we numerically confirmed that the self - energy for matsubara - frequencies @xmath50 can be decoupled as @xmath51 \label{eq : self - energy - form } , \end{aligned}\ ] ] and thus we show only the @xmath52-dependent part @xmath53 and @xmath54 in the following section . at sufficiently far from the vortex , only @xmath55 or @xmath56 survives . hereafter we assume that @xmath55 is a dominant part of the self - energy and survives in the bulk . as we can see in , cooper pair of chiral _ p_-wave superconductivity has internal angular momentum ( chirality ) . if there is a vortex , two types of vortices can exist in this system ; one type of vortex has vorticity ( the angular momentum of vortex ) parallel to the chirality , and the other type has vorticity anti - parallel to the chirality . in the present paper , we call the former `` parallel vortex '' and the latter `` anti - parallel vortex '' . in the ginzburg - landau(gl ) theory , an anti - parallel vortex was shown to be more stable than a parallel vortex@xcite . to solve , we used so - called riccati - parametrization method@xcite and solved the parametrized differential equation with a 4th- and 5th - order adaptive runge - kutta method . we used the cylindrical coordinate system and took 48 equally spaced points on the azimuthal coordinates . to improve the accuracy of numerical integration of the free - energy , we used the composite gauss - lobatto quadrature to choose discrete points @xmath57 on the radial line . we divided the closed interval @xmath58 $ ] into 16 subintervals , applied the 7-points gauss - lobatto formula to each subinterval and obtained 97 discrete points @xmath59 , and changed the variable as @xmath60 in order to make the sampling points denser near the center and more sparse far from the vortex . we calculated the self - energy from the quasiclassical green s functions via and iterated the above until the self - energy sufficiently converged . after obtaining converged solution , we calculated the free - energy of vortices using . we changed initial profiles of the vortices so that the initial dominant- and induced- vortices were at separate positions , and repeated the same procedure as the above . finally , we compared the resultant profiles and their free - energies to discuss stability . and @xmath61 . left - top : amplitude of dominant part ( @xmath62 ) , right - top : phase of dominant part ( @xmath63 ) , left - down : amplitude of induced part ( @xmath64 ) , right - down : phase of induced part ( @xmath65 ) . ( color figure online),title="fig : " ] and @xmath61 . left - top : amplitude of dominant part ( @xmath62 ) , right - top : phase of dominant part ( @xmath63 ) , left - down : amplitude of induced part ( @xmath64 ) , right - down : phase of induced part ( @xmath65 ) . ( color figure online),title="fig : " ] and @xmath61 . left - top : amplitude of dominant part ( @xmath62 ) , right - top : phase of dominant part ( @xmath63 ) , left - down : amplitude of induced part ( @xmath64 ) , right - down : phase of induced part ( @xmath65 ) . ( color figure online),title="fig : " ] and @xmath61 . left - top : amplitude of dominant part ( @xmath62 ) , right - top : phase of dominant part ( @xmath63 ) , left - down : amplitude of induced part ( @xmath64 ) , right - down : phase of induced part ( @xmath65 ) . ( color figure online),title="fig : " ] as for the anti - parallel vortices , we only obtained circular axisymmetric vortices for all temperature that we studied ( @xmath66 , @xmath67 , @xmath68 , @xmath69 , and @xmath70 ) , regardless of the initial profiles ; in this case , the strong - coupling effect just modifies the shape of the vortex . on the other hand , at moderately low temperatures , a non - axisymmetric solution emerges for parallel vortices , when the initial vortex sufficiently breaks the axisymmetry . figure [ fig : profile - of - non - axisymmetric - parallel - vortex ] shows the non - axisymmetric profile of dominant and induced parts of off - diagonal self - energy at @xmath71 and @xmath72 . there the vortex of dominant component forms triangle and those of the induced component split into three . figure [ fig : current - in - non - axisymmetric - parallel - vortex ] shows the current density around the vortex . we can confirm that the electromagnetic quantities also break the axisymmetry . when we calculated parallel vortices at @xmath73 , we found only an axisymmetric vortex : both circular and non - circular initial configurations of self - energy produce the same result . we thus conclude that unusual parallel vortices may not exist at high temperatures . in figure [ fig : free - energy - for - parallel - vortex ] , we plot the free - energy of each vortex . we can see that at low temperatures , the non - axisymmetric vortex is more stable than symmetric one . we note that the symmetric anti - parallel vortex is more stable than the non - axisymmetric parallel vortex , at least under the parameters in this study . there are many studies of non - axisymmetric vortices in spin triplet superfluids or superconductors with an isotropic fermi surface . however , many of them have targeted vortices in the superfluid @xmath74he - b@xcite , or an _ f_-wave superconductor similar to the @xmath74he - b@xcite ; these studies therefore can not be compared with our work directly . tokuyasu , _ et al . _ have studied two - dimensional chiral _ p_-wave superconductor within the gl theory in the weak- to strong - coupling regimes@xcite . they have reported that non - axisymmetric vortices can emerge in some non - weak - coupling coefficients . however , the coefficients of the gl - functional of our target system fall into the same ones that we obtain in the weak - coupling limit ( the @xmath75 parameter in ref . @xcite is 0.5 in our system ) . thus , the origin of non - axisymmetric vortices in our work is different from that of the previous work . this is also consistent with the fact that non - axisymmetric vortices only exist at low temperatures in the present work . aoyama and ikeda have reported that a vortex of @xmath74he - a can be non - axisymmetric under the existence of anisotropic scatterers@xcite . their model is different from ours , and the relationship between their and our results considered an important but remaining issue . in this study , we numerically found that a non - axisymmetric vortex metastably exists in strong - coupling chiral _ p_-wave superconductors . this anomalous vortex is more stable than the axisymmetric parallel one at sufficiently low temperatures , but symmetric anti - parallel vortex is still most stable . the emergence of this anomalous vortex is a consequence of the strong - coupling effect because we did not obtain such a vortex with the conventional weak - coupling gap equation . to clarify the underlying energetics that makes the non - axisymmetric vortex metastable is an interesting issue . the total phase diagram of this system is also left as a future issue . d. j. scalapino , in _ superconductivity _ , edited by r. d. parks ( marcel dekker , inc . , new york , 1969 ) p. 449 . w. l. mcmillan , j. m. rowell , in _ superconductivity _ , edited by r. d. parks ( marcel dekker , inc . , new york , 1969 ) p. 561 . j. p. carbotte , rev . phys . * 62 * , 1027 ( 1990 ) . y. maeno , h. hashimoto , k. yoshida , s. nishizaki , t. fujita , j. g. bednorz , f. lichtenberg , nature ( london ) * 372 * , 532 ( 1994 ) . mackenzie , y. maeno , rev . * 75 * , 657 ( 2003 ) . m. sigrist , prog . suppl . * 160 * , 1 ( 2005 ) . y. maeno , s. kittaka , t. nomura , s. yonezawa , k. ishida , j. phys . jpn . * 81 * , 011009 ( 2012 ) . g. e. volovik , jetp lett . * 70 * , 609 ( 1999 ) . m. matsumoto , m. sigrist , physica b * 281&282 * , 973 ( 2000 ) . y. kato , j. phys . 69 * , 3378 ( 2000 ) . n. hayashi , y. kato , m. sigrist , j. low temp . phys . * 139 * , 79 ( 2005 ) . y. tanuma , n. hayashi , y. tanaka , a. a. golubov , phys . lett . * 102 * , 117003 ( 2009 ) . m. eschrig , j. a. sauls , new . j. phys . * 11 * , 075008 ( 2009 ) . n. kurosawa , n. hayashi , e. arahata , y. kato , j. low temp . phys . * 175 * , 365 ( 2013 ) . n. kurosawa , n. hayashi , y. kato , j. phys . 84 * , 114710 ( 2015 ) . k. k. tanaka , m. ichioka , s. onari , phys . b * 93 * , 094507 ( 2016 ) . e. v. thuneberg , phys . lett . * 56 * , 359 ( 1986 ) . e. v. thuneberg , phys . b * 36 * , 3583 ( 1987 ) . m. m. salomaa , g. e. volovik , phys . lett . * 56 * , 363 ( 1986 ) . m. fogelstrm , j. kurkijrvi , j. low temp . phys . * 98 * , 195 ( 1995 ) . y. tsutsumi , t. kawakami , k. shiozaki , m. sato , k. machida , phys . b * 91 * , 144504 ( 2015 ) . m. a. silaev , e. thuneberg , m. fogelstrm , phys . lett . * 115 * , 235301 ( 2015 ) .

we studied strong - coupling effect upon an isolated vortex in a two - dimensional chiral _ p_-wave superconductor . we solved the eilenberger equation for the quasiclassical green s functions and the liashberg equation with single mode einstein boson self - consistently . we calculated the free - energy of each obtained vortex , and found that a non - axisymmetric vortex metastably exists in some situation .

introduction methods results and discussion conclusion

arxiv

understanding the origin of unconventional superconductivity in strongly correlated electron materials continues to be at the forefront of modern condensed matter physics @xcite . in copper oxide @xcite , iron pnictide @xcite , and heavy fermion @xcite superconductors , the appearance of a neutron spin resonance below the superconducting transition temperature @xmath7 suggests that spin - fluctuation mediated pairing is a common thread for different families of unconventional superconductors @xcite . the neutron spin resonance is a collective magnetic excitation coupled to superconductivity with a temperature dependence similar to the superconducting order parameter @xcite . it is located near the antiferromagnetic ( af ) ordering wave vector @xmath8 of the undoped parent compound and its energy @xmath9 at @xmath8 is related to either @xmath7 @xcite or the superconducting energy gap @xmath10 @xcite . although it is generally accepted that the resonance is a signature of unconventional superconductivity @xcite , there is no consensus on its microscopic origin . a common interpretation of the resonance is that it is a spin - exciton , arising from particle - hole excitations involving momentum states near the fermi surfaces that possess opposite signs of the @xmath0 ( or @xmath1)-wave superconducting order parameter @xcite . alternatively it has also been proposed to be a magnon - like excitation @xcite . at present , there is no consensus on its microscopic origin @xcite . in hole - doped copper oxide superconductors , the magnetic excitations has an hourglass dispersion with a downward dispersion at energies below @xmath9 and an upward magnon - like dispersion at energies above @xmath9 @xcite . the resonance , on the other hand , obtained by subtracting the normal state magnetic excitations from those in the superconducting state , displays predominantly a downward dispersion @xcite . in the case of ni - underdoped bafe@xmath11as@xmath11 with coexisting af order and superconductivity @xcite , the resonance only reveals an upward magnon - like dispersion @xcite . in both cases the resonance is well described by the spin - exciton scenario , the opposite dispersions being a result of @xmath6 or @xmath12 symmetry of the superconducting order parameter @xcite . for the heavy fermion superconductor cecoin@xmath4 ( @xmath13 k ) @xcite , the resonance appears below @xmath7 at @xmath14 mev and the commensurate af wave vector @xmath15 in reciprocal space @xcite . since cecoin@xmath4 has a superconducting gap with @xmath6-wave symmetry as determined from scanning tunneling microscopy ( stm ) experiments @xcite , the resonance is expected to show a downward dispersion similar to the cuprates within the spin - exciton picture @xcite . alternatively , the resonance , with its three - dimensional character @xcite , could be a magnon - like excitation of @xmath16 electrons that becomes visible due to its reduced decay rate in the superconducting state @xcite . in this case , the resonance is not a signature of @xmath6-wave superconductivity , but a measure of the hybridization between @xmath16 electrons and conduction electrons and its associated pairing - sensitive landau damping @xcite . yb@xmath17coin@xmath18 . * ( a ) crystal structure of ce@xmath2yb@xmath3coin@xmath18 . ( b ) @xmath19 $ ] scattering plane , where @xmath20 is measured from @xmath8 via @xmath21 . the red and green arrows represent scans along @xmath22 $ ] and @xmath23 $ ] centered at @xmath8 , respectively . ( c ) @xmath24 $ ] scattering plane . here scans along @xmath25 $ ] centered at @xmath8 can be carried out as indicated by the blue arrow . ( d ) dispersion of the resonance along @xmath23 $ ] . the axis above the figure is @xmath26 in r.l.u . whereas the axis at the bottom is @xmath20 in @xmath27 . an isotropic dispersion @xmath28 ( @xmath29 mev , @xmath30 mev@xmath31 ) is shown as a cyan solid line , where @xmath10 represents a spin gap and @xmath32 is the effective spin wave velocity . the horizontal bars represent experimentally observed peak full - width - at - half - maximums ( fwhm ) . the dashed vertical lines indicate the ordering wave vector of the so - called @xmath33 phase at @xmath34 with @xmath35 @xcite . ( e ) and ( f ) are similar to ( d ) but are for dispersions along @xmath25 $ ] and @xmath22 $ ] , respectively . ( g ) the fermi surfaces of cecoin@xmath18 where the blue and red shading represent the _ d_-wave symmetry of the superconductivity order parameter . the black arrow is @xmath8 that connects parts of fermi surfaces with sign - reversed superconductivity order parameters . ( h ) color - coded calculated intensity along the @xmath36 $ ] direction by considering the resonance mode to be a spin - exciton . ( i ) calculated intensity for the spin - exciton along the @xmath37 $ ] direction . ( j ) comparison of dispersions of the resonance in ce@xmath38yb@xmath17coin@xmath18 ( solid cyan line ) and spin waves in cerhin@xmath18 ( dashed purple and orange lines ) @xcite . ( k ) calculated intensity of the resonance along the @xmath36 $ ] direction assuming it is a magnon - like excitation . dispersion of the magnon - like excitations are obtained from fits to experimental data and the intensity is affected by damping due to the particle - hole continuum . ( l ) calculated intensity for the magnon - like excitation along the @xmath37 $ ] direction . ] when la is substituted for ce in ce@xmath2la@xmath3coin@xmath4 @xcite , superconductivity and the energy of the resonance are both rapidly suppressed while @xmath39 remains approximately constant , where @xmath40 is the boltzmann constant @xcite . at the same time , the energy width of the resonance broadens with increasing la - doping @xcite . when yb is doped into cecoin@xmath4 to form ce@xmath2yb@xmath3coin@xmath4 superconductivity is suppressed much slower @xcite . with increasing yb , de haas - van alphen and angle resolved photo emission studies find a change in the fermi - surface topology between yb nominal doping levels of @xmath41 and 0.2 @xcite . in addition , london penetration depth measurements suggest that the superconducting gap changes from nodal to nodeless around a similar yb doping level @xcite , arising possibly from composite electron pairing in a fully gapped superconductor for @xmath42 @xcite . if the resonance in cecoin@xmath4 is a spin - exciton , it should be dramatically affected by the yb - doping induced changes in fermi surface topology and superconducting gap . on the other hand , if the resonance is a magnon - like excitation , it should be much less sensitive to yb - doping across @xmath43 and display a upward dispersion similar to spin waves in antiferromagnetically ordered nonsuperconducting cerhin@xmath4 characteristic of a robust effective nearest - neighbor exchange coupling , regardless of its itinerant electron or local moment origin @xcite . here we use inelastic neutron scattering to demonstrate that the resonance in the heavy fermion superconductor ce@xmath2yb@xmath3coin@xmath4 with @xmath5 and @xmath44 k , respectively [ methods section and supplementary figure 1 ] @xcite has a dominant ring - like upward dispersion that is robust against yb - doping and the concomitant changes in electronic structure , a feature not present in the spin - exciton scenario . moreover , a downward dispersion expected in the spin - exciton scenario is not observed . the robust upward dispersion of the resonance suggests it may have a magnon - like contribution @xcite . specifically we find that the resonance in ce@xmath38yb@xmath17coin@xmath4 displays an upward dispersion along the @xmath23 $ ] , @xmath25 $ ] , and @xmath22 $ ] directions as shown in fig . 1(d ) , 1(e ) , and 1(f ) , respectively . upon increasing yb - doping to @xmath45 , the energy of the resonance at @xmath8 decreases corresponding to the reduction in @xmath46 [ supplementary figure 2 ] , but the overall dispersion and location of the mode in reciprocal space remain unchanged . upward dispersions similar to ce@xmath38yb@xmath17coin@xmath4 are also found in undoped cecoin@xmath4 and ce@xmath47yb@xmath48coin@xmath4 [ supplementary figures 3 , 4 and 5 ] . using the electronic structure and the momentum dependence of the @xmath6-wave superconducting gap determined from stm for cecoin@xmath4 [ fig . 1(g ) ] @xcite , we calculate the feedback of superconductivity on the magnetic excitations within the spin - exciton scenario [ supplementary note 1 , supplementary figures 6 , 7 and 8 ] . the resulting wave vector dependence of the spin - exciton along the @xmath36 $ ] and @xmath37 $ ] directions , which are shown in fig . 1(h ) and fig . 1(i ) , respectively , are inconsistent with the experimentally determined upward dispersion ( solid lines ) . similar dispersive resonance in cecoin@xmath4 and ce@xmath47yb@xmath48coin@xmath4 [ figure 3 , supplementary figures 3 , 4 and figure 5 ] are seen in spite of possible changes in the fermi surface and superconducting gap symmetry on moving from @xmath49 to 0.3 @xcite , also inconsistent with the expectation that a spin - exciton should depend sensitively on the fermi surface . we thus conclude that the upward dispersing resonance mode in ce@xmath38yb@xmath17coin@xmath4 can not be interpreted as a spin - exciton arising from the feedback of unconventional @xmath0-wave superconductivity @xcite . on the other hand , the similarity of the resonance s dispersion along the @xmath23 $ ] direction with the spin - wave dispersion in af ordered nonsuperconducting cerhin@xmath4 along the same direction @xcite [ fig . 1(j ) ] suggests the upward dispersing resonance may be magnon - like . in this scenario , the magnetic resonance arises since the opening of the superconducting gap leads to strong suppression of landau damping for preexisting magnon - like excitations , as shown in figs . 1(k ) and 1(l ) [ supplementary note 2 , supplementary figures 9 , 10 and 11 ] . this is , therefore , the first experimental observation of a magnetic resonance in an unconventional superconductor that can not be interpreted as a spin - exciton . yb@xmath17coin@xmath18 in the @xmath19 $ ] scattering plane . * ( a ) color - coded intensity of magnetic excitations along @xmath23 $ ] centered at @xmath8 at 0.6 k obtained from fits to data in ( c ) . ( b ) constant - energy scans along @xmath23 $ ] centered at @xmath8 with @xmath50 mev . the solid symbols are data well below @xmath7 ( 0.6 k ) where two peaks can be resolved whereas open symbols are obtained above @xmath7 ( 2.3 k ) showing a single peak centered at @xmath8 . the solid line is a fit to the data at 0.6 k with two gaussian functions whereas the dashed line is a fit to a single gaussian function for the data at 2.3 k. data at the two temperatures are fit simultaneously to have the same linear background . ( c ) constant - energy scans along @xmath23 $ ] at 0.6 k. for clarity , scans with @xmath51 and @xmath52 mev are respectively shifted upwards by @xmath53 and @xmath54 . the solid lines are fits to either one or two gaussian functions with a linear background . ( d ) constant-@xmath26 scans at @xmath8 . the arrows represent energies at which constant - energy scans are shown in ( c ) . all vertical error bars in the figure represent statistical error of 1 standard deviation . ] using a tetragonal unit cell with @xmath55 , and @xmath56 for ce@xmath38yb@xmath17coin@xmath4 [ fig . 1(a ) ] , we define the momentum transfer @xmath26 in three - dimensional reciprocal space in @xmath27 as @xmath57 , where @xmath58 , @xmath59 , and @xmath60 are miller indices and @xmath61 , @xmath62 , @xmath63 . the experiments are carried out using the @xmath19 $ ] and @xmath24 $ ] scattering planes to study the dispersions of the resonance along @xmath23 $ ] , @xmath25 $ ] , and @xmath22 $ ] [ figs . 1(b ) and 1(c ) ] . figure 2(a ) shows the color - coded plot of the spin excitations at 0.6 k obtained from fits to the raw data at energies @xmath64 and 1 mev along @xmath23 $ ] for ce@xmath38yb@xmath17coin@xmath18 [ fig . while the data show a weak commensurate peak at @xmath65 mev , we see a clear commensurate resonance at @xmath66 mev and upward dispersing incommensurate peaks for energies @xmath67 mev . figure 2(b ) shows constant - energy scans at @xmath68 mev below and above @xmath7 . at @xmath69 k , we see a broad peak centered at the commensurate af wave vector @xmath8 . upon cooling to below @xmath7 at @xmath70 k , the commensurate peak becomes two incommensurate peaks which disperse outward with increasing energy [ fig . 2(c ) ] . figure 2(d ) shows constant-@xmath26 scans at @xmath8 for temperatures @xmath71 and 2.3 k. similar to previous work on pure cecoin@xmath4 @xcite , the data reveal a clear resonance at @xmath66 mev below @xmath7 , and no peak in the normal state above @xmath7 . $ ] scattering plane for ce@xmath38yb@xmath17coin@xmath4 . * constant - energy map at @xmath72 mev at ( a ) 1 k and ( b ) 2.4 k. a @xmath73-dependent background has been subtracted . ( c ) cuts obtained from ( a ) and ( b ) by binning data with @xmath74 ; solid lines are fits to the data using either a single or two gaussian functions . since a background has already been subtracted in maps in ( a ) and ( b ) , no background is assumed in the fits . similarly , ( d ) , ( e ) and ( f ) are for @xmath75 mev , ( g ) , ( h ) and ( i ) are for @xmath76 mev , ( j ) , ( k ) and ( l ) are for @xmath77 mev and ( m ) , ( n ) and ( o ) are for @xmath78 mev . all vertical error bars in the figure represent statistical error of 1 standard deviation . ] to further illustrate the dispersive nature of the resonance , we show in figure 3 maps in the @xmath19 $ ] scattering plane of the spin excitations at different energies above and below @xmath79 obtained on macs for ce@xmath38yb@xmath17coin@xmath18 . in the probed reciprocal space , we see clear spin excitations around @xmath8 which disperse outward with increasing energy . at an energy ( @xmath80 mev ) below the resonance , spin excitations are commensurate below [ fig . 3(a ) ] and above [ fig . 3(b ) ] @xmath7 . the constant - energy cuts of the data along the @xmath23 $ ] direction confirm this conclusion [ fig . 3(c ) ] . figures 3(d ) , 3(e ) , and 3(f ) show similar scans at @xmath81 mev and indicate that the scattering become broader in reciprocal space . upon moving to @xmath68 mev 3(g ) , 3(h ) , 3(i ) ] , 1.0 mev [ figs . 3(j ) , 3(k ) , 3(l ) ] , and 1.2 mev [ figs . 3(m ) , 3(n ) , 3(o ) ] , we see clear ring - like scattering dispersing away from @xmath8 with increasing energy in the superconducting state . the normal state scattering is commensurate at all energies , and this is most clearly seen in the constant - energy cuts along the @xmath23 $ ] direction . based on the difference of data at 2.1 k and 1 k in figs . 3 , one can compose the dispersions of the resonance along the @xmath23 $ ] [ fig . 1(d ) ] and @xmath22 $ ] [ fig . 1(f ) ] directions . by plotting the dispersion in @xmath27 away from @xmath8 [ @xmath20 as defined in fig . 1(b ) ] , we see that the resonance disperses almost isotropically along these two directions . yb@xmath17coin@xmath18 in the @xmath24 $ ] scattering plane . * ( a ) constant - energy scan along @xmath25 $ ] centered at @xmath8 at 0.5 k for @xmath82 mev . the solid line is a fit to a single gaussian with a linear background . ( b ) similar to ( a ) but for @xmath83 mev . ( c ) constant - energy scan along @xmath25 $ ] centered at @xmath8 , obtained by subtracting data at 2.3 k from data at 0.5 k for @xmath81 mev . the solid line is a fit to a gaussian function with zero background . ( d ) similar to ( c ) , but for @xmath68 mev and the solid line is a fit to two gaussian functions . ( e ) similar to ( d ) , but for @xmath84 mev . the arrow points to @xmath85 , where measurement of the temperature dependence was carried out , shown in ( h ) . ( f ) similar to ( d ) and ( e ) , but for @xmath77 mev . ( g ) constant-@xmath26 scan at @xmath8 obtained by subtracting the 2.3 k data from the 0.5 k data . the solid line is a gaussian function centered at @xmath86 mev with zero background . arrows represent energies at which constant - energy scans are shown in ( a)-(f ) . ( h ) temperature dependence of scattering intensity at @xmath85 for @xmath84 mev . the solid line is a fit to @xmath0-wave superconductivity order parameter with constant background . the superconducting critical temperature @xmath7 obtained from the fit is 2.0(1 ) k. all vertical error bars in the figure represent statistical error of 1 standard deviation . ] in cuprate superconductors such as yba@xmath11cu@xmath87o@xmath88 @xcite , yba@xmath11cu@xmath87o@xmath89 @xcite , and la@xmath90ba@xmath91cuo@xmath92 @xcite , spin excitations above the resonance form a ring - like upward dispersion in the @xmath93 plane slightly softened from the spin waves in their af ordered parent compounds @xcite . to conclusively determine if the resonance dispersion is also ring - like in the @xmath93 plane in ce@xmath38yb@xmath17coin@xmath4 , we aligned the single crystals in the @xmath94\times [ 0,k,0]$ ] ( @xmath24 $ ] ) scattering plane to measure the dispersion of the resonance along @xmath25 $ ] centered at @xmath8 . figure 4(a)-4(f ) summarizes the constant - energy scans at @xmath95 mev along @xmath25 $ ] . while the scattering is clearly commensurate at @xmath96 mev below the resonance at @xmath66 mev [ fig . 4(a ) and 4(b ) ] , it becomes incommensurate above the resonance at @xmath97 mev with an upward dispersion as a function of increasing energy [ figs . 4(d ) , 4(e ) , and 4(f ) ] . figure 1(e ) summarizes the dispersion of the resonance in @xmath27 away from @xmath98 along @xmath25 $ ] . figure 4(g ) shows the difference of the constant-@xmath26 scans below and above @xmath7 at @xmath8 , again revealing a strong peak at the resonance energy of @xmath66 mev similar to fig . finally , fig . 4(h ) shows temperature dependence of the scattering at an incommensurate wave vector @xmath99 and @xmath84 mev , which reveals a clear superconducting order - parameter - like increase below @xmath7 and indicates that the incommensurate part of the resonance is also coupled to superconductivity . yb@xmath48coin@xmath18 and cecoin@xmath4 . * ( a ) difference of constant-@xmath26 scans at @xmath8=(0.5,0.5,0.5 ) for 0.3 k and 2 k displaying a resonance mode at @xmath100 mev for ce@xmath47yb@xmath48coin@xmath18 . filled symbols are obtained with fixed scattered neutron energy @xmath101 mev and open symbols are for @xmath102 mev scaled up by 4 times . all of the data in the rest of figure are obtained with @xmath101 mev . the solid line is a guide to the eye . ( b ) temperature dependence of the resonance mode in ce@xmath47yb@xmath48coin@xmath18 for @xmath103 mev and @xmath8=(0.5,0.5,0.5 ) , the solid line is a fit to @xmath0-wave superconducting gap , with @xmath104 k. dispersion of the resonance along ( c ) @xmath23 $ ] and ( d ) @xmath22 $ ] for ce@xmath47yb@xmath48coin@xmath18 . dispersions of the resonance for cecoin@xmath4 along @xmath23 $ ] and @xmath22 $ ] are showin in ( e ) and ( f ) , respectively . the solid cyan lines in ( c)-(f ) are dispersions of the resonance obtained ce@xmath38yb@xmath17coin@xmath18 . the horizontal bars represent experimentally observed peak full - width - at - half - maximums ( fwhm ) . all vertical error bars in the figure represent statistical error of 1 standard deviation . ] to determine how yb - doping , and in particular the possible changes in the fermi surface topology and superconducting gap structure between yb - doping of @xmath41 and @xmath43 affect the behavior of the resonance @xcite , we carried out additional inelastic neutron scattering experiments on cecoin@xmath4 and ce@xmath47yb@xmath48coin@xmath18 at macs . figure 5(a ) shows temperature differences of constant-*q * scans at @xmath8 below and above @xmath7 in ce@xmath47yb@xmath48coin@xmath4 , which reveals a clear resonance at @xmath100 mev . figure 5(b ) plots the temperature dependence of the resonance , displaying a superconducting order - parameter - like increase in intensity below @xmath7 . from wave vector scans along the @xmath23 $ ] and @xmath22 $ ] directions at different energies below and above @xmath7 for ce@xmath47yb@xmath48coin@xmath18 [ supplementary figure 5 ] , we can establish the dispersions of the resonance along these two directions as shown in figures 5(c ) and 5(d ) , respectively . similarly , figures 5(e ) and 5(f ) compare dispersions of the resonance for cecoin@xmath4 [ supplementary figure 4 ] and ce@xmath38yb@xmath17coin@xmath4 along the @xmath23 $ ] and @xmath22 $ ] directions , respectively . from figs . 5(c)-5(f ) , we see that the dispersions of the resonance are essentially yb - doping independent . however , the bottom of dispersive resonance at @xmath8 moves down in energy with increasing yb - doping and @xmath9 is proportional to @xmath105 , similar to la - doped cecoin@xmath4 @xcite . from the dispersions of the resonance along @xmath23 $ ] [ fig . 1(d ) ] , @xmath25 $ ] [ fig . 1(e ) ] , and @xmath22 $ ] [ fig . 1(f ) ] for ce@xmath38yb@xmath17coin@xmath18 , we see that the mode disperses isotropically in reciprocal space away from @xmath8 , which is inconsistent with the resonance being a spin - exciton [ see figs.1(h ) and 1(i ) ] , but resembles a magnon - like excitation with a dispersion resembling spin waves in cerhin@xmath4[fig . 1(j ) , supplementary note 3 and supplementary figure 12 ] that becomes undamped in the superconducting state @xcite . however , the fact that cecoin@xmath4 is a multiband system complicates the identification of the resonance s origin . while we assumed here that the main contribution to the resonance arises from the quasi - localized @xmath16-levels identified via quasi - particle interference ( qpi ) spectroscopy in stm experiment @xcite , it is of course possible that there exist further electronic bands that become superconducting and contribute to the resonance ( either directly or through a renormalization of the magnetic interaction ) but were not detected via qpi spectroscopy . clearly , further studies are necessary to investigate this possibility . moreover , in a recent work on pure cecoin@xmath4 , it was suggested that the resonance in the energy range of 0.4 - 0.7 mev is incommensurate along the @xmath23 $ ] direction with wave - vector @xmath106 where @xmath107 r.l.u . since the incommensurate wave vectors of the resonance appear to be close to the in - plane magnetic field induced incommensurate static magnetic order at @xmath106 with @xmath35 ( the so - called @xmath33 phase ) [ see the vertical dashed lines in fig . 1(d ) ] @xcite , and since it was suggested that the fluctuating moment of the resonance is entirely polarized along the @xmath32-axis similar to the ordered moment of the @xmath33 phase @xcite , the resonance has been described as a dynamical precursor of the @xmath33 phase @xcite . experimentally , we did not observe incommensurate excitations at @xmath108 mev , nevertheless our data suggest a smaller splitting than in previous work if the excitations at @xmath109 are incommensurate [ supplementary note 4 and supplementary figure 13 ] . furthermore , the @xmath33 phase precursor interpretation of the resonance is also inconsistent with the observed ring - like dispersion at @xmath110 mev . it is possible that there are more than one contribution to the resonance in cecoin@xmath4 given its electronic complexity . in the present work , we identify the upward dispersing magnon - like contribution being dominant , but do not rule out finer features at lower energies with @xmath111 mev which can only be resolved with better resolution . our data and previous work on cecoin@xmath4 @xcite are consistent with each other , both showing no signature of a downward dispersion . further insight into the nature of the resonance in cecoin@xmath4 can be gained by considering its behavior in an applied magnetic field . previous neutron scattering experiments by stock _ _ @xcite observed that the resonance in the superconducting state of cecoin@xmath4 splits into two modes if a magnetic field is applied along the @xmath112 $ ] direction . this splitting into two modes by an in - plane field is rather puzzling , since for a system with a heisenberg spin symmetry , a splitting into three modes is expected . moreover , if the resonance in cecoin@xmath4 was entirely polarized along the @xmath32-axis @xcite , application of an in - plane magnetic field should not split the resonance into the doublet observed experimentally @xcite . however , this observation can be explained if the system possess a magnetic anisotropy with a magnetic easy - plane [ indicated by the green ellipse in fig . 6(a ) ] that is perpendicular to the direction of the applied magnetic field [ red arrow in fig . since the magnetic field applied by stock _ et al . _ @xcite lies in the @xmath113 $ ] direction , this implies that the easy plane is spanned by the unit vectors in the @xmath114 $ ] and @xmath115 $ ] directions . this leads us to suggest that the resonance in cecoin@xmath4 should also have a component along the @xmath115 $ ] direction in addition to the @xmath32-axis component similar to the resonance in electron - doped iron pnictides @xcite . such in - plane spin excitation anisotropy can occur due to the presence of spin - orbit coupling , and does not break the four - fold rotational symmetry of the underlying lattice @xcite . present experimental results do not rule the presence of such a mode , although it is also challenging to experimentally confirm its presence [ supplementary note 5 , supplementary figures 14 and 15 ] . to quantitatively understand the effect of a magnetic field on spin excitations , we consider the hamiltonian ( see supplementary eq . 1 in ref . @xcite ) @xmath116 with the three terms representing the magnetic interactions between the @xmath16-electron moments , the magnetic anisotropy of the system and the interaction with the external magnetic field , respectively . here , we define the direction of the magnetic field along the @xmath113 $ ] direction as the @xmath117-axis in spin space . we assume @xmath118 , such that the system possesses a hard magnetic axis along @xmath113 $ ] and an easy plane [ green ellipse in fig . [ fig : im_chi_magfield](a ) ] perpendicular to it . this hamiltonian implies that the effective interaction for the longitudinal , non - spin - flip scattering mode ( parallel to the applied field ) is given by @xmath119 , while the interaction for the transverse mode is given by @xmath120 with @xmath121 being the fourier transform of @xmath122 in eq . ( 1 ) . in the vicinity of the af wave - vector @xmath8 , where @xmath123 , we thus obtain @xmath124 since @xmath118 for an easy - plane perpendicular to the @xmath113 $ ] direction . this implies that the effective interaction at @xmath8 for the longitudinal , non - spin - flip scattering mode ( parallel to the applied field ) is smaller than in the two transverse , spin - flip scattering modes which lie in the easy plane . as a result , the longitudinal mode will be located at energies higher than the transverse modes . in particular , for sufficiently large @xmath125 the longitudinal mode can be located above the onset energy , @xmath126 , for the particle - hole continuum in the superconducting state , and thus would not emerge as a resonance peak . hence , only the two transverse modes within the easy plane contribute to the resonance peak . the application of a magnetic field perpendicular to the easy - plane of the system then splits the two transverse modes of the resonance peak in energy ( while not affecting the longitudinal mode ) , with the energy splitting increasing linearly with the magnetic field , as shown in fig . 6(b ) , thus explaining the experimental observation in ref . @xcite . if spin excitations in cecoin@xmath4 are only polarized along the @xmath32-axis with the existence of an easy axis rather than an easy plane @xcite , application of a magnetic field along the direction perpendicular to the easy axis along the @xmath113 $ ] direction , the transverse mode along the easy axis shifts down with increasing field , but does not split . similarly , when a field is applied along the easy axis direction ( @xmath32-axis field ) , the two transverse modes are located at higher energies , while the longitudinal mode , which is located at lower energies , does not split in the magnetic field . the presence of a longitudinal spin excitations along the @xmath115 $ ] direction is also consistent the magnetic field effect work of ref . @xcite , where the resonance is believed to be a composite excitation which contains three excitation channels involving both transverse and longitudinal modes . while unconventional superconductivity in copper oxide , iron pnictide , and heavy fermion superconductors appears with the suppression of the static af order in their parent compounds , dispersive magnon - like excitations persist in the doped superconductors @xcite . our discovery that the resonance itself in ce@xmath2yb@xmath3coin@xmath4 shows robust ring - like upwards dispersion suggests instead of a spin - exciton in a @xmath0-wave superconductor @xcite , the resonance may be a magnon - like excitation revealed in the superconducting state @xcite . since the presence of a propagating spin resonance is characteristic of a nearby af state , we propose that the magnon - like resonance mode in ce@xmath2yb@xmath3coin@xmath4 is the strong - coupling analogue of a weak coupling spin - exciton . this would imply that the nature of the magnetic resonance spin - exciton versus magnon - like excitations represents a new criterion to distinguish between more weakly and more strongly coupled unconventional superconductors . * sample preparation * single crystals of ce@xmath2yb@xmath3coin@xmath18 ( @xmath127 = 0 , 0.05 , 0.3 ) were prepared by indium self - flux method . details of sample preparation and characterizations have been previously reported , lattice parameters for ce@xmath2yb@xmath3coin@xmath18 remain similar to pure cecoin@xmath18 for all reported doping levels @xcite . we use the nominal doping throughout the paper to be consistent with earlier work @xcite , while the actual doping is @xmath1281/3 of the nominal doping @xcite . supplementary fig . 1(a ) shows the out - of - phase ac magnetic susceptibility ( 15.9 hz ) measured on ce@xmath2yb@xmath129coin@xmath4 samples with @xmath127 = 0.05 and 0.3 from the same growth batches used for neutron scattering experiments . bulk superconductivity appear at @xmath46 = 2.25 k and @xmath130 k respectively , whereas @xmath131 k in pure cecoin5@xcite . hundreds of ce@xmath2yb@xmath129coin@xmath4 single crystals with total masses of 0.8 g , 2.5 g and 1.4 g respectively for @xmath127 = 0 , 0.05 and 0.3 were co - aligned on several aluminum plates using cytop as hydrogen - free glue [ supplementary fig . the plates are then mounted in either the @xmath132\times[0,0,l]$ ] ( @xmath19 $ ] ) [ supplementary fig . 1(c ) ] or the @xmath94\times[0,k,0]$ ] ( @xmath24 $ ] ) scattering plane [ supplementary fig . the total thickness of samples on co - aligned plates is 1 - 2 mm , minimizing neutron absorption due to indium . absorption becomes most significant when the incident or the scattered neutron beam becomes perpendicular to @xmath114 $ ] , which does not occur for reciprocal space regions shown in this work . * experiment details and analysis * neutron scattering experiments were carried out on the panda cold three - axes spectrometer at heinz maier - leibnitz zentrum and the multi - axis crystal spectrometer ( macs ) at the nist center for neutron research . the experiments on panda used a be filter 180 mm in length after the sample which is highly effective in removing contamination from higher order neutrons , both the analyzer and the monochromator are doubly focused to maximize neutron flux at the sample . vertical focusing of the analyzer is fixed whereas horizontal focusing is variable . both the horizontal and vertical focusing of the monochromator are variable . the variable focusings are adjusted depending on the neutron wavelength based on empirically optimized values . the panda experiment in the @xmath19 $ ] scattering plane used fixed @xmath133 @xmath27 ( @xmath134 mev ) and the experiment in the @xmath24 $ ] scattering plane used fixed @xmath135 @xmath27 ( @xmath136 mev ) . the macs experiments in the @xmath19 $ ] scattering plane used be filters both before and after the sample with fixed @xmath101 mev . macs consists of 20 spectroscopic detectors each separated by 8@xmath137 . by rotating the sample and shifting all of the detectors to bridge the 8@xmath137 gaps , a map in terms of sample rotation angle and scattering angle at a fixed energy transfer can be efficiently constructed . a significant portion of the reciprocal space in the scattering plane can be covered , which further allows cuts along the high symmetry directions . 90@xmath138 collimators are used between the sample and each individual analyzers . the analyzers are vertically focused while the monochromator is doubly focused . for the neutron scattering results on panda , a linear background is assumed for all measured constant - energy scans while no background is used for scans obtained by subtraction data above @xmath7 from those obtained below @xmath7 . the constant energy scans are then simply fit to either one or two gaussian peaks . for the neutron scattering results obtained on macs , maps of large portions of the scattering plane for several energies transfers were collected both below and above @xmath7 . a @xmath73-dependent background is obtained by masking the signal near @xmath139 and is then fit to a polynomial . the signal with @xmath140 @xmath27 is masked throughout the analysis . the fit background is then subtracted from the map and the data is folded into the first quadrant of the scattering plane to improve statistics . the results for ce@xmath38yb@xmath17coin@xmath18 are shown in figure 3 and supplementary figure 3 . cuts along @xmath23 $ ] are obtained by binning data with @xmath141 and fit with a single or two gaussian peaks . cuts along @xmath22 $ ] are obtained by binning data with @xmath142 and fit by a sum of lorentzian peaks , accounting for the ce@xmath143 magnetic form factor @xmath144 and the polarization factor assuming excitations are dominantly polarized along the @xmath32-axis similar to previous work @xcite . the possible presence of excitations polarized along the @xmath115 $ ] direction is discussed in supplementary note 5 . the function used to fit scans along @xmath22 $ ] can be written as @xmath145 where @xmath146 is either a single lorentizan peak centered at @xmath147 or two lorentzian peaks equally displaced from @xmath147 . the peaks along @xmath22 $ ] are significantly broader compared to those along @xmath23 $ ] , and remains non - 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the neutron spin resonance is a collective magnetic excitation that appears in copper oxide , iron pnictide , and heavy fermion unconventional superconductors . although the resonance is commonly associated with a spin - exciton due to the @xmath0(@xmath1)-wave symmetry of the superconducting order parameter , it has also been proposed to be a magnon - like excitation appearing in the superconducting state . here we use inelastic neutron scattering to demonstrate that the resonance in the heavy fermion superconductor ce@xmath2yb@xmath3coin@xmath4 with @xmath5 has a ring - like upward dispersion that is robust against yb - doping . by comparing our experimental data with random phase approximation calculation using the electronic structure and the momentum dependence of the @xmath6-wave superconducting gap determined from scanning tunneling microscopy for cecoin@xmath4 , we conclude the robust upward dispersing resonance mode in ce@xmath2yb@xmath3coin@xmath4 is inconsistent with the downward dispersion predicted within the spin - exciton scenario .

introduction results discussion

arxiv

the minimal supersymmetric extension of the standard model ( mssm ) is the most studied scenario beyond the sm , and the lightest neutralino is a favourite candidate for dark matter ( dm ) , when r - parity is conserved . searches for events characterised by the emission of a single hard jet , used as a signature of the hard scattering process , are usually considered as a probe of direct production of invisible dark matter particles in @xmath0 collisions . the study of monojet signature was pioneered by the tevatron experiments @xcite . interpretations of the results of direct searches for new particles at the lhc are often performed in the context of simplified scenarios . highly constrained susy models , such as the constrained mssm ( cmssm ) @xcite , have been studied in the past . currently the attention has shifted towards simplified models @xcite , where only few degrees of freedom , such as the neutralino mass and the mass splitting to the lightest susy particles , define the relevant phenomenology . in this study , we investigate the implications of monojet searches in the context of the simplified models in the mssm with neutralino dm by reinterpreting the lhc run 1 results in a quantitative way , and compare them to the constraints obtained from direct susy searches in the jet / lepton + met channels . the lack of signals of low energy susy in the lhc run 1 data already sets strong constraints on the mass spectrum of the susy particles , in the constrained mssm models . more general mssm scenarios with less ad - hoc universality assumptions , such as the phenomenological mssm ( pmssm ) @xcite , where constraints on the colored states do not affect non - colored sparticles , or scenarios with long decay chains or compressed spectra still remain largely viable @xcite . compressed scenarios are particularly interesting since the observed dark matter relic density can be achieved in these scenarios thanks to the enhanced effective cross sections due to co - annihilations . in particular , small mass splittings between squarks and gluino with the lightest neutralino can lead to final states with less energetic jets or leptons , thus reducing the detection efficiency and signal acceptance at the lhc . monojet searches are particularly sensitive to such scenarios , and we shall show that they are indeed powerful in constraining the mssm , provided all the involved processes are correctly taken into account . monojet signals in the context of the mssm have already been discussed in specific scenarios @xcite . the connections of monojet and dark matter searches at the lhc are discussed in section 2 . section 3 describes the way monojet searches can be affected in the mssm and presents the numerical set - up . the implications of the monojet searches in simplified mssm scenarios are presented in section 4 . section 5 addresses the sensitivity of the monojet searches at higher center of mass energies and luminosities . the conclusions are given in section 6 . monojet searches at the lhc consist in looking for events with one high-@xmath1 jet and missing energy , and are therefore particularly well - suited for the search of dark matter particles . a schematic representation of monojet events in simplified scenarios in which a dark matter candidate and a mediator are added to the standard model is given in fig . [ fig : standard_monojets ] . , width=132 ] the interpretation of monojet searches has been often performed in the context of effective scenarios , where operators linking quarks and gluons to two dm particles are considered . hence the constraints obtained on the dark matter particle mass are dependent on the operator under consideration , and can be compared directly to the results of dm direct detection experiments by computing the scattering cross section of dm with standard matter @xcite . more recently , the validity of the effective approach has been questioned @xcite , and instead simplified scenarios with different configurations of dark matter candidates and mediators have been suggested to probe dark matter at the lhc @xcite . the constraints obtained from monojet searches are dependent on the natures , masses and couplings of the dark matter candidates and mediator particles . for a specific set - up , it is possible to reinterpret the results in terms of scattering cross sections of dm with protons and compare it to direct detection experiment results . in the following we discuss the case of mssm with r - parity conservation and neutralino dark matter . monojets in this scenario can be generated by the final states with two neutralinos and one hard jet , as in fig . [ fig : standard_monojets ] , but more importantly two neutralinos , one hard jet and additional soft jets or particles invisible in the detectors . such final states occur in particular when two squarks or gluinos are produced in addition to a hard jet , as shown in fig . [ fig : mssm_monojets ] . this happens in particular in scenarios with compressed spectra , where the direct susy searches are less sensitive but the cross section for the monojet topologies is enhanced by the strong production of degenerate squarks or gluinos . contrary to the case of simplified or effective approaches , there is no correlation in the mssm between the monojet production cross section , which can probe the strong sector , and the neutralino scattering cross section with matter , which is sensitive to the electroweak sector , so that monojet searches can not be considered anymore as dark matter searches , but as complementary channels to the direct susy searches and to dark matter cosmological and astrophysical observables . in this analysis , we use madgraph 5 @xcite to compute the full @xmath2 matrix elements corresponding to all the combinations of @xmath3 , @xmath4 and @xmath5 , where @xmath6 refers to a squark of any type and generation , @xmath7 to the gluino , @xmath8 to any type of slepton , @xmath9 to any electroweakino , and @xmath10 to a hard jet . here we do not restrict ourselves to initial state radiation of a monojet . to generate events we adopt the cteq6l1 parton distribution functions @xcite . hadronisation is performed using pythia 8 @xcite , and detector effects are simulated with delphes 3.0 @xcite . the exclusion by the monojet searches is assessed based on the atlas @xcite and cms @xcite analyses , using the same cuts , selection efficiencies , acceptances and backgrounds , and predictions for higher energies are obtained by rescaling the background and assessing the sensitivity without modifying the experimental set - up applied in the 8 tev analyses . in this sense , our analysis is rather conservative as no optimisation is considered . also , systematic uncertainties have been shown to have an important effect on the limits that can be derived using the monojet signatures @xcite . here we account for these systematics by adding a 30% uncertainty on the cross sections . signal selection cuts corresponding to each of the analyses are applied to the simulated signal events . the number of sm background events in the signal regions are taken from the estimates reported by the experiments . when the experimental analysis investigates several signal regions , such as the atlas and cms monojet analyses , we calculate the region giving the largest signal - to - background ratio and we only use that region for determining the exclusion . the 95% confidence level ( c.l . ) exclusion in presence of background only is determined using the cls method @xcite . , width=162 ] in addition , we compute the relic density with superiso relic @xcite , as well as dark matter direct detection observables with micromegas @xcite . we compare the results to the dark matter density measurement of planck @xcite and to the results of lux @xcite for dm direct detection . finally , the electroweak observables are computed with a modified version of superiso @xcite . we consider various sets of simplified models and investigate the complementarity with the traditional jets / leptons + met susy searches . these models are characterised by a light neutralino accompanied by at least one additional heavier sparticle , while the other sparticle masses are much heavier . in practice , the light sparticles are lighter than about 1 tev , and the other masses are adjusted in the range 1040 tev in order to obtain a correct light higgs mass of 125 gev . the trilinear couplings of the third generation fermions are chosen in order to have no mixing , and @xmath11 is set to 10 , an intermediate value which allows us to be consistent with the flavor constraints . the five sets of models correspond to one light neutralino and : a light gluino , degenerate scalar quarks , a light scalar bottom quark , a light scalar top quark , and light neutralino 2 and chargino 1 . for the five scenarios , in addition to the discussion of the lhc supersymmetry and monojet searches , we also checked that the @xmath12 boson mass and electroweak oblique parameters are consistent with the lep measurements @xcite . concerning dark matter direct detection , we found that for all these scenarios , the neutralino - nucleon spin - independent scattering cross section is always smaller than @xmath13 pb , which is well below the lux limits @xcite , but within reach of the expected sensitivity of lz @xcite . a general feature exhibited by the scenarios we investigated is the significant improvement in sensitivity in the regions with small mass splittings . these regions are especially important since they correspond to the parameter range where coannihilation processes bring the neutralino relic density in agreement with the cmb data , as highlighted by the red lines on our plots . of data ( dark blue ) . the black lines correspond to the atlas supersymmetric direct search limit , and the red lines to the relic density value as measured by planck.[fig : mssm : gluino],title="fig : " ] + of data ( dark blue ) . the black lines correspond to the atlas supersymmetric direct search limit , and the red lines to the relic density value as measured by planck.[fig : mssm : gluino],title="fig : " ] + the first mssm simplified scenario we consider has @xmath14 and @xmath15 as main parameters , resulting in a pure bino neutralino and a gluino . the other masses are set to 40 tev , apart from the stop sector parameters which are adjusted to obtain a light higgs mass of 125 gev . the gluino is assumed to decay exclusively to the lightest neutralino and two light quarks . the values of @xmath14 and @xmath15 are varied between 0 and 1.5 tev . this scenario is of interest since it is well probed by the susy direct searches . because the lightest neutralino is a pure bino , its interaction with matter is suppressed and it can not be detected by direct dark matter detection experiments . regarding the relic density , strong co - annihilations with the gluino are necessary to obtain a relic density in agreement with the planck limits . results are presented in fig . [ fig : mssm : gluino ] , in the ( @xmath16 ) mass plane for the 8 tev run , as well as predictions for 14 tev with 300 fb@xmath17 of data . the observed limit from the atlas run 1 searches for direct gluino production in the jets + met channel @xcite is also shown for comparison , as well as the region corresponding to the observed dark matter density . as can be seen , monojet searches are more constraining in the parameter region where the gluino and the lightest neutralino have almost degenerate masses , which also corresponds to the region where the relic density is close to or smaller than the observed value . the constraints from monojet searches on the neutralino mass can reach 600 gev when this mass splitting is small . we observe that the monojet searches can marginally improve the constraints from the susy searches for mass splittings up to 50 gev as the direct susy searches in this scenario are very strong . at the 14 tev run , the monojet searches could probe neutralino masses up to 1.1 tev . in the degenerate squark scenario the lightest neutralino is a pure bino , and all the eight first and second generation scalar quarks are taken to be light and degenerate in mass . the squarks decay exclusively to a quark and the lightest neutralino . this model has two parameters : @xmath14 and the mass of the degenerate squarks , allowed to vary in the ranges @xmath18 $ ] tev . this scenario is probed well by the lhc susy searches since all the eight first and second generation squarks participate to the cross sections . again , since the neutralino is a bino , dark matter detection experiments are not sensitive enough to probe it , and a correct relic density requires a small mass splitting with the neutralino in order to have adequate co - annihilations . results are shown in fig . [ fig : mssm : squark ] , in the ( @xmath19 ) mass plane . the limits from the atlas run 1 direct squark searches in the jets + met channel @xcite are also presented , in addition to the thin region corresponding to the planck dark matter density . the most constrained region corresponds to that with the squarks and @xmath20 nearly degenerate in mass . in this region , the constraints on the @xmath20 mass go up to 400 gev . comparing the exclusions , we see that the monojet searches provide additional constraints to the direct susy searches in the region where the mass splitting is below 50 gev . the 14 tev run will probe neutralino masses close to 850 gev . of data ( dark blue ) . the black lines correspond to the atlas supersymmetric direct search observed limit , and the red lines to the relic density value as measured by planck.[fig : mssm : squark],title="fig : " ] + of data ( dark blue ) . the black lines correspond to the atlas supersymmetric direct search observed limit , and the red lines to the relic density value as measured by planck.[fig : mssm : squark],title="fig : " ] + of data ( dark blue ) . the black solid lines correspond to the atlas supersymmetric direct search observed limit , the dotted lines to the atlas monojet search observed limit , and the red lines to the relic density value as measured by planck.[fig : mssm : sbottom],title="fig : " ] + of data ( dark blue ) . the black solid lines correspond to the atlas supersymmetric direct search observed limit , the dotted lines to the atlas monojet search observed limit , and the red lines to the relic density value as measured by planck.[fig : mssm : sbottom],title="fig : " ] + the next scenario has a pure bino neutralino and a right - handed sbottom decaying exclusively to a bottom quark and a neutralino . the neutralino and sbottom masses are varied in the range @xmath21 $ ] tev . the other masses are set to 40 tev , apart from the stop sector where the parameters are adjusted to obtain a light higgs mass of 125 gev . again , dark matter can not be detected because of the elusive nature of the neutralino , and the correct dark matter density can be achieved thanks to co - annihilation with the sbottoms . figure [ fig : mssm : sbottom ] summarises the results in the ( @xmath22 ) mass plane . the bounds from the atlas run 1 sbottom searches in the 2 @xmath23jets + met and monojet channels @xcite are also shown , as well as the region where the correct relic density is reached . we notice that the constraints on the @xmath20 mass goes beyond 250 gev , while the direct sbottom searches probe neutralino masses up to 280 gev , and the constraints are improved by the monojet searches for mass splittings below 40 gev . as can be seen from the figure , our results at 8 tev are rather similar to the observed atlas monojet search results , which can be considered as a validation of our analysis . at 14 tev , neutralino masses up to 520 gev can be probed . we now consider a scenario with a wino - bino neutralino associated to a chargino close in mass , and a heavier scalar top quark . we define two separate regions : if the mass splitting of the squark with the chargino is smaller than the top mass , the scalar top decays exclusively to a bottom quark and the chargino , if the mass splitting is larger than the top mass , the stop decays exclusively to a top quark and the neutralino . the chargino subsequently decays to an off - shell @xmath12 boson and a neutralino , while the top quark can decay to an on - shell @xmath12 boson and a @xmath24 quark . the neutralino and stop masses vary in the range @xmath21 $ ] tev . the other masses are set to 40 tev , apart from the second stop . the parameters @xmath14 and @xmath25 are adjusted in order to obtain a correct relic density , following the relic density line described in the next subsection . the mass splitting with the stop is the only relevant parameter . this scenario is particularly interesting in the context of monojets because sizeable regions of its parameter space are not accessible to the standard stop searches . regarding dark matter searches , we checked that scattering cross section of the neutralino with matter is one order of magnitude below the current experimental sensitivity . the neutralino - chargino set - up of this scenario is a specific case of the scenario described in the next subsection , so we refer the reader to the next subsection for more discussions about dark matter . concerning the lightest higgs mass , a correct value can be obtained by adjusting the stop 2 mass in the range 110 tev and keeping no mixing in the stop sector . results are shown in fig . [ fig : mssm : stop ] , in the ( @xmath26 ) mass plane . the envelopes of the limits from the atlas run 1 direct stop searches in the 2 top quarks + met , 2 @xmath23jets + met , one isolated lepton + jets + met , 2 leptons + met , jets + met and monojet channels @xcite are shown for comparison , for the two separate regions corresponding to mass splitting below and above the top mass . the monojet searches at 8 tev only probe the region where the decay of the stop to a top and a neutralino is closed , corresponding to mass splitting below the top mass . in this region , the constraints are comparable to those published by the atlas collaboration . in the region where the stop can decay to a top and a neutralino , the monojet searches loose their sensitivity at 8 tev since a top quark will manifest itself as an additional high-@xmath1 jet . the lep constraints obtained in chargino searches @xcite are also shown for comparison . at 14 tev , neutralino masses up to 550 gev will be probed in the small mass splitting region , and the region where the stop can decay to the top quark and the neutralino will also be reached , so that mass splitting up to 450 gev can be probed . for mass splittings above @xmath27 , the monojet reach at 14 tev is considerably worse than published 8 tev stop search limits . of data ( dark blue ) . the dotted lines correspond to a mass splitting equal to the top mass . on the left side of this line , the stop decays to a bottom and a chargino , and on the right side the stop decays to a top and a chargino . the black lines correspond to the atlas supersymmetric direct search observed limits . the horizontal gray lines correspond to the lep chargino search limit.[fig : mssm : stop],title="fig : " ] + of data ( dark blue ) . the dotted lines correspond to a mass splitting equal to the top mass . on the left side of this line , the stop decays to a bottom and a chargino , and on the right side the stop decays to a top and a chargino . the black lines correspond to the atlas supersymmetric direct search observed limits . the horizontal gray lines correspond to the lep chargino search limit.[fig : mssm : stop],title="fig : " ] + the last simplified scenario we consider has @xmath14 and @xmath25 as the only low energy parameters . the other masses are set to 40 tev , apart from the stop sector where the parameters are adjusted to obtain a light higgs mass of 125 gev . this scenario results in three light particles : two neutralinos , which can be bino - like , wino - like or bino - wino mixed states , and a wino chargino . the @xmath28 is assumed to decay exclusively to the lightest neutralino and a ( on- or off - shell ) light higgs boson @xmath29 , and the chargino to the lightest neutralino and a ( on- or off - shell ) @xmath12 boson . the value of the @xmath14 and @xmath25 parameters are varied between 0 and 500 gev . results are shown in fig . [ fig : mssm : neutralino ] , in the ( @xmath30 ) mass plane for the lhc 8 tev run as well as the projection for the 14 tev run with 300 fb@xmath17 of data . for comparison , the observed limit from the atlas run 1 direct neutralino / chargino searches in the 2 and 3 leptons + met and 1 lepton + @xmath29 + met @xcite is also displayed . of data ( dark blue ) . the black lines correspond to the atlas supersymmetric direct search observed limit , and the red lines to the relic density value as measured by planck . the gray lines correspond to the lep chargino search limit.[fig : mssm : neutralino],title="fig : " ] + of data ( dark blue ) . the black lines correspond to the atlas supersymmetric direct search observed limit , and the red lines to the relic density value as measured by planck . the gray lines correspond to the lep chargino search limit.[fig : mssm : neutralino],title="fig : " ] + the monojet search is particularly constraining in the region where the @xmath31 and @xmath20 have similar masses . in this region , the constraints on the neutralino mass can reach 90 gev , and are complementary to the direct search limits . the monojet searches are particularly efficient in probing mass splittings below 40 gev . the lep constraints obtained in chargino searches @xcite are also shown for comparison , and they supersede the 8 tev monojet search limits . at 14 tev , the constraints will improve and neutralino masses up to 250 gev can be reached , beyond the lep limits . + + + monojet searches will remain a powerful tool for discovery at @xmath0 colliders of increasing energy and luminosity . in order to assess the evolution of their sensitivity with energy and luminosity , we repeat our study for center of mass energies of 8 , 13 , 14 , 30 , 50 and 100 tev for six different simplified mssm models : a pure bino neutralino ; a mixed state bino / wino in which there are two light neutralinos and one light chargino ; the following cases with mass splitting of 10 gev : a light gluino and a light bino neutralino , eight degenerate light squarks and a bino , a light sbottom and a bino ; and finally a light stop and bino - wino neutralino and chargino with mass splitting slightly smaller than the top quark mass . the mass splittings for the above - mentioned scenarios have been chosen to maximise the number of monojet events , but also to ensure the consistency between the susy model and the dm relic density constraints , requiring small mass splittings needed for co - annihilations ( see for example @xcite ) . the calculation of the mass reach as a function of the luminosity and energy requires a detailed study accounting for the sm backgrounds , which goes beyond the scope of this paper . however , it is interesting to study the scaling of the product of the monojet production cross section times acceptance and efficiency as a function of the neutralino mass and the collider energy . the acceptance is defined by @xmath32-dependent lower cuts on the jet @xmath1 and missing @xmath33 , scaled from typical values adopted in the 8 tev searches and given by @xmath34 tev)@xmath35 gev . the results are shown in fig . [ fig : mssm : energy_dep ] . for the bino case , we do vary the mass of the other susy particles between 5 and 50 tev since the monojet cross section is sensitive to it . the limits obtained for 8 tev give the current status of these searches and their extrapolation to 300 fb@xmath17 of data for the lhc 14 tev run are given for comparison . although the change in cross section times efficiency from 8 to 14 tev as a function of the mass is relatively small , the increase in mass coverage afforded by 14 tev is very significant . this motivates a possible increase of the energy up to 30 tev , in principle compatible with the radius of the lhc tunnel and dipoles of new technology , and beyond . the pure bino case remains out of reach due to its small cross section but a collider with an energy at the order of 100 tev and high luminosity would possibly provide enough statistics for probing neutralino masses in all the other scenarios up to more than 3 tev . this upper limit is particularly interesting since a relic density compatible with the cmb data can be reached for wino and higgsino neutralinos of masses between 1 and 3 tev in absence of co - annihilations with sfermions , a window which could tantalisingly be accessible at a 100 tev collider . the search for monojets is a powerful tool to explore new processes at hadron colliders . to illustrate this in a quantitative way , we have considered simplified mssm scenarios in which only a few relevant degrees of freedom are considered . we showed that direct searches in the jets / leptons + met final states and monojets are highly complementary , the latter improving the sensitivity in regions with small mass splittings . such regions are highlighted by dm relic density involving co - annihilation processes . recasting the monojet searches in the mssm , it is important to consider all the relevant topologies , namely processes involving squarks and gluinos escaping the detection in addition to the usual wimp - wimp - jet topologies , as the former result in large cross sections at the lhc and can be dominant when the squark / gluino mass becomes nearly degenerate with the lightest neutralino . we find that the complementarity of the monojet and direct susy searches is particularly striking in the case of the light neutralino and chargino scenario . small mass splittings in the gaugino sector naturally arise when the lightest neutralino is a pure wino or higgsino , resulting in weaker constraints from direct multi - 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we explore the implications of monojet searches at hadron colliders in the minimal supersymmetric extension of the standard model ( mssm ) . to quantify the impact of monojet searches , we consider simplified mssm scenarios with neutralino dark matter . the monojet results of the lhc run 1 are reinterpreted in the context of several mssm simplified scenarios , and the complementarity with direct supersymmetry search results is highlighted . we also investigate the reach of monojet searches for the run 2 , as well as for future higher energy hadron colliders .

introduction dark matter searches at the lhc monojet searches in the mssm mssm simplified scenarios monojet searches at higher energies conclusion

arxiv

large - scale evolutionary equations for many - body systems arise ubiquitously in numerical modeling . the cases of particular interest and difficulty involve many configuration coordinates . for instance , the time - dependent _ schroedinger _ equation describes the wavefunction , depending on all positions of all quantum particles or states of spins . another important example is the simulation of the joint probability density function either in continuous ( _ fokker - planck _ equation ) or discrete ( _ master _ equation ) variables . in case of @xmath0 configuration variables , solutions of these problems are @xmath0-variate functions . on the discrete level , one may typically assume that finite sets of @xmath1 admissible values are introduced for each coordinate independently ( e.g. a standard tensor product discretization grid ) . thereby , we do not discriminate the variables from the very beginning . however , the total amount of entries , defining the multivariate function , scales as @xmath2 . even if the _ dimension _ @xmath0 is of the order of hundreds and @xmath3 ( a modest size for spin dynamics problems ) , this becomes an enormously large number , and straightforward computations are unthinkable . to cope with such _ high - dimensional _ problems , one has to employ _ ( data-)sparse _ techniques , i.e. describe the solution by much less unknowns than @xmath2 . different state of the art approaches were developed for this task . among the most successful ones we may identify monte carlo methods @xcite , sparse grids @xcite , and tensor product representations . in this paper , we follow the latter framework . _ tensor product methods _ rely on the idea of separation of variables : a @xmath0-variate array ( or _ tensor _ ) may be defined or approximated by sums and products of univariate vectors . extensive information can be found in recent reviews and books , e.g. @xcite . a promising potential of tensor product methods stems from the fact that each univariate _ factor _ requires only @xmath1 elements to store instead of @xmath2 . if a tensor can be approximated up to the required accuracy with a moderate amount of such terms , the memory and complexity savings may be outstanding . there exist different tensor product _ formats _ , i.e. rules how to map univariate factors to the initial array . in case of two dimensions , one ends up with the well - known low - rank dyadic factorization of a matrix . this straightforward sum of direct products of vectors in higher dimensions is called cp format , and traces back to @xcite . however , the error function recast to the entries of the cp factors may not have a minimizer @xcite . therefore , even if all elements of a tensor are given , it is difficult to detect its cp rank . certain heuristics are available , for example , one may increase the rank one by one in a try - and - dispose als procedure @xcite or greedy algorithms @xcite . nevertheless , such methods typically exhibit a fast saturation of the convergence for rather modest ranks , and more accurate calculations become struggling . a family of reliable tools exploits recurrent two - dimensional factorizations to make the computations stable . in this work , we focus on the simplest member of this family , rediscovered several times under different names : _ valence bond states _ @xcite , _ matrix product states _ ( mps ) @xcite and _ density matrix renormalization group _ ( dmrg ) @xcite in condensed matter quantum physics , and _ tensor train _ ( tt ) @xcite in numerical linear algebra . this format possesses all power of the recurrent model reduction concept , but the description of algorithms may benefit from some transparency and elegance . for higher flexibility in particular problems , one may use more general tree - based constructions , such as the _ ht _ @xcite or _ extended tt / qtt - tucker _ @xcite formats . the dmrg is not only the name of the representation , but also a variety of computational tools . it was originally developed to find ground states ( lowest eigenpairs ) of high - dimensional hamiltonians of spin chains . the main idea behind the dmrg is the alternating optimization of a function ( e.g. rayleigh quotient ) on tensor format blocks in a sequence . it was noticed that this method may manifest a remarkably fast convergence @xcite , and later extensions to the energy function followed @xcite . besides the stationary problems , the same framework was applied to the dynamical spin schroedinger equation . two conceptually similar techniques , the _ time - evolving block decimation _ ( tebd ) @xcite and the _ time - dependent dmrg _ ( tdmrg ) @xcite take into account the nearest - neighbor form of the hamiltonian to split the operator exponent into two parts using the trotter decompositions . for each part , the exact exponentiation may be performed , but at the cost of increased sizes of tensor format factors . to reduce the storage , the truncated singular value decomposition is employed . thus , the method introduces two types of error : the truncated part of the trotter series , and the truncated part of the tensor format . if many time steps are required , the error may accumulate in a very unwanted manner : it lacks a reasonable separation of variables , and hence inflates the tensor format storage of the solution ( see e.g. @xcite ) . to stick the evolution to the manifold , generated by the tensor format , the so - called _ dirac - frenkel _ principle may be exploited @xcite . this scheme projects the time derivative onto the tangent space of the tensor product manifold , and formulates the dynamical equations for the factor elements directly . the storage of the format is now fixed , but approximation errors become generally uncontrollable . in addition , the projected dynamical equations may be ill - conditioned . as an alternative approach , one may consider time just as another variable , since the dimension contributes linearly to the complexity of tensor product methods , and solve the global system for many time layers simultaneously @xcite . in this work we follow this way . contrarily to @xcite , we use the spectral differentiation in time on the chebyshev grid , see @xcite . this makes the time discretization error negligible , and we show that a long - time dynamics is possible without explosion of the tensor format storage . the linear system arising from this scheme is always non - symmetric and requires a reliable solution algorithm in a tensor format . the traditional dmrg may suffer from a stagnation at a local minimum , far from the requested error level . recently , the _ alternating minimal energy _ ( amen ) method was proposed @xcite , which augments the tensor format of the solution in the dmrg technique by the tensor format of the global residual , mirroring the classical steepest descent iteration . this endows the method with the rank adaptivity and a guaranteed global convergence rate . importantly , the practically manifesting convergence appears to be much faster than the theoretical predictions , which yields a solution with a nearly - optimal tensor product representation for a given accuracy . another problem reported for tdmrg ( it takes place for the techniques in @xcite as well ) is the corruption of system invariants . even if the storage remains bounded during the dynamics , the magnitude of the error may rise . though we may be satisfied with the resulting approximation of the whole solution , it is worth sometimes to preserve a linear or quadratic function of the solution exactly ( see e.g. a remark in @xcite ) . in this paper we address this issue for linear functions and the second norm of the solution by including the vectors , defining the invariants , into the amen enrichment scheme . in the next section we formulate the ode problem , investigate its properties related to the first- and the second - order invariants , show the galerkin model reduction concept and how the invariants may be preserved in the reduced system , and suggest the spectral discretization in time . section [ sec : tensor ] gives a brief introduction to tensor product formats and methods , and finally , the new tamen algorithm ( the name is motivated by tdmrg ) is proposed and discussed . section 4 demonstrates supporting numerical examples , and section 5 contains concluding remarks . our central problem , considered in particular in the numerical examples , is the homogeneous linear system of odes , @xmath4 in section [ sec : spectral ] and in the final version of the algorithm , we will extend to the general quasi - linear form @xmath5 . analogously , the inhomogeneous case @xmath6 may be taken into account with a few technical changes . nevertheless , basic features may be illustrated already on the simple linear system , and we will keep it in focus in the first part of the paper . throughout the paper , @xmath7 and other quantities denoted by small letters will be considered as @xmath8 vectors , such that the _ dot _ ( inner , scalar ) product @xmath9 may be written as @xmath10 . the time discretization relies on both the finite approximation of the time derivative and boundary conditions for the cauchy problem . a simple way to derive them is presented below . given @xmath11 ( not necessarily linear ) together with @xmath12 , we introduce a new variable @xmath13 , and obtain @xmath14 to discretize this equation , we use the chebyshev spectral differentiation scheme @xcite . the base _ chebyshev _ nodes on the interval @xmath15 $ ] are written as @xmath16 , and after rescaling onto @xmath17 $ ] we obtain @xmath18 , @xmath19 . since @xmath20 starts from @xmath21 , the point @xmath22 is excluded , in accordance with the zero dirichlet boundary condition . now , we represent any function in the form @xmath23 , where @xmath24 be the lagrange interpolation polynomial built on @xmath25 , i.e. @xmath26 . therefore , the time derivative can be approximated by the matrix - by - vector product , @xmath27 where @xmath28 $ ] is the so - called _ chebyshev differentiation matrix_. taking into account the form of the lagrange polynomials , the elements @xmath29 can be calculated , @xmath30 \dfrac{1+\delta_{i,\i}}{1+\delta_{j,\i } } \dfrac{(-1)^{i+j}}{\hat t_i- \hat t_j } , & i \neq j , \\[1em ] \dfrac{2\i^2 + 1}{6 } , & i = j=\i . \end{array}\right.\ ] ] note that in the right - hand side we use the dimensionless points @xmath16 . substituting back @xmath13 , obtain the following discretized equation , @xmath31 the accuracy of the chebyshev differentiation is given by the next statement . [ st : cheb ] suppose @xmath32 , defined on @xmath33 $ ] , is analytically extensible to the complex ellipse @xmath34 with @xmath35 . then the error of the chebyshev derivative converges exponentially , @xmath36 if the ode solution is not smooth in time , more sophisticated hp - techniques may be required @xcite . in many cases , however , the chebyshev interpolation is preferable , since it allows to work with pointwise samples of functions instead of galerkin coefficients , and increases the sparsity of involved matrices . the chebyshev differentiation matrices can be also used for spatial variables in e.g. the fokker - planck equation , see @xcite . in many practical models , the right - hand side of the ode system is _ quasi - linear _ , i.e. @xmath37 . in case of a mild non - linearity , the straightforward _ iteration may exhibit a satisfactory convergence . given the initial vector @xmath38 , composed from the stacked samples @xmath39 at the chebyshev nodes in time , we write a counterpart of as the following linear system , @xmath40 where @xmath41 denotes the `` reversed '' kronecker product , @xmath42 $ ] , @xmath43 is the identity matrix of size @xmath44 , and @xmath45 . the reversed rule of the kronecker product is introduced for more convenient connection with tensor product schemes , see the next section . if the residual @xmath46 is too large , one may put @xmath47 and solve again , performing several picard steps . obviously , if @xmath48 does not depend on @xmath7 , the very first iteration gives the exact solution . if in addition , the matrix @xmath48 is stationary , the block - diagonal matrix in may be written as the kronecker product @xmath49 , and the simplified system reads @xmath50 here we denoted @xmath51 , the right - hand sides stacked . note that if @xmath52 , i.e. the ode system is stable , the spectrum of @xmath53 consists of values @xmath54 , and lies essentially in the right half of the complex plane . it is important for the convergence of the tensor product iterative algorithms , cf . the analysis of the minimal residual method @xcite . our goal will be to seek an ode solution in a compressed data - sparse form . a particular question of interest is the following : if the system preserves some quantities in time , is it possible to maintain this property in the data - sparse algorithms , which are based on the galerkin projection approach ? one of the most ubiquitous conserving quantities are the linear function of the solution , and the second norm . given some _ detecting _ vector @xmath55 , the linear function can be written as @xmath56 . it corresponds , for example , to the probability normalization in the fokker - planck equation : @xmath7 has the meaning of the discretized probability distribution , and it holds that @xmath57 , for @xmath55 being a vector of all ones . among the second - order invariants , we investigate the euclidean ( frobenius ) norm of the solution , @xmath58 . the conservation condition @xmath59 is a well - known property of the schroedinger equation @xmath60 , where @xmath61 is the imaginary unity , and @xmath62 . the following well - known algebraic properties of the system matrix guarantee conservation of a linear function or the second norm . [ st : inv1 ] if a matrix @xmath63 possesses a vector @xmath55 in the co - kernel , i.e. @xmath64 , the ode system conserves the linear function @xmath65 . [ st : inv2 ] the condition @xmath66 yields the conservation of the frobenius norm of the solution , @xmath59 . generally , the opposite is not true : a linear function may persist even if the detecting vector does not belong to the co - kernel of @xmath48 . however , in many practical examples ( fokker - planck , schroedinger equations ) , it is the property @xmath67 ( or @xmath68 ) that available at the problem formulation stage . so , we will focus on these stronger conditions , and investigate how they can be brought into the galerkin projection . an abstract model reduction may be written as follows . given an orthogonal set of columns @xmath69 , @xmath70 , one considers instead of the large system a reduced ode , @xmath71 numerical treatment of this equation is cheap if the basis size is small , @xmath72 . the approximate solution of the initial problem writes as @xmath73 . many approaches exist to determine the basis sets @xmath74 , see e.g. the reviews @xcite . the well - known krylov method for the computation of the matrix exponential @xcite belongs to this class as well . another celebrated technique is the proper orthogonal decomposition @xcite , which extracts principal components from a set of _ snapshots _ @xmath75 using the singular value decomposition . the accuracy @xmath76 of the reduced model depends on the approximation quality of the basis set . in this paper , we employ the alternating tensor product optimization scheme to calculate a sequence of bases similar to the proper orthogonal decomposition adaptively , and both the implementation and the convergence properties will be discussed in section [ sec : tensor ] . however , an invariant linear function of the solution can be preserved under the galerkin projection independently on the particular basis . suppose we are given vectors @xmath77 , such that @xmath78 . let us include them into the basis : we concatenate @xmath79 and @xmath74 , and perform the orthogonalization , @xmath80 since the first @xmath81 columns of @xmath82 belong to the kernel of @xmath83 , the reduced matrix writes @xmath84 where we denote @xmath85 . now , derive the reduced solution in the new set , given as @xmath86 . the recursion step for the exponential series establishes as follows . @xmath87 for any @xmath88 hence we obtain @xmath89 therefore , since the first line contains only the identity w.r.t . the @xmath90-part , the solution writes in the form @xmath91 with the linear invariants @xmath92 preserved . the skew - symmetry , yielding the conservation of the second norm , is even easier to consider , since it is maintained under any galerkin projection . indeed , @xmath93 so , if @xmath68 , the same holds for the reduced system , and @xmath94 . moreover , since @xmath74 is orthogonal , it holds @xmath95 . thus , it is enough to guarantee @xmath96 . one way to do this is to expand the basis @xmath74 by @xmath97 in the same way as @xmath98 were incorporated in . however , it would inflate the storage in a tensor product scheme exponentially with time . since no further invariants are considered , we may adopt the rescaling . given @xmath99 , we keep the top part , representing the exact values of the first - order invariants , and update only the bottom as follows . we are looking for the value @xmath100 , satisfying @xmath101 and derive @xmath102 due to the orthogonality , it always holds that @xmath103 , and hence @xmath100 is well - defined as soon as @xmath104 . in numerical practice , however , one should be careful if @xmath105 becomes close to the machine precision . note that the galerkin reduction may be combined with the chebyshev discretization in time straightforwardly . given the orthogonal basis set @xmath74 , we assemble and solve the following @xmath106 system , @xmath107 where @xmath108 . both linear and quadratic invariants may be taken into account in the same way as shown in and , respectively . in the end of the previous section we saw that the chebyshev discretization of the ode may result in a matrix , given by a sum of two kronecker products . note that the kronecker product is a heavy memory consuming operation : if @xmath63 and @xmath109 contain @xmath110 entries , the product @xmath111 is defined already by @xmath112 elements . the ultimate goal thus may be formulated as follows : _ never expand kronecker products_. in the rest of the paper , we represent or approximate both the matrix and the solution by a multilevel summation of the kronecker products . by _ tensor _ , we mean nothing else but an array with three or more indices , and denote it as @xmath113 \in\c^{n_1\times\cdots\times n_d}$ ] . a tensor may come from a discretized multidimensional pde , for example : suppose a function @xmath114 is discretized by sampling at grid nodes @xmath115 , then the sampled values may be collected into a tensor @xmath116 . however , when we pose a linear system problem , or an ode , @xmath116 should be considered as a vector , cf . . we will denote the same data , re - arranged as a vector , by @xmath117 the _ multi - index _ operation @xmath118 stands for renumeration of the elements of @xmath116 . we use the rule @xmath119 consistent with the reversed kronecker product from the previous section : suppose @xmath120_{i_k=1}^{n_k},$ ] @xmath121 then @xmath122 the tensor train ( tt ) , or matrix product states ( mps ) representation for a tensor @xmath116 ( resp . vector @xmath7 ) is defined as follows , @xmath123 the summation indices @xmath124 are called the _ indices , and their ranges @xmath125 are the _ tensor train _ ranks ( tt ranks ) . we keep @xmath126 and @xmath127 for uniformity of presentation , but agree that @xmath128 . the right - hand side consists from the tt _ blocks _ @xmath129 , and is denoted as @xmath130 . note that each tt block depends only on one initial index @xmath131 , thus , the tt format is a generalization of the direct product . introducing the asymptotic bounds @xmath132 , @xmath133 , we may estimate the memory compression : @xmath134 entries of @xmath116 reduce to @xmath135 elements of the format @xmath136 . a matrix @xmath48 , corresponding to the solution @xmath7 , may be similarly seen as a @xmath137-dimensional tensor @xmath138 . however , since usually @xmath48 is a full - rank matrix , the straightforward @xmath137-dimensional tt is inefficient , as it contains the rank @xmath139 in the middle . instead , the _ matrix _ tt format is written with the index permutation , @xmath140 note that if all @xmath141 , this construction resolves to the kronecker product of matrices , @xmath142 . a pleasant confirmation of consistency is for example the identity matrix , having tt ranks 1 in this form . we may not limit ourselves with @xmath128 , and introduce a _ subtrain _ with nontrivial border indices , defined as follows , @xmath143 note that for @xmath144 or @xmath145 , one of the border indices vanishes . therefore , such cases will be convenient to denote as the _ interface _ matrices , @xmath146 we may also agree that @xmath147 , and @xmath148 . the interface matrices help us to show an important linearity of the tt map w.r.t . each tt block @xmath149 . indeed , construct the _ frame _ matrix , @xmath150 which does not contain @xmath151 , then it is easy to see that @xmath152 . note that the vector notation @xmath153 and the tensor notation @xmath129 share the same data , similarly to @xmath7 and @xmath116 . different notations are introduced to make the matrix products of the form @xmath154 consistent . a powerful approach for solution of various equations is the optimization of a certain function . for example , a typical problem arising in quantum physics is the calculation of the ground state , i.e. the lowest eigenpair of a symmetric matrix . it may be posed as the minimization of the _ rayleigh quotient _ @xmath155 . to seek the solution in the tt format , the density matrix renormalization group ( dmrg ) formalism was proposed in @xcite and extensively developed since then . in particular , it was generalized to the _ energy function _ @xmath156 @xcite to solve a linear system @xmath157 with the symmetric positive definite matrix @xmath48 and the right - hand side @xmath158 given in the tt format , @xmath159 , @xmath160 . if we restrict the solution to the tt format @xmath161 with _ fixed _ tt ranks @xmath162 , the exact minimization formulates as @xmath163 this highly nonlinear and nonconvex problem can rarely be solved at once . instead , the dmrg , or als ( alternating linear scheme ) algorithm performs a sequence of local steps , optimizing over each tt block @xmath149 while the others are fixed , @xmath164 the active blocks are selected in an iterative ( or _ sweeping _ ) manner , @xmath165 , and so on until convergence . the frame matrices and linearity of the tt map reduce to the _ local _ linear system , @xmath166 it is important that the frame matrix can be also represented as a matrix tt format with the tt ranks not larger than those of @xmath116 . indeed , introduce the reshapes @xmath167 and @xmath168 . for the rest @xmath169 , define the fictitious indices @xmath170 and assume that @xmath171 . then @xmath172 with the tt ranks @xmath173 . therefore , @xmath174 and @xmath175 in can be assembled efficiently using the matrix products in the tt format ( see @xcite ) . the tt map is not unique : if @xmath176 for any nonsingular @xmath177 of consistent sizes , it holds @xmath178 . therefore , we may ensure the orthogonality of @xmath179 and @xmath180 , and hence of the frame matrix @xmath181 . as a result , the _ condition numbers _ of the local systems satisfy @xmath182 i.e. is conditioned not worse than the initial problem @xmath157 . the orthogonalization requires qr decompositions of @xmath183 or @xmath184 matrices , containing the elements of @xmath149 , and is never a bottleneck in tt computations . so , we will omit it in algorithms , and assume implicitly that the frame matrices are always orthogonal at the moment they are required for . however , the alternating sweeping in a prescribed tt format suffers from several drawbacks . first , the tt ranks must be properly chosen a priory , which is a difficult task in a general problem . second , even with a correctly given initial guess , the iteration may stagnate at a spurious local minimum of @xmath185 constrained to the tt elements , far away from the optimal accuracy level for the given ranks . the first remedy to this situation was the so - called _ two - site _ dmrg @xcite . it optimizes not over one @xmath186-th block , but over two blocks simultaneously : we merge @xmath187 , perform the update , and then compute the svd to separate back @xmath188 . in this operation , the tt rank @xmath125 is likely to change , and the convergence may be improved . nevertheless , the latter takes place not always , for non - symmetric linear systems even the two - site dmrg may demonstrate no convergence . besides , we have to solve a ( more difficult ) two - dimensional system at each step . the new family of algorithms , the so - called _ alternating minimal energy _ ( amen ) @xcite performs an explicit enrichment of tt blocks by the residual information similarly to . suppose we have solved for @xmath189 and are holding the new @xmath190 and all the other solution blocks @xmath191 are old . compute the low - rank tt approximation of the residual , @xmath192 and perform the enrichment of the first block ( followed by the orthogonalization ) , @xmath193 the next interface @xmath194 and frame @xmath195 matrices contain the residual components @xmath196 , and if the approximation quality @xmath197 is ensured , the global convergence rate may be proved . in practice it is usually sufficient to perform the approximation of the residual via the simple alternating least squares algorithm , starting from a low - rank initial guess @xmath198 . this approach is heuristic , since no accuracy is guaranteed for the fixed - rank als . nevertheless , even with as small enrichment ranks as @xmath199@xmath200 , the algorithm converges very satisfactory , while the complexity may be substantially reduced , compared to the accurate svd - based calculation . the enrichment in the transition @xmath201 may be similarly written for the general step @xmath202 . for more details and theoretical analysis of the amen scheme we refer to @xcite . here , the final algorithm will be formulated in the next subsection directly for the temporal system . the time - dependent version of the amen algorithm agglomerates both the residual - based enrichment and the augmentation by the constraint vectors related to the linear system invariants . besides , the second norm correction takes place in the last step . we are given the @xmath203-dimensional system , and apply the amen algorithm for it . in the @xmath186-th step , we are solving the local system and obtain @xmath204 . to treat it as the `` first '' block and associate the residual and the enrichment , we consider the _ @xmath186-th reduced _ system @xmath205 where @xmath206 and @xmath207 are the matrix and the right - hand side of , @xmath208 , \quad g = x_0 \otimes ( se)+f.\ ] ] now it holds @xmath209 , i.e. the @xmath186-th block is the first block of the @xmath186-th reduced system . therefore , we may compute the reduced residual and use it for the enrichment , @xmath210 the block @xmath211 may be derived simultaneously with the als update of the global residual approximation @xmath212 : we need to project @xmath213 onto the right interface @xmath214 of @xmath215 . besides , assuming that the constraint vectors are also given in the tt formats , @xmath216 , we may include them in the enrichment at the very same step , and write @xmath217 with column - orthogonal @xmath149 , @xmath218 . the _ partial projections _ @xmath219 , bringing the vectors @xmath98 to the tt format of @xmath7 , read @xmath220 . in practice , they can be extracted from the @xmath221-factor of the qr decomposition in with no additional calculations , @xmath222 compared to , it appears to be more accurate to put the solution @xmath204 at the first place in the enrichment and orthogonalization procedures . the augmentation is performed for @xmath165 , i.e. the spatial part only . note that it adapts the rank @xmath223 as well , i.e. the rank for the space - time separation . the last block @xmath224 corresponds to the temporal variable , and contains the second norm correction , if necessary . the latter , however , must be reformulated a bit , to account for the @xmath79-part staying after the @xmath7-part in . note that in the @xmath0-th step , the partial projections @xmath225 turn to the standard galerkin projections of the vectors @xmath98 onto the spatial part of the solution , @xmath226 . aggregate them into a matrix , and find its qr decomposition , @xmath227 now , the projection @xmath228 extracts exactly the coefficients of @xmath229 in @xmath97 , and we may rewrite as follows , @xmath230 after that , the right - hand side for the local problem with @xmath231 writes @xmath232 . a few words must be devoted to the solution of local systems for the spatial tt blocks , and the last system . for the inner blocks , the local system size is @xmath233 , which may become too large for the direct gaussian elimination already for @xmath234 . as an alternative , we may use an iterative solver for this step , since the matrix @xmath174 inherits the tt structure of @xmath48 , and the fast matrix - vector product is available @xcite . in particular , we employ the bicgstab algorithm ( see e.g. @xcite ) with no preconditioner . however , the stopping threshold for the iterative solver enters as an additional parameter . the first idea is to take the same @xmath235 as is used for the svd approximations . it appears though that some problems require higher accuracy . we define thus a _ local accuracy gap _ @xmath236 , and solve the local systems with the residual tolerance @xmath237 . the last ( temporal ) system is solved directly , in order to restore the invariants with the machine precision . fortunately , this is usually not an issue , since the size @xmath238 of this system is small . the whole procedure summarizes to algorithm [ alg : tamen ] . note that it always performs the _ forward _ sweep , i.e. the dimension index runs through @xmath239 . it differs from the traditional als and dmrg schemes ( for symmetric problems ) , where the two - side iteration is conducted . however , it was found that with non - symmetric systems , the error may increase when we change the direction . therefore , since the orthogonalization steps 24 in alg . [ alg : tamen ] are not the bottleneck , we prefer to update the solution on the forward sweep only . temporal points @xmath240 ; matrix @xmath241 $ ] , right - hand side @xmath242 , initial guesses for the solution @xmath161 and the residual @xmath243 in @xmath203-dimensional tt formats ; initial state @xmath97 and detection vectors @xmath244 in @xmath0-index tt formats ; truncation threshold @xmath235 and local accuracy gap @xmath245 . updated solution @xmath7 , residual @xmath246 in the tt formats . prepare @xmath247 $ ] and @xmath248 in the tt format . make @xmath249 and @xmath250 row - orthogonal . initialize @xmath251 , @xmath252 . form @xmath253 and @xmath254 as in , solve @xmath255 up to the residual @xmath237 . reduce the rank by svd , @xmath256 , @xmath257 . update the global residual @xmath258 . update the reduced residual @xmath259 . assemble @xmath260 . compute the qr decomposition @xmath261 , extract @xmath262 . compute the qr decomposition @xmath263 . form the reduced temporal system with @xmath264 . correct the norm according to . solve and return @xmath265 . we have implemented the algorithm [ alg : tamen ] in matlab , and conducted simulations on a linux machine with 2.0 ghz intel xeon cpu using one thread . the code is available at http://github.com/dolgov/tamen[`http://github.com/dolgov/tamen ` ] . as the first example of the ode with the skew - symmetric matrix , consider the transport equation in the periodic domain @xmath266 ^ 2 $ ] with the central difference discretization scheme , @xmath267 where @xmath268 is the mesh step of the uniform grid @xmath269 , @xmath270 , @xmath271 . the pure convection is a notoriously fragile problem , since inaccurate discretizations may cause large spurious oscillations . in this test , we select a smooth initial state @xmath272 , and consider large grids , @xmath273@xmath274 , such that the spatial part is properly resolved , and we may focus on the time integration scheme . we choose this model example for demonstration purposes _ deliberately_. it allows a comparison with a known analytical solution , and contains both types of invariants considered in the paper . moreover , fine grids still make the problem challenging , cf . large cpu times of the straightforward matrix exponentiation in table [ tab : conv_timeerror ] . the initial state is a rank-1 @xmath275-dimensional tt tensor if we separate @xmath276 and @xmath277 . however , to achieve higher cost reduction , we employ the so - called qtt format @xcite : we choose @xmath278 , and decompose each index @xmath131 to the binary digits , @xmath279 after that , all tensors are reshaped to the new indexing and compressed into the @xmath280-dimensional tt format , for example , @xmath281 the matrix in is exactly ( and constructively , see @xcite ) representable in the qtt format with the maximal tt rank @xmath282 , but @xmath283 does not possess an exact decomposition anymore , and the accuracy threshold @xmath235 plays a nontrivial role . and @xmath284 vs. time and accuracy.,title="fig : " ] and @xmath284 vs. time and accuracy.,title="fig : " ] .convection example . cpu times ( seconds ) and errors in different methods and parameters . [ cols="^,^,^,^,^,^,^ " , ] with and without additional enrichments . degeneracy of the normalization @xmath285 ( right ) is shown only for the test with enrichments.,title="fig : " ] with and without additional enrichments . degeneracy of the normalization @xmath285 ( right ) is shown only for the test with enrichments.,title="fig : " ] ( left ) and maximal errors in @xmath286 ( right),title="fig : " ] ( left ) and maximal errors in @xmath286 ( right),title="fig : " ] as the initial state , we choose the multinomial function according to @xcite , @xmath287 where @xmath288 , and @xmath289 is the heaviside function . though even infinite copy numbers are potentially allowed , the probability function @xmath290 vanishes in the limit @xmath291 . in practice , we have to deal with a finite problem , so we restrict the copy numbers to finite values . to ensure that the truncated part outside is negligible , we take @xmath292 . moreover , we adjust the propensities of generation reactions as follows : @xmath293 together with the natural condition @xmath294 for @xmath295 , we obtain the _ normalization conservation _ property @xcite , @xmath296 , where @xmath297 is a vector of all ones . therefore , our first constraint vector @xmath298 . besides , as one of statistical outputs , we may be interested in the _ mean copy numbers _ , computed as @xmath299 where @xmath300 are the all - ones vectors of size @xmath301 . to make the computations of more accurate , we also include @xmath302 in the enrichment set , which reads therefore @xmath303 . in fig . [ fig : cme - time - rank ] , we investigate the tt ranks of the solution and the cpu times of the calculations with the following parameters : the tensor truncation threshold @xmath304 , local residual gap @xmath305 , number of chebyshev points in time @xmath306 , the residual tt rank in tamen @xmath307 , and the time grid is exp - uniform in accordance with @xcite , @xmath308 , @xmath309 , such that @xmath310 for the step @xmath311 . to cope with large grid sizes ( @xmath312 ) , we employ the qtt format , as in the first example . we remind that the crank - nicolson calculations in @xcite required about one hour on the same computer . from fig . [ fig : cme - time - rank ] we may observe that the straightforward tamen algorithm requires less time , but the enrichments @xmath79 make it larger . in fig . [ fig : cme - meanconc ] , we show the evolution of the mean copy numbers in time , and compare them with the reference values @xmath313 , computed with smaller tolerance @xmath314 . we may notice that the enrichments improve the accuracy significantly . we would like to emphasize that the artifacts in the left plane of fig . [ fig : cme - meanconc ] do not reflect explicitly the error in the solution @xmath290 , rather than in the means . recall that the maximal value of @xmath315 is @xmath316 . the exact solution would have a fast decay of the elements , which compensates large values of the index in . however , the approximate solution may conceal this decay by oscillations at the magnitude @xmath317 . taking into account @xmath304 , we may conclude that @xmath318 may be of the order of @xmath319 , as appears in fig . [ fig : cme - meanconc ] . the same consideration holds for @xmath320 in the end of the dynamics . nevertheless , if we keep @xmath302 in the tt format for @xmath290 exactly , the inner products in recover satisfactory accuracy . as in the previous example , the degeneracy of the normalization @xmath285 stays below @xmath321 in the enriched version of the algorithm ( see fig . [ fig : cme - time - rank ] ) . for the sake of clarity , we do not plot this quantity for the algorithm without enrichments , since it grows up to @xmath322 . we have proposed and studied the alternating iterative algorithm for approximate solution of ordinary differential equations in the mps / tt format . the method combines advances of dmrg techniques and classical iterative methods of linear algebra . started from the solution at the previous time interval as the initial guess , it often converges in 24 iterations , and delivers accurate solution even for strongly non - symmetric matrices in the right - hand side of an ode . another important ingredient is the spectral discretization scheme in time . the high - order approximation allows to simulate systems with purely imaginary spectrum without blowing the solution storage up , due to the absence of a poorly - separable noise , an unfortunate phenomenon in low - order schemes . the method possesses a simple mechanism how to bring linear conservation laws into the reduced tensor product model exactly , provided the generating vectors admit low - rank representations . the second norm of the solution can be also preserved easily . the numerical experiments reveal a promising potential of this method in long time simulations with the chemical master and similar equations . nevertheless , several further research directions open . the second norm conservation benefits from the orthogonality properties of the tensor format . is it possible to maintain general quadratic and high - order invariants ? we saw that accurate solution of the reduced systems in the tensor product scheme may be crucial for the robustness of the whole process . to what extent can we relax this demand ? are there reliable ways to precondition the local problems ? stiff problems may require either small time steps or large numbers of chebyshev points in time . are there ways to refine temporal grids adaptively inside the tensor format ? we are planning to address some of these questions in future work . another part of research will involve verification of the technique in a broad range of applications . recently , the amen algorithm for linear systems was employed in the simulation of a nuclear magnetic resonance experiment for large proteins @xcite . the tt formalism allows to consider the whole quantum hilbert space with a controllable accuracy an unprecedented flexibility in nmr calculations . in future , we plan to extend the proposed approach to more complicated time - dependent nmr problems . concerning the non - linear modeling , it is intriguing to revisit the simulations of plasma @xcite .

we propose an algorithm for solution of high - dimensional evolutionary equations ( odes and discretized time - dependent pdes ) in tensor product formats . the solution must admit an approximation in a low - rank separation of variables framework , and the right - hand side of the ode ( for example , a matrix ) must be computable in the same low - rank format at a given time point . the time derivative is discretized via the chebyshev spectral scheme , and the solution is sought simultaneously for all time points from the global space - time linear system . to compute the solution adaptively in the tensor format , we employ the alternating minimal energy algorithm , the dmrg - flavored alternating iterative technique . besides , we address the problem of maintaining system invariants inside the approximate tensor product scheme . we show how the conservation of a linear function , defined by a vector given in the low - rank format , or the second norm of the solution may be accurately and elegantly incorporated into the tensor product method . we present a couple of numerical experiments with the transport problem and the chemical master equation , and confirm the main beneficial properties of the new approach : conservation of invariants up to the machine precision , and robustness in long evolution against the spurious inflation of the tensor format storage . _ keywords : _ high dimensional problems , tensor train format , mps , als , dmrg , ode , conservation laws , dynamical systems . _ msc2010 : _ 15a69 , 33f05 , 65f10 , 65l05 , 65m70 , 34c14 .

introduction ordinary differential equations tensor product representations and methods numerical experiments conclusion

arxiv

the study of spin - dependent transport phenomena has recently attracted much attention.@xcite it has opened the way for the field of spintronics,@xcite literally spin electronics , where new device functionalities exploit both the charge and spin degrees of freedom . there are various approaches in this sphere . for instance , the use of a quantum - dot setup@xcite has been proposed , which can be operated either as a spin filter to produce spin - polarized currents or as a device to detect and manipulate spin states . spintronics in graphene has also been proposed@xcite recently . in the following we address this issue in a new type of materials , graphene nanodisks.@xcite graphene nanostructure@xcite such as graphene nanoribbonsfujita , ezawaprb , brey , rojas , son , barone , kim , avouris , xu , ozyilmaz and graphene nanodisks@xcite could be basic components of future nanoelectronic and spintronic devices . graphene nanodisks are nanometer - scale disk - like materials which have closed edges . they are constructed by connecting several benzenes , some of which have already been manufactured by soft - landing mass spectrometry.rader,kim,berger there are varieties of nanodisks , among which zigzag trigonal nanodisks with size @xmath0 are prominent because they have @xmath0-fold degenerated zero - energy states . we have already shown@xcite that spins make a ferromagnetic order and that the relaxation time is finite but quite large even if the size @xmath0 is very small . we refer to this property as quasi - ferromagnet . furthermore we have argued@xcite that a nanodisk behaves as if it were a quantum dot with an internal degree of freedom , where the conductance exhibits a peculiar series of coulomb blockade peaks . in this paper we make an investigation of the spin current in the zero - energy sector of the trigonal zigzag nanodisk . we first analyze how the spin of a nanodisk filters the spin of the current by assuming that the nanodisk is a rigid ferromagnet . the fact that the direct and exchange coulomb interactions are of the same of magnitude plays an important role . however the nanodisk is not a rigid ferromagnet but a quasi - ferromagnet . hence an intriguing reaction phenomenon occurs : namely , the spin of the nanodisk can be controlled by the spin of the current . using these properties we propose some applications for spintronic devices , such as spin memory , spin amplifier , spin valve , spin - field - effect transistor and spin diode . this paper is organized as follows . in sec.[secnanodisk ] , we summarize the basic notion of the nanodisk . the low - energy physics is described by electrons in the @xmath0-fold degenerated zero - energy sector , which form a quasi - ferromagnet due to a large exchange interaction . in sec.secspinfilter , we investigate the spin - filter effects based on the master equation . in sec.[secreaction ] , we analyze the reaction to the spin of the nanodisk from the spin of electrons in the current . we discuss the relaxation process using the landau - lifshitz - gilbert equation . in sec.secapplication , we propose various spintronic devices and explore their properties . graphene nanodisks are graphene derivatives which have closed edges . the hamiltonian is defined by@xmath1where @xmath2 is the site energy , @xmath3 is the transfer energy , and @xmath4 is the creation operator of the @xmath5 electron at the site @xmath6 . the summation is taken over all nearest neighboring sites @xmath7 . owing to their homogeneous geometrical configuration , we may take constant values for these energies , @xmath8 and @xmath9ev . we have searched for zero - energy states or equivalently metallic states in graphene nanodisks with various sizes and shapes.@xcite the emergence of zero - energy states is surprisingly rare . among typical nanodisks , only trigonal zigzag nanodisks have degenerate zero - energy states and show metallic ferromagnetism , where the degeneracy can be controlled arbitrarily by designing the size . trigonal zigzag nanodisks are specified by the size parameter @xmath0 as in fig.[fignanodisk](a ) . the size-@xmath0 nanodisk has @xmath0-fold degenerated zero - energy states,@xcite where the gap energy is as large as a few ev . hence it is a good approximation to investigate the electron - electron interaction physics only in the zero - energy sector , by projecting the system to the subspace made of those zero - energy states . the zero - energy sector consists of @xmath0 orthonormal states @xmath10 , @xmath11 , together with the su(n ) symmetry . we can expand the wave function of the state @xmath10,@xmath12 in terms of the wannier function @xmath13 localized at the site @xmath6 . the amplitude @xmath14 is calculable by diagonalizing the hamiltonian ( [ hamiltb ] ) . is defined in this way . the number of carbon atoms is given by @xmath15 . ( b ) the nanodisk - lead system . the nanodisk with @xmath16 is connected to the right and left leads by tunneling coupling @xmath17 and @xmath18 . the chemical potential is @xmath19 or @xmath20 at each lead . , scaledwidth=46.0% ] we include the coulomb interaction between electrons in the zero - energy sector.@xcite it is straightforward to rewrite the coulomb hamiltonian as@xmath21 , \label{hamilferro}\end{aligned}\]]where @xmath22 and @xmath23 are the coulomb energy and the exchange energy between electrons in the states @xmath10 and @xmath24 . here , @xmath25 is the number operator and @xmath26 is the spin operator,@xmath27with @xmath28 the annihilation operator of electron with spin @xmath29 in the state @xmath10 : @xmath30 is the pauli matrix . the remarkable feature is that there exists a large overlap between the wave functions @xmath31 and @xmath32 , @xmath33 , since the state @xmath10 is an ensemble of sites as in ( [ eqa ] ) and identical sites are included in @xmath34 and @xmath24 . consequently , the dominant contributions come from the on - site coulomb terms not only for the coulomb energy but also for the exchange energy : indeed , it follows that @xmath35 in the on - site approximation . we thus obtain@xmath36where@xmath37with the coulomb potential @xmath38 . the coulomb energy @xmath39 is of the order of @xmath40ev because the lattice spacing of the carbon atoms is @xmath41 in graphene . since the exchange energy @xmath23 is as large as the coulomb energy @xmath22 , the spin stiffness @xmath23 is quite large furthermore , we have checked@xcite numerically that all @xmath42 are of the same order of magnitude for any pair of @xmath43 and @xmath44 , implying that the su(n ) symmetry is broken but not so strongly in the hamiltonian ( [ hamilferro ] ) . it is a good approximation to start with the exact su(@xmath0 ) symmetry . then , the zero - energy sector is described by the su(n ) heisenberg - hubbard model,@xmath45with @xmath46 . we rewrite @xmath47 as @xmath48 + \left ( \frac{u}{2}-\frac{j}{4}\right ) n_{% \text{tot}}^{2 } \notag \\ & + \left ( \frac{u}{2}+\frac{j}{4}\right ) \sum_{\alpha } n\left ( \alpha \right ) \notag \\ = & -js_{\text{tot}}^{2}+\left ( \frac{u}{2}-\frac{j}{4}\right ) n_{\text{tot}% } ^{2}+\left ( \frac{u}{2}+j\right ) n_{\text{tot } } \label{hamildx}\end{aligned}\]]in terms of the total spin,@xmath49and the total electron number,@xmath50at the half - filling , the eigenstate of the hamiltonian @xmath47 is labeled as @xmath51 , where @xmath52 is the total spin - angular momentum and @xmath53 is its z - component . based on this hamiltonian we have shown@xcite that the relaxation time of the ferromagnetic - like spin polarization is quite large even if the size of trigonal zigzag nanodisks is quite small . such a nanodisk may be called a quasi - ferromagnet . we refer to the total spin @xmath54 of a nanodisk as the nanodisk spin . we consider a system made of a nanodisk connected by two metallic leads [ fig.[fignanodisk](b ) ] . the model hamiltonian is given by@xmath55where @xmath47 is the hamiltonian ( [ hamild ] ) of a nanodisk , and @xmath56 is the lead hamiltonian @xmath57the hamiltonian @xmath58 describes a noninteracting electron gas in the leads with @xmath59 , while @xmath60 and @xmath61 are the transfer hamiltonian between the left ( l ) and right ( r ) leads and the nanodisk , respectively,[hamilt]@xmath62with @xmath63 the tunneling coupling constant : we have assumed that the spin does not flip in the tunneling process . the nanodisk - lead system looks similar to that of the @xmath0-dot system.tarucharev however , there exists a crucial difference . on one hand , in the ordinary @xmath0-dot system , an electron hops from one dot to another dot . on the other hand , in our nanodisk system , the index @xmath64 of the hamiltonian runs over the @xmath0-fold degenerate states and not over the sites . according to the hamiltonian ( [ hamilt ] ) , an electron does not hop from one state to another state . hence , it is more appropriate to regard our nanodisk as a one - dot system with an internal degree of freedom . the aim of this paper is to investigate a nanodisk as a spin filternittaprl02 based on the hamiltonian ( [ totalhamil ] ) . the setup we consider is a lead - nanodisk - lead system [ fig.[fignanodisk](b ) ] , where an electron makes a tunnelling from the left lead to the nanodisk and then to the right lead . we investigate how the electron spin is affected by the nanodisk spin during the transport process . the dynamics of the nanodisk system is described by the master equation , @xmath65 , \]]where @xmath66 represents the probability to find the system in the state @xmath51 , and @xmath67 is the transition rate between the states @xmath68 and @xmath69 . the master equation describes a stochastic evolution in the space spanned by the states @xmath68 . jumps between different states are assumed to be markovian . the stationary solution is given by the detailed balance condition @xmath70 . since the hamiltonian @xmath47 only depends on @xmath71 , the probability @xmath72 has no dependence on the spin component @xmath53 . then it is convenient to do a sum over @xmath53 , @xmath73with @xmath74 . the master equation is rewritten as@xmath75 .\]]in the following we denote @xmath76 as @xmath77 for simplicity . in this paper we consider the small coupling limit , @xmath78 , @xmath79l , r , where the dominant process is the sequential tunneling : it is of the order of @xmath80 , while the cotunneling process of the order of @xmath81 . in the sequential - tunneling regime the transition rate , @xmath82 , is obtained by the fermi s golden rule , beenakker , recher , golovach@xmath83 \delta _ { n^{\prime } , n-1}. \label{transrate}\end{aligned}\]]let us explain notations . first,@xmath84 \right\ } ^{-1}\]]is the fermi function at temperature @xmath85 , with @xmath86 the chemical potential at the right or left lead , @xmath87r , l . we set @xmath88second , @xmath89 is the energy difference between the two states @xmath90 and @xmath91,@xmath92third,@xmath93is the tunneling rate through the right or left lead , where @xmath94 is the density of states at the fermi level @xmath95 in the leads , which is a constant , and @xmath96it follows that the tunneling rate @xmath97 is a constant for the states @xmath91 and @xmath90 connected by a markov step , and zero otherwise . the current through the nanodisk can be written as @xmath98 , \label{spincurre}\]]where @xmath99 runs over the states @xmath91 and @xmath100 . thus the current @xmath101 depend on the spin configuration of electrons in the nanodisk . we have argued that the nanodisk is a quasi - ferromagnet . for the sake of simplicity , we start with the simplification that the ground state is a ferromagnet with polarized up - spins . when the transition interaction is small enough , a single electron tunnels at once , namely , the electron number in the nanodisk increases or decreases by one , @xmath102 . first of all , we notice that , when @xmath103 , the tunneling of spin - up electrons is blocked by the pauli s exclusion principle . the markov chain of spin configurations contain only a finite number of states . for instance it is @xmath104 in the case of @xmath105 nanodisk . we number them as indicated in fig.[figstate ] . an arrow ( @xmath106 ) indicates each markov step between two states with different spin configurations . the region surrounded by red solid lines shows the most dominant process for spin - polarized current . in this approximation there is no down - spin polarized current . the region surrounded by green dotted lines yields the next dominant contribution , which allows a down - spin polarized current as well.,scaledwidth=40.0% ] we calculate the energy @xmath107 of various states @xmath108 in this chain . according to the above numbering convention ( fig.[figstate ] ) , they are@xmath109based on the effective hamiltonian ( [ hamildx ] ) they are calculated as follows,@xmath110and so on . the order of the energies is@xmath111the probability to find the state @xmath68 contains the boltzman factor @xmath112 . hence , the most dominant markov chain consists only of the states @xmath113 and @xmath114 , and it is denoted as @xmath115 . this process yields the spin - up current . we consider the second dominant process . though @xmath116 , the state @xmath117 can only be reached via the state @xmath118 , which allows the spin - down current as well as the spin - up current . first we analyze the most dominant process , which involves only the ground state and the first excited state ( fig.[figstate ] ) . the currents are given by@xmath119 \notag \\ = & \frac{i_{0}}{2}\frac{\sinh ( \delta \mu /k_{\text{b}}t)}{\cosh ( \delta \mu /k_{\text{b}}t)+\cosh ( ( u-\mu ) /k_{\text{b}}t ) } , \\ i_{\downarrow } = & 0,\end{aligned}\]]where@xmath120 in terms of the tunneling rate @xmath97 given by ( [ tunnerate ] ) . the above expression can be further simplified in the following two limits . in the high - temperature limit ( @xmath121 ) the sequential tunneling current takes a simple form @xmath122 , \end{aligned}\]]where @xmath89 is defined in ( [ gap ] ) . on the other hand , in the low - temperature limit ( @xmath123 ) the current is given by@xmath124\]]with the step function @xmath125the current @xmath126 flows in the triangle domain ( @xmath127 ) in the @xmath128 plain . and @xmath129 in the @xmath130 nanodisk . we have set @xmath131 . the horizontal @xmath132-axis is the chemical potential @xmath133 , and the horizontal @xmath134-axis is the bias voltage @xmath135 . ( a ) the vertical @xmath136-axis is the current . ( b ) more currents flow in brighter regions . ( 1 ) the current @xmath137 induced by the process @xmath138 , which produces the up - spin polarized current . ( 2 ) the current @xmath139 induced by the process @xmath140 , which produces the up - spin polarized current . ( 3 ) the sum of the above two processes ; @xmath141 . ( 4 ) the current @xmath142 induced by the process @xmath143 , which produces the down - spin polarized current ; @xmath144 . ( 5 ) the difference @xmath145 between the up - spin and down - spin current , @xmath146.,scaledwidth=46.0% ] in order to search for the down - spin polarized current @xmath147 , it is necessary to analyze the second dominant process . it includes four states @xmath148 , as shown in the fig.[figstate ] . in this approximation , the transition matrix becomes a tridiagonal matrix , and we can solve the master equations exactly . the nonequilibrium density of states is calculated as@xmath149where the index @xmath150 stands for the four states with the energy @xmath151 . we can explicitly calculate @xmath152 as@xmath153with @xmath154 . the current @xmath155 is calculable with the use of ( [ spincurre ] ) and ( [ transrate ] ) , whose result we show in the fig.[figspinfilter ] . in particular we have@xmath156 \notag \\ & \times \left [ \theta \left ( \mu _ { l}-\delta _ { 32}\right ) -\theta \left ( \mu _ { r}-\delta _ { 32}\right ) \right ] \end{aligned}\]]in the zero - temperature limit . the current @xmath147 flows in the triangle domain ( @xmath157 and @xmath158 ) in the @xmath128 plain . in the conventional spin filter the ferromagnet is very rigid . on the other hand , the life time is finite in the case of a nanodisk , though it is quite long in spite of its small size . we expect a reaction to the nanodisk spin from the spin - polarized current , provided the nanodisk spin is not controlled externally , say , by magnetic field . this effect is very interesting , since it is an intrinsic nature to quasi - ferromagnets . we inject an electron into a nanodisk . let @xmath159 or @xmath160 be the energy increase when the spin of an injected electron is parallel or anti - parallel to the nanodisk spin . according to the hamiltonian ( [ hamild ] ) they are @xmath161when the direction between the nanodisk spin and the electron spin is @xmath162 , the energy increase is given by@xmath163where @xmath164 represents the probability of finding the injected electron to have spin @xmath165 . based on the spin - rotational symmetry we may write the effective hamiltonian as@xmath166where @xmath167 and @xmath168 is the normalized spin with @xmath169 . we introduce the gilbert damping term phenomenologically,@xmath170where @xmath64 is a dimensionless constant ( @xmath171 ) . using the variational method for the hamiltonian @xmath172 , we obtain the landau - lifshitz - gilbert equation,@xmath173where @xmath174 is the effective magnetic field produced by the injected electron spin,@xmath175with the gyromagnetic ratio @xmath176 . for definiteness we now inject the up - spin electron to the nanodisk . the effective magnetic field is@xmath177we investigate the dynamics of the normalized spin @xmath178 of the nanodisk under this field . in the polar coordinate , setting @xmath179we rewrite the landau - lifshitz - gilbert equation ( [ llg ] ) as@xmath180this is equivalent to@xmath181which we can solve explicitly,@xmath182 , \notag \\ \phi \left ( t\right ) & = \frac{\gamma b_{\text{eff}}}{1+\alpha ^{2}}t , \label{angle}\end{aligned}\]]where @xmath183 is an integration constant , and @xmath184 is the relaxation time given by@xmath185it is proportional to @xmath186 and hence proportional to the disk size @xmath0 because of ( [ effecb ] ) . the initial phase @xmath187 is related to the parameter @xmath183 as@xmath188 . \label{initialphase}\]]thus , @xmath189 corresponds to @xmath190we use the parameter @xmath183 instead of the initial phase @xmath191 for simplicity . and the vertical axis is the angle @xmath192 . the thin curve is the asymptotic function @xmath193 $ ] of the bold curve for @xmath194 . the dotted line is the asymptotic value for @xmath195.,scaledwidth=33.0% ] the time scale for the direction of the nanodisk spin to align with that of the spin - polarized current is @xmath184 , where the effective magnetic field ( [ effecb ] ) is proportional to the injected current @xmath196 . in other words , we can control the polarization of the nanodisk spin by using the spin - polarized current . we note that this is possible since the nanodisk is a quasi - ferromagnet . indeed , there exist no effective magnetic field ( [ effecb ] ) in the conventional ferromagnet . we summarize the spin properties of a nanodisk and an incoming electric current . first of all , being a quasi - ferromagnet , the nanodisk has a definite polarization . with respect to the incoming electric current there are three cases . ( 1 ) the polarized current , where all electrons have a definite polarization , rotates the nanodisk spin to that of the incoming current , as we have shown in sec.[secreaction ] . ( 2 ) the unpolarized current , where the polarization of each electron is completely random , does not induce any effective magnetic field . hence it is filtered so that the outgoing current is polarized to that of the nanodisk . ( 3 ) the partially polarized current , where the polarization of each electron is at random but the averaged polarization has a definite direction , induces a net effective magnetic field . hence it rotates the nanodisk spin to that of the incoming current , and then is filtered so that the outgoing current is completely polarized to the averaged polarization of the incoming current . furthermore , it is possible to control the nanodisk spin externally by applying magnetic field . then the outgoing current has the same polarization as that of the nanodisk , irrespective of the type of incoming current . using these properties we propose some applications of graphene nanodisks for spintronic devices . the first example is a spin memory.@xcite for a good memory device three conditions are necessary : ( i ) it keeps a long life time information ; ( ii ) information stored in the memory can be read out without changing the information stored ; ( iii ) it is possible to change the information arbitrarily . first , since the life time of the nanodisk quasi - ferromagnet is very long compared to the size@xcite,@xmath197we may use the nanodisk spin as an information . next , we can read - out this information by applying a spin - unpolarized current . the outgoing current from a nanodisk is spin - polarized to the direction of the nanodisk spin . thus we can obtain the information of the nanodisk spin by observing the outgoing current . finally , the direction of the nanodisk spin can be controlled by applying a spin - polarized current into the nanodisk . thus , the nanodisk spin satisfies the conditions as a memory device . the important point is that the size is of the order of nanometer , and it is suitable as a nanodevice . the second example is a spin amplifier . we inject a partially - polarized - spin current , whose average direction we take to be up for definiteness . thus , @xmath198 . on the other hand , the direction of the nanodisk spin is arbitrary . since spins in the nanodisk feel an effective magnetic field proportional to @xmath199 , they are forced to align with that of the partially - polarized - spin current after making damped precession . by using ( [ angle ] ) the time dependence is given by @xmath200 } } , \\ i_{\downarrow } \left ( t\right ) & = i_{\downarrow } ^{\text{in}}\sin \frac{% \theta \left ( t\right ) } { 2}=i_{\downarrow } ^{\text{in}}\frac{1}{\sqrt{1+\exp % \left [ 2t/\tau _ { \text{filter}}\right ] } } , \end{aligned}\]]where we have set @xmath189 , which means @xmath201 : see fig.[figamprelax ] . the outgoing current is initially given by@xmath202as a function of the incoming current @xmath203 . the time scale is given by the relaxation time ( [ relaxtime ] ) . and the vertical axis is the current @xmath126 and @xmath147 . we have set @xmath204 , @xmath205 . the currents saturate after enough time ( @xmath206 for @xmath126 , @xmath207 for @xmath147 ) , and the amplification ratio is @xmath208 in this example.,scaledwidth=35.0% ] after enough time @xmath209 , all spins in the nanodisk take the up direction and hence the outgoing current @xmath210 is the perfectly up - polarized one , @xmath211consequently , the small difference @xmath212 is amplified to the large current @xmath213 . the amplification ratio is given by , which can be very large . this effect is very important because the signal of spin will easily suffer from damping by disturbing noise in leads . by amplifying the signal we can make circuits which are strong against noises . the third example is a spin valve , or giant magnetoresistance effect [ fig.[figspinvalve]].@xcite we set up a system composed of two nanodisks sequentially connected with leads . we apply external magnetic field , and control the spin direction of the first nanodisk to be @xmath214and that of the second nanodisk to be @xmath215we inject an unpolarized - spin current to the first nanodisk . the spin of the lead between the two nanodisks is polarized into the direction of @xmath216 . subsequently the current is filtered to the up - spin one by the second nanodisk . the outgoing current from the second nanodisk is@xmath217since we can arrange the angle @xmath218 externally , we can control the magnitude of the up - polarized current from zero to one . in this sense the system act as a spin valve . the forth example is a spin - field - effect transistor@xcite [ fig.figspintrans ] . we again set up a system composed of two nanodisks sequentially connected with leads . we now apply the same external magnetic field to both these nanodisks , and fix their spin direction to be up , @xmath215as an additional setting , we use a lead between the two nanodisks possessing a strong rashba - type spin - orbit coupling@xcite , @xmath219spins make precession while they pass through the lead . the spin - rotation angle is given by@xcite@xmath220where @xmath221 is the electron effective mass in the lead and @xmath222 is the length of the lead . we can control @xmath223 by changing the coupling strength @xmath224 externally by applying electric field.nittag the outgoing current from the second nanodisk is@xmath225since we can arrange the angle @xmath223 by applying electric field and control the magnitude of the up - polarized current from zero to one , we expect the system acts as a spin - field - effect transistor as in the conventional case . the fifth example is a spin diode [ fig.[figspindiode ] ] . we use a system similar to the spin - field - effect transistor but with the following differences . first , two nanodisks have different sizes . when the left nanodisk is larger than the right nanodisk , the relaxation time of the left nanodisk @xmath226 is larger than that of the right nanodisk @xmath227,@xmath228second , the applied magnetic field is taken so small that the nanodisk spin can be controlled by a polarized current . for definiteness we take the direction of the magnetic field to be up . third , the lead has the rashba - type interaction such that the rotation angle is @xmath229 with small @xmath230 , say , @xmath231 . when no currents enter the nanodisk , the direction of two nanodisk spins is identical due to a tiny external magnetic field . when we inject the current in this state , the net outgoing current is very small,@xmath232this is the `` off '' state of spin diode . , the current flows from the left lead to the right lead ( @xmath233 ) , or in the opposite way ( @xmath234 ) . the incoming current is unpolarized , which is made polarized by the first nanodisk . the electron spin in the central lead is rotated by the rashba - type interaction.,scaledwidth=43.0% ] let us inject an unpolarized pulse square current to the system , starting at @xmath235 and finishing at @xmath236@xmath237where @xmath238 denotes the spin . the system become the `` on '' state by the pulse . when the bias voltage is such that @xmath239 , the current flows into the left nanodisk and then into the right nanodisk . the left nanodisk acts as a spin filter . the current in the central lead is initially up - polarized but is rotated by the angle @xmath223 due to the rashba - type coupling effect . then it enters the right nanodisk . since the relaxation time is @xmath240 , the total spin - dependent charge carried by the current is given by@xmath241with@xmath242 \right ] , \\ q_{\downarrow } \left ( t\right ) = & i^{\text{in}}\tau _ { \text{r}}\sinh ^{-1}% \left [ \exp \left [ -\left ( t - t_{0}\right ) /\tau _ { \text{r}}\right ] \right ] .\end{aligned}\]]on the other hand , when @xmath243 , the current enter the right nanodisk and goes out from the left nanodisk . since the relaxation time is @xmath244 , the total spin - dependent charge is given by the above formulas but with the replacement of @xmath240 by @xmath245 . because the size of two nanodisk are different , these two currents behave in a different way . and they are very different for the same incoming pulse wave due to the difference in the nanodisk size . we have set @xmath247 , and @xmath248.,scaledwidth=43.0% ] we define the direction dependent total charge @xmath249 and @xmath250 , where @xmath251 is the total charge with the spin @xmath238 when charges flow from left to right , while @xmath252 is the total charge when charges flow from right to left . we find the relation@xmath253from ( [ dioderelax ] ) , which implies the up ( down ) component increases ( decreases ) from the initial value . . the horizontal axis is the pulse width @xmath254 . we plot @xmath255 for various initial values @xmath256 $ ] . we have set @xmath257.,scaledwidth=40.0% ] we define the rectification coefficient by@xmath258which is approximately equals to @xmath259we illustrate this rectification coefficient as a function of the pulse width @xmath254 for various initial phases @xmath191 or equivalently @xmath183 in fig.[figpulsepeak ] . each curve has a peak structure . hence , when the relaxation - time ratio @xmath260 and the initial phase @xmath261 are given , it is possible to optimize the width @xmath262 so that the rectification coefficient @xmath255 is maximized . this maximized value of @xmath255 diverges as @xmath263 . however , the relaxation time diverges as well . it would be efficient to take the initial phase @xmath261 to make @xmath264 for a spin diode . we have studied the electromagnetic properties of the zigzag trigonal nanodisk by projecting the system to the zero - energy sector . we may regard it as a quasi - ferromagnet characterized by the exchange energy as large as the coulomb energy . the system is well approximated by the su(n ) heisenberg - hubbard model . the relaxation time is finite but quite large even if the size is very small . being a ferromagnet , it can be used as a spin filter . namely , only electrons with spin parallel to the spin of the nanodisk can go through it . additionally , it has a novel feature that it is not a rigid ferromagnet . the incoming spin - polarized current can rotate the nanodisk spin itself . combining the advantages of both these properties , we have proposed a rich variety of spintronic devices , such as spin memory , spin amplifier , spin valve , spin - field - effect transistor and spin diode . graphene nanodisks could well be basic components of future nanoelectronic and spintronic devices .

graphene nanodisk is a graphene derivative with a closed edge . the trigonal zigzag nanodisk with size @xmath0 has @xmath0-fold degenerated zero - energy states . a nanodisk can be interpletted as a quantum dot with an internal degree of freedom . the grand state of nanodisk has been argued to be a quasi - ferromagnet , which is a ferromagnetic - like states with a finite but very long life time . we investigate the spin - filter effects in the system made of nanodisks and leads based on the master equation . the finite - size effect on spin filter is intriguing due to a reaction from the polarization of incoming current to a quasi - ferromagnet . analyzing the relaxation process with the use of the landau - lifshitz - gilbert equation , we explore the response to four types of incoming currents , namely , unpolarized current , perfectly polarized current , partially polarized current and pulse polarized current . we propose some applications for spintronics , such as spin memory , spin amplifier , spin valve , spin - field - effect transistor and spin diode .

introduction nanodisk quasi-ferromagnets spin filter finite-size effects: reaction to quasi-ferromagnet spintronic devices and applications conclusions

arxiv

random matrix theory ( rmt ) @xcite was first applied to physical systems in the context of nuclear physics @xcite and has found application in condensed matter physics especially in the study of disordered systems . it has also been shown to play a crucial role in understanding how isolated quantum systems thermalize @xcite . in contrast to classical mechanics there is no notion of a phase space in quantum mechanics . hence , the concept of quantum ergodicity ( thermalization ) is not very well understood , and is presently a very active area of research @xcite.it is believed that isolated quantum systems that thermalize generally do not have dynamics strongly constrained by conservation laws and are thus not integrable . these non - integrable systems can be characterized by random matrix ensembles depending on the symmetry of their hamiltonians . integrable models have infinite conserved quantities in the thermodynamic limit @xcite , as a consequence of which they display no level repulsion and obey a poissonian level spacing distribution given by @xmath0 , where @xmath1 is energy spacing measured in units of the mean level spacing . in contrast a non - integrable system has a finite number of conserved quantities even in the thermodynamic limit . once , one has accounted for the corresponding symmetries , the rest of the energy spectrum displays level repulsion with @xmath2 as @xmath3 . depending on the symmetries of the system , @xmath4 can have the following forms : 1 . @xmath5 for the gaussian orthogonal ensemble ( goe ) , 2 . @xmath6 for the gaussian unitary ensemble ( gue ) ( where time reversal symmetry is broken ) 3 . @xmath7 for gaussian symplectic ensemble(gse ) ( where time reversal symmetry is preserved but spin rotation symmetry is broken ) . in the presence of disorder , the situation is different . the disorder renders the system non - integrable by destroying conservation laws that may have existed in its absence . however , it is possible that for a sufficiently strong value of disorder , the level spacing distribution is poissonian indicative of localization in the system . for non - interacting disordered systems ( without spin orbit coupling ) , it is known that in one and two dimensions , even an infinitesimal amount of disorder is sufficient to localize all states in the thermodynamic limit @xcite . in three dimension even in the thermodynamic limit one needs to have a finite amount of disorder to localize all states . significantly less is understood about the nature of localization for interacting disordered systems . the one parameter scaling theory for non - interacting systems @xcite is expected not to apply in this case and there is much debate over whether there is a finite amount of disorder required for localization and what its dependence on interaction strength and dimensionality is @xcite . we do not attempt to join the debate in this paper focussing instead on systems with weak enough disorder that localization does not occur . in a previous work with ramaswamy we investigated how integrability in a one - dimensional interacting system is destroyed by perturbations @xcite . we found that the scale of the perturbation that caused a crossover to non - integrability goes to zero with increasing system size as a power law in the system size whose exponent is independent of microscopic details . we conjectured that the value of this exponent was dependent only on the random matrix ensemble describing the non - integrable system which was of the goe type . in this paper , we elaborate on that claim by studying systems described by different random matrix ensembles and and investigate the crossovers among them . we also investigate the effect of dimensionality on the finite - size scaling of the perturbation that causes the crossover . to this end , it is most convenient for us to look at models which have disorder in addition to interactions . in this paper , we have looked at two disordered models , 1 ) a one dimensional interacting model of hard - core bosons and 2 ) a three dimensional model of non - interacting particles with spin - orbit - coupling(soc ) . these models allow us to realize phases with poissonian , goe , gue and gse level spacing distributions and thus study the crossovers among them . our main result is that the scale of perturbations responsible for the crossovers among the different classes of systems goes to zero with increasing system size as a power law , with an exponent that appears to depend only on the random matrix ensembles of the classes independent of microscopic details . we consider the one - dimensional heisenberg spin-1/2 chain containing @xmath8 spins with random on site magnetic field in the @xmath9-direction . the hamiltonian for this system is : @xmath10 where @xmath11 is the spin operator at site @xmath12 and @xmath13 is the nearest neighbor exchange constant . @xmath14 is a random magnetic field in the @xmath9 direction , which is uniformly distributed in the interval @xmath15 $ ] . for this model @xmath16 , where @xmath17 is the time reversal operator and therefore , @xmath18 the presence of a magnetic field breaks the time reversal symmetry . however , the antiunitary operator @xmath19 commutes with @xmath20 . the operator @xmath21 reverses the sign of @xmath22 and @xmath23 but not of @xmath24 @xcite . @xmath20 thus preserves an unconventional time reversal symmetry and the level spacing distribution for @xmath20 is goe type in the presence of a large enough random magnetic field that is not so large as to localize all states . introducing a three site interaction @xcite , the hamiltonian becomes , @xmath25 } , \label{eqn : hamiltonian_broken t}\ ] ] which breaks time reversal symmetry and unlike for the hamiltonian of eqn . [ eqn : hamiltonian ] , this time reversal symmetry violation can not be compensated by any anti - unitary spin reversal operator . the three spin term can be written as , @xmath26}&=&i j_{t}s_{j}^{z}\epsilon_{jkl}s_{k}^{+}s_{l}^{- } \nonumber \\ & = & i j_{t}s_{j}^{z}s_{j+1}^{+}s_{j+2}^{-}-i j_{t}s_{j}^{z}s_{j+2}^{+}s_{j+1}^{- } \label{eqn : triple product term}\end{aligned}\ ] ] and using a holstein and primakoff transformation @xcite this model can be mapped onto one of hardcore bosons . the spin operators in terms of the bosonic operators are @xmath27 where , @xmath28 and @xmath29 are the creation and annihilation operators for the hard core bosons and @xmath30 is the number density operator . the resultant hamiltonian , which we have studied is thus , @xmath31 if @xmath32 , the model reduces to the integrable @xmath33 model of hardcore boson @xcite . non - zero @xmath14 makes the model non - integrable and also destroys its lattice translational symmetry . we have used exact diagonalization techniques to obtain all the eigenvalues of the hamiltonian and have able to reach up to @xmath34 sites at half filling . the results we report have been averaged over different realizations of disorder to achieve convergence . we consider a three - dimensional disordered system @xcite with soc on a cubic lattice . the hamiltonian of this model is given by : @xmath35 where @xmath36 labels the sites of the lattice and @xmath37 labels the spin . @xmath38 labels nearest neighbor pairs @xmath36 and @xmath12 . @xmath39 is @xmath40 random on - site potential which does not contain the spin index and thus does not violate time reversal invariance . @xmath39 is chosen from the interval @xmath41 of uniformly distributed random variables . the nearest neighbor hopping @xmath42 has the following form : @xmath43 where , @xmath44 is the spin orbit coupling strength and @xmath45,@xmath46 and @xmath47 are independent uniform random variables taken from the interval @xmath48 . for nonzero @xmath44 , this model breaks spin rotation symmetry and hence its eigenvalues give a gse type level spacing distribution . when @xmath44 is zero , the model belongs to the goe class . a large value of @xmath49 will cause localization and the again energy eigenvalues will obey a poissonian level spacing distribution . we will restrict ourself to a region where model exhibits no localization . a study of the crossover from poissonian to gse level spacing statistics can be studied even in two dimensions for the model of eqn . [ eq : socham ] . however , the crossover from poissonian to goe statistics can not be examined in this model in two dimensions since this requires the soc to be turned off which will cause localization ( and no phase with goe statistics ) for any amount of disorder . like the system described by eqn . [ eqn : hamiltonian ] , this model too has disorder and hence no lattice translation symmetry . we thus write down the hamiltonian in a real space basis to perform exact diagonalization to obtain all the energy eigenstates . we have been able to diagonalize systems with @xmath50 sites and a maximum value of @xmath51 . once again , we average over different realizations of disorder to obtain good statistics for the quantities of interest . setting @xmath52 in eqn . [ eqn : hamiltonian_hardcore ] we obtain an integrable model of hard - core bosons with poissonian level spacing statistics @xcite . increasing the value of @xmath49 keeping @xmath53 , a crossover from poissonian to goe level spacing statistics is observed as shown in figure [ fig : level spacing poi - goe ] . for model ii , one can see a crossover from poisson to goe level spacing statistics as we increase the value of @xmath49 in eqn . [ eq : socham ] provided @xmath44 ( the soc strength ) in eqn . [ eq : intform ] is set to zero . this is shown in figure [ fig : poi_goe_3d ] . the intermediate distribution can be to a brody distribution @xcite . [ cols="^,^ " , ] for model ii , we have observed that the integrability breaking parameter which is responsible for the poissonian to goe / gse crossover goes to zero as a power law with exponent 1 and for the goe to gse crossover this exponent becomes 3/2.(figure [ fig : level powerlaw goe - gue and poi - goe ] ) we have also studied a different version of model ii to investigate the poisson to gue and goe to gue crossovers . here , we have taken the nearest neighbor hoping matrix @xmath54 to have the following form , @xmath55 this model breaks time reversal symmetry . increasing @xmath44 and @xmath49 simultaneously we have observed a poissonian - gue crossover and then setting @xmath56 and increasing @xmath44 , a goe to gue crossover . the exponents of the power law scaling of the crossover value of the perturbation are 1 and 3/2 for the poissonian - gue and goe - gue cases respectively . why do we obtain the same exponent for the poissonian to gue and poissonian to gse crossovers as we do for poissonian to goe ? a possible answer is that the poissonian to gue and poissonian to gse crossovers can be thought of as a crossover first from poissonian to goe , which introduces level repulsion and then a crossover from goe to gue and gse through the breaking of additional symmetries ( time reversal for goe - gue and spin rotation invariance for goe - gse ) . for this interpretation to be correct , the fall off of the crossover value with system size for goe - gue and goe - gse has to be faster than for poissonian - goe . this is indeed the case for all the cases we have considered as can be seen from figure [ fig : level powerlaw goe - gue and poi - goe ] . a second important question is why the poissonian - goe / gue / gse crossover scaling exponents are different for the two different models ? the reason does not seem to be that the models have different dimensionality . for instance , the poissonian - goe crossover for model ii is the same in both two and three dimensions . we conjecture that this difference can be attributed to the fact that model i contains interactions among the elementary degrees of freedom while model ii does not . this has a bearing on the integrable limits of these models as well in that the elementary degrees of freedom are correlated even in the integrable limit of model i and not model ii . this difference could well be responsible for the different exponents in the two models . to further investigate this and also the role of dimensionality , it would be desirable to obtain crossovers among all the different categories of ensembles in different dimensions . however , it is not possible for us to realize a goe phase in the non - interacting model in two dimensions owing to the fact that localization becomes operative in the absence of spin - orbit coupling and time reversal breaking in two dimensions . on the other hand , it is difficult to study model i in more than one dimension since the presence of interactions greatly reduces the system sizes we can study in higher dimensions . a through investigation of the effect of interactions and dimensionality on the crossovers among different ensembles will require a study of more models which we will defer to a later work . we have investigated the finite scaling of perturbations which cause crossovers among different random matrix ensembles in two different models : a one dimensional model of interacting hard core bosons ( or equivalently spin 1/2 ) objects and a disordered model of non - interacting particles with disorder and spin - orbit coupling . obtaining the level spacing distribution using numerical exact diagonalization , we have found that the scaling is a power law for crossovers among all the different ensembles ( poissonian , goe , gue and gse ) that can be realized in these models and obtained the corresponding exponents . we find that the scaling from poissonian to any of the other ensembles is dominated by the poissonian - goe crossover which introduces level repulsion while the symmetry breaking that causes the goe - gue and goe - gse crossovers is a sub - dominant effect . we also conjecture that there is a difference in the scaling depnding on whether the elementary degrees of freedom interact or not . the authors would like to acknowledge many useful discussions with sriram ramaswamy . sm thanks the department of science and technology , government of india . rm thanks tapan mishra for discussions and acknowledges support from ugc fellowship . 99 m.l.mehta , _ random matrices _ ( elsevier / academic press , 2004 ) f.j.dyson , journal of mathematical physics * 3 * , 140 ( 1962 ) m.l.mehta , nuclear physics * 18 * , 395 ( 1960 ) g.montambaux , d.poilblanc , j.bellissard and c.sire , phys . rev . lett . * 70 * , 497 ( 1993 ) m.rigol , v.dunjko and o.m . , nature * 45 * , 854 ( 2008 ) j.m.deutsch , phys . a * 43 * , 2046 ( 1991 ) m.srednicki , phys . rev . e * 50 * , 888 ( 1994 ) b.s.shatry , phys . lett . * 56 * , 1529 ( 1986 ) p.w.anderson , phys . 109 * , 1492 ( 1958 ) e.abrahams , p.w.anderson , d.c.licciardello and t.v.ramakrishnan , phys . lett . * 42 * , 673 ( 1979 ) v.oganesyan and david.a.huse , phys . b * 75 * , 155111 ( 2007 ) p.h.song and d.l.shepelyansky , phys . rev . b * 61 * , 15546 ( 2000 ) r.modak , s.mukerjee and s.ramaswamy , arxiv preprint arxiv:1309.1865 ( 2013 ) f.haake , _ quantum signatures of chaos _ ( springer , 3rd ed . 2010 ) y.avishai and j.richert and r.berkovits , phys . b * 66 * , 052416 ( 2002 ) m.a.cazalilla , r.citro , t.giamarchi , e.orignac , m.rigol , reviews of modern physics * 83 * , 1405 ( 2011 ) m.rigol , phys a * 80 * , 053607 ( 2009 ) s.n.evangelou , phys . rev . lett . * 75 * , 2550 ( 1995 ) r.sepehrinia , phys . b * 81 * , 045104 ( 2010 ) t.a.brody , j.flores , j.b.french , p.a.mello , a.pandey and s.s.m.wong , rev . mod . phys . * 53 * , 385 ( 1981 ) d.a.rabson , b.n.narozhny and a.j.mills , phys . b * 69 * , 054403 ( 2004 )

using numerical diagonalization we study the crossover among different random matrix ensembles [ poissonian , gaussian orthogonal ensemble ( goe ) , gaussian unitary ensemble ( gue ) and gaussian symplectic ensemble ( gse ) ] realized in two different microscopic models . the specific diagnostic tool used to study the crossovers is the level spacing distribution . the first model is a one dimensional lattice model of interacting hard core bosons ( or equivalently spin 1/2 objects ) and the other a higher dimensional model of non - interacting particles with disorder and spin orbit coupling . we find that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place . this exponent is independent of microscopic details of the perturbation . we also find that the crossover from the poissonian ensemble to the other three is dominated by the poissonian to goe crossover which introduces level repulsion while the crossover from goe to gue or goe to gse associated with symmetry breaking introduces a subdominant contribution . we also conjecture that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system .

introduction models methodology conclusions acknowledgments references

arxiv

the past few years have seen extensive investigations of the @xmath1 meson producing reactions at close to threshold energies . the experiments are aimed either at directly searching @xcite for the possible existence of eta - mesic nuclei as a result of the strong attractive nature of the @xmath1-n interaction @xcite or studying the final state eta - nuclear interaction to eventually conclude on the existence of eta - mesic nuclear states @xcite . a common feature of the data on @xmath1 production near threshold is the strong deviation of the cross sections from phase space . it can be understood as a manifestation of the strong attractive @xmath1-n interaction ( arising basically due to the proximity of the @xmath1-n threshold to the @xmath9 resonance n@xmath10(1535 ) ) . experiments on @xmath1 production have been performed in nucleon - nucleon collisions and have been extended to proton - deuteron and deuteron - deuteron collisions too @xcite . though the focus of reactions such as the p + d @xmath11 p + d + @xmath1 and @xmath12 is on investigating possibilities of bound states of eta mesons and 2 - 3 nucleon systems , theoretical studies of the reaction mechanism revealed interesting features too @xcite . in the p - d collisions , the production near threshold is found to be dominated by a two - step mechanism where the large momentum transfer in producing the @xmath1 meson is shared by three nucleons . these findings naturally led to the curiosity of what happens when the @xmath1 interacts with more than three or four nucleons in the final nucleus . with this motivation , measurements of the @xmath0 reaction were carried out by the turin group in 1993 @xcite at an incident energy of 683 mev . a theoretical study of this reaction along with others of the type , @xmath13 and @xmath14 was performed in @xcite . part of the emphasis of this work was on obtaining the right form factors for @xmath2li ( and @xmath2be ) and the interactions of the mesons with the nuclei in the final states were neglected . the interest in the @xmath0 reaction revived once again by the recent proposal of studying this reaction at cosy , jlich , at an incident energy of 673 mev @xcite . having performed detailed theoretical studies of the p + d @xmath11 p + d + @xmath1 and @xmath12 reactions @xcite and the @xmath1 meson interactions with the deuteron , @xmath5he and @xmath15he nuclei @xcite , we now develop a model to study the interaction of @xmath1 mesons with a @xmath5he-@xmath15he cluster , namely , the @xmath2be nucleus . the @xmath1-@xmath2be interaction is then incorporated in a theoretical calculation of cross sections for the @xmath0 reaction , with four possible low - lying states of the @xmath2be nucleus . an analysis of its effects on the @xmath0 cross sections presented here , should be useful in motivating further experimental studies of this reaction . based on literature which supports considering the @xmath2be nucleus as a bound state of an alpha ( @xmath15he ) and @xmath5he @xcite , we model the @xmath1-@xmath2be final state interaction in the form of a three body problem of the @xmath1-@xmath5he-@xmath15he interaction . regarding the @xmath3li too as a cluster of an alpha and a deuteron @xcite , the @xmath0 reaction is considered to proceed through the @xmath12 reaction with the @xmath4 remaining a spectator . besides , the present work focuses on the low energy region of @xmath1 production where ( a ) the @xmath12 production amplitude is large ( i.e. the p and d interact strongly to produce an @xmath1 ) and ( b ) the cluster picture of low lying levels of @xmath2be and @xmath3li is reasonably good . there exists in principle , the possibility of considering the deuteron as a spectator . however , a reaction of the type @xmath16 followed by the combination of @xmath17 and a deuteron to form the @xmath2be nucleus , does not agree with the cluster model approach here ( since @xmath2be is hardly known to be a cluster of d + @xmath18li , where @xmath18li is in fact unstable ) . the possibility of an intermediate @xmath19 reaction , with further steps of the final state @xmath20 from this reaction combining with the spectator deuteron , i.e. , @xmath21 forming a @xmath5he which eventually combines with the final state @xmath4 ( from the above reaction ) to form @xmath2be still remains . however , this is not a practical option with no information available on the @xmath22 reaction . as mentioned above , we consider the @xmath2be nucleus as a two body system made up of a @xmath5he and @xmath15he nucleus and construct an elastic transition matrix for the three body problem of an @xmath1 meson , @xmath5he nucleus and an @xmath4 ( @xmath15he ) . the individual scattering of the @xmath1 meson on @xmath5he and @xmath4 is evaluated using the @xmath23-matrices constructed earlier by the present authors @xcite . these t - matrices are numerically evaluated using few body equations and include the off - shell rescattering of the @xmath1 on the nucleons inside @xmath5he and @xmath15he . the @xmath1-@xmath5he t - matrix is tested to reproduce the @xmath12 cross section reasonably well @xcite . to formulate the three body problem of the @xmath1-@xmath5he-@xmath15he interaction , let us define @xmath24 and @xmath25 to be the coordinates of the @xmath5he and @xmath15he nuclei respectively , with respect to the mass-7 centre of mass system . defining the internal jacobi coordinate of the relative distance between the @xmath5he and @xmath15he nuclei as @xmath26 , one can see that @xmath27 and @xmath28 with @xmath29 and @xmath30 . the @xmath1-@xmath2be t - matrix is then written as , @xmath31\ ] ] where @xmath32 and @xmath33 are the medium modified t - matrices for the off - shell @xmath1 scattering on the bound @xmath5he and @xmath15he respectively . @xmath34 represents the cluster wave function of @xmath2be with angular momentum @xmath35 and @xmath36 , where , @xmath37 is the total @xmath1-nucleus energy in the centre of mass and @xmath38 is the energy required for the break up of @xmath2be @xmath39he+@xmath15he . the in - medium @xmath1-@xmath5he and @xmath1-@xmath15he t - matrices are written using a faddeev type decomposition @xcite , namely , @xmath40 where @xmath41 and the indices @xmath42 and @xmath43 correspond to the @xmath5he and @xmath15he t - matrices respectively . the @xmath44 s represent the single scattering terms and are the matrices for purely elastic @xmath1-@xmath5he and @xmath1-@xmath15he scattering . they have the form , @xmath45 \ , , \ ] ] with @xmath46 and @xmath41 as mentioned above . at the low energies considered here , the @xmath1-n interaction is dominated by the @xmath9 resonance n*(1535 ) and hence we perform a partial wave expansion and retain only @xmath47-waves , which reduces ( [ eta3he4he ] ) to the following two equations for @xmath1-@xmath5he ( @xmath48 ) and @xmath1-@xmath15he ( @xmath49 ) : @xmath50 @xmath51 where @xmath52 in eq . ( [ tetan ] ) has reduced to @xmath53 written in terms of the spherical bessel functions @xmath54 and @xmath55 . replacing the equation for @xmath49 in @xmath48 , we obtain a recursive relation for @xmath48 as follows , @xmath56 once @xmath48 is evaluated from ( [ eta3he ] ) , it can be replaced into the equation for @xmath49 and the two can be substituted in ( [ etabetmat ] ) to evaluate the @xmath1-@xmath2be t - matrix . thus ( [ etabetmat ] ) is evaluated retaining only @xmath47-waves in @xmath48 and @xmath49 . @xmath48 is evaluated numerically using the @xmath1-@xmath5he and @xmath1-@xmath15he t - matrices , @xmath57 and @xmath58 respectively , as inputs . @xmath57 and @xmath58 are themselves evaluated numerically using few body equations and an input coupled channel @xmath1-n t - matrix . details of this formalism can be found in @xcite . the two models of the elementary @xmath59 t - matrix used here will be discussed in the next subsection . the coupled channel @xmath1-n t - matrix which is required for the evaluation of the @xmath1-@xmath5he and @xmath1-@xmath15he t - matrices , @xmath57 and @xmath58 , is taken from two different models available in literature . in @xcite , a transition matrix including the @xmath60n and @xmath1n channels with the n*(1535 ) resonance playing a dominant role was constructed . it consisted of the meson - n * vertices and the n * propagator as given below : @xmath61 with , @xmath62 , where @xmath63 @xmath64 are the self energy contributions from the @xmath65 and @xmath59 loops . in @xcite elastic and inelastic @xmath1-deuteron scattering as well as @xmath1 photoproduction on the deuteron was studied using this @xmath59 model . we shall use two parameter sets available , one which leads to a scattering length of @xmath66 = ( 0.75 , 0.27 ) fm and another which leads to @xmath66 = ( 0.88 , 0.41 ) fm . we shall refer to this model henceforth as model a. model b used in the present work is taken from @xcite . the t - matrix for @xmath67 is written in a separable form as , @xmath68 where the on - shell part , @xmath69 , is described in an effective range approximation as , @xmath70 the off - shell form factors have the form , @xmath71 , where @xmath72 is the length parameter in this model . the parameter sets in this model are obtained from a fit to the @xmath73 , @xmath74 , @xmath75 and @xmath76 data . the parameters required in eqs . ( [ green1 ] ) and ( [ green2 ] ) above can be found in @xcite . we choose four parameter sets with @xmath59 scattering lengths of ( 0.88,0.25 ) fm , ( 0.77 , 0.25 ) fm , ( 0.51,0.26 ) fm and ( 0.4 , 0.3 ) fm . the choice of kinematic variables in a multiple scattering formalism where the many body scattering matrix is written in terms of a two body matrix is not unique in literature . the two most commonly used approaches are ( i ) the fixed scatterer approximation ( fsa ) and ( ii ) the on - energy shell impulse approximation ( oei ) @xcite ( related to yet another approach , namely , the fixed impulse approximation ( fia ) @xcite ) . the difference between these approaches ( which becomes important at intermediate and high energies ) lies in the fact that in the fsa , the struck nucleon is assumed to recoil with the target as a whole , while in the oei , it recoils freely . this means that in the fsa , the two - body operator does not follow from the equation for a free two - body t - matrix , but rather contains the mass of the nucleus ( in which the two - body system is embedded ) in the kinetic energy . though this seems to be mathematically sound , such a two body t - matrix can have physically undesired features . for example , in @xcite it was found from a phase shift study that such a two - body t - matrix may not display the resonant behaviour which it would be expected to . in the same work , in connection with @xmath60-nucleus scattering , it was found that considering a free two - body matrix simulated the contribution of continuum states ( otherwise neglected in that work ) and brought theory in closer agreement with data . though the differences arising from the particular approaches used above may not be crucial at the low energies considered in the present work , the many - body problem considered here is a bit more complicated and hence a small discussion is in order . in the works mentioned above , one studies the differences of the approaches involved , in a multiple scattering of a hadron on the individual nucleons in the nuclear target . here however , the problem appears to be that of one multiple scattering problem embedded inside another . the @xmath1 mesons scatter off the @xmath5he and @xmath15he nuclei as in a three body multiple scattering problem ( of the @xmath1-@xmath5he-@xmath15he system ) . however , the individual @xmath1-@xmath5he and @xmath1-@xmath15he scatterings are represented by t - matrices for the multiple scatterings of the @xmath1 on the three and four nucleons inside @xmath5he and @xmath15he . we choose then to work in a framework where we start by evaluating the momenta @xmath77 and @xmath78 in the @xmath1-@xmath2be centre of mass frame and then evaluate the individual @xmath1-@xmath5he and @xmath1-@xmath15he t - matrices at a lorentz boosted energy - momentum in the @xmath1-@xmath5he and @xmath1-@xmath15he centre of mass systems . the energy in the in - medium propagators ( inside @xmath2be ) is however taken to be in the @xmath1-@xmath2be centre of mass system . since the energy spacing between the first four low - lying levels of the @xmath79 nucleus is small , we include the contribution from these four levels . we consider the angular momentum states with @xmath80 and @xmath81 corresponding to the @xmath82 and @xmath83 levels respectively . the cluster wave functions for @xmath2be and those for @xmath3li required in the production amplitude of the @xmath0 reaction are ( 1 ) generated using a wood - saxon potential @xcite and ( 2 ) taken from a green s function monte carlo ( gfmc ) variational calculation with the urbana potential @xcite . in the gfmc case , the wave function for @xmath2li is used to represent the @xmath2be one . the @xmath4 and deuteron cluster in @xmath3li is assumed to be in the @xmath84 state . assuming that the beam proton interacts with a loosely bound deuteron in the @xmath3li nucleus to produce an @xmath1 meson and @xmath5he ( with the @xmath4 remaining a spectator ) the production amplitude for the @xmath0 reaction can be written in terms of that for the @xmath12 process . the off shell @xmath1 meson thus produced then rescatters on @xmath2be ( i.e. the @xmath5he-@xmath15he cluster ) . this production of the @xmath1 and its final state interaction ( fsi ) with @xmath2be is represented schematically in fig . 1 , and the corresponding transition matrix is written as , @xmath85 where @xmath86 and @xmath87 are the initial and final momenta in the centre of mass system . @xmath88 , @xmath89 and @xmath90 are the spin projections of the proton , @xmath91 and @xmath79 respectively . cluster model for @xmath1 production in the @xmath0 reaction . diagram ( a ) corresponds to the direct on - shell @xmath1 production and ( b ) to an @xmath1 which is first produced off - shell and rescatters via the @xmath1 + @xmath2be @xmath92 + @xmath2be process to become on - shell . , title="fig:",width=264,height=188 ] cluster model for @xmath1 production in the @xmath0 reaction . diagram ( a ) corresponds to the direct on - shell @xmath1 production and ( b ) to an @xmath1 which is first produced off - shell and rescatters via the @xmath1 + @xmath2be @xmath92 + @xmath2be process to become on - shell . , title="fig:",width=264,height=188 ] + the production matrix for a relative angular momentum @xmath35 between the @xmath93 and @xmath4 is written as , @xmath94 where , @xmath95 is the momentum of the incoming proton and @xmath96\vec{k}_p + \vec{p}_1 $ ] of the deuteron in @xmath3li . @xmath97 is the spin projection of @xmath5he . the on - shell @xmath1 meson momentum is denoted as @xmath98 and the off - shell one as @xmath99 . @xmath100\vec{k}_{\eta } + \vec{p}_2 $ ] or @xmath100\vec{q } + \vec{p}_2 $ ] is hence the momentum of on- or off - shell @xmath5he in @xmath2be . since the @xmath4 particle remains a spectator , its momentum in @xmath3li and @xmath2be is required to be the same and @xmath101\vec{k}_p - \vec{p}_1 \ , = \ , -[4/7]\vec{k}_{\eta } ( \vec{q } ) - \vec{p}_2 $ ] ( for on - shell ( off - shell ) @xmath1 production ) . thus @xmath102\vec{k}_p\,-\,[4/7]\vec{k}_{\eta } ( \vec{q})$ ] ( where @xmath103 and @xmath104 are the fermi momenta inside @xmath3li and @xmath2be respectively ) . the integration in ( [ prodtmat2 ] ) should in principle have been over both the fermi momenta , @xmath103 and @xmath104 , however , the above relation between them renders the integration over @xmath104 in ( [ prodtmat2 ] ) redundant . further , when one evaluates the unpolarized cross sections , one sums over the spins in the final state and averages over those in the initial state . as a result , in such a calculation , some sums in ( [ prodtmat1 ] ) and ( [ prodtmat2 ] ) become redundant . the @xmath105matrix for the process , @xmath106 , is written in a two - step model from our earlier work @xcite . considering the complexity of the present calculations which include the off - shell rescattering as given by the second term in ( [ prodtmat1 ] ) with the @xmath1-@xmath2be fsi and the fact that the two - step model of the @xmath12 is itself quite involved , we neglect the effect of fermi motion on @xmath106 and hence take it out of the integral over @xmath107 in ( [ prodtmat2 ] ) . the momentum space wave functions are expressed in terms of fourier transforms of their radial forms . thus the integral in momentum space is transformed to that in coordinate space . all this simplifies eq . ( [ prodtmat2 ] ) to a good extent and it can be written as , @xmath108 the @xmath3li-@xmath2be transition form factor for angular momentum states with @xmath80 and @xmath81 . the solid line corresponds to the gfmc variational wave functions with the urbana potential for @xmath80 and the dashed , dot - dashed and double dot - dashed lines are those generated using a woods - saxon potential ( for @xmath80 and @xmath81).,width=264,height=264 ] is the transition form factor for @xmath3li @xmath109 @xmath2be , with momentum transfer @xmath110 for example in the off - shell case . though not written explicitly , the transition form factor depends on the total angular momentum @xmath111 since the radial wave function in @xmath2be depends , even if mildly , on @xmath111 . in fig . 2 we present the two form factors with @xmath112 ( @xmath113 ) and @xmath114 ( @xmath115 ) required in the present work , using two different prescriptions of the nuclear wave functions mentioned in the previous section . with the @xmath80 levels , @xmath116 and @xmath117 being very close to each other , the difference between the two @xmath80 form factors is not visible in the figure . the form factor enters ( [ prodtmat1 ] ) via ( [ equation8 ] ) as ( [ prodtmat2 ] ) is simplified to ( [ equation8 ] ) due to the neglect of the fermi motion in the @xmath118 t - matrix . it is evaluated at @xmath119 in the first term of ( [ prodtmat1 ] ) whereas over a range of momenta in the second term of ( [ prodtmat1 ] ) where it appears inside the integral . the dotted vertical lines in fig . 2 indicate the relevant range corresponding to the beam energies of the present work . we shall see later how this difference between the two types of form factors affects the cross sections . in what follows , we shall present the cross section calculations for the @xmath0 reaction with an emphasis on the @xmath1-@xmath2be fsi . the transition matrices for the elementary processes @xmath60 + n @xmath120 + n and @xmath1 + n @xmath120 + n which enter as inputs to the production and fsi matrices are chosen from coupled channels calculations . the calculations in figs 3 - 7 are done within model a with a choice of parameters which corresponds to a scattering length of @xmath121 fm . within the two - step model of the @xmath12 reaction we use here , the data on this reaction were reproduced well with this choice of the scattering length . in fig . 8 , a comparison of the total ( inclusive ) cross sections within models a and b of the @xmath67 t - matrix and using different sets of scattering lengths is made . total cross sections as a function of the proton beam energy for different states of the @xmath2be nucleus . the dashed lines are plots without the inclusion of the @xmath1-@xmath2be final state interaction . the wave functions for @xmath3li and @xmath2be are generated using the woods - saxon potential.,width=377,height=377 ] same as fig . 3 except for the fact that the wave functions for @xmath3li and @xmath2be are from the gfmc variational method with the urbana potential.,width=377,height=188 ] .beam and excess energies for different levels of the @xmath2be nucleus [ cols= " < , < , < , < , < " , ] the smallest scattering length @xmath66 used here , leads to a positive real part of the @xmath1-@xmath2be scattering length , @xmath122 . however , this choice of parameters ( model f in @xcite ) is mentioned as an unconventional solution obtained after dropping the photoproduction data from the fits . the first four entries in table ii corresponding to large @xmath66 , display @xmath122 to be large with negative real parts . apart from the commonly known condition that the real part of the scattering length should be negative @xcite , in the third reference in @xcite , the authors found that the condition for the existence of a bound state is that @xmath123 for an eta - nucleus scattering length of ( @xmath124 ) . with these two conditions it seems that the first few entries in table ii for the large @xmath59 scattering lengths support the possibility of eta - mesic states . in @xcite , while investigating the connection between the @xmath1-@xmath5he scattering lengths and the corresponding binding energies and widths , the authors also use the above conditions but mention that in reality none of the above can be taken as a sufficient condition . the bottomline is then that it would indeed be premature to base the conclusions only on the signs or magnitudes of the scattering lengths . one should rather perform a better analysis for the search of @xmath1-@xmath2be states using the present @xmath1-@xmath2be model and a time delay analysis as in @xcite or a k - matrix analysis as in @xcite before drawing definite conclusions . the present work aimed at performing a thorough investigation of the effects of the @xmath1-@xmath2be interaction in the @xmath0 reaction near threshold . the work was partly motivated by the recent revival of interest in this reaction by the cosy - gem collaboration @xcite after the first measurement in 1993 . this work also comes as a sequel to our various earlier studies on @xmath1 meson production in light nuclei . a two - step model for the @xmath12 reaction including the @xmath1-@xmath5he interaction was tested earlier with data by the present authors @xcite . this model is used as an input to develop a cluster model approach for the @xmath0 reaction near threshold . the @xmath1-@xmath2be interaction is included in a multiple scattering formalism for an @xmath1 scattering on a @xmath5he-@xmath15he cluster inside @xmath2be . the @xmath1-@xmath5he and -@xmath15he scatterings are themselves included using few body equations . the calculations are done for four low - lying levels of @xmath2be . to the best of our knowledge , this is the most detailed study of the @xmath1-@xmath2be interaction in the @xmath0 reaction performed so far . the interesting two hump structure in the summed total cross section ( summed over @xmath111 ) hints toward a very strong near threshold effect of the @xmath1-@xmath2be interaction ( especially of the off - shell rescattering of the @xmath1 on @xmath2be ) which is worth verifying experimentally in future . + + * acknowledgements * + the authors gratefully acknowledge v. jha for the help in computing the cluster wave functions using the woods - saxon potential and k. p. khemchandani for her help with the @xmath125he programs and many useful discussions . the authors are also thankful to the anonymous referee for his / her constructive criticism . the work done by two of the authors , nju and bkj , was supported by the ramanna fellowship , awarded by the department of science and technology , government of india , to bkj . + 99 m. pfeiffer _ et al_. , phys . lett . * 92 * , 252001 ( 2004 ) ; g. a. sokol _ et al . _ , . lett . * 102 * , 71 ( 2000 ) . r. s. bhalerao and l. c. liu , phys . * 54 * ( 1985 ) 865 ; l.c . liu and q. haider , phys . c * 34 * , 1845 ( 1986 ) ; q. haider and l.c . liu , phys . c * 66 * , 045208 ( 2002 ) . n. g. kelkar , k. p. khemchandani and b. k. jain , j. phys . g : nucl . part . phys . * 32 * , l19 ( 2006 ) ; n. g. kelkar , k. p. khemchandani and b. k. jain , j. phys . * g 32 * , 1157 ( 2006 ) ; n. g. kelkar , phys . lett . * 99 * , 210403 ( 2007 ) . s. wycech , anthony m. green and j.a . niskanen , phys . * c 52 * , 544 ( 1995 ) ; v.a . tryasuchev , phys . atom . 60 * , 186 ( 1997 ) ( yad . fiz . * 60 * , 245 ( 1997 ) ) ; c.y . song , x.h . zhong and l. li , p.z . ning , europhys . * 81 * , 42002 ( 2008 ) ; d. jido , e.e . kolomeitsev , h. nagahiro and s. hirenzaki , nucl . phys . * a 811 * , 158 ( 2008 ) ; h. nagahiro , d. jido and s. hirenzaki , arxiv:0811.4516 . j. berger _ et al . _ , phys . rev . lett . * 61 * , 919 ( 1988 ) ; b. mayer _ et al_. , phys . c * 53 * , 2068 ( 1996 ) ; a. wroska _ et al . _ , a * 20 * , 640 ( 2005 ) ; h. , -h . , adam _ et al . c * 75 * , 014004 ( 2007 ) . khemchandani , n.g . kelkar and b.k . jain , nucl . phys . * a 708 * , 312 ( 2002 ) ; _ ibid _ , phys . rev . c * 68 * , 064610 ( 2003 ) ; n.j . upadhyay , k.p . khemchandani , b.k . jain and n.g . kelkar , phys . c * 75 * , 054002 ( 2007 ) ; k. p. khemchandani , n. g. kelkar and b. k. jain , phys . c * 76 * , 069801 ( 2007 ) . scomparin e. _ et al . _ , j. phys . * 19 * l 51 , ( 1993 ) . j. s. al - khalili , m. b. barbaro and c. wilkin , j. phys . g * 19 * , 403 ( 1993 ) . m. ulicny _ et al . _ , aip conf * 603 * , 543 ( 2001 ) ; h. machner , acta phys . slov . * 56 * , 227 ( 2006 ) ; e - print : nucl - ex/0511034 . t. kajino , t. matsuse and a. arima , nucl . a * 413 * , 323 ( 1984 ) . j. v. noble , phys . c * 9 * , 1209 ( 1974 ) ; a. c. merchant and n. rowley , phys . b * 150 * , 35 ( 1985 ) . s. a. rakityansky _ c * 53 * , 2043 ( 1996 ) ; s. a. rakityansky _ _ , few body syst . * 9 * , 227 ( 1995 ) . a. fix and h. arenhvel , eur . j a * 9 * , 119 ( 2000 ) . a. m. green and s. wycech , phys . c * 71 * , 014001 ( 2005 ) . e. kujawski and e. lambert , annals of physics * 81 * , 591 ( 1973 ) . r. crespo , a. m. moro and i. j. thompson , nucl . phys . a * 771 * , 26 ( 2006 ) . v. b. belyaev , s. a. rakityansky and j. wrzecionko , nucl . a * 368 * , 394 ( 1981 ) . b. buck and a. c. merchant , j. phys . g * 14 * l211 ( 1988 ) ; v. i. kukulin , v. g. neudatchin and yu . f. smirnov , nucl . phys . a * 245 * ( 1975 ) 429 . j. l. forest _ et al . _ , phys . rev . * c 54 * , 646 ( 1996 ) . c. j. joachain 1975 _ quantum collision theory _ ( north - 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the @xmath0 reaction has been investigated with an emphasis on the @xmath1 meson and @xmath2be interaction in the final state . considering the @xmath3li and @xmath2be nuclei to be @xmath4-d and @xmath4-@xmath5he clusters respectively , the reaction is modelled to proceed via the @xmath6 \ , \rightarrow \ , ^3{\rm he } [ \alpha ] \ , + \ , \eta$ ] reaction with the @xmath4 remaining a spectator . the @xmath1 meson interacts with @xmath2be via multiple scatterings on the @xmath5he and @xmath4 clusters inside @xmath2be . the individual @xmath1-@xmath5he and @xmath1-@xmath4 scatterings are evaluated using few body equations for the @xmath1 - 3n and @xmath1 - 4n systems with a coupled channel @xmath1-n interaction as an input . calculations including four low - lying states of @xmath2be lead to a double hump structure in the total cross section corresponding to the @xmath7 and @xmath8 angular momentum states . the humps arise due to the off - shell rescattering of the @xmath1 meson on the @xmath2be nucleus in the final state .

eta meson interactions with light nuclei cluster model approach results and discussions summary

arxiv

massive multiple - input multiple - output ( mimo ) is known to achieve high capacity performance with simplified transmit precoding / receive combining design @xcite-@xcite . most notably , simple linear precoding schemes , such as zero - forcing ( zf ) , are virtually optimal and comparable to nonlinear precoding like the capacity - achieving dirty paper coding ( dpc ) in massive mimo systems @xcite . however , to exploit multiple antennas , the convention is to modify the amplitudes and phases of the complex symbols at the baseband and then upcovert the processed signal to around the carrier frequency after passing through digital - to - analog ( d / a ) converters , mixers , and power amplifiers ( often referred to as the radio frequency ( rf ) chain ) . outputs of the rf chain are then coupled with the antenna elements . in other words , each antenna element needs to be supported by a dedicated rf chain . this is in fact too expensive to be implemented in massive mimo systems due to the large number of antenna elements . on the other hand , cost - effective variable phase shifters are readily available with current circuitry technology , making it possible to apply high dimensional phase - only rf or analog processing @xcite-@xcite . phase - only precoding is considered in @xcite , @xcite to achieve full diversity order and near - optimal beamforming performance through iterative algorithms . the limited baseband processing power can further be exploited to perform multi - stream signal processing as in @xcite , where both diversity and multiplexing transmissions of mimo communications are addressed with less rf chains than antennas . @xcite then takes into account more practical constraints such as only quantized phase control and finite - precision analog - to - digital ( a / d ) conversion . works in @xcite-@xcite , however , do not consider the multiuser scenario and are not aimed to maximize the capacity performance in the large array regime . in this paper , we consider the practical constraints of rf chains and propose to design the rf precoder by extracting the phases of the conjugate transpose of the aggregate downlink channel to harvest the large array gain in massive mimo systems , inspired by @xcite . low - dimensional baseband zf precoding is then performed based on the equivalent channel obtained from the product of the rf precoder and the actual channel matrix . this hybrid precoding scheme , termed pzf , is shown to approach the performance of the virtually optimal yet practically infeasible full - complexity zf precoding in a massive multiuser mimo scenario . furthermore , hybrid baseband and rf precoding has been considered for millimeter wave ( mmwave ) communications in works @xcite-@xcite . they share the idea of capturing dominant " paths of mmwave channels using rf phase control and the rf processing is constrained , more or less , to choose from array response vectors . we will also show in the simulation the desirable performance of our proposed pzf scheme in mmwave channels . we consider the downlink communication of a massive multiuser mimo system as shown in fig . [ fig : system ] , where the base station ( bs ) is equipped with @xmath1 transmit antennas , but driven by a far smaller number of rf chains , namely , @xmath2 . this chain limitation restricts the maximum number of transmitted streams to be @xmath2 and we assume scheduling exactly @xmath2 single - antenna users , each supporting single - stream transmission . as discussed , the downlink precoding is divided among baseband and rf processing , denoted by @xmath3 of dimension @xmath4 and @xmath5 of dimension @xmath6 , respectively . notably , both amplitude and phase modifications are feasible for the baseband precoder @xmath3 , but only phase changes can be made to the rf precoder @xmath5 with variable phase shifters and combiners @xcite . thus each entry of @xmath5 is normalized to satisfy @xmath7 where @xmath8 denotes the magnitude of the @xmath9th element of @xmath5 . we adopt a narrowband flat fading channel and obtain the sampled baseband signal received at the @xmath10th user @xmath11 where @xmath12 is the downlink channel from the bs to the @xmath10th user , and @xmath13 denotes the signal vector for a total of @xmath2 users , satisfying @xmath14 = \frac{p}{k } { \bf i}_k$ ] where @xmath15 is the transmit power at the bs and @xmath16 $ ] is the expectation operator . to meet the total transmit power constraint , we further normalize @xmath3 to satisfy @xmath17 . @xmath18 denotes the additive noise , assumed to be circular symmetric gaussian with unit variance , i.e. , @xmath19 . then the received signal - to - interference - plus - noise - ratio ( sinr ) at the @xmath10th user is given by @xmath20 where @xmath21 denotes the @xmath22th column of @xmath3 . if gaussian inputs are used , the system can achieve a long - term average ( over the fading distribution ) spectral efficiency @xmath23.\end{aligned}\ ] ] in massive mimo systems , zf precoding is known as a prominent linear precoding scheme to achieve virtually optimal capacity performance due to the asymptotic orthogonality of user channels in richly scattering environment @xcite . it is typically realized through baseband processing , requiring @xmath1 rf chains performing rf - baseband frequency translation and a / d conversion . this tremendous hardware requirement , however , restricts the array size from scaling large . to alleviate the hardware constraints while realizing full potentials of massive multiuser mimo systems , we propose to apply phase - only control to couple the @xmath2 rf chain outputs with @xmath1 transmit antennas , using cost - effective rf phase shifters . low - dimensional multi - stream processing is then performed at the baseband to manage inter - user interference . the proposed low - complexity hybrid precoding scheme , termed phased - zf ( pzf ) , can approach the performance of the full - complexity zf precoding , which is , as stated , practically infeasible due to the requirement of supporting each antenna with a dedicated rf chain . the spectral efficiency achieved by the proposed pzf scheme is then analyzed . the structure shown in fig . [ fig : system ] is exploited to perform the proposed hybrid baseband and rf joint processing , where the baseband precoder @xmath3 modifies both the amplitudes and phases of incoming complex symbols and the rf precoder @xmath5 controls phases of the upconverted rf signal . we propose to perform phase - only control at the rf domain by extracting phases of the conjugate transpose of the aggregate downlink channel from the bs to multiple users . this is to align the phases of channel elements and can thus harvest the large array gain provided by the excessive antennas in massive mimo systems . to clarify , denote @xmath24 as the @xmath9th element of @xmath5 and we perform the rf precoding according to @xmath25 where @xmath26 is the phase of the @xmath9th element of the conjugate transpose of the composite downlink channel , i.e. , @xmath27 $ ] . here we implicitly assume perfect channel knowledge at the bs which can potentially be obtained , e.g. , through uplink channel estimation combined with channel reciprocity in time division duplex ( tdd ) systems @xcite . we note that efficient channel estimation techniques leveraging hybrid structures and rigorous treatment of frequency selectivity are an ongoing research topic of great practical interest . then at the baseband , we observe an equivalent channel @xmath28 of a low dimension @xmath29 where @xmath30^h$ ] is the composite downlink channel . hence multi - stream baseband precoding can be applied to @xmath31 , where simple low - dimensional zf precoding is performed as @xmath32 where @xmath33 is a diagonal matrix , introduced for column power normalization . with this pzf scheme , to support simultaneous transmission of @xmath2 streams , _ hardware complexity is substantially reduced , where only @xmath2 rf chains are needed , as compared to @xmath1 required by the full - complexity zf precoding_. * quantized rf phase control : * according to , each entry of the rf precoder @xmath5 differs only in phases which assume continuous values . however , in practical implementation , the phase of each entry tends to be heavily quantized due to practical constraints of variable phase shifters . therefore , we need to investigate the performance of our proposed pzf precoding scheme in this realistic scenario , i.e. , phases of the @xmath34 entries of @xmath5 are quantized up to @xmath35 bits of precision , each quantized to its nearest neighbor based on closest euclidean distance . the phase of each entry of @xmath5 can thus be written as @xmath36 where @xmath37 is chosen according to @xmath38 where @xmath39 is the unquantized phase obtained from . then the baseband precoder is computed by with the quantized @xmath5 . in this part , we analyze the spectral efficiency achieved by our proposed pzf and full - complexity zf precoding in the limit of large transmit antenna size @xmath1 assuming rayleigh fading . closed - form expressions are derived , revealing the roles different parameters play in affecting system capacity . denoting the @xmath10th column of @xmath5 by @xmath40 , we obtain @xmath41 { \bf w s } + n_k\end{aligned}\ ] ] based on . as described in section [ sec : scheme ] , @xmath40 is designed by extracting the phases of @xmath42 , we thus have the diagonal term @xmath43 where @xmath44 denotes the @xmath45th element of the vector @xmath42 . under the assumption that each element of @xmath42 is independent and identically distributed ( i.i.d . ) complex gaussian random variable with unit variance and zero mean , i.e. , @xmath46 , we conclude that @xmath47 follows rayleigh distribution with mean @xmath48 and variance @xmath49 . when @xmath1 tends to infinity , the central limit theorem indicates @xmath50 for the off - diagonal term , i.e. , @xmath51 , we have @xmath52 , where @xmath53 gives the complex conjugation . its distribution is characterized in the lemma below . [ lemma ] in rayleigh fading channels , the off - diagonal term @xmath54 is distributed according to @xmath55 . the proof is achieved by analyzing the real and imaginary parts of @xmath56 separately , followed by proving their independence . the proof is straightforward by definitions , and hence details are left out due to space limit . based on lemma [ lemma ] , we derive that the magnitude of the off - diagonal term , i.e. , @xmath57 follows the rayleigh distribution with mean @xmath48 and variance @xmath49 . compared with the diagonal term @xmath58 given by , it is safe to say _ the off - diagonal terms are negligible when the transmit antenna number @xmath1 is fairly large_. this implies that the inter - user interference is essentially negligible even without baseband processing at large @xmath1 ! however , we note when @xmath1 assumes some medium high values , the residual interference may still deteriorate the system performance . therefore we apply in our proposed scheme zf processing at the baseband to suppress it as in . we reason that even with zf processing at the baseband , the spectral efficiency achieved is still less than it would be if the off - diagonal terms @xmath56 s were precisely zero . in other words , the spectral efficiency achieved by pzf is upper bounded by @xmath59 with @xmath60 $ ] , which can be characterized by the following theorem using the limit equivalence type of argument @xcite . [ theorem ] the spectral efficiency achieved by the proposed low - complexity pzf precoding scheme is tightly upper bounded by @xmath61 where @xmath62 the per - user upper bound is derived as @xmath63 \nonumber \\ % & = { \mathbb e}\left[\log_2\left(\left(1+\frac{\pi n_t}{4}\frac{p}{k}\right)\frac{1+\frac{p}{k}\left ( y + \frac{\sqrt{\pi n_t}}{2}\right)^2}{1+\frac{\pi n_t}{4}\frac{p}{k}}\right)\right ] \nonumber\\ & = \log_2\left(1+\frac{\pi}{4}\frac{pn_t}{k } \right ) + \underbrace{{\mathbb e}\left[\log_2\left(\frac{1+\frac{p}{k}\left ( y + \frac{\sqrt{\pi n_t}}{2}\right)^2}{1+\frac{\pi n_t}{4}\frac{p}{k}}\right)\right]}\limits_{\delta } \nonumber\end{aligned}\ ] ] where @xmath64 with @xmath65 . one may prove @xmath66 by showing @xmath67 is both upper and lower bounded by zero in the limit . an upper bound can be directly proved by applying the jensen s inequality . proof of the lower bound is involved . briefly , by defining @xmath68 and @xmath69 , we have @xmath70 + \lim\limits_{a\rightarrow\infty}\log_2\frac{a^2}{\frac{1}{\rho } + a^2 } \nonumber \\ & = \lim\limits_{a\rightarrow\infty}\frac{2a\log_2e}{\sqrt{2\pi}\sigma}\int_{0}^{+\infty } ( \ln x)e^{-\frac{a^2(x-1)^2}{2\sigma^2}}\left(1 + e^{-\frac{2a^2x}{\sigma^2 } } \right ) dx \nonumber\\ & \mathop \geq\limits^{(a ) } \lim\limits_{a\rightarrow\infty}\frac{2a \log_2e}{\sqrt{2\pi}\sigma}\left(1 + e^{-\frac{2a^2\xi}{\sigma^2 } } \right)\int_0 ^ 1 ( \ln x ) e^{-\frac{a^2(x-1)^2}{2\sigma^2}}dx \nonumber \\ & = \lim\limits_{a\rightarrow\infty } \frac{2ae^{-\frac{a^2}{2\sigma^2}}}{\sqrt{2\pi}\sigma\ln2}\left(1 + e^{-\frac{2a^2\xi}{\sigma^2 } } \right ) \int_0 ^ 1 ( \ln x ) e^{\frac{a^2x}{2\sigma^2 } } dx \nonumber\\ & \mathop \geq\limits^{(b ) } \lim\limits_{a\rightarrow\infty } \frac{2ae^{-\frac{a^2}{2\sigma^2}}}{\sqrt{2\pi}\sigma\ln2}\left(1 + e^{-\frac{2a^2\xi}{\sigma^2 } } \right)\frac{2\sigma^2 \left(1-e^{\frac{a^2}{2\sigma^2}}\right)}{a^2 } = 0 \nonumber\end{aligned}\ ] ] where ( a ) holds by shortening the integral range and then applying the mean value theorem for integral with @xmath71 . ( b ) is valid by using @xmath72 for @xmath73 . considering that the off - diagonal terms @xmath54 s are essentially negligible when @xmath1 is large , we expect the derived closed - form upper bound to be very tight in the large antenna regime . this is further verified in the simulation results as shown in fig . 2 . thus the closed - form upper bound serves as a good approximation of the spectral efficiency achieved by the proposed pzf precoding scheme at large @xmath1 . the full - complexity zf precoding vector ( with unit norm ) for the @xmath10th stream follows by projecting @xmath42 onto the nullspace of @xmath74^h$ ] . in the spectral efficiency analysis , we exploit the property that users channels are asymptotically orthogonal in massive multiuser mimo systems @xcite . it indicates full - complexity zf precoding converges to conjugate beamforming with _ inter - user interference forced to zero _ , achieving @xmath75 then according to , we obtain the spectral efficiency of full - complexity zf precoding in the limit of large @xmath1 as @xcite @xmath76 \nonumber \\ = & ke^{\frac{k}{p } } \log_2e \sum\limits_{n=1}^{n_t } e_n\left(\frac{k}{p } \right)\end{aligned}\ ] ] by acknowledging that @xmath77 follows chi - squared distribution with @xmath78 degrees of freedom and @xmath79 is the exponential integral of order @xmath80 . we numerically compare our proposed pzf precoding scheme in fig . [ fig : maincomparescheme ] along with its quantized version against the full - complexity zf scheme , which is deemed virtually optimal in the large array regime but practically infeasible due to the requirement of @xmath1 costly rf chains . it is observed that the proposed pzf precoding performs measurably close to the full - complexity zf precoding , with less than @xmath81 db loss but substantially reduced complexity . as for the heavily quantized phase control , we find that with @xmath82 bits of precision , i.e. , phase control candidates of @xmath83 , the proposed scheme suffers negligible degradation , say less than @xmath81 db . the derived analytical spectral efficiency expressions and are also plotted in fig . [ fig : maincomparescheme ] . we observe that the derived closed - form expressions are quite accurate in characterizing spectral efficiencies achieved by the proposed pzf precoding and full - complexity zf precoding schemes throughout the whole signal - to - noise ( snr ) is the common average snr received at each antenna with noise variance normalized to unity.]range , thus providing useful guidelines in practical system designs . apart from ideal i.i.d . rayleigh fading channels , our proposed pzf scheme can also be applied to the mmwave communication which is known to have very limited multipath components . to capture this poor scattering nature , in the simulation , we adopt a geometric channel model @xcite-@xcite @xmath84 where each user is assumed to observe the same number of propagation paths , denoted by @xmath85 , the strength associated with the @xmath86th path seen by the @xmath10th user is represented by @xmath87 ( assuming @xmath88 ) , and @xmath89 is the random azimuth ( elevation ) angle of departure drawn independently from uniform distributions over @xmath90 $ ] . @xmath91 is the array response vector depending only on array structures . here we consider a uniform linear array ( ula ) whose array response vector admits a simple expression , given by ( * ? ? ? * eq . ( 6 ) ) where @xmath92 is the normalized antenna spacing . we compare in fig . [ fig : mmwavecomparescheme ] our proposed pzf scheme against the beamspace mimo ( b - mimo ) scheme proposed in @xcite , which essentially steers streams onto the approximate strongest paths ( using dft matrix columns ) at the rf domain and performs low - dimensional baseband zf precoding based on the equivalent channel . for fair comparison , the bs is also assumed to have a total of @xmath2 chains . the b - mimo scheme achieves desirable performance in line - of - sight ( los ) channel but fails to capture sparse multipath components in non - los channels . in this paper , we have studied a large multiuser mimo system under practical rf hardware constraints . we have proposed to approach the desirable yet infeasible full - complexity zf precoding with low - complexity hybrid pzf scheme . the rf processing was designed to harvest the large power gain with reasonable complexity , and the baseband precoder was then introduced to facilitate multi - stream processing . its performance has been characterized in a closed form and further demonstrated in both rayleigh fading and poorly scattered mmwave channels through computer simulations . f. rusek , d. persson , b. k. lau , e. g. larsson , t. l. marzetta , o. edfors , and f. tufvesson , `` scaling up mimo : opportunities and challenges with very large arrays , '' _ ieee sig . process . mag . _ , vol . 30 , no . 1 , 4060 , jan . 2013 . v. venkateswaran and a. j. van der veen , `` analog beamforming in mimo communications with phase shift networks and online channel estimation , '' _ ieee trans . sig . process . _ , vol . 58 , no . 8 , pp . 41314143 , aug . w. roh _ et al . _ , `` millimeter - wave beamforming as an enabling technology for 5 g cellular communications : theoretical feasibility and prototype results , '' _ ieee commun . mag . _ , vol . 52 , no . 2 , pp . 106113 , feb . o. e. ayach , s. rajagopal , s. abu - surra , z. pi , and r. w. heath , jr . `` spatially sparse precoding in millimeter wave mimo systems , '' _ ieee trans . wireless commun . _ , vol . 13 , no . 3 , pp . 14991513 , mar . 2014 . j. choi , v. raghavan , and d. j. love , `` limited feedback design for the spatially correlated multi - antenna broadcast channel , '' in _ proc . ieee global telecommun . ( globecom ) _ , dec . 2013 , pp . 34813486 . alouini and a. j. goldsmith , `` capacity of rayleigh fading channels under different adaptive transmission and diversity - combining techniques , '' _ ieee trans . veh . _ , vol . 48 , no . 4 , pp . 11651181 , july 1999 .

massive multiple - input multiple - output ( mimo ) is envisioned to offer considerable capacity improvement , but at the cost of high complexity of the hardware . in this paper , we propose a low - complexity hybrid precoding scheme to approach the performance of the traditional baseband zero - forcing ( zf ) precoding ( referred to as full - complexity zf ) , which is considered a virtually optimal linear precoding scheme in massive mimo systems . the proposed hybrid precoding scheme , named phased - zf ( pzf ) , essentially applies phase - only control at the rf domain and then performs a low - dimensional baseband zf precoding based on the effective channel seen from baseband . heavily quantized rf phase control up to @xmath0 bits of precision is also considered and shown to incur very limited degradation . the proposed scheme is simulated in both ideal rayleigh fading channels and sparsely scattered millimeter wave ( mmwave ) channels , both achieving highly desirable performance . massive mimo , hybrid precoding , millimeter wave ( mmwave ) mimo , rf chain limitations .

introduction system model hybrid precoding in massive mimo systems simulation results conclusion

arxiv

transit transmission spectroscopy is an observational technique that can be used to characterize a planet s atmosphere as it transits its host star . this technique has been used to identify absorption features in some exoplanet atmospheres @xcite , but many planets have flat , featureless spectra @xcite . flat spectra can be explained by either a high mean molecular weight ( and thus small scale height ) atmosphere or by the presence of high - altitude clouds or hazes , as is common in planets in our own solar system , and has been inferred for the atmospheres of some hot jupiters @xcite . for gj 1214b , even high mean molecular weight atmospheres have recently been ruled out , leaving very high altitude clouds or hazes as the only physically plausible explanation for the planet s spectrum @xcite . the james webb space telescope ( jwst ) and large ground based telescopes such as the european extremely large telescope ( e - elt ) will open up new avenues for characterizing transiting planets . absorption features for an earth - like or super - earth planet could be detected in the near future with 200 hrs of jwst in - transit observations @xcite , or with @xmath120 hrs of e - elt in - transit observations @xcite . if there are clouds or hazes present in an atmosphere , they will limit the atmospheric levels that can be probed , making the targets less desirable for characterization . therefore , it would be beneficial to have a method that could relatively rapidly discriminate between haze - free and hazy planets , which are not easily characterized even in extended transit transmission observations @xcite . here we examine whether refractive effects on transit transmission spectroscopy could provide a more efficient way of discriminating between hazy , cloudy and clearsky ( free of clouds and hazes ) atmospheres . @xcite propose that measurements of absorption wing steepness , or a comparison of the depths of multiple absorption bands , could be used to distinguish between cloudy / hazy and clearsky planets . since both methods require relatively detailed characterization of absorption features , these techniques may not discriminate between a hazy and haze - free planet before considerable amounts of telescope time are used . in contrast , the refractive signal is independent of absorption features , and could be binned over a wide range of wavelengths , increasing detectability . while refraction can set a mid - transit maximum transit pressure ( or minimum tangent altitude ) that can be probed by transit transmission spectroscopy @xcite refraction provides the deepest probe of an atmosphere pre- and post - transit @xcite , when it also generates a refractive halo around the exoplanet , increasing the observed flux @xcite . @xcite derive analytic expressions for the halo brightness for both transparent atmospheres and atmospheres with extinction from rayleigh scattering . @xcite and @xcite examine the concept further by generating spectra of refracted light for the earth and for venus . here we expand on previous work by showing that a detection of refracted light in a transit light curve pre - ingress and post - egress would preclude hazy atmospheres , because hazes tend to obscure the layers of the atmosphere that refract light to a distant observer . we show that this signal could be more readily detectable than spectral absorption features in some cases and could be valuable for selecting targets for more extended follow - up observations . we used the refraction code that is described in detail in @xcite to calculate refraction angles for a suite of planetary atmospheres . briefly , refraction is governed by a set of differential equations that we solve at each step along the path through the atmosphere using a runge - kutta integration scheme . given the planetary radius , surface gravity , atmospheric composition and pressure - temperature profile , the model calculates the angle of deflection due to refraction for a range of tangent altitudes . we generated refractive light curves to calculate the amount of out - of - transit refracted light . we first determined whether or not each portion of the atmosphere ( given as an altitude and angle along the annulus of the atmosphere ) is illuminated at each time during the transit event , from half a transit length prior to ingress to half a transit length after egress , and then integrated over the entire atmosphere to generate the light curve . we quantified the signal of refracted light as the difference in the average value of the transit light curve between two stages of the transit event . we chose a quarter of a transit length as the time bin to maximize the signal to noise ratio ( s / n ) for the majority of cases we examined . as can be seen in figure [ fig : lightcurve ] , most of the refracted flux is seen in the quarter of a transit prior to ingress , so dividing the transit into longer stages would reduce the time - averaged signal . stages with shorter durations could increase the time - averaged signal , but would have greater noise levels because of the shorter integration time . refracted light brightness is more strongly peaked just outside of transit for planets with equilibrium temperatures ( t@xmath2 , see @xcite for definition ) @xmath1600 k , but we find that even for these cases adopting a time bin of 5% of the transit length results in poorer s / n for t@xmath3600 k and an increase in s / n by only a factor of @xmath42 for planets with greater temperatures we used a suite of planetary atmospheres to calculate the refracted light signal . these are shown in table [ tab : params ] . we have selected a combination of solar system analogs as well possible super - earth and mini - neptune atmospheres to cover a wide range of potential planetary atmospheres . we assumed the h@xmath5-dominated atmospheres have a solar h / he ratio ( 90% h , 10% he ) for simplicity , but the small change in the refractive index for different h / he ratios should have a negligible effect on our results . for the super - earth and mini - neptune planets , we ran our models on 4 test cases to span the most likely bulk atmospheric compositions : 100% n@xmath5 , solar composition , 100% h@xmath5o , and 100% co@xmath5 . out - of - transit refracted light must be deflected by a large enough angle to be scattered into the beam to a distant observer . the characteristic angle of deflection ( in radians ) is @xmath4r@xmath6/d , ( where r@xmath6 is the stellar radius and @xmath7 is the planet - star distance ) which is also half the angular size of the star , as seen by the planet . for example , half a transit length prior to ingress , on the trailing side of the planet , light originating at the near and far limb of the star would have to be refracted by r@xmath6/d and 3r@xmath6/d respectively , to reach a distant observer . more than half a transit length prior to ingress , the required refraction angles would increase , and closer to ingress they would decrease . based on the qualitative description given above , the brightness of the refracted light signal depends on the angles of refraction at each altitude in an atmosphere and the planet - star geometry . the deflection of light by a planetary atmosphere can be calculated by our model from the atmospheric scale height , the planetary radius ( r@xmath8 ) , and the index of refraction of the atmosphere . for each test case , r@xmath8 and the refractive index are given . the scale height is determined from the surface gravity , mean molecular weight of the atmosphere , and t@xmath2 . surface gravity is given for each test case , and the mean molecular weight is determined by the composition . we ran our model simulations over a grid of isothermal atmospheres with t@xmath2 from 100 to 1000 k , covering a wide range of atmospheric scale heights . we chose to use isothermal atmospheres for simplicity after testing other temperature profiles with realistic tropospheric lapse rates and stratospheric temperature inversions and finding no significant difference in our results . the planet - star geometry is determined by r@xmath8 , r@xmath6 , the impact parameter ( @xmath9 ) , and @xmath7 . the impact parameter is the sky - projected distance at conjunction , in units of stellar radius @xcite . to cover the full range of planet - star geometries , we ran our simulations over a range of values for @xmath9 , planetary albedo , and stellar types from m9 to f5 , constraining the stellar radius and luminosity . lcccc earth & 6371 & n@xmath5 & 1.00029 & 9.8 + super - earth & 12742 & n@xmath5 & 1.00029 & 9.8 + mini - neptune & 12742 & h@xmath5 & 1.00012 & 9.8 + h@xmath5o super - earth & 12742 & h@xmath5o & 1.00026 & 9.8 + co@xmath5 super - earth & 12742 & co@xmath5 & 1.00044 & 9.8 + neptune & 24622 & h@xmath5 & 1.00012 & 11.1 + saturn & 58232 & h@xmath5 & 1.00012 & 10.44 + jupiter & 69911 & h@xmath5 & 1.00012 & 24.8 + [ tab : params ] because our model does not explicitly calculate the effect of cloud and aerosol opacity , we simulated the effect of a cloud or haze layer by truncating the depth of the measurable atmosphere at a characteristic pressure layer . to determine appropriate pressure cutoff layers for the three main aerosol cases under consideration , we used our modeling results and examples of clouds and hazes in our own solar system to select pressure cutoffs at 1 bar ( clearsky case ) , 0.1 bars ( cloudy case ) and 1 mbar ( hazy case ) . we chose 1 bar as our clearsky pressure cutoff because at pressures @xmath101 bar , our modeling indicates that atmospheres within the range of compositions under consideration are optically thick near 1 @xmath11 m ( the central wavelength for our transit simulations ) when only rayleigh scattering is included . for the pressure cut - off for cloudy atmospheres we chose 0.1 bars , which is a characteristic lower pressure limit for the tropopause for atmospheres of a range of different compositions @xcite , and the majority of clouds are found within a planet s troposphere . lastly , we chose 1 mbar as the hazy pressure cutoff because hazes are typically generated via photochemistry in the upper atmosphere at pressures near 1 mbar . for example , at 1 @xmath11 m venus is optically thick ( @xmath12=1 ) in transit transmission at 90 km ( @xmath40.1 mbar ) @xcite and titan is optically thick at @xmath4240 km ( @xmath00.5 mbar ) @xcite . because hazes form at pressures @xmath01 mbar in both a warm co@xmath5-dominated atmosphere and a cold n@xmath5-dominated atmosphere , we chose 1 mbar as a reasonable cutoff for hazes over the parameter space we explore here . we used the publicly available exposure time calculators ( etcs ) to estimate the s / n for detecting refracted light in a transit light curve with jwst and e - elt . we used the etcs to estimate the noise at 1 @xmath11 m with spectral resolving power ( r ) equal to 100 for stellar types from f5v to m9v . we chose the lowest resolving power available ( r=100 for the jwst etc ) because this technique does not require high resolving power , and could even be performed with broadband filter photometry if necessary . the input stellar spectra were phoenix nextgen spectra with solar metallicities @xcite with the star placed at a distance of 10 pc . given the parts per million ( ppm ) flux difference for refracted light and the estimated s / ns from the etcs , we calculated the out - of - transit integration time required to detect refracted light at a s / n@xmath13 over all t@xmath2 values for each planetary atmosphere and stellar type . lccccccccccc earth & 400 & 5780 & clearsky & 0.15 & 0.13 & 4.77 & 1.0 & 0.3 & 999.00 & 781.2 & 231.9 + super - earth & 450 & 5780 & clearsky & 0.15 & 0.29 & 0.91 & 1.0 & 0.2 & 999.00 & 166.9 & 34.8 + mini - neptune & 250 & 5780 & clearsky & 0.15 & 1.98 & 0.02 & 1.0 & 1.2 & 27.65 & 2.0 & 2.4 + h@xmath5o super - earth & 400 & 5780 & clearsky & 0.15 & 0.41 & 0.45 & 1.0 & 0.3 & 640.78 & 73.9 & 21.9 + co@xmath5 super - earth & 600 & 5780 & clearsky & 0.15 & 0.22 & 1.60 & 1.0 & 0.1 & 999.00 & 393.7 & 34.6 + neptune & 250 & 5780 & clearsky & 0.15 & 3.79 & 0.01 & 1.0 & 1.2 & 7.58 & 1.0 & 1.2 + saturn & 300 & 5780 & clearsky & 0.15 & 10.98 & 0.01 & 1.0 & 0.7 & 0.90 & 1.0 & 0.7 + jupiter & 350 & 5780 & clearsky & 0.15 & 6.39 & 0.01 & 1.0 & 0.4 & 2.66 & 1.0 & 0.4 + [ tab : results ] table [ tab : results ] shows the ppm flux change , and the required integration time , number of transits and total time ( from first transit to last ) for detecting refracted light for each test case over the suite of parameters . the results shown here are for an albedo of 0.15 , but results for other albedos are available online . our results indicate that saturn analog planets exhibit the most detectable refracted light of any of the cases because saturn has a radius close to jupiter s radius and a lower surface gravity , which increases the atmospheric scale height at a given temperature . the amplitude of the refracted light signal ( as defined in section [ sec : modeldesc ] ) is no larger than half a scale height for all cases we have explored here . the maximum flux amplitude for planets orbiting sun - like stars is 10 ppm for a 300 k saturn analog . the other h@xmath5 cases have maximum amplitudes of 6 , 4 , and 2 ppm for the jupiter , neptune , and mini - neptune cases , respectively . the greatest ppm signals are for planets orbiting around m9v stars , for which the signals can increase by nearly two orders of magnitude to 950 ppm for a 200 k saturn analog . figure [ fig : inttime]a shows the jwst out - of - transit integration time required to detect refracted light for the 4 h@xmath5-dominated atmospheres : jupiter , saturn , neptune analogs and the ` mini - neptune ' , all without clouds or hazes . for many of the saturn and jupiter - analog cases , refracted light could be detected in @xmath010 hours of jwst time . this integration time can be achieved in 1 transit for jupiter and saturn - analog planets with t@xmath3600 k orbiting f , g and k stars and in @xmath05 transits for t@xmath3400 k orbiting m dwarfs . for cases in which multiple transits are required , the total time from first transit to last is @xmath01 year , and typically @xmath06 months . figure [ fig : inttime]a shows our results for @xmath9=0.0 , with observing times required increasing by 1% for @xmath9=0.2 , 10% for of @xmath9=0.6 , and 30% for @xmath9=0.9 . -dominated atmospheres . the results assume that observations are made at 1 @xmath11 m with r=100 , for a planet at a distance of 10 pc . the planets with the most detectable refracted light signal are those with t@xmath3600 k for solar - type stars and @xmath0400 k for m dwarfs . for many jupiter and saturn - analog cases , refracted light could be detected with @xmath010 hour of jwst time . * b ) * e - elt integration time required to detect refracted light for n@xmath5 and h@xmath5 atmospheres , assuming that 50 wavelength bins can be summed over at r=100 . refracted light is most detectable for the non - hazy atmospheres , and therefore could be used to distinguish between hazy and non - hazy worlds . , width=340 ] figure [ fig : inttime]b shows the e - elt integration time required to detect refracted light for super - earth and mini - neptune atmospheres with @xmath9=0.0 . we calculated the signal levels for n@xmath5 , h@xmath5o , co@xmath5 and h@xmath5 atmospheres , but only a comparison of n@xmath5 and h@xmath5 atmospheres is shown here . we find that refracted light could be detectable in @xmath010 hrs of e - elt time for many of the clearsky atmospheres , and even some cloudy atmospheres . in contrast , detecting refracted light for a hazy exoplanet would require @xmath1100 hours for all the planetary atmospheres we considered . here we have assumed that it is possible to bin over at least 50 spectral resolution elements . the justification for this is found in figure [ fig : lightcurve - diff ] , which shows the wavelength - dependent refracted light signal for 2 r@xmath13 planet with an earth - like atmosphere . a larger change in effective radius at a given wavelength means a stronger flux from refraction prior to ingress or after egress . between 0.8 and 1.35 @xmath11 m - shortward of a major h@xmath5o absorption feature and where rayleigh scattering opacities are small - there are @xmath450 spectral resolution elements that could be summed . for earth - analog atmospheres , there is a relatively large flux difference at all these wavelengths . therefore , we consider binning over multiple spectral resolution elements to decrease the integration time to be a valid approach , at least for n@xmath5-dominated planets like earth . the greatest amplitude of refracted flux for the n@xmath5 super - earth cases around a sun - like star is 0.12 ppm for a 400 k planet . for the cloudy case , the maximum amplitude is 0.06 ppm at 200 k. the cloudy h@xmath5 cases have amplitudes between 0.2 and 1.0 ppm , but only for the very cold ( @xmath0200 k ) cases . the amplitudes for the hazy h@xmath5 cases are all below 0.025 ppm , and below 0.005 ppm for t@xmath14150 k. the number of transits required to detect refracted light with e - elt is 1 for clearsky n@xmath5 super - earths with @xmath15800 k orbiting f , g and k stars and @xmath03 for t@xmath3500 k for those orbiting m dwarfs . as with the jupiter and saturn analogs , 3 transits is , from first transit to last , much less than a year and typically @xmath03 months . cloudy atmospheres with t@xmath3250 k could exhibit detectable refracted light signals , but hazy atmospheres have largely undetectable refracted light signals except for some very cold ( t@xmath2=100 k ) cases . super - earth with an earth - like atmosphere , represented as the change in effective radius ( km ) . the spectra shown are the differences in the spectra between stage 2 and stage 1 of the transit ( see figure [ fig : lightcurve ] ) . this figure shows that the refracted light signal could be detected over a wide wavelength range , and that it should be possible to bin over multiple spectral resolution elements to reduce the integration time needed to detect refracted light.,width=302 ] a detection of refracted light implies a haze - free atmosphere because refracted light is much more detectable for a clearsky atmosphere than for a hazy one ( see figure [ fig : inttime]b ) . however , discriminating between cloudy and hazy worlds could be more challenging . for example , for a 600 k n@xmath5 super - earth orbiting a sun - like star ( and for the majority of parameter space ) , a null detection of refracted light would be consistent with either a cloudy or hazy atmosphere , with no apparent way to differentiate between the two . on the other hand , for a 250 k n@xmath5 super - earth orbiting a sun - like star , both the clearsky and cloudy cases are consistent with a detection of refracted light . to disambiguate these results , one would need to quantify the refracted light , which would require more observing time . overall , a detection of refracted light is indicative of a non - hazy atmosphere and , for some regions of parameter space , quantifying the refracted light flux could aid in uniquely discriminating between cloudy , hazy and clearsky atmospheres . the refracted light brightness is strongest for planets with t@xmath2 between 150 - 350 k , and is undetectable for very high temperature planets . for hot , close - in planets , the planet - star distance ( @xmath7 ) is small , meaning that the characteristic deflection angle r@xmath6/d is large , and that large refraction angles are required to produce a strong refracted light signal . for t@xmath14800 k , angles this large would require probing pressures greater than 1 bar , where most atmospheres should be opaque , meaning that atmospheric opacity results in low refracted light signals . for the coldest ( t@xmath3150 k ) planets , @xmath7 is large and the refraction angles for clearsky atmospheres are often much larger than r@xmath6/d . this results in more refracted light being observed further away from ingress and egress , increasing the average flux in stage 1 relative to stage 2 and reducing the overall detectability . ( see figure [ fig : lightcurve ] ) . in the near future , e - elt could be used to identify non - hazy potentially habitable planets , which have 180@xmath0t@xmath3260 k ( @xcite , ravi kopparapu , private communication ) . as shown in table [ tab : results ] and figure [ fig : inttime ] , refracted light could be detectable with one transit with e - elt , or @xmath05 hrs of out - of - transit e - elt time for potentially habitable n@xmath5 dominated super - earths orbiting f , g and k stars . for planets orbiting m dwarfs , the required number of transits is typically less than two , with a total integration time of @xmath05 hrs . @xcite estimate that it could take up to 10 transits to detect h@xmath5o and co@xmath5 for earth - like planets orbiting f , g and k stars with e - elt using filter photometry , and @xcite find that it would take @xmath120 hours of e - elt time to detect o@xmath5 for earth - like planets orbiting m dwarfs using high resolution ( r@xmath110000 ) spectroscopy . these estimates are larger than the amount of out - of - transit e - elt time necessary to detect refracted light for potentially habitable planets . therefore , because refracted light could be more detectable than spectral absorption features , looking for refracted light to distinguish between hazy and non - hazy exoplanets could be a useful tool in selecting exoplanets for extended follow - up observations . increases in out - of - transit flux due to refraction prior to ingress and subsequent to egress could be detectable with @xmath010 hours of out - of - transit observing time for saturn and jupiter - sized planets with jwst and for super - earths / mini - neptunes with e - elt . detecting refracted light would be indicative of a haze - free atmosphere , and a quantification of the amount of refracted light could aid in distinguishing between cloudy and clearsky atmospheres for planets with equilibrium temperatures @xmath0300 k. because refracted light can , in some cases , be detectable with less than a few hours of out - of - transit observing time , this method could be an economical way of determining if an exoplanet is haze - free and therefore a good target for extended follow - up observations . this work was performed by the nasa astrobiology institute s virtual planetary laboratory , supported by the nasa astrobiology institute under cooperative agreement solicitation nnh05zda001c .

we propose a method to distinguish between cloudy , hazy and clearsky ( free of clouds and hazes ) exoplanet atmospheres that could be applicable to upcoming large aperture space and ground - based telescopes such as the james webb space telescope ( jwst ) and the european extremely large telescope ( e - elt ) . these facilities will be powerful tools for characterizing transiting exoplanets , but only after a considerable amount of telescope time is devoted to a single planet . a technique that could provide a relatively rapid means of identifying haze - free targets ( which may be more valuable targets for characterization ) could potentially increase the science return for these telescopes . our proposed method utilizes broadband observations of refracted light in the out - of - transit spectrum . light refracted through an exoplanet atmosphere can lead to an increase of flux prior to ingress and subsequent to egress . because this light is transmitted at pressures greater than those for typical cloud and haze layers , the detection of refracted light could indicate a cloud- or haze - free atmosphere . a detection of refracted light could be accomplished in @xmath010 hours for jovian exoplanets with jwst and @xmath05 hours for super - earths / mini - neptunes with e - elt . we find that this technique is most effective for planets with equilibrium temperatures between 200 and 500 k , which may include potentially habitable planets . a detection of refracted light for a potentially habitable planet would strongly suggest the planet was free of a global cloud or haze layer , and therefore a promising candidate for follow - up observations .

introduction methods results discussion conclusions acknowledgments

arxiv

image understanding @xcite is becoming one of the most important problems in computer vision and many research efforts have been devoted to this topic . while object recognition @xcite and scene recognition @xcite have been extensively studied in the task of image classification , event recognition @xcite in still images received much less research attention , which also plays an important role in semantic image interpretation . as shown in figure [ fig : example ] , the characterization of event is extremely complicated as the event concept is highly related to many other high - level visual cues , such as objects , scene categories , human garments , human poses , and other context . therefore , event recognition in still images poses more challenges for the current state - of - the - art image classification methods , and needs to be further investigated in the computer vision research . convolutional neural networks ( cnns ) @xcite have recently enjoyed great successes in large - scale image classification , in particular for object recognition @xcite and scene recognition @xcite . for event recognition , much fewer deep learning methods have been designed for this problem . our previous work @xcite proposed a new deep architecture , called _ object - scene convolutional neural network _ ( os - cnn ) , for cultural event recognition . os - cnns are designed to extract useful information for event understanding from the perspectives of containing objects and scene categories , respectively . os - cnns are composed of two - stream cnns , namely object nets and scene nets . object nets are pre - trained on the large - scale object recognition datasets ( e.g. imagenet @xcite ) , and scene nets are based on models learned from the large - scale scene recognition datasets ( e.g. places205 @xcite ) . decomposing into object nets and scene nets enables us to use the external large - scale annotated images to initialize os - cnns , which may be further fine tuned elaborately on the event recognition dataset . finally , event recognition is performed based on the late fusion of softmax outputs of object nets and scene nets . following the research line of os - cnns , in this paper , we try to further explore different aspects of os - cnns and better exploit os - cnns for better event recognition . specifically , we design four types of investigation scenarios to study the performance of os - cnns . in the first scenario , we directly use the softmax outputs of cnns as recognition results . in the next three scenarios , we treat cnns as feature extractors , and use them to extract both _ global _ and _ local _ features of an image region . global features are more compact and aim to capture the holistic structure , while local features focus on describing the image details and local patterns . our experimental results indicate these two kinds of features are complementary to each other and robust for event recognition . based on our empirical explorations with os - cnns , we come up with our solution for the cultural event recognition track at the iccv chalearn looking at people ( lap ) challenge @xcite and we secure the third place . the rest of this paper is organized as follows . in section [ sec : os - cnn ] , we will give a brief introduction to os - cnns , including network architectures and implementation details . after that , we will introduce our extensive explorations with os - cnns for event recognition in section [ sec : feature ] . we then report our experimental results in section [ sec : exp ] . finally , we conclude our method and present the future work in section [ sec : con ] . in this section , we will first briefly introduce the architecture of _ object - scene convolutional neural networks _ ( os - cnns ) , which was proposed in our previous work @xcite . then , we will present the implementation details of os - cnns , including network structures , data augmentations , and learning policy . event is a relatively complicated concept in computer vision research and highly related with other two problems : object recognition and scene recognition . the basic idea behind os - cnn is to utilize two separate components to perform event recognition from the perspectives of occurring objects and scene context . specifically , os - cnns are composed of object nets and scene nets , as shown in figure [ fig : os - cnn ] . * object nets . * object net is designed to capture useful information of objects to help event recognition . intuitively the occurring objects are able to provide useful cues for event understanding . for instance , in the cultural event of australia day as shown in figure [ fig : example ] , australian flag will be a representative object . as the main goal of object net is to deal with object cues , we build it based on recent advances on large - scale object recognition , and pre - train the network on the public imagenet models . then , we further fine tune the model parameters on the training dataset of cultural event recognition by setting the output number as @xmath0 ( cultural event recognition dataset containing 100 classes ) . * scene nets . * scene net is expected to extract scene information of image to assist event understanding . in general , the scene context will be helpful for recognizing the event category in the image . for example , in the cultural event of sapporo snow festival as shown in figure [ fig : example ] , outdoor will be usually the scene category . specifically , we pre - train the scene nets by using the models learned on the dataset places205 , which contains 205 scene classes and 2.5 millions images . similar to object nets , we then fine tune the network weights of scene nets on the event recognition dataset , where we set network output number as @xmath0 . based on the above analysis , recognizing cultural event will benefit from the transferred representations learned for object recognition and scene recognition . thus , we will fuse the network outputs of both object nets and scene nets as the prediction of os - cnns . in this subsection , we will describe the implementation details of training os - cnns , including network structures , data augmentations , and learning policy . * network structures . * network structures are of great importance for improving the performance of cnns . in the past several years , many successful network architectures have been proposed for object recognition , such as alexnet @xcite , clarifainet @xcite , overfeat @xcite , googlenet @xcite , vggnet @xcite , msranet @xcite , and inception2 @xcite . some good practices can be drawn from the evolution of network architectures : smaller convolutional kernel size , smaller convolutional stride , more convolutional channel , deeper network structure . in this paper , we choose the vggnet-19 as our main investigated structure due to its good performance in object recognition , which is composed of 16 convolutional layers and 3 fully connected layers . the detailed description about vggnet-19 is out of the scope of this paper and can be found in @xcite . * data augmentations . * by data augmentation , we mean perturbing an image by transformations that leave the underlying class unchanged . typical transformations include corner cropping , scale jittering , and horizontal flipping . specifically , during the training phase of os - cnns , we randomly crop image regions ( @xmath1 ) from 4 corners and 1 center of the whole image . meanwhile these cropped regions undergo horizontal flipping randomly . furthermore , we use three different scales to resize training images , where the smallest size @xmath2 of an image is set to @xmath3 . it should be noted that data augmentation is a method applicable to both training images and testing images . during training phase , data augmentation will generate additional training examples and reduce the influence of over - fitting . for testing phase , data augmentation will help to improve the classification accuracy . the augmented samples can be either regarded as independent images or combined into a single representation by pooling or stacking operations . in the current implementation , during the test phase , we use sum pooling to aggregate these representations of augmented samples into a single representation . * learning policy . * effective training methods are very crucial for learning cnn models . as the training dataset of cultural event recognition is relatively small compared with imagenet @xcite and places205 @xcite , we resort to pre - training os - cnns by using these public available models trained on imagenet and places205 . specifically , we pre - train object nets with public vggnet-19 model , which achieved the top performance at ilsvrc2014 . for scene net , we use the model released by @xcite to initialize the network weights , which has obtained the best performance on the places205 dataset so far . the network weights are learned using the mini - batch stochastic gradient descent with momentum ( set to 0.9 ) . at each iteration , a mini - batch of 256 images is constructed by random sampling . the dropout ratios for fully connected layers are set as @xmath4 . as we pre - train network weights with imagenet and places205 models , we set a smaller learning rate for fine tuning os - cnns : learning rate starts with @xmath5 , decreases to @xmath6 after 5k iterations , decreases to @xmath7 after 10k iterations and the training process ends at 12k iterations . to speed up the training process , we use a multi - gpu extension version @xcite of caffe toolbox @xcite , which is publicly available online . we have introduced the architectures and implementation details about os - cnns in the previous section . in this section , as shown in figure [ fig : pipeline ] , we will focus on describing the explorations of os - cnn activations from different layers and try to improve the recognition performance . the simplest way to utilize os - cnns for cultural event recognition is directly using the outputs ( softmax layer ) of cnn networks as final prediction results . specifically , given an image @xmath8 , its recognition score is calculated as follows : @xmath9 where @xmath10 and @xmath11 are the prediction scores of object nets and scene nets , @xmath12 and @xmath13 are the fusion weights of object nets and scene nets . in the current implementation , fusion weights are set to be equal for object nets and scene nets . another way to deploy os - cnns for cultural event recognition is to treat them as generic feature extractors and use them to extract the global representation of an image region . we usually extract the activations of * fully connected layers * , which are very compact and discriminative . in this case , we only use the pre - trained models without fine - tuning . specifically , given an image region @xmath14 , we extract this global representation based on os - cnns as follows : @xmath15,\ ] ] where @xmath16 and @xmath17 are the cnn activations from pre - trained object nets and scene nets , @xmath18 and @xmath19 are the fusion weights of object nets and scene nets . in current implementation , the fusion weights are set to be equal for object nets and scene nets . in previous scenario , os - cnns are only pre - trained on large scale dataset of object recognition and scene recognition , and directly applied to the smaller event recognition dataset . however , it was demonstrated that fine - tuning a pre - trained cnns on the target data can improve the performance a lot @xcite . we consider fine - tuning the os - cnns on the event recognition dataset and the resulted image representations become dataset - specific . after fine - tuning process , we obtain the following global representation with the fine - tuned os - cnns : @xmath20,\ ] ] where @xmath21 and @xmath22 are the cnn activations from the fine - tuned object nets and scene nets , @xmath18 and @xmath19 are the fusion weights of object nets and scene nets . in current implementation , the fusion weights are set to be equal for object nets and scene nets . in previous two scenarios , we extract a global representation of an image region with os - cnns . although this global representation is compact and discriminative , it may lack the ability of describing local patterns and detailed information . inspired by the recent success on video - based action recognition with deep convolutional descriptors @xcite , we investigate the effectiveness of * convolutional layer * activations . convolutional layer features have been also demonstrated to be effective in image - based tasks , such as object recognition @xcite , scene recognition @xcite and texture recognition @xcite . in this scenario , os - cnns are first pre - trained on large - scale imagenet and places205 datasets , and then fine - tuned on the event recognition dataset , just as in scenario 3 . specifically , given an image region @xmath8 , we first extract the convolutional feature maps of os - cnns ( activations of convolutional layers ) @xmath23 , where @xmath24 is feature map size and @xmath25 is feature channel number . each activation value in the convolutional feature map corresponds to a local receptive field in the original image , and therefore we call these activations of convolutional layers as os - cnn local representations . after extracting os - cnn local representations , we utilize two normalization methods , namely _ channel normalization _ and _ spatial normalization _ proposed in @xcite , to pre - process these convolutional feature maps into transformed convolutional feature maps @xmath26 . more details regarding these two normalization methods are out scope of this paper and can be found in @xcite . the normalized cnn activation @xmath27 at each postion @xmath28 is called as the _ transformed deep - convolutional descriptor _ ( these two kinds of normalization methods have turned out to be effective for improving the performance of cnn local representations in @xcite . moreover , the combination of them can obtain higher performance . therefore , we will use both normalization methods in our experimental explorations . finally , we employ fisher vector @xcite to encode these tdds into a global representation due to its good performance in object recognition @xcite and action recognition @xcite . in particular , according to our previous comprehensive study on encoding methods @xcite , we first use pca to reduce the dimension of tdd to @xmath29 . then each tdd is soft - quantized with a gaussian mixture model ( gmm ) with @xmath30 components ( @xmath30 set to 256 ) . the first and second order differences between each tdd @xmath31 and its gaussian center @xmath32 are aggregated in the block @xmath33 and @xmath34 , respectively . the final fisher vector representation is yielded by concatenating these blocks together : @xmath35.\ ] ] for os - cnns , the fisher vector of local representation is defined as follows : @xmath36,\ ] ] where @xmath37 is the fisher vector representation from object nets , @xmath38 is the fisher vector representation from scene nets , @xmath18 and @xmath19 are their fusion weights and set to be equal to each other in the current implementation . all the representations @xmath39 in previous three scenarios are used to construct a linear classifier @xmath40 , where @xmath41 is the weight of linear classifier . in our implementation , we choose libsvm @xcite as the classifier to learn the weight @xmath41 , where the parameter @xmath42 , balancing regularizer and loss , is set as @xmath43 . it is worth noting that all these representations are first normalized before fed into svm for training . for os - cnn global representations , we use @xmath44-normalization , and for os - cnn local representations , we use intra normalization and power @xmath44-normalization . .event recognition performance of os - cnn global and local representations on the validation data . [ cols="^,^,^,^",options="header " , ] * datasets . * the iccv chalearn lap challenge 2015 @xcite contains a track of cultural event recognition and provides an event recognition dataset . this dataset contains images collected from two image search engines ( google images and bing images ) . there are totally 100 event classes ( 99 event classes and 1 background class ) from different countries and some images are shown in figure [ fig : example ] . from these samples , we see that cultural event recognition is really complicated , where garments , human poses , objects and scene context all constitute the possible cues to be exploited for event understanding . this dataset is divided into three parts : development data ( 14,332 images ) , validation data ( 5,704 images ) , and evaluation data ( 8,669 images ) . as we can not access the label of evaluation data , we mainly train our models on the development data and report the results on the validation data . * evaluation protocol . * the principal quantitative measure is based on precision recall curve . they use the area under this curve as the computation of the average precision ( ap ) , which is calculated by numerical integration . finally , they average these per - class ap values across all event classes and employ the mean average precision ( map ) as the final ranking criteria . hence , in our exploration experiments , we report our results evaluated as ap value for each class and map value for all classes . * settings . * in this exploration experiment , we use the vggnet-19 as the os - cnn network structure . we extract activations from two fully connected layers ( ` fc6 ` , ` fc7 ` ) as os - cnn global representations , and activations from four convolutional layers ( ` conv5 - 1 ` , ` conv5 - 2 ` , ` conv5 - 3 ` , ` conv5 - 4 ` ) as os - cnn local representations . it should be noted that we choose the activations after rectified linear units ( relus ) . we use @xmath44-normalization to further process os - cnn global representations for better svm training . for fisher vector representation of os - cnn local representation , we employ intra - normalization and power @xmath44-normalization , as suggested by @xcite . * analysis . * we first report the numerical results in table [ tbl : result ] . from these results , several conclusions can be drawn as follows : * we see that the object nets outperform scene nets on the task of cultural event recognition , which may imply that object cues play more important roles than scene cues for cultural event understanding . * we observe that os - cnns are effective for event recognition as it extract both object and scene information from the image . they achieve superior performance to object nets and scene nets , no matter what scenario is adopted . * we can notice that combining fine tuned features with linear svm classifier ( scenario 3 ) is able to obtain better performance than direct using the softmax output of cnns ( scenario 1 ) . this result may be ascribed to the fact that cnns are easily over - fitted to the training samples when the number of training images is relatively small . * comparing fine - tuned features ( scenario 3 ) with pre - trained features ( scenario 2 ) , we may conclude that fine tuning on the target dataset is very useful for improving recognition performance , which agrees with the findings of @xcite . * comparing the local representations ( scenario 4 ) and global representations ( scenario 3 ) of cnns , we see that global representation achieve slightly higher recognition accuracy . * we further combine the global representation ( ` fc7 ` ) with local representation ( ` conv5 - 3 ` ) of cnns and find that this combination is capable of boosting final recognition performance . this performance improvement indicates that different layers of cnns capture different level abstraction of original image . these feature activations from different layers are complementary to each other . we also plot the ap values for all event classes in figure [ fig : ap ] . from these ap values , we see that the events of ` monkey buffet festival ` and ` battle of the oranges ` achieve the highest performance ( 100% ) . this result may be ascribed to the fact that there are specific objects in these two event categories . at the same time , we notice that some event classes obtain very low ap values , such as ` halloween festival of the dead ` , ` fiesta de la candelaria ` , ` apokries ` , and ` viking festival ` . the ap values of these cultural event classes are below 50% . in general , there are no specific objects and scene context in these difficult event classes , and besides these classes are easily confused with other classes from the perspective of visual appearance , as observed from figure [ fig : result_example ] . we visualize several recognition examples in figure [ fig : result_example ] . in the row 1 , we give eight examples that are successfully predicted by our method , from classes like ` keene pummpking ` , ` boryeong mud ` , ` afrikaburn ` and so on . meanwhile , we also provide some failure cases with high confidence from our method in the rows 2,3,4 . from these wrong predicted examples , we see that these failure cases are rather reasonable and there exists great confusion between some cultural event classes . for example , the event classes of ` dia de los muertos ` and ` halloween festival of the dead ` share similar human make - up and garments . the event classes of ` up helly aa ` and ` viking festtival ` share similar human dresses and containing objects . the event classes of ` harbin icen and snow festival ` and ` sapporo snow festival ` share similar scene context and color appearance . the event classes of ` chinese new year ` and ` pingxi lattern festival ` share similar containing objects . in summary , these examples in figure [ fig : result_example ] indicate that the concept of event is really complicated and there only exist slight difference between some event classes . for final evaluation , we merge the development data ( 14,332 images ) and validation data ( 5,704 images ) into a single training dataset ( 20,036 images ) and re - train our os - cnn models on this new dataset . our final submission results to the iccv chalearn lap challenge are based on our re - trained model . according to the above experimental explorations , we conclude that the os - cnn global and local representations are complementary to each other . thus , we choose to combine activations from ` fc7 ` and ` conv5 - 3 ` layers , to keep a balance between performance and efficiency . meanwhile , our previous study demonstrated that googlenet is complementary to vggnet @xcite . hence , we also extract a global representation by using the os - cnns of googlenet in our challenge solution . in summary , our challenge solution is composed of three representations : ( i ) os - cnn vggnet-19 local representations , ( ii ) os - cnn vggnet-19 global representations , and ( iii ) os - cnn googlenet global representations . the challenge results are summarized in table [ tbl : challenge ] . we see that our method is among the top performers and our map is very close to the best performance of this challenge ( 84.7% vs. 85.4% ) . regarding computational cost , our implementation is based on cuda 7.0 and matlab 2013a , and it takes about 1s to process one image in our workstation equipped with 8 cores cpu , 48 g ram , and tesla k40 gpu . in this paper , we have comprehensively studied different aspects of os - cnns for better cultural event recognition . specifically , we investigate the effectiveness of cnn activations from different layers by designing four types scenarios of adapting os - cnns to the task of cultural event recognition . from our empirical study , we demonstrate that the cnn activations from convolutional layers and fully connected layers are complementary to each other , and the combination of them is able to boost recognition performance . finally , we come up with a solution by using os - cnns at the iccv chalearn lap challenge and secure the third place . in the future , we may consider how to incorporate more visual cues such as human poses , garments , object and scene relationship in a systematic manner for event recognition in still images . this work is supported by a donation of two tesla k40 gpus from nvidia corporation . meanwhile this work is partially supported by national natural science foundation of china ( 91320101 , 61472410 ) , shenzhen basic research program ( jcyj20120903092050890 , jcyj20120617114614438 , jcyj20130402113127496 ) , 100 talents program of cas , and guangdong innovative research team program ( no.201001d0104648280 ) .

event recognition from still images is one of the most important problems for image understanding . however , compared with object recognition and scene recognition , event recognition has received much less research attention in computer vision community . this paper addresses the problem of cultural event recognition in still images and focuses on applying deep learning methods on this problem . in particular , we utilize the successful architecture of _ object - scene convolutional neural networks _ ( os - cnns ) to perform event recognition . os - cnns are composed of object nets and scene nets , which transfer the learned representations from the pre - trained models on large - scale object and scene recognition datasets , respectively . we propose four types of scenarios to explore os - cnns for event recognition by treating them as either `` end - to - end event predictors '' or `` generic feature extractors '' . our experimental results demonstrate that the global and local representations of os - cnns are complementary to each other . finally , based on our investigation of os - cnns , we come up with a solution for the cultural event recognition track at the iccv chalearn looking at people ( lap ) challenge 2015 . our team secures the third place at this challenge and our result is very close to the best performance .

introduction os-cnns revisited exploring os-cnns experiments conclusions acknowledgement

arxiv

it is widely assumed that the two - dimensional ( 2d ) @xmath6@xmath7 model @xcite is _ the _ correct model to describe the low - energy physics of the cuo@xmath0 planes.@xcite consequently , many authors believe that the high - temperature superconductivity in the cuprates _ can be _ explained by this model and it is merely the computationally challenging character of the model which leads to the lack of the understanding of the superconducting ground state ( see e.g. refs . ) . similarly , it has been suggested that the @xmath6@xmath7 model defined on the ladder ( called ladder @xmath6@xmath7 model in what follows ) is the right model to describe the low energy physics relevant for the cu@xmath0o@xmath1 coupled ladder planes of sr@xmath2ca@xmath3cu@xmath4o@xmath5 ( scco).@xcite this is a very attractive theoretical idea because : ( i ) the ladder @xmath6@xmath7 model is much easier to solve than its 2d counterpart and it has a superconducting ground state for some specific range of parameters,@xcite ( ii ) a superconducting ground state ( under pressure of 3 gpa ) was found@xcite in the ladder planes of scco for @xmath8 . this may suggest that indeed the @xmath6@xmath7 model contains the essential physics needed to explain the superconductivity , at least in the ladders . in this paper we would like to question the above point of view . as we show below , the ladder @xmath6@xmath7 model is too oversimplified and thus not sufficient to describe the low energy physics of the ladder planes in scco . actually , this can already be inferred by comparing the experimental observations in scco with the theoretical predictions for the ladders:@xcite ( i ) a charge density wave ( cdw ) ground state with period @xmath9 and @xmath10 was observed , but ( ii ) no cdw state with even period has been found,@xcite whereas ( iii ) the ladder @xmath6@xmath7 model may have a cdw ground state only with an even period.@xcite therefore , we investigate this problem here by a systematic derivation of the proper @xmath6@xmath7 model for the coupled ladder system , extended for topological reasons by the interladder repulsive term , and discuss a simple solution of this model . the paper is organized as follows . we introduce the charge transfer hamiltonian in sec . [ sec:2 ] . next , in sec . [ sec:3 ] we derive the low energy @xmath6@xmath7 hamiltonian which contains the kinetic energy and the superexchange , similar to the ladder @xmath6@xmath7 model , and the intraladder and interladder repulsion terms . in sec . [ sec:4 ] we examine the role of the coulomb intersite repulsion . finally , we present a numerical solution of the model in sec . [ sec:5 ] and draw conclusions in sec . [ sec:6 ] . the paper is supplemented by two appendices where some of the mathematical details of the model derivation are discussed . as the starting point we choose the multiband charge transfer hamiltonian introduced before for the cu@xmath0o@xmath1 coupled ladder geometry in scco.@xcite the model in hole notation reads , @xmath11 @xmath12 the model ( 1 ) was adopted to the present ladder geometry@xcite from the charge transfer models introduced before for cuo@xmath0 planes,@xcite and cuo@xmath13 chains@xcite in high temperature superconductors . the parameters are : the energy for oxygen @xmath14 ( @xmath15 or @xmath16 with creation operators @xmath17 and @xmath18 ) orbital @xmath19 ( the so - called charge transfer energy measured with respect to the energy of @xmath20 copper orbitals ) , the @xmath21-@xmath22 hopping @xmath23 between the nearest neighbor copper and oxygen sites , the on - site coulomb repulsion @xmath24 ( @xmath25 ) on the copper ( oxygen ) sites , and @xmath26 a realistic value of hund s exchange on oxygen ions@xcite ( for a complete set of realistic parameters see sec . [ sec : para ] ) . besides , in principle the actual electron energy at bridge orbital ( rung position ) of the ladder with creation operators @xmath27 is approximately 10 % smaller than the one at other oxygen positions.@xcite however , it was shown@xcite that this difference does not have any important physical consequences and therefore we will neglect it here . note also that the phases of the @xmath28 orbitals were explicitly taken into account in the hopping elements @xmath29 ( for clarity the phases are shown only in fig . [ fig:1 ] ) , the index @xmath30 denotes the right or left leg of the ladder ( @xmath31 and @xmath32 ) , @xmath33 for @xmath34 , and the upper ( lower ) sign stands for terms with @xmath35 ( @xmath36 ) . o@xmath1 ( white / gray ) ladders in scco : ( a ) orbitals in charge transfer model ( [ eq : ct ] ) ; ( b ) intraladder ( interladder ) bonds in the effective extended @xmath6@xmath7 model , see eq . ( [ eq : tj ] ) , shown by solid ( dashed ) lines . , width=317 ] the charge transfer model ( 1 ) includes seven orbitals per cu@xmath0o@xmath1 ladder unit cell @xmath37 ( see fig . [ fig:0 ] ) : two cu(@xmath38 ) orbitals on the @xmath39 leg , two o(@xmath40 ) orbitals on the @xmath39 leg , two o(@xmath41 ) side orbitals on the @xmath39 leg , and one o(@xmath42 ) bridge orbital on the rung . although it seems that the model is quasi one - dimensional ( 1d ) , the density operators @xmath43 and @xmath44 stand for the oxygen hole densites in the neighboring ladders and make it implicitly 2d as the interladder coupling couples the ladders , so the model extends over the entire cu@xmath0o@xmath1 plane . in what follows we will derive the low - energy version of hamiltonian ( [ eq : ct ] ) which is valid in the so - called charge transfer regime @xmath45 , i.e. for the typical values of model ( [ eq : ct ] ) parameters : @xmath46 , @xmath47 , and @xmath48 , see ref . and sec . [ sec : para ] below . the effective @xmath6@xmath7 hamiltonian consists then of four terms , and may be thus also called @xmath6@xmath7@xmath49 model , @xmath50 which are separately derived and discussed below . we begin with the superexchange term @xmath51 which is the only important term in eq . ( [ eq : tj ] ) at half - filling . in this case ( i.e. with one hole per copper site ) the charge transfer model ( [ eq : ct ] ) can be easily reduced to the low energy heisenberg model for spins @xmath52 using the perturbation theory to fourth order in @xmath23:@xcite @xmath53 here , tilde in @xmath54 implies that the hole double occupancies are excluded . the superexchange constant contains contributions due to charge excitations on copper sites and on the intermediate oxygen site for a cu o cu bond , and for finite @xmath25 reads:@xcite @xmath55 one may wonder whether the geometry of coupled ladders could influence the above result . indeed , there exists a @xmath56 superexchange process between the holes on two neighboring ladders . however , according to the goodenough - kanamori rules@xcite such a superexchange process is much weaker than the superexchange generated by charge excitations along the @xmath57 path in the single ladder and can be neglected . c c c c & @xmath58 & @xmath59 & single oxygen + singlet & @xmath60 & @xmath61 & @xmath62 triplet & @xmath63 & @xmath64 & @xmath63 when the ladder is away from half - filling , the perturbation theory gets complicated . therefore , following the zhang and rice construction,@xcite we first define a phase coherent _ symmetric plaquette state_@xcite @xmath65 which is formed by the four oxygen orbitals surrounding the central copper site @xmath66 : @xmath67 where the upper ( lower ) sign stands for @xmath35 ( @xmath36 ) . when a hole in this state forms a singlet state with the hole at the central copper site , it has a large negative binding energy of @xmath60 , see caption of table i for definition of @xmath68 hoppings and for more details . actually , this binding energy is not only much larger than the individual effective hopping terms ( which is of the order of @xmath69 or @xmath70 , see ref . ) but it is also considerably larger than the binding energy of some other possible bound states , see table i. note that finite @xmath25 , not considered by zhang and rice,@xcite results in finite @xmath71 hopping but does not qualitatively change the large binding energy of a symmetric singlet state ( [ eq : pi ] ) . the mere problem with the states defined by eq . ( [ eq : pi ] ) is that they are not orthogonal . it can be checked that in the case of the ladder geometry the following superposition of the symmetric plaquette states forms a complete and orthogonal basis for a low - energy hilbert subspace : @xmath72 where @xmath73 . then the zhang - rice ( zr ) singlets for the ladder are : @xmath74 although the binding energy is slightly reduced after this orthogonalization , the change is not significant : if the energy splitting between the orthogonalized zr singlets and triplets is defined as @xmath75 ( we consider a simplified case @xmath76 and @xmath77 ) , then @xmath78 both in the 1d ( @xmath79 ) and in the 2d case ( @xmath80 ) , see ref . . having shown that the zr singlets in the single ladder do not differ much from those which arise in the 2d cuprates,@xcite we can now safely apply all the arguments used in ref . to derive the effective hopping of zr singlets following from finite @xmath23 . thus , we obtain , @xmath81 where once again @xmath82 is a fermion operator in the restricted space . while we do not show here the detailed expression for the effective hopping @xmath6 of zr singlets , note that it is considerably smaller than @xmath23 ( ca . 50%).@xcite note also that having two zr singlets at the same site costs energy @xmath83 ( if @xmath84 , see ref . ) and therefore we used the tilde operators above to exclude these local configurations of two zr singlets . the coulomb interaction on oxygen sites @xmath25 , neglected in ref . , plays a minor role in the stability of the zr singlets ( see e.g. finite @xmath71 for finite @xmath25 in table i ) , but this issue is more subtle.@xcite actually , due to finite @xmath25 the two neighboring nonorthogonal zr singlets repel each other when two holes occupy a common oxygen site ( see figs . [ fig:1 ] and [ fig:2 ] ) . whereas the significance of the interladder repulsion is discussed in the next subsection , let us concentrate first on the repulsion between the zr singlets _ within a single ladder _ ( see fig . [ fig:1 ] ) , and calculate repulsion @xmath86 between two orthogonalized zr singlets within the ladder @xmath87 . let us note that the ` mixed terms ' such as @xmath88 , which could _ a priori _ destroy the zr singlets , fortunately turn out to be much smaller than the respective binding energy . ( [ eq : zhangrice ] ) , after a somewhat lengthy but straightforward calculation ( for more details see appendix [ app : a ] ) , one finds the following values for the intraladder interaction along the leg and the rung : @xmath89 we have verified that the interaction between the second nearest neighbors is ca . 15 times smaller and can be safely neglected ( the longer - range interaction is even smaller , cf . appendix [ app : a ] ) . hence , one finds that the interaction among the nearest neighbor zr singlets is almost isotropic . thus , we can write the effective hamiltonian for the repulsion between zr singlets ( as shown in fig . [ fig:1 ] ) @xmath90 where @xmath91 . let us note that the ratio @xmath92 is approximately @xmath93 smaller than the naively estimated nonorthogonal value @xmath94 . we also checked that dimensionality drives the following trend in the ratios @xmath95 : @xmath96 , @xmath97 , @xmath98 ( considered here ) , and @xmath99 , for a single rung , the 1d case , a ladder , and the 2d case , so @xmath95 increases with the increasing number of neighbors.@xcite this follows because in lower dimensions the charge escapes easier from the orbitals @xmath100 and @xmath101 ( providing the dominating contribution ) , while in the 2d case _ all _ the orbitals suffer from the orthogonality problem . . , scaledwidth=47.0% ] finally , we calculate the _ interladder _ repulsion between the zr singlets due to on - site repulsion @xmath25 in orbitals belonging to two neighboring ladders : @xmath103 a _ bar _ sign over @xmath104 denotes the singlet formed on the neighboring ladder . besides , since the neighboring ladder is misaligned by a lattice constant @xmath105 with respect to the one considered , we label the zr singlets on the neighboring ladder by @xmath106 ( for the copper - copper lattice constant equal to @xmath107 ) . next , using eq . ( [ eq : zhangrice ] ) one finds after a somewhat tedious but straightforward calculation ( for more details see appendix [ app : b ] ) the following value for the interladder interaction between the closest sites belonging to the neighboring ladders ( see fig . [ fig:2 ] ) , @xmath108 while other ( neglected ) longer - range repulsive terms are at least one order of magnitude smaller , cf . appendix [ app : b ] . thus , the repulsion between holes on the neighboring ladders reads : @xmath109 where @xmath110 operator is related to the zr singlets as before and @xmath111 . we again neglected all spin - flip terms which are small in comparison with the zr binding energy and give zero when ` sandwiched ' in the singlet states . besides , the numerical prefactor ( equal to @xmath112 ) is here slightly enhanced with respect to the expected @xmath113 value ( unlike in the intraladder case ) . this is because a significant fraction of charge escapes from the @xmath100 and @xmath101 orbitals to the @xmath114 orbitals due to the orthogonalization procedure . c c c c & @xmath23 & 1.3 & @xmath6 & 0.54 @xmath19 & 3.6 & @xmath7 & 0.24 @xmath25 & 4.0 & @xmath115 & 0.11 @xmath24 & 10.5 & @xmath116 & 0.27 @xmath117 & 0.8 & & the calculated parameters of the effective @xmath6@xmath7@xmath49 model ( [ eq : tj ] ) derived above are shown in table [ tab:2 ] . this calculation is based on the cuprate charge transfer model parameters from ref . ( the value of @xmath117 is taken from ref . ) which might be considered as the most widely accepted choice of the cuprate parameters , cf . refs . and . due to the same cu o distances in scco as in cuo@xmath0 planes in e.g. la@xmath0cuo@xmath118 , we can can adopt these parameters also to the present case . let us note that on the one hand , it should be emphasized that the _ interladder _ coupling @xmath116 is of the order of @xmath7 for the realistic parameters@xcite and therefore _ can not be neglected_. on the other hand , the value of @xmath115 is two and a half times smaller and therefore we suggest that , if necessary , this interaction could be skipped in the first - order calculations . one may wonder whether the intersite coulomb repulsion @xmath119 in the charge transfer model could alone lead to a significant repulsion ( i.e. , of the order of the estimated value of @xmath116 ) between the zr singlets in the neighboring ladders . this term , which stands for the repulsion between charges situated in the nearest neighbor copper @xmath20 and oxygen @xmath120 orbitals , was neglected in ref . and in the above analysis , cf . ( [ eq : ct ] ) . in fact , including this term may e.g. lead to a significant renormalization of the parameters of the 2d @xmath6@xmath7 model.@xcite indeed , one finds that the repulsion between a hole in a symmetric state @xmath121 and a copper hole in state @xmath122 situated on the nearest neighbor sites in two neighboring ladders is of the order of @xmath123 . although typically @xmath119 is smaller than @xmath23 , e.g. @xmath124 ev @xcite or even @xmath125 ev,@xcite this contribution might still be significant and , in principle , should not be entirely neglected . however , the key observation is that this term leads to roughly equally large energy cost if : ( i ) _ either _ the two zr singlets are situated on the above mentioned sites @xmath37 and @xmath126 and repel each other due to @xmath119 , _ or _ ( ii ) the two zr singlets are situated far away from each other on two ladders and ( due to @xmath119 ) merely feel the repulsion with the neighboring copper hole on the neighboring ladder . hence , including the intersite repulsion @xmath119 increases the total energy of the system but almost does not contribute to the energy difference between the two above situations , measured by the value of the interladder repulsion @xmath116 . it is only a small residual repulsion due to the orthogonalization procedure , see eq . ( [ eq : plaql ] ) , which may change the value of @xmath116 by a small fraction . this has been also confirmed by the results of ref . , where @xmath119 leads indeed to a very small repulsion between holes in the 2d @xmath6@xmath7 model ( ca . @xmath127 ) . now we shall verify whether the derived @xmath6@xmath7@xmath49 model eq . ( [ eq : tj ] ) supports the cdw states observed in scco and to understand to what extent the interladder term ( [ eq : hv2 ] ) influences the stability of the cdw state . as a thorough investigation is beyond the scope of this work and left for future studies , we solve the model ( [ eq : tj ] ) in the simplest possible way . thus , we first introduce the gutzwiller factors , @xmath128 and @xmath129 , which renormalize the kinetic ( @xmath130 ) and interaction ( @xmath131 ) terms ; for their justification see e.g. refs . . here @xmath132 denotes the average number of @xmath21 holes per site in the effective model ( [ eq : tj ] ) , i.e. , @xmath133 . second , we use the mean - field approximation for the interaction terms . next , we diagonalize the effective one - particle hamiltonian introducing the classical fields @xmath134 where @xmath135 is the cdw order parameter and @xmath136 is the cdw period . furthermore , @xmath137 are defined as in eq . ( [ eq : p ] ) but with @xmath138 replaced by @xmath139 [ @xmath140 for @xmath141 this assumption minimizes the classical energy cost of the interladder repulsion @xmath116 . in fig . [ fig:3 ] we show the cdw order parameter @xmath22 calculated self - consistently as a function of the interladder interaction @xmath116 for the three experimentally interesting doping levels:@xcite @xmath142 which corresponds to @xmath143 holes per copper site in charge transfer model ( [ eq : ct ] ) , @xmath144 corresponding to @xmath145 , and @xmath146 corresponding to @xmath147 . the results demonstrate that the interladder interaction plays indeed a crucial role in the stability of the cdw while the intraladder one is rather unimportant . we have found that the cdw state with period @xmath148 ( @xmath149 ) is stable for @xmath142 ( @xmath146 ) for the rather realistic values of the parameters @xmath150 , @xmath151 and @xmath152 . please note , that : ( i ) we adopted here a somewhat smaller value of @xmath150 which is closer to a typical value for cuprates,@xcite and ( ii ) values of @xmath116 exceeding @xmath153 t can be obtained using for example the set of parameters suggested in ref . . this finding explains well the experimental results of ref . . besides , the cdw ordered state is also stable for period @xmath154 which was not observed.@xcite we expect that the stability of the cdw state with this period is a shortcoming of the above simplified solution which does not capture well the frustration between two possible cdw patterns in the neighboring ladders which occurs for period @xmath154.@xcite in summary , we derived the @xmath6@xmath7 model which describes the low energy physics of cu@xmath0o@xmath1 coupled ladders . apart from the ` standard ' superexchange @xmath155 and kinetic energy terms @xmath156 , the model contains also the repulsion between the nearest neighbor holes in a ladder @xmath157 and in the two neighboring ladders @xmath158 , and hence is also referred to as a @xmath6@xmath7@xmath49 model [ see eq . ( [ eq : tj ] ) ] . we showed that the latter @xmath116 term is roughly two and a half times larger than @xmath115 and ( contrary to @xmath115 ) can not be skipped in fact it is crucial to explain the onset of the odd period cdw state in scco . we emphasize that this particular extra term is restricted to the copper oxides in which oxygen is coordinated by three copper ions in the same plane . therefore , it is not present in the cuo@xmath0 planes,@xcite or in cu o chain@xcite in copper oxides , but ( apart from the discussed scco case ) could become relevant for the coupled chains of srcuo@xmath0 ( provided they are hole - doped ) . furthermore , it is both a many - body term _ and _ of the order of @xmath7 , contrary to various corrections to a 1d or 2d @xmath6@xmath7 model.@xcite besides , we also verified that the coulomb intersite interacton @xmath119 alone ( not included in the presented derivation ) can not lead to a significant interladder repulsion @xmath116 . the simple mean - field solutions of the @xmath6@xmath7@xmath49 model derived here provides evidence in favor of the experimental observations of the onset of the odd cdw state in the ladder planes of the scco . this is further supported by the recent density matrix renormalization group calculations @xcite where a ladder @xmath6@xmath7 model with the interladder coupling @xmath116 ( denoted as @xmath159 in ref . ) was studied : also there the cdw with odd period is stabilized due to the presence of the interladder interaction @xmath116 . finally , let us note that the other hole - doped ladder compound la@xmath160sr@xmath3cuo@xmath161 is also characterized by a large ( but different than the one discussed here ) interladder coupling.@xcite therefore , we argue that it is currently a challenge for the condensed matter community to search for a hole - doped ladder compound which could indeed be modelled by the ladder @xmath6@xmath7 hamiltonian [ given by eqs . ( [ eq : hj ] ) and ( [ eq : ht ] ) , i.e. , without additional intersite repulsion terms ] . we thank alexander chernyshev for insightful discussions and maria daghofer for the critical reading of the manuscript . we acknowledge financial support by the foundation for polish science ( fnp ) and the polish ministry of science and education under project no . n202 068 32/1481 . k. w. thanks university of british columbia for the kind hospitality . here we show how to calculate the repulsion between orthogonalized zr singlets within the ladder due to the on - site interaction @xmath25 in @xmath162 . thus , one needs to determine the following matrix elements : @xmath163 let us note that the mixed terms such as for example @xmath164 vanish in the zr singlet basis they could _ a priori _ lead to the destruction of the zr singlets , but fortunately they are much smaller than the respective binding energy . _ intraladder repulsion along the leg. _ first , we calculate the matrix elements of @xmath162 between the orthogonal plaquette states eq . ( [ eq : plaql ] ) along the leg : @xmath165 and @xmath166 while the same spin elements are zero , @xmath167 one can evaluate numerically the above expressions . it occurs that the largest positive element is the nearest neighbor interaction @xmath168 while following eq . ( [ eq : negative ] ) the absolute value of the largest negative element , which corresponds to spin - flip nearest neighbor interaction , is the same . furthermore , the second largest element is the next nearest neighbor interaction and is over 20 times smaller , which means that it can be safely neglected . second , we calculate the matrix elements of @xmath162 between the nearest neighbor zr singlets , defined by eq . ( [ eq : zhangrice ] ) . this introduces a factor @xmath105 to the above estimations of the repulsion between orthogonal plaquette states : it is because there is a @xmath169 probability to have opposite spins on a particular shared oxygen site occupied by two holes from two different zr singlets . note that the spin - flip - plaquette terms do not give any contribution to the repulsion between zr singlets , although they could in principle destabilize the zr states themselves . fortunately , this is not possible since the binding energy of the zr singlets is much larger . thus altogether , we obtain for the repulsion along the same leg @xmath170 _ intraladder repulsion along the rung. _ following a similar scheme , one can calculate the repulsion between zr singlets on different legs . one obtains the following matrix elements of @xmath162 between the orthogonal plaquette states eq . ( [ eq : plaql ] ) on different legs @xmath171 and @xmath172 and @xmath173 evaluating numerically the above expressions one obtains that the largest element is the nearest neighbor repulsion this time between the orthogonal plaquette states on the same rung : @xmath174 while the second largest element ( the next nearest neighbor interaction ) is over 15 times smaller and can be neglected . finally , following the same steps as those leading from ( [ eq : phileg ] ) to ( [ eq : psileg ] ) , we obtain the repulsion between the nearest neighbor zr singlets [ defined by eq . ( [ eq : zhangrice ] ) ] along the same rung which is twice reduced : @xmath175 here the task is to calculate the repulsion between two zr singlets centered at the neighboring copper positions of two ladders ( and thus sharing the same oxygen sites but _ not _ the @xmath22 orbitals , see fig . [ fig:2 ] ) due to the on - site repulsion on oxygen sites . however , again we will calculate the repulsion between arbitrarily located zr singlets and only then we will show which elements are negligible . note that the plaquette states on two ladders are orthogonal to each other although they still have to be orthogonalized for the same ladder ( as in appendix [ app : a ] ) . explicitly one needs to calculate the following matrix elements : @xmath176 and @xmath177 _ interladder repulsion between plaquettes with the same spin. _ we calculate the matrix elements of @xmath178 between the orthogonal plaquette states eq . ( [ eq : plaql ] ) with the same spin but situated on different legs : @xmath179 while for the same legs we obtain @xmath180 as it might have been expected , it occurs that the biggest term is the repulsion between orthogonal plaquette states with the same spin situated on the closest possible sites in the neighboring ladders ( see fig . [ fig:2 ] ) : @xmath181 and all other terms are of the order of @xmath182 and can be neglected . _ interladder repulsion between plaquettes with opposite spin. _ a very similar calculation as above , but performed for the orthogonal plaquette states eq . ( [ eq : plaql ] ) with opposite spins leads to the repulsion between orthogonal plaquette states with opposite spins and situated on the closest possible sites in the neighboring ladders : @xmath183 while again all other longer - range repulsive terms can be neglected . finally , combining eqs . ( [ eq : phiphisame])-([eq : phiphiopposite ] ) with the definition of the zr singlet ( [ eq : zhangrice ] ) we obtain the value of the repulsion between the two zr singlets ( cf . similar discussion in appendix [ app : a ] ) on the closest possible sites in the neighboring ladders to be @xmath184 t. a. maier , m. jarrell , t. c. schulthess , p. r. c. kent , and j. b. white , * 95 * , 237001 ( 2005 ) ; m. ogata and h. fukuyama , rep . prog . phys . * 71 * , 036501 ( 2008 ) ; l. spanu , m. lugas , f. becca , and s. sorella , * 77 * , 024510 ( 2008 ) . v. j. emery , * 58 * , 2794 ( 1987 ) ; a. m. ole , j. zaanen , p. fulde , physica b&c * 148 * , 260 ( 1987 ) ; c. m. varma , s. schmitt rink , and e. abrahams , solid state commun . * 62 * , 681 ( 1987 ) ; j. dutka and a. m. ole , * 42 * , 105 ( 1990 ) . in la@xmath160sr@xmath3cuo@xmath161 the interladder coupling even leads to the antiferromagnetic order for @xmath186 , see : s. matsumoto , y. kitaoka , k. ishida , k. asayama , z. hiroi , n. kobayashi , and m. takano , * 53 * , r11942 ( 1996 ) .

starting from the proper charge transfer model for cu@xmath0o@xmath1 coupled ladders in sr@xmath2ca@xmath3cu@xmath4o@xmath5 we derive the low energy hamiltonian for this system . it occurs that the widely used ladder @xmath6@xmath7 model is not sufficient and has to be supplemented by the coulomb repulsion term between holes in the neighboring ladders . furthermore , we show how a simple mean - field solution of the derived @xmath6@xmath7 model may explain the onset of the charge density wave with the odd period in sr@xmath2ca@xmath3cu@xmath4o@xmath5 . _ published in phys . rev . b * 81 * , 214522 ( 2010 ) _

introduction the charge transfer hamiltonian the effective @xmath6@xmath7 hamiltonian role of the intersite coulomb repulsion @xmath119 numerical results conclusions derivation of the intraladder repulsion term @xmath157 derivation of the interladder repulsion term @xmath158

arxiv

to date , most surveys searching for extragalactic water masers have targeted spiral galaxies and radio - quiet agn in the local universe . likely , this is the reason why , with the exception of a type 2 quasar at @xmath3 ( ( * ? ? ? * barvainis & antonucci 2005 ) ) , the majority of the known extragalactic water masers have been found in seyfert 2 or liner galaxies at low redshift ( @xmath4 ) . observations of objects at higher redshifts are limited in part by the range of frequencies available but mainly by the sensitivity of current radio telescopes . to overcome this limitation , we have been carrying out a survey of water masers in known gravitationally lensed quasars with the effelsberg and arecibo telescopes ( ( * ? ? ? * mckean et al . 2011 ; mckean et al . in prep ) ) . observing gravitationally lensed quasars allows us to use the magnification provided by the lensing galaxy to increase the measured flux density of the background agn , strongly reducing the integration time necessary to detect the signal originating from distant objects ( @xmath5 ) . this potentially allows us to discover water masers at cosmological distances and to study the parsec - scale environment of agn in the early universe . our first confirmed high - redshift water maser was found toward the lensed quasar mg j0414 + 0534 at @xmath6 ( ( * ? ? ? * impellizzeri et al . 2008 ) ) . the line was originally detected with the effelsberg radiotelscope and subsequently confirmed with the evla that found the emission to be coincident with the lensed images of the quasar ( a1 and a2 ) . the apparent isotropic luminosity of the line ( @xmath210,000l@xmath7 , after correcting for the estimated lens magnification ) makes the water maser in mg j0414 + 0534 not only the most distant but also one of the most luminous water masers ever detected and suggests that the emission is associated with the agn . based on the large line width of the effelsberg and evla spectra , our initial hypothesis on the origin of the maser was that the emission is associated with the jet(s ) of the radio loud quasar . in order to reveal possible variations in the maser flux density , typical of the masers produced by the interactions between a molecular cloud and a radio jet , and determine if a correlation exists between the maser and the continuum emission , we monitored the line and the radio continuum in mg j0414 + 0534 with the 300-m arecibo telescope during a time interval of 15 months . here we report the results of this monitoring campaign and discuss them within the framework of the two main scenarios for the origin of the maser , i.e. jet - maser and disk - maser emission ( for more details on this work , see ( * ? ? ? * castangia et al . 2011 ) ) . we adopt a cosmology with @xmath8 , @xmath9 and @xmath10kms@xmath0mpc@xmath0 . in the following , the quoted line velocities are defined w.r.t . the optical redshift of mg j0414 + 0534 ( @xmath11=2.639 ; ( * ? ? ? * lawrence et al . 1995 ) ) , using the optical velocity definition in the heliocentric frame . we have monitored the redshifted ( rest frequency : 22ghz ) radio continuum and maser emission in mg j0414 + 0534 for @xmath215 months at @xmath26 week intervals and found that both are surprisingly stable . absolute deviations of the continuum flux from the mean are on average comparable with the flux calibration uncertainty ( 7% ) . the 6ghz ( observed frequency ) continuum flux density of mg j0414 + 0534 thus remained nearly constant for the duration of the entire monitoring period , with an average flux density of [email protected] , the error being the standard deviation of the mean . the line peak flux density is also surprisingly stable throughout the period of the observations . small fluctuations are not exceeding the limits of uncertainty ( between 10% and 50% ) . from the analysis of the 11 epochs of the monitoring , we can place an upper limit on the velocity drift of the line peak of 2kms@xmath0yr@xmath0 . . the root - mean - square ( r.m.s . ) noise level of the spectrum is 0.2mjy per channel . the velocity scale is relative to redshift 2.639 using the optical velocity definition in the heliocentric frame . the red cross marks the systemic velocity and the associated uncertainty . the blue and the black crosses indicate the peaks of the co emission ( ( * ? ? ? * barvainis et al . 1998 ) ) and the hi absorption components ( ( * ? ? ? * moore et al . 1999 ) ) , respectively , with their errors.,width=326 ] a significant change in the line profile seems to have occurred between the effelsberg and evla observations and the first epoch of the arecibo monitoring campaign . indeed , the line appears to be much broader in the first arecibo spectrum ( taken in october 2008 ) w.r.t . the previous observations . the full width at half maximum ( fwhm ) of the gaussian profile fitted to the line is 174@xmath125kms@xmath0 , i.e. more than a factor of two larger than the fwhm of the effelsberg and evla spectra ( ( * ? ? ? * impellizzeri et al . 2008 ) ) . as a consequence , we measure an unlensed isotropic luminosity of 26,000l@xmath7 , that makes the maser in mg j0414 + 0534 the most luminous that is currently known . furthermore , in october 2008 we tentatively detected a weaker satellite line at + 470kms@xmath0 ( fig.[fig : fig_oct ] ) that , however , was not confirmed by the spectra of the other epochs . this second feature , detected with a signal - to - noise ratio of three , is displaced by about 800kms@xmath0 from the main line and is five time less luminous . in february 2009 we performed deeper observations aimed at confirming the presence of this feature . no emission line other than the main one at the velocity of about @xmath13300kms@xmath0 was detected above a 3@xmath14 noise level of 0.3mjy per 19.2kms@xmath0 channel . however , a weak feature is seen in the spectrum at the velocity of about + 490kms@xmath0 ( see fig.[fig : fit4 ] , lower panel ) . the satellite line remains undetected also in the spectrum produced by averaging all of the epochs with the same weights ( fig [ fig : fit4 ] , upper panel ) . nonetheless , we note that the range between 200 and 500kms@xmath0 looks spiky and that , interestingly , one of these spikes is at the position of the satellite line . averaging the spectra using different weights ( e.g. 1/r.m.s@xmath15 or the integration time ) does not change the shape of the resulting spectrum significantly . this may indicate that many weak lines are present in the range 200500kms@xmath0 and that in october 2008 we saw one of these lines flaring . the high snr of the february 2009 spectrum ( @xmath213 ; see fig [ fig : fit4 ] , lower panel ) reveals that the main line has a complex profile that is likely the result of the blending of at least four components with line widths between 30 and 160kms@xmath0 . in order to inspect the variability of the individual velocity features , we produced a spectrum by averaging with equal weights the last three epochs of the monitoring campaign ( september and november 2009 and january 2010 ) . the resulting spectrum ( fig [ fig : fit4 ] , middle panel ) has an r.m.s comparable with that of the february 2009 observation . comparing the gaussian peak velocities , we find that the velocities of components i and ii did not change , while the velocities of components iii and iv have marginally increased by + 15@xmath123kms@xmath0 and + 10@xmath123kms@xmath0 , respectively . since velocity drifts have been observed only in very narrow lines ( fwhm @xmath2 14kms@xmath0 ) , we think that the change in the peak velocities of these features maybe due to variations of a large number of sub - components , simulating a change in the radial velocity as observed most notably in ngc 1052 ( ( * ? ? ? * braatz et al . 1996 ) ) . our monitoring data are partially consistent with our initial hypothesis that the emission is associated with the prominent relativistic jets of the quasar ( ( * ? ? ? * impellizzeri et al . 2008 ) ) . first of all , even when the maser line profile is resolved into multiple velocity components , individual emission features have line widths between 30 and 160kms@xmath0 that resemble those of known h@xmath1o masers associated with radio jets ( e.g. mrk 348 ; ( * ? ? ? * peck et al . 2003 ) ) . our non - detection of a radial acceleration of the main maser peak is also compatible with the jet - maser scenario . the extreme stability of the main line peak and the continuum flux density in mg j0414 + 0534 resulting from our study , seems to exclude a jet - maser scenario similar to that in mrk 348 ( ( * ? ? ? * peck et al . 2003 ) ) and ngc 1068 ( ( * ? ? ? * gallimore et al . 2001 ) ) , while the reported significant variations in the line profile of our target may hint at similarities with the case of ngc 1052 ( ( * ? ? ? * ; * ? ? ? * braatz et al . 1996 and 2003 ) ) . we note , however , that the number of sources in which the maser emission is confidently associated with the jet(s ) is very low and that more of these masers should be studied in detail in order to investigate the properties of these kind of sources . furthermore , the tentative detection of the redshifted feature in the october 2008 spectrum is compatible with the disk - maser hypothesis . if the main maser line and the satellite line at + 470kms@xmath0 are considered as the blueshifted and redshifted lines from the tangentially seen part of an edge - on accretion disk in keplerian rotation , then the radius at which the emission originates is given by @xmath16 , where @xmath17 is the gravitational constant , @xmath18 is the black hole mass , and @xmath19 is the rotational velocity at radius @xmath20 . from the difference between the line of sight velocities of the main and satellite maser lines ( @xmath21 ) , we obtain @xmath22 @xmath23kms@xmath0 . adopting the black hole mass of @xmath24m@xmath7 calculated by ( * pooley et al . ( 2007 ) ) for mg j0414 + 0534 , and assuming an edge - on orientation ( @xmath25 = 90@xmath26 ) for the accretion disk , we get a radius of @xmath27 30pc . this value is fairly large compared to the radii at which maser emission is found in the accretion disks of nearby radio quiet agn ( typically , 0.1 to 1 pc ) . we should keep in mind however , that mg j0414 + 0534 is a radio loud quasar , while known disk - maser hosts are mainly radio quiet seyfert or liner galaxies with a mass of the nuclear engine that is two orders of magnitude lower ( @xmath28m@xmath7 ; ( * ? ? ? * kuo et al . 2011 ) ) . in conclusion , although we have been able to provide useful elements to determine the nature of the maser in mg j0414 + 0534 , our current data are presently insufficient to confidently rule out whether the maser emission is due to a jet - cloud interaction or a rotating circumnuclear disk . vlbi observations and longer time - scale single - dish monitoring will be essential to shed light on the origin of the h@xmath1o maser in this distant quasar .

we monitored the 22ghz maser line in the lensed quasar mg j0414 + 0534 at z=2.64 with the 300-m arecibo telescope for almost two years to detect possible additional maser components and to measure a potential velocity drift of the lines . the main maser line profile is complex and can be resolved into a number of broad features with line widths of 30 - 160kms@xmath0 . a new maser component was tentatively detected in october 2008 at a velocity of + 470kms@xmath0 . after correcting for the estimated lens magnification , we find that the h@xmath1o isotropic luminosity of the maser in mg j0414 + 0534 is @xmath226,000 solar luminosities , making this source the most luminous ever discovered . both the main line peak and continuum flux densities are surprisingly stable throughout the period of the observations . an upper limit on the velocity drift of the main peak of the line has been estimated from our observations and is of the order of 2kms@xmath0 per year . we discuss the results of the monitoring in terms of the possible nature of the maser emission , associated with an accretion disk or a radio jet . this is the first time that such a study is performed in a water maser source at high redshift , potentially allowing us to study the parsec - scale environment around a powerful radio source at cosmological distances .

introduction results discussion and concluding remarks

arxiv

the hubble tarantula treasury project ( http ) is a photometric survey at high spatial resolution of the tarantula nebula ( 30dor ) , from near ultraviolet ( nuv ) to near infrared ( nir ) wavelengths ( sabbi et al . its purpose is to reconstruct the region s star - formation history in space and time on a parsec scale over a total extent of @xmath4pc@xmath5 . the ultimate goal is to establish the strength , duration , and spatial scale of the star - formation episodes and their possible mutual relationships . an initial study limited to the central ngc2070 cluster ( cignoni et al . 2015 ) confirms that over the past @xmath6myr the cluster experienced a prolonged activity of star formation ( e.g. , walborn & blades 1997 ; walborn et al . 1999 ) , with several episodes ( de marchi et al . 2011a ) , culminating in a peak @xmath7myr ago . besides high - quality photometry ( sabbi et al . 2015 ) , these studies rely on our ability to securely measure the intrinsic physical properties of stars , i.e. their true colours and luminosities , since these are crucial to extract reliable masses , ages and other physical parameters to track the star - formation process . knowledge of the properties and amount of the interstellar extinction is thus of paramount importance , and equally fundamental is knowing how to apply this information to correct the photometry of individual stars . this is particularly crucial in an environment such as the tarantula nebula , due to its complex structure and to the presence of a considerable amount of atomic and molecular gas ( e.g. , indebetouw et al . 2013 ; yeh et al . 2015 ) and dust ( e.g. , meixner et al . 2013 ) , resulting in a patchy and uneven level of extinction across the nebula . high resolution _ hubble space telescope _ ( hst ) studies of the massive stellar tarantula clusters ngc2070 and hodge301 have long shown a wide spread of extinction values ( e.g. , hunter et al . 1995a ; grebel & chu 2000 ; de marchi et al . 2011a ) , particularly in the central ngc2070 cluster , where @xmath8 varies by more than 3mag over regions of @xmath9pc across , as shown by maz apellniz et al . ( 2014 ) and de marchi & panagia ( 2014 ) . more importantly , both maz apellniz et al . ( 2014 ) and de marchi & panagia ( 2014 ) have independently shown from spectroscopy and photometry that the extinction law for the central ngc2070 cluster is very different from that typical of the diffuse galactic interstellar medium ( ism ) , with a @xmath10 higher ratio of total - to - selective extinction , namely @xmath11 instead of @xmath12 . this finding is particularly intriguing because the tarantula nebula is routinely considered an ideal test case ( the `` starburst rosetta '' , walborn 1991 ) for regions of strong star formation at greater distances , where observations can not reveal individual objects and one must rely on integrated properties . therefore , understanding whether these apparently anomalous extinction properties are just peculiar to the central ngc2070 cluster or are a common feature throughout the tarantula complex is fundamental for our understanding and interpretation of the integrated star formation diagnostics and of the chemical evolution in more distant galaxies . in this paper , we extend the work of de marchi & panagia ( 2014 ) to the entire tarantula nebula . the traditional approach to determine the extinction properties is the `` pair method '' , whereby the spectrum of a reddened star is compared with that of a reference , un - extinguished object of the same spectral type ( e.g. , johnson 1968 ; massa , savage & fitzpatrick 1983 ; cardelli , sembach & mathis 1992 ) . this requires high quality spectra , extending from the nuv to the nir , that are difficult to obtain in the crowded 30dor regions and are necessarily limited to the brightest and hence most massive stars ( e.g. , fitzpatrick & savage 1984 ; gordon et al . 2003 ; maz apellniz et al . 2014 ) . following this method results in a sparse coverage of the tarantula nebula , preferentially limited to the areas of more recent star formation . conversely , by making use of multi - band photometry of red giant stars in the red clump ( rc ) phase ( e.g. , paczynski & stanek 1998 ; cole 1998 ; girardi et al . 1998 ; gao et al . 2009 ; wang et al . 2013 ) , the method developed by de marchi & panagia ( 2014 ; see also de marchi , panagia & girardi 2014 ) allows us to obtain a rich and uniform coverage spread over several thousand lines of sight in the tarantula region , resulting in a self - consistent absolute extinction curve of high statistical significance over the entire field and wavelength range of the observations . the structure of the paper is as follows . in section 2 we describe the http observations relevant for this study . section 3 is devoted to the identification of rc stars through an innovative use of unsharp - masking techniques . in section 4 we derive the absolute extinction towards rc stars and the corresponding extinction law . in section 5 we present the reddening distribution in this field and discuss how this information should be used to correct the photometry of individual objects . a summary and our conclusions follow in section 6 . the observations are part of the http survey , described in detail in sabbi et al . ( 2013 , 2015 ) . they cover a region of @xmath13 including the 30dor nebula , corresponding to @xmath4pc@xmath5 at the distance of the large magellanic cloud ( lmc ) . throughout this paper we will adopt a distance modulus @xmath14 as obtained by panagia et al . ( 1991 ; see also panagia 2005 ) for sn1987a , located in the vicinity of the tarantula nebula . the observations were obtained with the _ advanced camera for surveys _ ( acs ) and _ wide field camera 3 _ ( wfc3 ) instruments on board the _ hst _ in a set of broad and narrow bands over the range @xmath15@xmath16 m ( respectively f275w , f336w , f555w , f658n , f775w , f110w , and f160w ) . the photometric reduction and the corresponding catalogue are presented in sabbi et al . ( 2015 ) . that paper also illustrates how the two cameras were used to cover the entire field and it provides a detailed list of the exposure times reached in each field . the latter typically amount to 1164s in the f275w band , 1402s in f336w , 2270s in f555w , 2220s in f658n , 2329s or 2639s in f775w , 1298s in f110w , and 1598s in f160w observations in the f775w band were taken with both the acs and wfc3 , covering adjacent regions ( see sabbi et al . 2015 for details ) . both cameras feature a filter with that name , but although rather similar their overall response in those bands is not quite the same . in figure[fig1 ] , we show the differences between the f775w magnitudes of objects in a strip of @xmath17 that was observed in this band with both cameras . the selected magnitude range , @xmath18 , is relevant for the rc stars discussed in this work . the thin solid line shows the mean magnitude difference between the two bands , corresponding to @xmath19mag . the difference is smaller than the typical photometric uncertainty for these objects , namely @xmath20mag ( the dashed lines mark the corresponding @xmath21 band ) . the root mean square deviation with respect to the mean is @xmath22mag , whereas the same root mean square deviation with respect to zero is just a millimagnitude larger , or @xmath23mag , making a correction not necessary . close inspection of the trend seen in the figure might suggest that there is a small colour term , since for this population the magnitude correlates directly with the colour . however , most of the stars in the range @xmath24 have uncertainties estimated by sabbi et al . ( 2015 ) to be @xmath25mag due to saturation ( open circles in figure[fig1 ] ) . for this reason , the apparent deviation is not significant . indeed , as we will show in section4 , there is no detectable systematic difference between the slopes of the reddening vectors in the northern and southern portions of the field , covered respectively by the wfc3 and acs . therefore , in the context of this work it is not necessary to apply a colour - term correction to the photometry and in the following we will not distinguish between the two f775w bands . note that in figure[fig1 ] there are a few stars with differences exceeding @xmath26 ( dashed lines ) . even though their number is not statistically significant , these objects could be variable stars in the field , and the magnitude difference may originate because the acs and wfc3 observations were not taken simultaneously , or they might be blends or stars with nearby neighbours in projection . selecting from the sabbi et al . ( 2015 ) catalogue all the stars with combined photometric uncertainty @xmath27mag in the f555w and f775w bands ( @xmath28 objects ) , we obtain the colour magnitude diagram ( cmd ) shown in figure[fig2 ] . following romaniello ( 1998 ) , the combined uncertainty @xmath29 is defined as : @xmath30 where @xmath31 and @xmath32 are the uncertainties in each individual band . can be generalised for any combination of bands . ] besides a rather broad upper main sequence ( ums ) , several times wider than the photometric uncertainty @xmath33mag , the most prominent feature of figure[fig2 ] is an elongated stellar sequence almost parallel to the main sequence ( ms ) itself but well separated from it . to help characterise its nature , we show as a red circle the location of the `` nominal rc '' , defined as the theoretical rc of stars of the lowest metallicity applicable to this field and for ages in the range @xmath34gyr . de marchi , panagia & girardi ( 2014 ) have shown that a metallicity @xmath35 is appropriate for the old stars ( @xmath36gyr ) in 30dor . .apparent magnitudes @xmath37 of the rc and corresponding @xmath38 spread in all bands , already including the effects of the distance and extinction by intervening mw dust in the foreground . [ cols="<,<,^,^",options="header " , ] [ tab3 ] in figure[fig6 ] we also show as a long - dashed line the extinction curve measured by de marchi & panagia ( 2014 ) with the same method in a smaller region of the tarantula nebula , namely the central @xmath39 radius around r136 . the agreement between the extinction curves over the whole tarantula field and the r136 area is very good , over the common wavelength range . this is remarkable because de marchi & panagia ( 2014 ) covered a much smaller region ( sampling only about 140 rc stars instead of the roughly 3500 objects in this work ) , and the two studies did not use the exact same set of filters : instead of f775w , de marchi & panagia ( 2014 ) used f814w . the excellent agreement indicates that the method is solid . the thick short - dashed line in figure[fig6 ] displays the canonical extinction law for the galactic diffuse ism , from the work of fitzpatrick & massa ( 1990 ; see also fitzpatrick 1999 ) for @xmath40 . when expressed in units of @xmath41 , as per eq.[eq3 ] , this corresponds to @xmath42 , which is a value significantly smaller than the @xmath43 that we measure in the tarantula ( see table[tab3 ] ) . gordon et al . ( 2003 ) have studied the extinction properties towards eight different lines of sight associated with the lmc2 superbubble near the tarantula nebula . unfortunately , none of them are included within the field of view of our observations , so it is not possible to make a direct comparison . the closest objects are sk@xmath44 , located some @xmath45 ne of r136 , and sk@xmath46 , about @xmath47 sw of it . these and the other more distant lines of sight probed by gordon et al . ( 2003 ) sample regions of considerably more diffuse ism than those characteristic of the tarantula nebula itself . it is , therefore , not surprising that the @xmath48 values measured spectroscopically by gordon et al . ( 2003 ) for these stars , e.g. @xmath49 for sk@xmath44 or @xmath50 for sk@xmath51 , do not match the value @xmath52 measured by de marchi & panagia ( 2014 ) for the r136 region and confirmed by these observations for the tarantula nebula at large . in fact , at optical wavelengths the extinction curve obtained by gordon et al . ( 2003 ) from the lmc2 supershell sample is very similar to the galactic extinction law of fitzpatrick & massa ( 1990 ; see de marchi & panagia 2014 for a direct comparison ) , which we already concluded does not agree with the extinction curve in the tarantula ( see figure[fig6 ] ) . combining hst optical photometry ( de marchi et al . 2011a ) with spectroscopy and near - infrared ( nir ) photometry from the vlt - flames tarantula survey ( evans et al . 2011 ) , maz apellniz et al . ( 2014 ) derived the extinction law for the r136 cluster . their study of a sample of 83 stars of spectral types o and b with the bayesian code chorizos ( maz apellniz 2004 ) concluded that , inside the cluster , the value of @xmath53 is larger than in the galactic ism . they find @xmath54 when all 83 objects are considered . limiting the sample to the 50 objects with the smallest uncertainties , the same average value is found but the spread is reduced , namely , @xmath55 . these values are in excellent agreement with those shown in table[tab3 ] for the entire tarantula nebula . an immediate implication of our extinction law for the young stars in the tarantula nebula is that their intrinsic brightness has been so far systematically underestimated . with a median colour excess for ums stars of @xmath56 , the difference between the tarantula and galactic extinction laws implies that one would obtain systematically fainter intrinsic fluxes by a factor of @xmath57 , on average , and by more than a factor of 2 for the most extinguished 10% of the stars . as an example , crowther et al . ( 2010 ) derived new luminosities and masses for the most massive members of r136 , correcting their photometry with the extinction law of fitzpatrick & savage ( 1984 ) , namely @xmath58 . even approximating this value to 4 , as crowther et al . ( 2010 ) have done , the difference with our slope is large ( @xmath59 , see table[tab3 ] ) , and implies that the luminosities and masses of the stars are in fact considerably higher than crowther et al.s ( 2010 ) estimates . for instance , the luminosity of r136c grows from @xmath60 to @xmath61 , which according to their models brings the mass of the star from 220to more than 300 . as de marchi & panagia ( 2014 ) concluded , the extinction law in these regions is `` flatter '' than in the galactic ism , i.e. less steep in logarithmic terms . as one can see from figure[fig6 ] , at optical wavelengths in linear terms the extinction law in these regions is almost exactly parallel to the galactic curve . the thin short - dashed line is the portion of the standard galactic law shortwards of @xmath62 m with a vertical offset of @xmath63 and it matches the measured extinction curve surprisingly well ( the match is in fact so good that the thin short - dashed line is often hard to discern ) . the difference between the galactic extinction law and the one in the tarantula is shown by the green solid line in figure[fig6 ] . at wavelengths shorter than @xmath64 m the difference is practically constant ( see also table[tab3 ] ) . a noticeable feature is the small dip at @xmath65m@xmath66 or @xmath67 m , which is a likely consequence of the lack of @xmath68-band observations in our photometry . while the galactic extinction law features a small knee at this wavelength ( see thick short - dashed line ) , our interpolation is rather smooth in this range since we have no data points between the f336w and f555w filters . also , we do not regard as significant the apparent decline of the curve in the nuv because of the larger photometric uncertainties and filter red leak at these wavelengths , as mentioned above . the practically constant difference between the galactic and tarantula extinction curves in the optical indicates that the dust is in fact of the same or similar type but that in the tarantula nebula there is an additional component . since in the optical the contribution of this component is grey , i.e. it does not appear to depend on the wavelength , its most likely origin is the presence of a larger fraction of large grains than in the diffuse ism in the galaxy and lmc . this is the accepted interpretation for the high ratios of total - to - selective extinction observed in some galactic environments ( see e.g. strom , strom & yost 1971 ; jones 1972 ; dunkin & crawford 1998 ; skorzynski , strobel & galazutdinov 2003 ) . the nir domain provides further indication that , except for the extra grey component , the extinction law in the tarantula nebula is similar to that of the diffuse ism in the galaxy or lmc . at wavelengths longer than @xmath64 m , the tarantula extinction law tapers off as @xmath69 , following almost exactly the observed properties of the galactic extinction law ( e.g. cardelli , clayton & mathis 1989 ; wang et al . the dot - dashed line shown in figure[fig6 ] is the portion of the galactic extinction law longwards of @xmath62 m multiplied by a factor of 2 , and it offers a remarkably good fit to our observations in the @xmath70 and @xmath71 bands . therefore , there is no reason to believe that the nature of the tarantula grains should be drastically different from that of the diffuse galactic ism . a detailed analysis of the grain properties as a function of the location inside the nebula will be presented in a forthcoming work ( de marchi , panagia , et al . 2015 , in prep . ) . however , as de marchi & panagia ( 2014 ) have already pointed out , simple considerations can provide valuable insights into the properties of the additional dust component present in the 30dor regions . it is well known ( _ e.g. , _ van de hulst 1957 ; greenberg 1968 ; draine & lee 1984 ) that , at wavelengths short enough , the extinction (= absorption + scattering ) cross section of a grain of radius @xmath72 tends asymptotically to the geometric cross section @xmath73 . at longer wavelengths the cross section is smaller than @xmath74 and becomes proportional to the grain volume . conveniently enough , the transition occurs approximately at @xmath75 and , for a given grain size , one would expect a sort of a step function behaviour with the transition occurring rapidly around @xmath76 . to account for the observed mw extinction law s steady increase with wave number over a wide wavelength range ( @xmath77m@xmath78 m ) , it is generally assumed that there is a distribution of grain sizes of the type @xmath79 , with @xmath80 and the grain radius @xmath72 ranging from @xmath81 m to @xmath82 m ( mathis , rumpl & nordsieck 1977 ; draine & lee 1984 ) . with @xmath80 , at wavelengths longer than @xmath83 the extinction is dominated by the largest grains and is proportional to the total mass in grains . taking the galactic extinction law as a reference template , the fact that in the nir the absolute value of the extinction in the tarantula is about twice as large as it is in the mw ( see figure[fig6 ] ) implies that the mass fraction in large grains is about twice as high as in the mw . therefore , the extinction law inside the tarantula nebula can be represented with the sum of two components : one being the standard galactic extinction law and the other being made up only of large grains , which are similar in type to those found in the diffuse galactic ism . de marchi & panagia ( 2014 ) concluded that , for the central regions of 30dor , the most likely origin for the higher relative abundance of large grains is the selective injection of `` fresh '' large grains into the mw mix . the same conclusions can now be extended to the tarantula nebula at large . the two other ways to explain an excess of large grains would be selective destruction of small grains , or selective condensation of material on the surface of small grains , but both would imply a decrease in the number of small grains and hence an extinction law that is flatter than the mw s at uv wavelengths . in fact , measurements towards the stars of the magellanic clouds reveal a steeper rise in the uv extinction curve compared to mw objects ( _ e.g. _ fitzpatrick 1998 and references therein ) . note that , as mentioned above , the apparent decline of the curve at nuv wavelength in figure[fig6 ] is not significant , due to the large photometric uncertainties and filter red leak at those wavelengths . the selective addition of new large grains to the mix can easily account for the presence of the extra grey component in the extinction curve , without conflicting against measurements at other wavelengths . actually , gall et al . ( 2014 ) recently revealed rapid formation of large , @xmath16m - size dust grains in the dense circumstellar medium around sn2010jl in the metal - poor galaxy ugc5189 ( newton & puckett 2010 ) . their observations with the _ very large telescope _ reveal that the extinction curve around the supernova evolves rapidly and turns into a mix of grey - extinction dust grains and mw dust grains . the extinction contribution of the grey dust is about 40% in the @xmath84 band . also in the lmc , recent _ herschel _ and _ alma _ observations of sn1987a ( matsuura et al . 2011 ; indebetouw et al . 2014 ) indicate that a substantial amount ( @xmath85 ) of large grains ( @xmath86 m ) is being produced in the ejecta . a similar amount of dust is expected in sn2010ji if the dust production continues to follow the trend observed so far ( gall et al . these recent findings make injection of large grains by supernova explosions an exciting possibility for the extinction law in the tarantula as well . indeed , very large , grey dust grains recently received much attention in the literature as there is ample evidence for such @xmath16m - sized grains in the galactic ism ( _ e.g. _ , wang , li & jiang 2015a ) . star formation has been active for at least 30myr in the tarantula nebula , and possibly longer , as witnessed by the presence of both young and older generations of stars in ngc2070 ( walborn & blades 1997 ; de marchi et al . 2011a ; cignoni et al . 2015 ) , in hodge301 ( grebel & chu 2000 ) , and in ngc 2060 ( mignani et al . 2005 ) . if all typeii supernova explosions result in an output comparable to that of sn1987a and sn2010jl , the excess of large grains should have built up considerably over time and will reach a peak after about @xmath87myr , which is the lifetime of the 8 stars at the lower mass limit of supernova type ii progenitors . even though the large grains are eventually destroyed in the hot gas behind shock fronts in supernova remnants ( draine 2009 ; dwek & scalo 1980 ; dwek 1998 ) , with a total mass in excess of @xmath88 in this starburst region ( bosch et al . 2001 ; andersen et al . 2009 ) the expected supernova rate is above @xmath89yr@xmath66 ( cervio et al . 2001 ) , implying a sustained injection of large grains into the ism . as an order of magnitude , one would expect in a typical 10myr time frame about 1000 type ii sne , corresponding to up to @xmath90of large grains . with the quoted total mass ( @xmath91 ) and metallicity ( @xmath92 ; e.g. hill , andrievsky & spite 1995 ; geha et al . 1998 ) for these regions , the resulting mass in large grains compares favourably with the expected @xmath93 fraction of metals locked in grains ( _ e.g. _ savage & sembach 1996 ) . to confirm whether this interpretation of the observed extinction law is indeed correct and to understand whether differences with the diffuse galactic ism are mainly in the fraction of large grains , further studies are required . spectroscopic uv observations of early - type stars in suitable locations inside the tarantula nebula in the range @xmath94are needed to probe the distribution of small grains and to measure in which proportion they are present . these observations are possible with the _ cosmic origin spectrograph _ ( cos ) on board the hst . so far we have seen that , across the entire tarantula nebula , the extinction law implies a steeper reddening vector than in the diffuse galactic ism : with @xmath95 , the reddening vector in the cmds is consistently @xmath93 steeper . since this is not the case in more diffuse ism regions in the lmc , the extinction must be related to the intense star formation witnessed by the tarantula complex . at the same time , the properties of the extinction law , and hence those of the grains , show relatively small variations across the @xmath96 region that we studied . indeed , a value of @xmath97 is also found in regions devoid of massive stars or in hot x - ray super - bubbles ( wang & helfand 1991 ) , where @xmath8 is generally lower ( see sabbi et al . therefore , this effect is not limited just to the regions of most recent star formation , but also to those where star formation peaked some 2030myr ago ( e.g. , hodge301 and ngc2060 ) . if the overabundance of large grains is due to injection by type ii supernovae , as in the case of sn1987a ( matsuura et al . 2011 ; indebetouw et al . 2014 ) and sn2010jl ( gall et al . 2014 ) , the ism enrichment in large grains will be progressive . it will begin with the explosion of the most massive progenitors of the first generation of stars , a few myr into the star - formation episode , and will continue to increase for about @xmath87myr , i.e. the lifetime of the 8myr stars at the lower mass limit of type ii progenitors . at this stage , the excess of large grains should be highest and it will begin to decrease progressively as the grains are destroyed in the environment . they must be relatively easy to destroy , since the regions around the tarantula show a rather standard extinction curve ( gordon et al . this would be easy to understand if the new large grains were mostly made of ices , which sublimate at low temperatures without affecting appreciably the underlying grain distribution . a grey extinction component caused by @xmath16m - size ice grains is also compatible with the mid - ir extinction properties of the galactic ism ( wang , li & jiang 2015b ) . constraints on the timescale of these phenomena can be set by comparing the extinction properties and ages of the populations inside the tarantula nebula with those in the surrounding regions . obviously , our findings can have important implications for the study of the star formation properties in galaxies . beyond the nearest universe , star formation properties such as the star formation rate or stellar masses are derived from diagnostics of hii regions ( e.g. kennicutt 1998 ) . their integrated colours and spectra are dominated by the energy of massive stars and are significantly affected by extinction , i.e. by both the amount and the properties of dust grains . as we have concluded , in regions undergoing massive star formation the properties of the dust grains appear to change from those characteristic of the diffuse galactic ism . even though the changes might be short lived and might last only some @xmath87myr , they affect the hii regions when these are most easily detectable in distant galaxies . therefore , assuming typical ism conditions in these regions could result in severely inaccurate total masses and star formation rates . if the @xmath98 value measured in the tarantula nebula is anywhere typical of massive star forming regions and the reddening is high , assuming the classical @xmath99 value could result in fluxes that are about a factor of 2 too faint . the outcome would be a seriously underestimated star formation rate leading to a distorted view of the formation and of the chemical evolution of galaxies ( e.g. matteucci 2012 ) . in this section we study the reddening distribution across the field in order to derive a reddening map , and discuss how to use it to correct the photometry of individual stars . having determined the slope of the reddening vector in all observed bands ( table[tab2 ] ) , we can measure the total extinction towards objects whose nominal location in the cmd can be determined unambiguously , using them as probes of the extinction along their respective lines of sight . these include not only rc stars , but also the objects in the ums , since that portion of the cmd is not shared with stars in other evolutionary phases . it is crucial to understand that the extinction map resulting from this collection of lines of sight is by definition a two - dimensional projected distribution . on the other hand , in order to correct the photometry of other objects in the field one would need a three - dimensional distribution , both of the stars and of the absorbing material , to account for their location along the line of sight . one could be tempted to interpolate between lines of sight , but this would not necessarily result in a more accurate photometry . in fact , it is likely to introduce larger uncertainties . as we will show , one has instead to use additional information ( e.g. spatial distribution ) to determine a meaningful correction for stars other than the reddening probes . the objects serving as reddening probes are those between the dashed lines shown in the cmd of figure[fig7 ] . for the rc , we have used as limits the envelopes around the extended rc after unsharp masking ( see figure[fig4 ] ) , considering only objects redder than @xmath100 since that is the minimum colour consistent with the uncertainty around the nominal rc colour in these bands , taking into account the galactic extinction component towards the lmc ( see table[tab1 ] ) we have further restricted the selection to stars more massive than 10 with combined photometric uncertainty @xmath101 , and have excluded objects with @xmath102 since they could be pms stars ( see section3 ) . in total , about 3500 objects were selected in this way , corresponding to a projected average rc stars density of @xmath6arcmin@xmath103 . to guarantee a similar density of ums stars over the same field , we selected a somewhat larger number of objects with colours @xmath104 , namely @xmath105 , since many are clustered around r136 . also in this case , only objects with @xmath101 were considered . all stars serving as reddening probes are shown in colour in figure[fig7 ] . for rc stars , the colour excess is calculated as colour difference from @xmath106 , i.e. the nominal rc colour in these bands ( see table[tab1 ] ) . for ums stars , however , we use as a reference the isochrone shown in figure[fig7 ] , namely the zero age main sequence ( zams ) from the models of marigo et al . ( 2008 ) , extending up to 60 , and obtained specifically for the hst filters used here and a metallicity of @xmath107 as appropriate for 30dor and the young lmc population in general ( e.g. hill , andrievsky & spite 1995 ; geha et al . 1998 ) . we have assumed a distance modulus @xmath108 ( panagia et al . 1991 ; panagia 2005 ; walborn & blades 1997 ) and have already included the intervening galactic extinction along the line of sight , i.e. @xmath109 or @xmath110 as indicated above . each ums star is then translated back to the zams , along the direction of the reddening vector , and the colour excess is computed . for stars that , once translated to the isochrone , would be brighter than @xmath111 and as such more massive than @xmath112 , we have assumed an intrinsic colour @xmath113 . note that most of the ums stars are likely members of binary systems , so their intrinsic @xmath114 colour can be redder than that of the zams and our procedure could overestimate their reddening . on the other hand , the typical mass ratio for stars above @xmath115like those in our sample is @xmath116 ( kiminki & kobulnicky 2012 ; sana et al . model calculations of ms and pms stars show that for @xmath117 and an age of 1myr or older the @xmath114 colour difference for these massive stars would amount to at most @xmath118mag and as such it can be ignored . also rc stars could belong to binary systems , but since the nominal rc location is determined from the cmds after unsharp masking ( see figure[fig5 ] ) , the presence of binaries does not affect the colour excess that we derive . in fact , the excellent match between the observed rc location in the cmd and the theoretical models of girardi & salaris ( 2001 ) indicates that possible lower - mass companions do not appreciably alter the colour of the systems . this is not surprising given the relatively rapid evolution of rc stars along the red giant branch . the resulting reddening distribution towards the selected stars is shown schematically in figure[fig8 ] , panels _ b ) _ and _ c ) _ , where the median and the 17 and 83 percentile values of @xmath119 are displayed across the field of view in cells of @xmath120 or @xmath9pc on a side . for reference , in panel _ a ) _ we mark the positions of the three clusters r136 , hodge301 , and ngc2060 , as well as the location of the 30dor west field studied by de marchi et al . each cell in the panels contains on average @xmath121 stars of either type ( rc or ums ) , so the percentile values reported in the figure are fully statistically significant , with the exception of some cells along the borders that are only partly covered by the observations . the extent of the shading inside each cell marks the portion actually covered by the observations , and the level of grey in panels _ b ) _ and _ c ) _ reflects the median reddening . the same scale of grey is used in both panels . the purpose of figure[fig8 ] is to provide a map of the typical @xmath119 values and of their variations across the field , but because of the large dispersions it should not be used as an `` extinction map '' to correct for reddening the photometry of individual objects . this point is particularly important to understand , because the apparent similarity of the extinction towards rc and ums stars might be deceiving . indeed , the median reddening value in fully populated cells and most of those along the borders indicate that there is comparable extinction towards rc and ums stars inside the same cell . it would , therefore , be tempting to conclude that on the @xmath9pc scale of a cell the extinction is known and that one might use figure[fig8 ] as a `` three - dimensional reddening map '' and take the median reddening values to correct the photometry of all stars inside the corresponding cell . as mentioned above , this could introduce large uncertainties , since there are systematic differences in the distribution of reddening values between cells , even if the median values are similar as a first example , we show in figure[fig9 ] the result of applying to all rc stars inside a cell the median @xmath119 value for that cell . note that the correction for reddening is applied on purpose only to rc stars , not also to ms objects , so one can directly compare figure[fig9 ] with figure[fig4 ] . it is immediately clear that the spread of @xmath119 values inside each cell is so broad that using a single value of @xmath119 for all the rc stars in that cell does not bring them all back to their nominal undispersed location and the spread remains . in other words , without further corrections one would not be able to derive sensible intrinsic physical parameters of rc stars from the cmd of figure[fig9 ] , as was the case for figure[fig5 ] . furthermore , figure[fig8 ] reveals that inside the same cell , rc and ums stars have very different reddening distributions and it is not possible to use one type of stars to correct for extinction towards the other type . this is immediately clear from figure[fig10 ] , offering a direct comparison between the percentile reddening levels measured towards ums and rc stars . different colours refer to different levels : blue for 17% , yellow for 50% , and red for 83% . the dots in figure[fig10 ] correspond to the cells as marked in figure[fig8 ] . within @xmath122mag ( @xmath38 ; solid lines ) , there is a fair correlation between the reddening distribution derived from rc and ms stars inside the same cell . this means that , in general , inside a 40pc - wide cell , the amount of extinction towards rc and ums stars has a similar distribution , within the quoted @xmath122mag . it is also clear , however , that the reddening towards rc stars begins at systematically lower values than that towards ums stars , by about @xmath123mag . this is consistent with the projected spatial distribution of rc and ums stars observed across the region : while rc stars are uniformly distributed , ums stars are clearly clumped ( see sabbi et al . this suggests that ums stars are distributed over a smaller extent also along the line of sight and probe a more limited volume of the ism . this is not surprising , since the young ums stars tend to be clustered together in more compact groups , while the much older rc stars are distributed more uniformly along the line of sight . for example , in their study of the tomography of the lmc , haschke , grebel & duffau ( 2012 ) concluded that young populations ( in that work traced by cepheids ) have a considerably smaller extent along the line of sight than older populations . figure[fig10 ] also reveals that , even inside the same cell , in some cases ums stars probe systematically more extinguished regions of the ism , as indicated by the dots in the upper left portion of the figure . it appears that , while the 83 percentile value of the reddening towards rc stars in those cells is in the range @xmath124 , for ums objects it is @xmath125 . a similar conclusion had already been reached by zaritsky ( 1999 ) , whose analysis of the reddening distribution towards hot ( @xmath126k ) and cold ( 5500k @xmath127 6500k ) stars contained in the magellanic clouds photometric survey ( harris , zaritsky & thompson 1997 ) revealed that ums stars are on average more extinguished than red giants . although this is not always the case , even around the tarantula nebula ( see de marchi et al . 2014 ) , in comparison to rc objects , ums stars are by their very nature systematically more closely associated with the higher density ism regions in which they formed . therefore , a higher reddening level is to be expected towards ums objects , particularly when star formation has only recently started . in their study of the main body of the lmc , zaritsky et al . ( 2004 ) concluded indeed that , on average , hot stars appear to be preferentially located in dusty regions . in summary , our observations leave no doubt that there is a considerable amount of dust distributed between stars along the line of sight in this young complex . it has long been known from radio and ir observations of galactic star forming regions that distributions of this type are common in the milky way ( e.g. panagia 1974 ; natta & panagia 1976 ; and references therein ) . it is important to understand whether this is unique to the local universe or it is typical of massive star forming regions in general , because it appears to be is in contrast with what has been concluded by calzetti , kinney & storchi bergmann ( 1996 ) for extra - galactic star forming regions . the tarantula complex offers the best environment to study in detail different phases and conditions of star formation , with no ambiguity about the distance or contamination by background sources . although the general trend in figures[fig8 ] and [ fig10 ] shows that a fair level of correlation exists between the reddening probed by rc and ums stars , it also reveals that the value of @xmath119 is subject to wide variations over the field . the typical difference between the 17 and 83 percentile values ( blue and red dots in figure[fig10 ] , respectively ) for ums stars is @xmath128mag , indicating typical fluctuations of the order of @xmath129mag in @xmath119 ( @xmath38 ) towards the young stars inside any given cell . therefore , using a single value of @xmath119 to correct the photometry of all stars in any given cell will introduce large uncertainties on the stellar parameters ( @xmath130 ) , with important implications for the masses and ages derived through comparison with theoretical isochrones . in particular , for a typical pms star of 1 , an uncertainty of @xmath129mag in @xmath8 leads to an uncertainty of a factor of about 2 on the age and of about @xmath131 on the mass , which grows to 2 or more for objects younger than @xmath132myr . such uncertainties dominate over those caused for instance by accretion - induced variability or unresolved binaries ( e.g. gouliermis 2012 ) . therefore , since one of the goals of the http ( sabbi et al . 2013 ) is to determine the physical properties of the young populations in these regions ( see e.g. cignoni et al . 2015 ) , including the pms stars ( de marchi , panagia et al . 2015 , in prep . ) , it is essential to apply the most appropriate correction for reddening separately on each individual object . thus , the simple projected two - dimensional distribution provided by the reddening map ( in fact , a higher - resolution version of figure[fig8 ] ) must be complemented by additional information on the properties and spatial distribution of the objects , in order to constrain the position of the stars with respect to the extinguishing material along the line of sight . in the following , we discuss the conditions under which it is possible to apply an extinction correction to individual young objects in the cmd and how to do it . we can distinguish several cases , as follows . 1 . massive stars of spectral types o and b ( blue dots in figure[fig7 ] ) spend most of their life on the ums ( e.g. , marigo et al . 2008 ) , so the reddening correction can be determined for each object individually as per the procedure described above , since the extinction law is very robust . 2 . for young pms stars that are identified through their @xmath133 excess emission ( de marchi et al . 2010 , 2011b ) and whose spatial distribution projected on the sky overlaps with that of ums stars , one can reasonably assume that they are physically co - located with the ums objects also along the line of sight . in this case , the extinction appropriate for each pms star can be derived as the average of the extinction towards a subset of ums stars in their vicinity . for example , this `` nearest neighbours '' approach has been used to correct for extinction the pms in the core of 30dor ( de marchi et al . 2011a ) , and in the ngc346 star forming cluster in the small magellanic cloud ( de marchi et al . 2011b ) . in both cases , using between 5 and 20 nearest neighbours on the entire photometric catalogue resulted in a tight ums in the cmd . it is reasonable to adopt this approach also for stars that share the same photometric properties as young pms objects without necessarily displaying @xmath133 excess emission , namely objects that occupy a similar position in the broad - band cmd and whose projected spatial distribution follows that of ums stars . these objects could indeed be pms stars not actively accreting at the time of observation , as it is known that pms objects show large variations in their @xmath133 emission over hours or days ( e.g. fernandez et al . 1995 ; smith et al . 1999 ; alencar et al . 3 . for older ( @xmath134myr ) pms objects with @xmath133 excess emission but with a spatial distribution different from that of ums stars , it is not safe to assume that the ism probed by the two types of objects be similar . the ums stars probe the most recent episode of star formation , associated with a denser ism , where one would expect higher extinction values than towards older pms objects . while one could still use the nearest ums stars to estimate the amount of reddening , it would likely be too high and after correction some stars might appear bluer than the ms . additional constraints need to be applied in this case . a possible approach , followed for instance by de marchi et al . ( 2011a ) for the population of @xmath135myr old pms stars in 30dor , is to de - redden all these objects by the same amount , chosen in such a way to guarantee a statistically acceptable distribution of colours , i.e. that no more than @xmath136 of the objects are bluer than the ms after reddening correction . alternatively , one could estimate the extinction towards older pms stars individually using as nearest neighbours the rc objects in their vicinity . as shown above , rc stars are more uniformly distributed and sample a less dense ism , resulting in typically lower median extinction values towards those lines of sight ( see figures[fig8 ] and [ fig10 ] ) . needless to say , both approaches imply a larger uncertainty on the physical parameters that one can derive for these pms objects , and they must be properly taken into account in any further analysis . 4 . in all other cases , and for a more solid result in case ( iii ) , one needs spectral information . using large ground - based facilities it is already possible to obtain the spectral type of a sample of bright pms stars in less crowded regions in the magellanic clouds ( e.g. kalari et al . 2014 ) , providing reliable intrinsic colours and effective temperatures for at least the most massive of these objects . in the magellanic clouds , it will be possible to extend this investigation to older pms stars of solar mass with the _ james webb space telescope _ ( gardner et al . 2006 ) . following this approach one obtains a much more reliable extinction correction than the one achievable using pixel - to - pixel maps of the reddening of the gas in the star forming region , estimated from ratios of h recombination lines at different wavelengths ( e.g. natta & panagia 1984 ; calzetti et al . 1996 ; pasquali et al . 2011 ; pang , pasquali & grebel 2011 ; zeidler et al . 2015 ) . the reason is easily understood and can be summarised as follows . 1 . a relationship between the colour excess of the gas and the observed and theoretical flux ratio of line pairs can only be defined in the ideal ( yet unlikely ) case that the dust is all in the foreground and is homogeneously distributed ( see calzetti et al . if it is not , natta & panagia ( 1984 ) have shown that the solution is not unique and may span large uncertainties . 2 . the extinction curve specific to the environment must be known ( see calzetti et al . 1996 ) . as we show in this work , it is not safe to adopt the extinction law of the galactic diffuse ism in star forming regions . extinction is the sum of absorption and scattering , which affect light differently depending on whether the photons originate from stars or from an extended gas cloud . while for a star both absorption and scattering result in the loss of a photon from the beam , in a nebula scattering simply diffuses the photon within the nebula and only absorption will kill it . therefore , to interpret observed h line intensity ratios properly , a full knowledge of the dust properties including both absorption and scattering cross sections is required . the map of gas line ratios is valid for a specific optical depth , that of the gas . unless stars share a similar location along the line of sight , using these maps for the stars is a delusive strategy and the resulting reddening correction will be systematically wrong . since the above conditions are normally not satisfied in massive clusters or regions of extended star formation ( and surely not in galaxies ; see , e.g. , penner et al . 2015 ) , it is somewhat nave and exceedingly simplistic to rely on maps of the gas line ratios for a quantitative measure of the reddening towards individual stars . the very fact that the extinction towards the gas is often found to be larger than that of the ums stars indicates that the thickness of the dust layers in front of stars are not the same as the ones attenuating the gas . elementary logic dictates that loose matter located behind a source can not produce reliable , if any , information about what lies in front of that source . instead , the approach based on the _ nearest neighbours with similar age and spatial distribution _ that we describe here provides what is presently the most robust quantitative estimate of the reddening towards individual stars in the tarantula . eventually , an even more accurate reddening value for individual stars in the http catalogue is expected to be obtained through a bayesian study of the spectral energy distribution of individual objects ( arab , gordon , et al . 2015 , in preparation ) , including stars outside the ums and rc loci . we have studied the properties of the interstellar extinction over a field of @xmath137 ( @xmath138pc@xmath5 ) in the tarantula nebula , imaged with the _ hst _ as part of the http ( sabbi et al . the photometric catalogue contains more than 820000 stars observed at nuv , optical and nir wavelengths through the filters f275w , f336w , f555w , f658n , f775w , f110w , and f160w ( sabbi et al . 2015 ) . since in these regions the levels of extinction are considerable and very uneven , rc stars are found to be spread across the cmd defining a tight band . this has allowed us to accurately derive the absolute extinction @xmath139 and the extinction law @xmath140 in the range @xmath141 m , from more than 3500 rc stars . the main results of this work can be summarised as follows . 1 . the cmds obtained from the observations reveal a prominent elongated sequence , almost parallel to the ms , made up of several thousand rc stars affected by various amounts of extinction ( figure[fig2 ] ) . application of the unsharp - masking kernel to the cmds reduces the contrast of the low - frequency component , resulting in a vastly improved definition of the sharp , elongated rc feature ( figures[fig4 ] and [ fig5 ] ) . 2 . from the best linear fit to the elongated rc , we obtain a fully empirical determination of the slopes of the reddening vector in all combinations of bands . the reddening vector appears to have a similar slope over the entire field of view , within the uncertainty , although the se quadrant of the nebula reveals a systematically lower value of @xmath140 by about 12% . the excellent match between the head of the elongated rc sequence and the position of the un - extinguished rc predicted by theoretical models ( girardi & salaris 2001 ) is an independent validation of the models . the reddening slopes immediately provide the ratio @xmath140 of total - to - selective extinction in the specific _ hst _ bands , with high accuracy . knowledge of the un - extinguished position of the rc in the cmds readily gives the absolute extinction @xmath139 in all bands towards more than 3500 stars . interpolation at the wavelengths of the standard @xmath68 , @xmath84 , and @xmath142 bands provides the extinction curves in the canonical forms @xmath143 or @xmath144 in the range @xmath145 m . the latter form is more accurate because our photometry includes observations in bands very close to the johnson cousin @xmath84 and @xmath142 filters . the slope of the reddening vector in the tarantula nebula is considerably steeper , in all bands , than in the galactic diffuse ism , i.e. the value of @xmath146 is systematically higher in 30dor ( figure[fig6 ] ) than in the mw . we measure @xmath147 and @xmath43 instead of the canonical @xmath148 and @xmath42 found in the galaxy ( e.g. cardelli et al . 1989 ; fitzpatrick & massa 1990 ) . on the other hand , our @xmath146 values are in excellent agreement with those measured in the central ngc2070 cluster by de marchi & panagia ( 2014 ) from _ hst _ photometry of rc stars , that is @xmath149 , and by maz apellniz et al . ( 2014 ) from spectro - photometry of ob - type objects in the same field , namely , @xmath55 . 5 . an immediate implication of our extinction law is that the masses derived until now from the photometry of ums objects have been systematically underestimated , by a factor of @xmath57 , on average , and by more than a factor of 2 for the most extinguished 10% of the stars . for instance , the luminosity of r136c grows from @xmath60 to @xmath61 , which according to the models of crowther et al . ( 2010 ) brings the star from 220 to more than 300 . if the extinction law that we measure in the tarantula nebula is typical of massive star forming regions in galaxies , current star formation rates of galaxies derived from diagnostics of hii regions will have to be seriously revised upwards . 6 . at optical wavelengths , the extinction law @xmath150 is best represented by the galactic curve shifted vertically by an offset of @xmath63 . for @xmath151 m , the best match is the galactic law multiplied by a factor of 2 ( both curves fall off with wavelength as @xmath152 ) . we interpret this as indication that the tarantula extinction curve is due to dust similar to that of the diffuse ism in the galaxy , but that it contains a larger fraction of large grains ( about a factor of 2 ) . we show that this scenario is consistent with type ii supernova explosions injecting `` fresh '' large grains into an otherwise mw - like mix , as recently revealed by observations of sn1987a and sn2010jl . uv observations , e.g. with cos on board the hst , are needed to verify the evolution of the population of grains also at the small end of the size distribution . since these extinction properties are consistently found across the entire tarantula nebula but not in the more diffuse regions in its surroundings , they must be related to the recent intense star formation episodes inside the nebula itself . assuming that type ii sne are the source of the extra large grains , their excess should reach a peak after @xmath153myr ( i.e. the lifetime of the least massive type ii sn progenitors ) , before the grains are destroyed in the environment . the lack of an excess of large grains in the surroundings of the tarantula suggests that these grains are relatively easy to destroy , making ices in the sne ejecta their likely source . knowing the slope of the reddening vector for all bands , we can measure the total extinction towards all objects whose nominal cmd location can be determined unambiguously . we select 3700 ums objects and 3500 rc stars to derive uniform , densely populated maps ( @xmath154 stars per arcmin@xmath5 ) of the extinction towards both young and old objects . even though there is a fair correlation between rc and ums reddening over scales of @xmath9pc , reddening towards rc stars begins at systematically @xmath123mag lower values and ums stars have on average @xmath123mag more extinction than rc stars . not surprisingly , this indicates that ums objects sample smaller volumes along the line of sight and probe a more limited region of the ism . we address the use of extinction maps for reddening correction in regions of high and variable extinction . we show that it is not sufficient to rely on the projected position of the objects on the sky and that additional information , such as age and spatial distribution , must be used to compensate for the missing knowledge of the line - of - sight distribution of the stars under study . we warn against the large uncertainties inherent in applying extinction corrections based on simple line ratios of the diffuse gas . instead , an approach based on the nearest rc or ums neighbours with similar age and spatial distribution provides a more robust quantitative estimate for individual stars . an important conclusion that we draw from this work is that , in regions of intense star formation , the ism undergoes fundamental and rapid changes as fresh large grains are selectively injected into it by type ii supernovae and are later destroyed . this results in profoundly different extinction properties in these areas for periods of 50 100myr , with @xmath48 values in the range @xmath155 that must be taken into account in the study of cosmological sources . understanding how these changes correlate with star formation and the timescale on which they proceed in galaxies is fundamental : not only for the study of young resolved stellar populations in nearby galaxies , but also to decipher the properties of star formation and chemical evolution of galaxies in the early universe . we are grateful to an anonymous referee , whose insightful comments have helped us to improve the presentation of this work . support for hst programme # 12939 was provided by nasa to the us team members through a grant from the space telescope science institute , which is operated by aura , inc . , under nasa contract nas 526555 . np acknowledges partial support by stsci ddrf grant d0001.82435 .

we report on the study of interstellar extinction across the tarantula nebula ( 30doradus ) , in the large magellanic cloud , using observations from the hubble tarantula treasury project in the @xmath0 m range . the considerable and patchy extinction inside the nebula causes about @xmath1 red clump stars to be scattered along the reddening vector in the colour magnitude diagrams , thereby allowing an accurate determination of the reddening slope in all bands . the measured slope of the reddening vector is remarkably steeper in all bands than in the the galactic diffuse interstellar medium . at optical wavelengths , the larger ratio of total - to - selective extinction , namely @xmath2 , implies the presence of a grey component in the extinction law , due to a larger fraction of large grains . the extra large grains are most likely ices from supernova ejecta and will significantly alter the extinction properties of the region until they sublimate in @xmath3myr . we discuss the implications of this extinction law for the tarantula nebula and in general for regions of massive star formation in galaxies . our results suggest that fluxes of strongly star forming regions are likely to be underestimated by a factor of about 2 in the optical . hertzsprung russell and colour magnitude diagrams dust , extinction magellanic clouds

introduction observations extinction across the tarantula nebula summary and conclusions acknowledgments

arxiv

the ls - category of a topological space is an invariant of the homotopy type of the space introduced in the early 1930 s by lusternik and schnirelmann @xcite . it is a numerical invariant that measures the complexity of the space , in particular if @xmath0 is a compact manifold the ls - category of @xmath0 provides a lower bound for the number of critical points of any smooth function on the manifold . the ls - category of a space @xmath1 is defined as the least number of open subsets , contractible in @xmath1 , required to cover @xmath1 . for a survey in ls - category see @xcite . there are many generalizations of the original concept adapted to various contexts such as the _ fibrewise category _ , introduced by i.m . james and j.r . morris @xcite , the _ equivariant category _ , by e. fadell @xcite and the _ transverse _ and _ tangential _ categories for foliated manifolds @xcite . our aim is to develop a lusternik - schnirelmann theory in the context of lie groupoids . our main application will be in the setting of orbifolds as groupoids . since an orbifold is defined as a morita equivalence class of certain type of groupoids @xcite our notion of ls - category for lie groupoids ought to be invariant under morita equivalence . in this spirit , the right notion of morphism between lie groupoids is that of a generalized map @xcite . we develop a notion of homotopy between generalized maps . we start with a notion of strong homotopy between strict morphisms which is not morita invariant , and introduce the notion of _ essential homotopy equivalence _ that simultaneously weakens a strict homotopy and generalizes an essential equivalence . we will say that two lie groupoids @xmath2 and @xmath3 have the same _ morita homotopy type _ if there exists a third groupoid @xmath4 and essential homotopy equivalences @xmath5 we prove that the bicategory @xmath6 of lie groupoids , strict morphisms and strict homotopies admits a bicalculus of fractions that formally inverts the essential homotopy equivalences @xmath7 . using the techniques developed in @xcite we obtain a new bicategory of lie groupoids @xmath8 where the essential homotopy equivalences were formally inverted . morita homotopy equivalences introduced above amount to _ isomorphisms _ in this bicategory . our notion of _ morita homotopy _ corresponds to the 2-arrows in the bicategory @xmath8 . this deformation within the groupoid is closely related to the notion of @xmath3-path developed by haefliger @xcite . we propose a modified version of a @xmath3-path ( between two objects ) that we call a _ multiple @xmath3-path _ ( between two orbits ) . a morita homotopy determines multiple @xmath3-paths between the orbits . a multiple @xmath3-path is defined as a generalized map @xmath9 where @xmath10 and @xmath11 are certain groupoids associated to the interval @xmath12 $ ] . multiple @xmath3-paths will play a key role in defining the integral curves of a gradient vector field in a lie groupoid . based on the morita homotopy notion , we define a @xmath3-categorical subgroupoid in the sense of lusternik and schnirelmann . a subgroupoid @xmath13 is @xmath3-categorical if it can be deformed by a morita homotopy into a transitive groupoid . the _ groupoid ls - category of @xmath3 _ , @xmath14 , is the minimal number of @xmath3-categorical subgroupoids required to cover @xmath3 . when @xmath3 is the unit groupoid over a manifold @xmath0 , this number specializes to the classical ls - category of the manifold @xmath0 . we prove that @xmath14 is an invariant of the morita homotopy type . therefore , @xmath14 is invariant under morita equivalence and yields a well defined invariant for orbifolds . let @xmath3 be an orbifold groupoid defining the orbifold @xmath15 . the orbifold ls - category of @xmath15 is defined as the groupoid ls - category of @xmath3 , @xmath16 . the orbifold ls - category is a homotopy invariant that detects part of the complexity of the orbifold . it detects the existence of obstructions given by the twisted sectors to contract an open set , but somehow misses the `` weight '' of such obstructions . we introduce a variant of the orbifold ls - category that we call the _ weighted orbifold ls - category _ , @xmath17 . this numerical invariant also generalizes the classical ls - category for manifolds , if the orbifold is a manifold @xmath1 then @xmath18 . we prove that the weighted orbifold category is an invariant of orbifold homotopy type which relates to the orbifold category of the inertia orbifold . finally we give a version of the lusternik - schnirelmann theorem for lie groupoids . if @xmath19 is a generalized map satisfying certain _ deformation conditions _ ( d ) then @xmath20 where @xmath21 is the full subgroupoid over the critical points at the level @xmath22 . we show that if @xmath3 is an _ orbifold groupoid _ defining a compact orbifold , then the deformation conditions ( d ) are satisfied and we have that the ls - theorem holds for orbifolds : if @xmath15 is a compact orbifold and @xmath23 is an orbifold map , then @xmath24 has at least @xmath25 critical points . when counting a finite number of critical points in an orbifold , we face the dilemma of how to count . let @xmath2 be the critical groupoid . if @xmath2 is a finite groupoid , we can chose one isolated critical point @xmath26 in each orbit and we have that @xmath2 is equivalent to a groupoid whose set of objects is @xmath27 and the set of arrows is the disjoint union of the isotropy groups @xmath28 where the order of @xmath29 is @xmath30 and the number of conjugacy classes in @xmath29 is @xmath31 ( then @xmath32 if @xmath29 is abelian ) . by the one hand we have that the cardinality of a groupoid of baez and dolan @xcite and the euler characteristic of a finite category of leinster @xcite both give as a cardinality of @xmath2 the rational number @xmath33 . on the other hand the string theoretic euler characteristic @xcite gives as a cardinality of @xmath2 the integer number @xmath34 . our weighted orbifold category is a lower bound for this latter cardinality . this paper is organized as follows . section 2 presents the notions of strong and morita equivalence for strict morphisms between lie groupoids . we describe in this section some examples that will be studied further throughout the paper . section 3 is where we develop the notion of strict homotopy associated to a subdivision of the interval @xmath35 $ ] . we recall the notion of @xmath3-path and introduce the multiple @xmath3-paths . we introduce the notion of essential homotopy equivalence and homotopy pullback of groupoids . section 4 is where we show that the bicategory of lie groupoids , strict morphisms and strict homotopies constructed in section 3 admits a bicalculus of fractions that inverts the essential homotopy equivalences . we define our notion of morita homotopy as the 2-arrows in this bicategory of fractions . in section 5 we propose a notion of ls - category for lie groupoids . we say that a subgroupoid is @xmath3-categorical , in the sense of lusternik and schnirelmann , if there is a morita homotopy between the inclusion and a morphism with image in a single orbit . we define the ls - category for lie groupoids and prove that it is invariant under morita homotopy . in particular , it is invariant under morita equivalence and yields a well defined number for orbifolds . in section 6 we treat in detail this ls theory for orbifolds . we describe the obstructions to deform an open set in terms of the orbifold structure of the twisted sectors and give estimates for the orbifold ls - category in terms of the categories of the twisted and untwisted sectors . we also propose the notion of weighted orbifold ls - category and prove that it is invariant under morita homotopy equivalence . section 7 is where we give the preliminaries and prove the lusternik - schnirelmann theorem for orbifolds . given a generalized map @xmath19 we introduce a groupoid version of the classical deformation conditions . when the groupoid defines an orbifold , we can define the @xmath3-flow of the gradient of @xmath36 and its integral curves will be given by multiple @xmath3-paths . these integral @xmath3-curves provide the tool to prove that the deformation conditions are satisfied for smooth maps on orbifold groupoids . we show that both @xmath14 and @xmath37 satisfy some monotonicity , additivity , deformation invariance and continuity properties required to prove the theorem . i am grateful to andr haefliger , for suggesting to me the idea of developing a lusternik - schnirelmann theory for orbifolds . my thanks also to john oprea and dorette pronk for their valuable comments and help . recall that a _ groupoid _ @xmath3 is a small category in which each arrow is invertible @xcite @xcite . our notation for groupoids is that @xmath38 is the space of objects and @xmath39 is the space of objects , with source and target maps @xmath40 , multiplication @xmath41 , inversion @xmath42 , and object inclusion @xmath43 . a lie groupoid is a groupoid for which @xmath39 and @xmath38 are manifolds , @xmath44 and @xmath45 are surjective submersions , and @xmath46 , @xmath47 and @xmath48 are smooth @xcite . the set of arrows from @xmath49 to @xmath50 is denoted @xmath51 . the set of arrows from @xmath49 to itself , @xmath52 , is a group called the _ isotropy _ group of @xmath53 at @xmath49 and denoted by @xmath54 . the _ orbit _ of @xmath49 is the set @xmath55 . the orbit space @xmath56 of @xmath3 is the quotient of @xmath57 under the equivalence relation : @xmath58 iff @xmath49 and @xmath50 are in the same orbit . we will describe now various lie groupoids that will appear throughout the paper . 1 . _ unit groupoid . _ consider the groupoid @xmath3 associated to a manifold @xmath0 with @xmath59 . this is a lie groupoid whose arrows are all units , called the unit groupoid and denoted @xmath60 . pair groupoid . _ let @xmath3 with @xmath61 as before and consider @xmath62 . this is a lie groupoid with exactly one arrow from any object @xmath49 to any object @xmath50 , called the pair groupoid and denoted @xmath63 . point groupoid . _ let @xmath53 be a lie group . let @xmath64 be a point . consider the groupoid @xmath3 with @xmath65 and @xmath66 . this is a lie groupoid with exactly one object @xmath64 and @xmath53 is the manifold of arrows in which the maps @xmath44 and @xmath45 coincide . we denote the point groupoid by @xmath67 . translation groupoid . _ let @xmath68 be a lie group acting ( on the left ) on a manifold @xmath0 . consider the groupoid @xmath3 with @xmath61 and @xmath69 . this is a lie groupoid with arrows @xmath70 from any object @xmath49 to @xmath71 , called the translation or action groupoid and denoted @xmath72 . _ holonomy groupoid . _ let @xmath73 be a foliated manifold . consider the groupoid @xmath3 with @xmath61 and whose arrows from @xmath49 to @xmath50 on the same leaf @xmath74 are the holonomy classes of paths in @xmath75 from @xmath49 to @xmath50 . there are not arrows between points in different leaves . this is a lie groupoid called the holonomy groupoid and denoted @xmath76 . we will denote @xmath77 the trivial groupoid with one object and one arrow , @xmath78 . from now on all groupoids will be assumed to be lie groupoids . a _ morphism _ @xmath79 of groupoids is a functor given by two smooth maps @xmath80 and @xmath81 that together commute with all the structure maps of the groupoids @xmath2 and @xmath3 . a _ natural transformation _ @xmath82 between two morphisms @xmath83 is a smooth map @xmath84 with @xmath85 such that for any arrow @xmath86 in @xmath87 , the identity @xmath88 holds . we write @xmath89 . a morphism @xmath79 of groupoids is an _ equivalence _ of groupoids if there exists a morphism @xmath90 of groupoids and natural transformations @xmath82 and @xmath91 such that @xmath92 and @xmath93 . sometimes we will refer to this notion of equivalence as a @xmath94 equivalence . a morphism @xmath95 of groupoids is an _ essential equivalence _ of groupoids if * @xmath96 is essentially surjective in the sense that @xmath97 is a surjective submersion where @xmath98 is the pullback along the source @xmath99 ; * @xmath96 is fully faithful in the sense that @xmath87 is given by the following pullback of manifolds : @xmath100^{{\epsilon } } \ar[d]_{(s , t ) } & g_1 \ar[d]^{(s , t ) } \\ k_0\times k_0 \ar[r]^{{\epsilon}\times { \epsilon } } & g_0\times g_0}\ ] ] the first condition implies that for any object @xmath101 , there exists an object @xmath102 whose image @xmath103 can be connected to @xmath50 by an arrow @xmath104 . the second condition implies that for all @xmath105 , @xmath96 induces a diffeomorphism @xmath106 between the submanifolds of arrows . for general categories the notions of equivalence and essential equivalence coincide . this applies to the particular case in which the categories are groupoids . but when some extra structure is involved ( continuity or differentiability ) these two notions are not the same anymore . an essential equivalence implies the existence of the inverse functor using the axiom of choice but not the existence of a _ smooth _ functor . @xcite every equivalence of lie groupoids is an essential equivalence . the converse does not hold for lie groupoids . _ morita equivalence _ is the smallest equivalence relation between lie groupoids such that they are equivalent whenever there exists an essential equivalence between them . two lie groupoids @xmath2 and @xmath3 are _ morita equivalent _ if there exists a lie groupoid @xmath4 and essential equivalences @xmath107 this defines an equivalence relation that we denote @xmath108 . in this case , it is always possible to chose the equivalences @xmath96 and @xmath109 being surjective submersions on objects @xcite . [ tear ] consider the lie groupoid @xmath3 whose manifold of objects is the disjoint union of two open disks , @xmath110 as shown in figure [ t ] . the disk @xmath111 is acted on by @xmath112 . the manifold of arrows is the disjoint union of three copies of @xmath111 , two annuli @xmath113 and @xmath114 and a copy of @xmath115 , @xmath116 . the source map @xmath117 is given by the projection @xmath118 , the inclusions @xmath119 and @xmath120 and on the other annulus @xmath114 is given by the triple covering @xmath121 . target map @xmath122 coincides with source map on @xmath115 whilst in @xmath123 it is given by the action of @xmath124 , @xmath125 and in the annuli @xmath113 and @xmath114 is given by the triple covering @xmath126 and the inclusion @xmath119 respectively . now consider the translation groupoid @xmath127 defined by the action of @xmath128 on @xmath129 given by @xmath130 . the manifold of objects is @xmath131 and the manifold of arrows is @xmath132 . if we think of @xmath129 as the union of two solid tori as shown in figure [ solid ] , we have that the orbits of this action are circles of length @xmath133 for points on the two cores of the tori and circles of length @xmath134 elsewhere . this is a seifert fibration on @xmath129 . the inclusion of two transverse disks to this seifert fibration determines a functor @xmath135 which is an essential equivalence . thus the groupoids @xmath2 and @xmath3 are morita equivalent . we will define now two notions of homotopy that will generalize the notions of strong and morita equivalence respectively . given a subdivision @xmath136 of the interval @xmath35 $ ] consider a groupoid @xmath137 whose manifold of objects is given by the following disjoint union : @xmath138\ ] ] an element in the connected component @xmath139 $ ] will be denoted by @xmath140 . then the manifold of objects is @xmath141 , i=1 , \ldots , n\}$ ] . the manifold of arrows of @xmath137 is given by the disjoint union : @xmath142\right)\bigsqcup\{r_1,\cdots , r_{n-1 } , r'_1,\cdots , r'_{n-1}\}\ ] ] where @xmath143 $ ] is the set of unit arrows and for each point @xmath144 in the subdivision @xmath145 two arrows were added : @xmath144 and its inverse arrow @xmath146 such that the source of @xmath144 is @xmath147 and its target is @xmath148 . @xmath149}="3 " ; ( 35,0)*{}="4 " ; ( 28,0)*{[}="5 " ; ( 42,0)*{]}="6 " ; " 3";"5 " * * \crv{(24,4 ) & ( 25,4 ) } ; " 3";"5 " * * \crv{(24,-4 ) & ( 25,-4 ) } ; ( 24.5,3)*{\scriptstyle > } + ( 0,3)*{\scriptstyle r_i } ; ( 24.5,-3)*{\scriptstyle < } + ( 0,-3)*{\scriptstyle r^{-1}_i } ; ( 7,-4)*{\scriptstyle r_{i-1 } } ; ( 21,-4)*{\scriptstyle r_i } ; ( 14,2.5)*{\scriptstyle } ; ( 28,-4)*{\scriptstyle r_{i } } ; ( 42,-4)*{\scriptstyle r_{i+1 } } ; ( 56,2.5)*{\scriptstyle } ; " 2 " ; " 3 " * * \dir{- } ; " 5 " ; " 6 " * * \dir{- } ; { \ar@{}"4 " ; " 5 " } ; ( 0,0)*{\cdots } ; ( 49,0)*{\cdots } ; \endxy\ ] ] we call @xmath150 the trivial groupoid over @xmath151 and @xmath77 the trivial groupoid over @xmath152 . let @xmath153 be morphisms . we will say that @xmath154 is _ homotopic _ to @xmath155 if there exists a subdivision @xmath145 and a morphism @xmath156 such that @xmath157 and @xmath158 . this defines a relation of equivalence between morphisms that we will call _ homotopy equivalence _ and that sometimes we will refer to it as @xmath94 or @xmath159 homotopy equivalence depending what feature we want to emphasize since we will introduce a weaker version ( in the same sense that an essential equivalence weakens a strong equivalence ) and a generalized version of this notion ( in the same sense that a morita equivalence generalizes an essential equivalence ) . 1 . a natural transformation @xmath160 determines an _ homotopy _ @xmath156 over the subdivision @xmath161 . the homotopy is given by @xmath162 on objects ; @xmath163 on units arrows of @xmath137 and @xmath164 for the arrow @xmath165 . + therefore , there exist a subdivision @xmath145 and a morphism @xmath166 such that @xmath167 , @xmath168 and @xmath169 , @xmath170 . 2 . a morphism @xmath171 , with @xmath10 being the unit groupoid over the interval @xmath35 $ ] , such that @xmath172 and @xmath173 determines a homotopy @xmath156 over the subdivision @xmath174 . the original morphism @xmath171 does not define an equivalence relation , since transitivity fails . an ordinary homotopy @xmath171 such that @xmath175 and @xmath176 determines an homotopy @xmath156 over the subdivision @xmath177 . the morphism @xmath171 defines an equivalence relation but fails to be invariant of morita equivalence ( it is not even invariant of strong equivalence for groupoids ) . [ inj ] let @xmath178 be a homotopy of the inclusion , @xmath179 . then , there is an injection @xmath180 for all @xmath181 . the homotopy @xmath166 restricted to the connected component @xmath182 $ ] determines an ordinary homotopy , then @xmath183 injects into @xmath184 for all @xmath185 $ ] . the isotropy groups @xmath186 and @xmath187 coincide , since there is an arrow @xmath188 from @xmath189 to @xmath190 . then , we have a finite number of injections : @xmath191 and @xmath183 injects into @xmath192 for all @xmath193 $ ] with @xmath194 . we recall now the definition of @xmath3-path , due to haefliger @xcite . for another approach see also @xcite . a _ @xmath3-path _ from @xmath49 to @xmath50 over the subdivision @xmath136 of the interval @xmath195 $ ] is a sequence : @xmath196 where 1 . for all @xmath197 the map @xmath198\to g_0 $ ] is a path with @xmath199 and @xmath200 2 . for all @xmath201 the arrow @xmath202 satisfies : + @xmath203 + @xmath204 @xmath205 our notion of homotopy @xmath156 between @xmath157 and @xmath158 determines for each @xmath102 a @xmath3-path over the subdivision @xmath145 between the objects @xmath206 and @xmath207 in @xmath38 . in fact , it determines many more paths in @xmath39 other than the one defined by the unit arrows . if we think of a @xmath3-path as a morphism from some version of the interval @xmath208 to @xmath3 , we have that the haefliger @xmath3-paths correspond to morphisms @xmath209 , where the groupoid @xmath137 is the one constructed above . [ gpath ] a homotopy @xmath156 determines a @xmath3-path @xmath210 where @xmath211 is the trivial groupoid over @xmath102 . note that thinking of a @xmath3-path as a morphism @xmath109 from @xmath137 to @xmath3 , the image of the arrows in @xmath137 by @xmath109 is almost entirely contained in @xmath212 since most of the arrows in @xmath137 are units , with the only exception of the arrows @xmath213 and their inverses . in other words , we can not fill the space @xmath39 with paths of arrows given by the @xmath3-paths . this is going to be a very relevant problem when we try to integrate a vector field . motivated by these considerations , we propose the notion of _ multiple _ @xmath3-path . to define the several branches coming into a multiple @xmath3-path we need to introduce several copies of each subinterval in the subdivision @xmath145 . we define a groupoid @xmath214 associated to the interval @xmath208 and the subdivision @xmath145 in the following way . given a subdivision @xmath136 of the interval @xmath35 $ ] consider the manifold of objects given by the disjoint union : @xmath215_j\ ] ] where the extra copies of each interval are indexed by @xmath216 . @xmath149}="3 " ; ( 35,0)*{}="4 " ; ( 49,0)*{[}="5 " ; ( 63,0)*{]}="6 " ; ( 7,-4)*{\scriptstyle 0 } ; ( 21,-4)*{\scriptstyle r_1 } ; ( 14,2.5)*{\scriptstyle } ; ( 28,0)*{\scriptstyle \sqcup } ; ( 40,0)*{\scriptstyle \sqcup } ; ( 49,-4)*{\scriptstyle r_{n-1 } } ; ( 63,-4)*{\scriptstyle 1 } ; ( 56,2.5)*{\scriptstyle } ; " 2 " ; " 3 " * * \dir{- } ; " 5 " ; " 6 " * * \dir{- } ; { \ar@{}"4 " ; " 5 " } ; ( 34,0)*{\cdots } ; ( 7,-20)*{[}="12 " ; ( 21,-20)*{]}="13 " ; ( 35,-10)*{}="14 " ; ( 49,-20)*{[}="15 " ; ( 63,-20)*{]}="16 " ; ( 7,-24)*{\scriptstyle 0 } ; ( 21,-24)*{\scriptstyle r_1 } ; ( 14,2.5)*{\scriptstyle } ; ( 14,-4)*{\scriptstyle \sqcup } ; ( 14,-16)*{\scriptstyle \sqcup } ; ( 14,-10)*{\vdots } ; ( 56,-4)*{\scriptstyle \sqcup } ; ( 56,-16)*{\scriptstyle \sqcup } ; ( 56,-10)*{\vdots } ; ( 35,-10)*{\vdots } ; ( 49,-24)*{\scriptstyle r_{n-1 } } ; ( 63,-24)*{\scriptstyle 1 } ; ( 56,2.5)*{\scriptstyle } ; " 12 " ; " 13 " * * \dir{- } ; " 15 " ; " 16 " * * \dir{- } ; { \ar@{}"14 " ; " 15 " } ; \endxy\ ] ] an element in the connected component @xmath139_j$ ] will be denoted by @xmath217 . then the manifold of objects is @xmath218 , j=1 , \ldots , m_i , i=1 , \ldots , n\}.\ ] ] the manifold of arrows of @xmath214 is generated by the disjoint union of : 1 . @xmath219_j$ ] the set of unit arrows ; 2 . @xmath220 the set of arrows connecting the jumps at the subdivision points @xmath144 , i.e. the source and target of the arrow @xmath144 are @xmath221 and @xmath222 and 3 . @xmath223_j$ ] the set of arrows between the different copies @xmath139_j$ ] of each subinterval , i.e. the source and target of the arrow @xmath224_j$ ] are @xmath225 and @xmath226 . @xmath227}="1 " ; ( 7,-5)*{}="2 " ; ( 2,-10)*{\scriptstyle r_{ij } } ; ( 8,-5)*{}="20 " ; ( 9,-5)*{}="30 " ; ( 15,-10)*{\cdots } ; ( 21,-5)*{}="40 " ; ( 21,-5)*{}="3 " ; ( 35,0)*{}="4 " ; ( 28,0)*{[}="5 " ; ( 42,0)*{]}="6 " ; " 0 " ; " 1 " * * \dir{- } ; " 5 " ; " 6 " * * \dir{- } ; { \ar@{}"4 " ; " 5 " } ; " 1";"5 " * * \crv{(24,4 ) & ( 25,4 ) } ; " 1";"5 " * * \crv{(24,-4 ) & ( 25,-4 ) } ; ( 24.5,3)*{\scriptstyle > } + ( 0,3)*{\scriptstyle r_i } ; ( 24.5,-3)*{\scriptstyle < } + ( 0,-3)*{\scriptstyle r^{-1}_i } ; ( 7,-20)*{[}="10 " ; ( 21,-20)*{]}="11 " ; ( 7,-15)*{}="12 " ; ( 8,-15)*{}="22 " ; ( 9,-15)*{}="32 " ; ( 21,-15)*{}="42 " ; " 10 " ; " 11 " * * \dir{- } ; { \ar@{->}"2 " ; " 12 " } ; { \ar@{->}"20 " ; " 22 " } ; { \ar@{->}"30 " ; " 32 " } ; { \ar@{->}"40 " ; " 42 " } ; \endxy\ ] ] [ mpath ] a multiple @xmath3-path over a subdivision @xmath145 is a morphism @xmath228 . note that a @xmath3-path in the sense of haefliger is a multiple @xmath3-path over the same subdivision by taking @xmath229 for all subintervals and @xmath230 on objects ; @xmath231 on arrows @xmath144 and @xmath232 for unit arrows . we can think of a multiple @xmath3-path as a @xmath3-path between _ orbits _ or as a path between orbit subgroupoids . in this spirit , we will say that the initial subgroupoid of the path is @xmath233 and the end subgroupoid is @xmath234 . where @xmath235 and @xmath236 are the _ full _ subgroupoids over the orbits of @xmath237 and @xmath238 , which in general will not be trivial groupoids . consider the lie groupoid @xmath3 given in example [ tear ] where @xmath110 . we show in figure [ g ] the image in @xmath38 of a haefliger @xmath3-path between the points @xmath49 and @xmath50 given by the map @xmath209 for the subdivision @xmath239 . in figure [ multi ] we see the image of a multiple @xmath3-path between the orbits of @xmath49 and @xmath50 given by @xmath240 over the same subdivision . the manifold of objects of @xmath241 is now a disjoint union of three copies of the interval @xmath242 $ ] plus one copy of the interval @xmath243 $ ] . a homotopy @xmath156 , when restricted to the full subgroupoid @xmath244 over an orbit @xmath245 , defines a multiple @xmath3-path between the orbit subgroupoids @xmath246 and @xmath247 . if @xmath166 is a homotopy between @xmath154 and @xmath155 , then for all @xmath102 , the @xmath145-homotopy defines a multiple @xmath3-path between the orbits of @xmath206 and @xmath207 . sometimes we will denote a multiple @xmath3-path by @xmath248 where @xmath249 is a @xmath3-path and @xmath250 is a @xmath3-path for each @xmath251 and each @xmath252 . a morphism @xmath253 of groupoids is an _ essential homotopy equivalence _ of groupoids if there exists an @xmath145-homotopy equivalence @xmath254 and an essential equivalence @xmath255 such that @xmath256 . @xmath257^{\eta}\ar[rd]_{h}&&{{\mathcal g}}\\ & { { \mathcal l}}\ar[ur]_{{\epsilon } } & } \ ] ] this implies that for any object @xmath101 , there exists an object @xmath102 whose image @xmath258 can be connected to @xmath50 by a concatenation of paths and arrows . also @xmath259 induces a homotopy equivalence @xmath260 between the orbit spaces . [ isotropy ] if @xmath253 is an essential homotopy equivalence , then there is an injection between the corresponding isotropy groups @xmath261 and @xmath262 . [ ex ] consider the inclusion functor @xmath263 where @xmath264 is the seifert fibration of the mbius band @xmath0 . let @xmath265 be a fix point in the central fiber in @xmath0 . consider the inclusion functors @xmath266 and @xmath267 . the functor @xmath268 is a homotopy equivalence with homotopic inverse the constant map and @xmath96 is an essential equivalence . we have that @xmath269 is an essential homotopy equivalence . note that in this case @xmath259 is neither a ( strong ) homotopy equivalence nor an essential equivalence . it is clear that essential equivalences as well as ( strong ) homotopy equivalences are essential homotopy equivalences . [ bf1 ] 1 . an essential equivalence is an essential homotopy equivalence . a homotopy equivalence is an essential homotopy equivalence . [ bf4 ] if @xmath270 and @xmath259 is an essential homotopy equivalence , then @xmath271 . since @xmath272 , we have that @xmath273 . then @xmath274 because @xmath96 is an essential equivalence . if @xmath275 is the homotopic inverse of @xmath268 , we have that @xmath276 . thus , @xmath277 . let @xmath79 and @xmath278 be morphisms of lie groupoids and @xmath145 a subdivision of the interval @xmath35 $ ] . let @xmath279 be the space of @xmath3-paths over the subdivision @xmath145 . the _ groupoid homotopy pullback _ @xmath280 is the lie groupoid whose manifold of objects is @xmath281 and whose manifold of arrows is @xmath282 and source and target maps are given by : @xmath283 and @xmath284 . the groupoid homotopy pullback is well defined ( up to homotopy ) whenever one of the maps @xmath154 or @xmath155 is homotopic to a submersion on objects . we will show the construction of the manifold of objects @xmath285 for a subdivision @xmath286 by a sequence of pullbacks and homotopy pullbacks of manifolds . we recall first the definition of homotopy pullback of manifolds @xcite . given two maps @xmath287 and @xmath288 , the _ homotopy pullback _ is the space @xmath289 the following diagram commutes up to homotopy and it is universal ( up to homotopy ) with this property . @xmath290^{p_1 } \ar[d]_{p_2 } & x \ar[d]^{f } \\ y\ar[r]^{g } & z}\ ] ] [ manifold ] if @xmath24 is homotopic to a submersion , then the homotopic pullback @xmath291 is a manifold homotopic to the ordinary pullback . consider the following _ ordinary _ pullbacks of manifolds : @xmath292^{p_1 } \ar[d]_{p_2 } & k_0 \ar[d]^{\phi } \\ g_1\ar[r]^{s } & g_0 } \mbox { and } \xymatrix { j_0\times_{g_0}g_1 \ar[r]^{p'_1 } \ar[d]_{p'_2 } & j_0 \ar[d]^{\psi } \\ g_1\ar[r]^{t } & g_0}\ ] ] and the following _ homotopy _ pullback of manifolds : @xmath293^ { } \ar[dd]_{p } & & k_0\times_{g_0}g_1 \ar[d]^{p_2 } \\ & & g_1\ar[d]^{t}\\ j_0\times_{g_0}g_1\ar[r]^{p'_2 } & g_1\ar[r]^{s } & g_0}\ ] ] we have that @xmath285 is the manifold @xmath294 analogously , we can define the manifold of arrows @xmath295 as the homotopy pullback @xmath296 if @xmath297 is an arrow from @xmath298 to @xmath299 and @xmath300 is an arrow from @xmath301 to @xmath302 , the composition of arrows is given by @xmath303 @xmath304 we can generalize the construction to obtain the groupoid homotopy pullback @xmath305 corresponding to the subdivision @xmath136 by iterating @xmath306 homotopy pullbacks . consider the homotopy pullback @xmath307 of length @xmath306 given by the following diagram : @xmath308^ { } \ar[ddddd ] _ { } & k_0\times^hg^{n-2}_1 \ar[dddd]^{}\ar[r]^{}&\cdots\ar[r]^{}&k_0\times^hg_1\ar[d]^{}\ar[r]^{}&k_0\ar[d]^ { } \\ & & & g_1\ar[d]^{t}\ar[r]^{s}&g_0\\ & & { \:\:\ : \:\:\ : } \ar[r]^{s}&g_0&\\ & & \ddots&&\\ & g_1\ar[d]^{t}\ar[r]^{s}&{\:\:\ : \:\:\ : } & & \\ g_1\ar[r]^{s}&g_0 & & & } \ ] ] then , we have that @xmath309 and @xmath310 we will show that the following diagram of groupoids commutes up to strong homotopy : @xmath311^{\pi_1 } \ar[d]_{\pi_3 } & { { \mathcal k}}\ar[d]^{\phi } \\ { { \mathcal j}}\ar[r]^{\psi } & { \mathcal g}}\ ] ] consider the homotopy @xmath312 given by @xmath313 on objects and @xmath314 on arrows ( see remark [ gpath ] ) . we have that @xmath315 and @xmath316 the homotopy pullback of groupoids satisfies the following universal property . for any groupoid @xmath317 and morphisms @xmath318 and @xmath319 with @xmath320 there exists a morphism @xmath321 such that both triangles commute up to homotopy : @xmath322_{\gamma } \ar@/^/[drr]^{\delta } \ar@{.>}[dr]|-{\xi } \\ & { { \mathcal p}}_s \ar[d]^{\pi_3 } \ar[r]_{\pi_1 } & { { \mathcal k}}\ar[d]_{\phi}\\ & { { \mathcal j}}\ar[r]^{\psi } & { \mathcal g}}\ ] ] we define @xmath321 by @xmath323 on objects and @xmath324 on arrows , where @xmath325 is the trivial groupoid over @xmath326 . note that the groupoid homotopy pullback @xmath327 is defined for each subdivision @xmath145 of the interval @xmath35 $ ] . if @xmath136 is a subdivision of @xmath35 $ ] , then for all subdivision @xmath328 we have : 1 . if @xmath329 , then @xmath330 . 2 . there exists a morphism @xmath331 such that @xmath332 and @xmath333 . since @xmath334 , then @xmath335 and the following diagram commutes up to @xmath336-homotopy : @xmath337_{\pi_3 } \ar@/^/[drr]^{\pi_1 } \ar@{.>}[dr]|-{\xi } \\ & \ar@{}[dr ] |{\downarrow_{s'}}{{\mathcal p}}_{s ' } \ar[d]_{\pi'_3 } \ar[r]^{\pi'_1 } & { { \mathcal k}}\ar[d]_{\phi}\\ & { { \mathcal j}}\ar[r]^{\psi } & { \mathcal g}}\ ] ] [ order ] if there is a subdivision @xmath145 such that the following diagram commutes up to @xmath145-homotopy : @xmath338^{\simeq_{s}}{{\mathcal k}}\ar[dr]_{{\epsilon } } \ar[r]^{\eta } & { \mathcal g}\\ & { \mathcal l}\ar[u]_{h}}\ ] ] where @xmath96 is an essential equivalence and @xmath268 is a homotopy equivalence , then @xmath253 is an essential homotopy equivalence . let @xmath275 be the homotopic inverse of @xmath268 . then , the following triangle commutes up to @xmath145-homotopy : @xmath339_{\simeq_{s } } \ar[dr]^{{{\scriptstyle i d } } } \ar[d]_{g } & \\ { \mathcal l}\ar[r]_{h } & { \mathcal g}}\ ] ] consider the following fibered product of groupoids : @xmath338 |{\sim}{\mathcal g}\times_{{\mathcal l}}{{\mathcal k}}\ar[d]_{g ' } \ar[r]^{{\epsilon}'}&{\mathcal g}\ar[d]^{g}\\ { { \mathcal k}}\ar[r]_{{\epsilon } } & { \mathcal l}}\ ] ] the square above commutes up to a natural transformation , then it commutes up to @xmath145-homotopy and has the universal property . since @xmath340 , we have that the following diagram commutes up to @xmath145-homotopy : @xmath341_{{{\scriptstyle i d } } } \ar@/^/[drr]^{\eta}\ar@{.>}[dr]|-{\xi } \\ & \ar@{}[dr ] |{}{\mathcal g}\times_{{\mathcal l}}{{\mathcal k}}\ar[d]_{g ' } \ar[r]^{{\epsilon}'}&{\mathcal g}\ar[d]^{g}\ar[dr]^{{{\scriptstyle id}}}&\\ & { { \mathcal k}}\ar[r]_{{\epsilon } } & { \mathcal l}\ar[r]_{h}&{\mathcal g}}\ ] ] therefore , @xmath342 and @xmath343 . we have that : @xmath344 and also : @xmath345 by proposition [ bf4 ] , we have that @xmath346 and @xmath347 is the homotopic inverse of @xmath348 . then @xmath349 with @xmath350 and essential equivalence and @xmath351 a homotopy equivalence . [ above ] if @xmath96 is an essential equivalence and @xmath275 is an @xmath145-homotopy equivalence , then the homotopic pullback @xmath311^{{\epsilon } ' } \ar[d]_{g ' } & { { \mathcal j}}\ar[d]^{g } \\ { \mathcal l}\ar[r]^{{\epsilon } } & { \mathcal g}}\ ] ] exists and @xmath348 is an @xmath145-homotopy equivalence as well . since @xmath96 is an essential equivalence , the pullback @xmath305 exists . let @xmath352 be the homotopic inverse of @xmath275 , then @xmath353 and @xmath354 . consider the following pullbacks : @xmath355^{{\epsilon } '' } \ar[d]_{f ' } & { \mathcal g}\ar[d]^{f } \\ { { \mathcal p}}_s \ar[r]^{{\epsilon } ' } \ar[d]_{g ' } & { { \mathcal j}}\ar[d]^{g } \\ { \mathcal l}\ar[r]^{{\epsilon } } & { \mathcal g}}\ ] ] by the universal property of the large square , there exists @xmath356 such that the following diagram commutes up to @xmath145-homotopy : @xmath357_{{{\scriptstyle i d } } } \ar@/^/[drr]^{{\epsilon}}\ar@{.>}[dr]|-{\xi } \\ & \ar@{}[dr ] |{}{{\mathcal p}}'_s \ar[d]^{g'f ' } \ar[r]^{{\epsilon}''}&{\mathcal g}\ar[d]^{{{\scriptstyle id}}}\\ & { \mathcal l}\ar[r]_{{\epsilon } } & { \mathcal g}}\ ] ] then @xmath358 and @xmath359 . we will see that @xmath360 is the homotopic inverse of @xmath348 . we have that @xmath361 by proposition [ bf4 ] , @xmath362 and @xmath348 is a homotopy equivalence . [ bf2 ] if @xmath259 and @xmath363 are essential homotopy equivalences , then @xmath364 is an essential homotopy equivalence as well . we have that @xmath365 and @xmath366 with @xmath96 and @xmath109 being essential equivalences and @xmath268 and @xmath24 homotopy equivalences . consider the following homotopic pullback , where @xmath145 is a refinement of @xmath145 and @xmath336 : @xmath311^{g ' } \ar[d]_{{\epsilon } ' } & { { \mathcal a}}\ar[d]^{{\epsilon } } \\ { { \mathcal b}}\ar[r]^{g } & { \mathcal g}}\ ] ] where @xmath275 is the homotopic inverse of @xmath24 . by lemma [ above ] , we have that @xmath348 is a homotopy equivalence . then , the following diagram commutes up to @xmath145-homotopy : @xmath367_{h}\ar[rr]^{\eta}&&{\mathcal g}\ar[rr]^{\nu}&&{\mathcal l}\\ & { { \mathcal a}}\ar[ur]_{{\epsilon}}&&{{\mathcal b}}\ar[ur]_{\sigma}\ar[ul]^{g}&\\ & & { { \mathcal p}}_s \ar[ul]^{g'}\ar[ur]_{{\epsilon } ' } & & } \ ] ] let @xmath368 be the homotopic inverse of @xmath348 . then , @xmath369 where @xmath370 is an essential equivalence and @xmath371 is a homotopy equivalence . we can pull back the decomposition given in proposition [ order ] and we have the following [ bf3 ] if @xmath372 is an essential homotopy equivalence , then there is a subdivision @xmath145 such that the morphism @xmath373 is an essential homotopy equivalence as well . [ mhe ] a lie groupoid @xmath2 is _ morita homotopy equivalent _ to @xmath3 if there exists a lie groupoid @xmath374 and essential homotopy equivalences @xmath375 note that it is possible to choose these essential homotopy equivalences being homotopic to surjective submersions on objects . by using two homotopy pullbacks of groupoids and proposition [ bf3 ] , it is easy to see that morita homotopy equivalence is a relation of equivalence . we will introduce in this section a bicategory whose objects are lie groupoids and where morita homotopy equivalence amounts to equivalence of objects . we follow a bicategorical approach as emphasized by landsman in @xcite . recall that a bicategory @xmath376 consists of a class of objects , morphisms between objects and 2-morphisms between morphisms together with various ways of composing them . a 2-morphism is a _ 2-equivalence _ if it is invertible in @xmath376 whereas a morphism is an _ equivalence _ if it is invertible up to a 2-equivalence in @xmath376 . we have that @xmath377 is a 2-equivalence in @xmath376 if there exists @xmath53 such that @xmath378 and @xmath379 , and @xmath380 is an equivalence in @xmath376 if there exists @xmath155 such that @xmath381 and @xmath382 . we can compose 2-morphisms in two ways called horizontal and vertical composition.vertical composition is strictly associative whereas horizontal composition is only associative up to 2-equivalence ( associator ) . the unit laws for morphisms hold up to 2-equivalences ( left and right unit constrains ) . associator and unit constrains are required to be natural with respect to their arguments and verify certain axioms @xcite . given a bicategory @xmath376 and a subset @xmath383 of morphisms satisfying certain conditions , there exists a bicategory @xmath384 having the same objects as @xmath376 but inverse morphisms of morphisms in @xmath113 have been added as well as more 2-morphisms . this new bicategory is called a _ bicategory of fractions _ of @xmath376 with respect to @xmath113 and was constructed by pronk in @xcite . the bicategory of fractions @xmath384 is characterized by the universal property that any morphism @xmath385 sending elements of @xmath113 into equivalences factors in a unique way as @xmath386 @xmath387 \ar^{u\;\;\;\;}[r ] & { { \mathfrak b}}(a^{-1})\ar^{\tilde f}[ld ] \\ { { \mathfrak d } } & } \ ] ] the following conditions are needed on @xmath113 to admit a bicalculus of right fractions @xcite : 1 . all equivalences are in @xmath113 . if @xmath154 and @xmath155 are in @xmath113 , then @xmath388 . 3 . for all @xmath389 and @xmath390 with @xmath391 there exists an object @xmath392 and morphisms @xmath393 and @xmath394 with @xmath395 such that the following square commutes up to a 2-equivalence : @xmath396 \ar^{\psi}[r ] & { { \mathcal j}}\ar^{\eta}[d ] \\ { { \mathcal k}}\ar^{\phi}[r ] & { \mathcal g}}\ ] ] 4 . if @xmath397 is a 2-morphism with @xmath391 then there exists a morphism @xmath395 and a 2-morphism @xmath398 such that @xmath399 . 5 . if there is a 2-equivalence @xmath400 with @xmath395 then @xmath391 . the set of lie groupoids , morphisms and ( homotopy classes of ) strong homotopies as 2-morphisms forms a bicategory @xmath6 . strong homotopies @xmath401 and @xmath402 can be composed vertically by taking the subdivision @xmath403 . we take homotopy classes of homotopies to assure associativity of the vertical composition . we will formally invert the set of essential homotopy equivalences @xmath7 . the set @xmath7 of essential homotopy equivalences allows a bicalculus of right fractions . condition bf1 follows from proposition [ bf1 ] . condition bf2 follows from proposition [ bf2 ] . condition bf3 follows from proposition [ bf3 ] . condition bf4 follows from proposition [ bf4 ] . finally , if @xmath404 is a strong homotopy equivalence and @xmath259 is an essential homotopy equivalence , then @xmath405 and @xmath363 is an essential homotopy equivalence as well . therefore , there exists a bicategory of fractions @xmath8 inverting the essential homotopy equivalences . the objects of @xmath8 are lie groupoids . the 1-morphisms from @xmath2 to @xmath3 are formed by pairs @xmath406 @xmath407 such that @xmath259 is an essential homotopy equivalence . the 1-morphisms @xmath406 in this bicategory will be called _ generalized homotopy maps_. if @xmath96 is an essential equivalence , then @xmath408 is called a generalized map . a 2-morphism from @xmath406 to @xmath409 is given by the following diagram : @xmath410^{\phi}="0 " \ar[dl]_{\eta } = " 2"&\\ { { { \mathcal k}}}&{{\mathcal l } } \ar[u]_{u } \ar[d]^{v}&{{\mathcal g}}\\ & { { { \mathcal j}}'}\ar[ul]^{\eta'}="3 " \ar[ur]_{\phi'}="1 " & \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8"_{h^s } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^{f^s } } \ ] ] where @xmath374 is a lie groupoid , @xmath48 and @xmath411 are essential homotopy equivalences and @xmath412 and @xmath413 are strong homotopy equivalences . the notion of _ homotopy _ we propose corresponds to 2-morphisms in this bicategory @xmath8 . two generalized homotopy maps @xmath406 and @xmath409 are _ morita homotopic _ if there is a 2-morphism between them . in this case , we write @xmath414 and we say that there is a _ morita homotopy _ between @xmath406 and @xmath409 . in particular , when @xmath259 and @xmath415 are essential equivalences , we have a notion of homotopy for generalized maps and when they are identities , we have a notion of homotopy for strict maps . two objects @xmath2 and @xmath3 are equivalent in @xmath8 if there are morphisms @xmath406 from @xmath2 to @xmath3 and @xmath416 from @xmath3 to @xmath2 such that @xmath417 is morita homotopic to the identity @xmath418 and @xmath419 . a morphism @xmath406 is an equivalence in @xmath8 if and only if @xmath154 is an essential homotopy equivalence . in this case , the inverse of @xmath406 is the morphism @xmath420 . in other words , the definition of morita homotopy equivalence in subsection [ mhe ] amounts to equivalence of objects in the bicategory @xmath8 . so , we write @xmath421 for equivalence of objects in @xmath8 . morita homotopy type _ of @xmath3 is the class of @xmath3 under the equivalence relation @xmath422 . [ mht ] the holonomy groupoid @xmath423 of the seifert fibration on the mbius band has the same morita homotopy type than the point groupoid @xmath424 since we have @xmath425 where @xmath259 and @xmath363 are essential homotopy equivalences . the homotopic inverse is the generalized map @xmath426 . note that @xmath427 is morita equivalent to @xmath423 since the inclusion @xmath428 is an essential equivalence , but the groupoids @xmath2 and @xmath3 are not morita equivalent . we show now that the morita homotopy type is invariant under morita equivalence . if @xmath429 , then @xmath421 . if @xmath2 and @xmath3 are morita equivalent , then there is lie groupoid @xmath374 and essential equivalences @xmath375 the morphisms @xmath259 and @xmath363 are also essential homotopy equivalences ( see proposition [ bf1 ] ) . then the morphisms @xmath430 and @xmath431 are inverse up to equivalence and @xmath2 is equivalent to @xmath3 in the bicategory @xmath8 . a morita homotopy equivalence @xmath421 induces a morita equivalence between the homotopy groupoids @xmath432 . if @xmath433 is a morita equivalence , then @xmath433 is a @xmath238-homotopy equivalence in the sense of @xcite . we will give an explicit description of the horizontal and vertical composition in this bicategory . the horizontal composition of 1-morphisms @xmath434 and @xmath435 is given by the diagram @xmath436\ar[dl]&&\\ & { { \mathcal j}}\ar[dl]_{\eta}\ar[dr]^{\phi}\rrtwocell<\omit>{<0>s}&&{{\mathcal j}}'\ar[dl]_{\nu}\ar[dr]^{\psi}&\\ { { \mathcal k}}&&{\mathcal g}&&{\mathcal l}}\ ] ] and the vertical composition of the 2-arrows @xmath437^{\phi}="0 " \ar[dl]_{\eta } = " 2"&\\ { { { \mathcal k}}}&{{\mathcal l } } \ar[u]_{u } \ar[d]^{v}&{{\mathcal g}}\\ & { { { \mathcal d}}}\ar[ul]^{\eta'}="3 " \ar[ur]_{\phi'}="1 " & \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8"_{s } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^{s } } \qquad \xymatrix { & { { { \mathcal d}}}\ar[dr]^{\phi'}="0 " \ar[dl]_{\eta ' } = " 2"&\\ { { { \mathcal k}}}&{{\mathcal l } ' } \ar[u]_{u ' } \ar[d]^{v'}&{{\mathcal g}}\\ & { { { \mathcal e}}}\ar[ul]^{\eta''}="3 " \ar[ur]_{\phi''}="1 " & \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8"_{s ' } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^{s ' } } \ ] ] is given by the diagram @xmath438^{\phi}="2 " \ar[ddlll]_{\eta}="0 " & & & \\ & & & { \mathcal l}\ar[dr]^{v}\ar[u]_{u}&&&\\ { { \mathcal k } } & & { { \mathcal p}}_{s''}\ar[dr]^ { } \ar[ur]_{}\rrtwocell<\omit>{<0>{\;\;s''}}&&{{\mathcal d}}&&{\mathcal g}\\ & & & { \mathcal l}'\ar[d]^{v ' } \ar[ur]_{u ' } & & & \\ & & & { { \mathcal e}}\ar[uulll]^{\eta''}="1 " \ar[uurrr]_{\phi''}="3 " & & & & \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^{s '' } \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8"_{s '' } } \ ] ] where @xmath439 . if the morphisms @xmath408 and @xmath440 are generalized maps , the composition given by the fibered product groupoid @xmath441 is _ morita homotopic _ to this horizontal composition in the bicategory @xmath8 . since @xmath442 we have that @xmath443 and by the universal property of the homotopy pullback , there exists a map @xmath444 such that the following diagram commutes up to homotopy : @xmath445_{p_3 } \ar@/^/[drr]^{p_1}\ar@{.>}[dr]|-{\xi}&&\\ & { { \mathcal p}}_s\ar[d]^{\pi_3}\ar [ r]^{\pi_1}&{{\mathcal j } } ' \ar[d]_{\nu}\\ & { { \mathcal j}}\ar[r]^{\phi}&{\mathcal g}}\ ] ] therefore there is a 2-morphism between the generalized homotopy maps @xmath441 and @xmath446 given by the following diagram : @xmath447^{\xi}\ar[dr]^{p_1}="0 " \ar[dl]_{p_3 } = " 2"&&\\ { { { \mathcal k}}}&{{{\mathcal j } } } \ar[l]_{\eta } & & { { \mathcal j}}'\ar[r]^{\psi}&{\mathcal l}\\ & & { { \mathcal p}}_s\ar[ul]^{\pi_3}="3 " \ar[ur]_{\pi_1}="1 " & \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8 " _ { } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^ { } } \ ] ] then , when the generalized homotopy maps are generalized maps , we can set the groupoid fibered product as our chosen homotopy pullback . we can think of a multiple @xmath3-path as a generalized map @xmath448 since the map @xmath96 defined in the obvious way is an essential equivalence . then , two multiple @xmath3-paths @xmath448 and @xmath449 are _ homotopic _ if there is a commutative diagram : @xmath450^{\sigma}="0 " \ar[dl]_{{\epsilon}}="2"&\\ { { { \mathcal i}}}&{{{\mathcal i}}'_{s '' } } \ar[u]_{u } \ar[d]^{v}&{{\mathcal g}}\\ & { { { \mathcal i}}_{s'}}\ar[ul]^{{\epsilon}'}="3 " \ar[ur]_{\sigma}="1 " & \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8"_{h ' } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^{h } } \ ] ] where @xmath451 a submanifold @xmath452 is _ invariant _ if @xmath453 . the _ restricted groupoid @xmath13 _ to an invariant submanifold @xmath452 is the full groupoid over @xmath454 . in other words , @xmath455 and @xmath456 . we write @xmath457 . a restricted groupoid @xmath458 over an orbit @xmath459 will be called an _ orbit groupoid _ and denoted by @xmath460 , where @xmath461 is the isotropy group of @xmath49 for any @xmath462 . a _ generalized constant map _ from @xmath2 to @xmath3 is a generalized map @xmath463 such that the image of the functor @xmath22 is contained in an orbit groupoid . we have that @xmath464 with @xmath461 for some @xmath465 . the _ restriction _ @xmath466 of a generalized map @xmath467 to the restricted groupoid @xmath468 is the composition of the generalized map @xmath408 and the inclusion functor @xmath469 : @xmath470^{p_1}\ar[dl]_{p_3}&&\\ & { { \mathcal v}}\ar[dl]_{{{\scriptstyle id}}}\ar[dr]^{i_{{{\mathcal v}}}}&&{{\mathcal j}}\ar[dl]_{{\epsilon}}\ar[dr]^{\phi}&\\ { { \mathcal v}}&&{{\mathcal k}}&&{\mathcal g}}\ ] ] where @xmath471 is the fibered product groupoid . the _ product _ @xmath472 of two generalized maps @xmath467 and @xmath473 is given by the generalized map @xmath474 we will define now a notion of @xmath3-contraction within the groupoid @xmath3 . we keep the classical lusternik - schnirelmann terminology of _ categorical _ for contractible subspaces in a given space . [ d ] for an invariant open set @xmath452 , we will say that the restricted groupoid @xmath13 is @xmath3-_categorical _ if the inclusion functor @xmath475 is morita homotopic to a generalized constant map @xmath476^{i_{{{\mathcal u}}}}="0 " \ar[dl]_{{{\scriptstyle id}}}="2"&\\ { { { \mathcal u}}}&{{\mathcal l } } \ar[u]_{u } \ar[d]^{v}&{{\mathcal g}}\\ & { { { \mathcal u}}'}\ar[ul]^{{\epsilon}}="3 " \ar[ur]_{c}="1 " & \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8"_{h ' } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^{h } } \ ] ] where @xmath48 and @xmath411 are essential homotopy equivalences . let @xmath477 be the holonomy groupoid of the seifert fibration @xmath478 on the klein bottle @xmath68 . consider the invariant open set @xmath479 , a mbius band . then the restricted groupoid over @xmath454 is @xmath480 . consider the essential homotopy equivalence @xmath263 given by the inclusion as in example [ ex ] . let @xmath481 be the action groupoid given by the action of @xmath482 on the interval @xmath208 by reflection . the inclusion functor @xmath483 is an essential equivalence . we have the following diagram @xmath484^{i_{{{\mathcal u}}}}="0 " \ar[dl]_{{{\scriptstyle id}}}="2"&\\ { { \mbox{\rm hol}}(m,{{\mathcal f}}_s)}&\bullet^{{{\mathbb{z}}}_2 } \ar[u]_{\eta } \ar[d]^{i}&{{\mbox{\rm hol}}(k,{{\mathcal f}}_s)}\\ & { { { \mathbb{z}}}_2}\ltimes i\ar[ul]^{{\epsilon}}="3 " \ar[ur]_{c}="1 " & \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8"_{h ' } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^{h } } \ ] ] here @xmath485 is a constant map on objects @xmath486 where @xmath49 is any point on a fiber with holonomy @xmath487 and @xmath22 is a constant map in each of the connected components of the manifold of arrows @xmath488 , @xmath489 and @xmath490 . here the manifold of arrows for the groupoid @xmath491 is @xmath492 . then @xmath493 and @xmath479 is categorical . given a categorical subgroupoid @xmath13 , we have that the inclusion @xmath494 composed with an essential homotopy equivalence @xmath48 factors through an orbit groupoid up to homotopy : @xmath495^{i_{{{\mathcal u}}}}="0 " & \\ { { \mathcal l } } \ar[u]_{u } \ar[d]^{v}&{{\mathcal g}}\\ { { { \mathcal u}}'}\ar[ur]_{c}="1 " \ar[r]^{}&{{{\mathcal o}}^k}\ar[u]^ { } \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8 " _ { } } \ ] ] in other words , if @xmath13 is @xmath3-categorical then there exists a groupoid @xmath374 and a group @xmath68 such that the diagram @xmath496^{u}\ar[rd]^{}&{{{\mathcal u } } } \ar[r]^{i_{{{\mathcal u}}}}&{{\mathcal g}}\\ & { { { \mathcal o}}^k}\ar[ur]^ { } & } \ ] ] commutes up to homotopy . recall that a lie groupoid is _ transitive _ if the map @xmath497 is a surjective submersion . @xcite a lie groupoid @xmath3 is morita equivalent to a point groupoid if and only if @xmath3 is transitive . since an orbit groupoid is transitive , for @xmath13 to be @xmath3-categorical we can substitute the constant generalized map in definition [ d ] by a generalized map with image contained in a point groupoid . the ls category , @xmath14 , of a lie groupoid @xmath3 is the least number of @xmath3-categorical subgroupoids required to cover @xmath3 . if @xmath3 can not be covered by a finite number of @xmath3-categorical subgroupoids , set @xmath498 . this number is an _ invariant of morita equivalence _ that generalizes the ordinary ls category of a manifold . if @xmath60 is the unit groupoid , then @xmath499 , where @xmath500 in the second term means the ordinary ls category . with these definitions we choose the _ point groupoids _ as our contractible groupoids . we make this choice in spite of the fact that the fundamental group of the point groupoid is not necessary trivial since @xmath501 if @xmath68 is discrete . our choice is inspired by the @xmath317-category of clapp and puppe @xcite and justified by the equivariant ls theory for group actions . the point groupoid can be seen as the translation groupoid of the ( ineffective ) action of @xmath68 on a point @xmath64 . since the equivariant ls category of a @xmath68-manifold , @xmath502 , coincides with the ordinary category @xmath503 when the action is trivial , we have that the equivariant category @xmath504 . in our theory , a @xmath3-categorical groupoid factors through a point groupoid . for all groups @xmath68 we have : @xmath505 where the first term is the _ groupoid category _ of the point groupoid , the second is the _ equivariant category _ of the action of @xmath68 on a point , and the third , the ordinary category of a point . in section [ 6 ] we will give a weighted version of the ls groupoid category that takes into account the fact that point groupoids are not contractible in the sense mentioned above . we will show that the ls category of a groupoid is an invariant of morita homotopy type , and then , invariant under morita equivalence . [ morita ] if @xmath506 then @xmath507 . we will prove that if @xmath2 dominates @xmath3 , then @xmath508 . let @xmath467 and @xmath509 be two generalized maps such that the composition @xmath510 is morita homotopic to the identity in @xmath3 . we have the following diagram : @xmath511^{{{\scriptstyle id}}}="0 " \ar[ddll]_{{{\scriptstyle id}}}="2"&&\\ & & & & \\ { { \mathcal g}}\rrtwocell<\omit>{<0>s}&&{{{\mathcal e } } } \ar[uu ] _ { } \ar[dd]^{}\rrtwocell<\omit>{<0>s}&&{{\mathcal g}}\\ & { { \mathcal j}}'\ar[ul]^{\delta}&&{{\mathcal j}}\ar[ur]^{\phi}&\\ & & { { { \mathcal j}}\times_{{{\mathcal k}}}{{\mathcal j}}'}\ar[ul]^{p_3}="3 " \ar[ur]_{p_1}="1 " & & } \ ] ] let @xmath512 be a @xmath2-categorical subgroupoid , then we have the following diagram : @xmath513^{i_{{{\mathcal k}}}}="0 " \ar[dll]_{{{\scriptstyle id}}}="2"&&\\ { { { \mathcal u}}}\rrtwocell<\omit>{<0>s'}&&{{\mathcal l } } \ar[u ] _ { } \ar[dd]^{}\rrtwocell<\omit>{<0>s'}&&{{{\mathcal k}}}\\ & & & { \bullet^k}\ar[ur]&\\ & & { { { \mathcal u}}'}\ar[uull]^{z}="3 " \ar[ur]_{}="1 " & & } \ ] ] let @xmath514 be the groupoid given by @xmath515 in the following pullback : @xmath516^{p'_1}\ar[d]^{p'_3}&{{{\mathcal u } } } \ar[d]^{i_{{{\mathcal u}}}}\\ { \mathcal g}&{{{\mathcal j}}'}\ar[r]^{\psi}\ar[l]^{\delta}&{{\mathcal k}}}\ ] ] in diagram ( [ 1 ] ) , take the restriction to @xmath517 , i.e. compose both maps with the inclusion functor @xmath518 @xmath519^{i_{{{\mathcal v } } } } \ar[dl]_{{{\scriptstyle id}}}&&\\ & { { { \mathcal v}}}\ar@{=}[dd]\ar[dr]^{i_{{{\mathcal v}}}}\ar[dl]_{{{\scriptstyle id}}}&&{{\mathcal g } } \ar[dr]^{{{\scriptstyle i d } } } \ar[dl]_{{{\scriptstyle id}}}&\\ { { \mathcal v}}&&{{\mathcal g}}\rtwocell<\omit>{<0>s}&{{\mathcal e}}\rtwocell<\omit>{<0>s}\ar[u]_{}\ar[d]_{}&{\mathcal g}\\ & { { { \mathcal v}}}\ar[ur]^{i_{{{\mathcal v}}}}\ar[ul]^{{{\scriptstyle id}}}&&{{{\mathcal j}}\times_{{{\mathcal k}}}{{\mathcal j } } ' } \ar[ur]_{\phi p_1 } \ar[ul]^{\delta p_3}&\\ & & { { { \mathcal j}}\times_{{{\mathcal k}}}{{\mathcal j}}'\times_{{\mathcal g}}{{\mathcal v}}}\ar[ur]^ { } \ar[ul ] _ { } & & } \ ] ] now take the restriction of @xmath509 to @xmath517 @xmath520^{p_1}\ar[dl]^{p_3}&&\\ & { { \mathcal v}}\ar[dl]_{{{\scriptstyle id}}}\ar[dr]^{i_{{{\mathcal v}}}}&&{{\mathcal j}}'\ar[dl]_{\delta}\ar[dr]^{\psi}&\\ { { \mathcal v}}&&{\mathcal g}&&{{\mathcal k}}}\ ] ] @xmath521 . ( of the lemma ) use that @xmath454 is invariant and the essential equivalence @xmath522 induces a diffeomorphism between the isotropy groups . since @xmath521 we obtain the generalized map @xmath523 from diagram ( [ 4 ] ) . now we compose this map with the inclusion functor @xmath524 and with the generalized map @xmath467 , we have : @xmath525\ar[dll]&&\\ & & { { \mathcal u}}\times_{{{\mathcal u}}}{{\mathcal j}}'\times_{{\mathcal g}}{{\mathcal v}}\ar[dl]_{}\ar[dr]^{}&&&&{{\mathcal j}}\ar[ddll]_{{\epsilon}}\ar[ddrr]^{\phi}&&\\ & { { \mathcal j}}'\times_{{\mathcal g}}{{\mathcal v}}\ar[dl]_{}\ar[dr]^{}&&{{\mathcal u}}\ar[dl]_{}\ar[dr]^{}&&&&&\\ { { \mathcal v}}&&{{\mathcal u}}&&{{\mathcal k}}&&&&{\mathcal g}}\ ] ] we will show that the generalized map @xmath526 obtained by this composition is homotopic to the map @xmath527 obtained from the following compositions @xmath528^{\pi_1}\ar[dll]&&\\ & & { { \mathcal u}}'\times_{{{\mathcal u}}}{{\mathcal j}}'\times_{{\mathcal g}}{{\mathcal v}}\ar[dl]_{}\ar[dr]^{}&&&&{{\mathcal j}}\ar[ddll]_{{\epsilon}}\ar[ddrr]^{\phi}&&\\ & { { \mathcal j}}'\times_{{\mathcal g}}{{\mathcal v}}\ar[dl]_{}\ar[dr]^{}&&{{\mathcal u}}'\ar[dl]_{z}\ar[dr]^{c}&&&&&\\ { { \mathcal v}}&&{{\mathcal u}}&&{{\mathcal k}}&&&&{\mathcal g}}\ ] ] let @xmath529 , we have the following [ lema ] the image of @xmath529 is contained in an orbit groupoid . moreover , if @xmath530 then @xmath531 where @xmath532 . ( of the lemma ) use the fact that @xmath96 induces a homeomorphism between the orbit spaces and a diffeomorphism between the isotropy groups . then @xmath533 is a generalized constant map . by using diagram ( [ 2 ] ) we have that @xmath534 then we have a 2-morphism @xmath535^{}="0 " \ar[dl]_{}="2"&\\ { { { \mathcal v}}}&{{\mathcal l } } \ar[u ] _ { } \ar[d]^{}&{{\mathcal g}}\\ & { { { \mathcal j}}\times_{{{\mathcal k } } } { { \mathcal j}}'\times_{{\mathcal g}}{{\mathcal v}}}\ar[ul]^{}="3 " \ar[ur]_{}="1 " & \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8"_{s } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^{s } } \ ] ] given by diagram ( [ 3 ] ) and another 2-morphism @xmath536^{}="0 " \ar[dl]_{}="2"&\\ { { { \mathcal v}}}&{{\mathcal l } ' } \ar[u ] _ { } \ar[d]^{}&{{\mathcal g}}\\ & { { { \mathcal j}}\times_{{{\mathcal k}}}{{\mathcal u}}'\times_{{{\mathcal u}}}{{\mathcal j}}'\times_{{\mathcal g}}{{\mathcal v}}}\ar[ul]^{}="3 " \ar[ur]_{c'}="1 " & \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8"_{s ' } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^{s ' } } \ ] ] given by equation ( [ 7 ] ) . the vertical composition of these 2-morphisms ( [ 8 ] ) and ( [ 9 ] ) gives a 2-morphism between the inclusion and a generalized constant map : @xmath537^{i_{{{\mathcal v}}}}="2 " \ar[ddll]_{{{\scriptstyle id}}}="0 " & & & \\ & & { \mathcal l}\ar[dr]^{v}\ar[u]_{}&&&\\ { { \mathcal k } } & \rtwocell<\omit>{<0>\;\;\scriptstyle s '' } & { { \mathcal p}}_{s''}\rtwocell<\omit>{<0>\;\;\scriptstyle s''}\ar[d]^ { } \ar[u]_{}&{{\mathcal d}}\rtwocell<\omit>{<0>\;\;\scriptstyle s''}&&{\mathcal g}\\ & & { \mathcal l}'\ar[d]^ { } \ar[ur]_{u ' } & & & \\ & & { { \mathcal j}}\times_{{{\mathcal k}}}{{\mathcal u}}'\times_{{{\mathcal u}}}{{\mathcal j}}'\times_{{\mathcal g}}{{\mathcal v}}\ar[uull]^{}="1 " \ar[uurrr]_{}="3 " & & & & } \ ] ] where @xmath538 . therefore , @xmath517 is categorical for @xmath3 . if @xmath13 is @xmath3-categorical , we have that the following diagram commutes up to homotopy : @xmath476^{i_{{{\mathcal u}}}}="0 " \ar[dl]_{{{\scriptstyle id}}}="2"&\\ { { { \mathcal u}}}&{{\mathcal l } } \ar[u]_{u } \ar[d]^{v}&{{\mathcal g}}\\ & { { { \mathcal u}}'}\ar[ul]^{{\epsilon}}="3 " \ar[ur]_{c}="1"\ar[r]^{}&{{\bullet^k}}\ar[u]^ { } \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8"_{h ' } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^{h } } \ ] ] then , for all @xmath539 the isotropy group @xmath540 injects into @xmath183 for @xmath541 and into @xmath542 for @xmath543 by proposition [ inj ] and remark [ isotropy ] . we have that @xmath544 . in particular , if the isotropy groups are finite , we have that @xmath545 divides @xmath546 for all @xmath547 . for instance , a categorical subgroupoid @xmath13 can not factor through a trivial group @xmath68 except in case that all the points in @xmath454 have trivial isotropy . let @xmath477 be the holonomy groupoid of the seifert fibration @xmath478 on the klein bottle @xmath68 . we can cover @xmath68 by two mbius bands @xmath0 as before . since the isotropy at the points in the center fiber is @xmath482 these points can not be moved by a homotopy to a point in the neighborhood with trivial isotropy . therefore this covering is minimal and the groupoid category is @xmath548 . we recall now the description of orbifolds as groupoids due to moerdijk and pronk @xcite . orbifolds were first introduced by satake @xcite as a generalization of a manifold defined in terms of local quotients . the groupoid approach provides a global language to reformulate the notion of orbifold . we follow the exposition in @xcite and @xcite . a groupoid @xmath3 is _ proper _ if @xmath497 is a proper map and it is a _ foliation _ groupoid if each isotropy group is discrete . an _ orbifold _ groupoid is a proper foliation groupoid . for instance the holonomy group of a foliation @xmath549 is always a foliation groupoid but it is an orbifold groupoid if and only if @xmath549 is a compact - hausdorff foliation . given an orbifold groupoid @xmath3 , its orbit space @xmath56 is a locally compact hausdorff space . given an arbitrary locally compact hausdorff space @xmath1 we can equip it with an orbifold structure as follows : an _ orbifold structure _ on a locally compact hausdorff space @xmath1 is given by an orbifold groupoid @xmath3 and a homeomorphism @xmath550 . if @xmath551 is an essential equivalence and @xmath552 is the induced homeomorphism between orbit spaces , we say that the composition @xmath553 defines an _ equivalent _ orbifold structure . an _ orbifold _ @xmath15 is a space @xmath1 equipped with an equivalence class of orbifold structures . a specific such structure , given by @xmath3 and @xmath554 is a _ presentation _ of the orbifold @xmath15 . if two groupoids are morita equivalent , then they define the same orbifold . therefore any structure or invariant for orbifolds , if defined through groupoids , should be invariant under morita equivalence . an _ orbifold map @xmath555 is given by an equivalence class of generalized maps @xmath408 from @xmath2 to @xmath3 between presentations of the orbifolds such that the diagram commutes : @xmath556 & x}\ ] ] a specific such generalized map @xmath408 is called a _ presentation _ of the orbifold map @xmath24 . our notion of morita homotopy gives a notion of homotopy for orbifolds : an _ orbifold homotopy _ between the orbifold maps @xmath557 is given by a morita homotopy between the presentations @xmath408 of @xmath24 and @xmath440 of @xmath275 . in other words , an orbifold homotopy is given by the following diagram @xmath558^{\phi}="0 " \ar[dl]_{\eta}="2"&\\ { { { \mathcal k}}}&{{\mathcal l } } \ar[u]_{u } \ar[d]^{v}&{{\mathcal g}}\\ & { { { \mathcal k}}''}\ar[ul]^{\nu}="3 " \ar[ur]_{\psi}="1 " & \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8"_{h ' } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^{h } } \ ] ] where @xmath2 to @xmath3 are presentations of the orbifolds @xmath559 and @xmath15 . since the groupoid @xmath374 has the same _ morita homotopy type _ as @xmath2 , we can always take a presentation of the orbifold homotopy @xmath560 given by a strict homotopy @xmath561 , where @xmath374 and @xmath2 define the same orbifold homotopy class of orbifolds . [ ex1 ] consider the orbifold @xmath15 having as a presentation groupoid the holonomy groupoid associated to the seifert fibration on the mbius band , @xmath423 , its underlying space is @xmath562 . the orbifold @xmath15 has the same orbifold homotopy type as the orbifold @xmath559 represented by the point groupoid @xmath424 . the morita homotopy equivalence is given by an orbifold map @xmath563 with presentation : @xmath425 as in example [ mht ] . the homotopic inverse @xmath564 is given by the generalized map @xmath426 . if @xmath3 is a presentation for the orbifold @xmath15 , we define the ls category of @xmath15 as the groupoid category of @xmath3 , @xmath16 . by proposition [ morita ] the ls category of a groupoid is invariant under morita equivalence , then this definition is independent of the chosen presentation for @xmath15 . the ls category of an orbifold @xmath15 is an invariant of the orbifold homotopy type which generalizes the classical ls category in case that the orbifold is a manifold . if the orbifold is _ effective _ the ls category for orbifolds as groupoids coincide with the ls category for orbifolds as defined by the author in @xcite using local charts . since a transitive groupoid is morita equivalent to a point groupoid , we have that @xmath565 if the underlying topological space @xmath566 of the orbifold @xmath15 is a point . for instance , we have that in example [ ex1 ] the orbifold category of @xmath15 is @xmath565 since the presentation @xmath423 has the same morita homotopy type of a point groupoid . in this case , the ordinary category of the underlying topological space @xmath562 , coincides with the orbifold category of @xmath15 , though in general , it is just a lower bound : let @xmath15 be an orbifold and @xmath1 its underlying topological space . then @xmath567 . let @xmath3 be a presentation groupoid for the orbifold @xmath15 , then @xmath56 is homeomorphic to @xmath1 . we will show that @xmath568 . let @xmath452 be an invariant open set such that @xmath13 is @xmath3-categorical . since @xmath44 and @xmath45 are open maps , we have that @xmath569 is an open map too . therefore , @xmath570 is an open set in @xmath56 . the @xmath145-homotopy between the inclusion and the constant map induces a homotopy @xmath571 in the orbit spaces . since @xmath137 is morita equivalent to the interval @xmath208 regarded as a unit groupoid , we have that @xmath572 . thus @xmath570 is categorical for @xmath573 , in the ordinary sense . the groupoids @xmath3 and @xmath2 in example [ tear ] define the _ teardrop _ orbifold @xmath15 . for the groupoid @xmath3 , we have a @xmath3-categorical covering given by @xmath574 and @xmath575 and for @xmath2 we can consider the @xmath2-categorical covering given by @xmath576 and @xmath577 where @xmath578 and @xmath579 are the cores of the solid tori . we have that @xmath580 . we recall now the notion of inertia groupoid @xcite and its decomposition in twisted and untwisted sectors . we show that the orbifold category of the twisted sectors provides a lower bound for the orbifold category of the orbifold . let @xmath3 be an orbifold groupoid . the inertia groupoid @xmath581 is a lie groupoid whose manifold of objects is given by @xmath582 we can think of the elements in @xmath583 as loops in @xmath3 . the manifold of arrows @xmath584 is given by the following pullback of manifolds : @xmath585^ { } \ar[d ] _ { } & s_{{\mathcal g}}\ar[d]^{\beta } \\ g_1\ar[r]^{s } & g_0}\ ] ] where @xmath586 sends a loop @xmath587 to its source ( or target ) @xmath588 . then an arrow between the loop @xmath587 and @xmath589 is given by an arrow @xmath590 such that @xmath591 . the source and target maps @xmath592 are given by @xmath593 and @xmath594 . if @xmath108 , then @xmath595 and there is a well defined notion of inertia for orbifolds . the orbifold @xmath596 is determined by the inertia groupoid @xmath597 corresponding to any presentation @xmath3 of @xmath15 . if @xmath598 is a translation groupoid , then the inertia groupoid is also a translation groupoid given by @xmath599 and the action of @xmath53 given by @xmath600 . [ dihedral ] 1 . the inertia orbifold for the teardrop orbifold @xmath15 in example [ tear ] is given by the disjoint union of itself and @xmath601 copies of the point @xmath602 . then the orbifold category of the inertia orbifold is given by @xmath603 . + 2 . given the action of the dihedral group @xmath604 on the sphere @xmath605 where @xmath109 acts by reflection and @xmath606 by rotation about the @xmath607-axis , consider the translation groupoid @xmath608 defining the orbifold @xmath15 . both poles @xmath602 and @xmath145 in @xmath609 are fixed by the action , then they have isotropy @xmath610 . there are four great circles whose points have isotropy @xmath487 generated by each of the symmetries @xmath611 and @xmath612 . the manifold of objects @xmath583 of the inertia groupoid @xmath581 is given by the disjoint union of @xmath605 plus @xmath613 copies of each @xmath602 and @xmath145 corresponding to the rotations @xmath614 and @xmath615 and the four great circles with non - trivial isotropy as shown in figure [ d ] + + the orbifold @xmath15 defined by @xmath3 is a disk orbifold with silvered boundary and two dihedral singular points on the boundary . the inertia orbifold @xmath616 given by the action of @xmath617 on @xmath583 is the disjoint union of the orbifolds shown in figure [ d4 ] : + + consider the covering @xmath618 by subgroupoids of @xmath3 given by @xmath619 and @xmath620 . this covering is categorical and @xmath580 . the category of each of the 1-dimensional orbifolds in the decomposition is @xmath601 . then the category of the inertia orbifold is @xmath621 . let @xmath622 be a @xmath3-saturated connected component in @xmath583 . if @xmath623 is the full subgroupoid over @xmath622 , we have that the partition of @xmath624 induces a decomposition of the inertia groupoid : @xmath625 as well as a decomposition of the inertia orbifold , @xmath626 . the components @xmath627 besides @xmath38 are called _ twisted sectors _ whilst @xmath38 is the _ untwisted sector_. the corresponding @xmath628 and @xmath629 are also called twisted sectors . if @xmath630 is a twisted sector of the orbifold @xmath15 , then @xmath631 . if @xmath13 is a @xmath3-categorical subgroupoid for @xmath3 , consider the subgroupoid @xmath632 in @xmath623 given by the following fibered product of groupoids : @xmath633 _ { } \ar[r]^{p_1}&{\mathcal g}^i \ar[d]^{\beta_i}\\ { { \mathcal u}}\ar[r]_{i_{{{\mathcal u } } } } & { \mathcal g}}\ ] ] where @xmath634 is the restriction to @xmath623 of @xmath635 given by @xmath636 on objects and @xmath637 on arrows . the inclusion functor @xmath475 is morita homotopic to a generalized constant map @xmath638 . consider the fibered product @xmath639 . we have that the inclusion @xmath640 is morita homotopic to a generalized constant map @xmath641 . in the classical lusternik - schnirelmann theory for topological spaces , we count the amount of categorical sets required to cover the space . there are no distinctions between the categorical sets since all of them factor through a point . in the groupoid context , we count subgroupoids which factor through point groupoids . we are going to distinguish our point groupoids by their _ weights _ and count categorical subgroupoids together with the weights of their associated point groupoids . this idea is inspired by the work of tom leinster on the euler characteristic of a finite category @xcite and based on the string - theoretic orbifold euler characteristic @xcite . let @xmath3 be an orbifold groupoid and @xmath13 be a @xmath3-categorical subgroupoid . let @xmath68 be a group of _ minimal order _ such that @xmath13 factors through @xmath642 . we define the weight of @xmath13 as the number @xmath643 of conjugacy classes in @xmath68 , @xmath644 . if @xmath68 is abelian , then the weight of @xmath13 is the order of @xmath68 , @xmath645 . a covering @xmath646 of @xmath3 is _ minimal _ if @xmath647 is @xmath3-categorical and @xmath648 . given a minimal covering @xmath646 and its associated set of weights @xmath649 we propose the following the _ weighted _ ls category , @xmath37 , of a lie groupoid @xmath3 is the least value of the sum : @xmath650 for all @xmath646 minimal coverings . 1 . consider the translation groupoid @xmath651 defining the teardrop orbifold @xmath15 as in example [ tear ] . for the categorical covering @xmath618 by subgroupoids of @xmath3 given by @xmath576 and @xmath577 we have that the associated set of weights is @xmath652 . this covering is minimal and gives the least sum , then @xmath653 . 2 . for the translation groupoid @xmath608 , consider the categorical covering @xmath618 by subgroupoids of @xmath3 given by @xmath619 and @xmath620 as in example [ dihedral ] ( 2 ) . both subgroupoids @xmath13 and @xmath517 factor through the group @xmath654 . the conjugation classes in @xmath654 are given by @xmath655 , then the class number is @xmath656 . therefore , a the set of weights associated to this covering is @xmath657 . we have that @xmath658 . we will show that the weighted category of an orbifold groupoid is an invariant of morita homotopy type , then in particular yields a well defined invariant for orbifolds : @xmath659 for any groupoid presentation @xmath3 . moreover , the weighted category for orbifolds is an invariant of orbifold homotopy type . if @xmath2 and @xmath3 are orbifold groupoids with @xmath421 , then @xmath660 . let @xmath661 and let @xmath646 be a minimal @xmath2-covering realizing the least sum , then @xmath662 where @xmath649 is the set of weights associated to @xmath646 . we construct a @xmath3-minimal covering @xmath663 as in the proof of proposition [ morita ] . by lemma [ lema ] we have that @xmath649 is also a set of weights for @xmath663 and it is minimal with this property . then @xmath664 . we conjecture that the weighted category of an orbifold groupoid coincides with the ( unweighted ) category of its inertia groupoid , @xmath665 . classically , the ls category of a manifold is a lower bound for the number of critical points of a smooth function under certain conditions . we propose a generalization of the notion of critical point for generalized maps and prove that the ls theorem holds for _ orbifold groupoids_. let @xmath670 be an arrow @xmath673 . from the following diagram @xmath674^{\phi_1 } \ar[d]_s \ar@<1ex>[d]^t & g_1\ar[d]_s \ar@<1ex>[d]^t\\ l_0 \ar[r]^{\phi_0 } & g_0}\ ] ] we have that @xmath675 . then @xmath676 since @xmath677 is critical for @xmath678 . since @xmath44 is a submersion , we have that @xmath679 has maximal rank and then @xmath680 . thus , @xmath275 is critical for @xmath672 . moreover , this implies that @xmath681 . since @xmath682 , then @xmath683 . we have that @xmath684 since @xmath45 has maximal rank . then @xmath685 is critical for @xmath678 . if @xmath690 is a natural transformation between @xmath154 and @xmath155 , the following diagrams are commutative @xmath691_s \\ l_0 \ar[r]^{\phi_0}\ar[ru]^{t } & g_0}\qquad\xymatrix { & g_1\ar[d]_t \\ l_0 \ar[r]^{\psi_0}\ar[ru]^{t } & g_0}\ ] ] and we have that @xmath692 and @xmath693 . if @xmath49 is critical for @xmath154 , then @xmath694 and since @xmath44 has maximal rank , @xmath695 . then @xmath696 and @xmath49 is critical for @xmath155 . since the following diagram commutes up to natural transformations @xmath705^{\phi}="0 " \ar[dl]_{{\epsilon}}="2"&\\ { { { \mathcal j}}}&{{{\mathcal a } } } \ar[u]_{\alpha } \ar[d]^{\beta}&{{\mathcal g}}\\ & { { \mathcal l}'}\ar[ul]^{{\epsilon}'}="3 " \ar[ur]_{\phi'}="1 " & \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{}"7 " ; " 8"_{\sim_{t } } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{}"5 " ; " 6"^{\sim_{t ' } } } \ ] ] we have that @xmath706 for all orbit subgroupoids @xmath707 . since @xmath699 is critical for @xmath408 , we have that there exists a subgroupoid @xmath701 for @xmath154 with @xmath702 and @xmath708 . consider the subgroupoid @xmath709 given by the following fibered product of groupoids : @xmath710^{p_1}\ar[d]_{p_3 } & { { \mathcal o}}'^{k'}\ar[d]^{i_{{{\mathcal o}}'^{k ' } } } \\ { { \mathcal a}}\ar[r]^{\alpha } & { \mathcal l}}\ ] ] then @xmath711 and @xmath712 since @xmath586 has maximal rank . taking @xmath713 we have that @xmath460 is critical for @xmath704 . let @xmath3 be a lie groupoid and @xmath714 be an invariant open set . consider the full subgroupoid @xmath715 over @xmath0 . the _ groupoid ls - category of @xmath715 in @xmath3 , @xmath716 , _ is the least number of @xmath3-categorical subgroupoids required to cover @xmath715 . the subgroupoid @xmath715 is @xmath3-deformable into @xmath717 in @xmath3 if there are groupoids @xmath718 and @xmath374 such that the following diagram commutes : @xmath719^{i_{{{\mathcal m}}}}="0 " \ar[dl]_{{{\scriptstyle id}}}="2"&\\ { { { \mathcal m}}}&{{\mathcal l } } \ar[u]_{u } \ar[d]^{v}&{{\mathcal g}}\\ & { { { \mathcal m}}'}\ar[ul]^{\eta}="3 " \ar[ur]_{\phi}="1"\ar[r]_{}&{{\mathcal n}}\ar[u]_{i_{{{\mathcal n } } } } \ar@{}"0";"1"|(.4){\,}="7 " \ar@{}"0";"1"|(.6){\,}="8 " \ar@{=>}"7 " ; " 8"_{h^s } \ar@{}"2";"3"|(.4){\,}="5 " \ar@{}"2";"3"|(.6){\,}="6 " \ar@{=>}"5 " ; " 6"^{f^s } } \ ] ] 1 . if @xmath720 , then @xmath721 . 3 . if @xmath715 is @xmath3-deformable into @xmath717 , then @xmath721 . if @xmath723 is an invariant closed set , then there exists an invariant open set @xmath454 with @xmath724 such that @xmath725 . we will say that the subgroupoid @xmath13 is a neighborhood of @xmath726 . if @xmath3 is an orbifold groupoid defining the orbifold @xmath15 , then for each @xmath727 there exist arbitrary small neighborhoods @xmath454 of @xmath49 for which @xmath13 is isomorphic to the action groupoid @xmath728 @xcite . the subgroupoid @xmath13 determines an open set @xmath729 , for which @xmath730 is the quotient of @xmath454 by the action of the finite group @xmath54 . let @xmath731 be a morphism of lie groupoids , where @xmath732 is the unit groupoid over the real numbers . let @xmath2 be the critical subgroupoid of @xmath154 , i.e. the subgroupoid given by the union of all critical orbit subgroupoids : @xmath733 let @xmath734 be the set of critical values and @xmath735 the set of regular values . consider the full subgroupoids @xmath736 $ ] and @xmath737 . we will impose the following @xmath3-deformation conditions on the groupoid @xmath3 and the morphism @xmath154 : 1 . for all @xmath22 in the interior of the set of regular values @xmath738 there exists an @xmath739 such that @xmath740 is @xmath3-deformable into @xmath741 . 2 . for any isolated critical value @xmath22 and @xmath13 neighborhood of @xmath21 , there is an @xmath739 such that @xmath742 is @xmath3-deformable into @xmath741 . if @xmath743 then there is an @xmath739 such that @xmath3 is @xmath3-deformable into @xmath744 . 1 . the function @xmath46 is ( weakly ) increasing ; 2 . @xmath46 is locally constant in the interior of the set of regular values @xmath738 ; 3 . at any isolated critical value @xmath22 of @xmath154 the function @xmath46 jumps by @xmath748 at most and 4 . when @xmath743 then @xmath749 . given an orbifold @xmath15 and an orbifold map @xmath751 , we will say that a point @xmath752 is _ critical _ if there is a presentation groupoid @xmath3 for @xmath15 and a presentation generalized map @xmath19 for @xmath751 such that @xmath49 is the image in the quotient @xmath753 of some critical orbit subgroupoid @xmath754 for the generalized morphism @xmath408 . we define the _ tangent groupoid _ @xmath756 of the groupoid @xmath3 as a groupoid whose manifold of objects and arrows are @xmath757 and @xmath758 respectively . source and target are given by the differentials of @xmath759 which are also submersions : @xmath760_{ds } \ar@<1ex>[d]^{dt}\\ tg_0}\ ] ] the _ length _ @xmath779 of a @xmath3-curve @xmath109 is the sum of the lengths of the paths @xmath780 . we have that @xmath781 the sum of the lengths of any other branch of the multiple @xmath3-path will give the same length . the _ distance _ between two orbits @xmath459 and @xmath782 is the infimum of the the lengths of @xmath3-paths between the orbifold subgroupoids @xmath460 and @xmath783 . this distance defines a metric which is complete if the orbifold is compact . let @xmath1 be a @xmath3-vector field on @xmath3 . we will say that @xmath109 is an _ integral @xmath3-curve _ for @xmath1 if @xmath784 . if @xmath237 is in @xmath767 , we call @xmath785 the initial subgroupoid of the integral @xmath3-curve . for a compact orbifold groupoid , we have that for each orbit subgroupoid @xmath460 there is a maximal integral @xmath3-curve @xmath786 for @xmath1 whose domain is given by @xmath787 and the initial subgroupoid is @xmath460 . given a morphism @xmath794 where @xmath3 is an orbifold groupoid defining a compact orbifold , consider the @xmath3-vector field @xmath795 given by the gradient @xmath796 . then the flow of the @xmath3-vector field @xmath795 gives the morita homotopy required by the @xmath3-deformation conditions . then an orbifold groupoid @xmath3 defining a compact orbifold @xmath15 is in the hypothesis of corollary [ ls ] and the statement of the theorem follows .

we propose a new homotopy invariant for lie groupoids which generalizes the classical lusternik - schnirelmann category for topological spaces . we use a bicategorical approach to develop a notion of contraction in this context . we propose a notion of homotopy between generalized maps given by the 2-arrows in a certain bicategory of fractions . this notion is invariant under morita equivalence . thus , when the groupoid defines an orbifold , we have a well defined ls - category for orbifolds . we prove an orbifold version of the classical lusternik - schnirelmann theorem for critical points .

introduction recollections on lie groupoids and equivalences homotopy the morita homotopy bicategory of lie groupoids lusternik-schnirelmann category orbifolds as groupoids critical points

arxiv

studying the kinematics of the satellites of the milky way ( mw ) allows us to adddress various fundamental issues such as the origin and evolution of this satellite system ( and the mw itself ) , the role of tidal interactions in the evolution of the local group ( lg ) and the matter distribution of the latter ( including that of the dark matter , thus allowing tests of some cosmological predictions @xcite ) . this requires tracing their positions back in time , by integrating their orbits , which , in turn , requires knowing their current positions and their full 3-d space velocities . while radial velocities for lg galaxies are known to better than @xmath35 km s@xmath5 , and their distances to @xmath310% , the biggest source of uncertainty rests on their proper motions ( pms ) . in the year 2000 we started a ground - based program aimed at determining the absolute pm of three southern dwarf spheroidal ( dsph ) galaxies , carina , fornax and sculptor , with respect to background quasars ( qsos ) , used as inertial reference points . three to four epochs of homogeneous ccd data were obtained using a single telescope+detector setup over a period of eight years : ours is the first entirely optical ccd / ground - based pm study of an external galaxy other than the magellanic clouds . in mndez et al . ( 2010 , hereafter paper i ) , we presented a detailed description of our methods , as well as our first results for the pm of fornax , based on one qso field ( qj0240 - 3434b ) . in this paper we present our final pm for fornax based on measurements from five qso fields in the background of fornax . in section [ obs ] we present a summary description of our observational material , in section [ pms ] we explain how we obtained our pms , and , finally , in section [ comp ] we compare our results to previous studies and present our main conclusions . the fornax dsph galaxy , at a distance of 147 kpc ( e.g. , pietrzyski et al . 2009 ) , is relatively isolated , luminous and well - resolved into individual stars . it seems to be dark matter dominated , and has an estimated total mass of @xmath6 m@xmath7 ( e.g. , walker et al . 2007 ) . unlike other dwarf galaxies , fornax harbours five globular clusters @xcite , and it appears to have a complex stellar substructure in the form of shell - like features indicative of recent merger activity ( coleman et al . 2004 , coleman & da costa 2005 , olszewski et al . furthermore , results by @xcite suggest that the ancient stellar population in the centre of fornax is not in equilibrium ( apparent as a non - gaussian , double peaked velocity distribution ) , which also points to a relatively recent accretion of external material , such as gas accretion due to the merger with another smaller stellar system . all observations were carried out with the `` superb seeing imager '' , susi2 , attached to one of the nasmyth focii of the eso 3.5 m ntt telescope at la silla observatory in chile . the overall characteristics of the detector are fully described in @xcite and @xcite . we followed exactly the same observational , reduction , and calibration procedures for all of our qso fields as described in paper i. therefore , in as much as possible , our dataset is homogeneous from this point of view ( for certain limitations to this statement , see section [ pms ] ) . susi2 is a mosaic of two 2k@xmath84k eev 44 - 82 chips ( called # 45 , on the west side , and # 46 , on the east side , for the adopted rotator angle of 0 ) . as explained in paper i , we always placed our qsos at a nominal pixel position near @xmath9 , close to the middle of chip # 46 , and only data from this chip was subsequently used for our astrometry . all of the astrometric observations , including those required to compute the differential chromatic constants ( dcr - see paper i , section 3.3 . ) , were acquired through a bessel r filter , whereas for the blue frames needed to construct color - magnitude diagrams ( cmds ) we used the bessel b filter , both of which are part of the standard set of filters available for susi2 . our initial list of qsos in the background of fornax comprised all eleven distinct qsos ( one , reported in paper i , forms a gravitational lens pair ) reported by @xcite ( their table 5 ) and @xcite ( his table 1 ) . unfortunately , several of them proved to be useless for astrometry : deep , good - seeing ( [email protected] arcsec ) images taken during the first epoch revealed that they were either too faint for our required astrometric @xmath10 ( @xmath11 integrated over the psf fitting radius ; see paper i , section 3.2 and figure 4 ) , had a noticeably elongated structure ( and therefore had a different , usually more extended , psf than that of the stars ) , had a very nearby ( usually stellar - like ) bright companion , or were definitely blended with field stars . as explained at length in paper i , any of these features render these targets unsuitable for high - accuracy relative astrometry . these problematic qsos were dropped from the observing list in subsequent epochs , and we concentrated on `` clean '' ( as far as we could determine with a seeing of @xmath30.5 arcsec ) qsos . our final list of five qsos , from which we were able to determine the pm of fornax , is given in table [ tbl1 ] , where we also show the full list of known qsos behind fornax from the above - cited references . in figure [ fields ] we show the stellar configuration in the immediate surroundings of each of our five selected qsos . all of these images were acquired on the first epoch , when we consistently had very good seeing ( [email protected] arcsec ) . the only potentially problematic case was that of qj0239 - 3420 which has a faint companion to the nw , at a distance of @xmath12 arcsec . fortunately , this nearby source is very dim ( with a star / qso peak brightness ratio smaller than 0.1 ) , and far enough ( farther than @xmath13 , adopted as the psf fitting radius , see paper i , section 3.2 and figure 4 ) that , even on our worse frames ( with a fwhm of @xmath14 arcsec ) , it did not pose a problem for the astrometric solution , and so it was fully included in our analysis below . table [ tbl2 ] contains a summary of the observational material that was , in the end , used to compute our pms . we note that more data was acquired for these qso fields , but proved to be useless on account of bad seeing , poor image quality ( deteriorating the astrometric solution ) , or bad sky transparency . as explained in paper i , at the start of our program we adjusted the exposure time on the basis of seeing ( typical values were between 300 s and 900 s ) , but later , it was decided to use a fixed integration time of 900 s in all cases for simplicity . to determine the pms we used the same procedure described at length in paper i. a flow - chart summary of the full process is presented in figure [ flow ] . we refer the reader to paper i for further details on the methodology and precise meaning of all steps . even though in our pm solutions we only included frames acquired within @xmath15 hr . from the meridian ( see table [ tbl2 ] ) , all of our coordinates were ( pre)-corrected for ( continuous ) atmospheric refraction and dcr as described in paper i. we also excluded from the pm solution all frames with a @xmath16 arcsec , as they clearly deteriorated the linear fit of baricentric position _ vs. _ epoch diagram ( see , e.g. , fig . 12 in paper i ) , where these frames stand - out due to their large scatter . all pm data were treated as homogenously as possible , including the following constraints : * all stars with @xmath17 @xmath1 in our initial local reference system ( lrs , basically a set of `` high - quality '' reference stellar images , eventually bona - fide fornax stars - see paper i ) , indicating that they are either foreground galactic stars , or that they have a high ( pseudo)-pm , possibly due , e.g. , to a faint unresolved companion or another problem in the image ) , were purged from the lrs . the full geometric registration ( and pms ) for the remaining lrs stars and the qso were re - computed in an iterative process , * all lrs stars that exhibited a registration residual @xmath18 of the formal rms of the two - dimensional geometric registration in the @xmath19 or @xmath20 coordinates in at least four ( not necessarily consecutive ) frames , were eliminated from the lrs . for registration we used a full 3@xmath21 order polynomial fit , which has been justified in paper i , * all lrs stars with @xmath22 @xmath1 were excluded from the lrs . the cut value , @xmath23 , was estimated for each field based on the distribution of pm errors _ vs. _ magnitude ( see table [ tbl3 ] ) , * lrs stars exhibiting a pm error larger than that of the bulk of the lrs stars at a given magnitude ( even if they , individually , had @xmath24 ) were eliminated . this cut was done visually , on plots of pm error vs. magnitude for each qso field , * finally , we expunged objects that in the cmd appeared not to belong to fornax . this photometric cleansing was done by first calibrating our instrumental photometry following the procedure described in paper i ( section 4 ) , and then by comparing our resulting cmd with that of @xcite , which defines the main features of the fornax cmd . in table [ tbl3 ] we show a summary of what results from applying the above criteria to the initial set of lrs stars for each of our five qso fields . after applying all previously described cuts , we verified that we ended with a uniform @xmath25 distribution of the lrs stars ( see , e.g. , figure 15 in paper i ) , and with a reasonable distribution in magnitude and color . the resulting cmds for the lrs stars and the respective qso are shown in figure [ cmd ] . we note that in paper i we calibrated our photometry approximately by adopting a color of @xmath26 for the red - clump from @xcite and the qso blue magnitude from the works by @xcite and @xcite . however , we noticed that , probably as result of uncertainties in the blue photographic magnitudes for the qsos ( and/or possible qso variability ) , the ordinate in these figures had a large zero - point variation from field to field , and hence the red - clump did not fall at the same apparent @xmath27 magnitude for the five fields , as is expected . we therefore decided to adopt a calibration based on the color of the red - clump ( as before ) , but fixing , instead , the magnitude of the red - clump at @xmath28 , also from the photometry by stetson ( 1997 , his figure 7 ) . in figures [ posepoch ] and [ vpd ] we show , respectively , the barycentric position _ vs. _ epoch and the vector - point diagrams for our five fields , for the final lrs stars and the corresponding background qso . after processing all of the qso fields following the protocols described in the previous paragraphs , we computed pms for the five qso fields presented in the upper part of table [ tbl1 ] . the results are summarized in table [ tbl4 ] , and are plotted in figure [ pmind ] . table [ tbl4 ] gives the pm in ra and dec as well as the overall rms of the linear fit of the barycentric coordinates _ vs. _ epoch . we recall that the slope of this linear fit gives us the pm of the fornax stars with respect to the qso , as explained in paper i. figure [ pmind ] clearly shows that our five measurements scatter more widely than would be expected from their error bars , and therefore computing a weighted mean does not seem appropriate for these data . instead , and not knowing the source of the additional uncertainty affecting our proper motion measurements , we opted for computing an unweighted mean over the resulting values for all five qso fields . from table [ tbl4 ] we thus arrive at the following values for the pm of the fornax dsph galaxy ( and the standard deviation of the mean ) : @xmath29 @xmath1 , and @xmath30 @xmath1 . only two astrometric determinations of the pm for the fornax dsph galaxy are available , namely that by @xcite , based on a combination of ground - based plates and hubble - wfpc data , and that based exclusively on hst data by piatek et al . ( 2007 , hereafter pi07 ) , who present revised values to those reported earlier ( pre - ccd charge transfer inefficieny corrections ) in @xcite . figure [ pmind ] compares our results from individual qsos , to those obtained by pi07 . we have three qso fields in common with pi07 ; their values , along with our measurements , are given in table [ tbl5 ] . from this table we see that the difference of the mean pms derived from the qsos in common between these two studies is less than @xmath31 of our error in the mean in both , ra and dec . on the other hand , the overall weighted mean value reported by pi07 , based on 4 qsos ( their table 3 ) of @xmath32 @xmath1 , and @xmath33 @xmath1 differs by about @xmath31 from our overall unweighted - mean pm values . even though our results are not statistically inconsistent with those of pi07 , it is apparent from figure [ pmind ] that our individual results do exhibit a much larger scatter than their results ( @xmath34 @xmath1 and @xmath35 @xmath1 vs. @xmath36 @xmath1 and @xmath37 @xmath1 for pi07 ) . we have no explanation for these differences . the pm values published so far , along with our own value are given in table [ tbl6 ] . more recently , @xcite have used a non - astrometric method called `` perspective rotation '' which is also included in the table ( for details about this method see paper i ) , because it provides a completely independent measurement of the tranverse motion of fornax ( albeit with a larger error than the more recent astrometric determinations ) . all of these values are plotted , for comparison , in figure [ pmwei ] . from the plot and figure we can see that , in general , there is a good agreement between all measurements ; indeed , none of them depart by more than @xmath38 from the straight average of the values in table [ tbl6 ] , with our result being however the most extreme in this sense . also , averaging the results from all of the authors , we find that the pm in dec shows a larger scatter ( @xmath39 @xmath1 ) than that in ra ( @xmath40 @xmath1 ) . this would suggest that ( some of ) these measurements are affected by unaccounted systematic effects . of the obvious culprits , we can mention that ground - based astrometric data are affected by dcr . in the case of fornax data ( dec@xmath41 ) acquired from the southern hemisphere , dcr should however mostly affect ra pms ( we stress that our data have been corrected for this effect as far as possible , see paper i ) . on the other hand , hst data , while not affected by dcr , can be affected by `` charge transfer inefficiency '' ( cti ) in the ccd detectors due to the very low sky background which is insufficient to fill - in the empty charge traps on the detector . these traps evolve in time because they are produced by in - flight radiation damage , and induce systematic ( time dependent ) position shifts in the detected sources ( for details see , e.g. , @xcite , especially his figure 10 ) . hst data has also been corrected for this effect , again , as best as they could ( compare , e.g. , table 3 from @xcite ( cti - uncorrected ) and pi07 ( cti - corrected ) ) . we note however that , since the hst cameras were not oriented exactly along dec in the parallel readout direction , which is the direction mostly affected by cti ( see , e.g. , figures 2 , 3 and 4 in pi07 ) , it is difficult to ascribe the dec scatter to this problem . also , as mentioned previously , hst data shows a much smaller intrinsic scatter than our measurements ( figure [ pmind ] ) , thus suggesting small remaining systematics due to cti . one of the ultimate goals of these studies is a determination of a reliable orbit for fornax . it is interesting to compare the impact of the different values given in table [ tbl6 ] on the kinematical and orbital parameters that can be derived from these measurements . table [ tbl7 ] gives the heliocentric pms and the corresponding tangential velocities in galactic coordinates for the different values reported in table [ tbl6 ] . from this table , it is clear that our pm value will render one of the most energetic orbits yet derived for fornax . for illustration purposes , and also as a key ingredient to compute and interpret the orbit of fornax from the above motions in galactic coordinates , we have computed velocities in the heliocentric ( hc ) and galactocentric ( gc ) reference systems . for a detailed description of these various reference systems , and the equations relating them , the reader is referred to @xcite . they are shown in table [ tbl8 ] , along with model - independent kinematical and orbital parameters derived from the pm values taken from different studies . given the large values of @xmath42 in comparison with the @xmath43 derived from all pm measurements , it is clear that fornax must be close to perigalaticon at this time . also , given the rather small ratio of @xmath44 it is clear that there must be significant excursions of fornax away from the galactic plane , regardless of the adopted pm values . we note again that , of all measurements available , our pm measurement yields the most extreme orbit . we have computed galactic orbits for fornax by integrating back in time the equation of motion under a realistic three - component ( disk , halo and spheroid ) model galactic potential @xcite , and using the current position and velocity ( and their uncertainties , in a monte carlo scheme ) as initial conditions . details of the integrator , and the monte carlo simulations , are discussed in @xcite . a particularly important feature of the integrator used is its care to conserve energy and total angular momentum , obvious features that are howevever sometimes tricky to achieve over the full orbit , particularly when the number of integration steps ( @xmath3several thousands in our case ) is large . the adopted gravitational potential is strictly axi - symmetric , and it does not include the galactic bar , nor the mw spiral pattern . recent numerical simulations by @xcite , that include these non axi - symmetric components , indicate however that their effect , in particular in the orbits of galactic globular clusters ( and therefore also in the case of the more external satellite galaxies ) , is minimal , and it should not alter our conclusions importantly . the results of these integrations are shown in table [ tbl9 ] and in figure [ xyz ] . in table [ tbl9 ] the meaning of the different columns is rather obvious , except perhaps for the last two columns that correspond to the crossing ( vertical ) velocity of fornax through the galactic plane ( @xmath45 ) and the total speed at perigalacticon ( @xmath46 ) . as it can be seen , regardless of what pm values one adopts , comparing the @xmath46 values to the @xmath42 in table [ tbl8 ] confirms that the current fornax position lies indeed near perigalacticon ( see also figure [ dgct ] , top panel ) . actually , all pms indicate that the minimum distance , projected into the galactic plane , happened 200 to 300 myr ago ( figure [ dgct ] , bottom panel ) . this is a very interesting quantity , because @xcite have found evidence of at least three distinct stellar components in fornax : a young population ( few 100 myr old ) concentrated in the centre of the galaxy , visible as a main sequence in the cmd ; an intermediate age population ( 2 - 8 gyr old , possibly related to a shell structure in fornax , described in the next paragraph ) ; and an ancient population ( @xmath47 gyr ) . more recently , @xcite have also found ( see , e.g. , their figure 2 ) evidence of enhanced star formation in the range 200 - 300 myr . one could therefore conclude that the latest episode of star formation on fornax may have been indeed triggered by its perigalacticon passage . when tracing the galaxy back in time , the uncertainty in the computed position increases as time goes on , for a given uncertainty in the initial conditions , and , as we extrapolate further back in time , different pm values lead to quite different orbits . this is clearly shown in figure [ dgct ] , which shows the distance of fornax from the galactic center ( @xmath48 , top panel ) and the same distance projected onto the galactic plane ( @xmath49 , bottom panel ) as a function of time from now . initially all pm values produce a similar @xmath48 , @xmath49 _ vs. _ time , but the solutions diverge afterwards . as mentioned before , our value renders the most extreme solution , basically indicating that in a hubble time fornax has not completed an orbit yet . this is at odds with all of the other solutions which , while being different among themselves , do indicate nevertheless several perigalacticon passages in the last 10 gyr . if , as argued before , perigalacticon has had an influence on enhancing star formation , the stellar population results by @xcite favour a rather long orbital period , thus supporting our solution . we note here that the 2 gyr population belonging to the shell structure found by @xcite , @xcite , and @xcite has been interpreted by these authors as the product of a merger with a smaller , gas - rich system , and may not bear any relation to the interaction between fornax and the mw . we are thus seemingly left with two significant star formation episodes , one at ( 200 - 300 myr ) , close to perigalacticon , and another one at age @xmath47 gyr , which again argues for a longer orbital period ( interestingly , from figure [ dgct ] we see that @xmath50 gyr mark the most recent apogalactic position for fornax for the pi07 and @xcite orbits ) . our extended orbit implies that fornax would belong to one of the `` hypervelocity '' satellites of the mw , as argued by @xcite , @xcite , @xcite ( results all based on the same hst data set ) . one could perhaps wonder that the late infall nature of fornax s orbits ( assuming our pms ) is possibly related to the fact that it is the only satellite dwarf galaxy of the mw ( mateo 1998 ) , along with the spatially dissipated sagittarius dwarf ( law & majewski 2010 ) , that harbour a globular cluster population . a final word of caution regarding the above discussion is that , from table [ tbl9 ] , we see that current pm measurements for fornax implies that the range of derived orbital parameters is quite broad , and it seems adventurous to extract strong conclusions from them . as already mentioned , of particular concern is the possible existence of yet unaccounted systematic effects in the pm measurements . only high - accuracy future astrometric satellite measurements , like gaia , with expected uncertainties of a few @xmath51arc - sec ( average over many stars ) could help resolve this issue . ram and ec acknowledge support by the fondo nacional de investigacin cientfica y tecnolgica ( fondecyt project no . 1070312 ) , the chilean centro de astrofsica ( fondap project no . 15010003 ) and the chilean centro de excelencia en astrofsica y tecnologas afines ( pfb 06 ) . mhp acknowledges support by project # 4721 - 09 from universidad de tarapac . cg acknowledges support by the instituto de astrofsica de canarias ( p3 - 94 ) and by the ministry of education and research of the kingdom of spain ( aya2004 - 06343 ) . ram acknowledges support on various stages of reduction process by dr . matias radiszcz . we are greatful to the eso opc for their continued support of this long - term program , as well as to the la silla scientists , engineers and operations staff for their continuous help in the course of the program , especially dr . michael f. sterzik and mr . federico fox . we are also greatful to dr . dana casseti - dinescu , who ran the orbit simulations for our data using her code . we are also very gratefull to an anonymous referee for many useful comments . allen , c. , moreno , e. , & pichardo , b. 2008 , , 674 , 237 battaglia , g. , et al . 2006 , , 459 , 423 besla , g. , kallivayalil , n. , hernquist , l. , robertson , b. , cox , t. j. , van der marel , r. p. , & alcock , c. 2007 , , 668 , 949 bovy , j. , hogg , d. w. , & rix , h .- w . 2009 , , 704 , 1704 bristow , p. , kerber , f. , & rosa , m. r. 2006 , the 2005 hst calibration workshop : hubble after the transition to two - gyro mode , 299 ( document available at http://www.stsci.edu/hst/hst_overview/documents/calworkshop/workshop2005/papers/cws05proc.pdf ) brunthaler et al . 2011 , arxiv:1102.5350 coleman , m. , da costa , g. s. , bland - hawthorn , j. , martnez - delgado , d. , freeman , k. c. , & malin , d. 2004 , , 127 , 832 coleman , m. g. , & da costa , g. s. 2005 , , 22 , 162 cokunov glu , b. , et al . 2011 , , 412 , 1237 costa , e. , mndez , r. a. , pedreros , m. h. , moyano , m. , gallart , c. , nol , n. , baume , g. , & carraro , g. 2009 , , 137 , 4339 costa , e. , mndez , r. a. , pedreros , m. h. , moyano , m. , gallart , c. , nol , n. 2011 , , in press dehnen , w. , & binney , j. j. 1998 , , 298 , 387 dinescu , d. i. , keeney , b. a. , majewski , s. r. , & girard , t. m. 2004 , , 128 , 687 dodorico , s. , 1998 , the eso messenger , 91 , 14 dodorico , s. , beletic , j. w. , amico , p. , hook , i. , marconi , g. , & pedichini , f. 1998 , , 3355 , 507 hodge , p. w. 1961 , aj , 66 , 83 johnston , k. v. , spergel , d. n. , & hernquist , l. 1995 , , 451 , 598 kallivayalil , n. , van der marel , r. p. , & alcock , c. 2006 , , 652 , 1213 kirby , e. n. , cohen , j. g. , smith , g. h. , majewski , s. r. , sohn , s. t. , & guhathakurta , p. 2011 , , 727 , 79 law , d. r. , & majewski , s. r. 2010 , , 718 , 1128 mateo , m. 1998 , , 36 , 435 mndez , r. a. , costa , e. , pedreros , m. h. , moyano , m. , altmann , m. , & gallart , c. 2010 , , 122 , 853 . paper i miyamoto , m. , & soma , m. 1993 , , 105 , 691 olszewski , e. w. , mateo , m. , harris , j. , walker , m. g. , coleman , m. g. , & da costa , g. s. 2006 , , 131 , 912 piatek , s. , et al . 2002 , , 124 , 3198 piatek , s. , pryor , c. , bristow , p. , olszewski , e. w. , harris , h. c. , mateo , m. , minniti , d. , & tinney , c. g. 2007 , , 133 , 818 . pi07 piatek , s. , pryor , c. , & olszewski , e. w. 2008 , , 135 , 1024 pietrzyski , g. , grski , m. , gieren , w. , ivanov , v. d. , bresolin , f. , & kudritzki , r .- 2009 , , 138 , 459 shaya , e. , et al . 2009 , astro2010 : the astronomy and astrophysics decadal survey , 2010 , 274 stetson , p. b. 1997 , baltic astronomy , 6 , 3 tinney , c. g. , da costa , g. s. , & zinnecker , h. 1997 , , 285 , 111 tinney , c. g. 1999 , , 303 , 565 van den bergh , s. 1999 , the stellar content of local group galaxies , iau symposium , 192 , 3 walker , m. g. , mateo , m. , olszewski , e. w. , gnedin , o. y. , wang , x. , sen , b. , & woodroofe , m. 2007 , , 667 , l53 walker , m. g. , mateo , m. , & olszewski , e. w. 2008 , , 688 , l75 ccccccc qj0238 - 3443 & 02:38:43.8 & -34:43:53 & 20.2 & 22.9 & 222 + qj0238 - 3440 & 02:38:55.6 & -34:40:45 & 20.1 & 19.0 & 223 + qj0239 - 3420 & 02:39:49.0 & -34:20:00 & 20.6 & 07.3 & 344 + qj0240 - 3434b & 02:40:08.2 & -34:34:22 & 19.9 & 07.6 & 166 & reported in paper i + qj0240 - 3438 & 02:40:38.7 & -34:38:58 & 20.2 & 14.5 & 146 + qj0239 - 3425 & 02:39:32.9 & -34:25:25 & 20.4 & & & no finding chart available + qj0239 - 3421 & 02:39:36.9 & -34:21:30 & 19.9 & & & fuzzy , elongated qso image , with a fwhm 48% larger than that of stars in the fov + qj0240 - 3434a & 02:40:07.7 & -34:34:20 & 19.1 & & & blended to foreground star + qj0240 - 3437 & 02:40:19.0 & -34:37:20 & 17.9 & & & qso exhibits complex structure + qj0241 - 3420 & 02:41:57.9 & -34:20:49 & 20.4 & & & no finding chart available , far from fornax s center + qj0242 - 3426 & 02:42:06.5 & -34:26:12 & 21.8 & & & faint and far from fornax s center + qj0242 - 3424 & 02:42:19.9 & -34:24:20 & 21.5 & & & faint and far from fornax s center + qj0238 - 3443 & 2000.98 ... 2008.82 & 4 & 09 & 09 & 0.68 ... 2.99 + qj0238 - 3440 & 2000.98 ... 2008.82 & 4 & 19 & 13 & 0.36 ... 3.86 + qj0239 - 3420 & 2000.99 ... 2007.87 & 3 & 14 & 12 & 0.55 ... 3.70 + qj0240 - 3434b & 2000.61 ... 2007.85 & 3 & 15 & 13 & -0.65 ... -4.03 + qj0240 - 3438 & 2000.61 ... 2008.83 & 4 & 20 & 10 & -0.74 ... -3.62 + cccccc qj0238 - 3443 & 295 & 226 & 0.78 & 3 & 12 + qj0238 - 3440 & 217 & 175 & 0.60 & 6 & 7 + qj0239 - 3420 & 337 & 250 & 0.60 & 12 & 11 + qj0240 - 3434b & 260 & 217 & 0.50 & 11 & 7 + qj0240 - 3438 & 156 & 123 & 0.60 & 0 & 9 + ccccc qso i d & @xmath52 & @xmath53 & @xmath54 in ra & @xmath54 in dec + & [ @xmath1 ] & [ @xmath1 ] & [ mas ] & [ mas ] + qj0238 - 3443 & @xmath56 & @xmath57 & 1.5 & 1.2 + qj0238 - 3440 & @xmath58 & @xmath59 & 1.6 & 2.0 + qj0239 - 3420 & @xmath60 & @xmath61 & 1.8 & 2.8 + qj0240 - 3434b & @xmath62 & @xmath63 & 0.9 & 1.3 + qj0240 - 3438 & @xmath64 & @xmath65 & 2.8 & 2.7 + unweighted mean & @xmath66 & @xmath67 & & + ccccc qso i d & @xmath52 & @xmath53 & @xmath52 & @xmath53 + & [ @xmath1 ] & [ @xmath1 ] & [ @xmath1 ] & [ @xmath1 ] + & & + & & + qj0238 - 3443 & @xmath68 & @xmath69 & @xmath56 & @xmath57 + qj0240 - 3434b & @xmath70 & @xmath71 & @xmath62 & @xmath63 + qj0240 - 3438 & @xmath72 & @xmath73 & @xmath64 & @xmath65 + unweighted mean & @xmath74 & @xmath75 & @xmath76 & @xmath75 + cccccc @xcite & @xmath77 & @xmath78 & @xmath79 & @xmath80 & plates , 48 galaxies and 8 qsos + @xcite & @xmath81 & @xmath82 & @xmath83 & @xmath84 & hst+pc2+stis , 4 qso fields + this work & @xmath85 & @xmath67 & @xmath86 & @xmath87 & ntt+susi2 , 5 qso fields + @xcite & @xmath88 & @xmath89 & @xmath90 & @xmath91 & radial velocities , `` perspective rotation '' + cccccc reference & @xmath92 & @xmath93 & @xmath94 & @xmath95 & @xmath96 + & [ @xmath1 ] & [ @xmath1 ] & [ km s@xmath5 ] & [ km s@xmath5 ] & [ km s@xmath5 ] + @xcite & @xmath98 & @xmath99 & @xmath100 & @xmath101 & @xmath102 + @xcite & @xmath103 & @xmath104 & @xmath105 & @xmath106 & @xmath107 + this work & @xmath108 & @xmath109 & @xmath110 & @xmath111 & @xmath112 + @xcite & @xmath113 & @xmath114 & @xmath115 & @xmath116 & @xmath117 + @xcite & @xmath118 & @xmath119 & @xmath120 & @xmath121 & @xmath122 & @xmath123 & @xmath124 & 35 & 38 & -3.9 + @xcite & @xmath125 & @xmath126 & @xmath80 & @xmath127 & @xmath128 & @xmath129 & @xmath130 & 23 & 31 & 6.1 + this work & @xmath131 & @xmath132 & @xmath133 & @xmath134 & @xmath135 & @xmath136 & @xmath137 & 66 & 53 & 13.3 + @xcite & @xmath138 & @xmath139 & @xmath140 & @xmath141 & @xmath142 & @xmath143 & @xmath144 & 18 & 27 & 1.1 + @xcite & @xmath145 & @xmath146 & @xmath147 & @xmath148 & @xmath149 & @xmath150 & @xmath151 & @xmath152 & @xmath153 + @xcite & @xmath154 & @xmath155 & @xmath156 & @xmath157 & @xmath158 & @xmath159 & @xmath160 & @xmath161 & @xmath162 + this work & @xmath163 & @xmath164 & @xmath165 & @xmath166 & @xmath167 & @xmath168 & @xmath169 & @xmath170 & @xmath171 + @xcite & @xmath172 & @xmath173 & @xmath174 & @xmath175 & @xmath176 & @xmath177 & @xmath178 & @xmath179 & @xmath180 +

we present the first entirely ground - based astrometric determination of the proper motion for the fornax local group dwarf spheroidal satellite galaxy of the milky way , using ccd data acquired with the eso 3.5 m ntt telescope at la silla observatory in chile . our unweighted mean from five quasar fields in the background of fornax , used as fiducial reference points , leads to @xmath0 @xmath1 , and @xmath2 @xmath1 . a detailed comparison with all previous measurements of this quantity seems to imply that there is still no convincing convergence to a single value , perhaps indicating the existence of unnacounted systematic effects in ( some of ) these measurements . from all available proper motion and radial velocity measurements for fornax , we compute fornax s orbital parameters and their uncertainty using a realistic galactic potential and a monte carlo simulation . properties of the derived orbits are then compared to main star formation episodes in the history of fornax . all published proper motion values imply that fornax has recently ( 200 - 300 myr ago ) approached perigalacticon at a distance of @xmath3150 kpc . however , the derived period exhibits a large scatter , as does the apogalacticon . our orbit , being the most energetic , implies a very large apogalactic distance of @xmath4 kpc . if this were the case , then fornax would be a representative of an hypervelocity mw satellite in late infall .

introduction observations proper motions discussion and outlook

arxiv

low surface brightness ( lsb ) galaxies are characterized by low star formation rate @xcite and low disk surface density @xcite . the spiral structure in lsbs is often incipient or fragmentary and usually faint and difficult to trace @xcite , and they generally do not host any strong large - scale spiral structure , the kind we see in case of normal hsb galaxies like our milky way . we note that , we are interested only in small lsbs , which are more abundant , and do not include the giant lsbs like malin 1 . some of the giant lsbs show fairly strong , large - scale spiral structure as in ugc 6614 @xcite and in the inner regions they are dynamically similar to their high surface brightness ( hsb ) counterparts @xcite . @xcite has applied the technique of density - wave theory to put constraint on the decomposition of the rotation curve in lsbs . however we note that in the sample considered by @xcite contains giant lsb ( e.g. , ugc 6614 ) which often show large - scale spiral structure ( e.g. , see * ? ? ? the lsbs are dark matter dominated from the very inner regions @xcite . within the optical disk , the dark matter constitutes about @xmath1 per cent of the total mass of lsbs , whereas for the hsbs the contributions of the dark matter halo mass and stellar mass are comparable @xcite . thus , the lsbs constitute a natural laboratory to study the effect of dark matter halo on different aspects of galactic dynamics . several past studies have shown the effect of dominant dark matter halo in the suppression of global non - axisymmetric bar modes @xcite , in making the galactic disks superthin @xcite and in prohibiting the swing amplification mechanism from operating , thus explaining the lack of small - scale spiral structure as noted observationally @xcite . according to the density wave theory , the grand - design spiral arms are the high density regions of a rigidly rotating spiral density wave , with a well defined pattern speed , that are self - consistently generated by the combined gravity of the unperturbed disk and the density wave @xcite . for a recent review on this see @xcite . in a recent work @xcite ( hereafter paper 1 ) investigated the role of a dominant dark matter halo on the global spiral modes within the framework of the density wave theory by treating the galactic disk as a fluid . using the input parameters of a superthin lsb galaxy ugc 7321 and the galaxy , they found that for ugc 7321 , the dark matter halo has a negligible effect on arresting the global spiral modes when the disk is modeled as a fluid . this is in contrast to the results for small - scale spiral features where the dark matter was shown to suppress the small - scale , swing - amplified spiral structures almost completely @xcite . ghosh et al . ( paper 1 ) argued that that since lsbs are relatively isolated , tidal interactions are less likely to occur compared to those for the high surface brightness ( hsb ) galaxies . thus even though the global spiral modes are formally permitted in the fluid disk plus dark matter halo model , it was argued that it is the lack of tidal interaction that makes it difficult for the global spiral structure to develop in these galaxies . in this paper we address the effect of dark matter halo on global @xmath0 modes by modeling the galactic disk more realistically as a @xmath2 system . a tidal encounter is likely to give rise to global modes , as has been seen in simulations , as in m51 ( see e.g. , * ? ? ? we use the dispersion relation for a @xmath2 disk to construct global standing - wave like solutions by invoking the bohr - sommerfeld quantization condition ( for details see paper 1 ) . note that fluid disks allow wavelike solutions at small wavelengths , since fluid pressure provides the restoring force ; but collisionless disks suppress wavelike modes for very small wavelengths ( e.g. see * ? ? ? the 2 contains formulation of the problem and the input parameters . in 3 we present the wkb analysis and the relevant quantization rule . 4 * 5 * contain the results and discussion , respectively while 6 contains the conclusions . we model the galactic disk as a collisionless system characterized by an exponential surface density @xmath3 , and one - dimensional velocity dispersion @xmath4 . for simplicity , the galactic disk is taken to be infinitesimally thin . in other words , we are interested in perturbations that are confined to the mid - plane ( @xmath5 ) . the dark matter halo is assumed to be non - responsive to the gravitational perturbations of the disk . we have used cylindrical coordinates @xmath6 in our analysis . in this subsection , we describe the models that we have used for the study of effect of dark matter halo on the global spiral modes . the dynamics of the disk is calculated first only under the gravity of the disk ( referred to as disk - alone case ) and then under the joint gravity of disk and the dark matter halo ( referred to as disk plus halo case ) . we took an exponential stellar disk with central surface density @xmath7 and the disk scalelength @xmath8 , which is embedded in a concentric dark matter halo whose density follows a pseudo - isothermal profile characterized by core density @xmath9 and core radius @xmath10 . the net angular frequency , @xmath11 and the net epicyclic frequency , @xmath12 for a galactic disk embedded in a dark matter halo , concentric to the galactic disk , are given as : + @xmath13 the expressions for @xmath14 , @xmath15 in the mid - plane ( @xmath5 ) for an exponential disk and @xmath16 , @xmath17 for a pseudo - isothermal halo in the mid - plane ( @xmath5 ) have been calculated earlier ( see paper 1 for details ) . in the early - type galaxies , the bulge component dominates in the inner regions , and hence exclusion of the bulge component from the models for early - type galaxies will underestimate the rotation curve in the inner regions . in paper 1 , it was shown that for the galaxy , inclusion of bulge yielded more realistic results ( for details see 5.1 in paper 1 ) . also note that ugc 7321 has no discernible bulge @xcite . therefore , only for our galaxy we have included bulge in both the disk - alone and disk plus dark matter halo models . we adopt a plummer - kuzmin bulge model for our galaxy which is characterized by total bulge mass @xmath18 and bulge scalelength @xmath19 . the expressions for @xmath20 , @xmath21 in the mid - plane ( @xmath5 ) for such a bulge having a plummer - kuzmin profile are given in paper 1 . therefore , for the galaxy , these @xmath20 and @xmath21 terms will be added in quadrature to the corresponding terms due to disk and dark matter halo . the input parameters for different components of ugc 7321 @xcite and the galaxy @xcite are summarized in table 1 . [ cols="^,^,^,^,^,^,^ " , ] shows that the modes do not extend up to the resonance points , consistent with the fact that @xmath22 is not equal to zero anywhere for modes in fig . [ fig3].,title="fig:",width=326,height=230 ] shows that the modes do not extend up to the resonance points , consistent with the fact that @xmath22 is not equal to zero anywhere for modes in fig . [ fig3].,title="fig:",width=336,height=230 ] from table 2 , it is evident that the dark matter halo has a negligible effect on the global spiral modes for the galaxy , the difference can be seen only in the change of the specific values of pattern speed @xmath23 . this result is at par with our expectation , since the dark matter halo is known to be not important in the inner regions of the galaxy ( e.g. , * ? ? ? * ; * ? ? ? * ) . these results are consistent with the results of paper 1 where the galactic disk was modeled as a fluid disk . the key difference with results from paper 1 is that the specific values of pattern speed @xmath23 which give the global modes are different ( for comparison , see table 3 in paper 1 ) . @xcite found a range of @xmath24 km s@xmath25 kpc@xmath25 for the pattern speed of spiral arms for the galaxy . on comparison the pattern speed values obtained in this paper lie outside the observed range of pattern speed . the shape of the closed contours in this work ( figs . 1 - 3 ) is different from that in paper 1 . a curve of constant @xmath23 had nearly identical @xmath22 values at the turning points of a closed curve of _ type b _ in paper 1 , whereas the contours look slanted in the present paper , meaning that the @xmath22 values at the turning points are substantially different . this is due to the difference in the dispersion relation for the two cases . a more detailed interpretation is not possible in a simple analytical form , since the dispersion relation for the collisionless case ( equation ( [ disp - equation ] ) ) has a transcendental form . in this paper for simplicity we have not included the low velocity dispersion component , namely , gas in the system which could further modify the obtained range of pattern speeds . note that any late - type spiral galaxy contains non - negligible amount of gas , and it has been shown that the gas plays a significant role in various dynamical issues ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the amount of gas present in ugc 7321 is quite small in comparison to that in hsb galaxies @xcite . therefore , the results for ugc 7321 obtained in this paper - in particular the suppression of global modes by the dominant dark matter halo is unlikely to change significantly if gas is included in the calculation . the effect of gas on the global spiral modes in a gravitationally coupled two - component ( stars and gas ) system , as applicable to the gas - rich hsb galaxies , will be taken up in a future paper . the results presented here are based on the assumption that the grand - design spiral structure seen in disk galaxies is due to the density waves , as proposed by @xcite . however this hypothesis has not yet been fully confirmed observationally . for example , the angular off - set in age of the stellar population , as predicted by the classical density wave theory has not been found in studies by @xcite in their sample of late - type disk galaxies . also the study of @xmath26 observation of ngc 1068 using generalized tremaine - weinberg method by @xcite has revealed that the pattern speed of the spiral structure of this galaxy varies rapidly with radius , indicative of short - lived spiral features . a study of the gas content and the @xmath27 line of sight velocity distribution in ngc 6754 by @xcite revealed a different sense of streaming motion in the trailing and leading edge of this galaxy , indicating that the spiral arms of this galaxy is likely to be a transient . on the other hand , it is also worth noting that @xcite have recently presented a study of measuring pitch angle from the images taken in different wavelengths for a large sample of disk galaxies and found that the pitch angle varies from one wavelength to another as predicted by the classical density wave theory , and thus furnishes strong observational evidence for the validity of the density - wave theory of spiral structure in disk galaxies . generally the various @xmath28-body studies so far do not show the evidence of long - lived spiral structure . several @xmath28-body simulations have shown that the spiral arms are transient and get wound up quickly @xcite , which goes against the classical density wave picture . interestingly an opposite trend is shown in a recent work by @xcite who reported long - lived spiral structure in a @xmath28-body study of models for galaxies with intermediate bulges . we have investigated the effect of a dominant dark matter halo on the possible existence of global spiral arms while modeling the galactic disk as a _ collisionless _ system . the wkb dispersion relation and the bohr - sommerfeld quantization condition were used to obtain discrete global spiral modes present in any model . we have analysed a superthin lsb galaxy ugc 7321 , and the galaxy , for this work . we found that for ugc 7321 both the disk - alone and the disk plus dark matter halo cases did not yield any discrete global spiral modes . even an increase in the stellar central surface density by a factor of few failed to produce any global spiral modes . thus our findings provide a natural explanation for the observed dearth of strong large - scale spiral structure in the lsbs . our results differ from those obtained in paper 1 where the galactic disk was treated as a fluid , where it was found that the dominant dark matter halo had a negligible constraining effect on the existence of global spiral modes in these galaxies . this difference is due to the different dispersion relation for fluid and collisionless systems , which played a pivotal role in determining the discrete global mode(s ) present in a model . thus the correct collisionless treatment for stars as done here has led us to a result for the non - existence of global spiral modes which is in agreement with the observations . as a check , for the galaxy we carried out a similar modal analysis and found that the dark matter halo has a negligible effect on the global spiral modes , as expected since the galaxy is not dark matter dominated in the inner regions , in contrast to ugc 7321 . thus , the dark matter halo that dominates from the innermost regions is shown to suppress the growth of local , swing - amplified non - axisymmetric features @xcite as well as the global spiral modes as shown here . * acknowledgements : * we thank the anonymous referee for the constructive comments that have helped to improve the paper . cj would like to thank the dst , government of india for support via j.c . bose fellowship ( sb / s2/jcb-31/2014 ) . banerjee a. , jog c.j . 2013 , mnras , 431 , 582b

low surface brightness ( lsb ) galaxies are dominated by dark matter halo from the innermost radii ; hence they are ideal candidates to investigate the influence of dark matter on different dynamical aspects of spiral galaxies . here , we study the effect of dark matter halo on grand - design , @xmath0 , spiral modes in a galactic disk , treated as a collisionless system , by carrying out a global modal analysis within the wkb approximation . first , we study a superthin , lsb galaxy ugc 7321 and show that it does not support discrete global spiral modes when modeled as a disk - alone system or as a disk plus dark matter system . even a moderate increase in the stellar central surface density does not yield any global spiral modes . this naturally explains the observed lack of strong large - scale spiral structure in lsbs . an earlier work @xcite where the galactic disk was treated as a fluid system for simplicity had shown that the dominant halo could not arrest global modes . we found that this difference arises due to the different dispersion relation used in the two cases and which plays a crucial role in the search for global spiral modes . thus the correct treatment of stars as a collisionless system as done here results in the suppression of global spiral modes , in agreement with the observations . we performed a similar modal analysis for the galaxy , and found that the dark matter halo has a negligible effect on large - scale spiral structure . galaxies : kinematics and dynamics , galaxies : spiral , galaxies : structure , galaxies : individual : ugc 7321 , galaxies : halos , instabilities

introduction formulation of the problem discussion conclusion references

arxiv

over the past decade , the interest for parametric systems based on josephson junctions has revived substantially since the pioneering discoveries@xcite@xcite@xcite@xcite , due to their implementation in various amplification schemes used to detect weak microwave photons in quantum devices@xcite@xcite@xcite@xcite@xcite . one appeal of these circuits is associated with the presence of multistable regimes , naturally occuring in nonlinear systems@xcite . sharp transitions separate these regimes ( in phase space ) , making the devices very useful as sensitive probes of quantum dynamics . when engineering quantum systems in circuit quantum electrodynamics ( cqed ) architectures , the power of the microwave signal to be measured often reaches the single - photon regime , and consequently the limiting factor of experiments is often the ability to detect and amplify these weak signals with sufficient signal - to - noise ratio . + a requirement for the implementation of quantum information processing is to read out the states of quantum bits ( qubits ) with high fidelity on short time scales compared with the qubit coherence times . in order to coherently manipulate a superconducting qubit as well as to protect it from noise , it is placed in an engineered electromagnetic environment often realized by a superconducting resonator . the combined qubit@xmath3resonator system can then be described by the jaynes - cummings hamiltonian@xcite . when the qubit transition frequency @xmath4 is far detuned from the resonator s angular frequency @xmath5 , compared to the qubit@xmath3resonator coupling rate @xmath6 , @xmath7 , the resonator picks up a dispersive frequency shift @xmath8 , with a sign depending on the qubit state . this provides a way to non - destructively probe the qubit dynamics through the resonator response and has been extensively used as a qubit readout method . however , the measurement fidelity is often limited by the weak response signal relative to the noise added from the cryogenic high - electron mobility transistor ( hemt ) amplifier . + the need to overcome this measurement obstacle has boosted the interest for parametric amplifiers@xcite@xcite , which offer a large signal gain and a possibility to continuously probe the system without adding a large amount of noise . building a parametric amplifier requires a nonlinearity , which in a resonant circuit has the desirable consequence of introducing instabilities and bifurcation points . these add degrees of freedom and complexity to the resonator dynamics , which can be used to implement more efficient readout schemes . the natural and well known candidate as a nonlinear element in superconducting circuits is the josephson junction , due to its low dissipation and nonlinear inductance . an example of a josephson - based device utilizing this nonlinearity is the josephson bifurcation amplifier ( jba ) @xcite@xcite . it consists of a @xmath9 resonator with a duffing nonlinearity , realized by placing a josephson junction in its current node , and has been used to perform single - shot read out of a transmon qubit@xcite . in this readout scheme , the two qubit states are brought into correspondence with two oscillation states of the hysteretic bistable system . by probing the resonator close to its bifurcation threshold , the dispersive shift from the qubit state is used to push the resonator into its bistable state where a sharp jump in amplitude is observed for one of the qubit states but not the other . this has enhanced the readout contrast sufficiently to obtain a fidelity of 94@xmath10 using a single - shot sample - and - hold pulse sequence of the resonator probe . + in this work , we investigate the experimental manifestation of two types of nonlinearities occuring in a nonlinear resonator with a parametrically flux - modulated boundary condition . in addition to the duffing nonlinearity present in the jba , the magnetic - flux modulation of the josephson inductance adds an additional degree of freedom , and a nonlinearity to the system dynamics . the flux modulation enters into the josephson energy term of the resonator s boundary condition in the form of a mixing product with the field inside the resonator@xcite , @xmath11 , where @xmath12 is the josephson energy , @xmath13 and @xmath14 are the magnetic flux and flux quantum , respectively , and @xmath15 denotes the phase across the josephson junctions directly related to the field in the resonator . considering the taylor expansions of the flux- and phase contributions to the mixing product@xcite , the number of terms entering into the dynamics is set by the microwave pump strength and the number of photons in the resonator . + our measured devices consist of a distributed @xmath16 coplanar waveguide resonator of length @xmath17 , with a flux - tunable inductance realized by terminating one end to ground via two parallel josephson junctions forming a dc - superconducting quantum interference device , ( dc - squid)@xcite@xcite@xcite@xcite , see fig . [ fig1](a ) . by threading the squid loop with magnetic flux , the electrical length of the resonator is tuned through the changing josephson inductance @xmath18 where @xmath19 is the critical current of the squid . to operate the device parametrically , we modulate the flux around a static dc - bias point , @xmath20 , by coupling the squid to an on - chip microwave pump line@xcite , yielding a total flux @xmath21 . if this flux pumping is done at around twice the fundamental resonator frequency , @xmath22 , parametric oscillations build up the field exponentially in time inside the resonator , above the parametric threshold@xcite@xcite . the amplitude of the field in the resonator is eventually limited by the duffing nonlinearity . the parametric pumping of the boundary condition is the same as that in the dynamical casimir effect experiment@xcite . however , when the field is confined inside a resonator , certain conditions are imposed on the pump - resonator detuning @xmath23 and the effective strength of the pump @xmath24 to observe parametric effects . since oscillations occur only within a limited region in the [ @xmath23,@xmath24]-plane , it is possible to use such a parametric oscillator as a new member of the family of dispersive read - out techniques for superconducting qubits . in this case , the parametric resonator would work as a threshold detector , in which the two qubit states would be encoded into one oscillating- and one quiet state . this technique would relate to the bifurcation amplifier@xcite in the sense of utilizing the cavity pull to push the system into a bistable oscillating state . however , in contrast to the jba , the resonator can be left empty for one of the two states where the parametric pumping does not build up an oscillating field in the resonator . depending on the choice of operation point , the choice of `` quiet state '' can be tailored to reduce back action on the qubit@xcite , e.g. by encoding the ground state of the qubit into the oscillating state of the resonator . + the motivation for this work is to facilitate future designs of pumped nonlinear systems by developing an understanding of these two leading nonlinearities . in particular , when the device is operated as a parametric amplifier , a large bandwidth is preferable . this has the unwanted consequence that the system needs to be parametrically pumped at higher pump strength , introducing higher - order terms in the pump expansion , which we need to account for when operating the system . + the structure of the paper is as follows . in sections 2 and 3 , we introduce the frequency tunability of the resonator with applied magnetic flux and the theoretical framework for the field inside the resonator , respectively . next , in section 4 , the duffing nonlinearity is described and extracted , whereas section 5 is devoted to the pump - induced nonlinearity . the first step in characterizing our system s dynamics is to find its fundamental frequency s dependence on the applied dc - flux bias , @xmath26 . we measured the devices using a vector network analyzer ( vna ) connected to a microwave reflectometry setup , depicted in fig . [ fig1](a ) , in a dilution refrigerator with a base temperature of 20 mk . the shape of the frequency tuning curve as a function of applied magnetic flux is governed by the participation ratio of the squid s nonlinear josephson inductance @xmath27 in eq . ( [ eqsquidinductance ] ) to the geometrical resonator inductance , @xmath28 , where @xmath29 is the inductance per unit length of the resonator and @xmath17 its length@xcite@xcite . the frequency is well approximated by @xmath30 where @xmath31 denotes the bare resonant frequency , in the absence of the josephson contribution to its total inductance . since the nonlinearity originates from the squid inductance@xcite , the frequency - flux curvature governs much of the rich nonlinear dynamic properties of the system ; @xmath32 and @xmath33 should therefore be the main design aspects to consider . to investigate where the nonlinearities enter into the system response , we measured two samples with parameters listed in table [ tab1 ] . we extracted resonant frequencies and are plotted as a function of magnetic flux in fig . [ fig1](b ) . .extracted resonator parameters for the two measured samples . @xmath34 and @xmath35 are the bare- and zero - flux resonant frequencies , respectively . @xmath32 denotes the inductive participation ratio and @xmath19 is the critical current of the squid . [ cols="^,^,^,^,^ " , ] [ tab2 ] the next nonlinear effect enters the dynamics when the parametric pumping gets sufficiently strong for higher order terms of the mixing product expansion to affect the resonator . to investigate this nonlinearity , we minimize the duffing nonlinearity by turning off the probe signal . we then parametrically pump the flux around a bias point a bit higher up on the flux curve where the duffing influence is weaker , compare table [ tab2 ] . the first higher - order term is proportional to the square of the pump strength and has the effect of shifting the resonator down in frequency as a consequence of rectification in the flux@xmath3frequency transfer function . we reveal this effect by detecting the region of parametric instability in the parameter - plane spanned by the pump@xmath3resonator detuning @xmath23 and the effective pump strength @xmath24 , see fig . [ fig2 ] . the energy of the field inside the resonator originates from the pump , and starts to build up exponentially in time when @xmath24 is sufficiently strong to compensate for the total damping rate of the resonator : @xmath36 . after pumping for some time , the field saturates to a steady state set by the duffing nonlinearity at the given point in the ( @xmath37)-plane , which we expect to shift the resonator frequency out from the degenerate parametric pumping condition , @xmath22 . + in this section , we will investigate the pump conditions that need to be fulfilled to observe parametric oscillations . the boundaries represent the bifurcation threshold at which the resonator enters into the parametric bistable regime , where oscillations in one of two metastable states of the system hamiltonian occur@xcite . the thresholds are obtained analytically by finding the steady - state , zero - field solutions to the intracavity field differential equation ( [ eqlangevin])@xcite@xcite . this yields a threshold symmetric in @xmath23 , plotted as the gray dashed line in fig . [ fig2](a ) and defined by the relation @xmath38 however , this symmetric region does not take into account the pump - induced frequency shift , adding to the detuning of the boundary . this is clearly observed in experiments , see fig . [ fig2](b ) . this effect is a result of the higher pump strength needed to drive parametric oscillations when the first derivative of the frequency - flux curve in fig . [ fig1](a ) is small compared with the more linear response closer to @xmath39 . this introduces a quadratic , higher order pump term in eq . ( [ eqparareg1 ] ) , and the resonator becomes red detuned , which can be understood as a rectification from the deviation from pure sinusoidal pumping of its frequency . we characterize this effect using a dimensionless parameter @xmath2@xcite @xmath40 when this pump - induced shift is taken into account , we obtain two solutions for the enclosed parametric region , together forming the skewed threshold to the parametric oscillation region in fig . the lower and upper parametric instability boundaries follow the relations @xmath41 where the parameter @xmath2 can be approximated in terms of characteristic resonator parameters , @xmath42 where @xmath43 . we can develop an intuition for this shift by finding the pump power at which the onset of parametric instability is obtained . this takes place when the pump exactly compensates for the damping and the resonator is empty of photons . the slight shift of the parametric threshold from zero detuning tells us that this is a different effect than the duffing nonlinearity , which is proportional to the field inside the resonator . instead , the higher - order pump shift can be understood by considering the curvature of the flux - tuned frequency curve in fig . [ fig3](a ) , plotted for three different values of @xmath44 . the effective , pump - shifted resonator frequency is lower than the actual static bias point if the second derivative of the curve starts to dominate over the first derivative . another way to think about this effect is rectification , since the resonator , on average , spends longer time at a lower effective frequency upon parametric pumping due to the steeper curvature on the low - frequency side of the static flux bias point . this is in agreement with the approximation of @xmath2 , which diverges as we approach zero flux bias and goes to zero at @xmath45 . this also agrees with the lower pump strength required to drive parametric oscillations for dc - flux bias points at lower frequencies . [ fig3](b ) , we summarize the flux bias dependence of the duffing nonlinearity parameter , @xmath1 in eq . ( [ eqalpha ] ) , described in sec . [ sec : duffing ] and the pump - induced nonlinearity parameter , @xmath2 in eq . ( [ eqbeta ] ) . in fact , it is possible to cancel both the rectification and the skewed threshold by adding a dc - component and a second pump tone with a frequency of @xmath46 . this is shown in the appendix a. ) for a bare non - tunable resonator ( @xmath47 ) ( black dashed line ) and three values of the inductive participation ratio @xmath44 = 0.040 , 0.090 , and 0.14 , in dashed blue , solid black , and dashed red , respectively . * ( b ) * magnetic - flux dependence of the normalized duffing nonlinearity parameter , @xmath48 from eq . ( [ eqalpha ] ) ( left axis ) and pump - induced frequency shift parameter , @xmath49 from eq . ( [ eqbeta ] ) ( right axis ) , shown in red and blue regions , respectively . the three different traces correspond to the same values of @xmath44 as in ( a ) . ] in conclusion , we have shown how the two different nonlinear effects manifest themselves in superconducting parametric resonators and presented methods to quantify both of them . first , we extracted the duffing nonlinearity associated with the current flowing through the squid by fitting the nonlinear frequency shift as a function of the probe power , using extracted damping rates of the resonator . second , we studied the parametric response in the absence of a probe signal for two different magnetic - flux points . we conclude that the duffing nonlinearity dominates at @xmath50 , whereas the pump - induced nonlinearity dominates as @xmath51 as a consequence of the fact that the system there needs to be pumped more strongly in order to drive the parametric oscillations , introducing higher order pump terms into the system response . finally , we see that the interplay between these two nonlinear effects is governed by the inductive josephson participation ratio , @xmath32 . for a small josephson contribution , the system is more robust against the duffing shift , but more susceptible to pump - induced nonlinearity and vice versa . with this interplay in mind , more advanced circuits with tailored nonlinear dynamics can be realized . the frequency of a tunable resonator is not linear with respect to the applied magnetic flux . thus , when supplying a sinusoidal flux , the nonlinear element will experience additional frequency pumping on top of the harmonic base tone . in this appendix , we will demonstrate how this pump - induced deviation from a sinusoidal pump tone effectively can be canceled out by adjusting the pump accordingly . we start out by approximating the frequency tuning using polynomial taylor expansion around the static dc - flux bias , @xmath52 @xmath53 using a pure harmonic pump tone @xmath54 , yields a frequency - flux relation on the form @xmath55 we see that apart from the fundamental frequency , a dc - rectification contribution , _ i.e. _ a pump - induced frequency shift proportional to the square of the pump flux amplitude , @xmath56 , is added . we also get a second rf tone at twice the pump frequency . + next , we will use this knowledge to adjust the parametric pump signal in such a way that these two higher order pump effects are canceled out . consider a pump signal ansatz on the following form @xmath57 where @xmath58 . next , we insert the flux ansatz ( [ eqapp3 ] ) into the frequency relation in ( [ eqapp1 ] ) @xmath59 @xmath60 @xmath61 using harmonic balance , the second order tone can be canceled if we satisfy the following condition @xmath62 by inserting the second order cancelation condition in ( [ eqapp6 ] ) into the actual pump signal , neglecting higher order , as well as , fast rotating terms@xcite , we get @xmath63 finally , the dc - rectification component can be evaluated from the ansatz in eq . ( [ eqapp3 ] ) @xmath64 now , we substitute back the relation for the flux @xmath58 into eq . ( [ eqapp8 ] ) @xmath65 in conclusion , to cancel out the second order pump - induced frequency shift , the parametric pumping should be done using the following adjusted signal @xmath66 let us now evaluate the cancelation scheme for a tunable resonator with the resonance frequency well approximated by eq . ( [ eqcavityfrequency ] ) , where the inductive participation ratio of the system is small ( @xmath67 ) . to find the compensation terms , we derive the first and second derivatives of the frequency with respect to magnetic flux , plotted in fig . [ fig4 ] , together with the ratio of the second derivative to the first , given by @xmath68 in conclusion , the pumping needed to compensate for the second order pump - induced nonlinearity in a tunable resonator can be written on the following form @xmath69 @xmath70 , plotted for the same three values of the inductive participation ratio @xmath32 = 0.040 , 0.090 , and 0.14 , plotted with dashed blue , solid black , and dashed red lines , respectively . * ( b ) * the second derivative of the frequency with respect to magnetic flux , @xmath71 . * ( c ) * the ratio of the second derivative in * ( b ) * to the first in * ( a ) * , given in equation ( [ eqapp14 ] ) ] . the samples were fabricated in the mc2 clean room facilities at chalmers university of technology , sweden and measured at university of queensland , australia . we acknowledge financial support from the european research council , erc no . 247208 , the australian research council centre of excellence in engineered quantum systems , arc no . ce110001013 , and the swedish foundation for international cooperation in research and higher education , stint no . ig2009 - 2023 . we gratefully thank c. m. wilson , l. tornberg , i.c . hoi , and m. simoen for fruitful discussions . 10 wahlsten s _ et al . _ 1977 _ appl . phys . lett . _ * 30 * 298 yurke b _ et al . _ 1988 _ phys . _ , * 60 * 764 - 767 yurke b _ et al . _ 1989 _ phys . * 39 * 2519 landau l d and lifshitz e m 1976 _ mechanics _ vol . 1 , 3rd edn . , ( oxford : pergamon ) dykman m 1998 _ phys . e _ , * 57 * 5202 dykman m 2012 _ fluctuating nonlinear oscillators _ ed m dykman ( oxford : oxford university press ) pp 165 - 197 siddiqi i _ et al . _ 2004 _ phys . lett . _ * 93 * 207002 vijay r _ _ 2009 _ rev . inst . _ * 80 * 111101 wustmann w and shumeiko v s 2013 _ phys . b _ * 87 * 184501 strogatz s h 1994 _ nonlinear dynamics and chaos _ ( reading , ma : addison - wesley ) jaynes e t and cummings f w 1963 _ proc . ieee _ * 51 * 89 - 109 yamamoto t _ _ 2008 _ appl . lett . _ * 93 * 042510 sundquist k _ _ 2013 _ appl . lett . _ * 103 * 102603 castellanos - beltran m a and lehnert k w 2007 _ app . lett . _ * 91 * 083509 mallet f _ et al . _ 2009 _ nature physics _ * 5 * 791 - 795 wallquist m , shumeiko v s and wendin g 2006 _ phys . b _ * 74 * 224506 sandberg m _ 2009 _ physica scripta _ * t137 * 014018 sandberg m _ _ 2008 _ appl . * 92 * 203501 wilson c m _ _ 2010 _ phys . lett . _ * 105 * 233907 palacios - laloy a _ et al . _ 2008 _ j. low temp . phys . _ * 151 * 1034 - 42 picot t _ _ 2008 _ phys . b _ , * 78 * 132508 wilson c m _ _ 2011 _ nature _ * 479 * 376 - 379 manucharyan v e _ _ 2007 _ phys . b _ , * 76 * 014524 bertet p _ _ 2012 _ fluctuating nonlinear oscillators _ ed m dykman ( oxford : oxford university press ) pp 1 - 32

we experimentally study the behavior of a parametrically pumped nonlinear oscillator , which is based on a superconducting @xmath0 resonator , and is terminated by a flux - tunable squid . we extract parameters for two devices . in particular , we study the effect of the nonlinearities in the system and compare to theory . the duffing nonlinearity , @xmath1 , is determined from the probe - power dependent frequency shift of the oscillator , and the nonlinearity , @xmath2 , related to the parametric flux pumping , is determined from the pump amplitude for the onset of parametric oscillations . both nonlinearities depend on the parameters of the device and can be tuned _ in - situ _ by the applied dc flux . we also suggest how to cancel the effect of @xmath2 by adding a small dc flux and a pump tone at twice the pump frequency .

introduction tunability of the fundamental frequency: + a measurement of the josephson inductance pump-induced nonlinearity conclusions optimization of parametric pumping acknowledgements references

arxiv

precise measurements of neutron @xmath0-decay observables determine fundamental parameters of the weak interaction and contribute to tests of the standard model @xcite . because the momentum transfer in neutron @xmath0-decay ( @xmath5 kev ) is small compared to the @xmath6 mass , the decay can be modeled as a four - fermion contact interaction with an amplitude under the standard model given by @xmath7 where @xmath8 is the fermi weak coupling constant , @xmath9 is the weak - quark - mixing ckm matrix element , and @xmath10 is the leptonic weak vector and axial - vector current . in its most general form , the hadronic weak vector and axial - vector ( @xmath11 ) current includes six form factors @xcite , @xmath12u_n , \ ] ] where @xmath13 is the four - momentum transfer ; @xmath14 is the nucleon mass ; and @xmath15 , @xmath16 , @xmath17 , @xmath18 , @xmath19 , and @xmath20 are the vector , weak magnetism , induced scalar , axial vector , induced tensor , and induced pseudoscalar form factors , respectively . in the limit of @xmath21 , the hadronic weak current is dominated by the weak vector and axial vector coupling constants of the nucleon , defined to be the values of the vector and axial vector form factors at @xmath22 , @xmath23 and @xmath24 . under the conserved vector current ( cvc ) hypothesis of the standard model and the assumption of isospin symmetry , the vector coupling constant is @xmath25 ( independent of the nuclear medium ) . isospin - symmetry - breaking effects to the value of @xmath26 in neutron @xmath0-decay have been calculated in chiral perturbation theory , with the correction to @xmath26 found to be at a negligible @xmath27 level @xcite . also per the cvc hypothesis , the weak magnetism coupling constant , @xmath28 , which appears at recoil order in the vector current , is related to the proton and neutron anomalous magnetic moments by @xmath29 . in contrast to the vector current , the axial - vector current is renormalized by the strong interaction such that the value of @xmath30 must be determined experimentally and also by lattice quantum chromodynamics ( qcd ) calculations . any contribution from the induced pseudoscalar coupling constant , @xmath31 , to neutron @xmath0-decay observables is expected to be negligibly small , with the contribution of @xmath31 to the energy spectrum calculated to be of order @xmath32 @xcite . the two remaining terms , the induced scalar , @xmath17 , in the vector current , and the induced tensor , @xmath19 , in the axial - vector current , are termed second - class currents due to their transformation properties under @xmath33-parity . under the requirement of @xmath33-parity symmetry , both @xmath34 . however , @xmath33-parity symmetry is violated within the standard model due to differences in the @xmath35 and @xmath36 quarks charges and masses ( i.e. , isospin symmetry breaking effects ) . an estimate for @xmath37 including su(3 ) breaking effects suggested @xmath38 in neutron @xmath0-decay to be @xmath39 @xcite , and an evaluation of @xmath40 using qcd sum rules found @xmath41 @xcite . finally , recent lattice qcd studies of su(3 ) breaking in semi - leptonic decays find small , @xmath42 , values for both @xmath17 and @xmath19 in neutron @xmath0-decay , but the results are statistically limited and consistent with zero at 12 standard deviations @xcite . however , despite these hints for non - zero values of these second - class currents , their contributions to neutron @xmath0-decay observables are again expected to be negligibly small , as they also appear at order @xmath43 in the energy spectrum @xcite . therefore , under the assumption that any such contributions from @xmath31 , @xmath44 , and @xmath37 are negligibly small relative to the current level of experimental precision , it is clear that a description of neutron @xmath0-decay under the cvc hypothesis of the standard model requires the specification of only two parameters , @xmath9 and @xmath30 , given the high precision results for @xmath8 achieved in muon decay @xcite . both @xmath9 and @xmath30 can be accessed via measurements of two different types of neutron @xmath0-decay observables : the lifetime , and angular correlation coefficients in polarized and unpolarized @xmath0-decay . the first of these , the lifetime , as calculated from the amplitude and integration over the allowed phase space , is of the form @xcite @xmath45 where @xmath46 is the electron mass and the parameter @xmath47 is defined to be the ratio of the axial vector and vector coupling constants , @xmath48 . the numerical value for the phase space factor of @xmath49 @xcite includes corrections for the fermi function , the finite nucleon mass , the finite nucleon radius , and the effect of recoil on the fermi function . the factor @xmath50 denotes the total effect of all electroweak radiative corrections , including the @xmath51 outer ( long - distance loop and bremsstrahlung effects ) and inner ( short distance , including axial - vector - current , loop effects ) radiative corrections ; an @xmath52 correction resulting from factorization of the fermi function ; and @xmath52 leading - log and next - to - leading - log corrections ( for lepton and quark loop insertions in the photon propagator ) @xcite . the total electroweak radiative correction has been calculated to be @xmath53 @xcite where the @xmath54 uncertainty was reduced by a factor of two ( from its previous value of @xmath55 @xcite ) after the development of a new method for calculating hadronic effects in the matching of long- and short - distance contributions to axial - vector current loop effects ( primarily from the @xmath56 box diagram ) . the second type of observable , angular correlation coefficients , parametrize the angular correlations between the momenta of the decay products and the spin of the initial - state neutron . in general , the directional distribution of the electron and antineutrino momenta and the electron energy in polarized @xmath0-decay is of the form @xcite @xmath57 , \label{eq : w_phase_space_distribution}\end{aligned}\ ] ] where @xmath58 ( @xmath59 ) and @xmath60 ( @xmath61 ) denote , respectively , the electron s ( antineutrino s ) total energy and momentum ; @xmath62 ( @xmath63 kev + @xmath46 ) is the electron endpoint energy ; and @xmath64 is the neutron polarization . the angular correlation coefficients @xmath65 ( @xmath66-@xmath67-asymmetry ) , @xmath68 ( @xmath0-asymmetry ) , and @xmath69 ( @xmath67-asymmetry ) are , to lowest order , functions only of @xmath47 where , under a @xmath70 sign convention , @xmath71 the contributions of terms in eq.([eq : w_phase_space_distribution ] ) proportional to the fierz interference term @xmath72 and the time - reversal - odd triple - correlation - coefficient @xmath73 are at recoil order for standard model interactions @xcite , and are negligible at the current level of experimental precision . note that to our knowledge , there are no published direct measurements of @xmath72 in neutron @xmath0-decay . as already noted , at lowest order @xmath74 , @xmath1 , and @xmath75 are functions only of @xmath47 . however , recoil - order corrections , including the effects of weak magnetism and @xmath26-@xmath30 interference , introduce energy - dependent corrections to the asymmetry , and are of @xmath76(1% ) for @xmath65 and @xmath68 @xcite and @xmath76(0.1% ) for @xmath69 @xcite . for @xmath68 , the recoil - order corrections are of the explicit functional form @xcite @xmath77 where @xmath78 , @xmath79 , @xmath80 , and the @xmath81 ( @xmath82 ) coefficients are functions only of @xmath47 and @xmath83 ( assuming @xmath84 , and negligible contributions from @xmath31 ) . under these assumptions , @xmath85 , @xmath86 , and @xmath87 . note that the @xmath88 dependence of the form factors does not appear until next - to - leading recoil order @xcite . in addition to the above recoil - order corrections , there is a small energy - dependent radiative correction ( for virtual and bremsstrahlung processes ) to polarized asymmetries , resulting in a @xmath89 correction to @xmath68 @xcite . after application of these recoil - order and radiative corrections , a value for @xmath47 can be extracted from @xmath74 , @xmath1 , and @xmath75 . note , however , that for a given ( relative ) statistical precision , the sensitivity of @xmath1 to @xmath47 is slightly higher than that of @xmath74 , and a factor of @xmath90 greater than that of @xmath75 , where at leading order the relative uncertainties compare as @xmath91 thus , measurements of the angular correlation coefficients determine a value for @xmath47 ( or @xmath30 , assuming @xmath25 under the cvc hypothesis ) , a fundamental parameter in the nucleon weak current . a precise value for @xmath30 is also important in many other contexts . in hadronic physics studies of the spin structure of the nucleon @xcite , the bjorken sum rule relates the difference in the first moments of the proton and neutron spin - dependent @xmath92 structure functions ( i.e. , isovector channel ) , as probed in polarized deep inelastic electron scattering , to @xmath30 . in qcd , the assumption of a partially conserved axial - vector current ( pcac ) , valid in the limit of a massless pion ( identified as the goldstone boson of the spontaneously broken chiral symmetry ) , leads to the goldberger - treiman relation @xcite , relating the value of @xmath30 to the pion decay constant @xmath93 , the weak pion - nucleon - nucleon coupling constant @xmath94 , and the nucleon mass . the value of @xmath30 is also important in astrophysical processes , including calculations of solar fusion cross sections and rates , in particular , of the @xmath95 fusion reaction , impacting the solar neutrino flux for this process @xcite . high - precision experimental results for @xmath30 also serve as an important benchmark for theoretical calculations of @xmath30 , both in fundamental lattice qcd calculations @xcite and in relativistic constituent quark model calculations @xcite . a precise value for @xmath30 is also important as a phenomenological input parameter ( together with other low energy constants , such as the pion decay constant @xmath93 , the nucleon mass , etc . ) to effective field theory calculations involving the axial vector current @xcite . although not a fundamental weak interaction parameter by itself , a precise value for the lifetime is important for big bang nucleosynthesis calculations , impacting the neutron - to - proton ratio and hence the primordial @xmath96he abundance at the time of freeze - out , when the weak reaction rate became less than the hubble expansion rate @xcite . the value of the lifetime is also important for the interpretation of data from neutrino oscillation experiments employing antineutrinos from reactors @xcite , which typically search for the reaction @xmath97 in detectors . the cross section for this reaction is inversely proportional to the neutron lifetime ; therefore , an accurate and precise experimental value for the lifetime is needed for an interpretation of measured detector antineutrino reaction rates in terms of the underlying neutrino oscillation physics . measurements of the lifetime and a value for @xmath47 from measurements of angular correlation coefficients permit the extraction of a value for @xmath9 solely from neutron @xmath0-decay observables according to eq . ( [ eq : lifetime - vud ] ) . although a value for @xmath9 from neutron @xmath0-decay @xcite is not yet competitive with the definitive value deduced from measurements of @xmath98 values in superallowed @xmath99 nuclear @xmath0-decay @xcite , the appeal of such an extraction is that it does not require corrections for isospin - symmetry - breaking and nuclear - structure effects . ultimately , when the precision on a neutron - based value for @xmath9 approaches the precision of the @xmath99 result , the two values must agree in the absence of new physics . however , given that the neutron - based value for @xmath9 is not yet competitive , one can treat the @xmath99 value for @xmath9 as a fixed input parameter , and instead perform a robust test of the consistency of the various measured neutron @xmath0-decay observables under the standard model . in particular , results for @xmath30 extracted from correlation coefficient measurements can then be directly compared with results from measurements of the neutron lifetime @xmath100 . finally , measurements of the angular correlation coefficients themselves are sensitive to beyond - the - standard - model physics , such as scalar and tensor interactions @xcite . with the projected improvements to the experimental precision in future years , neutron @xmath0-decay measurements will be sensitive to any such sources of new physics at energy scales rivaling those probed directly at the large hadron collider @xcite . lllccc experiment & years published & type & polarization & @xmath1 result & notes + perkeo @xcite & 1986 & cold neutron beam & @xmath101 & @xmath102 & + pnpi @xcite & 1991 , 1997 & cold neutron beam & @xmath103 & @xmath104 & + ill - tpc @xcite & 1995 , 1997 & cold neutron beam & @xmath105 & @xmath106 & + perkeo ii @xcite & 1997 , 2002 & cold neutron beam & @xmath107 & @xmath108 & + ucna @xcite , this work & 2009 , 2010 & stored ultracold neutrons & @xmath109 & @xmath110 & + perkeo ii @xcite & 2012 & cold neutron beam & @xmath111 & @xmath112 & + + [ tab : summary_measurements ] the remainder of this article is organized as follows . in section [ sec : status_measurements ] , we summarize the current status of measurements of @xmath1 . we then outline the experimental motivation for a measurement of @xmath1 with ultracold neutrons in section [ sec : ucna_experiment ] , and then present a detailed description of the ucna ( `` ultracold neutron asymmetry '' ) experiment @xcite at the los alamos national laboratory . our measurement procedures and experimental geometrical configurations are reported in section [ sec : measurements ] . results from our calibration and analysis procedures are discussed in section [ sec : analysis ] . details of our procedure for the extraction of asymmetries are presented in section [ sec : asymmetry ] , and the corrections to measured asymmetries for various systematic effects are discussed in section [ sec : corrections ] . systematic uncertainties are summarized in section [ sec : uncertainties ] , and our final results for @xmath1 are then reported in section [ sec : summary_final_results ] . we then conclude with a brief summary of the physics impact of our work in section [ sec : conclusions ] . the data presented here were obtained during data - taking runs in 20082009 and published rapidly in 2010 @xcite ; in this article we provide a more detailed account of the experiment and analysis procedures . the current status of published results @xcite for the neutron @xmath0-asymmetry parameter @xmath1 is summarized in table [ tab : summary_measurements ] and shown in fig.[fig : status_beta_asymmetry ] . other than the ucna experiment , all of the experiments have been performed with beams of polarized cold neutrons , with reported values for the polarization ranging from @xmath103 @xcite to @xmath114 @xcite . magnetic solenoidal spectrometers providing @xmath115 solid angle acceptance for detection of the decay electrons were employed in the perkeo @xcite and perkeo ii @xcite experiments at the institut laue - langevin ( ill ) . in contrast , the solid angle was defined by the geometric acceptance in an experiment at the petersburg nuclear physics institute ( pnpi ) @xcite in which the decay electrons and protons were detected in coincidence in detectors surrounding the beam decay region , and in an experiment at the ill @xcite which utilized a time projection chamber for reconstruction of the electron track . the current world average value for @xmath116 includes the most recent perkeo ii result reported in @xcite which accounted for correlations of systematic errors in the two separately published perkeo ii results @xcite . ] @xcite , but excludes the ucna proof - of - principle result @xcite . note that the current error bar of @xmath117 includes the particle data group s @xmath118 scaling @xcite . the need for this expanded error bar suggests an incomplete assessment of the systematic errors in one or more of the cold - neutron - based experiments . the ucna experiment , installed in area b of the los alamos neutron science center ( lansce ) at the los alamos national laboratory ( lanl ) , was designed to perform the first - ever measurement of the neutron @xmath0-asymmetry parameter with ultracold neutrons ( ucn ) , and to - date is the only experimental measurement of any neutron @xmath0-decay angular correlation coefficient performed with ultracold neutrons ( ucn ) . ucn are defined to be neutrons with kinetic energies sufficiently low ( @xmath119 nev , corresponding to speeds @xmath120 m s@xmath121 ) such that they undergo total external reflection at any angle of incidence from an effective potential barrier ( a volume average of fermi potentials @xmath122 ) at the surfaces of certain materials @xcite . thus , ucn can be stored in material - walled vessels , whereas cold neutrons ( kinetic energies 0.0525 mev , speeds 1002200 m s@xmath121 ) must be transported along neutron guides at reflection angles less than the guide critical angle , resulting in short residency times in an apparatus . a schematic diagram of the ucna experiment is shown in fig.[fig : ucna_schematic ] , and the basic principle of the experiment is as follows . spallation neutrons resulting from the interaction of a pulsed ( typically 0.2 hz ) 800 mev proton beam with a tungsten target were moderated in cold polyethylene to the cold neutron regime , and then downscattered to the ucn regime in a solid deuterium ( sd@xmath123 ) crystal . the ucn were then transported along a series of ucn guides through a 7.0-tesla solenoidal polarizing magnet , where the spin - dependent @xmath124 potential ( @xmath125 nev t@xmath121 ) served as a spin - state selector for magnetic moments @xmath126 oriented parallel to the direction of the longitudinal magnetic field @xmath127 . the polarized ucn were then transported along non - magnetic ucn guides through an adiabatic - fast - passage ( afp ) spin - flipper 1.0-tesla field region , used to prepare ucn with spins either parallel or anti - parallel to the magnetic field . the ucn were then directed to the center of a 12.4-cm diameter , 3-m long cylidrical decay storage volume located within the warm bore of a 1.0-t solenoidal spectrometer . emitted @xmath0-decay electrons then spiraled ( with a maximum larmor diameter of 7.76 mm for 782 kev endpoint electrons emitted perpendicular to the 1.0-t field ) along the magnetic field lines towards one of two electron detectors located on both ends of the spectrometer . in principle , the @xmath0-asymmetry @xmath68 can be extracted from measurement of the @xmath128 angular distribution by forming an energy - dependent `` measured asymmetry '' , @xmath129 , of the detectors ( background - subtracted ) count rates , @xmath130 where @xmath131 denote the energy - dependent count rates observed in the two detectors , @xmath132 denotes the neutron polarization , @xmath0 denotes the electron velocity in units of @xmath133 , and @xmath134 is the average value of @xmath135 integrated over the detectors angular acceptance for that particular value of @xmath58 . note that for nominal values of @xmath136 , @xmath137 , @xmath138 , and @xmath139 , the experimental measured asymmetry is of order @xmath140 . in practice , the asymmetry is extracted from ratios of the two detectors energy - dependent count rates for the two neutron spin states with polarizations oriented parallel and anti - parallel to the magnetic field via a `` super - ratio '' technique . here , the super - ratio , @xmath141 , is defined in terms of the measured energy - dependent detector count rates for the two spin states , @xmath142 , to be @xmath143 with the energy - dependent measured asymmetry , @xmath129 , then calculated from the super - ratio according to @xmath144 the merit of this super - ratio technique is that effects due to differences in the two detectors efficiencies and spin - dependent differences in the efficiencies for transport of the two ucn spin states into the spectrometer cancel to first order . in a binned analysis , energy - dependent detection efficiencies also largely cancel in the super - ratio , and are negligible for the energy bin sizes used in this work . the motivation for the development of the ucna experiment was several - fold . first , the use of ucn in a neutron @xmath0-asymmetry experiment controls key neutron - related systematic corrections and uncertainties , including the neutron polarization and neutron - generated backgrounds . as discussed in detail later in this article , the polarization has been demonstrated to be @xmath145% at the 68% c.l . , with the precision , at present , limited only by statistics . further , neutron - generated backgrounds have been constrained to be negligible , a direct result of the relatively small number of neutrons present in the apparatus at any time , the small probability for their capture and subsequent generation of accompanying irreducible gamma ray backgrounds , and the fact that nearly all of the neutrons present in the apparatus are located within the spectrometer s decay volume . in the ucna experiment , a relatively large fraction , @xmath146 , of the ucn stored in the decay volume contribute to the measured decay rate , whereas in cold neutron beam experiments typically only @xmath147 of the neutrons passing through the apparatus contribute to the decay rate @xcite . therefore , control of neutron - generated backgrounds is expected to be intrinsically more challenging in cold neutron beam experiments . second , as described in detail elsewhere @xcite , the electron detector system developed for the ucna experiment , consisting of a low - pressure multiwire proportional chamber ( mwpc ) backed by a plastic scintillator , provides position sensitivity , suppresses ambient gamma - ray backgrounds , and permits the reconstruction of low - energy - deposition electron backscattering events . the ucna experiment is the first neutron @xmath0-asymmetry experiment to employ a mwpc , providing the experiment with two critical advantages . first , the position information permits the definition of a fiducial volume on an event - by - event basis . second , the position information also permits for an event - by - event correction for the scintillator s position - dependent energy response . we now provide a more detailed description of the primary components of the ucna experiment . a detailed description of the design principles and performance of the lanl sd@xmath123 ucn source is given elsewhere @xcite ; therefore , we provide only a brief description here . protons from the 800 mev lansce accelerator were delivered in pulsed modes beam bursts separated by 0.05 s , with 5.2 s between each pulse s leading edge burst . ] at a repetition rate of 0.2 hz to a tungsten spallation target , which was surrounded by a room - temperature beryllium reflector . with the spallation target operated in this pulsed mode , prompt beam related backgrounds can be eliminated with simple timing cuts , with negligible loss of duty factor for the @xmath0-decay measurements performed with the ucn stored in the electron spectrometer . the spallation neutrons were moderated in cold - helium - gas - cooled polyethylene ( maintained at a temperature of @xmath148 k for time - averaged proton beam currents of 5.8 @xmath149a ) located between the tungsten target and the beryllium reflector . the moderated cold neutrons were then downscattered to the ucn regime in a @xmath150 l cylindrical volume of ortho - state sd@xmath123 @xcite embedded at the bottom of a vertically - oriented cylindrical liquid - helium - cooled aluminum cryostat coated with @xmath151ni , presenting a nominal effective potential of 342 nev to the emerging ucn flux . the sd@xmath123 was maintained at temperatures @xmath152 k during the proton beam pulses on the spallation target . a butterfly - style `` flapper '' valve coated with @xmath151ni was located immediately above the sd@xmath123 volume . this `` flapper '' valve opened and subsequently closed ( with opening and closing response times of about 0.1 s ) with each proton pulse , in order to increase the storage lifetime of the ucn in the volume of the source above the sd@xmath123 volume . a typical ucn lifetime with the flapper open was @xmath153 s , whereas the lifetime with the flapper closed was @xmath154 s. the flapper leads to a corresponding increase in the ucn density . the ucn were then extracted from the source along horizontally - oriented 10.16-cm diameter stainless steel guides ( presenting a nominal potential of 189 nev ) through the biological shielding surrounding the source and out into the experimental area . as shown in fig . [ fig : ucna_schematic ] , this system of guides through the biological shielding included two @xmath155 bends to eliminate neutrons with kinetic energies above the stainless steel guide potential . the maximum ucn density at the biological shield exit that we have obtained is @xmath156 @xmath157 @xcite , but for this work ( runs in 20082009 ) typical densities were @xmath158 @xmath157 . after exiting the biological shield , the ucn were transported along stainless steel guides through a gate valve , which served to separate the ucn source from the experiment , thus permitting measurements of backgrounds in the electron spectrometer detectors with the proton beam still operating in its normal pulsed mode , but no accompanying ucn transport to the spectrometer . a 6.0-t superconducting solenoidal pre - polarizing magnet ( ppm ) was located immediately downstream of this gate valve . the ppm was included in the experiment design in order to minimize ucn transport losses through a thin ( 0.0254-mm thick ) zr foil which served to separate the vacuum in the sd@xmath123 source from the downstream vacuum in the remainder of the experiment . note that the ucn population was polarized after transport through the ppm s longitudinal magnetic field . to preserve this initial polarization , 10.16-cm diameter electropolished cu guides ( nominal potential of 168 nev ) were installed downstream of the ppm magnet . the ucn were then transported along these guides through a `` switcher '' valve , which allowed the downstream guides comprising the @xmath0-asymmetry measurement to be connected to either the upstream guides from the ucn source , or to a @xmath159he ucn detector @xcite used , as described later , for measurements of the depolarized population . these electropolished cu guides then transported the ucn through the primary 7.0-t polarizing magnet ( called the afp magnet ) . a 100-cm long quartz guide section coated with diamondlike carbon ( dlc ) @xcite passed through the center of a resonant ( 1.0-t ) `` bird - cage '' r.f . cavity @xcite , used for adiabatic fast passage ( afp ) spin - flipping of the ucn . downstream of this dlc - coated quartz guide , another section of 10.16-cm diameter cu guide transitioned to a 7-cm ( vertical ) @xmath160 4-cm ( horizontal ) rectangular cu guide , which transported the ucn through a horizontal penetration in the 1.0-t solenoidal electron spectrometer coil into the decay trap . permanent magnets were attached to the outer surfaces of the rectangular guide in order to suppress majorana spin - reorientations @xcite of neutrons passing through `` field zeros '' in the 1.0-t solenoidal spectrometer s field . the ucn rate along the transport guide system was monitored with @xmath159he ucn detectors @xcite at two key locations : at the gate valve ( for monitoring of the sd@xmath123 source performance ) , and slightly downstream of the apf spin flipper ( for monitoring of the afp spin - flipping efficiency ) . these @xmath159he ucn detectors were coupled to the guide system via small ( 0.64-cm diameter ) holes in the bottom of the guides . note that the ucn density in the spectrometer for the spin - flipped state was smaller than that for the unflipped state , because of losses ( after the 2.0-t @xmath161 energy boost associated with the spin flip ) in the transport guides located between the afp spin - flip region and the electron spectrometer . the measured @xmath0-decay rates for the spin - flipped state were @xmath162% smaller than those for the non - spin - flipped state . the maximum neutron @xmath0-decay rates measured in the spectrometer during the 20082009 runs correspond to a stored density of approximately 1 @xmath157 in the decay trap . major sources of loss are transport through the high field regions in the ppm and in the afp magnet . the transport though the ppm is about 25% . approximately half of the loss is due to polarization of the neutrons , and the other half is due to ucn absorption in the zr foil and non - specular scattering on the ucn guide walls in the high field region of the magnet which leads to enhanced wall losses . there is an approximate 15% loss in ucn density in the transition from the stainless steel to the copper guides because of the lower fermi potential of the copper . transmission through the afp magnet is about 60% , again due to non - specular scattering in the high field region . there is a 50% loss in density in loading the decay trap because the loading time ( which is determined by the aperture of the above - described 7-cm @xmath160 4-cm rectangular guide ) and decay trap lifetime are nearly the same . finally , there is an approximate factor of two loss in the transport from the biological shield exit to the decay trap due to guide losses . ( the typical loss per bounce in the guide system is @xmath163 which is dominated by gaps in the guide couplings . ) thus , all of these factors combined account for the reduction in the ucn density from its intial value of @xmath164 @xmath157 at the biological shield exist to @xmath165 @xmath157 in the spectrometer decay trap during runs in 20082009 . the decay trap consisted of a 300-cm long , 12.4-cm diameter electropolished cu tube situated along the warm bore axis of the 1.0-t solenoidal electron spectrometer . the vacuum pressure in the ucn guides downstream of the zr foil in the ppm and in the decay trap was typically @xmath166 torr . the ends of the decay trap were closed off with variable thickness mylar end - cap foils , whose inside surfaces were coated with 300 nm of be ( nominal 252 nev potential ) which served to increase the ucn storage time in the decay trap ( and , hence , the @xmath0-decay rate ) . an additional important feature of the end - cap foils is that they eliminated the possibility for neutron @xmath0-decay events in the region of the spectrometer where the field is expanded from 1.0-t to 0.6-t ( discussed later in section [ sec : experiment_electron_spectrometer_scs ] ) . collimators with inner radii of 5.84 cm mounted on the two ends of the decay trap suppressed events originating near the decay trap walls and also functioned as mounts for the end - cap foils . as discussed in more detail later , the thicknesses of the mylar end - cap foils were varied from 0.7 @xmath149 m to 13.2 @xmath149 m to study key systematics related to electron energy loss in , and backscattering from , these foils . the ucn density in the decay trap was monitored by a @xmath159he ucn detector coupled to a small ( 0.64-cm diameter ) hole in the bottom of the decay trap center , as indicated schematically in fig.[fig : ucna_schematic ] . a detailed description of the 7.0 t polarizing magnet and the afp spin - flip system is given elsewhere @xcite ; therefore , we provide only a brief description of this system here . the solenoidal superconducting magnet which serves as the primary ucn polarizer ( the afp magnet ) and provides the requisite environment for an adiabatic fast - passage ( afp ) spin flipper was designed by american magnetics using a cryostat supplied by ability engineering . it possesses a 194.9 cm long , 12.7 cm diameter warm bore and provides both a peak field of 7.0-t near the entrance as well as a 44.5-cm long precision gradient spin - flip region with an average field of 1.0 t , chosen to reduce neutron spectral differences between the flipped and unflipped spin states in the electron spectrometer decay trap volume . when energized to 96.45 a , the main coil of this magnet produces both the maximum polarizing field as well as the 1 t field , with an average gradient of @xmath167 t cm@xmath121 through the latter . ten superconducting shim coils centered on the uniform field region and spaced every 5.1 cm provide the ability to further tailor the uniform field in order to optimize performance of the spin flipper . due to the high field in the spin flip region , the spin flipper was constructed using an efficient high - pass birdcage resonant cavity geometry @xcite . for the ucna experiment , this configuration was realized with eight cu tubes ( rungs ) arranged in a cylindrical geometry and connected at the top by 820 pf american technical ceramics chip capacitors and at the bottom by a sliding cu tuning ring whose position determined the inductance presented by the cu tubes . when excited by an r.f . signal such a geometry is resonant , with the fundamental mode corresponding to a discretized sinusoidal distribution of current in the rungs . this current distribution provides a transverse r.f . field , one rotating component of which is utilized for afp spin flipping , providing efficient spin reversal over a wide band of neutron speeds @xcite . the specific operation frequency was adjusted by moving the tuning ring , and the cavity formed after tuning the spin flipper to operate at @xmath169 mhz was @xmath170 cm long ( 8.74 cm diameter ) , coaxial with a 7 cm diameter dlc - coated quartz ucn guide . the ucna spin flipper was typically operated with 40 w of input power , which necessitated an impedance - matching system comprised of a calculated length of drive line and three jennings vacuum variable capacitors . water cooling was also required , and was accomplished by flowing chilled , filtered tap water serially through the rungs . in order to provide for stable electrical operation and to prevent r.f.radiation from inducing noise elsewhere in the experiment , the birdcage cavity was driven in a balanced mode and electrically shielded by a grounded al cylindrical enclosure which also provided a vacuum seal around the dlc - coated quartz guide . the interior of this enclosure connected through four bellows to the outside of the afp magnet so that the actual r.f . cavity remained at atmosphere while the afp magnet bore and the guide system were under vacuum . this arrangement also provided feedthroughs for the r.f . drive line , water cooling lines , an rtd temperature sensor , and an r.f . field sensor loop . initial characterization of the spin flipper was performed in a crossed polarizer analyzer geometry as described in @xcite , which determined the average spin flip efficiency to be @xmath171 . during the actual running of the ucna experiment during the years 20082009 , tuning of the spin flipper , as well as run - to - run monitoring of its performance , was accomplished using a @xmath159he ucn monitor located just downstream of the afp magnet , @xmath165 m below a @xmath172 cm hole in the bottom of the guide ( the location of this ucn monitor is indicated schematically in fig.[fig : ucna_schematic ] ) . this detector had a magnetized fe foil covering the detector acceptance which provided for spin state selection . the electron spectrometer system , consisting of a 1.0-t superconducting solenoidal magnet and a multiwire proportional chamber ( mwpc ) and plastic scintillator electron detector package , is described in detail elsewhere @xcite . nevertheless , for completeness , we discuss the primary components of this system here . as described below , the electron spectrometer system was designed both to suppress the total electron backscattering fraction and to reconstruct low - energy - deposition backscattering events . the primary components of the two identical mwpc and plastic scintillator detector packages are shown in fig.[fig : mwpc_scintillator_schematic ] . the spectrometer magnet @xcite is a warm - bore 35-cm diameter , 4.5-m long superconducting solenoid ( hereafter , scs magnet ) . the coil , which was designed and fabricated by american magnetics , inc . , consists of a main coil winding with a single persistence heater switch , 28 shim coil windings ( each with individual persistence heater switches ) , and three rectangular 7-cm @xmath160 4-cm radial penetrations ( two providing horizontal access , and one providing vertical access , to the warm bore ) . these penetrations are located at the center of the coil . the magnet s 1600-l capacity liquid helium cryostat was designed and fabricated by meyer tool and manufacturing , inc . the magnet s full energized field strength of 1.0 t requires a current of 124 a in the main coil winding . note that the magnet s shim coils were not energized during the data - taking runs reported in this article . an important feature of the scs magnet design was that the field is expanded , as indicated schematically in fig.[fig : ucna_schematic ] , from 1.0 t in the decay trap region to 0.6 t at the location of the mwpc and plastic scintillator electron detectors , to suppress large - pitch - angle backscattering . in particular , the field - expansion ratio of 0.6 maps pitch angles of @xmath173 in the 1.0-t region to pitch angles of @xmath174 in the 0.6-t region . another important feature of the magnet design concerned the field uniformity in the decay trap region . electrons emitted with momentum @xmath175 , with @xmath176 ( @xmath177 ) the initial transverse ( longitudinal ) momentum component , in some local field @xmath75 will be reflected from field regions @xmath69 if @xmath178 , thus contributing to a false asymmetry . by this same process , electrons emitted at large pitch angles in the vicinity of a local field minimum will be trapped . the scs field profile measured during the data - taking period reported in this article is shown in fig . [ fig : scs_field_uniformity ] . as can seen there , the field was uniform to the level of @xmath179 over the decay trap length , but included a @xmath180 t `` dip '' near the center of the decay trap . note that the field uniformity shown here was degraded from that originally published in @xcite ; this was the result of damage to the shim coil persistence heater switches during multiple magnet quenches . the impact of this field non - uniformity on the measured asymmetry is discussed later in this article . some of the most important features of the mwpc @xcite are as follows . first , because an mwpc is relatively insensitive to gamma rays , requiring a coincidence between the mwpc and the scintillator greatly suppressed gamma - ray backgrounds , a dominant background source in previous experiments ( see , e.g. , background spectrum in @xcite ) . second , the mwpc permitted reconstruction of an event s transverse @xmath181 position . this permitted the definition of a fiducial volume and the subsequent rejection of events occurring near the decay trap walls . such electrons can scatter from the decay trap walls , leading to a distortion in the energy spectrum and/or a bias to the asymmetry . the @xmath181 position information from the mwpc also permitted a characterization of the scintillator s position - dependent response , as the scintillator was viewed by four photomultiplier tubes ( discussed in the next section ) . the 64-wire anode plane was strung with 10-@xmath149 m diameter gold - plated tungsten wires , and the two cathode planes ( oriented at @xmath173 relative to each other ) were each strung with 64 50-@xmath149 m diameter gold - plated aluminum wires . the wire spacing on both the anode and cathode planes was 2.54 mm , yielding an active area of @xmath182 @xmath183 . this area in the 0.6-t field - expansion region mapped to a @xmath184 @xmath183 = @xmath185 @xmath183 square in the 1.0-t region , thereby providing full coverage of the 12.4-cm diameter decay trap volume . as demonstrated previously @xcite , the center of the event ( i.e. , the center of the charge cloud resulting from the electron s larmor spiral in the mwpc gas ) could be reconstructed with an accuracy of better than 2 mm , sufficient for the definition of a fiducial volume . third , to suppress `` missed backscattering events '' ( i.e. , those events depositing no energy above threshold in any detector element along the electron s trajectory prior to backscattering ) , the entrance window separating the mwpc fill gas from the spectrometer vacuum was designed to be as thin as possible . fourth , because of this thin entrance window requirement , the fill gas pressure was required to be as low as possible . the chosen fill gas , c@xmath186h@xmath187 ( 2,2-dimethylpropane , or `` neopentane '' ) , a low-@xmath188 heavy hydrocarbon , was shown to yield sufficient gain at a pressure of 100 torr and a bias voltage of 2700 v. at this pressure , the minimum window thickness ( over the mwpc s 15-cm diameter entrance and exit windows ) shown to withstand this 100 torr pressure differential with minimal leaks from pinholes was 6 @xmath149 m of aluminized mylar . note that the front window was further reinforced by kevlar fibers . the plastic scintillator detector was a 15-cm diameter , 3.5-mm thick disk of eljen technology ej-204 scintillator . this 15-cm diameter mapped to a 11.6-cm diameter disc in the 1.0-t decay trap region , providing nearly full coverage of the decay trap volume . the range of an endpoint energy electron in the plastic was 3.1 mm ; therefore , the 3.5-mm thickness was sufficient for a measurement of the full @xmath0-decay energy spectrum , while minimizing the ambient gamma ray background rate . with the scintillator located in the 0.6-t field - expansion region at a distance of 2.2 m from the center of the scs magnet , light from the disc was transported over a distance of @xmath165 m along a series of uvt light guides to photomultiplier tubes which were mounted in a region where the magnetic field was @xmath189 t. the light guide system , shown schematically in fig.[fig : mwpc_scintillator_schematic ] , consisted of twelve rectangular strips ( 39-mm wide @xmath160 10-mm thick uvt ) coupled to the edge of the scintillator disc with optical grease . these twelve rectangular strips were then bent through @xmath173 over a 35-mm radius , transported over a distance of @xmath165 m away from the scintillator , and then adiabatically transformed into four @xmath190 mm@xmath191 rectangular clusters , with 5.08-cm diameter burle 8850 photomultiplier tubes ( pmts ) glued to each of these four rectangular clusters . therefore , each pmt effectively viewed one @xmath192 quadrant of the scintillator face . the magnetic shielding for each of the pmts consisted of an array of active and passive components , including ( moving from the outside to inside ) steel and medium - carbon - steel shields , a bucking solenoidal coil wound on the surface of a thin @xmath149-metal foil , and a @xmath149-metal cylinder . magnetic end caps were not required . the vacuum housing enclosing the scintillator , light guides , and pmts was maintained at @xmath193 torr of nitrogen , and was separated from the mwpc volume ( with its 100 torr of neopentane gas ) by the mwpc exit window . the nitrogen volume pressure was maintained at a somewhat lower pressure than the mwpc pressure to ensure that the mwpc exit window bowed out , or away , from the mwpc interior , to avoid contact with the mwpc wire planes . the scintillators were calibrated periodically with conversion electron sources , including commercially - available @xmath194cd ( 63 kev , 84 kev ) , @xmath195ce ( 127 kev , 160 kev ) , @xmath196sn ( 364 kev , 388 kev ) , @xmath197sr ( 499 kev ) , and @xmath198bi ( 481 kev , 975 kev , 1047 kev ) conversion - electron sources , and a custom - prepared @xmath199 in ( 162 kev , 186 kev , 189 kev , 190 kev ) conversion - electron source ( via implantation of @xmath196 in onto an al substrate and subsequent irradiation in a reactor @xcite ) . these calibrations were conducted _ in - situ _ using a vacuum load - lock source insertion system which permitted insertion and removal of calibration sources with the electron spectrometer under vacuum . the insertion point for these sources was through one of the superconducting solenoid magnet s horizontal rectangular penetrations at the center of the coil . note that this source insertion system permitted the sources to be positioned only along the horizontal axis of the decay trap s circular geometry ; however , as described later , the position dependence of the energy calibration over the full circular geometry was achieved by comparing the reconstructed neutron @xmath0-decay endpoint in a large number of binned positions over the scintillator face . the pmt gains were monitored on an approximate daily basis with a @xmath196sn source using this source insertion system . fits to the minimum - ionizing peak of cosmic - ray muons served as a run - to - run gain monitor . the electron spectrometer was surrounded with a cosmic - ray muon veto system which consisted of the following components . first , as shown in fig . [ fig : mwpc_scintillator_schematic ] , a 15-cm diameter , 25-mm thick plastic scintillator ( the `` backing veto '' ) was located immediately behind each of the spectrometer scintillators . second , a large - scale plastic scintillator and sealed drift tube veto counters @xcite surrounded the electron spectrometer magnet . the frontend electronics for the experiment consisted of a vme - based system for the event trigger logic ( via discriminators and programmable logic units ( plus ) ) and for the readout of scalers , analog - to - digital convertors ( adcs ) and time - to - digital convertors ( tdcs ) . a nim - based system coupled to the vme system was employed for the implementation of a `` busy logic '' , which served to veto event triggers arriving during adc / tdc conversion times ( i.e. , during these modules busy states ) . this busy logic also prevented re - triggering by correlated scintillator afterpulses ( mostly occuring over a @xmath165 @xmath149s window @xcite ) , and was implemented with a lecroy 222 gate generator in latch mode . a simplified schematic diagram of the trigger logic is shown in fig.[fig : trigger_logic_schematic ] . for each detector package , a trigger was defined by a two - fold pmt coincidence trigger above the discriminator threshold for each pmt ( nominally , set at 0.5 photoelectrons ) . the resulting two - fold pmt trigger rate in each scintillator was @xmath200 s@xmath121 ( primarily from low - energy background gamma rays ) ; the singles rates in each pmt as determined by counting in scalers were typically @xmath2011000 s@xmath121 ( from both dark noise and low - energy backgrounds ) . the main event trigger was then defined to be the ` or ` of the two detectors two - fold pmt triggers and other experiment triggers ( e.g. , from the @xmath159he ucn monitor detectors ) . the logic for the two - fold pmt coincidence triggers and the main event trigger was performed with caen v495 dual plus . those main event triggers not vetoed by the busy logic then triggered gate / delay generators for the readout of the adcs , tdcs , and scaler modules . the total number of two - fold pmt coincidence triggers were counted in scalers as a monitor of the daq dead time . caen v775 tdc modules were used for the relative measurement of the time - of - flight between the two detectors two - fold pmt coincidence triggers . this relative timing information provided for the identification of the detector with the earlier arriving trigger , important , as discussed later , for the assignment of the initial direction of incidence for electron backscattering events triggering both scintillators . these tdcs were also used to record the timing information from the plastic - scintillator - based muon veto detectors . a global event - by - event time stamp was defined by the counting of a 1 mhz clock in a caen v830 scaler . caen v792 charge - integrating adc ( qadc ) modules , triggered for readout by a @xmath202 ns gate from a caen v486 gate / delay generator , provided a measurement of the total charge measured in each pmt . the analog signals from the cosmic - ray muon backing vetos were also read out by these qadc modules . peak - sensing caen v785 adc ( padc ) modules , triggered for readout by a @xmath203 @xmath149s gate from a caen v462 gate generator , digitized the mwpc anode and cathode - plane signals . note that the anode signal that was read out was the summation ( i.e. , single channel per anode plane ) of the signals on all 64 of the wires comprising the anode plane . the 64 wires on each of the two cathode planes were read - out in groups of four ( i.e. , 16 channels per cathode plane ) ; hereafter , we will simply call each of these four - wire groups a `` wire '' . analog signals from the @xmath159he ucn monitors and the drift tube cosmic - ray muon veto counters were also read out with padcs . the data acquisition ( daq ) system was based on the ` midas ` package @xcite , with a dedicated linux - based workstation for implementation of the frontend electronics acquisition code and a separate dedicated linux - based workstation for run control and online analysis . the frontend acquisition code accessed the vme crate via a struck pci / vme interface . the ` midas ` raw data banks were subsequently decoded into ` cernlib ` ` paw ` @xcite and ` root ` @xcite file formats for data analysis . a separate data acquisition system , based on the ` pcdaq ` software package @xcite was used to monitor the proton beam charge incident upon the ucn source s tungsten spallation target and to asynchronously monitor environmental variables in the experimental area . the incident proton flux was measured using an integrating current toroid mounted around the proton beam line 8 m upstream of the tungsten target , just before the proton beam entered the biological shield . as noted earlier in section [ sec : experiment_source_guides ] , each proton beam pulse consisted of five 625 @xmath149s beam bursts separated by 0.05 s , with 5.2 s between each pulse s leading edge burst . the proton charge integrating system measured only the charge of the first of the five beam bursts in each pulse ; the resulting value was then scaled by five to yield the total proton charge delivered during each of the 0.2 hz beam pulses . the environmental monitoring system asynchronously read and stored up to 96 variables , on a typical time scale of 0.2 s to 1.0 s between readings . variables measured included cryogenic temperatures in the ucn source ( read by lakeshore 218 temperature monitors ) , pressures in the different segments of the ucn guide system ( read by capacitance manometer , thermocouple , and cold - cathode ion vacuum gauges ) , ambient temperature in the experimental area , and liquid helium levels and gas pressures throughout the cryogenic systems . the environmental data were time - stamped for later comparison to the @xmath0-decay data acquired with the main data acquisition system . in this section we provide a detailed description of our measurement procedures for @xmath0-decay and ambient background runs ; the various geometrical configurations of the experiment during our @xmath0-decay runs ; and our procedures for , and results from , measurements of the neutron polarization . the data taking during normal @xmath0-decay production running was organized into octets , each consisting of a- and b - type quartet run sequences . the structure of these quartet and octet run sequences , shown in table [ tab : octet_structure ] , was such that the neutron spin state ( hereafter designated @xmath205 or @xmath206 , with @xmath205(@xmath206 ) corresponding to the loading of ucn with afp - spin - flipper - on ( -off ) spin states into the electron spectrometer ) was toggled according to a @xmath207 spin - sequence ( for octets in which a - type runs preceded b - type runs ) or a @xmath208 ( i.e. , complement ) spin - sequence , with the order of @xmath0-decay and ambient background run pairs toggled for a particular spin state within each a - type or b - type run sequence . within each octet , the decision for whether the a - type runs would precede or follow the b - type runs was made randomly . the notation in table [ tab : octet_structure ] is such that b@xmath209 and @xmath210 denote , respectively , ambient background and @xmath0-decay runs for the two spin states . the notation for depolarization runs is such that d@xmath211 , for example , denotes a measurement of the depolarized spin - state population for which the spin - state was polarized in the @xmath205 spin - state during the preceding @xmath0-decay run . .run structure for the octet data taking sequence , consisting of a- and b - type quartets . see text for details . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^ " , ] lcccc + & a [ % ] & b [ % ] & c [ % ] & d [ % ] + @xmath212 & @xmath213 & @xmath214 & @xmath215 & @xmath216 + @xmath217 & @xmath218 & @xmath219 & @xmath220 & @xmath221 + @xmath222 & @xmath223 & @xmath224 & @xmath225 & @xmath226 [ tab : systematic_corrections_uncertainties ] our systematic corrections and uncertainties are summarized in table [ tab : systematic_corrections_uncertainties ] , where we have categorized the effects as either geometry - dependent ( i.e. , effects which varied with the decay trap end - cap foil and mwpc window thicknesses , measured detector thresholds , etc . ) , or geometry - independent ( e.g. , ucn polarization , dead time effects , etc . ) in the rest of this section we discuss each of these systematic effects ( in the order in which they appear in table [ tab : systematic_corrections_uncertainties ] ) in more detail . nearly all dead time effects cancel in the super - ratio technique . indeed , in order for there to be any bias to the asymmetry resulting from dead time effects in the background - subtracted @xmath0-decay rates , there must be a difference in the two detectors dead times , and there must be a difference in a particular detector s dead time for the two neutron spin states . thus , these effects are expected to be quite small . nevertheless , as previously noted in section [ sec : experiment_electronics ] , the dead time of the daq system was monitored by counting , in scalers , the total number of detector two - fold pmt triggers , including those that were vetoed by the daq `` busy logic '' during the @xmath203 @xmath149s readout gates for the padc modules . however , to avoid spurious ( and correlated ) trigger chains from scintillator afterpulses ( as noted earlier in section [ sec : experiment_electronics ] ) distorting the determination of the dead time , the dead time was determined only from the scaler counts of detector triggers that occurred during triggers from the opposite - side detector or from other experimental triggers , such as the ucn monitors . the dead time , as extracted from the correlation between the daq trigger rate and the fraction of these `` missed triggers '' , was found to be @xmath227 @xmath149s , which is consistent with the nominal @xmath203 @xmath149s system dead time ( associated with the gate for the padc readout ) . further , the difference in the fraction of `` missed triggers '' for ( up to ) a 20 s@xmath121 trigger rate difference between the two spin states is no larger than @xmath189% . considered together , any possible bias to the asymmetry was no greater than 0.01% , which is the error we quote in table [ tab : systematic_corrections_uncertainties ] . alternatively , another possible way dead time effects could bias the asymmetry is in the background subtraction procedure , resulting from differences in the daq total trigger rates during @xmath0-decay and background runs . however , these effects tend to cancel in the super - ratio , as the four background - subtracted @xmath0-decay rates appearing in the super - ratio would be expected to be biased in the same direction . further , the effect is minimized as the signal - to - background ratio increases . under the conservative assumption of a 200 s@xmath121 daq trigger rate difference ( e.g. , from differences in the scintillator trigger rates , ucn monitor trigger rates , etc . ) for @xmath0-decay versus background runs , and a signal - to - background ratio greater than 5 , any such systematic bias to the asymmetry from dead time effects is @xmath228% . figure [ fig : energy_reconstruction_error_envelope ] showed the error envelope for the uncertainty in the visible energy calibration . to estimate the systematic error associated with possible errors in our energy calibration , we generated a large number ( 200 per geometry ) of random error curves that were constrained to fit within the limits of this error envelope . we then extracted from these error curves their contributions to an error in the asymmetry , resulting from an incorrect reconstruction of the electron energy , and hence @xmath229 . from these calculations we concluded that the maximum ( i.e. , worst case ) error , resulting from the case where the error curves for the two detectors are identical , is a fractional 0.47% uncertainty in the asymmetry for the analysis energy window of 275625 kev . as a conservative estimate of the systematic uncertainty associated with our energy calibration , we then assign this worst - case error of 0.47% to be the systematic uncertainty associated with possible errors in our energy calibration . as discussed earlier in section [ sec : analysis_initial_energy_reconstruction ] , the default @xmath230 parametrization we employed was based on a fit to the scintillator visible energy @xmath231 only ; by constrast , an alternative @xmath230 fit included both @xmath231 and the calibrated mwpc energy @xmath232 . to study the sensitivity of the reconstructed asymmetry to these two different @xmath230 parametrizations , we extracted values for the energy - corrected asymmetry @xmath233 for these two different fits . the difference between these two methods , averaged over the entire data set , was 0.2% . this is small relative to the 0.47% systematic uncertainty associated with the energy calibration , and we also noted in section [ sec : analysis_initial_energy_reconstruction ] that this alternative @xmath230 parametrization based on both @xmath231 and @xmath232 is subject to ( uncorrectable ) overflow of the mwpc anode readout . another source of a systematic error resulting from the energy reconstruction as discussed in detail in section [ sec : asymmetry_endpoint_distributions ] ( and shown in fig.[fig : kurie_fit_distributions ] ) was the observed systematic @xmath23414 kev difference between the fitted endpoints and the monte carlo prediction . we investigated the systematic uncertainty to the extracted asymmetry due to this systematic difference by extracting a `` stretching factor '' , @xmath235 , where @xmath236 denotes the monte carlo predicted endpoint and @xmath237 the fitted endpoint , for each run . the data were then re - analyzed by applying on an event - by - event basis this `` stretching factor '' to the reconstructed energy @xmath230 , thus effectively forcing the fitted endpoints to match the monte carlo predicted endpoints . the asymmetries extracted from the `` streched '' data differed by @xmath238% from the ( original ) `` unstretched '' data which , again , is much less than the 0.47% error associated with the energy calibration . finally , to account for a slight mismatch ( @xmath150 kev ) between the monte carlo and measured energy spectra ( this is visible in the final @xmath230 spectrum later in fig.[fig : final_spectra_asymmetries ] ) we fitted the monte carlo visible energy spectra to the measured visible energy spectra , and then extracted values for the asymmetry assuming these modified values for the visible energy . the bias to the asymmetry was @xmath239% averaged over all four geometries which , again , is much less than the 0.47% the energy calibration uncertainty . as discussed in section [ sec : analysis_position_cuts ] , we required backscattering events to satisfy a default vertex cut of @xmath240 mm . we studied the impact of this cut on the asymmetry by varying this cut from 10 mm to 40 mm ; the effect on the asymmetry was @xmath241% , indicating a negligible systematic effect . as was also discussed there , our fiducial cut required the position ( radius ) of the event on the primary triggering scintillator side to satisfy @xmath242 mm . to examine whether there was any position bias , we extracted the asymmetry in successive annuli via cuts on @xmath243 in six different annular bins , ranging from [ 0,400 ] mm@xmath191 to [ 2025,2500 ] mm@xmath191 the asymmetries in all of these annuli were in statistical agreement , with no statistical evidence for any systematic difference with position . recall also in section [ sec : analysis_position_cuts ] we noted the possibility for the definition of four different coordinate systems . to determine whether there was any bias resulting from the choice of the coordinate system ( for example , a consideration could be whether there were any systematic variations in the backscattering fractions in the vicinity of the fiducial cut ) , we studied the variation of the asymmetry with the choice of coordinate system , and for fiducial volume radius cuts of 45 mm and 50 mm . the rms spread in the asymmetries for the different coordinate system choices was 0.24% for the 45 mm radius cut and 0.21% for the 50 mm radius cut . although the rms spread for the 50 mm radius cut was actually somewhat smaller ( suggesting that employing a larger fiducial volume would have introduced no bias to the asymmetry ) , we nevertheless chose the 45 mm radius cut as our ( conservative ) definition of the fiducial volume , and thus assigned a 0.24% systematic uncertainty to the definition of the fiducial volume . as noted in section [ sec : experiment_calibration_gain ] , the pmt gains were monitored on a run - to - run basis using the minimium - ionizing peak from cosmic - ray muon events . nevertheless , any residual uncompensated run - to - run gain fluctuations could bias the asymmetry on a run - to - run basis ; however , any such short - term run - to - run fluctuations will average away according to the usual @xmath244 statistics assuming the long - term gain corrections are accurate . we estimated the level of any such run - to - run residual gain errors by extracting the level of fluctuations in the run - to - run fitted values for the @xmath0-decay spectrum endpoint . these were typically of order @xmath245% in each detector , with the gain fluctuations in the two detectors only slightly correlated relative to each other . [ correlated gain fluctuations are significantly more problematic than are anti - correlated gain fluctuations . ] conservatively assuming the worst - case sensitivity for gain fluctuations in one of the geometries to be representative of the entire data set , we quote a systematic uncertainty of 0.20% for uncompensated gain fluctuations . the detector rates ( and , hence , asymmetries ) were ultimately calculated from the number of events passing the analysis cuts normalized to the detectors respective live times ; the concept of the detector live time was discussed in detail previously in section [ sec : analysis_data_quality_cuts ] . as discussed there , we defined a run s live time to be the fraction of that run surviving all of the global data quality cuts . however , as we noted there , it was necessary to apply a correction for the geometry b live times due to the large fraction ( up to @xmath246% ) of events suffering from an event - by - event tdc event counter problem . the correction factors for each run were determined using events identified as gamma rays , which were statistically independent of the neutron @xmath0-decay events and also provided higher statistics ( event rates up to 100 s@xmath121 in each detector ) than the neutron @xmath0-decay events themselves for the calculation of the correction factors . the resulting correction factors , defined to be the ratio of the number of gamma ray events surviving the event - by - event tdc event counter cut to the total number of gamma ray events , were then computed on a run - by - run basis for each detector . only the geometry b live times were corrected according to this procedure . nevertheless , to assess the systematic error associated with our definition of and calculation of the live time , we extracted values for the asymmetries for all four geometries with and without application of these live time correction factors ( the correction factors for geometries a , c , and d were small , with the values for the asymmetries differing by @xmath247% under the two scenarios ) . averaged over all four geometries , the difference between the asymmetries extracted under these two different scenarios was 0.24% , which is the value for the systematic uncertainty associated with this effect we quote in table [ tab : systematic_corrections_uncertainties ] . our monte carlo calculations of the corrections for backscattering and the @xmath248 acceptance discussed in section [ sec : corrections_overview ] assumed a uniform magnetic field in the decay trap region . we studied the impact of the actual measured nonuniformity in the spectrometer magnetic field shown previously in fig . [ fig : scs_field_uniformity ] in monte carlo . qualitatively , the impact of the @xmath246 gauss `` field dip '' in the central decay - trap region is such that electrons from decays occuring in this `` field dip '' region are either reflected ( analogous to backscattering ) or are trapped ( for large pitch angles ) . we studied these effects in our ` geant4 ` monte carlo simulation program by implementing the magnetic field profile shown in fig.[fig : scs_field_uniformity ] directly in the simulation . neutron @xmath0-decay events were then generated uniformly along the length of the decay trap . in the monte carlo , @xmath249% of the events incident initially on one of the two detectors were reflected from the field dip , with an average @xmath250 of @xmath251 . because this small fraction of events carries little @xmath248 `` analyzing power '' , the resulting bias to the asymmetry is negligible . the fraction of electrons trapped by the field dip was @xmath252% , again with an average @xmath229 of @xmath251 . the remaining 97.1% of the events were not impacted by the field dip . assuming that the electrons trapped by the field dip eventually scatter from residual gas molecules , the impact is a dilution to the asymmetry . the calculated dilution to the asymmetry was @xmath253% . in lieu of applying a correction to the asymmetry , we assigned a @xmath254 systematic uncertainty to this effect . we also note that our monte carlo calculations found that the time for a trapped electron to scatter from residual gas for a vacuum pressure of @xmath255 torr is @xmath256 ms , with a small distortion to their energy distribution of @xmath257 kev . note that our monte carlo results for the fraction of events trapped by the field dip and their average value of @xmath229 are consistent with the following simple estimates . as discussed earlier in section [ sec : experiment_electron_spectrometer_scs ] , electrons emitted with some momentum @xmath258 , with @xmath176 ( @xmath177 ) the initial transverse ( longitudinal ) momentum component , in some local field @xmath75 will be reflected from higher field regions @xmath69 if @xmath259 ( thus , only the pitch angle @xmath260 of the emitted electron is relevant , not the magnitude of the momentum ) . taking @xmath261 t and @xmath262 t for the measured 2009 field profile ( here , @xmath69 is taken to be the average of the local maxima at @xmath263 cm and @xmath264 cm ) shown in fig.[fig : scs_field_uniformity ] , one finds electrons with pitch angles @xmath265 will be trapped in the field dip region . approximating the initial angular distribution of emitted electrons as isotropic ( reasonable , given that the @xmath0-asymmetry is an @xmath266 effect ) , one finds that the fraction of electrons emitted in the local field dip region @xmath261 t which will be trapped is @xmath267 then , assuming a uniform distribution of events along the 300-cm long decay trap , the fraction of events emitted in the @xmath148-cm long field dip region is @xmath268 , implying the total fraction of events generated over the length of the decay trap which will be trapped in the field dip region is @xmath269 , which is consistent with the monte carlo result of 2.6% . for a nominal value of @xmath138 , @xmath270 = 0.02 $ ] for these trapped events , again , consistent with the monte carlo result . we estimated the effect of a possible systematic uncertainty resulting from fluctuations in the muon - veto efficiency by extracting values for the asymmetries with and without application of the muon - veto detector cuts . averaged over geometries , the variations in the asymmetry were at the 0.3% level . we note that the assignment of this 0.3% uncertainty is quite conservative . a linear drift in the muon veto cut efficiency would be equivalent to a linear drift in the backgrounds , and as discussed in section [ sec : asymmetry_asymmetry_extraction ] , linear background drifts cancel under the octet - based super - ratio asymmetry structure . as already dicussed in detail , ambient backgrounds were measured and subtracted on a run - by - run basis . however , a possible source of irreducible backgrounds was neutron capture on materials near the electron detectors , generating prompt gamma rays with energies up to 7.9 mev , 7.1 mev , 6.8 mev , 4.9 mev , or 8.2 mev for capture on @xmath271cu , @xmath272cu , @xmath273be , @xmath274c , or @xmath275c , respectively , the elements of which the decay trap and end - cap foils were primarily composed . such backgrounds can not , of course , be subtracted . this background was expected to be significantly suppressed in the ucna experiment as compared to previous cold neutron beam experiments because , as discussed earlier in section [ sec : experiment_overview ] , the fraction of neutrons present in the apparatus which contribute to the decay rate is orders of magnitude larger in the ucna experiment than in previous cold neutron beam experiments , and also because of the small probability for capture and upscatter by ucn stored in the decay trap . we carried out three different approaches to our assessment of the contamination level from any such backgrounds . the idea of our first approach is as follows . if a gamma ray emitted from a neutron capture subsequently interacted with the scintillator , the mwpc should not have recorded any energy deposition if the gamma ray forward compton scattered in the scintillator . further , as calculated in simulations , there is a factor of 1020 suppression in the fraction of gamma ray events incident on the electron detectors triggering both the scintillator and mwpc as compared to those triggering only the scintillator . therefore , any such neutron - generated backgrounds should appear as non - zero residuals in a comparison of background - subtracted scintillator spectra [ i.e. , ( @xmath0-decay run @xmath206 background run ) spectra ] formed with and without application of a mwpc - scintillator coincidence cut . in particular , an excess would be expected in the background - subtracted spectrum constructed without the requirement of a mwpc - scintillator coincidence cut as compared to the background - subtracted spectrum obtained with the requirement of a mwpc - scintillator coincidence cut . now consider the following model . under application of a mwpc coincidence cut , let @xmath276 where @xmath277 denotes the resulting background - subtracted scintillator event rate , @xmath278 and @xmath279 denote , respectively , the underlying @xmath0-decay + background and background event rates , respectively , and @xmath280 denotes the mwpc cut efficiency . we then write a similar expression for the background - subtracted scintillator event rates obtained without application of a mwpc cut as @xmath281 where now @xmath282 and @xmath283 denote the signal and background rates during the @xmath0-decay and background runs from gamma ray events which would otherwise fail the mwpc cut . note that in the absence of any neutron - generated gamma rays , the statistical averages of @xmath282 and @xmath283 should be identical . the difference between @xmath284 and @xmath277 is then @xmath285 where @xmath286 and @xmath287 denote , for gamma ray and neutron @xmath0-decay events , respectively , the difference between the background - subtracted scintillator rates with and without application of a mwpc coincidence cut . thus , this model then requires an estimate for the mwpc cut efficiency . we extracted a value for our mwpc cut efficiency by examining the mwpc anode spectrum for @xmath196sn source calibration data . after placing a fwhm cut on the scintillator visible energy spectrum , we then fitted the resulting mwpc spectrum ( such as shown , for example , in fig.[fig : anode_cathode_spectra ] ) to a landau distribution , and its pedestal to a gaussian . we then calculated the fraction of the landau distribution falling below the cut line , which should provide an estimate of the mwpc cut efficiency . these values were found to be 99.93(3)% and 99.95(3)% for the east and west detector , respectively . we do note that this provides for an estimate of the mwpc efficiency only at the @xmath196sn energy . the residual @xmath288 ( no mwpc cut @xmath206 mwpc cut ) rates integrated over the analysis energy window ( and after all analysis cuts ) ranged from @xmath289 s@xmath121 to @xmath290 s@xmath121 for the two detectors and two spin states for all four geometries , representing @xmath291 of the @xmath0-decay rates . after accounting for the factor of 1020 supression for the fraction of events which would trigger both the scintillator and mwpc , the estimated contamination fractions for the actual @xmath0-decay analysis employing the mwpc - scintillator coincidence cut are then on the order of @xmath292 . propagation of the measured contamination fractions ( for each detector and spin state ) through the super ratio then led to a systematic bias to the asymmetry of order @xmath251% . in our second approach , we extrapolated the residual background ( i.e. , after background subtraction ) above the @xmath0-decay endpoint into the signal region . above the endpoint , the residual background - subtracted rates in the 25 kev @xmath230 bins were typically of order @xmath292 s@xmath121 or less . under the assumption that the neutron - generated background is independent of energy ( e.g. , as was employed in the analysis of @xcite ) , we then extrapolated these above - the - endpoint rates into the analysis energy window . the resulting contamination fractions , @xmath293@xmath294 , were similar to the analysis in our first approach comparing background - subtracted scintillator spectra obtained with and without a mwpc cut . these contamination fractions were again propagated through our super - ratio asymmetry analysis , and the systematic bias was again found to be of order @xmath251% . finally , in our third approach , we carried out a `` beta - blocker '' measurement in which a 6.35-mm thick piece of acrylic was placed between the decay trap and the mwpc in the field - expansion region of the spectrometer , as indicated schematically in fig . [ fig : beta_blocker_schematic ] . the idea for this measurement was two - fold : ( 1 ) the acrylic was sufficiently thick to stop the endpoint @xmath0-decay electrons , thereby `` blocking '' the @xmath0-decay signal of interest ; and ( 2 ) the acrylic then served as a `` source '' of compton - scattered electrons , resulting from interactions of neutron - generated gamma rays with the acrylic . measurements were conducted with this piece of acrylic at two different positions , a and b , as shown in fig.[fig : beta_blocker_schematic ] , in front of one of the detectors . the motivation for doing so was that a comparison of the results from positions a and b should , in principle , permit a decomposition of the measured detector signal into contributions from compton - scattered electrons originating in this acrylic piece ( the solid angle for which was clearly smaller in position b as compared to position a ) , and direct neutron - generated gamma - ray interactions in the plastic scintillator detector ( which should not have varied with the position of the acrylic piece ) . note that this measurement is subject to some model dependence , including an assumption for the source positions along the decay trap of the neutron - generated backgrounds ( which determines the ratio of the solid angles for positions a and b ) . our resulting estimates for the contamination fraction , as extracted from our measurements of the residual ( background - subtracted ) rates with the acrylic piece located at both positions a and b ( under the assumption that the ratio of the a and b solid angles for production of compton - scattered electrons was 20:1 ) , were of order @xmath294 . then accounting for the factor of 1020 suppression for gamma ray events triggering both the scintillator and mwpc ( in the actual geometry ) leads to an estimate for the contamination fraction on the order of @xmath292 , consistent with the other two approaches . the ucn polarization systematic was discussed earlier in section [ sec : measurements_polarization ] . because the measured depolarization was consistent with zero at the @xmath295 level ( i.e. , @xmath296 ) , we did not apply a correction for the polarization , and instead quote a one - sided systematic uncertainty in @xmath1 of @xmath297 resulting from the constraint @xmath298 . a potential systematic effect would arise from any spin - state - correlated systematic gain shifts , such as from rate - dependent gain shifts . however , any such gain shifts shared by both detectors cancel to first order in the super ratio asymmetry and are thus expected to be small . as noted earlier in section [ sec : experiment_source_guides ] , during the operation of the experiment , the total daq recorded data rates ( i.e. , the `` online '' @xmath0-decay rates integrated over all energies with few cuts ) during @xmath0-decay runs for the non - flipped spin state were typically @xmath299 s@xmath121 greater than those recorded during measurements of the flipped spin state . by taking data with calibration sources with different activities , we were able to bound any such rate - dependent gain shifts to then be @xmath300%/(@xmath299 s@xmath121 ) , which corresponds to a systematic uncertainty in the asymmetry of @xmath301% . our working assumption was that application of the @xmath304 and @xmath305 scale factors to the ` geant4 ` backscattering distributions calibrated our monte carlo calculations of our @xmath212 backscattering corrections to the asymmetry . to estimate the uncertainty in these now - calibrated corrections , we compared the ` geant4 ` results for the @xmath212 backscattering correction with the ` penelope ` results . [ note that the ` penelope ` calculations required @xmath304 and @xmath305 scale factors of 0.91.1 and 1.3 , respectively , somewhat smaller than those required by ` geant4 ` . ] these agreed to better than 22% for geometry a , and better than 6% for geometries b , c and d. we also note that the rms of the @xmath212 corrected asymmetries for analysis choices 15 was 0.10% , 0.13% , 0.30% , and 0.27% for geometries a , b , c , and d , respectively , providing a powerful check of the robustness of the calculation of the @xmath212 correction ( indeed , consistent with the robustness of the agreement between the measured and simulated asymmetries for the various analysis choices demonstrated previously in fig . [ fig : asymmetries_analysis_choice_geometries ] ) . we have taken a conservative approach to our estimate of the systematic uncertainty in our backscattering corrections , and quote a 30% relative uncertainty in the @xmath212 backscattering correction ( and , thus , in the asymmetry ) for all of the geometries . to estimate the systematic uncertainty in the @xmath217 angle effect correction , we varied the thickness of the decay trap end - cap foil thicknesses in the monte carlo . assuming 0.5 @xmath149 m to be a reasonable uncertainty in the foil thickness , the relative uncertainty in @xmath217 was found to be no larger than 25% further , we note that the ` geant4 ` and ` penelope ` results for @xmath217 agreed to better than @xmath162% for all of the geometries . therefore , we again quote a conservative 25% relative uncertainty in the @xmath217 angle effects correction for all of the geometries . figure [ fig : delta2_delta3_error_bands ] shows the resulting energy - dependent error bands for the combined @xmath212 and @xmath217 corrections for each of the geometries . the analysis energy window of 275625 kev referenced earlier in section [ sec : corrections_analysis_energy_window ] was chosen such that the total error in @xmath1 resulting from integration of these error bands over some energy window combined with the statistical error within that energy window was a global minimum . as discussed earlier in section [ sec : analysis_mwpc_efficiency ] , the `` standard '' mwpc cut for the separation of gamma rays and charged particles was a cut on a fixed padc channel number . however , as shown there , the mwpc exhibited a strong position - dependent response . therefore , for some particular energy deposition in the mwpc , employing such a standard cut resulted in a position - dependent efficiency for the identification of an event as either a gamma ray or charged particle event . we investigated the impact of this position - dependent efficiency on the asymmetry by comparing results for the asymmetry extracted from analyses employing the standard padc channel number cut with those obtained with fixed mwpc energy cuts ranging from 0.10.7 kev . the error bounds from this analysis appear as the mwpc efficiency systematic uncertainties in table [ tab : systematic_corrections_uncertainties ] . note that this effect is largest for geometry d , for which the differences between the two mwpcs efficiency curves was greatest , as shown previously in fig . [ fig : mwpc_efficiency_energy_deposition ] . our final results are shown in fig.[fig : final_spectra_asymmetries ] , where we compare the background - subtracted @xmath0-decay @xmath230 spectrum , summed over both detectors , averaged over the two neutron spin states , and averaged over all four of the geometries , with the monte carlo predicted @xmath230 spectrum . there , we also show our geometry - averaged energy - binned values for the @xmath0-asymmetry @xmath1 , and the final statistical result of @xmath306 . note that the central value for @xmath1 was insensitive to the choice of analysis energy window , with the variation less than 15% of the statistical uncertainty for other windows between 150 and 750 kev . each of the four geometries yielded a data set with a statistical error and a systematic error . the statistical errors for each of these geometries were , of course , independent . however , three of our dominant systematic uncertainties ( energy reconstruction , backscattering , and angle effects ) were all correlated . for example , a mistake in the decay trap end - cap window thickness would have biased the angle effects correction for all of the data sets . therefore , we assume all three of these systematic uncertainties are 100% correlated across the four different geometries . under this assumption , we then combined the results from the four geometries according to the following procedure which incorporates correlations properly in the construction of a global @xmath307 @xcite . each individual measurement @xmath308 gives a constraint of @xmath309 where @xmath310 and @xmath311 denote , respectively , the central value and statistical uncertainty for the @xmath312 measurement ( i.e. , geometry ) , and the @xmath313 denotes the correlated uncertainty due to the three systematic effects just discussed . we then constructed a @xmath307 as @xmath314 where @xmath308 and @xmath315 are the indices of the measurements , and @xmath316 is the covariance matrix with @xmath317 . as was shown in @xcite , the @xmath307 constructed this way satisfies a standard @xmath307 distribution , in which the value of @xmath1 follows from minimization of this @xmath307 , and the one - sigma uncertainty is obtained from the usual condition @xmath318 . using standard minimization techniques , the final combined result for @xmath1 we obtained is @xmath319 \nonumber\end{aligned}\ ] ] where the first ( second ) error represents the statistical ( systematic ) error . from this , we extract the following value for @xmath320 under the standard model , @xmath321 in this article we have presented a comprehensive and detailed description of the first precision result @xcite from the ucna experiment , an experiment designed to perform the first - ever measurement of the neutron @xmath0-asymmetry parameter @xmath1 with polarized ultracold neutrons . as demonstrated here , the use of ucn in a neutron @xmath0-asymmetry experiment controls key neutron - related systematic corrections and uncertainties , including the neutron polarization and neutron - generated backgrounds . our result for the neutron polarization was shown to only be statistics limited , and our neutron - generated backgrounds were negligible to the level of @xmath241% precision . all of our results reported here are consistent with our previously published proof - of - principle results obtained during data - taking runs in 2007 @xcite . to evaluate the immediate impact of this work , we first note , as shown earlier in fig . [ fig : status_beta_asymmetry ] , that our measurement agrees well with the most recent ( and most precise ) published result for the @xmath0-asymmetry @xmath1 from the perkeo ii experiment @xcite , but is in poorer agreement with the three other results @xcite employed by the particle data group in their averaging procedure . because of the differences between the perkeo ii and ucna experimental techniques , we believe this is a significant result . second , the value for @xmath322 that we extract from our measurement is @xmath323 . we compare our value for @xmath322 with results from a global fit under the standard model for @xmath322 values extracted from results for the @xmath0-asymmetry @xmath1 @xcite , a simultaneous measurement of @xmath1 and the neutrino asymmetry @xmath75 @xcite , and from individual measurements of the neutron lifetime @xcite and the current particle data group average value for the lifetime ( @xmath324 s @xcite ; recently updated for the corrected result of @xcite which supersedes the original result of @xcite ) , where the lifetime results assume the superallowed @xmath325 value for @xmath9 of @xmath326 @xcite in eq.([eq : lifetime - vud ] ) . the results of our global fit are displayed as an ideogram in fig . [ fig : ideogram_ga ] , where it can be seen that under the standard model the perkeo ii @xcite and ucna @xcite @xmath0-asymmetry experiments are in agreement with the three most recent ( or updated ) results for the neutron lifetime reported from experiments with stored ucn @xcite , but in poorer agreement with the other results for the lifetime employed by the particle data group in their averaging procedure obtained in experiments utilizing stored ucn @xcite and cold neutron in - beam @xcite techniques . indeed , from our result for @xmath1 alone of @xmath327 , we extract , according to eq . ( [ eq : lifetime - vud ] ) , a value for the neutron lifetime of @xmath328 in agreement with the measured values from the three most recent results reported from experiments using stored ucn @xcite . thus , we conclude that our @xmath0-asymmetry measurement already provides significant impact to a self - consistent evaluation of the landscape of neutron @xmath0-decay observables and the superallowed @xmath99 @xmath9 data set . the fact that the most recent values for the @xmath0-asymmetry and the neutron lifetime currently exhibit statistically significant deviations from their respective world averages prepared by the particle data group , but are seen to be in agreement with each other ( as shown in fig.[fig : ideogram_ga ] ) , motivates further refinement of the ucna technique , with its novel approach to key neutron - related systematic errors , in order to conduct a more precise evaluation of neutron @xmath0-decay observables . indeed , recently demonstrated improvements to our ucn source and refinements to the energy calibration and gain monitoring systems will permit the future collection of a data set with significantly improved statistical and systematic uncertainties . this work was supported in part by the department of energy office of nuclear physics ( grant number de - fg02 - 08er41557 ) , the national science foundation ( grant numbers nsf-0555674 , nsf-0855538 , nsf-0653222 , nsf-1005233 ) , and the los alamos national laboratory ldrd program . we gratefully acknowledge the support of the lansce and aot divisions of los alamos national laboratory .

we present a detailed report of a measurement of the neutron @xmath0-asymmetry parameter @xmath1 , the parity - violating angular correlation between the neutron spin and the decay electron momentum , performed with polarized ultracold neutrons ( ucn ) . ucn were extracted from a pulsed spallation solid deuterium source and polarized via transport through a 7-t magnetic field . the polarized ucn were then transported through an adiabatic - fast - passage spin - flipper field region , prior to storage in a cylindrical decay volume situated within a 1-t @xmath2 solenoidal spectrometer . the asymmetry was extracted from measurements of the decay electrons in multiwire proportional chamber and plastic scintillator detector packages located on both ends of the spectrometer . from an analysis of data acquired during runs in 2008 and 2009 , we report @xmath3 , from which we extract a value for the ratio of the weak axial - vector and vector coupling constants of the nucleon , @xmath4 . complete details of the analysis are presented .

introduction status of measurements of @xmath113 ucna experiment measurements, experimental geometries, and polarization summary of final results summary and conclusions

arxiv

the problem of the formation and evolution of dwarf galaxies has partly been motivated by the long - standing questions about possible connections and transformations between different types of dwarf galaxies . the two basic morphological types are : gas - poor dwarf spheroidals with very little present - day star formation , and gas - rich dwarf irregulars ( di s ) and blue compact dwarf galaxies which contain recent bursts of star formation . intrinsic properties ( e.g. , total mass ) and external forces ( e.g. , environment ) all have a role to play , but key processes may be difficult to identify and disentangle ( e.g. , @xcite ) . the study of dwarf galaxies in different environments of varying galaxy number density ( i.e. , field , groups , clusters ) provides valuable insights about the key parameters controlling evolution , and constraints to the various models for galaxy evolution . the centaurus a ( cen a ) group is a loose collection of galaxies , and contains a very rich population of galaxies with the largest dispersion in morphological types @xcite . the average distance of the cen a group is comparable to that of the m81 group ( cf . the cen a group may already be virialized as indicated by the relatively short crossing timescale ( @xmath7 23 gyr ; @xcite ) . the cen a group is separated into two `` subgroups '' ( * ? ? ? * their fig . 1 ) with one collection of galaxies surrounding ngc 5128 and the other collection surrounding the spiral galaxy m 83 . the m 83 collection is more compact and contains more late - type galaxies , whereas the ngc 5128 collection is more dispersed and contains fewer late - type galaxies . the ngc 5128 subgroup has a heliocentric velocity @xmath8 km s@xmath9 and an average distance @xmath10 mpc . there are 31 known members , of which ten ( 32% ) are classified as late - type dwarf galaxies ( rc3 morphological type @xmath11 ; @xcite ) . the m 83 subgroup has @xmath12 km s@xmath9 and an average distance @xmath13 mpc . there are 19 known members , of which 12 ( 63% ) are classified as late - type dwarf galaxies ( @xmath11 ) . the cen a group is uniquely dominated by ngc 5128 ( centaurus a ) , which is a large massive radio - loud elliptical galaxy ( e.g. , @xcite ) . the remaining massive members in the group all exhibit disturbed morphologies or abnormal properties , suggestive of a recent infall episode in which a population of gas - rich dwarf galaxies has been accreted into the group ( e.g. , @xcite ) . this is precisely the scenario presented by @xcite , who discovered a long , thin , blue arc in the northeast halo of ngc 5128 . this feature is thought to have once been a low - mass di which fell into the halo of the elliptical and is undergoing tidal disruption . this may also help explain why there are relatively few di s in the vicinity of ngc 5128 , compared to m 83 @xcite . additional studies of dwarf galaxies in the cen a group have been carried out in ( s. ct et al . , in prep . ) and in ( @xcite ) to examine the total and spatial distribution of gas and recent star - formation . @xcite studied the red giant stellar populations in two cen a dwarf elliptical galaxies , and found the fraction of intermediate - age stars to all stars is smaller than that found in local group dwarf ellipticals . @xcite identified 13 new dwarf elliptical galaxies , and confirmed the membership of two dwarf irregular galaxies in the group . they also identified am 1318@xmath0444 and eso 381@xmath0g018 as two new dwarf irregulars in the cen a group . @xcite examined three gas - rich dwarf spheroidals with old ( @xmath14 gyr ) stellar populations , and showed that the relatively high gas content could be explained by the low level of past star formation . oxygen abundances are a reliable measure of the present - day gas - phase metallicity within regions in gas - rich star - forming dwarf galaxies ( e.g. , @xcite ) . in fact , they provide useful constraints as the most recent and maximum metallicity to anchor the star - formation history . @xcite and @xcite obtained oxygen abundances for southern irregular and spiral galaxies , including galaxies in the cen a group and in the field , with a measured range of abundances from about ten to sixty per cent of the solar value . although these papers have been , until recently , the primary work regarding abundances in cen a late - type dwarf galaxies , the spectra were obtained with inherently non - linear detectors . in many cases , the character of the non - linearities were not understood until after publication ( e.g. , @xcite ) ; subsequent corrections for non - linearity were not possible . for a comparison of gas - phase abundances with published work on nearby dwarf galaxies ( e.g. , @xcite ) , we undertook a program of determining new and confirming previous nebular oxygen abundances for gas - rich dwarf galaxies in the local volume @xcite , which would help answer the question about whether recent chemical enrichment is sensitive to the ( group ) environment . here , we present spectra of 35 regions in eight cen a di s , as well as spectra of 13 regions in three additional nearby dwarf galaxies . the properties of galaxies in the present sample are listed in table [ table_gxylist ] . lcccccccccc & other & & @xmath15 & @xmath16 & @xmath17 & @xmath18 & & & & 12@xmath1 + galaxy & name(s ) & @xmath19 & ( km s@xmath9 ) & ( mag ) & ( jy km s@xmath9 ) & ( mpc ) & method & @xmath20 & md & log(o / h ) + ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) & ( 6 ) & ( 7 ) & ( 8) & ( 9 ) & ( 10 ) & ( 11 ) + + am 1318@xmath0444 & kk 196 & 10 & @xmath21 & 16.1 & & @xmath22 & trgb & @xmath23 & cen a & @xmath24 + am 1321@xmath0304 & kk 200 , kdg 15 & 9 & @xmath25 & 16.67 & 1.7 & @xmath26 & trgb & @xmath27 & m 83 & + eso 272@xmath0g025 & pgc 52591 & 8 & @xmath28 & 14.79 & 3.0 & 5.9 & hf & @xmath29 & cen a & @xmath2 + eso 274@xmath0g001 & uks 1510@xmath0466 & 7 & @xmath30 & 11.70 & 117@xmath31 & @xmath32 & trgb & @xmath33 & cen a & @xmath34 + eso 321@xmath0g014 & am 1211@xmath0375 & 10 & @xmath35 & 15.16 & 2.9 & @xmath36 & trgb & @xmath37 & cen a & + eso 324@xmath0g024 & am 1324@xmath0411 & 10 & @xmath38 & 12.91 & 52.1@xmath31 & @xmath39 & trgb & @xmath40 & cen a & @xmath41 + eso 325@xmath0g011 & am 1342@xmath0413 & 10 & @xmath42 & 13.99 & 25.4@xmath31 & @xmath43 & trgb & @xmath44 & cen a & @xmath45 + eso 381@xmath0g020 & am 1243@xmath0333 & 10 & @xmath46 & 14.44 & 36@xmath47 & @xmath48 & trgb & @xmath37 & m 83 & @xmath49 + ic 4247 & eso 444@xmath0g034 & 10 & @xmath50 & 14.41 & 3.0 & @xmath51 & trgb & @xmath52 & m 83 & @xmath53 + ic 4316 & eso 445@xmath0g006 & 10 & @xmath54 & 14.97 & 7.8@xmath31 & @xmath55 & trgb & @xmath40 & m 83 & @xmath56 + + ddo 47 & ugc 3974 & 10 & @xmath57 & 13.60 & 67.0@xmath58 & @xmath59 & trgb & @xmath60 & ngc 2683 & @xmath4 + ngc 3109@xmath61 & ddo 236 & 9 & @xmath62 & 10.26 & 1110@xmath63 & @xmath64 & trgb & @xmath65 & antlia & @xmath45 + sextans b@xmath61 & ddo 70 , ugc 5373 & 10 & @xmath66 & 11.85 & 102.4@xmath58 & @xmath67 & trgb & @xmath68 & mway & @xmath5 + galaxies are are listed in alphabetical order by their primary name . all properties are obtained from ned , unless otherwise noted . ( 1 ) and ( 2 ) : galaxy name used in the present work , and other names from ned , respectively . ( 3 ) : morphological type @xcite . col . ( 4 ) : heliocentric velocity . col . ( 5 ) : total apparent @xmath69 magnitude . ( 6 ) : total 21-cm flux ; additional reference : @xcite . ( 7 ) : measured or estimated distances . references : centaurus a group galaxies @xcite ; ddo 47 @xcite ; ngc 3109 @xcite ; sextans b @xcite , @xcite . ( 8) : method of determining distances : trgb tip of the red giant branch ; hf hubble flow . ( 9 ) and ( 10 ) : tidal index , and `` closest '' or main disturber ( md ) galaxy , respectively @xcite . ( 11 ) : nebular oxygen abundances derived in the present work . notes @xmath70 from @xcite . @xmath71 from @xcite . @xmath72 from @xcite . @xmath73 common group of dwarfs including ngc 3109 , sextans a , sextans b , and the antlia dwarf @xcite . @xmath74 from @xcite . the paper is organized as follows . observations and reductions of the data are presented in [ sec_obs ] , the measurements and analyses are given in [ sec_analysis ] , derivations of chemical abundances are described in [ sec_abund ] , individual galaxies are presented in [ sec_discuss ] , a discussion of environmental effects is given in [ sec_enveffects ] , and the conclusions are given in [ sec_concl ] . for the remainder of this paper , we adopt 12@xmath1log(o / h ) = 8.66 as the revised solar value for the oxygen abundance , and @xmath75 = 0.0126 as the revised solar mass fraction in the form of metals @xcite . long - slit spectroscopic observations were carried out on 2003 mar . 68 ut with the efosc2 imaging spectrograph on the 3.6-m telescope at eso la silla observatory . details of the instrumentation employed and the log of observations are listed in tables [ table_obsprops ] and [ table_obslog ] , respectively . observations were obtained just after new moon phase . conditions were mostly clear on the first two nights with patchy thin cloud towards the end of the second night ; the third night was clear and photometric . [ cols="<,^ " , ] see table [ table_abund1 ] for general comments . ddo 47 - shk91 refers to the identifications in @xcite ; see also fig . [ fig_dwarfs2 ] and table [ table_data7 ] . ngc 3109 - fig . [ fig_dwarfs2 ] and table [ table_data8 ] . sextans b - shk refers to the identifications in @xcite ; see also fig . [ fig_dwarfs2 ] and table [ table_data9 ] . notes : @xmath70 @xcite bright - line calibration . @xmath71 bright - line calibration : lower branch - @xcite ; upper branch - @xcite . @xmath72 slit orientation a. @xmath73 slit orientation b. @xmath74 `` narrow '' extraction aperture of total 10 pixels . @xmath76 `` wide '' extraction aperture of total 44 pixels . oxygen abundances in regions were derived using three methods : ( 1 ) the direct method ( e.g. , @xcite ) ; and two bright - line methods discussed by ( 2 ) @xcite , which is based on photoionization models ; and ( 3 ) @xcite , which is purely empirical . we briefly summarize these methods here ; further details of these methods are found in @xcite and @xcite . the `` direct '' conversion of emission line intensities into ionic abundances requires a reliable estimate of the electron temperature in the ionized gas . we adopt a two - zone model for regions , with a low- and a high - ionization zone characterized by temperatures @xmath77o@xmath78 and @xmath77o@xmath79 , respectively . the temperature in the o@xmath80 zone is measured with the emission line ratio @xmath81()/@xmath81 ( ) @xcite . the temperature in the o@xmath6 zone is given by @xmath82 where @xmath83 k @xcite . the total oxygen abundance by number is given by o / h = o@xmath6/h@xmath6 @xmath1 o@xmath80/h@xmath6 . for conditions found in typical regions and those presented here , very little oxygen in the form of neutral oxygen is expected , and is not included here . ionic abundances for singly- and doubly - ionized oxygen were computed using o@xmath6 and o@xmath80 temperatures , respectively , as described above . the o@xmath84 contribution is assumed negligible where he ii emission is absent . although he ii emission is reported in two regions ( see above ) , we have not included the small contribution by o@xmath84 to the total oxygen abundance . in four of the 48 region spectra , line fluxes were measured , and subsequent electron temperatures were derived . we derived direct oxygen abundances in regions with detections using the method described by @xcite . ionic and total abundances are computed using the emissivities from the five - level atom program by @xcite . as described above , we use the same two - temperature zone model and temperatures for the remaining ions . the error in @xmath85(o@xmath80 ) is derived from the uncertainties in the corrected emission - line ratios , and does not include any uncertainties in the atomic data , or the possibility of temperature variations within the o@xmath80 zone . the fractional error in @xmath85(o@xmath80 ) is applied similarly to @xmath85(o@xmath6 ) to compute the uncertainty in the latter . uncertainties in the resulting ionic abundances are combined in quadrature for the final uncertainty in the total linear ( summed ) abundance . the appropriate temperatures , abundances , and their uncertainties are listed in tables [ table_abund1 ] and [ table_abund2 ] . in regions without measurements , the bright - line method has been used to derive oxygen abundances , as the latter are usually given in terms of bright [ o ii ] and [ o iii ] emission lines . @xcite devised the @xmath86 indicator , defined by @xmath86 = [ @xmath81 ( ) + @xmath81()]/@xmath81 ( ) ; @xcite discussed the method further for low - metallicity galaxies . @xcite developed a grid of photoionization models and suggested using @xmath86 and an ionization proxy , represented by @xmath87 = @xmath81()/@xmath81 ( ) , to estimate oxygen abundances . to break the degeneracy in the bright - line method , we have used the ratio ( e.g. , @xcite ) to choose either the `` upper branch '' ( high oxygen abundance ) or the `` lower branch '' ( low oxygen abundance ) . in some instances , oxygen abundances with the mcgaugh method could not be computed , because the @xmath86 values were outside of the effective range for the models . @xcite proposed an empirical calibration at low metallicity with fits of oxygen abundance against bright oxygen lines . @xcite have shown that while the pilyugin method covers a large range in @xmath86 , the calibration applies mostly to regions with higher ionizations ; see also @xcite . oxygen abundances derived using the bright - line calibrations are listed in tables [ table_abund1 ] and [ table_abund2 ] . in the absence of , oxygen abundances derived using bright - line methods are in agreement with direct abundances to within @xmath88 0.2 dex . previously , @xcite reported spectra for six cen a late - type dwarf galaxies : ddo 161 ( ugca 320 ) , eso 324@xmath0g024 , eso 381@xmath0g020 , eso 383@xmath0g087 , eso 444@xmath0g084 , and ngc 5264 ( ddo 242 ) . only and were detected in eso 324@xmath0g024 , and an oxygen abundance was not derived . was detected in eso 383@xmath0g087 , and bright - line abundances were derived for the remaining four galaxies . we briefly comment on a number of cen a di s . our measured spectrum of am 1318@xmath0444 is indicative of a metal - poor region , and the resulting ( bright - line ) oxygen abundance is about one - sixth of the solar value . while the bright - line abundance for eso 274@xmath0g001 region no . 4 implies an unusually high oxygen abundance ( compared to the derived lower limit ) , our measured intensity ratios ( table [ table_data2 ] ) suggest that the nebula is a supernova remnant ; see also the discussion in @xcite . if the object in question is a supernova remnant , the measured @xmath81()/@xmath81 ( ) , @xmath81()/@xmath81 ( ) , and the model grid from ( * ? ? * their fig . 8) yield an estimate of 12@xmath1log(o / h ) @xmath89 for the oxygen abundance . as an edge - on dwarf galaxy , eso 274@xmath0g001 is reminescent of ngc 55 in the sculptor group and ngc 1560 in the ic 342 group . these galaxies may provide further opportunities to investigate the possibility of abundance gradients in low - luminosity late - type galaxies . in fact , @xcite reported an unusually high ( bright - line ) abundance for ngc 5264 in the cen a group . additional high signal - to - noise spectra would be valuable in confirming this result . for eso 324@xmath0g024 , we obtained a higher - quality spectrum compared to the observation described in @xcite , but we did not detect . the bright - line oxygen abundance derived for eso 381@xmath0g020 is 0.12 dex higher than the bright - line value determined in @xcite , but these values are well within the acceptable uncertainty ( @xmath7 0.2 dex ) associated with bright - line abundances . in all , the derived oxygen abundances for the present sample of cen a di s ( cf . table [ table_gxylist ] ) are in the range between about 10% to 50% of the solar value , which are in general agreement with the results obtained by @xcite and @xcite . @xcite obtained spectra for the region shk91 no . 18 in ddo 47 using the iids spectrometer on the kpno 2.1-m telescope . with their measurement , they derived an oxygen abundance of 12@xmath1log(o / h ) = @xmath90 . in our spectrum , we measured and derived an oxygen abundance of 12@xmath1log(o / h ) = @xmath4 , in agreement with @xcite . we have also measured he ii @xmath914686 . as he ii emission is indicative of the presence of o@xmath84 , the latter is generally a small contributor to the total oxygen abundance . for example , @xcite measured he ii emission in i zw 18 and found that the resulting o@xmath84 contribution was of order one to four percent . in region shk91 no . 18 , the he ii 4686 to flux ratio is about 10 percent , which implies that the resulting contribution by o@xmath84 to the total oxygen abundance could be of order 10% . for comparison with other di s , we have not included any o@xmath84 contribution in the total oxygen abundance for ddo 47 . the optical appearance ( e.g. , @xcite ) is suggestive of a low - luminosity spiral galaxy . from the data reported previously by @xcite , the authors suggested the presence of an abundance gradient in ngc 3109 , which had been previously hinted by @xcite ( @xcite , @xcite ) . we did not measure in any of the spectra shown here . the subsequent mean of six bright - line abundances is @xmath92 , which is larger than the @xcite value by about 0.2 dex , but is in better agreement with other dwarfs at comparable optical luminosity . since this galaxy is known to contain multiple regions and planetary nebulae ( i.e. , @xcite ) , deep spectra of these nebulae would be valuable to confirm the presence or absence of a radial gradient in oxygen abundance . published nebular abundances for sextans b have differed by as much as 0.5 dex . @xcite published a limit of 12@xmath1log(o / h ) @xmath93 7.38 from data obtained with the iids detector on the eso 3.6-m telescope . @xcite reported a higher lower limit to the oxygen abundance : 12@xmath1log(o / h ) @xmath93 7.56 . from data obtained with the ids on the 2.5-m int telescope , @xcite measured and subsequently derived an oxygen abundance 12@xmath1log(o / h ) = 8.11 . if the higher value is adopted , the present - day metallicity appears @xmath7 0.3 dex too high for its galaxy optical luminosity , compared to other dwarf irregulars at comparable luminosity . @xcite reported ntt spectra for one planetary nebula and six regions in sextans b , from which was detected in the planetary nebula and in three regions . they reported in shk91 no . 5 , but their spectrum is cut off below about 3800 , and there is no measured . alternatively , they used their measurement to derive the o@xmath6/h abundance with the method developed to derive oxygen abundances for metal - poor galaxies observed in the sloan digital sky survey @xcite . their resulting oxygen abundance for region no . 5 is 12@xmath1log(o / h ) = @xmath94 . @xcite claimed that the recent chemical enrichment was spatially inhomogeneous over length scales as large as 1 kpc , based on a dispersion of 0.31 dex for direct abundances in three regions . with fors2 on vlt , @xcite measured spectra of regions and planetary nebulae , and detected in three regions , including no . 5 . their direct oxygen abundance is 12@xmath1log(o / h ) = @xmath95 , which agrees with the @xcite result . we measured only in region no . 5 , but our spectrum includes the full range of optical emission lines from to ( fig . [ fig_spectra ] , table [ table_data9 ] ) . our resulting direct abundance is 12@xmath1log(o / h ) = @xmath5 , which also agrees with the values reported by @xcite and @xcite . the data presented here shows that the derived bright - line abundances for regions shk91 nos . 1 , 2 , 5 , and 10 all agree to within @xmath88 0.1 dex ( table [ table_abund2 ] ) . although the direct abundance for shk91 no . 5 and the bright - line abundance for shk91 no . 10 differ by 0.25 dex , the difference is comparable to our adopted uncertainty of 0.2 dex for a bright - line abundance . if dwarf galaxies are more robust to the effects of internal processes , such as supernova feedback , external processes found within environments of galaxy groups may be more damaging to the `` health '' of gas - rich dwarf galaxies . galaxy - galaxy encounters create tidal interactions , which can remove stellar and/or gaseous material and eject the stripped material into the intragroup medium . disturbed galaxy morphologies are also indicative of these interactions . groups may contain hot ( @xmath96 k ) dense x - ray emitting gas , which could provide an agent for ram - pressure stripping , if group galaxies traverse the intragroup medium at high speeds . however , only the highest - mass groups have significant x - ray luminosities ( see , e.g. , @xcite ) . in most nearby groups ( at least within the local volume ) , it is difficult to measure strong x - ray emission . @xcite have suggested that ram - pressure stripping may have been responsible to explain the lower content in cen a dwarfs relative to dwarfs in the local group and the scl group . at present , we assume that the presence of x - ray gas is negligible , although the possibility of warm gas ( @xmath97 k ) is not entirely ruled out @xcite . higher star - formation rates can also be induced by tidal interactions , as molecular gas is subsequently compressed and shock - heated . a subsequent result would be a larger fraction of galaxies with strong emission - lines and/or blue colors in these environments . unfortunately , extensive galaxy surface photometry has not yet been obtained for cen a or sculptor ( scl ) group dwarf galaxies , although narrow - band images have been obtained for some of the di s in the two groups ( @xcite ; s. ct , personal communication ) . although we can not yet compare the star formation properties , we can compare the end results to their present - day chemical evolution . in the discussion which follows , we have also included additional spectra for cen a dwarfs from @xcite and scl group dwarfs from @xcite . we have adopted a distance of 3.0 mpc for the scl group dwarfs am0106@xmath0382 and eso348@xmath0g009 . the optical luminosity - metallicity relation has long been used as a diagnostic for the evolution of nearby star - forming di s ; e.g. , @xcite . to augment the local sample described by @xcite , additional galaxies with measurements and/or distances from stellar indicators are taken from the literature : ddo 167 @xcite , eso 489@xmath0g056 @xcite , ngc 1705 @xcite , ngc 3738 @xcite , ngc 4449 @xcite , ngc 6822 @xcite , and wlm @xcite . we have also included direct and bright - line abundances for additional cen a and scl group dwarf irregulars from @xcite and @xcite . the luminosity - metallicity relation is shown in fig . [ fig_diags]a . the solid line is a fit to the sample of nearby dwarfs as described above . cen a and scl group dwarfs with measurements are in best agreement with the fit and with the locus of nearby dwarfs that have measurements . although the spectra for ngc 5264 are of lower signal - to - noise , this galaxy exhibits an unusually high oxygen abundance for its luminosity , which suggests a more spiral - like nature to the galaxy and warrants future measurements . three cen a dwarfs with detections have oxygen abundances consistent for their optical luminosities . for the other cen a di s without measurements , their oxygen abundances generally appear higher than abundances for nearby di s at similar luminosities . we have found that oxygen abundances exhibit the least dispersion in the luminosity - metallicity relationship , and there is no apparent difference between di s in cen a and scl groups . it would be interesting to examine the relative dispersion in the luminosity - metallicity relation at near - infrared wavelengths ; see @xcite and references within . the gas content in galaxies may be sensitive to environmental effects , and we examine here how environment may affect chemical evolution directly . the simplest `` closed - box '' model @xcite can be written as @xmath98 , where @xmath99 is the oxygen mass fraction , @xmath100 is the yield by mass , and @xmath101 is the baryonic gas fraction equal to the ratio of the gas mass to the total mass in gas and stars . in fig . [ fig_diags]b d , we have plotted the -gas - to-@xmath69-light ratio and parameters related to gas fraction against oxygen abundance . in the absence of measured optical colours for cen a and scl di s to derive color - based stellar mass - to - light ratios described by @xcite , we compute stellar masses by simply assuming a constant stellar mass - to-@xmath69-light ratio equal to one . our adopted value is within the range of stellar mass - to-@xmath69-light ratios derived by , for example , @xcite and @xcite . we compute total gas mass as @xmath102 = 1.36 @xmath103 , which includes helium , but ignores molecular gas . under the assumption of closed - box evolution , we have also plotted various curves with yields varying by a factor of five . we find that nearby di s @xcite and ugc di s @xcite span the range of yields shown . by inspection , @xmath104 @xmath88 @xmath105 is a good description of the data shown , in general agreement with the results of @xcite and @xcite . the two cen a di s with direct abundances and low gas fractions are eso 272@xmath0g025 and eso 383@xmath0g087 . generally , cen a di s span the range of yields shown , whereas the small number of scl di s tend to cluster around the lower end of the yield range . however , these results are not definitive because of small - number statistics , and measurements of additional galaxies could strengthen possible differences in gas fractions between the two sets of di s . to explore further the effects of group environments on the properties of gas - rich dwarf galaxies , we examine various properties against tidal index and projected distance ; the latter two parameters are defined and compiled in @xcite and @xcite . briefly , the tidal index of a galaxy is the maximum value of an enhancement in mass density caused by all neighbouring galaxies . objects with negative tidal indices are isolated in the field , and objects with positive tidal indices are found within group environments , and are likely undergoing a tidal interaction . within each group , the projected distance of a galaxy , @xmath106 , is given by @xmath107 , where @xmath108 is the distance of the main ( or primary ) group member to the milky way , and @xmath109 is the angular distance ( in degrees ) of the galaxy to the main member . in fig . [ fig_ztidal ] , we have plotted various parameters against the tidal index and the projected distance of galaxies within each group . on the left side , we have plotted oxygen abundance , gas - to - light fraction , and the surface mass density as a function of tidal index in panels a , b , and c , respectively . values for the surface mass density were obtained from @xcite . we have compared cen a and scl group dwarfs against other dwarfs ( as in fig . [ fig_diags ] ) , ngc 1705 , and ddo 154 . there are no obvious trends in the first two panels . we noted above that eso 272@xmath0g025 and eso 383@xmath0g087 appeared to have low gas fractions for their oxygen abundances , compared to other dwarfs . both have negative tidal indices , which indicate only that there is no strong present - day tidal interactions . in panel c , the surface mass density appears to decrease with increasing positive tidal index , which is expected as strong tidal interactions remove gas from galaxies . we note that eso 272@xmath0g025 ( cen a di ) has an unusually low surface mass density for its tidal index ( @xmath110 , log @xmath111 @xmath88 6 ) . this suggests that this dwarf may have encountered a strong interaction , which has reduced its content and has induced present - day star formation ( i.e. , presence of regions ) . detailed spatially - resolved observations would be very timely to confirm this scenario . in figs . [ fig_ztidal]d g , we have plotted luminosity , -gas - to - light fraction , oxygen abundance , and the effective yield as a function of the projected distance , respectively . we have separated dwarfs in different groups by color : local group ( milky way @xmath1 m31 ) in blue , m81 group dwarfs in brown , ic342 @xmath1 maffei group dwarfs in dark green , and cvni group dwarfs in black . once more , there are no obvious trends ; there is large scatter present in the plots of oxygen abundance and yield versus distance . naturally , more data are required for examination and for improved statistics in an intergroup comparison . we note that eso 381@xmath0g020 exhibits an unusually high effective yield for its projected distance ( @xmath88 0.9 mpc ) , although the di does not appear to have any other unusual properties . although a number of cen a dwarfs in our sample have positive tidal indices , there is little to separate these galaxies from those with negative tidal indices in the luminosity - metallicity , metallicity - gas fraction , and metallicity - tidal index diagrams . there is also no clear trend between the current chemical enrichment ( as represented by either oxygen abundance or yield ) with the projected distance of galaxies within each group . because of low - number statistics , it is difficult to say at the present time whether any separation of cen a dwarfs and scl dwarfs in these diagnostics is meaningful . results of optical spectroscopy of regions from eight dwarf galaxies in the centaurus a group are presented . for eso272@xmath0g025 and eso324@xmath0g024 , direct oxygen abundances of 12@xmath1log(o / h ) = @xmath2 and @xmath3 are derived , respectively . bright - line abundances for the remaining galaxies are derived with the mcgaugh and the pilyugin calibrations . we have also considered data for additional cen a di s and sculptor group di s from the literature . for their galaxy luminosities , we have found that direct or oxygen abundances agree well with the luminosity - metallicity relationship for dwarf irregular galaxies . despite large tidal indices for a number of cen a dwarf galaxies , there is no difference between galaxies with positive tidal indices galaxies with negative tidal indices in the luminosity - metallicity , metallicity - gas fraction , and metallicity - tidal index diagrams . as expected in strong tidal interactions , the surface mass density appears to decrease with increasing positive tidal index . we have also examined global properties of dwarf galaxies based on their intragroup properties . there are no obvious trends in plots of luminosity , hi - to-@xmath69-light ratio , oxygen abundance , and yield against projected distance of galaxies within nearby groups . we also report spectra and abundances for three nearby dwarf irregular galaxies : ddo 47 , ngc 3109 , and sextans b. our direct oxygen abundance ( @xmath4 ) for ddo 47 agrees with the measurement previously reported by @xcite . for sextans b , our direct oxygen abundance ( @xmath5 ) is consistent with the low value reported by @xcite , and agrees with abundances reported by @xcite and @xcite . we thank the anonymous referee for comments which improved the presentation of the manuscript . h. l. is supported by the gemini observatory , which is operated by the association of universities for research in astronomy , inc . , on behalf of the international gemini partnership of argentina , australia , brazil , canada , chile , the united kingdom , and the united states of america . we are grateful to eso for awarded telescope time , and we thank lisa germany , george hau , and the staff at eso la silla for help with the observations . partial support for this work was provided by nasa through grant go-08192.97a from the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , under nasa contract nas5 - 26555 . d. b. z. acknowledges support from the national science foundation postdoctoral fellowship . e. k. g. appreciates support by the swiss national science foundation through grants 200021 - 101924 and 200020 - 105260 . h. l. is grateful for support from the max - planck - institute for astronomy where this project was begun , for partial support from a nasa ltsarp grant nag 5 - 9221 , and from the university of minnesota . h. l. also thanks stephanie ct and evan skillman for providing their images , and liese van zee for discussions regarding bright - line calibrations . some data were accessed as guest user , canadian astronomy data center , which is operated by the dominion astrophysical observatory for the national research council of canada s herzberg institute of astrophysics . this research has made use of the nasa / ipac extragalactic database , which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . grebel , e. k. , gallagher , j. s. , & harbeck , d. r. 2003 , , 125 , 1926 grossi , m. , disney , m. j. , pritzl , b. j. , knezek , p. m. , gallagher , j. s. , minchin , r. f. , & freeman , k. c. 2007 , , 374 , 107 hoffman , g. l. , salpeter , e. e. , farhat , b. , roos , t. , williams , h. , & helou , g. 1996 , , 105 , 269 lee , h. , skillman , e. d. , & venn , k. a. 2005 , , 620 , 223 lee , h. , skillman , e. d. , & venn , k. a. 2006a , , 642 , 813 lee , h. , skillman , e. d. , cannon , j. m. , jackson , d. c. , gehrz , r. d. , polomski , e. f. , & woodward , c. e. 2006b , , 647 , 970 magrini , l. , leisy , p. , corradi , r. l. m. , perinotto , m. , mampaso , a. , & vlchez , j. m. 2005 , , 443 , 115

we present results of optical spectroscopy of 35 regions from eight dwarf galaxies in the centaurus a group . is detected in eso272@xmath0g025 and eso324@xmath0g024 , and direct oxygen abundances of 12@xmath1log(o / h ) = @xmath2 and @xmath3 are derived , respectively . for the remaining galaxies , abundances are derived using common bright - line methods . to compare the influence of group environments on dwarf galaxies , we have also gathered data for additional dwarf irregular galaxies from the cen a and the sculptor groups from the literature . we have examined possible relationships between oxygen abundance , gas fraction , effective chemical yield , and tidal indices . despite large positive tidal indices for a number of cen a dwarfs in the present sample , there is no clear separation between galaxies with positive tidal indices and galaxies with negative tidal indices in the luminosity - metallicity , metallicity - gas fraction , and metallicity - tidal index diagrams . the surface mass density decreases with increasing positive tidal index , which is expected in strong tidal encounters . there are no strong trends between oxygen abundances or yields and projected distances of galaxies within their respective groups . we also present spectra for 13 regions in three nearby dwarf irregular galaxies : ddo 47 , ngc 3109 , and sextans b. for ddo 47 , the oxygen abundance ( @xmath4 ) for the region shk91 no . 18 agrees with recently published values . for sextans b , the oxygen abundance ( @xmath5 ) for region shk91 no . 5 agrees with published work in which o@xmath6 abundances were determined entirely from fluxes . [ firstpage ] galaxies : abundances galaxies : dwarf galaxies : evolution galaxies : interactions galaxies : irregular

introduction observations and reductions nebular abundances discussion of individual galaxies exploration of environmental effects conclusions acknowledgements

arxiv

gs182624 was discovered serendipitously in 1988 by the _ ginga _ satellite ( makino et al . 1988 ) at an average flux of 26 mcrab ( 140 kev ) and was fitted by a single power law spectrum with @xmath6 . whilst showing some evidence for variability during 198889 ( tanaka & lewin 1995 ; int zand 1992 ) , _ rosat _ pspc observations in 1990 and 1992 ( barret et al . 1995 ) found comparable flux levels and no x - ray bursts were detected during 8 hours exposure on the source . the spectrum was well fitted by a single power law with @xmath7 and an absorption column , @xmath8 @xmath2 . temporal analysis of both the _ ginga _ and _ rosat _ data yielded a featureless @xmath9 power spectrum extending from @xmath10500 hz ( tanaka & lewin 1995 ; barret et al . 1995 ) , with neither quasi - periodic oscillations ( qpos ) nor pulsations being detected . since there was no detection prior to _ ginga _ , the source was catalogued as an x - ray transient . its similarities to cyg x1 and gx3394 in the low state , both in spectrum and temporal behaviour ( hard x - ray spectrum and strong flickering ) , led to an early suggestion by tanaka ( 1989 ) that it was a soft x - ray transient with a possible black - hole primary . following its detection by _ cgro _ osse in the 60200 kev energy range , strickman et al . ( 1996 ) doubted the suggestion of a black - hole primary after examining the combined spectrum from both _ ginga _ and osse . they found that this required a model with an exponentially cut - off power law plus reflection term . the observed cut - off energy around 58 kev is typical of the cooler neutron star hard x - ray spectra . the suggestion that gs182624 contains a neutron star was also discussed in detail by barret et al . ( 1996 ) , where they compared the luminosity of the source with other x - ray bursters . the recent report of 70 x - ray bursts in 2.5 years by _ bepposax _ wfc ( ubertini et al . 1999 ) and an optical burst by homer et al . ( 1998 ) confirms the presence of a neutron star accretor . following the first _ pspc all - sky survey observations in september 1990 , and the determination of a preliminary x - ray position , a search for the counterpart yielded a time variable , uv - excess , emission line star ( motch et al . 1994 ; barret et al . the source had @xmath11 , and an uncertain v magnitude ( @xmath12 ) , due to contamination by a nearby star . subsequent high - speed ccd photometry by homer et al . ( 1998 ) yielded a @xmath5 2.1 hr optical modulation , but confirmation of its stability requires observation over a longer time interval . we therefore carried out an _ asca _ observation and simultaneous _ rxte_/optical observations of gs182624 in order to study its spectral behaviour and very short timescale variability , as well as the 2.1 hr optical modulation . in table 1 , we summarize the _ asca _ , _ rxte _ and saao observations used in this work . this paper is structured as follows . an outline of all the x - ray and optical observations is given in section 2 . in section 3 we report the spectral analysis of _ asca _ data for both persistent and burst emission . rxte_/optical observations , including the analysis of a simultaneous x - ray / optical burst are also presented . we discuss the implications of the x - ray bursts and constrain the nature of the source from the delay between the x - rays and optical in section 4 . we present the overall timing study of the _ asca _ and _ rxte_/optical observations in a companion paper ( homer et al . 1999 , henceforth paper ii ) . .a journal of the x - ray / optical observations of gs182624 [ cols="^,^,^,^,^ " , ] x - ray and optical bursts from gs182624 were previously reported by ubertini et al . ( 1999 ) , int zand et al . ( 1999 ) and homer et al . our x - ray / optical observations of gs182624 showed three x - ray bursts , of which two were observed by _ asca _ , whereas one was observed by _ rxte _ simultaneously with the optical . on the basis of their spectral properties and time profiles , all the bursts have a cooling trend during their decays and exhibit blackbody spectra with temperatures of a few kev . the time profiles show fast rise times of 79 s and long decay times ranging from 2060 s depending on the energy band . we therefore interpret the three bursts detected from gs182624 as type i bursts ( hoffmann et al . the relatively long rise time ( @xmath13 s ) , as compared to bursts in other systems , indicates that the burst front may have enough time to spread over the whole neutron star surface during the rise to burst and suggests that the burning is homogeneous over the surface of the neutron star . this is consistent with the fact that we see no evidence for pulsations during the burst . we note that x - ray burst rise times in other sources have been observed to be smaller than @xmath56 sec ( see e.g. sztajno et al . 1986 ; lewin et al . 1987 ) , making this system rather unique . the burst rise times derived by int zand et al . ( 1999 ) range from 58 s which are also consistent with our observations . for the simultaneous burst we may compare the ratio of the persistent to peak fluxes , in both the x - ray and optical , using the results of lawrence et al . they derived a simple power law relation between the changes in u , b and v band fluxes and the corresponding x - ray flux variations during a well - studied burst of x1636536 , with @xmath14 , where @xmath15 varies with passband . our burst shows that @xmath16 which is comparable to the value @xmath17 found for x1636536 in the @xmath18 band , which is the closest approximation to our white light passband . hence , we may imply that the reprocessed emission from the gs182624 burst is also approximately that from a blackbody ( with a temperature set by the degree of x - ray irradiation ) , where the optical passband is on the rayleigh - jeans tail ( lawrence et al . 1983 ) . the x - ray burst observed by rxte shows evidence for an increase in the burning area during the early rise phase , but no evidence for photospheric radius expansion . note that this is consistent with the fact that most x - ray bursts showing photospheric radius expansion have rise times less than @xmath51 sec ( see lewin et al . however , by assuming that our observed peak luminosity of @xmath19 erg @xmath2 s@xmath3 is near the eddington limit of @xmath20 ergs s@xmath3 for a 1.4 m@xmath21 neutron star , we can set an upper limit to the distance to gs1826@xmath2224 . we derive a maximum distance of @xmath23 kpc . this estimate is consistent with the upper limit from _ bepposax _ nfi observations ( @xmath24 kpc ; int zand et al . 1999 ) and the optical lower limit of 4 kpc ( barret et al . 1995 ) . the luminosity ratios are @xmath25 55 and @xmath5 50 for the bursts observed with _ asca _ and _ rxte _ , respectively . this is comparable with that found by _ wfc ( @xmath26 ; ubertini et al . 1999 ) . coupling this value with an estimated stable accretion rate of @xmath27 m@xmath21 yr@xmath3 ( ubertini et al . 1999 ) , the burst must involve a combined hydrogen - helium burning phase ( lewin , van paradijs & taam 1995 ) . this relatively long burst also resembles the theoretical results of x - ray bursts driven by rapid proton capture process , or rp - process ( see hanawa & fujimoto 1984 ; taam 1981 ; bildsten 1998 ; schatz et al . 1998 ) . pedersen et al . ( 1982 ) have shown that the optical burst mainly reflects the geometry of the system and that the contribution of intrinsic radiative processes is small . hence the correlated optical and x - ray bursts discussed above are useful as probes of the structure and geometry of the compact object surroundings . within the framework of the low - mass x - ray binary system , the reprocessing can occur in the accretion disk and the companion star . based on our observed mean delay of @xmath28 s for the optical burst with respect to low energy x - rays , we can then constrain the orbital period of the system . by kepler s law , the light travel time of 24 s corresponds to an orbital period of 1.65.5 hr if we assume a 1.4@xmath29 neutron star and a companion star mass of 0.11.1@xmath29 ( i.e. for a low - mass main sequence star and stable mass transfer ) . hence , this range of periods provides support for the @xmath30 hr orbital period proposed by homer et al . lastly , from only one simultaneous optical / x - ray burst , we can not draw a firm conclusion as to whether the optical burst is due to reprocessing in the disk or on the surface of the companion star . however , given that the source is a low inclination system ( @xmath31 70@xmath32 ; homer et al . 1998 ) and the ratio of smearing to delay is @xmath33 , the reprocessing is expected to be dominated by the accretion disk . it is important in future studies to search for the possibly variable delays if the reprocessing occurs on the surface of the companion star ( matsuoka et al . 1984 ) or the ` thick spot ' in the disk proposed by pedersen et al . whether the dominant reprocessor is the companion star or the ` thick spot ' , one expects the ratio of optical to x - ray flux in a burst to vary periodically . moreover , the optical delay would vary as a function of orbital phase as suggested by pedersen , van paradijs & lewin ( 1981 ) . ubertini et al . ( 1999 ) recently proposed a 5.76-hr quasi - periodicity in the occurrence of x - ray bursts in gs182624 , which makes this problem difficult to resolve with current low - earth - orbit satellites . however , with upcoming missions such as _ chandra _ and _ xmm _ , much longer continuous x - ray coverage will be possible , and together with ground - based telescopes will enable us to probe the structure of this source in much greater detail . we are grateful to darragh odonoghue ( saao ) for his advice on the the use of the uct - ccd for high - speed photometry and his help with the subsequent reductions . we also thank lars bildsten for valuable comments and fred marang ( saao ) for his support at the telescope , and the _ rxte _ soc team for their efforts in scheduling the simultaneous time . this paper also utilizes results provided by the asm/_rxte _ team . barret , d. , motch , c. , pietch , w. , 1995 , a&a , 303 526 barret , d. , mcclintock , j.e . , grindlay , j.e . , 1996 , apj , 473 , 963 bildsten , l. , 1998 , in the many faces of neutron stars , ed . a. alpar , l. buccheri , & j. van paradijs , ( dordrecht : kluwer ) bradt , h.v . , rothschild , r.e . , swank , j.h . , 1993 , a&as , 97 , 355 hanawa , t. , fujimoto , m.y . , 1984 , pasj , 36 , 199 hoffman , j.a . , marshall , h. , j.a . , lewin , w.h.g . , 1978 , nature , 271 , 630 homer , l. , charles , p.a . , odonoghue , d. , 1998 , mnras , 298 , 497 homer , l. , et al . , 1999 , in preparation ( paper ii ) hynes , r.i . , obrien , k. , horne , k. , chen , w. , haswell , c.a . , 1998 , mnras , 299 , l37 int zand , j.j.m . , 1992 , phd thesis , university of utrecht int zand , j.j.m . , heise , j. , kuulkers , e. , bazzano , a. , cocchi , m. , ubertini , p. , 1999 , a&a , 347 , 891 lawrence , a. , cominsky , l. , engelke , c. , et al . , 1983 , apj , 271 , 793 . lewin , w.h.g . , van paradijs , j. , taam , r.e . , 1995 , in _ x - ray binaries , ed . w.lewin , j.van paradijs & e.van den heuvel _ ( cambridge : cambridge univ . press ) , p.175 lewin , w.h.g . , penninx , w. , paradijs , j. , et al . , 1987 , apj , 319 , 893 london , r. , taam , r.e . , howard , m. , 1984 , apj , 287 , l27 makino , f. , et al . , 1988 , iauc 4653 matsuoka , m. , mitsuda , k. , ohashi , t. et al . , 1984 , apj , 283 , 774 motch , c. , et al . , 1994 , iauc 6101 odonoghue , d , 1995 , baltic astronomy , 4 , 519 pedersen , h. , van paradijs , j. , lewin , w.h.g . , 1981 , nature , 294 , 725 pedersen , h. , lub , j. , inoue , h. , et al . , 1982 , apj , 263 , 325 schatz , h. , aprahamian , a. , g@xmath34rres , j. , et al . , 1998 , physics reports , 294 , 167 stetson , p.b . , 1987 , pub . astron . soc . , 99 , 191 strickman , m. , et al . , 1996 , a&as , 120 , 217 strohmayer , t.e . , zhang , w. , swank , j.e . , 1997 , apj , 487 , l77 sztajno , m. , van paradijs , j. , lewin , w.h.g . , tr@xmath35mper , j. , stollman , g. , pietsch , w. , van der klis , m. , 1985 , apj , 299 , 487 sztajno , m. , van paradijs , j. , lewin , w.h.g . , langmeier , j. , tr@xmath35mper , j. , pietsch , w. , 1986 , mnras , 222 , 499 taam , r.e . , 1981 , apj , 247 , 257 tanaka , y. , 1989 , in two topics in x - ray astronomy , 23rd eslab symposium . , p.3 , esa : paris , bologna tanaka , y. , inoue , h. , holt , s.s . , 1994 , pasj , 46 , l37 tanaka , y. , lewin , w.g.a . , 1995 , in x - ray binaries , cambridge university press , eds . lewin w.h.g . , van paradijs , j. & van den heuvel , e.p.j . , p.126 ubertini , p. , et al . , 1999 , apj , 514 , l27 van paradijs , j. , lewin , w.h.g . , 1985 , a&a , 157 , l10 van paradijs , j. , sztajno , m. , lewin , w.h.g . , tr@xmath35mper , j. , vacca , w.d . , van der klis , m. , 1986 , mnras , 221 , 617

we report results from the first simultaneous x - ray ( _ rxte _ ) and optical ( saao ) observations of the low - mass x - ray binary gs182624 in june 1998 . a type - i burst was detected in both x - ray and optical wavelengths . its energy - dependent profile , energetics and spectral evolution provide evidence for an increase in the x - ray burning area but not for photospheric radius expansion . however , we may still derive an upper limit for its distance of @xmath0 kpc , assuming a peak flux of @xmath1 erg @xmath2 s@xmath3 . a @xmath4 s optical delay with respect to the x - ray burst is also observed and we infer that this is related to the x - ray reprocessing in the accretion disk into the optical . this provides support for the recently proposed orbital period of @xmath52 h. we also present an _ asca _ observation from march 1998 , during which two x - ray bursts were detected . accretion , accretion disks binaries : close stars : individual ( gs182624 ) x - rays : bursts

introduction discussion acknowledgements

arxiv

galaxy clustering has proven to be invaluable in assembling our current picture of a universe with a nearly scale - invariant spectrum of the primordial curvature perturbations @xcite . the principle tool in clustering studies has been the two - point correlation function , or in fourier space the power spectrum , determined under the assumption of statistical isotropy ( si ) . with the advent of new generations of galaxy surveys , as well as longer - term prospects for measuring the primordial mass distribution with 21-cm surveys of the epoch of reionization @xcite and/or dark ages @xcite , it is worthwhile to think about what can be further done with these measurements . many inflationary models introduce new fields that may couple to the inflaton responsible for generating curvature perturbations . the effects of these fields may then appear as local departures from si , or as non - gaussianity , in the curvature perturbation . for example , models with an additional scalar field introduce a nontrivial four - point correlation function ( or trispectrum , in fourier space ) @xcite , which we below will describe as local departures from statistical isotropy ; apart from this correlation , the scalar field may leave no visible trace . there may also be vector ( spin-1 ) fields @xmath0 @xcite or vector spacetime - metric perturbations brought to life in alternative - gravity theories @xcite that , if coupled to the inflaton @xmath1 ( e.g. , through a term @xmath2 ) may leave an imprint on the primordial mass distribution without leaving any other observable trace . similar correlations with a tensor ( i.e. , spin-2 ) field @xmath3 ( e.g. , @xmath4 ) can be envisioned . even in the absence of new fields , there are tensor metric perturbations ( gravitational waves ) that may be correlated with the primordial curvature perturbation @xcite . tensor distortions to the two - point correlation function ( `` metric shear '' ) may also be introduced at late times @xcite , and late - time nonlinear effects may induce scalar - like distortions to the two - point function @xcite . here we describe how the fossils of primordial tensor , vector , and scalar fields are imprinted on the mass distribution in the universe today . we express these relics in terms of two - point correlations that depart locally from si or off - diagonal correlations of the density - field fourier components . this formalism allows the correlations to be decomposed geometrically into scalar , vector , and tensor components . we write down the optimal estimators for these scalar , vector , and tensor correlations and quantify the amplitudes that can be detected if these perturbations have ( as may be expected in inflationary models ) nearly scale - invariant spectra . we begin with the null hypothesis that primordial density perturbations are statistically isotropic and gaussian . this implies that the fourier modes @xmath5 of the density perturbation @xmath6 ( at some fixed time ) have covariances , @xmath7 , where the kronecker ( dirac ) delta on the right - hand side is zero unless @xmath8 , @xmath9 is the matter power spectrum , and @xmath10 is the volume of the survey . in other words , the different fourier modes of the density field are uncorrelated under the null hypothesis . coupling of the inflaton to some other field produces non - gaussianity in the mass distribution that appears as off - diagonal ( i.e. , @xmath11 ) correlations of the density - field fourier components in the presence of a given realization of the new field . global si requires that a given fourier mode @xmath12 of wavevector @xmath13 and polarization @xmath14 ( about which we will say more below ) of the new field induces a correlation , @xmath15 where @xmath16 is shorthand for a kronecker delta that sets @xmath17 . note that @xmath12 here are the new - field fourier components during inflation when their effect on primordial perturbations is imprinted . the function @xmath18 is related to the density - density new - field bispectrum @xmath19 and new - field power spectrum @xmath20 through @xmath21 . global si requires that @xmath18 be a function only of @xmath22 , @xmath23 , and @xmath24 . the parameter @xmath14 labels the polarization state of the new field and @xmath25 its polarization tensor , a symmetric @xmath26 tensor . the most general such tensor can be decomposed into 6 orthogonal polarization states @xcite , which we label @xmath27 , that satisfy @xmath28 . these states can be taken to be two scalar modes @xmath29 and @xmath30 , two vector modes @xmath31 with @xmath32 , and two transverse traceless modes ( the `` tensor '' modes ) @xmath33 and @xmath34 . if @xmath13 is taken to be in the @xmath35 direction , then the @xmath36 polarization of the tensor mode has @xmath37 with all other components zero , and the @xmath38 polarization has @xmath39 with all other components zero . these two tensor modes are thus characterized by a @xmath40 or @xmath41 dependence , for @xmath42 and @xmath43 , respectively , on the azimuthal angle about the @xmath13 direction of the tensor mode . the first two columns in fig . [ fig : picture ] show the distortions induced to an otherwise isotropic two - point correlation function by correlation of the density field with a @xmath36 and @xmath38 polarized tensor mode . shown there is a quadrupolar distortion in the @xmath44-@xmath45 plane that then oscillates in phase as we move along the direction @xmath35 of the fourier mode . the first scalar mode has @xmath46 and as shown in fig . [ fig : picture ] represents an isotropic modulation of the correlation function as we move along the direction @xmath35 of the fourier wavevector . the other scalar ( or longitudinal - vector ) mode has @xmath47 which represents a stretching and compression along @xmath35 . both scalar modes represent local distortions of the two - point function that have azimuthal symmetry about @xmath13 . finally , the two transverse - vector modes have @xmath48 with all other components zero , and @xmath49 with all other components zero . these two modes represent stretching in the @xmath50 and @xmath51 directions , respectively , as shown in the last @xmath52 and @xmath53 columns in fig . [ fig : picture ] . these two transverse - vector modes have @xmath54 and @xmath55 dependences on the azimuthal angle @xmath56 about the direction of the fourier mode . the specific functional form of @xmath18 depends on the coupling of the new field ( scalar , vector , or tensor ) to the inflaton . global si requires , though , that @xmath18 will be the same for the two tensor polarizations and the same for the two vector polarizations ; i.e. , @xmath57 , and @xmath58 . the same is not necessarily true for the scalar perturbations . in fact , the polar - angle dependence that distinguishes the 0 and @xmath59 polarizations can be absorbed into @xmath60 and @xmath61 . thus , in practice , one can describe the most general scalar distortions to clustering in terms of either the 0 or the @xmath59 polarization by appropriate definition of @xmath60 or @xmath61 . ( this is the mixing between a scalar mode and a longitudinal - vector mode . ) we thus below merge these two polarizations into a single polarization which we label with a subscript @xmath62 . suppose now that a correlation such as that in eq . ( [ eqn : offdiagonal ] ) , for either a scalar , vector , or tensor distortion , is hypothesized . how would we go about measuring it ? according to eq . ( [ eqn : offdiagonal ] ) , each pair @xmath63 and @xmath64 of density modes with @xmath65 ( note that we have re - defined the sign of @xmath13 here ) provides an estimator , @xmath66^{-1},\ ] ] for the fourier - polarization amplitude @xmath12 . since @xmath67 , where @xmath68 is the measured matter power spectrum , including the signal @xmath9 and noise @xmath69 , the variance of this estimator is @xmath70 the minimum - variance estimator for @xmath12 is then obtained by summing over all these individual @xmath71 pairs with inverse - variance weighting : @xmath72 where the noise power spectrum , @xmath73^{-1 } , \label{eqn : noisepowerspectrum}\ ] ] is the variance with which @xmath74 is measured . this @xmath75 is a function only of the magnitude @xmath76 ( not its orientation ) as a consequence of global si , and for the same reason , @xmath77 , for both the signal and noise power spectra , and similarly @xmath78 . in general , the amplitudes @xmath12 arise as realizations of random fields with power spectra @xmath79 , for @xmath80 , which we write in terms of amplitudes @xmath81 and fiducial power spectra @xmath82 . we now proceed to write the optimal estimator for the amplitudes @xmath81 . each fourier - mode estimator @xmath74 for the appropriate polarizations ( @xmath62 for scalar , @xmath44 and @xmath45 for vector , and @xmath36 and @xmath38 for tensor ) provides an estimator , @xmath83^{-1 } \left[{v}^{-1 } \left|\widehat { h_p(\veck ) } \right|^2 - p_p^{n}(k ) \right ] , \label{eqn : individualestimator}\ ] ] for the appropriate power - spectrum amplitude . here we have subtracted out the noise contribution to unbias the estimator . if @xmath74 is estimated from a large number of @xmath63-@xmath64 pairs , then it is close to being a gaussian variable . if so , then the variance of the estimator in eq . ( [ eqn : individualestimator ] ) is , under the null hypothesis , @xmath84^{-2 } \left [ p_p^{n}(k)\right]^2.\ ] ] adding the estimators from each fourier mode with inverse - variance weighting leads us to the optimal estimator , @xmath85 ^ 2 } { 2\left[p_p^{n}(k ) \right]^2 } \left ( v^{-1 } \left|\widehat { { h}_p(\veck ) } \right|^2 - p_p^{n}(k ) \right ) , \label{eqn : tensorestimator}\ ] ] where @xmath86 ^ 2 / 2 \left[p_p^{n}(k ) \right]^2 . \label{eqn : noise}\ ] ] for the vector - power - spectrum amplitude @xmath87 we sum over @xmath88 and for the tensor - power - spectrum amplitude @xmath89 over @xmath90 . following the discussion above , the sum on @xmath14 is only for @xmath91 for @xmath92 . the estimator in eq . ( [ eqn : tensorestimator ] ) , along with the quadratic minimum - variance estimator in eq . ( [ eqn : minimvariance ] ) , demonstrates that the correlation of density perturbations with an unseen scalar , vector , or tensor perturbation appears in the density field as a nontrivial four - point correlation function , or trispectrum . the dependence of the trispectrum on the azimuthal angle about the diagonal of the fourier - space quadrilateral distinguishes the shape dependences of the trispectra for scalar , vector , and tensor modes . to specify this trispectrum more precisely , though , requires inclusion of the additional contribution induced by modes @xmath13 that involve the other two diagonals of the quadrilateral . likewise , if a signal is detected i.e . , if the null - hypothesis estimators above are found to depart at @xmath93 from the null hypothesis then the optimal measurement and characterization of the trispectrum requires modification of the null - hypothesis estimators in a manner analogous to weak - lensing estimators @xcite . we now evaluate the smallest amplitudes @xmath94 , @xmath95 , and @xmath96 that can be detected with a given survey . to do so , we take for our fiducial models nearly scale - invariant spectra @xmath97 , with @xmath98 . we moreover take the density - density new - field bispectrum to be of the form in ref . we then find that the integrand ( using @xmath99 ) in eq . ( [ eqn : noisepowerspectrum ] ) is dominated by the squeezed limit ( @xmath100 ) where @xmath101 . we then approximate @xmath102 for @xmath103 , where @xmath104 is the largest wavenumber for which the power spectrum can be measured with high signal to noise , and @xmath105 for @xmath106 . this then yields a noise power spectrum @xmath107 and @xmath108 . evaluating the integral in eq . ( [ eqn : noise ] ) , we find the scalar , vector , and tensor amplitudes detectable at @xmath109 ( for @xmath110 ) to be @xmath111 where @xmath112 and @xmath113 . the smallest detectable power - spectra amplitudes are thus inversely proportional to the number of fourier modes in the survey . we show the projected detection sensitivities for surveys with volumes of @xmath114 ^ 3 $ ] and @xmath115 ^ 3 $ ] in fig . [ fig : ston ] . , @xmath95 and @xmath96 , respectively , detectable at the 3@xmath116 level as a function of the maximum wavenumber @xmath104 of the survey . shown are results for survey volumes of @xmath115 ^ 3 $ ] and @xmath114 ^ 3 $ ] , or minimum wavenumbers @xmath117 $ ] and @xmath118 $ ] , respectively . , scaledwidth=48.0% ] for example , if we apply this estimate to a tensor field and assume this tensor field to be primordial gravitational waves , then a sensitivity to a tensor amplitude @xmath119 near the current upper limit requires @xmath120 . such a dynamic range is probably beyond the reach of galaxy surveys , but it may be within reach of the 21-cm probes of neutral hydrogen during the dark ages envisioned in refs . @xcite . of course , the signal could be larger if the inflaton is correlated with a scalar , vector , or tensor field that leaves no other trace . finally , several new tests for parity - violating early - universe physics can be developed from simple modification of the estimators above . to do so , we substitute the @xmath44 and @xmath45 polarizations , and @xmath36 and @xmath38 polarizations , with circular - polarization tensors @xmath121 and @xmath122 . the two right - most patterns shown in fig . [ fig : picture ] are the circular polarization patterns for tensor and vector modes . it may then be tested whether the power spectra for right- and left - circular polarizations are equal . for example , chiral - gravity models @xcite may predict such parity - violating signatures in primordial gravitational waves , and similar models with parity - violating vector perturbations are easily imaginable . of course , `` real - world '' effects like redshift - space distortions , biasing , and nonlinear evolution , must be taken into account before the estimators written above can be implemented , but there are well - developed techniques to deal with these issues @xcite . in summary , we have shown that the most general two - point correlation functions for the cosmological mass distribution can be decomposed into scalar , vector , and tensor distortions . we have presented straightforward recipes for measuring these distortions . such effects may arise if the inflaton is coupled to some new field during inflation . we have avoided discussion of specific models , but the introduction of new fields during inflation is quite generic to inflationary models . we therefore advocate measurement of these correlations with galaxy surveys , and in the future with 21-cm surveys , as a simple and general probe of new inflationary physics . m. tegmark _ et al . _ [ sdss collaboration ] , phys . rev . d * 69 * , 103501 ( 2004 ) [ astro - ph/0310723 ] ; d. j. eisenstein _ et al . _ [ sdss collaboration ] , astrophys . j. * 633 * , 560 ( 2005 ) [ astro - ph/0501171 ] ; s. cole _ et al . _ [ the 2dfgrs collaboration ] , mon . not . soc . * 362 * , 505 ( 2005 ) [ astro - ph/0501174 ] . s. furlanetto , s. p. oh and f. briggs , phys . rept . * 433 * , 181 ( 2006 ) [ arxiv : astro - ph/0608032 ] ; m. f. morales and j. s. b. wyithe , annu . astrophys . * 48 * , 127 ( 2010 ) [ arxiv:0910.3010 [ astro-ph.co ] ] . s. jester and h. falcke , new astron . rev . * 53 * , 1 ( 2009 ) [ arxiv:0902.0493 [ astro-ph.co ] ] . d. baumann and d. green , arxiv:1109.0292 [ hep - th ] . k. dimopoulos _ et al . _ , jcap * 0905 * , 013 ( 2009 ) [ arxiv:0809.1055 [ astro - ph ] ] ; a. golovnev and v. vanchurin , phys . rev . d * 79 * , 103524 ( 2009 ) [ arxiv:0903.2977 [ astro-ph.co ] ] ; n. bartolo _ et al . _ , jcap * 0911 * , 028 ( 2009 ) [ arxiv:0909.5621 [ astro-ph.co ] ] . r. w. hellings and k. nordtvedt , phys . d * 7 * , 3593 ( 1973 ) ; j. beltran jimenez and a. l. maroto , phys . d * 80 * , 063512 ( 2009 ) [ arxiv:0905.1245 [ astro-ph.co ] ] . j. m. maldacena , jhep * 0305 * , 013 ( 2003 ) [ astro - ph/0210603 ] . d. seery , m. s. sloth and f. vernizzi , jcap * 0903 * , 018 ( 2009 ) [ arxiv:0811.3934 [ astro - ph ] ] . s. dodelson , e. rozo and a. stebbins , phys . lett . * 91 * , 021301 ( 2003 ) [ astro - ph/0301177 ] . k. w. masui and u. l. pen , phys . lett . * 105 * , 161302 ( 2010 ) [ arxiv:1006.4181 [ astro-ph.co ] ] . u. -l . pen , r. sheth , j. harnois - deraps , x. chen and z. li , arxiv:1202.5804 [ astro-ph.co ] . d. m. eardley _ et al . _ , phys . lett . * 30 * , 884 ( 1973 ) . m. h. kesden , a. cooray and m. kamionkowski , phys . d * 67 * , 123507 ( 2003 ) [ astro - ph/0302536 ] . a. loeb and m. zaldarriaga , phys . lett . * 92 * , 211301 ( 2004 ) [ arxiv : astro - ph/0312134 ] . a. lue , l. -m . wang and m. kamionkowski , phys . * 83 * , 1506 ( 1999 ) [ astro - ph/9812088 ] ; t. takahashi and j. soda , phys . lett . * 102 * , 231301 ( 2009 ) [ arxiv:0904.0554 [ hep - th ] ] ; c. r. contaldi , j. magueijo and l. smolin , phys . * 101 * , 141101 ( 2008 ) [ arxiv:0806.3082 [ astro - ph ] ] . d. jeong , ph.d . thesis , university of texas at austin ( 2010 ) .

many inflationary theories introduce new scalar , vector , or tensor degrees of freedom that may then affect the generation of primordial density perturbations . here we show how to search a galaxy ( or 21-cm ) survey for the imprint of primordial scalar , vector , and tensor fields . these new fields induce local departures to an otherwise statistically isotropic two - point correlation function , or equivalently , nontrivial four - point correlation functions ( or trispectra , in fourier space ) , that can be decomposed into scalar , vector , and tensor components . we write down the optimal estimators for these various components and show how the sensitivity to these modes depends on the galaxy - survey parameters . new probes of parity - violating early - universe physics are also presented .

introduction

arxiv

chemistry of sulfur - containing species in space is especially interesting due to its chemical activity and relatively high abundance @xcite . it is known that abundances of sulfur - bearing molecules are strongly influenced by the presence of shocks @xcite . so far , 14 sulfur - bearing species have been identified in space ; cs , so , ns , sis , so@xmath5 , sh@xmath6 , ocs , ccs , c@xmath7s , h@xmath6cs , hncs , so@xmath6 , hcs@xmath5 , and ch@xmath7sh . however , the simplest sulfur compound , sh , has not been detected in spite of radioastronomical searches using the @xmath8-type doubling transitions @xcite . while sulfur chemistry in the circumstellar envelope of red giant stars has been studied extensively ( e.g. , * ? ? ? * ) , that in the atmosphere has been regarded as in the thermal equilibrium state @xcite . in the atmosphere of oxygen - rich giants , sh is the molecule first formed from the sulfur atom , at the temperature of about 2000 k. the sh abundance decreases when temperature is below @xmath9 k , and h@xmath6s and sis become the dominant species . on the other hand , @xcite report the detection of the @xmath10 infrared band of so@xmath6 in the spectra of oxygen - rich mira variables obtained by the short - wavelength spectrometer ( sws : * ? ? ? * ) on board the infrared space observatory ( iso : * ? ? ? their model analysis indicates that the molecules are located in the extended atmosphere , at about five stellar radii . the abundance of so@xmath6 is 510 orders of magnitude larger than the values in thermal equilibrium @xcite . the result strongly suggests the presence of non - equilibrium processes in the extended atmosphere , probably related to shocks due to stellar pulsation @xcite . in this paper , we report the detection of sh ro - vibrational transition lines in the published high - resolution infrared spectrum of the s - type star , r and . we discuss the physical and chemical conditions of the sh layer in the star . the data used in this study were obtained with the fourier - transform spectrometer at the kitt peak national observatory 4 m telescope , and were published in @xcite . r and is a mira variable with a period of 409 days and a visual amplitude of 9.1 mag @xcite . the spectral type ranges in s3,5e s8,8e . the observation of r and was performed at the optical variable phase of @xmath11 . unfortunately , the reduced data were lost after several updates of archive formats at the observatory . therefore , we re - digitized the data from the printed spectrum . the results may be as accurate as the original within @xmath12 cm@xmath2 in wavenumber and a few % in intensity . the spectrum was shifted by 5.2 kms@xmath2 to longer wavelengths to adjust the absorption minimum of hcl lines at their rest frequencies @xcite . no correction for terrestrial motion was applied . in their original paper , @xcite give identifications of oh , nh , ch , sio , cs , hcl , and atomic lines . the spectrum of r and , especially in the wavelengths shorter than the sio first - overtone bandheads , is dominated by hcl lines . oh and nh lines are detected , but much less prominently than in the oxygen - rich stars like @xmath14 ori and @xmath14 tau . there are many strong lines left unidentified . figure [ fig1 ] shows the spectrum of r and between 2700 and 2750 cm@xmath2 . the positions of sh and hcl ro - vibrational transitions are indicated . the frequencies are calculated from the molecular constants given by @xcite for sh , and are taken from hitran database 1996 edition @xcite for hcl . it is obvious that many unidentified strong lines are attributed to sh transitions . we assigned 39 , 41 , and 11 transitions in the @xmath15 , @xmath16 , and @xmath17 bands , respectively , between 2500 and 2778 cm@xmath2 . the lower state energies of these transitions are up to @xmath18 cm@xmath2 , indicating that the molecules are highly excited . the isotope @xmath19sh may also be detected . the frequencies of @xmath19sh lines are estimated from the molecular constants of sh using the relation of reduced mass @xcite . we analyze the observed sh line profiles with a simple model . the sh ( and also hcl ) lines in r and show inverse p - cygni profiles , indicating that the molecular layer is moving inward to the star due to stellar pulsation . we , therefore , apply a spherical shell model with a constant infall velocity . the molecules in front of the star cause absorption , while those extended in the blank sky contribute as emission . considering that the shell may be in the atmosphere , we adopt an exponential density law , @xmath20 , where @xmath21 is the scale height . the star is assumed to be a 3000 k blackbody . for simplification , a constant excitation temperature is adopted , and the energy population of the molecule is calculated by assuming local thermodynamic equilibrium ( lte ) . this may be justified if the molecular shell is thin and in the high density region near the photosphere . the line intensity of sh is calculated based on @xcite , which takes account of the herman - wallis effect . the spectrum is normalized by the stellar continuum . no smoothing is applied . the excitation temperature can be determined from the relative intensities of the lines at different energy levels ; higher temperatures excite the molecules to higher energy levels and increase the line strength from these levels . we find that 2200 k is most reasonable for the present case . a turbulent velocity of 6 kms@xmath2 and infall velocity of @xmath22 kms@xmath2 reproduce the line profiles . the model spectrum is shifted by @xmath23 kms@xmath2 . considering the terrestrial velocity of @xmath24 kms@xmath2 at transit of the observation , and the shift applied by @xcite , @xmath25 kms@xmath2 , a radial velocity of @xmath26 kms@xmath2 is obtained . the inner radius and the scale height of the sh shell is 1.0 and 0.08 times the stellar radius , respectively . the sh column density in the shell is @xmath27 @xmath28 . the uncertainties of the parameters , which changes the model spectrum by about 10 % , are @xmath29 k for excitation temperature , a factor of two for column density , @xmath30 r@xmath31 for scale hight , and @xmath32 kms@xmath2 for the velocities . the radial velocity has an error of @xmath33 kms@xmath2 due to the re - digitization process . the model spectrum is compared with the observation in figure [ fig2 ] . the wavelength regions are selected so that the contamination of other spectral lines is minimal , and that a wide range of energy levels is covered . strong lines in figure [ fig2 ] are listed in table [ tbl1 ] . the fit is satisfactory in most of the lines . fitting of @xmath19sh lines results in an isotopic abundance ratio of 510 % . we note that the same parameters also give reasonable fits for the hcl lines with a column density of @xmath34 @xmath28 . why are sh lines so prominent in r and , an s - type star ? s - type stars are characterized by their chemical anomaly of similar carbon and oxygen abundances . this leaves few c and o atoms for further chemical processes , after formation of co molecules at 45000 k. sh should be quite abundant in the atmosphere of oxygen - rich stars , but a minor product in a carbon - rich environment @xcite . since we do see sio lines , but no cs line , in the spectrum of r and , the star is slightly oxygen - rich . nevertheless , it is not expected that sh is much more abundant in s - type stars than o - rich stars . this also holds for hcl , which is the most abundant cl - bearing molecule in an oxygen - rich environment , although the absolute abundance is by one order of magnitude lower than sh . @xcite suggested that the atmosphere of s - type stars is more transparent than o - rich stars , so that the lines of minor species could be stronger due to larger path length . we especially emphasize the contribution of the h@xmath6o molecules . in the same paper , ridgway et al . reported that an enormous number of lines heavily blanket the 24002800 cm@xmath2 region of the @xmath35 cet spectrum . they suspected that these are highly excited water lines . this is supported by the iso / sws observation of @xmath35 cet @xcite . despite the relatively poor resolution of the sws spectrum , they demonstrate that the star shows highly excited h@xmath6o lines in the 3.54.0 @xmath36 m region , arising from hot ( 2000 k ) and optically thick molecular layer . probably , in o - rich stars , sh ( and hcl ) lines are weaker because of the shorter path length , and also heavily contaminated by the hot water lines . on the other hand , h@xmath6o molecules are not favored in the atmosphere of s - type stars . the thermal equilibrium abundance of h@xmath6o in s - type stars is 34 orders lower than in o - rich stars @xcite . the derived infall velocity and the excitation temperature of the sh layer are consistent with the measurement of co @xmath3 lines by @xcite at @xmath37 , indicating that the sh molecules are in the same region as hot co near the photosphere . comparison with the co column density , @xmath38 @xmath28 , given by @xcite , leads to sh / h @xmath39 . in thermal equilibrium at the temperature of 2200 k and gas pressure of @xmath40 , the sh abundance is @xmath41 @xcite . this value is insensitive to the c / o ratio as long as c / o @xmath42 . the present estimate of sh abundance in r and , although it is rather crude , is consistent with this calculated value . the sulfur chemistry in sh layer still follows thermal equilibrium , because of its high density and high temperature . we see no clear evidence of distinct velocity and/or different temperature component of sh lines in the spectrum . this implies that the sh molecules are distributed only in a thin layer in the atmosphere . in thermal equilibrium , sh is most abundant around 1800 k , and then is rather quickly transformed to h@xmath6s or sis below @xmath9 k @xcite . otherwise , non - equilibrium chemical reactions may lead to completely different compositions in the extended atmosphere , e.g. the enhancement of so@xmath6 in oxygen - rich stars @xcite . we could not find any clear indications of h@xmath6s or so@xmath6 in the spectrum of r and . the upper limit of the h@xmath6s column density is at least a factor of 10 larger than the observed amount of sh . a dioxide molecule so@xmath6 may not be abundant in the atmospheres of s - type stars . other possible candidates , so or sis , have no transition in the present wavelength coverage . beck , h. k. b. , gail , h. -p . , henkel , r. , & sedlmayr , e. , 1992 , , 265 , 626 benidar , a. , farrenq , r. , guelachvili , g. , & chackerian , jr . c. , 1991 , j. mol . , 147 , 383 de graauw , th . , et al . , 1996 , , 315 , l49 duari , d. , cherchneff , i. , & willacy , k. , 1999 , , 341 , l47 duley , w. w. , millar , t. j. , williams , d. a. , 1980 , , 192 , 945 hartquist , t. w. , oppenheimer , m. , & dalgarno , a. , 1980 , , 236 , 182 heiles , c. e. , & turner , b. e. , 1971 , , 8 , 89 herzberg , g. f. r. s. , 1950 , molecular spectra and molecular structure i. spectra of diatomic molecules ( princeton : d . van nostrand ) hinkle , k. h. , scharlach , w. w. g. , & hall , d. n. b. , 1984 , , 56 , 1 kessler , m. f. , et al . , 1996 , , 315 , l27 kholopov , p. n. , et al . , 1988 , general catalogue of variable stars . , ( moscow : nauka publishing house ) meeks , m. l. , gordon , m. a. , & litvak , m. m. , 1969 , science , 163 , 173 omont , a. , lucas , r. , morris , m. , & guilloteau , s. , 1993 , , 267 , 490 ram , r. s. , bernath , p. f. , engleman , jr . r. , & brault , j. w. , 1995 , j. mol . spectrosc . , 172 , 34 ridgway , s. t. , carbon , d. f. , hall , d. n. b. , & jewell , j. , 1984 , , 54 , 177 rothman , l. s. , et al . , 1992 , jqsrt , 48 , 469 tsuji , t. , 1964 , ann . tokyo astron . obs . , 2nd ser . 9 , 1 tsuji , t. , 1973 , , 23 , 411 tsuji , t. , 1999 , private communication woitke , p. , helling , ch . , winters , j. m. , & jeong , k. s. , 1999 , , 348 , l17 yamamura , i. , de jong , t. , onaka , t. , cami , j. , & waters , l. b. f. m. , 1999a , , 341 , l9 yamamura , i. , de jong , t. , & cami , j. , 1999b , , 348 , l55 rlllr 1 & 2532.1257 & 3/2 & @xmath43 p 3.5ff & @xmath44 + 1 & 2532.1486 & 3/2 & @xmath43 p 3.5ee & @xmath45 + 2 & 2534.7858 & 1/2 & @xmath17 r 7.5ee & @xmath46 + 2 & 2534.8657 & 1/2 & @xmath17 r 7.5ff & @xmath47 + 3 & 2546.2628 & 1/2 & @xmath16 r 1.5ee & @xmath48 + 4 & 2546.4908 & 1/2 & @xmath16 r 1.5ff & @xmath49 + 5 & 2546.9511 & 1/2 & @xmath17 r 8.5ee & @xmath50 + 5 & 2547.0067 & 1/2 & @xmath17 r 8.5ff & @xmath51 + 6 & 2549.2843 & 1/2 & @xmath43 p 2.5ff & @xmath52 + 7 & 2549.5750 & 1/2 & @xmath43 p 2.5ee & @xmath53 + 8 & 2641.3159 & 3/2 & @xmath17 r24.5ee & @xmath54 + 8 & 2641.3458 & 3/2 & @xmath17 r24.5ff & @xmath55 + 9 & 2642.8296 & 3/2 & @xmath43 r 1.5ee & @xmath56 + 9 & 2642.8395 & 3/2 & @xmath43 r 1.5ff & @xmath57 + 10 & 2644.6568 & 1/2 & @xmath43 r 1.5ee & @xmath58 + 10 & 2644.7048 & 3/2 & @xmath16 r 8.5ee & @xmath59 + 10 & 2644.7949 & 3/2 & @xmath16 r 8.5ff & @xmath60 + 10 & 2644.8958 & 1/2 & @xmath43 r 1.5ff & @xmath61 + 11 & 2659.3953 & 3/2 & @xmath43 r 2.5ee & @xmath62 + 11 & 2659.4143 & 3/2 & @xmath43 r 2.5ff & @xmath63 + 12 & 2661.5489 & 1/2 & @xmath16 r 9.5ee & @xmath64 + 12 & 2661.5866 & 1/2 & @xmath16 r 9.5ff & @xmath65 + 13 & 2661.9331 & 1/2 & @xmath43 r 2.5ee & @xmath53 + 13 & 2662.1518 & 1/2 & @xmath43 r 2.5ff & @xmath52 + 14 & 2709.9111 & 1/2 & @xmath43 r 5.5ee & @xmath66 + 14 & 2710.0575 & 1/2 & @xmath43 r 5.5ff & @xmath67 + 15 & 2710.9529 & 1/2 & @xmath16 r14.5ff & @xmath68 + 15 & 2711.0189 & 1/2 & @xmath16 r14.5ee & @xmath69 + 16 & 2770.9408 & 3/2 & @xmath43 r10.5ee & @xmath70 + 16 & 2771.0636 & 3/2 & @xmath43 r10.5ff & @xmath71 +

we report the identification of sh @xmath0 ro - vibrational lines in the published high - resolution infrared spectrum of the s - type star , r and . this is the first astronomical detection of this molecule . the lines show inverse p - cygni profiles , indicating infall motion of the molecular layer due to stellar pulsation . a simple spherical shell model with a constant infall velocity is adopted to determine the condition of the layer . it is found that a single excitation temperature of 2200 k reproduces the observed line intensities satisfactory . sh is located in a layer from 1.0 to @xmath1 stellar radii , which is moving inward with a velocity of 9 kms@xmath2 . these results are consistent with the previous measurements of co @xmath3 transitions . the estimated molecular abundance sh / h is @xmath4 , consistent with a thermal equilibrium calculation .

introduction identification of sh lines modeling discussion

arxiv

the charge and spin oscillatory interactions in metals has attracted considerable attention both on the theoretical and experimental sides @xcite . ruderman and kittle @xcite suggested that the spin oscillatory interaction in metals could provide a long - range interaction between nuclear spins in metals . afterwards , kasuya and yosida @xcite extended the theory to include the long - range interaction between magnetic impurities and thus the combined refers to rkky interaction . the recent discovery of graphene @xcite , the two - dimensional crystal of carbon atoms , has provided a new material with a peculiar structure for the charge and the spin interactions . this stable crystal has already attracted considerable attention because of its unusual effective many - body properties @xcite that follow from chiral nature of linearly dispersing low energy excitations described by pair of dirac cones at the @xmath3 and @xmath4 edges of the first brillouin zone . the rkky interaction in pristine graphene has been studied by several groups @xcite . due to the particle - hole symmetry of graphene , the rkky interaction induces ferromagnetic correlation between magnetic impurities on the same sublattice , while anti - ferromagnetic correlation between the ones on different sublattices . the dependence of the interaction on the distance @xmath2 between two local magnetic moments , at the dirac point , is found to be @xmath5 , whereas it behaves as @xmath6 in conventional two - dimensional ( 2d ) systems @xcite . such a fast decay rate denotes that the interaction is rather short - ranged . in doped graphene , on the other hand , the spacial dependence of the interaction is predicted to be similar to conventional 2d systems , but this still remains to be experimentally verified . due to the fact that the rkky interaction is originated by the exchange coupling between the impurity moments and the spin of itinerant electrons in the bulk of the system , spin polarization of electrons is expected to influence directly this interaction @xcite . in particular , combination of the spin - dependence with a dirac - like spectrum can mediate a much richer collective behavior of magnetic adatoms @xcite . this has been explained for surface states of a three dimensional topological insulator , on which magnetic impurities exhibit a frustrated rkky interaction with two possible phases of ordered ferromagnetic phase and a disordered spin glass phase @xcite . graphene , in particular , with imbalanced chemical potentials of spin - up and spin - down electrons , presents a unique spin chiral material in which the interplay between the spin polarization , gapless spectrum , and the chiral nature of electrons have been shown to result intriguing phenomena @xcite . in two - dimension graphene system , the polarization of the chemical potential can be tuned to be of order or even higher than the mean chemical potential , the condition which is not possible in ordinary conductors . the aim of the present study is to address the question of how this peculiarity can affect the collective coupling of magnetic impurities on the surface of graphene sheet with a finite polarization of the spin . in this work , we calculate the rkky interaction mediated by spin - polarized dirac fermions in a monolayer graphene using the green s function method . our theory for the spin polarization dependence of rkky interaction is motivated not only by fundamental transport considerations , but also by application and potential future experiments in graphene spintrotic field . with a spin - polarization along the @xmath0-axis , we show that the rkky interaction is anisotropic corresponding to a xxz model interaction between the two magnetic moments when their spin orientations are fixed . besides @xmath5 dependence of the interaction for undoped graphene , we show particularly that the interaction behaves like @xmath6 when the spin polarization is finite . in addition , a beating pattern for the interaction in the cases where impurities are located along certain directions is obtained near the resonance condition which is controlled by the chemical potential and the spin polarization . the paper is organized as the following . in sec . [ sect : theory ] we introduce the formalism that will be used in calculating the rkky interaction from the lattice green s function . in sec . [ sect : results ] we present our analytic and numeric results for the coupling strengths of the rkky interaction in both undoped and doped graphene sheets . sec . [ sect : conclusion ] contains discussions and a brief summary of our main results . we consider a spin - polarized graphene system identified by a spin dependent chemical potential @xmath7 ( @xmath8 ) , implying a mean chemical potential @xmath9 and the spin polarization @xmath10 . such a spin - polarization can be injected , for instance , by ferromagnetic electrodes on top of the graphene sheet @xcite . intrinsic ferromagnetic correlations are also predicted to exist in graphene sheets @xcite and nanoribbons with zigzag edges @xcite under certain conditions . the electronic structure of spin polarized graphene can be reasonably well described using a rather simple tight - binding hamiltonian , leading to analytical solutions for their energy dispersion and related eigenstates . the noninteracting nearest neighbor tight - binding hamiltonian for @xmath11-band electrons with spin @xmath12 , is determined by @xcite @xmath13 where @xmath14 annihilates an electron with spin @xmath12 on sublattice a(b ) of unit cell @xmath15 and @xmath16 ev denotes the nearest neighbor hopping parameter @xcite . the sum @xmath17 in eq . ( [ eq : tb ] ) runs over distinct nearest neighbors . the @xmath12-component of the noninteracting hamiltonian in momentum space is written as @xmath18 where the form factor in general case is @xmath19 , in which @xmath20 s are nearest neighbor position vectors . in this work , we are interested in the low - energy behavior , in which @xmath21 , where @xmath22 at the dirac point @xmath3 and @xmath23 at the another dirac point , @xmath4 , the chiral angle is @xmath24 , @xmath25 m / s is the fermi velocity with @xmath26 being the carbon - carbon distance in honeycomb lattice . our system incorporates two localized magnetic moments whose interaction is mediated through a spin polarized electron liquid . we assume that the graphene is spin polarized first , and then we add the magnetic moments . the contact interaction between the spin of itinerant electrons and two magnetic impurities with magnetic moments @xmath27 and @xmath28 , located respectively at @xmath29 and @xmath30 , is given by @xmath31 where @xmath32 is the coupling constant between conduction electrons and impurity , @xmath33 is the spin density operator @xcite with @xmath34 and @xmath35 being the position and vector of spin operators of @xmath15th electron . the rkky interaction which arises from the quantum effects is obtained by using a second order perturbation @xcite which reads as ( from now on we set @xmath36 ) @xmath37 n(\mu)~,\end{aligned}\ ] ] where @xmath38 denotes the fermi - dirac distribution function , @xmath39 is the vector of pauli matrices in the _ spin _ space , @xmath40 is a @xmath41 matrix of the single particle retarded green s function in spin space , and @xmath42 and @xmath43 refer to the sublattices where two impurities are placed and finally , the trace is taken over the spin degree of freedom . for spin unpolarized graphene , eq . ( [ eq : rkky ] ) simplifies to @xmath44 , where @xmath45 is the spin susceptibility of the itinerant electrons and determines the indirect interaction between two local moments . in order to calculate the interaction hamiltonian of eq . ( [ eq : rkky ] ) , the form of the electronic single particle green s function , @xmath46 is needed . to calculate the retarded green s function in real space , its fourier components in momentum space might be first obtained . due to the fact that our 2d dirac fermion system is noninteracting and thus the direction of spin remains unchanged , the retarded green s function @xmath47 are diagonal in the spin space [ green1 ] g^s_aa(*r*,0,)= ( e^i*k*+e^i*k * ) g_aa(-s_p ) , and @xmath48 moreover , @xmath49 and @xmath50 . here @xmath51 and @xmath52 , where @xmath53 and @xmath54 are the modified bessel functions of the second kind , @xmath55 is the angle of the position @xmath56 with respect to the @xmath57 direction and @xmath58 , in which @xmath59 is the area of the brillouin zone . by inserting the retarded green s functions given by eqs . ( [ green1 ] ) and ( [ green2 ] ) in eq . ( [ eq : rkky ] ) , and taking the trace over spin degree of matrices , the rkky hamiltonian simplifies to @xmath60~,\ ] ] which is the honored xxz model . here @xmath61 and @xmath62 , with @xmath63 , @xmath64 $ ] and @xmath65 $ ] . the different components of @xmath66 for impurities on the same sublattice read as [ eq : j ] i^aa_x&=2 m , + i^aa_z&=m , where @xmath67 , @xmath68 , @xmath69 , and @xmath70 . for impurities on different sublattices , one only needs to replace @xmath53 with @xmath71 in the above equations . note that @xmath72 and @xmath73 . we find analytic results for the @xmath0 component of the rkky exchange coupling strength @xmath74 , for both cases where magnetic moments are located on the same or different sublattices . for the same sublattice case , we begin by splitting the integral in the second line of eq . ( [ eq : j ] ) into the conduction and valance band contributions , and find @xmath75\nonumber\\ & + \sum_{s=\pm}\im m\left [ \int_0^{x_{\rm fs } } dx\ x^2 k_0 ^ 2(-ix)\right]~,\end{aligned}\ ] ] where @xmath76 . the first integral can be solved @xcite easily and the result is @xmath77 . the contribution from the second line of eq . ( [ eq : iz ] ) , can be also obtained by replacing @xmath78 $ ] with @xmath79 , and using the following relation _ 0^x_f dx x^2 sign(x ) j_0(|x|)y_0(|x|)=- m(x_f ) , where @xmath80 $ ] is meijer @xmath81-function @xcite . as a result , the function @xmath82 is given by [ eq : iz_aa ] i_z^aa(r)=_s= . to calculate the long - range behavior of the rkky interaction , the asymptotic behavior of the meijer @xmath81-function is needed . it is also easy to see that asymptotic behavior of @xmath83 at large @xmath1 is @xcite @xmath84/(8\sqrt{\pi}x)$ ] . it should be noticed that the @xmath83 tends to its long - range asymptotic expression for @xmath85 . therefore , @xmath86 for the long - range regime is simplified as @xmath87\ ] ] on the other hand , for the case that the impurities are located on two different sublattices we can follow the same procedure discussed above , while we use @xmath88 = 3\pi^2/32 $ ] , and @xmath89=\pi^2 sign(x)j_1(|x|)y_1(|x|)/2 $ ] to find _ 0^x_f dx x^2 sign(x ) j_1(|x| ) y_1(|x| ) = - m(x_f ) , where @xmath90 $ ] . finally the @xmath91 reads as [ eq : iz_ab ] i_z^ab(r)=_s= . the asymptotic behavior of @xmath92 at large @xmath1 is @xmath93/(8\sqrt{\pi}|x|)$ ] . therefore , the long - range behavior of @xmath94 is obtained as [ eq : iz_ab_lim ] i_z^ab(ra)_s= . it should be mentioned that we were not able to find simple analytic expressions for the in plane components of the exchange coupling , @xmath95 and in the next section we will present our numerical results of them . as a function of the distance @xmath2 when both impurities are located on the same sublattice for various values of the spin polarization , @xmath96 in units of ev . the chemical potential is set to zero . symbols refer to the analytical results of eq . ( [ eq : iz_aa_lim ] ) which are compared to the numerical evaluation of eq . ( [ eq : iz_aa ] ) , plotted as lines . for @xmath97 , @xmath98 is just a constant . ( b ) a comparison between the integral @xmath99 and @xmath100 as a function of the distance @xmath2 for various values of the spin polarization @xmath96 in units of ev . at finite @xmath96 , the integral @xmath74 has a quite different behavior , oscillating as a function of @xmath2 , with a period given by @xmath101 and a linearly growing amplitude . a comparison between @xmath2-dependence of the integral @xmath99 and that of @xmath100 , shows their difference at short distance while reaching each other as @xmath2 increases . [ fig : aaizef0 ] , title="fig : " ] as a function of the distance @xmath2 when both impurities are located on the same sublattice for various values of the spin polarization , @xmath96 in units of ev . the chemical potential is set to zero . symbols refer to the analytical results of eq . ( [ eq : iz_aa_lim ] ) which are compared to the numerical evaluation of eq . ( [ eq : iz_aa ] ) , plotted as lines . for @xmath97 , @xmath98 is just a constant . ( b ) a comparison between the integral @xmath99 and @xmath100 as a function of the distance @xmath2 for various values of the spin polarization @xmath96 in units of ev . at finite @xmath96 , the integral @xmath74 has a quite different behavior , oscillating as a function of @xmath2 , with a period given by @xmath101 and a linearly growing amplitude . a comparison between @xmath2-dependence of the integral @xmath99 and that of @xmath100 , shows their difference at short distance while reaching each other as @xmath2 increases . [ fig : aaizef0 ] , title="fig : " ] in this section , we present our main results for the rkky exchange coupling in the presence of a spin polarization dirac fermions along the @xmath0-axis by analyzing the above calculated integrals of @xmath102 and @xmath74 . we extend the previously studied @xcite results for dependencies on the distance @xmath2 and lattice direction @xmath55 to the case of @xmath103 , for two different regimes of undoped , ( @xmath104 ) and doped , ( @xmath105 ) graphene . versus distance between two impurities , when both impurities are located on the same sublattice for undoped graphene and for several values of @xmath96 in units of ev . finite @xmath96 , produces a linear increase of @xmath106 with a slope proportional to @xmath96 . ( b ) a comparison between @xmath102 and @xmath74 for different configurations and for @xmath107 ev . [ fig : aaixef0 ] , title="fig : " ] versus distance between two impurities , when both impurities are located on the same sublattice for undoped graphene and for several values of @xmath96 in units of ev . finite @xmath96 , produces a linear increase of @xmath106 with a slope proportional to @xmath96 . ( b ) a comparison between @xmath102 and @xmath74 for different configurations and for @xmath107 ev . [ fig : aaixef0 ] , title="fig : " ] for the @xmath74 component of interactions , we solve the two expressions in eqs . ( [ eq : iz_aa ] ) and ( [ eq : iz_ab ] ) , numerically and then compare the results with asymptotic results obtained from the analytical expressions given by eqs . ( [ eq : iz_aa_lim ] ) and ( [ eq : iz_ab_lim ] ) , respectively . generally , the results obtained from the two approaches match quite good in most of the case specially at long distances . the distance dependence of @xmath74 for both @xmath108 and @xmath109 cases are illustrated in fig . [ fig : aaizef0 ] for the undoped graphene . for an unpolarized graphene @xmath97 , @xmath110 is just a constant . at finite @xmath96 , the integral @xmath74 has quite different behavior exhibiting an oscillatory behavior as a function of @xmath2 , with a linearly growing amplitude and a period given by @xmath101 , as can be obtain directly from eq . ( [ eq : iz_aa_lim ] ) . this behavior of @xmath111 results in an oscillatory @xmath112 with a decreasing amplitude as @xmath6 , which mimics the behavior of the rkky coupling of an unpolarized doped graphene @xcite . we can understand this analogy by noting that the polarization induces spin - dependent doping of up and down spin dirac bands of an undoped sample by shifting their chemical potential from the dirac point . a comparison between @xmath2-dependence of the integral @xmath99 and that of @xmath100 for various values of @xmath96 in fig . [ fig : aaizef0](b ) , shows their difference at short distance while reaching each other as @xmath2 increases . as @xmath113 the coupling interactions tend to their values of the unpolarized case where @xmath114 is three times larger than @xmath114 , as is discussed in ref . [ ] . from the numerical calculations of the integrals appearing in eq . ( [ eq : j ] ) , we can also obtain the behavior of @xmath106 and @xmath115 for the rkky interaction coupling of the components of the magnetic moments which are perpendicular to the spin polarization axis . [ fig : aaixef0](a ) shows @xmath106 as a function of @xmath2 for the undoped graphene at different values of the spin polarization , @xmath96 . for @xmath97 , this function is a constant resulting a @xmath116 which decays as @xmath5 . a finite difference @xmath96 between the chemical potentials of spin up and spin down carriers , produces a linear increase of @xmath106 with a slope proportional to @xmath96 . thus , @xmath116 decays as @xmath6 . importantly , the sign of interaction @xmath116 is always positive which shows that the coupling between the perpendicular components of the moments remains ferromagnetic - like for all @xmath2s . to analyze the difference between the two configurations of @xmath108 and @xmath109 , in fig . [ fig : aaixef0](b ) we have compared @xmath106 and @xmath115 , which shows that despite the difference at short distances , they tend to each other at larger distances . at finite values of both the chemical potential and the spin polarization , more complicated behavior of the rkky coupling can be occurred . in this case , the behavior of @xmath74 is determined by a superposition of four sinusoidal functions with two different periods of @xmath117 and @xmath118 each occurring twice with amplitudes @xmath119 and @xmath120 , respectively . as the result , we observe that for certain values of @xmath121 and @xmath96 , oscillations of @xmath74 exhibit a beating pattern with two characteristic periods . [ fig : aaizef1](a ) shows this beating behavior of integral , @xmath99 as a function of the impurities distance along armchair direction ( where @xmath122 with @xmath123 being an integer number ) for @xmath124 ev and @xmath125 ev . fig . [ fig : aaizef1](b ) shows the similar behavior of integral @xmath99 as a distance along zigzag direction ( where @xmath126 for an integer @xmath127 ) for @xmath128 ev and @xmath129 ev . we have obtained a similar beating pattern for oscillations of @xmath100 , which also occurs for a certain values of @xmath121 and @xmath96 . as a function of the impurities distance along armchair direction when both impurities are located on the same sublattice . the existence of two different periods in doped polarized graphene for certain values of @xmath124 ev and @xmath125 ev is clear in this figure . ( b ) the integral @xmath99 as a distance along zigzag direction for @xmath128 ev and @xmath129 ev.,title="fig : " ] as a function of the impurities distance along armchair direction when both impurities are located on the same sublattice . the existence of two different periods in doped polarized graphene for certain values of @xmath124 ev and @xmath125 ev is clear in this figure . ( b ) the integral @xmath99 as a distance along zigzag direction for @xmath128 ev and @xmath129 ev.,title="fig : " ] versus distance between two impurities , when both impurities are on the same sublattice for doped graphene with the chemical potential @xmath130 ev and various value of @xmath96 in units of ev . ( b ) same as ( a ) but for fixed @xmath107 ev and various value of the chemical potential . [ fig : aaixef1 ] , title="fig : " ] versus distance between two impurities , when both impurities are on the same sublattice for doped graphene with the chemical potential @xmath130 ev and various value of @xmath96 in units of ev . ( b ) same as ( a ) but for fixed @xmath107 ev and various value of the chemical potential . [ fig : aaixef1 ] , title="fig : " ] the behavior of the perpendicular components @xmath102 for @xmath131 is also different from their linear behavior of the undoped case , as it is shown in fig . [ fig : aaixef1](a ) for the fixed value of @xmath132 ev and different values of @xmath96 . in this case @xmath133 exhibits oscillations with a linearly increasing amplitude whose slope is proportional to @xmath134 . fig . [ fig : aaixef1](b ) is the same as fig . [ fig : aaixef1](a ) , but this time @xmath96 is fixed and @xmath121 changes . [ cols="<,^,>",options="header " , ] in conclusion , we have studied the influence of spin polarization on rkky interaction in graphene . with a spin polarization along the @xmath0-axis , the induced interaction between two magnetic impurities is found to be described by an anisotropic xxz hamiltonian with an exchange coupling depending on the distance @xmath2 between the impurities and the doping level . for undoped but spin - polarized graphene , we have found that while the interaction between the @xmath1 components of the moments remains constant with ferromagnetic sign , for the @xmath0 components it oscillates with the distance @xmath2 . in unpolarized spin case , the rkky interaction induces ferromagnetic correlation between magnetic impurities on the same sublattice , while anti - ferromagnetic correlation between the ones on different sublattices @xcite . the dependence of the interaction on the distance @xmath2 between two local magnetic moments , at the dirac point , is found to be @xmath5 , whereas it behaves as @xmath6 in doped graphene sheet . besides @xmath5 dependence of the interaction for undoped graphene , we show particularly that the interaction behaves like @xmath6 when the spin polarization is finite . at finite value of both the chemical potential and the spin polarization , more complicated behavior of the rkky coupling can be occurred . we have found that both components of the interaction oscillate with @xmath2 . we have explored that for the chemical potentials @xmath121 close to the polarization @xmath96 , oscillations of the rkky interaction exhibit a beating pattern when the impurities are located along zigzag or armchair directions . the two characteristic periods of the beating oscillations are determined by inverse of the difference and the sum of the chemical potential and the spin polarization . since several works on rkky interaction in 2d graphene systems are available , a proper comparison with those results seems to be in order ( see table i ) . we are grateful to jahanfar abouie for useful discussions . this work is partially supported by ipm grant . m. polini , r. asgari , g. borghi , y. barlas , t. pereg - barnea , and a. h. macdonald , phys . b * 77 * , 081411(r ) ( 2008 ) ; e. h. hwang and s. das sarma , _ ibid _ * 77 * , 081412 ( 2008 ) ; a. qaiumzadeh and r. asgari , new j. phys . * 11 * , 095023 ( 2009 ) . f. de juan , a. g. grushin , and m. a. h. vozmediano , phys . b * 82 * , 125409 ( 2010 ) ; p. e. trevisanutto , c. giorgetti , l. reining , m. ladisa , and v. olevano , phys * 101 * , 226405 ( 2008 ) ; r. roldn , m. p. lpez - sancho , and f. guinea , phys . b * 77 * , 115410 ( 2008 ) . m. a. h. vozmediano , m. p. lpez - sancho , t. stauber , and f. guinea , phys . b * 72 * , 155121 ( 2005 ) ; d. a. abanin , a. v. shytov , and l. s. levitov , phys . rev . lett . * 105 * , 086802 ( 2010 ) ; a. m. black - schaffer , phys . rev . b * 82 * , 073409 ( 2010 ) ; _ ibid _ * 84 * , 125416 ( 2011 ) ; s. r. power , f. s. guimares , a. t. costa , r. b. muniz , and m. s. ferreira , _ ibid _ , * 84 * , 195411 ( 2012 ) ; h. lee , e. r. mucciolo , g. bouzerar , and s. kettemann , arxiv:1204.5006 . m. zareyan , h. mohammadpour , and a. g. moghaddam , phys . b * 78 * , 193406 ( 2008 ) ; y. asano , t. yoshida , y. tanaka , and a. a. golubov , phys . rev . b * 78 * , 014514 ( 2008 ) ; j. linder , t. yokoyama , d. huertas - hernando , and a. sudb , phys . lett . * 100 * , 187004 ( 2008 ) ; a. g. moghaddam and m. zareyan _ ibid _ * 105 * , 146803 ( 2010 ) ; d. a. abanin _ et al . _ , science * 332 * , 328 ( 2011 ) .

we study the ruderman - kittle - kasuya - yosida ( rkky ) interaction in the presence of spin polarized two dimensional dirac fermions . we show that a spin polarization along the @xmath0-axis mediates an anisotropic interaction which corresponds to a xxz model interaction between two magnetic moments . for undoped graphene , while the @xmath1 part of interaction keeps its constant ferromagnetic sign , its @xmath0 part oscillates with the distance of magnetic impurities , @xmath2 . a finite doping causes that both parts of the interaction oscillate with @xmath2 . we explore a beating pattern of oscillations of the rkky interaction along armchair and zigzag lattice directions , which occurs for some certain values of the chemical potential . the two characteristic periods of the beating are determined by inverse of the difference and the sum of the chemical potential and the spin polarization . = 0.5 cm

introduction method and theory numerical results and discussions summary and conclusions acknowledgments

arxiv

there is increasing evidence for the existence of an extended scattered disk ; a massive population of objects orbiting beyond the kuiper belt @xcite . these objects have orbits with substantial eccentricities and inclinations and are distinct from kuiper belt objects ( kbos ) in that their perihelia are little affected by gravitational perturbations from neptune . thus it appears that neptune can not be responsible for their unusual orbits , and several novel mechanisms to explain the origin of these object have been proposed @xcite . the total mass in these objects is poorly known because only a handful of members have been discovered . these include the recently detected object ( 90377 ) sedna ( @xmath14 ) , whose orbit has a semimajor axis of @xmath15 and a perihelion of @xmath16 @xcite . sedna appears to be extreme in several ways in addition to its unusual orbit . it is intrinsically bright , with an absolute magnitude of @xmath17 , implying that it is one of the largest known minor planets . unpublished reports also indicate that it is quite red , has a relatively high albedo , a weak opposition surge , and has a very long rotation period , with @xmath18 @xcite . the latter claim is especially interesting in light of the fact that a _ hubble space telescope _ snapshot of sedna revealed no evidence for a large companion that could have tidally decreased sedna s rotation period from typical solar system rotation periods of @xmath7 to a longer period of @xmath19 . here we present precise relative photometry of sedna that indicates a rotation period of @xmath7 , and rules out rotation periods longer than @xmath20 , under reasonable assumptions . the rotation period of sedna is likely within the range of typical solar system objects , obviating the need for a massive companion . ccccc ut 2004 oct 8 & 3286.83028 & -0.001 & 0.005 & 0.3759 + & 3286.84411 & -0.004 & 0.006 & 0.3758 + & 3286.84823 & -0.013 & 0.006 & 0.3757 + & 3286.85249 & -0.001 & 0.007 & 0.3757 + & 3286.85723 & -0.001 & 0.007 & 0.3757 + & 3286.86124 & 0.005 & 0.009 & 0.3757 + & 3286.86525 & -0.010 & 0.007 & 0.3756 + [ tab : data ] we observed sedna over eight nights in october 2004 ( ut 2004 oct 8,9,16 ) and january 2005 ( ut 2005 jan 7 - 9,11,15 ) . photometric data were obtained with the megacam ccd camera @xcite on the mmt 6.5 m telescope . the megacam instrument uses 36 2048x4608 ccds to cover a 24x24 field - of - view with a pixel scale of 0.08 . our primary science goal was to search for small kbos , but we chose to target the field of sedna to simultaneously acquire a precise light curve for this unusual object . the results of the kbo search will be presented elsewhere . conditions during the observations ranged from good to poor , with image fwhms in the range 0.7 - 1.9 . all data were taken with a sloan @xmath1-band filter with 2x2 image binning . exposure times were 300 - 450 seconds . the apparent motion of sedna during our observations was @xmath21 , so trailing losses are negligible . the images were further binned and then reduced in the usual manner . photometry was performed in two ways : using psf - fitting photometry with the daophot ii package @xcite , and using image - subtraction photometry with the isis 2.1 package @xcite . for the daophot reductions , relative photometry of sedna was derived using 10 - 50 reference stars . for moving objects , one must take care to consider background stars or galaxies that may be blended with the target in only a subset of exposures , potentially leading to artificial variability when using psf - fitting photometry . in fact , during the night of ut 2004 jan 8 , sedna was blended with a background object that was @xmath22 magnitudes fainter . image subtraction photometry eliminates any constant , stationary objects , and so removes such contamination . on the other hand , the quality of psf - fitting photometry can be comparable to image - subtraction photometry for uncontaminated objects in relatively sparse fields . furthermore , we have found that daophot can extract reliable measurements from very poor - quality frames , where isis fails . therefore , in order to provide the best possible photometry , we adopted a hybrid approach , combining psf - fitting photometry for the nights which showed no evidence for contaminating background objects ( ut 2004 oct 8 - 9,16 and ut 2005 jan 11 ) , and image - subtraction photometry for the remainder of the nights ( ut 2005 jan 7 - 9,15 ) . we stress that , for nights with no contamination , the light curves produced by the two methods are completely consistent . we used the daophot - reported errors for all data , as we judged these to be more reliable than isis - reported errors . due to sedna s proper motion , it was not possible to use the same reference stars or images and thus tie the photometry to the same zero point for the entire dataset . therefore the data consist of three ` chunks ' , corresponding to data taken on ut 2004 oct 8 - 9 , ut 2004 oct 16 , and ut 2005 jan 7 - 15 . each of these chunks have an independent zero point . although the relative offset and absolute photometric calibration of these chunks could be determined by various methods , these methods all require additional data . these data are not currently available . we therefore chose to present only relative photometry . this final photometry , consisting of 143 data points , is listed in table 1 , where we have subtracted the error - weighted mean instrumental magnitude from each chunk . we note that the apparent magnitude of sedna during our observations was approximately @xmath23 . figure [ fig : one ] shows the light curve for sedna , where each panel corresponds to a different night . the nights belonging to the three separate chunks are indicated ; each chunk has an independent zero point . the solid curve is a sinusoidal model , which is described below . several relatively model - independent conclusions can be drawn from the properties of the light curve . first , the rms deviation during the largest chunk spanning nine nights during ut 2005 jan 7 - 15 is only @xmath2 . in addition , these data show no evidence for significant curvature ; a simple second - order polynomial fit to the january data yields an upper limit to the coefficient of the quadratic term of @xmath24 . this implies that if the light curve amplitude is large , the rotation period must be long . for example , for a sinusoidal light curve , this corresponds a limit on the semi - amplitude of @xmath25 for large @xmath3 . second , the data during any given individual night spanning @xmath8 generally have very small rms deviations . for example , the rms for the night of ut 2004 oct 8 is only 0.7% . nevertheless , several nights show evidence for significant variability that is not seen in a comparison star of similar magnitude . in many cases , this variability is consistent with a simple linear trend , which argues that the period can not be @xmath8 . however , for a few nights , curvature is evident . for example , a second - order polynomial fit to the ut 2004 oct 8 data yields a @xmath26 detection of curvature with @xmath27 . similarly , a fit to the ut 2004 oct 9 data yields @xmath28 . the detection of significant curvature , the fact that the curvature on adjacent nights has opposite sign , and the fact that the difference in mean magnitudes between adjacent nights is @xmath0 , argue that the period must be @xmath7 . this assumes that the primary power in the intrinsic light curve occurs at only one period . we believe this is a reasonable assumption . cccccccc 10.273@xmath29 0.002 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 150.0 + 10.321@xmath29 0.002 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 150.0 + 17.991@xmath29 0.006 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 150.0 + 17.845@xmath29 0.006 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 150.2 + 18.139@xmath29 0.006 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 150.9 + [ tab : fits ] we fit the light curve to the seven - parameter model , @xmath30-k[\alpha(t_i)-\alpha_0]+f_{0,j } , \label{eqn : model}\ ] ] where @xmath31 is the flux at the time @xmath32 of observation @xmath33 , @xmath34 is the phase angle of sedna at this time , @xmath35 is the coefficient of the phase function . ] , @xmath36 is the flux zero point for chunk @xmath37 , and @xmath38 and @xmath39 are the error - weighted mean observation times and phase angles , respectively . note that we are fitting relative photometry , and thus @xmath40 , @xmath35 , and @xmath36 are dimensionless . in practice , we expand the sinusoidal term in into separate sine and cosine terms , and then perform a linear fit in flux to the coefficients of these terms , the phase angle term , and the constant terms . we then reconstruct the more physical parameters @xmath40 and @xmath41 from the coefficients of the sine and cosine terms . this has the advantages that the only non - linear variable that must be fitted is @xmath3 , and errors on the parameters @xmath42 , and @xmath36 can be determined analytically at fixed @xmath3 . we constrain @xmath35 to be within @xmath43 of the range @xmath44 , although the exact form of the contraint has little effect on the results . note that , aside from the phase angle term , is equivalent to a lomb - scargle periodogram with a floating mean @xcite . we search for fits in the range @xmath45 , with steps of @xmath46 . the resulting periodogram , here displayed as @xmath47 versus @xmath3 , is shown in figure [ fig : two ] . the best - fit has @xmath48 for @xmath49 , indicating a good fit . for reference , a constant flux fit to the data yields @xmath50 for @xmath51 . thus the detection of variability , as judged by the improvement in @xmath52 , is extremely significant . the parameters for the fit are @xmath10 and @xmath53 . the phase angle coefficient @xmath35 is poorly constrained , due to the fact that the separate chunks are not tied together , and thus the time baseline for determining @xmath35 is limited to the @xmath54 day span of our january data . figure [ fig : three ] shows the light curve phased to the best - fit period , with the constant flux and phase angle terms subtracted , along with the model fit . the model appears to describe the data reasonably well . flanking the best - fit period are additional fits separated by @xmath55 ( see fig . [ fig : two ] ) ; these correspond to fits in which there are one or more additional cycles between the october and january datasets , i.e.where @xmath56 for integer @xmath57 . in addition , there is a cluster of fits that is separated by @xmath58 from the best - fit period . these corresponds to fits in which there is one additional cycle between ut oct 9 and ut oct 16 . finally , there are also diurnal aliases near @xmath59 and @xmath60 ( and the associated aliases of these aliases ) . fits near the latter period are allowed at the @xmath26 level . we find a total of five fits that are statistically indistinguishable ( @xmath61 ) from the best fit . the parameters of these fits are given in table 2 . two of these fits have @xmath62 , and appear equally good by eye . the other three fits have @xmath63 . although these fits are statistically acceptable , they appear much less convincing upon inspection of the phased light curves , one example of which is shown in figure [ fig : four ] . the amplitude is relatively constant for all the acceptable fits , with @xmath64 . models with @xmath65 are ruled out at the @xmath66 level . refitting the data after subtracting the flux predicted by the best - fit model reveals no significant additional periodicities . as a sanity check , we repeated the analysis described above on a light curve constructed from comparison stars of similar magnitude as sedna . we find no evidence for variability at the level exhibited by sedna . the best fit has an improvement in @xmath52 over a constant flux model of @xmath67 for 4 additional degrees of freedom , with an amplitude of only @xmath68 . we have presented relative photometry of the unusual solar system object sedna , obtained with the mmt 6.5 m telescope over eight nights in two campaigns in october 2004 and january 2005 . the light curve during the longest contiguous stretch of nine days has a remarkably small rms of @xmath2 , and exhibits no significant curvature , which severely constrains the amplitude of any long - term variability to @xmath69 . the light curve during any individual night exhibits significant variability that is not seen in a comparison star of similar brightness . the photometry from several individual nights shows significant curvature over the span of @xmath5 hours . these properties indicate that the period of sedna is @xmath7 , and can not be larger than @xmath70 . a sinusoidal model fit to sedna yields a best - fit period of @xmath71 and semi - amplitude @xmath72 , with additional acceptable fits with flanking periods separated by @xmath12 , as well as another class of fits with @xmath13 , although these later fits appear less viable based on visual inspection . we note that , if the variability is due to an aspherical shape such as a triaxial ellipsoid , the true rotation period is twice the fitted period . there also exist fits at the diurnal aliases of the primary period with @xmath73 that are marginally acceptable at the @xmath74 level . fits with @xmath75 or @xmath65 are ruled out at the @xmath76 level . thus we conclude that the rotation period of sedna is most likely @xmath62 , although other periods can not be completely excluded . additional observations should be pursued to distinguish between the various viable fits found here , and so firmly identify the true rotation period of sedna . the best - fit rotation period of @xmath77 makes sedna entirely typical of the bulk of solar system objects , including main - belt asteroids @xcite , as well as the @xmath78dozen kbos with measured rotation rates @xcite . we conclude that there is no real evidence that the period of sedna is extraordinarily long ( @xmath65 ) or even unusual . therefore there is no compelling reason to invoke a massive companion to spin down sedna s rotation period . bsg was supported by a menzel fellowship from the harvard college observatory . kzs acknowledges support from the william f. milton fund . we would like to thank roman rafikov for helpful discussions , scott kenyon for reading the manuscript , and perry berlind , emeric le floch , casey papovich , jane rigby , and kurtis williams for assistance in acquiring additional data . ccccc ut 2004 oct 8 & 3286.83028 & -0.001 & 0.005 & 0.3759 + & 3286.84411 & -0.004 & 0.006 & 0.3758 + & 3286.84823 & -0.013 & 0.006 & 0.3757 + & 3286.85249 & -0.001 & 0.007 & 0.3757 + & 3286.85723 & -0.001 & 0.007 & 0.3757 + & 3286.86124 & 0.005 & 0.009 & 0.3757 + & 3286.86525 & -0.010 & 0.007 & 0.3756 + & 3286.86923 & -0.006 & 0.005 & 0.3756 + & 3286.87324 & 0.001 & 0.007 & 0.3756 + & 3286.87721 & -0.017 & 0.007 & 0.3755 + & 3286.88508 & -0.008 & 0.007 & 0.3755 + & 3286.88937 & -0.019 & 0.008 & 0.3754 + & 3286.89847 & -0.026 & 0.007 & 0.3754 + & 3286.90247 & -0.013 & 0.008 & 0.3753 + & 3286.90671 & -0.009 & 0.007 & 0.3753 + & 3286.91076 & -0.007 & 0.007 & 0.3752 + & 3286.91472 & -0.007 & 0.007 & 0.3752 + & 3286.91867 & -0.016 & 0.007 & 0.3752 + & 3286.92265 & -0.008 & 0.007 & 0.3751 + & 3286.92768 & -0.012 & 0.007 & 0.3751 + & 3286.93158 & -0.008 & 0.006 & 0.3750 + & 3286.93554 & -0.005 & 0.006 & 0.3750 + & 3286.93950 & -0.004 & 0.007 & 0.3749 + & 3286.94340 & -0.007 & 0.009 & 0.3749 + & 3286.94738 & -0.002 & 0.010 & 0.3749 + & 3286.95137 & -0.014 & 0.007 & 0.3748 + & 3286.95530 & 0.016 & 0.009 & 0.3748 + & 3286.95929 & -0.006 & 0.008 & 0.3748 + & 3286.97152 & -0.006 & 0.007 & 0.3747 + & 3286.97705 & 0.001 & 0.007 & 0.3746 + & 3286.98105 & 0.007 & 0.008 & 0.3746 + ut 2004 oct 9 & 3287.79633 & -0.011 & 0.011 & 0.3676 + & 3287.80032 & -0.016 & 0.011 & 0.3676 + & 3287.80436 & -0.010 & 0.009 & 0.3676 + & 3287.80875 & -0.003 & 0.011 & 0.3675 + & 3287.81288 & -0.029 & 0.009 & 0.3675 + & 3287.81709 & 0.006 & 0.009 & 0.3674 + & 3287.82135 & 0.012 & 0.010 & 0.3674 + & 3287.82553 & 0.000 & 0.010 & 0.3674 + & 3287.83422 & 0.009 & 0.009 & 0.3673 + & 3287.83827 & 0.037 & 0.010 & 0.3672 + & 3287.84231 & 0.011 & 0.009 & 0.3672 + & 3287.84635 & 0.010 & 0.008 & 0.3672 + & 3287.85118 & 0.000 & 0.007 & 0.3671 + & 3287.85518 & 0.014 & 0.008 & 0.3671 + & 3287.85914 & 0.013 & 0.008 & 0.3671 + & 3287.86317 & 0.012 & 0.009 & 0.3670 + & 3287.86728 & 0.007 & 0.009 & 0.3670 + & 3287.87132 & -0.001 & 0.009 & 0.3670 + & 3287.87678 & 0.022 & 0.009 & 0.3669 + & 3287.88130 & 0.020 & 0.008 & 0.3669 + & 3287.88542 & 0.012 & 0.009 & 0.3669 + & 3287.88954 & -0.007 & 0.008 & 0.3668 + & 3287.89360 & -0.005 & 0.011 & 0.3668 + & 3287.89764 & 0.008 & 0.009 & 0.3668 + & 3287.90167 & 0.019 & 0.009 & 0.3667 + & 3287.90569 & 0.020 & 0.010 & 0.3667 + & 3287.90969 & 0.001 & 0.009 & 0.3667 + & 3287.95197 & 0.015 & 0.009 & 0.3662 + & 3287.95628 & 0.014 & 0.008 & 0.3662 + & 3287.96039 & 0.017 & 0.009 & 0.3662 + & 3287.96446 & 0.024 & 0.009 & 0.3661 + & 3287.96847 & 0.026 & 0.010 & 0.3661 + & 3287.97248 & 0.013 & 0.010 & 0.3661 + & 3287.97643 & 0.011 & 0.008 & 0.3660 + & 3287.98805 & 0.001 & 0.008 & 0.3659 + & 3287.99222 & 0.025 & 0.009 & 0.3659 + & 3287.99626 & 0.010 & 0.009 & 0.3659 + & 3288.00030 & 0.020 & 0.008 & 0.3659 + & 3288.00428 & 0.008 & 0.010 & 0.3658 + & 3288.00826 & 0.008 & 0.010 & 0.3658 + & 3288.01231 & 0.013 & 0.011 & 0.3658 + ut 2004 oct 16 & 3294.92551 & -0.010 & 0.012 & 0.3049 + & 3294.93194 & 0.013 & 0.021 & 0.3048 + & 3294.93511 & -0.001 & 0.013 & 0.3048 + & 3294.93771 & 0.023 & 0.014 & 0.3048 + & 3294.94036 & 0.019 & 0.014 & 0.3048 + & 3294.94297 & 0.015 & 0.013 & 0.3047 + & 3294.94560 & 0.018 & 0.012 & 0.3047 + & 3294.94859 & -0.007 & 0.013 & 0.3047 + & 3294.95159 & -0.002 & 0.011 & 0.3047 + & 3294.95417 & 0.003 & 0.012 & 0.3046 + & 3294.95682 & 0.004 & 0.014 & 0.3046 + & 3294.95942 & 0.014 & 0.011 & 0.3046 + & 3294.96230 & 0.000 & 0.012 & 0.3045 + & 3294.96511 & -0.023 & 0.014 & 0.3045 + & 3294.96771 & -0.008 & 0.012 & 0.3045 + & 3294.97066 & -0.003 & 0.010 & 0.3045 + & 3294.97333 & -0.001 & 0.012 & 0.3044 + & 3294.97592 & -0.002 & 0.015 & 0.3044 + & 3294.97858 & -0.013 & 0.014 & 0.3044 + & 3294.98151 & 0.001 & 0.009 & 0.3044 + & 3294.98411 & -0.005 & 0.009 & 0.3043 + & 3294.98672 & -0.004 & 0.010 & 0.3043 + & 3294.98936 & -0.016 & 0.016 & 0.3043 + & 3294.99201 & -0.016 & 0.012 & 0.3043 + & 3294.99467 & 0.005 & 0.015 & 0.3042 + ut 2005 jan 7 & 3377.72877 & -0.019 & 0.014 & 0.5415 + & 3377.73359 & -0.012 & 0.010 & 0.5415 + & 3377.73765 & -0.021 & 0.011 & 0.5415 + ut 2005 jan 8 & 3378.60779 & -0.001 & 0.008 & 0.5463 + & 3378.61168 & 0.008 & 0.012 & 0.5463 + & 3378.61564 & 0.010 & 0.013 & 0.5463 + & 3378.61955 & 0.015 & 0.012 & 0.5464 + & 3378.62348 & 0.038 & 0.009 & 0.5464 + & 3378.62764 & 0.011 & 0.010 & 0.5464 + & 3378.63155 & 0.010 & 0.010 & 0.5464 + & 3378.63556 & 0.012 & 0.011 & 0.5464 + & 3378.63979 & 0.002 & 0.011 & 0.5465 + & 3378.64371 & 0.020 & 0.010 & 0.5465 + & 3378.64770 & 0.014 & 0.011 & 0.5465 + & 3378.65161 & 0.017 & 0.010 & 0.5465 + & 3378.65551 & 0.008 & 0.012 & 0.5466 + & 3378.65941 & -0.002 & 0.013 & 0.5466 + & 3378.66333 & 0.013 & 0.014 & 0.5466 + & 3378.66742 & 0.014 & 0.013 & 0.5466 + & 3378.67147 & -0.007 & 0.012 & 0.5466 + & 3378.67559 & -0.007 & 0.013 & 0.5467 + & 3378.67951 & 0.006 & 0.010 & 0.5467 + & 3378.68344 & 0.005 & 0.013 & 0.5467 + & 3378.68732 & 0.008 & 0.011 & 0.5467 + & 3378.69146 & -0.006 & 0.014 & 0.5468 + & 3378.69543 & 0.003 & 0.012 & 0.5468 + & 3378.69936 & -0.004 & 0.012 & 0.5468 + & 3378.70325 & -0.016 & 0.011 & 0.5468 + ut 2005 jan 9 & 3379.60272 & 0.021 & 0.020 & 0.5517 + & 3379.60662 & 0.013 & 0.025 & 0.5517 + & 3379.67326 & 0.015 & 0.010 & 0.5520 + & 3379.67991 & 0.006 & 0.011 & 0.5521 + & 3379.68560 & -0.010 & 0.009 & 0.5521 + & 3379.69299 & 0.001 & 0.012 & 0.5522 + & 3379.69874 & -0.001 & 0.008 & 0.5522 + & 3379.75010 & -0.019 & 0.009 & 0.5525 + & 3379.75624 & -0.020 & 0.009 & 0.5525 + & 3379.76249 & -0.007 & 0.010 & 0.5526 + & 3379.76823 & -0.013 & 0.009 & 0.5526 + & 3379.77411 & -0.014 & 0.009 & 0.5526 + & 3379.77987 & -0.003 & 0.010 & 0.5527 + ut 2005 jan 11 & 3381.75575 & -0.015 & 0.024 & 0.5629 + & 3381.75992 & 0.010 & 0.026 & 0.5629 + & 3381.76399 & -0.013 & 0.022 & 0.5629 + ut 2005 jan 15 & 3385.80239 & -0.019 & 0.013 & 0.5817 + & 3385.80874 & -0.032 & 0.011 & 0.5817 + [ tab : datafull ]

we present precise , @xmath0 , @xmath1-band relative photometry of the unusual solar system object ( 90377 ) sedna . our data consist of 143 data points taken over eight nights in october 2004 and january 2005 . the rms variability over the longest contiguous stretch of five nights of data spanning nine days is only @xmath2 . this subset of data alone constrain the amplitude of any long - period variations with period @xmath3 to be @xmath4 . over the course of any given @xmath5-hour segment , the data exhibits significant linear trends not seen in a comparison star of similar magnitude , and in a few cases these segments show clear evidence for curvature at the level of a few millimagnitudes per hour@xmath6 . these properties imply that the rotation period of sedna is @xmath7 , can not be @xmath8 , and can not be @xmath9 , unless the intrinsic light curve has significant and comparable power on multiple timescales , which is unlikely . a sinusoidal fit yields a period of @xmath10 and semi - amplitude of @xmath11 . there are additional acceptable fits with flanking periods separated by @xmath12 , as well as another class of fits with @xmath13 , although these later fits appear less viable based on visual inspection . our results indicate that the period of sedna is likely consistent with typical rotation periods of solar system objects , thus obviating the need for a massive companion to slow its rotation .

introduction observations and data reduction analysis summary and discussion

arxiv

pulsars are rapidly rotating , highly magnetized neutron stars , the remnants of massive stars after their death in supernova explosions . they are extremely valuable astronomical tools with many physical applications that have been used to , for example , constrain the equation of state of ultra - dense matter ( e.g. * ? ? ? * ; * ? ? ? * ) , test relativistic gravity ( e.g * ? ? ? * ; * ? ? ? * ) , probe plasma physics within the magnetosphere ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and gain a better understanding of the complete radio pulsar population ( e.g. * ? ? ? certain individual pulsar systems are especially well suited to studying these areas of astrophysics , and thus continued pulsar surveys to find these rare objects remain a major scientific driver in the field . radio pulsars are found primarily in non - targeted , wide - area surveys such as the pulsar - alfa ( palfa ) survey at 1.4ghz , which began in 2004 @xcite . palfa observations use the 7-beam arecibo l - band feed array ( alfa ) receiver of the arecibo observatory william e. gordon 305-m telescope and focus on the galactic plane ( @xmath7 ) in the two regions visible with arecibo , namely the `` inner galaxy '' region ( 32 @xmath8 77 ) , and the `` outer galaxy '' region ( 168 @xmath8 214 ) . for the first 5 years , palfa survey observations were made using the wideband arecibo pulsar processor ( wapp ) , a 3-level auto - correlation spectrometer with 100 mhz of bandwidth @xcite . since 2009 , the mock spectrometerastro / mock.shtml ] , a 16-bit poly - phase filterbank with 322 mhz of bandwidth , has replaced the wapp spectrometer as the data - recorder of the palfa survey . the increased bandwidth , poly - phase filterbank design , and increased bit - depth of the mock spectrometer have increased the sensitivity and robustness to interference of the palfa survey . for this reason , we are re - observing regions of the sky previously observed with the wapp spectrometers . the palfa consortium currently employs two independent full - resolution data analysis pipelines . the einstein@home - based pipeline ( e@h ) has already been described by @xcite : this pipeline derives its computational power by aggregating the spare cycles of a global network of pcs and mobile devices using the boinc platform , and is also searching data from the palfa survey for pulsars . in this work we describe the pipeline based on the presto suite of pulsar search programssransom / presto/ ] @xcite . in addition to these pipelines , we also employ a reduced - resolution `` quicklook '' pipeline , which is run on - site at arecibo shortly after observing sessions are complete and which enables a more rapid discovery and confirmation of strong pulsars @xcite . as of march 2015 , there have been 145 pulsars discovered in wapp and mock spectrometer observations with the various palfa data analysis pipelines . this is already a sizable increase on the known sample of 258 galactic radio pulsars in the full survey region found in other searches . the relatively high observing frequency and unparalleled sensitivity of arecibo , coupled with the high time and frequency resolution of palfa ( @xmath9s and @xmath10khz , respectively ) make it particularly well suited for detecting millisecond pulsars ( msps ) deep in the plane of the galaxy , such as the distant msps reported by @xcite and @xcite , the highly eccentric msp psr j1903 + 0327 @xcite , and faint , young pulsars ( e.g. * ? the huge instantaneous sensitivity of arecibo enables short integration times , which has been helpful in detecting relativistic binaries ( e.g. psr j1906 + 0746 ; * ? ? ? * ) by reducing the deleterious effect of time - varying doppler shifts of binary pulsars . the palfa survey has also proven successful at detecting transient astronomical signals . for example , the survey has led to the discovery of several rotating radio transient pulsars ( rrats ; * ? ? ? * ) , as well as frb 121102 , the first fast radio burst ( frb ) detected with a telescope other than the parkes radio telescope @xcite . while palfa is the most sensitive large - scale survey for radio pulsars ever conducted , it is not the only on - going radio pulsar survey . other major surveys are the htru - s @xcite , htru - n @xcite , and span512 @xcite surveys at @xmath21.4ghz , the gbncc @xcite and ao327 drift @xcite surveys at @xmath2350mhz , and the lofar surveys @xcite at @xmath2150mhz . the underlying distributions of the pulsar population can be estimated using simulation techniques ( e.g. * ? ? ? * ; * ? ? ? * ) . the large sample of pulsars found in non - targeted surveys are essential for these simulations however , for population analyses to be done accurately , the selection biases of each survey must be taken into account . while the sensitivity of pulsar search algorithms is reasonably well understood , the effect of radio frequency interference ( rfi ) on pulsar detectability has not been previously studied in detail . this paper reports on the current state of palfa s primary search pipeline , its discoveries , and its sensitivity . the rest of the article is organized as follows : the observing set - up is summarized in [ sec : observations ] . the details of the palfa ` presto`-based pipeline are described in [ sec : analysis ] . [ sec : results ] reports basic parameters of the pulsars found with the pipeline , and [ sec : injected ] details how the survey sensitivity is determined , including a technique involving injecting synthetic pulsars into the data . these accurate sensitivity limits are used to improve upon population synthesis analyses in [ sec : popsynth ] . the broader implications of the accurate determination of the survey sensitivity are presented in [ sec : discussion ] before the paper is summarized in [ sec : conclusion ] . the palfa survey observations have been restricted to the two regions of the galactic plane ( @xmath7 ) visible from the arecibo observatory , the inner galaxy ( 32 @xmath8 77 ) , and the outer galaxy ( 168 @xmath8 214 ) . integration times are 268 s and 180 s for inner and outer galaxy observations , respectively . to optimize the use of telescope resources , the palfa survey operates in tandem with other compatible projects using the alfa 7-beam receiver . in particular , we have reciprocal data - sharing agreements with collaborations that search for galaxies in the optically obscured ( `` zone of avoidance '' ) directions through the milky way @xcite and recombination - line studies of ionized gas in the milky way @xcite . the palfa project leads inner galaxy observing sessions , whereas our partners lead outer galaxy sessions . for the inner galaxy region , the pointing strategy has prioritized observations of the @xmath11region before densely sampling the galactic plane at larger galactic latitudes . to densely cover a patch of sky out to the alfa beam fwhm , three interleaved alfa pointings are required ( see * ? ? ? * for more details ) . in contrast , our commensal partners have focused outer galaxy observations in order to densely sample particular galactic longitude / latitude ranges . a sky map showing the observed pointing positions can be found in figure [ fig : skymap ] . observations conducted with alfa have a bandwidth of 322mhz centered at 1375mhz . each of the seven alfa beams is split into two overlapping 172-mhz sub - bands and processed independently by the mock spectrometersastro / mock.shtml ] . the sub - bands are divided into 512 channels , each sampled every @xmath265.5 @xmath12s . the observing parameters are summarized in table [ tab : obs_setup ] . the data are recorded to disk in 16-bit search - mode psrfits format @xcite . palfa survey data have been recorded with the mock spectrometers since 2009 . although , in 2011 our pointing grid was altered slightly to accommodate our commensal partners . this required some sky positions to be re - observed . prior to 2009 , survey observations were recorded with the wideband arecibo pulsar processors ( wapps ; see * ? ? ? * ; * ? ? ? the two data recording systems were run in parallel during 2009 to check the consistency and quality of the mock spectrometer data . an unpulsed calibration diode is fired during the first ( or sometimes last ) 510s of our integration . while this is primarily used by our partners , we have found the diode signals useful in calibrating observations for our sensitivity analysis ( see [ sec : sensitivity ] ) . the calibration signal is removed from the data prior to searching ( see [ sec : pipeline - preproc ] ) . the original 16-bit mock data files are compressed to have 4 bits per sample . these smaller data files are more efficient to ship and analyze thanks to reduced disk - space requirements . the 4-bit data files utilize the _ scales _ and _ offsets _ fields of the psrfits format to retain information about the bandpass shape despite the reduced dynamic range . the scales and offsets are computed and stored for every 1-s sub - integration . this reduction of bit - depth results in a total loss of only a few percent in the @xmath13of pulsar signals . the converted 4-bit psrfits data files are copied to hard disks , and couriered from arecibo to cornell university where they are archived at the cornell university center for advanced computing . meta - data about each observation , parsed from the telescope logs and the file headers , are stored in a dedicated database . as of 2014 november , a total of 87689 beams of mock spectrometer data have been archived . the break - down of observed , archived and analyzed sky positions for the two survey regions is shown in table [ tab : data ] . palfa observations more than one year old are publicly available . small quantities of data can be requested via the web . access to larger amounts of data is also possible , but must be coordinated with the collaboration because of logistics . additional details about the data management logistics and data preparation are in [ sec : pipeline - logistics ] and [ sec : pipeline - preproc ] . [ sec : pipeline ] the ` presto`-based pipeline has been used to search palfa observations taken with the mock spectrometers since mid-2011 for radio pulsars and transients . all processing is done using the guilliminsupercomputer of mcgill university s high performance computing center . while the pipeline described here was designed specifically for the palfa survey , it is sufficiently flexible to serve as a base for the data reduction pipeline of other surveys . for example , the span512 survey being undertaken at the nanay radio telescope uses a version of the palfa ` presto`pipeline described here tuned to their specific needs @xcite . the palfa pipeline source code is publicly available online . since the analysis began with the pipeline , there have been several major improvements , primarily focusing on ameliorating its robustness in the presence of rfi ( [ sec : pipeline - rfi ] ) , as well as post - processing algorithms for identifying the best pulsar candidates ( [ sec : pipeline - post ] ) . the palfa consortium is constantly monitoring the performance of the pipeline and the rfi environment at arecibo ( as described later , rfi is one of the major challenges ) , and looking for ways to further improve the analysis . here we report on the state of the software as of early-2015 . the pipeline overview presented here is grouped into logical components . in [ sec : pipeline - logistics ] we outline the significant data tracking and processing logistics required to automate the analysis . in [ sec : pipeline - preproc ] we detail the data file preparation required before searching an observation . in [ sec : pipeline - search ] we describe the techniques used to search for periodic and impulsive pulsar signals . in [ sec : pipeline - rfi ] we summarize the various complementary stages of rfi identification and mitigation . finally , in [ sec : pipeline - post ] and [ sec : cyberska ] we outline the tools used to help select and view pulsar candidates , as well as other on - line collaborative facilities used by the palfa consortium . figure [ fig : flowchart ] shows a flowchart summarizing the stages of the pipeline . the palfa search pipeline is designed to be almost entirely automated . this includes the logistics of data management required to maintain the analysis of @xmath14 observations on the guilliminsupercomputer at any given time . this is accomplished with a _ job - tracker _ database that maintains the status of processes that are downloading raw data , reducing data , and uploading results . the pipeline is configured to continually request and download raw data that have not been processed and delete the local copies of files that have been successfully analyzed . data files are copied to mcgill via ftp from the cornell university center for advanced computing ( cac ) . the multi - threaded data transfers from the cac to mcgill are sufficiently fast to maintain 10002000 jobs running simultaneously . when the transfer of an observation is complete , job entries are created in the pipeline s job - tracker database . as compute resources become available , jobs are automatically submitted to the super - computer s queue . when jobs terminate , the pipeline checks for results and errors . failed jobs are automatically re - submitted up to three times to allow for occasional hiccups of the guillimintask management system , or processing node glitches . if all three processing attempts result in failure , the observation is flagged to be dealt with manually . observations that are salvageable are re - processed after fixes are applied . the positions of un - salvageable observations are re - inserted into the observing schedule , along with those from observations severely contaminated with rfi . observations may be un - salvageable if they are aborted scans , contain malformed metadata , or their files have become corrupted . only @xmath20.15% of all observations have data files that can not be searched , and only @xmath24.5% of all observations are flagged to be re - observed due to excessive rfi . the results from successfully processed jobs are parsed and uploaded to a database at the cac , and the local copies of the data files are removed to liberate disk space enabling more observations to be requested , downloaded , and analyzed . the inspection of uploaded results is done with the aid of a web - application ( see [ sec : cyberska ] ) . before analyzing the data for astrophysical signals , the two mock sub - bands must be combined into a single ` psrfits`file . each of the two mock data files have 512 frequency channels , 66 of which are overlapping with the other file . for each sub - integration of the observation , the 478 low - frequency channels from the bottom sub - band and the 480 high - frequency channels from the top sub - band are extracted , concatenated together along with two extra , empty frequency channels for each sample , and written into a new full - band data file , consisting of 960 channels . the choice to discard part of both bands was made in order to mitigate the effect of the reduced sensitivity at the extremities of the mock sub - bands , which causes a slight reduction of sensitivity where they are joined together . the psrfits scales and offsets of the mock sub - bands are adjusted such that the data value levels of top and bottom bands are appropriately weighted with respect to each other . the combining of the two mock sub - bands is performed using ` combine_mocks`of ` psrfits_utils ` . next , the sub - integrations containing the calibration diode signal are deleted from the observation . the start time and length of the observation are updated accordingly . at this stage , prior to searching for periodic and impulsive signals , ` presto ` s ` rfifind`is run on the merged observation to generate an rfi mask . [ sec : masking ] for details . we will now cover the various steps required to search for pulsars and transients . because the dms of yet - undiscovered pulsars and transients are not known in advance , a wide range of trial dms must be used to maintain sensitivity to pulsars . for each trial dm value a dedispersed time series is produced by shifting the frequency channels according to the assumed dm value and then summing over frequency . when generating these time series , the motion of the earth around the sun is removed so that the data are referenced to the solar system barycenter , assuming the coordinates of the beam center . the palfa ` presto`pipeline searches observations for periodic and impulsive signals up to a dm of @xmath210000@xmath5 . we search to such high dms despite the maximum dm in our survey region predicted by the ` ne2001`model being @xmath21350@xmath5@xcite to ensure sensitivity to highly - dispersed , potentially extragalactic frbs ( e.g. * ? ? ? * ; * ? ? ? a dedispersion plan is determined by balancing the various contributions to pulse broadening that can be controlled : the duration of each sample ( including down - sampling ) , @xmath15 ; the dispersive smearing within a single channel , @xmath16 ; the dispersive smearing within a single sub - band due to approximating the dm , @xmath17 ; and the dispersive smearing across the entire observing band due to the finite dm step size ( i.e. if the dm of the pulsar is half - way between two dm trials ) , @xmath18 . additionally , pulses are broadened by interstellar scattering , @xmath19 , which can not be removed . the amount of scatter - broadening depends on the dm , observing frequency and line - of - sight . @xcite empirically determined the relationship as @xmath20 where @xmath19 is given in @xmath12s , and @xmath21 is the observing frequency in ghz . even for the same dm , @xmath22 are different for pulsars in different locations with a standard deviation of @xmath23 @xcite . because @xmath19 can not ( in practice ) be corrected , we ignore it when determining our dedispersion plan . the total correctable pulse broadening , @xmath24 , is estimated by summing the first four contributions in quadrature , @xmath25 all of these broadening terms vary with dm . the dedispersion plan is chosen to equate these four broadening effects roughly by adjusting the dm step - size and down - sampling factor as a function of dm . to reduce the number of dm trials , the step - size is never so small that @xmath26ms . the palfa survey dedispersion plan for mock spectrometer data was determined with a version of ` presto ` s ` ddplan.py`modified to allow for non - power - of - two down - sampling factors , and is shown in table [ tab : ddplan ] . the down - sampling factors are selected to be divisors of the number of spectra per sub - integration , 15270 . the amount of dispersive smearing incurred at the middle of the observing band , @xmath21375mhz , when using the dedispersion plan in table [ tab : ddplan ] , ranges from @xmath20.1ms for the lowest dms , to @xmath21ms for dms of a few 100@xmath5 , increasing to @xmath210ms for a dm of @xmath210000@xmath5 . above a dm of @xmath2500@xmath5scattering begins to dominate ( see fig . [ fig : ddplan ] ) . the more aggressive down - sampling at higher dms has the advantage of reducing the data size , making the analysis more efficient . also , at higher dms the step - size between successive dm trials is increased , further reducing the amount of processing . therefore , the extra computing required to go to high dms is relatively small compared to what is required to search for pulsars and transients at low dms . searching dms between 1000@xmath5and 10000@xmath5adds only @xmath25% the total data analysis time . dedispersion is done with ` presto ` s ` prepsubband ` , passing through the raw data 99 times , and resulting in 7292 dedispersed time series . in all cases ` prepsubband`internally uses 96 sub - bands , each of 10mhz , for its two - stage sub - band dedispersion process . time intervals containing strong impulsive rfi are removed by ` prepsubband ` , as prescribed by a rfi mask ( see [ sec : masking ] ) . a second set of dedispersed time series are created as before , but also applying a version of the zero - dm filtering technique described by @xcite that has been augmented to use the bandpass shape when removing the zero - dm signal from each channel . these zero - dm filtered time series are especially useful for single - pulse searching , which is described in [ sec : singlepulse ] . [ sec : zerodm ] for details on time - domain rfi mitigation strategies used . dedispersion makes up roughly 15 - 20% of the processing time . ( gray ) , as well as the optimal case ( dashed black ) where neither down - sampling nor smearing from dm errors are included . the optimal case including interstellar scattering is shown ( with @xmath27 ; thin dashed black ) assuming the empirical scattering dependence on dm of @xcite . while this dependence is likely reasonable for estimating the scattering of galactic sources , it is likely to grossly overestimate the scattering of extragalactic sources ( e.g. frbs ) . in all cases , the middle of the observing band is assumed ( @xmath21375mhz ) . discontinuities are due to down sampling . the horizontal lines ( red ) show the down sampled time resolution at various dms . [ fig : ddplan ] ] for every dedispersed time series , the discrete fourier transform ( dft ) is computed using ` presto ` s ` realfft ` . prior to searching the dft for peaks , it is normalized to have unit mean and variance . the normalization algorithm is designed mainly to suppress red noise ( i.e. low - frequency trends in the time series ; for more details see [ sec : deredden ] ) . also , fourier bins likely to contain interference are replaced with the median - value of nearby bins . details of the algorithm used to determine rfi - prone frequencies are described in [ sec : zapping ] . two separate searches of the dft are conducted using ` presto ` s ` accelsearch ` . both searches identify peaks in the dft down to a frequency of 0.125hz . the first , _ zero - acceleration _ , search is tuned to identify isolated pulsars . the power spectrum of the signal from an isolated pulsar will consist of narrow peaks at the rotational frequency of the pulsar and at harmonically related frequencies . the number of significant harmonics depends on the width of the pulse profile , @xmath28 , and the spin period , @xmath29 , as @xmath30 . to improve the significance of narrow signals , power from harmonics is summed with that of the fundamental frequency . the zero - acceleration search sums up to 16 harmonics , including the odd harmonics , in powers of 2 ( i.e. 1 , 2 , 4 , 8 , 16 harmonics ) . this harmonic summing procedure also improves the precision of the detected frequency . the second , _ high - acceleration _ search is optimized to find pulsars in binary systems . the time - varying line - of - sight velocity of such pulsars gives rise to a doppler shift that varies over the course of an observation . this smears the signal over multiple bins in the fourier domain . to recover sensitivity to binary pulsars we use the fourier - domain acceleration search technique described in @xcite . in short , the high - acceleration search performs matched - filtering on the dft using a series of templates each corresponding to a different constant acceleration . we search using templates up to 50 fourier bins wide , which corresponds to a maximum acceleration of @xmath21650for a 5-min observation of a 10-ms pulsar . only up to 8 harmonics are summed in the high - acceleration case because of its larger computational requirements . for each of the periodic signal candidates identified in both the zero- and high - acceleration searches we compute the equivalent gaussian significance , _ f@xmath31 , based on the probability of seeing a noise value with the same amount of incoherently summed power ( see * ? ? ? * for details ) . the zero- and high - acceleration candidate information is saved to separate lists for later post - processing ( see [ sec : sifting ] ) . typically , the zero - acceleration and high - acceleration searches make up between 2%-5% and @xmath230% of the overall computation time , respectively . each dedispersed time series is also searched with ` presto ` s ` single_pulse_search.py`for impulsive signals with a matched - filtering technique ( e.g. * ? ? ? multiple box - car templates corresponding to a range of durations up to 0.1s are used . candidate single - pulse events at least 5 times brighter than the standard deviation of nearby bins , _ @xmath32 , are recorded . diagnostic plots featuring only @xmath336_@xmath32candidate events are generated and archived for later viewing . in addition to the basic diagnostic plots , all of the @xmath335_@xmath32events are used in post - processing algorithms designed to distinguish astrophysical signals ( e.g. from pulsars / rrats and extragalactic frbs ) from rfi and noise . the algorithms employed by palfa are described elsewhere @xcite . the same searching and post - processing procedure is also applied to data from which the dm=0@xmath5time series has been subtracted , using an enhanced version of what was originally described in @xcite . [ sec : zerodm ] for more details about the time - domain rfi - mitigation techniques used . the single - pulse searching makes up approximately 20% of the computing time . as described above , the output of periodicity searching is a set of files , the zero- and high - acceleration candidate lists for each dm trial , containing the frequency of significant peaks found in the fourier transformed time series , along with other information about the candidate . in total , for all dms , there are typically @xmath34 period - dm pairs per beam . these signal candidates are _ sifted _ to identify the most promising pulsar candidates , match harmonically related signals , and reject rfi - like signals . the first stage of the sifting process is to remove short - period candidate signals ( @xmath35ms ) , which contribute a large number of false - positives , as well as to ensure no candidate signals with periods longer than the limit of our search ( @xmath36s ) are present . weak candidates with fourier - domain significances @xmath37 are also removed . furthermore , candidates with weak or strange harmonic powers are rejected if they match one of the following cases : 1 ) the candidate has no harmonics with @xmath38 ; 2 ) the candidate has 8 harmonics and is dominated by a high harmonic ( fourth or higher ) , having at least twice as much power as the next - strongest harmonic ; 3 ) the candidate has 4 harmonics and is dominated by a high harmonic ( third or higher ) , having at least three times as much power as the next - strongest harmonic . the next stage of sifting is to group together candidates with similar periods ( at most 1.1 fourier bins apart ) found in different dm trials . when a duplicate period is found , the less significant candidate is removed from the main list , and its dm is appended to a list of dms where the stronger candidate was detected . at this stage , for each periodic signal , there is a list of dms at which it was detected . the next step is to purge candidates with suspect dm detections . specifically , candidates not detected at multiple dms , candidates that were most strongly detected at dm @xmath39@xmath5 , and candidates that were not detected in consecutive dm trials are all removed from subsequent consideration . the steps described above are applied separately to candidates found in the zero- and high - acceleration searches . at this point , the two candidate lists are merged , and signals harmonically related to a stronger candidate are removed from the list . this process checks for a conservative set of integer harmonics , and small integer ratios between the signal frequencies . as a result , some harmonically related signals are occasionally retained in the final candidate list . the sifting process typically results in @xmath2200 good candidates per beam , of which @xmath40 are above the significance threshold for folding . the fraction of time spent on candidate sifting is negligible ( @xmath41% ) compared to the rest of the pipeline . the raw data are folded for each periodicity candidate with @xmath42 remaining after the sifting procedure using ` presto ` s ` prepfold ` . at most 200 candidates are folded for each beam . in more than 99% of cases this limit is sufficient to fold all @xmath42 candidates . if too many candidates have @xmath42 , the candidates with largest _ f@xmath31are folded . after folding , ` prepfold`performs a limited search over period , period - derivative , and dm to maximize the significance of the candidate . however , for candidates with @xmath43ms the search over dm is excluded because it is prone to selecting a strong rfi signal at low dm even if there is a pulsar signal present . furthermore , the optimization of the period - derivative is also excluded for @xmath44ms candidates . for each folded candidate a diagnostic plot is generated ( see * ? ? ? * for examples ) . these plots , along with basic information about the candidate ( optimized parameters , significance , etc . ) are placed in the palfa processing results database , hosted at the cornell center for advanced computing . the ` prepfold`binary output files generated for each fold are also archived at cornell . the binary output files created by ` prepfold`are used by a candidate - ranking artificial intelligence system , as well as to calculate heuristics for candidate sorting algorithms . details can be found in [ sec : pipeline - post ] . folding the raw data for up to 200 candidates per beam is a considerable fraction ( @xmath225% ) of the overall computing time . the sensitivity of arecibo and palfa can only be fully realized if interference signals in the data are identified and removed . to work toward this goal , the palfa pipeline includes multiple levels of rfi excision . each algorithm is designed to detect and mitigate a different type of terrestrial signal . because these interference signals are terrestrial they are not expected to show the @xmath45 frequency sweep characteristic of interstellar signals . unfortunately , some broadband terrestrial signals show frequency sweeps that can not be distinguished from astronomical signals by data analysis pipelines ( e.g. perytons * ? ? ? despite some non - astronomical signals remaining in the data , the suite of rfi - mitigation techniques described here are an essential part of the pipeline . all of the algorithms described here are applied to non - dedispersed , topocentric data . unfortunately , some of the electronics hardware at the arecibo observatory , specifically the alfa bias monitoring system , introduced strong periodic interference into our data . by the time the source of the interference was determined several months of observations had been affected . fortunately , we were able to develop a finely - tuned algorithm to excise the signal using our knowledge of the sub - pulse structure to identify and remove these intense bursts of interference . finely tuned algorithms such as this one have the advantage of more easily identifying specific rfi signals and only extracting the affected data . in this particular case , each 1-s burst of rfi is made up of a comb of @xmath210ms - long sub - pulses with repeating every @xmath250ms . by removing these bursts , our algorithm largely eliminates the broad peaks in the fourier domain that are introduced by the pernicious electronics , typically between 1 and 1000 hz ( i.e. exactly where we expect pulsars to be found ) . see figure [ fig : radar ] for an example . furthermore , by removing the interference pulses in the time domain , the power spectrum is cleaned without sacrificing any intervals of the fourier domain , as would be the case with the zapping algorithm described in [ sec : zapping ] . because the equipment causing the bursts of interference in our observations is not essential to data taking we have been able to shut it off during palfa sessions . , is applied ( labeled `` after '' ) . part of the time series is sacrificed , but the broad features in the frequency domain are completely removed . the rfi peak at 60hz that remains in the bottom panel is caused by the electrical mains and is later removed by zapping intervals of the power spectrum ( described in [ sec : zapping ] ) . the source of this interference signal has been identified and can be dealt with by shutting it off during palfa observations . [ fig : radar ] ] every observation is examined for narrow - band rfi signals using ` presto ` s ` rfifind ` , which considers 2-s long blocks of data in each frequency channel separately . for each block of data two time - domain statistics are computed : the mean of the block data value , and the standard deviation of the block data values . also , one fourier - domain statistic is computed for each block : the maximum value in the power spectrum . blocks where the value of one or more of these three statistics is sufficiently far from the mean of its respective distribution are flagged as containing rfi . for the two time - domain metrics , in the palfa survey the threshold for flagging a block is 10 standard deviations from the mean of the distribution , and for the fourier - domain metric , the threshold is 4 standard deviations from the mean . the resulting list of flagged blocks is used to mask out rfi . masked blocks are filled with constant data values chosen to match the median bandpass . channels that are more than 30% masked are completely replaced , as are sub - integrations that are at least 70% masked . on average , only @xmath25.75% of time - frequency space is masked by this algorithm , and @xmath293% of observations have mask - fractions less than 10% . observations where the mask - fraction is larger than 15% will be re - inserted into the list of sky positions to observe . these represent only @xmath21.1% of observations . the fraction of data masked for each beam , and a graphical representation of the mask are stored in the results database as diagnostics of the observation quality . generating the ` rfifind`mask makes up only @xmath21% of the total pipeline running time . it is possible for broad - band impulsive interference signals to be missed by the masking procedure described above if the signals are not sufficiently strong to be detected in individual channels . fortunately , the palfa pipeline makes use of a complementary algorithm designed to remove such signals from the data : a list of bad time intervals is determined by identifying samples in the dm=0@xmath5time series that are significantly larger ( @xmath46 ) than the surrounding data samples . the spectra corresponding to the bad time intervals are replaced by the local median bandpass . as previously mentioned , for single - pulse searching , the palfa pipeline also applies the ` presto`-implementation of the zero - dm filtering technique described in @xcite . this implementation enhances the original prescription by using the bandpass shape as weights when removing the dm=0@xmath5signal . the zero - dm filter greatly reduces the impact of rfi on single - pulse searching , facilitating low - dm rrats being distinguished from rfi . to illustrate the benefits of zero - dm filtering , figure [ fig : zerodm_comparison ] shows a comparison of the single - pulse events identified by ` single_pulse_search.py`in an observation of psr j1908 + 0734 with and without filtering . both by eye and algorithmically . the pulsar s dm=11@xmath5is indicated with the dashed red line . [ fig : zerodm_comparison ] ] in order to properly normalize the power spectrum and compute more correct false - alarm probabilities ( see * ? ? ? * ) , we use a power spectrum whitening technique to suppress frequency - dependent , and in particular `` red '' noise . the median power level is measured in blocks of fourier frequency bins and then multiplied by @xmath47 to convert the median level to an equivalent mean level assuming that the powers are distributed exponentially ( i.e. @xmath48 with 2 degrees - of - freedom ) . the number of fourier frequency bins per block is determined by the log of the starting fourier frequency bin , beginning with 6 bins and increasing to approximately 40 bins by a frequency of 6hz . above that frequency , where there is little to no `` colored '' noise , block sizes of 100 bins are used . the resulting filtered power spectrum has unit mean and variance . this process is accomplished with ` presto ` s ` rednoise`program . sufficiently bright periodic sources of rfi can be mistakenly identified as pulsar candidates by our fft search . to excise , or _ zap _ , these signals from our data we tabulate frequency ranges often contaminated by rfi . the fourier bins contained in this _ zap list _ are replaced by the average of nearby bins prior to searching . the rfi environment at arecibo is variable . the number , location , and width of interference peaks in the fourier transform of dm=0 @xmath5time series vary on a time scale of months to years . to demonstrate this , the fraction of fourier bins occupied by rfi as a function of epoch is illustrated in figure [ fig : zapfrac_summary ] . the median fraction of the fourier spectrum occupied by rfi for all mock spectrometer data for various intervals is : 2.9% ( 0 - 10hz ) , 5.1% ( 10 - 100hz ) , and 0.5% ( 100 - 1000hz ) . to account for this dynamic nature of the rfi , we compute zap lists for each mjd . ) has nearly always been turned off during palfa observations , significantly reducing the rfi in the 10 - 100hz interval . [ fig : zapfrac_summary ] ] to compute zap lists we exploit the fact that rfi signals are typically detected by multiple feeds in a single 5-min pointing , or persist for most of an observing session ( typically 13 hours ) . the strategy we employ here is similar to what was used in the parkes multibeam pulsar survey @xcite . fourier bins contaminated by rfi are determined by finding peaks in a _ median power spectrum _ , which is comprised of the bin - wise median of multiple dm=0@xmath5power spectra . this is done twice , using two different subsets of data : a ) all observations made with a given alfa feed on a given day ( to identify rfi signals that persist for multiple hours , or issues specific to the alfa receiver ) , and b ) all seven observations from a given pointing ( to identify shorter - duration periodic rfi signals that enter multiple feeds ) . the zap list for any given observation is the union of the lists for its pointing and its feed . with the advent of sophisticated candidate ranking and candidate classifying machine - learning algorithms ( see [ sec : pipeline - post ] ) , it is better to leave some rfi in the data than to remove large swaths of the fourier domain . to avoid excessive zapping we remove at most 3% from each frequency decade , up to a maximum of 1% globally , preferentially zapping bins containing the brightest rfi . in addition to being an essential part of the palfa rfi - mitigation strategy , zap lists have also proven to be a useful diagnostic for monitoring the rfi environment at arecibo . a series of 19 heuristic ratings are computed for each folded periodicity candidate produced by the data analysis pipeline . these ratings encapsulate information about the shape of the profile , the persistence and broadbandedness of the signal , whether the frequency of the signal is particularly rfi - prone , and whether the signal is stronger at dm=0@xmath5 . each of the ratings is uploaded to the results database , and is available for querying and sorting candidates ( see [ sec : cyberska ] ) . the ratings and brief descriptions are presented in table [ tab : ratings ] . the ratings are incorporated into candidate - selection queries along with standard parameters such as period , dm , and various measures of time - domain and frequency - domain significance . using ratings in this way allows users to constrain the candidates they view to have certain features they would require when selecting promising candidates by eye . alternatively , the ratings have been used in a decision - tree - based artificial intelligence ( ai ) algorithm , but this has since been supplanted by the more sophisticated `` pics '' algorithm described in [ sec : ai ] @xcite . the code to compute the ratings is compatible with the binary files produced by ` presto ` s ` prepfold`for each periodicity candidate . for each candidate a text file is written containing the name , version , description , and value for all ratings being computed . this task is performed as part of the data analysis pipeline . the rating information is later uploaded to the results database . in cases where a new rating is devised , or an existing rating is improved , the ` prepfold`binary files are fetched from the results archive , ratings are computed in a stand - alone process ( i.e. independent of the pipeline ) , and the values are inserted into the database . the values of improved ratings are inserted alongside values from old versions to permit detailed comparisons . all periodicity candidates are also assessed by the pulsar image - based classification system ( pics ; * ? ? ? * ) , an image - pattern - recognition - based machine - learning system for selecting pulsar - like candidates . the pics deep neural network enables it to recognize and learn patterns directly from 2-d diagnostic images produced for every periodicity pulsar candidate . the large variety of pulsar candidates used to train pics has developed its ability to recognize both pulsars and their harmonics . pics can reduce the number of candidates to be inspected by human experts by a factor of @xmath2100 while still identifying 100% of pulsars and 94% of harmonics to the top 1% of all candidates @xcite . since late 2013 , pics has been integrated directly into the palfa processing pipeline . it produces a single rating for each candidate , which is uploaded into the results database as a rating ( see [ sec : ratings ] ) . so far , this has aided in the discovery of 9 pulsars ( see [ sec : results ] ) . while palfa has been successful at finding moderately bright msps , the vast quantity of periodicity candidates close to the detection threshold at very short periods ( 2ms ) have made it more challenging to identify the _ faint _ msps in the palfa results database . to facilitate the process , a search for signals with compatible periods , dms and sky positions has been performed on the periodicity candidates in the database . by applying our coincidence matching algorithm to the complete list of folded candidates we are able to reliably probe lower @xmath13sthan would be reasonable to do thoroughly by manual viewing . this algorithm is complementary to our machine learning technique that operates on each candidate individually . the software developed to find matching candidates is available on the web for general use . large parts of the survey region have either been observed more than once or have been densely sampled ( see fig . [ fig : skymap ] ) , making it possible to match the detection of a pulsar from multiple observations confidently . for each observation , a list of beams from other pointings that fall within @xmath49 is generated . candidates from the different beams are matched by their dms and barycentric periods . allowances are made for slightly different dms and periods , as well as for harmonically related periods . multiple matches that include the same candidate are consolidated to form groups of more than two candidates . the results of this matching algorithm are examined with a dedicated , web - based interface . many known pulsars , especially high harmonics of very bright slow pulsars , have already been identified . as of 2015 january , our coincidence matching search has not yet resulted in the discovery of new pulsars , but it continues to be applied to the results database . this algorithm will be increasingly useful as more of the palfa survey region becomes densely sampled , and as more mock spectrometer observations cover positions previously observed with the wapp spectrometers . the palfa consortium has created and made use of several online collaborative tools on the cyberska portal @xcite , a website developed to help astronomers build tools and strategies for large - scale projects in the lead - up to the square kilometre array ( ska ) . the cyberska portal allows for third - party applications to be accessed directly without a need for separate user authentication . within this framework several palfa - specific applications were developed : _ candidate viewer _ the primary method for viewing and classifying palfa candidates is by using the cyberska _ candidate viewer _ application . it allows users to access the cornell - hosted results database using form - based , free - text , and saved queries . queries include basic observation and candidate information ( e.g. sky position , period , dm , significance ) , as well as ratings ( [ sec : ratings ] ) , and the pics classifications ( [ sec : ai ] ) . users are presented with a series of ` prepfold`diagnostic plots in sequence , one for each candidate matching the query . by inspecting the plots , as well as other relevant information provided , such as a histogram showing the number of occurrences of signals in the relevant frequency range as well as a summary plot showing all the beam s periodic signal candidates in a period - dm plot , the user can quickly classify candidates . classifications are saved to the database and can be easily retrieved . _ top candidates _ especially promising candidates found with the candidate viewer can be added to the _ top candidates _ application , which is designed to store the most likely pulsar candidates . the application also allows collaboration members to view and vote on which candidates should be subject to confirmation observations , as well as help organize and track these observations and their outcomes . _ survey diagnostics _ optimizing the use of telescope time and computing resources is extremely important for large - scale pulsar surveys such as palfa . the _ survey diagnostics _ application automatically compiles a set of information and a set of plots from various sources to help the project run smoothly . this includes the status of data acquisition and reduction , the severity of the rfi environment , and the quality of the data . the palfa survey has discovered 145 pulsars , including 19 msps and 11 rrats , and one frb , as of 2015 march . the ` presto`-based pipeline described in [ sec : pipeline ] has discovered 41 pulsars from their periodic emission , 5 rrats from their impulsive emission , and re - detected another 60 pulsars that were previously discovered with other palfa data analysis pipelines . the other pulsars found in the palfa survey were discovered with the different data analysis pipelines , such as the e@h and quicklook pipelines @xcite which use complementary rfi - excision and search algorithms , with dedicated transient searches , or in earlier observations with the wapp spectrometers using an earlier version of the pipeline described here . not all sky positions observed with the wapp spectrometers have been covered with the mock spectrometers yet . we report details for 41 of the periodicity - discovered pulsars found in mock spectrometer data with the pipeline described above . all but one of these discoveries are in the inner galaxy region . these pulsars were discovered by analyzing 85333 beams , covering a total of 134sq . deg . , which consists of 80sq . deg.in the inner galaxy region , and 54 sq . deg.in the outer galaxy region ( see table [ tab : data ] ) . basic parameters of the discoveries are in table [ tab : discoveries ] , and pulse profiles from the discovery observations are shown in figure [ fig : discoveries ] . eight of the 41 pulsars reported here are msps , including the most distant msp ( based on its dm ) discovered to date , psr j1850 + 0242 . the distance estimated from the dm of psr j1850 + 0242 , assuming the ne2001 model @xcite , is 10.4kpc , a testament to the ability of the palfa survey to find highly dispersed , short period pulsars . psr j1850 + 0242 , along with three of the other msps discoveries reported here are described in detail in @xcite . three more of the msps reported here will be included in stovall et al . ( in prep . ) . nine of the 41 pulsars reported here are in binary systems , including seven of the msps , and two slower pulsars , psrs j1932 + 17 ( @xmath50ms ) and j1933 + 1726 ( @xmath51ms ) , that were spun - up by the accretion of mass and transfer of angular momentum , the so - called `` recycling '' process @xcite . the timing analysis of psr j1933 + 1726 will be provided by stovall et al . ( in prep . ) timing solutions for six of the slow pulsars presented in this work , including the young psr j1925 + 1721 , will be published in a forthcoming paper along with the timing of other palfa - discovered pulsars ( lyne et al . , in prep . ) . in addition to the 41 periodicity pulsars detailed here , the ` presto`-based pipeline has found 5 rrats . the beams containing these rrats were identified using a post - processing algorithm originally developed for pulsar surveys at 350mhz with the green bank telescope ( see * ? ? ? * for details ) . discovery parameters and detailed follow - up observations for these rrats will be described elsewhere . the flux densities of the new discoveries were estimated using the radiometer equation @xcite , @xmath52 where relevant parameters are the pulse profile width , @xmath28 , the telescope gain , @xmath53 , the number of polarization channels summed , @xmath54 , the observation length , @xmath55 , the observing bandwidth , @xmath56 , the period of the pulsar , @xmath29 , the system and sky temperatures , @xmath57 and @xmath58 , respectively . the time - domain signal - to - noise ratio , ( s / n ) _ t@xmath59 , was measured from folded profiles using the area under the pulse and the off - pulse rms . in some cases , predominantly for long - period pulsars , the baseline of the pulse profile exhibited broad features , likely due to red noise . ( see some examples in fig . [ fig : discoveries ] . ) to more robustly estimate flux densities , we fit gaussian components to the pulse profile , including the broad off - pulse features . the integrated pulsar signal was determined from the on - pulse components , and the noise level of the profile was determined from the standard deviation of the residuals after subtracting all fitted components from the profile . the gain was scaled according to the angular offset of the pulsar from the beam center , @xmath60 , assuming an airy disk beam pattern with @xmath61 @xcite , as well as the dependence on the zenith angle , @xmath62 . the gain also took into account the alfa beam with which the pulsar was detected , @xmath63k / jy for the central beam , and @xmath64k / jy for the outer 6 beams @xcite . sky temperatures were scaled from the @xcite 408-mhz survey to 1400mhz using a spectral index of @xmath65 for the galactic synchrotron emission @xcite . the sky temperatures also include the 2.73k cosmic microwave background . the resulting phase - averaged flux density estimates of the palfa pulsars discovered with our pipeline range from 16@xmath12jy to 280@xmath12jy ( see table [ tab : discoveries ] ) , making them among the weakest detected pulsars in the galactic field , along with other palfa - discovered pulsars ( see fig . [ fig : smean_hist ] ) . ] in total , 83 pulsars for which 1400-mhz phase - averaged flux densities , @xmath66 , are reported in the atnf catalogue were detected with the mock spectrometers in 268 different palfa observations ( i.e. some known pulsars were re - detected multiple times ) . to confirm that our observing set - up is as sensitive as expected , we estimate the ( s / n ) _ t@xmath59at which our pipeline should blindly re - detect known pulsars in our observations and compare with the ( s / n ) _ t@xmath59measured from the profile of the corresponding candidate . the expected ( s / n ) _ t@xmath59values were estimated by inverting eq . [ eq : radiometer ] to solve for the signal - to - noise ratio using @xmath66 from the atnf catalogue . as in [ sec : faint ] the telescope gain is modeled as an airy disk with @xmath67 . by comparing expected and measured signal - to - noise ratios against pulsar spin period we find that longer - period pulsars show an increase scatter in ( s / n ) _ t@xmath59ratio as well as a bias towards larger ratios ( see fig . [ fig : knownpsr_snr ] ) . this is consistent with the reduced sensitivity to long - period pulsars due to red noise we find from our sensitivity analysis using synthetic pulsar signals ( see [ sec : injected ] ) . in addition to the 83 known pulsars with published @xmath66 detected with the palfa ` presto`pipeline , there are 50 more that do not have values for @xmath66 listed in the atnf catalogue . the complete list of 128 previously discovered pulsars blindly re - detected by the palfa ` presto`pipeline is in table [ tab : redetections ] . as a function of pulsar period . expected ( s / n ) _ t@xmath59values are calculated using the radiometer equation and measured flux densities at 1400mhz from the atnf catalogue . measured ( s / n ) _ t@xmath59values are computed from detections of known pulsars in palfa observations . the increased scatter and bias towards higher @xmath13ratios of longer - period pulsars are consistent with reduced sensitivity to these pulses due to red noise ( see [ sec : sensitivity ] and fig . [ fig : sensitivity_curves ] ) . known pulsars without reported flux densities and uncertainties are excluded , as are pulsars that have reported flux densities consistent with 0 mjy . also excluded from the plot are 15 known pulsars with published flux densities that were detected in observations pointed more than @xmath68 from the position of the pulsar . this is because the actual beam pattern differs considerably from the theoretical airy disk beam pattern beyond @xmath2@xmath68 , making it difficult to reliably estimate the expected ( s / n ) _ the dashed line indicates equality of the expected and measured ( s / n ) _ t@xmath59values , and the dotted lines are at a factor of two above and below equality . [ fig : knownpsr_snr ] ] in addition to the 268 detections of 128 separate known pulsars mentioned in [ sec : known ] , there were 7 instances in which a known pulsar was not detected by the search pipeline , despite being detected when subsequently folding the search data with the most recently published ephemeris . in all cases the data were badly affected by rfi ; there are strong signals within one fourier bin of the pulsar period . furthermore , these are long - period pulsars , which are more difficult to detect than expected due to red noise in the data . it is therefore not entirely surprising that these observations did not result in detections . a thorough analysis of the effects of rfi and red noise on the sensitivity to long period pulsars is therefore crucial , and forms the discussion of the following section . the sensitivity of pulsar observations is typically estimated using the radiometer equation ( eq . [ eq : radiometer ] ) . in principle , the effects of dm , period , and pulse width on sensitivity are adequately described by the radiometer equation . the expression derived by ( * ? ? ? * see their appendix a ) , includes a more complete description of pulse shape and the effect of dm , which causes distortions of the pulse profile . however , neither of these equations includes the effect of rfi . in this section , we describe a prescription for accurately modeling the sensitivity of pulsar search observations including the effect of rfi , as well as its dependence on period , dm , and pulse width . to estimate the survey sensitivity we injected synthetic pulsar signals into actual survey data , and attempted to recover the period and dm of the input signal using our pipeline . by using synthetic signals we can also better determine the selection effects imposed by our pipeline . for this work , a simple synthetic pulsar signal was constructed for a given combination of period , dm , phase - averaged flux density , and profile shape . once the relevant parameters were chosen ( see [ sec : injectsearch ] and table [ tab : injparams ] ) , a two - dimensional pulse profile ( intensity vs. spin phase and observing frequency ) was generated . the pulse profile of each frequency channel was smeared by convolving with a box - car whose phase width corresponded to the dispersion delay within the channel , as well as scattered by convolving with a one - sided exponential function with a characteristic phase width corresponding to the pulse broadening time scale . we determined the scattering time scale using eq . [ eq : scattering ] . care was taken to conserve the area under the profile during the convolutions . the scaling factor applied to the synthetic signals was determined by flux - calibrating the palfa observing system ( see [ sec : calibration ] ) . on 2013 december 21 , we observed the radio galaxy 3c 138 in order to calibrate the central beam of alfa . three observations using the standard survey set - up described in [ sec : observations ] were conducted , but with 5-min integrations , and with the calibration diode being pulsed on and off at 40 hz . the on - source scan of 3c 138 was preceded by an off - source scan 0.5to the north of 3c 138 and followed by a similar off - source scan 0.5to the south . the calibration observation data were converted to 4-bit samples , and the mock spectrometer sub - bands were combined ( see [ sec : pipeline - preproc ] ) . the data were folded at the modulation frequency of the calibrator diode using ` fold_psrfits`of ` psrfits_utils ` . next , the on - cal and off - cal levels in the on - source and off - source observations were used to relate the flux density of the calibration diode with the cataloged flux density of 3c 138 ( for details , see e.g. * ? ? ? the result is the flux density of the calibration diode as a function of observing frequency . in practice , this was done using ` fluxcal`of ` psrchive ` . the per - channel scaling factors between flux density and the observation data units were determined by applying the calibration solution along with the calibration diode signal . this procedure determines the absolute level of the injected signal corresponding to a target phase - averaged flux density , as well as the shape of the bandpass . artificial pulsar signals were injected into the data by summing the two - dimensional , smeared , scattered , and scaled synthetic pulse profile with the data at regular intervals corresponding to the period of the synthetic pulsar . the scaling was determined using the calibration procedure described in [ sec : calibration ] . the resulting data file , including the injected signal , was written out with 32-bit floating - point samples in ` sigproc```filterbank '' format to avoid having to quantize the weak synthetic pulsar signal . many synthetic signals with a broad range of parameters were required to build a comprehensive picture of the survey sensitivity ( see table [ tab : injparams ] ) . in total , 17 periods were selected between 0.77ms and 11s along with six dms ranging from 10 to 600@xmath5 . in all cases , the profile of the synthetic signal was chosen to have a single centered von mises component with a fwhm selected from 5 possible values between @xmath21.5% and @xmath224% of the period . the example profile in figure [ fig : inject_example ] shows the case where fwhm=2.6% . the synthetic signals were injected into 12 different observations to determine the survey sensitivity in a variety of rfi conditions . all 12 observations used in this analysis are from late 2013 and from the central beam of alfa . although the gains of the outer beams are lower than that of the central beam , the response of the observing system and pulsar search pipeline to rfi and red noise derived for the central beam should also apply to the outer beams . ms pulsar consisting of a single von mises component with fwhm=2.6% ( gray ) , and the same profile broadened according to dm=250@xmath5 . the broadening is caused by dispersive smearing within each channel and scattering according to eq . [ eq : scattering ] . note that the plot is zoomed into the region : @xmath69 . [ fig : inject_example ] ] the total number of combinations of synthetic signals and observations is @xmath336000 . multiple trials , each with a different amplitude , were constructed , injected , and searched to determine the sensitivity limit at each point in ( period , dm , pulse fwhm ) phase - space . to reduce the computational burden , not all possible combinations of parameters were used . in particular , only the profile with fwhm @xmath23% was injected into all 12 observations . the remaining four profiles shapes were only injected into a single observation . this still permits the determination of the dependence of @xmath70 on pulse width . it is well known @xcite that the minimum detectable flux density of a pulsar depends on the intrinsic width of its profile , as well as the dm , because dispersive smearing and scattering broaden the profile . it is also reasonable to expect a reduction of sensitivity due to rfi and red noise , even with the red noise suppression algorithms employed ( see [ sec : deredden ] ) . by recovering injected signals using the pipeline described in [ sec : analysis ] , we have determined the true sensitivity of the palfa survey , and its dependence on spin period and dm ( see fig . [ fig : sensitivity_curves ] ) . we found the commonly used version of the radiometer equation ( eq . [ eq : radiometer ] ; * ? ? ? * ) overestimates the survey sensitivity to long - period pulsars . for example , for @xmath712.0s pulsars with dm @xmath72@xmath5(the majority of the pulsars we expect to find with palfa ) , the degradation in sensitivity compared with the ideal case is a factor of @xmath21.12 . we have also confirmed the claim by @xcite that the @xcite radiometer equation underestimates the sensitivity to high - dm msps , by not correctly modeling the distortion of the profile due to smearing and scattering . the more accurate variant of the radiometer equation from @xcite better matches our measured sensitivity curves in the msp regime , thanks to its inclusion of the profile shape and distortions . however , the degraded sensitivity we find at long periods is still not properly modeled with these adjustments . red noise present in pulsar search data due to rfi , receiver gain fluctuations , and opacity variations of the atmosphere makes it difficult to detect long - period radio pulsars . our analysis has shown that for the palfa survey , at low dms , the reduction in sensitivity already affects pulsars with periods of @xmath40ms . fortunately , the effect is slightly less significant for pulsars with higher dms . this is evident in figure [ fig : sensitivity_curves ] . we have parameterized the sensitivity curves by fitting @xmath73 vs. dm with a quadratic function and modeling how these curves depend on period . to estimate @xmath70 at an arbitrary profile width , we first estimate @xmath70 at each of the five trial widths , then fit a quadratic function in @xmath73 vs. width , and use the parameters of the fit to calculate @xmath70 at the desired width . this ad - hoc scheme provides reliable estimates of @xmath70 within the intervals used for trial values of period , dm , and width , as well as for modest extrapolation . sensitivity maps for each of the five profile widths used are shown in figure [ fig : sensitivity_maps ] . we have used the sensitivity curves determined above ( see [ sec : sensitivity ] ) to re - evaluate the expected yield of the palfa survey by performing a population synthesis analysis with ` psrpoppy ` @xcite . galactic populations of non - recycled pulsars were simulated using the radial distribution from @xcite and a gaussian distribution of heights above / below the plane with a scale height of 330pc . the pulsar periods were described by a log - normal distribution with @xmath74 and @xmath75 @xcite . the pulse - width - to - period relationship was also taken from @xcite . we used a log - normal luminosity distribution described by the best - fit parameters found by @xcite , @xmath76 and @xmath77 . we created 5000 simulated pulsar populations , each containing enough pulsars such that a simulated version of the parkes multi - beam surveys detected 1038 pulsars , the number of non - recycled pulsars detected by the actual surveys . we then compared the pulsars in each of these populations against a list of palfa observations80% of the central beam , consistent with the gains reported by @xcite . ] , and estimated their significance using the radiometer equation . pulsars with @xmath78 were considered detected was chosen such that the minimum detectable flux density coincided with the measured sensitivity curves for a duty cycle of 2.6% . ] . next , we compared the flux - density for each `` detected '' pulsar against the parameterized palfa sensitivity curves to determine if the pulsar also has a sufficiently large flux density to lie above the measured sensitivity curves . for each pulsar , the measured sensitivity curves are shifted according to the zenith angle of the observation , the gain of the beam used , the sky temperature and the angular offset between the pulsar position and the beam center . we found @xmath6% of the simulated pulsars having fluxes above the theoretical sensitivity threshold derived from the radiometer equation ( eq . [ eq : radiometer ] ) are not sufficiently bright to be `` detected '' by our measured sensitivity limits for the palfa survey ( e.g. fig . [ fig : sensitivity_maps ] ) due to the residual effect of red noise and rfi following the extensive mitigation procedures described in [ sec : pipeline - rfi ] . the median period of the pulsars missed is @xmath79ms , which is considerably longer than the median period of the potentially detectable pulsars brighter than the radiometer - equation - based threshold , @xmath80ms ( see fig . [ fig : popsynth - pdist ] ) . our 5000 realizations of simulated galactic pulsar populations , adjusted for the reduced sensitivity to long - period pulsars , suggest @xmath81 un - recycled pulsars should be detected in palfa mock spectrometer observations , given the current processed pointing list . as of 2015 january , 241 un - recycled pulsars have been discovered / detected in palfa observations with the mock spectrometers . ) . + _ middle _ cumulative fraction of simulated pulsars ( thick black line ) , and pulsars missed ( thin red line ) as a function of pulse period . + _ bottom _ period distribution of potentially detectable simulated population of un - recycled pulsars averaged over 5000 realizations ( thick black line ) compared with the period distribution of pulsars expected to be missed due to red noise ( thin red line ) . the median spin period of the potential detectable pulsars ( @xmath82ms ) is shown by the dashed black line , and the median spin period ( @xmath83ms ) of the missed pulsars is shown by the dotted red line . [ fig : popsynth - pdist ] ] the number of un - recycled pulsar detections predicted for the palfa survey by @xcite is an overestimate for two reasons . first , their analysis used a threshold @xmath13@xmath84 . given the observing parameters assumed , a more appropriate threshold of @xmath13@xmath85 should have been used to correspond to the minimum detectable flux density we find ( @xmath86mjy ) . second , the analysis by @xcite did not include the effect of red noise , which we have shown reduces the number of pulsars expected to be found in the palfa survey by 35% . the detailed sensitivity analysis of [ sec : sensitivity ] confirms that , on average , the palfa survey is as sensitive to msps and mildly recycled pulsars as expected from the radiometer equation . however , the survey is less sensitive to long - period pulsars than predicted . the degradation in sensitivity is between 10% and a factor of 2 for the majority of pulsars we expect to find in the palfa survey ( spin periods between 0.1s and 2s and dm @xmath87@xmath5 ) , and up to a factor of @xmath88 in the worst case ( dm @xmath89@xmath5and p @xmath90s ; this fortunately corresponds to a parameter space that contains far fewer expected pulsars ) . the reduction of sensitivity is likely caused by red noise present in the observations . the empirical sensitivity curves we determined apply specifically to the palfa survey , its observing set - up , and the search algorithms used . because the effects of red noise on radio pulsar survey sensitivity have the potential to be significant , as in the case of palfa , we strongly suggest measuring the impact of red noise on other surveys by performing similar analyses to what we described in [ sec : injected ] . also , future population analyses should include these measured effects of red noise rather than assuming the theoretical radiometer equation ( e.g. * ? ? ? * ; * ? ? ? * ) when deriving spatial , spin , and luminosity distributions for the underlying galactic population of pulsars . what are the potential ramifications of reduced sensitivity to long - period pulsars being unaccounted for in population synthesis analyses ? first , the existence of radio - loud pulsars beyond the `` death line '' is important to our understanding of the radio emission mechanism in pulsars . for example , the existence of the 8.5-s psr j2144@xmath913933 contradicted several existing emission theories @xcite . the existence of a larger population of slowly rotating pulsars , particularly the discovery of pulsars so slow that existing theories can not explain their radio emission , would further constrain models . it is also possible there is a larger population of highly magnetized rotation - powered pulsars and quiescent radio - loud magnetars that have been missed by the lower than predicted sensitivity of pulsar surveys . radio emission from three of the four known radio - loud magnetars was detected following high - energy radiative events @xcite . however , the other radio - loud magnetar psr j1622@xmath914950 was discovered from its radio emission @xcite . there is no evidence that the turn - on of psr j1622@xmath914950 at radio wavelengths was preceded by a high - energy event . the possibility that radio emission from magnetars is not always accompanied by x - ray or @xmath92-ray emission means it is crucial to understand the biases against finding such long - period pulsars . characterizing , and hopefully uncovering a hidden population of radio - loud magnetars , as well as highly magnetized - rotation powered pulsars , will help clarify the relationship between these two classes of pulsars , as well as the influence of strong magnetic fields on emission properties ( e.g flux and spectral index variability ) . it may be possible to address the reduced sensitivity to long - period pulsars by utilizing algorithms that perform better in the presence of red noise , as well as algorithms that remove red noise without suppressing the pulsar signal . long - period pulsars may be found via their harmonics even if red noise obscures the signal in the fourier domain at the fundamental frequency of the pulsar . the detection of the pulsar signal will be reduced in two ways . first , the harmonic summing algorithm will exclude the power contained at the fundamental and low harmonic frequencies , which can contain large amounts of power , especially in the case of pulsars with wide profiles . second , by not being based at the fundamental frequency of the pulsar , the harmonic summing algorithm will skip slower , more significant harmonics in favor of weaker harmonics at higher frequencies . despite the reduction in sensitivity several pulsars have been found in the palfa survey thanks to their higher harmonic content . one suggested method of improving sensitivity to long - period pulsars is by using the fast - folding algorithm ( ffa ; see e.g. * ? ? ? * ; * ? ? ? * and references therein ) . the periodograms produced by the ffa , a time - domain algorithm , are generated from computing a significance metric from pulse profiles . thus , the broad profile features caused by red noise pose a problem for ffa - based searches . in short , the ffa is not immune to the degradation of sensitivity to long - period pulsars described above . however it does have the advantage of coherently summing _ all _ harmonics of a given period and greater period resolution than the dft . these two factors should make the ffa slightly more sensitive to long - period pulsars , especially those with narrow profiles , than the fourier transform techniques described in [ sec : periodicity ] , which is limited in the number of harmonics that can be summed ( typically incoherently ; * ? ? ? the ffa has only been used sparingly in large - scale pulsar searches ( e.g. * ? ? ? . a more systematic investigation and application of the ffa is warranted . another algorithm that might have better performance in the presence of red noise is the single - pulse search technique described in [ sec : singlepulse ] . single - pulse search algorithms are known to be more sensitive than standard fft techniques to long - period pulsars in short observations @xcite . this is because of the natural variability of pulsar pulses and small number of pulses . pulse - to - pulse variability was not included in the synthetic pulsar signals used in our sensitivity analysis and no single pulse searching was performed . it is likely that the sensitivity curves determined in this work are partially compensated by the single - pulse search techniques already in place , especially considering the recent suggestion that pulsars with @xmath93ms have a greater likelihood of being detected in single - pulse searches than faster pulsars @xcite . however , the extent of this compensation depends on the pulse - energy distributions of pulsars and the relative significances of their detections in periodicity and single - pulse searches . we described the ` presto`-based palfa pipeline , the primary data analysis pipeline used to search palfa observations made with the mock spectrometers . this pipeline has led to the discovery of 41 periodicity pulsars and 5 rrats , the re - detection of 60 pulsars previously discovered in the survey ( using other pipelines ) , and the detection of 128 previously known pulsars . the ` presto`-based pipeline described here consists of several complementary search algorithms and rfi - mitigation strategies . the performance of the pipeline was determined by injecting synthetic pulses into actual survey observations and recovering the signals . we have found that the palfa survey is as sensitive to fast - spinning pulsars as expected by the theoretical radiometer equation . however , in the case of long - period pulsars , we have found that there is a reduction in the sensitivity due to rfi and red noise in the observations . the actual detection threshold for pulsars with @xmath3s at @xmath4@xmath5is up to @xmath210 times higher than predicted by the theoretical radiometer equation . we have performed a population synthesis analysis using this empirical model of the survey sensitivity . our analysis indicates that @xmath6% of pulsars , with predominantly long periods , are missed by palfa , compared to expectations based on theoretical sensitivity curves derived using the radiometer equation . the magnitude of the effect of red noise on the palfa survey s sensitivity to long - period pulsars is surprising and should be taken into account in future population synthesis analyses . furthermore , the effect of red noise on other radio pulsar surveys should be quantified in a similar manner and be included in population synthesis analyses to ensure the distributions determined for the underlying pulsar population are robust . the presence of more long - period pulsars could have implications on the location of the pulsar death line , the structure of pulsar magnetospheres and radio emission mechanism , as well as the relationship between canonical pulsars , highly magnetized rotation - powered pulsars , radio - loud magnetars , and rrats . the arecibo observatory is operated by sri international under a cooperative agreement with the national science foundation ( ast-1100968 ) , and in alliance with ana g. mndez - universidad metropolitana , and the universities space research association . the cyberska project was funded by a canarie nep-2 grant . computations were made on the supercomputer guillimin at mcgill university , managed by calcul qubec and compute canada . the operation of this supercomputer is funded by the canada foundation for innovation ( cfi ) , nanoqubec , rmga and the fonds de recherche du qubec - nature et technologies ( frq - nt ) . we would like to recognize the help of bryan fong in developing the decision tree ai , mark tan for his contributions to the decision tree ai and survey diagnostics cyberska application , and the sequence factory for developing of the cyberska - integrated palfa applications . pl would like to thank david champion for helpful discussions . pl acknowledges the support of imprs bonn / cologne and fqrnt b2 . palfa work at cornell university is supported by nsf grant phy 1104617 . vmk receives support from an nserc discovery grant and accelerator supplement , centre de recherche en astrophysique du qubec , an r. howard webster foundation fellowship from the canadian institute for advanced study , the canada research chairs program and the lorne trottier chair in astrophysics and cosmology . jwth acknowledges funding from an nwo vidi fellowship and erc starting grant `` dragnet '' ( 337062 ) . pccf and lgs gratefully acknowledge financial support by the european research council for the erc starting grant beacon under contract no . pulsar research at ubc is supported by an nserc discovery grant and discovery accelerator supplement and by the canadian institute for advanced research . , j. m. 2002 , in astronomical society of the pacific conference series , vol . 278 , single - dish radio astronomy : techniques and applications , ed . s. stanimirovic , d. altschuler , p. goldsmith , & c. salter , 227250 , c. , andrecut , m. , brazier , a. , et al . 2011 , in astronomical society of the pacific conference series , vol . 442 , astronomical data analysis software and systems xx , ed . i. n. evans , a. accomazzi , d. j. mink , & a. h. rots , 669 lc + parameter & value + + + + sample time , @xmath15 ( @xmath12s ) & 65.476 + integration time@xmath94 , @xmath55 ( s ) & 268 ( inner galaxy , 32 @xmath8 77 ) + & 180 ( anti - center , 168 @xmath8 214 ) + + + + number of channels & 512 + low frequency ( mhz ) & 1364.290 + high frequency ( mhz ) & 1536.016 + + + + number of channels & 512 + low frequency ( mhz ) & 1214.290 + high frequency ( mhz ) & 1386.016 + + + + number of channels & 960 + low frequency ( mhz ) & 1214.290 + center frequency ( mhz ) & 1375.489 + high frequency ( mhz ) & 1536.688 + bandwidth , @xmath56 ( mhz ) & 322.398 + channel bandwidth , @xmath95 ( khz ) & 335.831 + + + @xmath94this includes the @xmath96s when the calibration diode is turned on , which is not usable for searching for pulsars . the interval of the observation containing this calibration signal is removed prior to our analysis ( see [ sec : pipeline - preproc ] ) . cccccc + & no . beams@xmath94 & no . unique & sky coverage & completeness@xmath97 , @xmath982 & completeness@xmath97 , @xmath985 + & & sky positions & ( sq . deg . ) & ( % ) & ( % ) + + + + observed & 40705 & 38479 & 94 & 69 & 32 + archived & 35030 & 33243 & 81 & 60 & 27 + analyzed & 33888 & 32499 & 80 & 58 & 27 + + + + observed & 60305 & 26194 & 64 & 30 & 18 + archived & 52659 & 21990 & 54 & 23 & 15 + analyzed & 51445 & 21899 & 54 & 23 & 15 + + x-1 cccccc + & dm step size & no . sub - bands & sub - band dm spacing & down - sample factor & approx . computing + & ( @xmath5 ) & & & ( @xmath5 ) & & ( % ) + + 0 - 212.8 & 0.1 & 2128 & 96 & 7.6 & 1 & 73.19 + 212.8 - 443.2 & 0.3 & 768 & 96 & 19.2 & 2 & 12.20 + 443.2 - 534.4 & 0.3 & 304 & 96 & 22.8 & 3 & 8.13 + 534.4 - 876.4 & 0.5 & 684 & 96 & 38.0 & 5 & 2.93 + 876.4 - 990.4 & 0.5 & 228 & 96 & 38.0 & 6 & 2.44 + 990.4 - 1826.4 & 1.0 & 836 & 96 & 76.0 & 10 & 0.73 + 1826.4 - 3266.4 & 2.0 & 720 & 96 & 144.0 & 15 & 0.24 + 3266.4 - 5546.4 & 3.0 & 760 & 96 & 228.0 & 30 & 0.08 + 5546.4 - 9866.4 & 5.0 & 864 & 96 & 360.0 & 30 & 0.05 + + ll + rating & description + + + + duty cycle & fraction of profile bins larger than half the maximum value of the profile + peak over rms & maximum value of the profile divided by the rms + + + + amplitude & amplitude of a single gaussian component fit to the profile + single component gof & goodness of fit of a single gaussian component fit to the profile + fwhm & full - width at half - maximum of a single gaussian component fit to the profile + no . components & number of gaussian components required to acceptably fit the profile + & ( up to 5 components ) + multi - component gof & goodness of fit of the multiple gaussian component fit ( up to 5 components ) + pulse width & ratio of narrowest component of the multiple gaussian fit compared to the + & pulse broadening ( excluding scattering ) + + + + period stability & fraction of good time intervals that deviate in phase by @xmath99 + frac . of good sub - ints & fraction of time intervals that contain the pulsar signal + sub - int . snr variability & the standard deviation of sub - integration @xmath13s + + + + frac . of good sub - bands & fraction of sub - bands that contain the pulsar signal + sub - band snr variability & the standard deviation of sub - band @xmath13s + + + + dm comparison & ratio of the standard deviation of the profile at dm=0@xmath5 + ( standard deviation ) & and at the optimal dm + dm comparison ( @xmath48 ) & ratio of the @xmath48 of the profile at dm=0@xmath5and at the optimal dm + dm comparison ( peak ) & ratio of the peak value of the profile at dm=0@xmath5and at the optimal dm + + + + known pulsar & a measure of how similar the candidate period and dm are to a nearby pulsar + & ( also checks harmonic relationships ) + mains rfi & a measure of how close the topocentric frequency is to 60hz , or a harmonic + beam count & the number of beams from the same pointing containing another candidate + & with the same period + + + note . see [ sec : ratings ] for more details on how ratings are used to select candidates . + @xmath94prior to computing ratings , the profile is normalized such that median level is 0 and the standard deviation is 1 . + l d4.2 d4.1 d2.2 d1.6 + name & & & & + & & & & + + j0557 + 1550@xmath97 & 2.55 & 102.7 & 8.34 & 0.050(6)^c + j1850 + 0242@xmath97 & 4.48 & 540.5 & 13.08 & 0.33 + j1851 + 0232 & 344.02 & 605.4 & 10.82 & 0.09 + j1853 + 03 & 585.53 & 290.2 & 14.28 & + j1854 + 00@xmath100 & 767.33 & 532.9 & 10.44 & + j1858 + 02 & 197.65 & 492.1 & 14.91 & + j1901 + 0235@xmath100 & 885.24 & 403.0 & 26.7 & + j1901 + 0300@xmath97 & 7.79 & 253.7 & 11.8 & 0.113(4)^c + j1901 + 0459 & 877.06 & 1103.6 & 10.93 & 0.10 + j1902 + 02@xmath100 & 415.32 & 281.2 & 7.58 & + j1903 + 0415@xmath100 & 1151.39 & 473.5 & 12.48 & + j1904 + 0451@xmath97 & 6.09 & 183.1 & 8.78 & 0.117(9)^c + j1906 + 0055 & 2.79 & 126.9 & 16.47 & 0.12 + j1906 + 0725 & 1536.51 & 480.4 & 7.13 & 0.05 + j1907 + 0256 & 618.77 & 250.4 & 12.07 & 0.19 + j1907 + 05 & 168.68 & 456.7 & 10.0 & + j1909 + 1148 & 448.95 & 201.9 & 15.93 & 0.06 + j1910 + 1027 & 531.47 & 705.7 & 9.29 & 0.06 + j1911 + 09 & 273.71 & 334.7 & 7.13 & + j1911 + 10 & 190.89 & 446.2 & 7.48 & + j1913 + 0617 & 5.03 & 155.8 & 9.81 & + j1913 + 1103 & 923.91 & 628.9 & 9.86 & 0.09 + j1914 + 0659 & 18.51 & 224.7 & 12.66 & 0.33 + j1915 + 1144 & 173.65 & 338.3 & 23.59 & 0.08 + j1915 + 1149 & 100.04 & 702.1 & 7.58 & + j1918 + 1310 & 856.74 & 247.4 & 6.56 & + j1921 + 16 & 936.43 & 204.7 & 8.13 & + j1924 + 1628@xmath100 & 375.09 & 542.9 & 21.12 & 0.09 + j1924 + 17 & 758.43 & 527.4 & 10.66 & + j1925 + 1721 & 75.66 & 223.7 & 16.06 & 0.09 + j1926 + 1613@xmath100 & 308.30 & 32.9 & 14.9 & + j1930 + 14@xmath100 & 425.71 & 209.2 & 12.15 & 0.04 + j1930 + 1723@xmath100 & 1609.72 & 231.7 & 9.68 & 0.12 + j1931 + 1440 & 1779.23 & 239.3 & 23.63 & 0.12 + j1932 + 17@xmath100 & 41.82 & 53.2 & 12.89 & + j1933 + 1726 & 21.51 & 156.6 & 7.28 & 0.04 + j1934 + 19 & 230.99 & 97.6 & 18.67 & 0.10 + j1936 + 20 & 1390.88 & 205.1 & 6.6 & + j1938 + 2012@xmath100 & 2.63 & 237.1 & 8.55 & 0.02 + j1940 + 2246 & 258.89 & 218.1 & 14.47 & 0.09 + j1957 + 2516 & 3.96 & 44.0 & 6.61 & 0.04 + + + @xmath94phase - averaged flux density . determined using the radiometer equation ( see [ sec : faint ] ) unless otherwise noted . + @xmath97pulsar was previously published by @xcite . + @xmath101flux calibrated using noise diode . value from @xcite . + @xmath102refined position not available . flux density could not be estimated . + @xmath100pulsar was first identified using the pics machine learning candidate selection system described in [ sec : ai ] . + + + name & & & & & + & & & & & + + + continued ... b1848 + 04 & 284.70 & 115.5 & 0.66(8 ) & 36.9 & + b1849 + 00 & 2180.20 & 787.0 & 2.2(2 ) & 64.1 & + b1853 + 01 & 267.44 & 96.7 & 0.19(3 ) & 99.7 & 0.323 + b1854 + 00 & 356.93 & 82.4 & 0.9(1 ) & 267.9 & 1.048 + b1855 + 02 & 415.82 & 506.8 & 1.6(2 ) & 470.2 & 2.288 + b1859 + 01 & 288.22 & 105.4 & 0.38(5 ) & 74.7 & 0.531 + b1859 + 03 & 655.45 & 402.1 & 4.2(4 ) & 1061.3 & 3.498 + b1859 + 07 & 644.00 & 252.8 & 0.9(1 ) & 339.1 & 1.830 + b1900 + 01 & 729.30 & 245.2 & 5.5(6 ) & 106.5 & + b1900 + 05 & 746.58 & 177.5 & 1.2(1 ) & 283.2 & 1.228 + b1900 + 06 & 673.50 & 502.9 & 1.1(1 ) & 21.5 & + b1901 + 10 & 1856.57 & 135.0 & 0.58(7 ) & 212.1 & 0.568 + b1903 + 07 & 648.04 & 245.3 & 1.8(2 ) & 91.2 & 1.892 + b1904 + 06 & 267.28 & 472.8 & 1.7(2 ) & 33.9 & + b1906 + 09 & 830.27 & 249.8 & 0.23(3 ) & 17.7 & 0.127 + b1907 + 02 & 989.83 & 171.7 & 0.63(7 ) & 37.7 & + b1907 + 10 & 283.64 & 150.0 & 1.9(2 ) & 365.2 & 2.591 + b1907 + 12 & 1441.74 & 258.6 & 0.28(4 ) & 28.2 & 0.196 + b1910 + 10 & 409.35 & 147.0 & 0.22(3 ) & 47.1 & 0.196 + b1911 + 09 & 1241.96 & 157.0 & 0.14(2 ) & 18.9 & 0.228 + b1911 + 11 & 601.00 & 100.0 & 0.55(7 ) & 85.4 & 0.301 + b1911 + 13 & 521.47 & 145.1 & 1.2(1 ) & 85.5 & 1.221 + b1913 + 10 & 404.55 & 241.7 & 1.30(14 ) & 416.8 & 0.905 + b1913 + 105 & 628.97 & 387.2 & 0.22(3 ) & 46.2 & 0.507 + b1913 + 167 & 1616.23 & 62.6 & & 16.1 & + b1914 + 09 & 270.25 & 61.0 & 0.9(1 ) & 298.6 & 0.721 + b1914 + 13 & 281.84 & 237.0 & 1.2(1 ) & 616.7 & 2.043 + b1915 + 13 & 194.63 & 94.5 & 1.9(2 ) & 1453.2 & 4.477 + b1916 + 14 & 1181.02 & 27.2 & 1.0(1 ) & 14.3 & 0.362 + b1919 + 14 & 618.18 & 91.6 & 0.68(8 ) & 217.6 & 1.060 + b1921 + 17 & 547.21 & 142.5 & & 126.6 & 0.408 + b1924 + 14 & 1324.92 & 211.4 & 0.48(6 ) & 126.6 & 0.860 + b1924 + 16 & 579.82 & 176.9 & 1.3(2 ) & 179.1 & 0.735 + b1925 + 18 & 482.77 & 254.0 & & 156.0 & 0.441 + b1925 + 188 & 298.31 & 99.0 & & 77.3 & 0.385 + b1929 + 15 & 314.36 & 140.0 & & 69.4 & 0.360 + b1929 + 20 & 268.22 & 211.2 & 1.2(4 ) & 457.9 & 1.099 + b1933 + 16 & 358.74 & 158.5 & & 73.0 & + b1933 + 17 & 654.41 & 214.6 & & 62.8 & 0.176 + b1937 + 21 & 1.56 & 71.0 & & 349.1 & 12.572 + b1937 + 24 & 645.30 & 142.9 & & 39.4 & + b1944 + 22 & 1334.45 & 140.0 & & 55.0 & 0.173 + b2002 + 31 & 2111.26 & 234.8 & 1.8(1 ) & 68.2 & + j0621 + 1002 & 28.85 & 36.6 & 1.9(3 ) & 11.4 & + j0625 + 10 & 498.40 & 78.0 & & 14.5 & 0.086 + j0631 + 1036 & 287.80 & 125.4 & & 175.3 & 0.941 + j1829 + 0000 & 199.15 & 114.0 & & 52.4 & 0.370 + j1843@xmath910000 & 880.33 & 101.5 & 2.9(3 ) & 38.5 & + j1844 + 00 & 460.50 & 345.5 & 8.6(9 ) & 1226.8 & 4.616 + j1849 + 0127 & 542.16 & 207.3 & 0.46(9 ) & 143.2 & 0.444 + j1849 + 0409 & 761.19 & 56.1 & & 29.0 & 0.312 + j1851 + 0118 & 906.98 & 418.0 & 0.10(2 ) & 27.9 & 0.118 + j1852 + 0305 & 1326.15 & 320.0 & 0.8(2 ) & 37.7 & 0.214 + j1853 + 0056 & 275.58 & 180.9 & 0.21(4 ) & 55.3 & 0.281 + j1853 + 0545 & 126.40 & 198.7 & 1.6(1.7 ) & 5.3 & + j1854 + 0317 & 1366.45 & 404.0 & 0.12(1 ) & 34.9 & 0.153 + j1855 + 0307 & 845.35 & 402.5 & 1.0(1 ) & 129.7 & 0.393 + j1855 + 0422 & 1678.11 & 438.0 & 0.45(9 ) & 104.0 & 0.245 + j1856 + 0102 & 620.22 & 554.0 & 0.4(1 ) & 66.3 & 0.195 + j1856 + 0404 & 420.25 & 341.3 & 0.48(1 ) & 40.4 & 0.276 + j1857 + 0143 & 139.76 & 249.0 & 0.7(2 ) & 37.2 & 0.486 + j1857 + 0210 & 630.98 & 783.0 & 0.30(6 ) & 40.2 & 0.236 + j1857 + 0526 & 349.95 & 466.4 & 0.66(8 ) & 145.5 & 0.645 + j1858 + 0215 & 745.83 & 702.0 & 0.22(4 ) & 42.8 & 0.280 + j1859 + 00 & 559.63 & 420.0 & 4.8(5 ) & 581.9 & 24.461 + j1859 + 0601 & 1044.31 & 276.0 & 0.30(4 ) & 15.9 & 0.126 + j1900 + 0227 & 374.26 & 201.1 & 0.33(7 ) & 111.6 & 0.414 + j1901 + 00 & 777.66 & 345.5 & 0.35(4 ) & 32.4 & + j1901 + 0254 & 1299.69 & 185.0 & 0.58(7 ) & 102.1 & 0.911 + j1901 + 0320 & 636.58 & 393.0 & 0.9(1 ) & 67.3 & 0.301 + j1901 + 0355 & 554.76 & 547.0 & 0.15(3 ) & 40.9 & 0.185 + j1901 + 0413 & 2663.08 & 352.0 & 1.1(2 ) & 161.9 & 0.521 + j1901 + 0435 & 690.58 & 1042.6 & & 106.9 & 4.244 + j1901 + 0510 & 614.76 & 429.0 & 0.66(8 ) & 47.6 & 0.498 + j1902 + 0248 & 1223.78 & 272.0 & 0.17(3 ) & 60.6 & 0.169 + j1903 + 0601 & 374.12 & 388.0 & 0.26(4 ) & 9.7 & + j1904 + 0412 & 71.09 & 185.9 & 0.23(5 ) & 68.4 & 0.271 + j1904 + 0800 & 263.34 & 438.8 & 0.36(5 ) & 11.2 & 0.285 + j1905 + 0600 & 441.21 & 730.1 & 0.42(5 ) & 85.6 & 0.401 + j1905 + 0616 & 989.71 & 256.1 & 0.51(6 ) & 43.5 & 0.236 + j1906 + 0912 & 775.34 & 265.0 & 0.32(6 ) & 34.0 & 0.149 + j1907 + 0249 & 351.88 & 261.0 & 0.5(1 ) & 124.3 & 0.478 + j1907 + 0345 & 240.15 & 311.7 & 0.17(3 ) & 21.5 & 0.133 + j1907 + 0534 & 1138.40 & 524.0 & 0.36(7 ) & 24.6 & 0.096 + j1907 + 0731 & 363.68 & 239.8 & 0.35(4 ) & 68.8 & 0.571 + j1907 + 0740 & 574.70 & 332.0 & 0.41(8 ) & 121.4 & 0.327 + j1907 + 0918 & 226.11 & 357.9 & 0.29(4 ) & 133.4 & 0.263 + j1907 + 1149 & 1420.16 & 202.8 & & 30.4 & 0.156 + j1908 + 0457 & 846.79 & 360.0 & 0.9(1 ) & 274.4 & 0.958 + j1908 + 0500 & 291.02 & 201.4 & 0.79(9 ) & 48.5 & + j1908 + 0734 & 212.35 & 11.1 & 0.54(6 ) & 36.0 & 0.205 + j1908 + 0839 & 185.40 & 512.1 & 0.49(1 ) & 114.4 & 0.403 + j1908 + 0909 & 336.55 & 467.5 & 0.22(4 ) & 110.7 & 0.340 + j1909 + 0616 & 755.99 & 352.0 & 0.33(7 ) & 10.3 & + j1909 + 0912 & 222.95 & 421.5 & 0.35(7 ) & 125.8 & 0.533 + j1910 + 0534 & 452.87 & 484.0 & 0.41(8 ) & 62.4 & 0.444 + j1910 + 0714 & 2712.42 & 124.1 & 0.36(5 ) & 137.3 & 0.287 + j1910 + 0728 & 325.42 & 283.7 & 0.8(1 ) & 189.8 & 0.887 + j1910 + 1256 & 4.98 & 38.1 & 0.5(1 ) & 139.7 & 0.497 + j1913 + 0832 & 134.41 & 355.2 & 0.6(1 ) & 187.9 & 0.999 + j1913 + 0904 & 163.25 & 95.3 & & 96.7 & 0.224 + j1913 + 1000 & 837.15 & 422.0 & 0.53(6 ) & 28.8 & 0.522 + j1913 + 1011 & 35.91 & 178.8 & 0.5(1 ) & 111.0 & 0.434 + j1913 + 1145 & 306.07 & 637.0 & 0.43(9 ) & 126.5 & 0.403 + j1913 + 1330 & 923.39 & 175.6 & & 213.6 & + j1914 + 0631 & 693.81 & 58.0 & 0.3(1 ) & 36.9 & 0.140 + j1915 + 0738 & 1542.70 & 39.0 & 0.34(4 ) & 109.1 & 0.254 + j1915 + 0752 & 2058.31 & 105.3 & 0.21(3 ) & 18.2 & 0.238 + j1915 + 0838 & 342.78 & 358.0 & 0.29(4 ) & 12.3 & + j1915 + 1410 & 297.49 & 273.7 & & 11.6 & 0.134 + j1916 + 0748 & 541.75 & 304.0 & 2.8(3 ) & 66.8 & + j1916 + 0844 & 440.00 & 339.4 & 0.44(5 ) & 89.9 & 0.526 + j1916 + 0852 & 2182.75 & 295.0 & 0.13(2 ) & 36.6 & 0.148 + j1920 + 1040 & 2215.80 & 304.0 & 0.57(7 ) & 24.5 & 0.092 + j1920 + 1110 & 509.89 & 182.0 & 0.39(8 ) & 22.9 & 0.288 + j1921 + 1544 & 143.58 & 385.0 & & 65.5 & 0.211 + j1922 + 1733 & 236.17 & 238.0 & & 435.6 & 1.157 + j1924 + 1639 & 158.04 & 208.0 & & 73.6 & 0.207 + j1926 + 2016 & 299.07 & 247.0 & & 12.0 & 0.122 + j1928 + 1923 & 817.33 & 476.0 & & 221.7 & 0.639 + j1929 + 1955 & 257.83 & 281.0 & & 25.1 & 0.421 + j1930 + 17 & 1609.69 & 201.0 & & 30.9 & + j1931 + 1952 & 501.12 & 441.0 & & 71.9 & 0.126 + j1935 + 2025 & 80.12 & 182.0 & & 79.6 & 0.527 + j1936 + 21 & 642.93 & 264.0 & & 13.6 & + j1938 + 2213 & 166.12 & 91.0 & & 20.4 & + j1946 + 2611 & 435.06 & 165.0 & & 232.0 & 0.697 + j1957 + 2831 & 307.68 & 139.0 & 1.0(2 ) & 34.4 & + lcccccc + parameter & + + & 0.766 & 1.102 & 2.218 & 5.218 & 10.870 & 18.505 + period , ms & 26.965 & 61.631 & 126.175 & 286.555 & 533.320 & 850.158 + & 1657.496 & 2643.410 & 3927.013 & 5580.899 & 10964.532 & + dm , @xmath5 & 10 & 40 & 150 & 325 & 400 & 600 + fwhm , % phase & 1.5 & 2.6 & 5.9 & 11.9 & 24.3 & + +

the on - going palfa survey at the arecibo observatory began in 2004 and is searching for radio pulsars in the galactic plane at 1.4 ghz . observations since 2009 have been made with new wider - bandwidth spectrometers than were previously employed in this survey . a new data reduction pipeline has been in place since mid-2011 which consists of standard methods using dedispersion , searches for accelerated periodic sources , and search for single pulses , as well as new interference - excision strategies and candidate selection heuristics . this pipeline has been used to discover 41 pulsars , including 8 millisecond pulsars ( msps ; @xmath0ms ) , bringing the palfa survey s discovery totals to 145 pulsars , including 17 msps , and one fast radio burst ( frb ) . the pipeline presented here has also re - detected 188 previously known pulsars including 60 found in palfa data by re - analyzing observations previously searched by other pipelines . a comprehensive description of the survey sensitivity , including the effect of interference and red noise , has been determined using synthetic pulsar signals with various parameters and amplitudes injected into real survey observations and subsequently recovered with the data reduction pipeline . we have confirmed that the palfa survey achieves the sensitivity to msps predicted by theoretical models . however , we also find that compared to theoretical survey sensitivity models commonly used there is a degradation in sensitivity to pulsars with periods @xmath1ms that gradually becomes up to a factor of @xmath210 worse for @xmath3s at @xmath4@xmath5 . this degradation of sensitivity at long periods is largely due to red noise . we find that @xmath6% of pulsars are missed despite being bright enough to be detected in the absence of red noise . this reduced sensitivity could have implications on estimates of the number of long - period pulsars in the galaxy .

introduction observations pulsar and transient search pipeline results assessing the survey sensitivity population synthesis analysis discussion conclusions

arxiv

in this study , we are interested in the ave with non - hermitian toeplitz matrix of the form @xmath0 where @xmath1 denotes the component - wise absolute value of the vector @xmath2 . a slightly more generalized form of the ave , @xmath3 was discussed in @xcite and investigated in a more general context in @xcite . moreover , the theoretical and numerical aspects of these problems have been extensively investigated in recent literature @xcite . generally speaking , the ave ( [ ku1 ] ) arises from quadratic programs , linear programs , bimatrix games and other problems , which can all be resulted in an linear complementarity problem ( lcp ) @xcite , and the lcp is equivalent to the ave ( [ ku1 ] ) . this means that the ave is np - hard in its general form @xcite . if @xmath4 , then generalized ave ( [ ku2 ] ) reduces to a system of linear equations @xmath5 , which have several applications in scientific computation @xcite . the recent research concerning the ave contents can be summarized as the following aspects , one is the theoretical analysis , which focuses on the theorem of alternatives , various equivalent reformulations , and the existence and nonexistence of solutions ; refer to @xcite for details . and the other is how to solve the ave numerically . in the last decade , based on the fact that the lcp can be reduced to the ave , which enjoys a very special and simple structure , a large variety of numerical methods for solving the ave ( [ ku1 ] ) can be found in the recent literature ; see e.g. @xcite and references therein . for example , a finite computational algorithm that is solved by a finite succession of linear programs ( slp ) in @xcite , and a semi - smooth newton method is proposed in @xcite , which largely shortens the computation time than the slp method . furthermore , a smoothing newton algorithm was presented in @xcite , which was proved to be globally convergent and the convergence rate was quadratic under the condition that the singular values of @xmath6 exceed 1 . this condition was weaker than the one applied in @xcite . during recent years , the picard - hss iteration method and nonlinear hss - like method are established to solve the ave in succession @xcite , respectively . the sufficient conditions to guarantee the convergence of this method and some numerical experiments are given to show the effectiveness of the method . however , the numbers of the inner hss iterative steps are often problem - dependent and difficult to be determined in actual computations . moreover , the iterative vector can not be updated timely . it has shown that the nonlinear hss - like iterative method is more efficient than the picard - hss iteration method in aspects of the defect mentioned above , which is designed originally for solving weakly nonlinear systems in @xcite . in order to accelerate the nonlinear hss - like iteration method , zhang @xcite had extended the preconditioned hss ( phss ) method @xcite to solve the ave and also exploit the relaxation technique to accelerate his proposed methods . meanwhile , numerical results also show the effectiveness of his proposed method in @xcite . in this paper , we consider the special case of @xmath6 involving the non - hermitian toeplitz structure . similar to the strategies of @xcite , two kinds of circulant and skew - circulant splitting ( cscs)-based methods are proposed to fast solve the ave ( [ ku1 ] ) . the rest of this paper is organized as follows . in section 2 we review the cscs iteration method and its relative topics . in section 3 , we devote to introduce two cscs - based iteration methods to solve ave ( [ ku1 ] ) and investigate their convergence properties , respectively . numerical experiments are reported in section 4 , to shown the feasibility and effectiveness of the cscs - based methods . finally , the paper closes with some conclusions in section 5 . here let @xmath7 be a non - hermitian toeplitz matrix of the following form @xmath8 i.e. , @xmath6 is constant along its diagonals ; see @xcite , and @xmath9 be a zero matrix , the general ave ( [ ku2 ] ) reduced to the system of linear equations @xmath10 it is well - known that a toeplitz matrix @xmath6 possesses a circulant and skew - circulant splitting @xcite @xmath11 , where @xmath12 and @xmath13 note that @xmath14 is a circulant matrix and @xmath15 is a skew - circulant matrix . a circulant matrix can be diagonalized by the discrete fourier matrix @xmath16 and a skew - circulant matrix can be diagonalized by a discrete fourier matrix with diagonal scaling , i.e. , @xmath17 . that is to say , it holds that @xmath18 where @xmath19 and @xmath20 is the imaginary unit @xcite . @xmath21 and @xmath22 are diagonal matrices formed by the eigenvalues of @xmath14 and @xmath15 , respectively , which can be obtained in @xmath23 operations by using the fft . moreover , ng @xcite established the following cscs iteration method to solve non - hermitian toeplitz system of linear equations ( [ ku3 ] ) . * algorithm 1 the cscs iteration method*. + _ given an initial guess @xmath24 , compute @xmath25 for @xmath26 using the following iterative scheme until @xmath27 converges , @xmath28 where @xmath29 is a positive constant and @xmath30 is the identity matrix . _ in the matrix - vector form , the cscs iteration can be equivalently rewritten as @xmath31 where @xmath32 it is easy to see that cscs is a stationary iterative method obtained from the splitting @xmath33 where @xmath34 on the other hand , we have @xmath35 here , @xmath36 is the iterative matrix of the cscs method . we remark that the cscs iteration method is greatly similar to the hss iteration method @xcite and its variants , see e.g. @xcite . when the circulant part @xmath14 and skew - circulant part @xmath15 of the coefficient matrix @xmath37 are both positive definite , ng proved that the spectral radius @xmath38 of the cscs iterative matrix @xmath36 is less than 1 for any positive iterative parameters @xmath29 , i.e. , the cscs iteration method unconditionally converges to the exact solution of @xmath39 for any initial guess @xmath40 ; refer to @xcite for details . motivated by the pioneer works of @xcite , we extend the classical cscs iteration method to two types of cscs - based methods for solving the ave ( [ ku1 ] ) . these methods will fully exploit the toeplitz structure to accelerate the computation speed and save storage . next , we will devote to constructing these two new methods , i.e. , the picard - cscs iterative method and nonlinear cscs - like iterative method . recalling that the picard iterative method is a fixed - point iterative method and the linear term @xmath41 and the nonlinear term @xmath42 are separated @xcite , the ave can be solved by using of the picard iterative method @xmath43 we assume that the toeplitz matrix @xmath6 is non - hermitian positive definite . in this case , the next iterate of @xmath44 can be approximately computed by the cscs iteration by making use of @xmath45 as following ( see @xcite ) @xmath46 where @xmath47 and @xmath48 are the matrices defined in the previous section , @xmath29 is a positive constant , @xmath49 a prescribed sequence of positive integers , and @xmath50 is the starting point of the inner cscs iteration at @xmath51th outer picard iteration . this leads to the inexact picard iteration method , called picard - cscs iteration method , for solving the system ( [ ku1 ] ) which can be summarized as following ( refer to @xcite ) . * algorithm 2 the picard - cscs iteration method * + _ let @xmath52 be a non - hermitian toeplitz matrix ; @xmath14 and @xmath15 are the circulant and skew - circulant parts of @xmath6 given in ( [ ku4x ] ) and ( [ ku4y ] ) and they are both positive definite . given an initial guess @xmath53 and a sequence @xmath49 of positive integers , compute @xmath44 for @xmath54 , using the following iteration scheme until @xmath55 satisfies the following stopping criterion : _ * set @xmath56 ; * for @xmath57 , solve the following linear systems to obtain @xmath58 : @xmath59 where @xmath29 is a given positive constant . * set @xmath60 . the advantage of picard - cscs iterative method is obvious . first , the two linear subsystems in all inner cscs iterations have the same shifted circulant coefficient matrix @xmath61 and shifted skew - circulant coefficient matrix @xmath62 , which are constant with respect to the iteration index @xmath51 . second , the exact solutions can be efficiently achieved via ffts in @xmath23 operations @xcite . hence , the computations of the picard - cscs iteration method could be much cheaper than that of the picard - hss iteration method . the next theorem provides sufficient conditions for the convergence of the picard - cscs method to solve system ( [ ku1 ] ) . [ theorem1 ] let @xmath52 be a non - hermitian toeplitz matrix ; @xmath14 and @xmath15 are the circulant and skew - circulant parts of @xmath6 given in ( [ ku4x ] ) and ( [ ku4y ] ) and they are both positive definite . let also @xmath63 . then the ave ( [ ku2 ] ) has a unique solution @xmath64 , and for any initial guess @xmath53 and any sequence of positive integers @xmath65 , the iteration sequence @xmath66 produced by the picard - cscs iteration method converges to @xmath64 provided that @xmath67 , where @xmath68 is a natural number satisfying @xmath69 * proof*. the proof uses arguments similar to those in the proof of the convergence theorem of the picard - hss iteration method ; see @xcite . in fact , we only need to replace the hermitian matrix @xmath70 and the skew - hermitian matrix @xmath15 of the convergence theorem of the picard - cscs iteration method by the circulant matrix @xmath14 and the skew - circulant matrix @xmath15 , and then obtain the convergence theorem of the picard - cscs iteration method . according to theorem [ theorem1 ] , we see that the picard - cscs iteration method to solve the ave ( [ ku2 ] ) is convergent if the matrix @xmath71 is positive definite , @xmath72 ( see @xcite for the definition of @xmath73 ) and the sequence @xmath74 , is defined as in theorem [ theorem1 ] . similar to @xcite , the residual - updating form of the picard - cscs iteration method can be written as following . * algorithm 3 the picard - cscs iteration method * ( residual - updating variant ) + _ let @xmath52 be a non - hermitian toeplitz matrix ; @xmath14 and @xmath15 are the circulant and skew - circulant parts of @xmath6 given in ( [ ku4x ] ) and ( [ ku4y ] ) and they are both positive definite . given an initial guess @xmath53 and a sequence @xmath49 of positive integers , compute @xmath44 for @xmath54 , using the following iteration scheme until @xmath55 satisfies the following stopping criterion : _ * set @xmath75 and @xmath76 ; * for @xmath57 , solve the following linear systems to obtain @xmath77 : @xmath78 where @xmath29 is a given positive constant . * set @xmath79 . in the picard - cscs iteration , the numbers @xmath80 of the inner cscs iterative steps are often problem - dependent and difficult to be determined in actual computations @xcite . moreover , the iterative vector can not be updated timely . thus , to avoid the defection and still preserve the advantages of the picard - cscs iterative method , based on the nonlinear fixed - point equations @xmath81 we propose the following nonlinear cscs - like iteration method . * algorithm 4 the nonlinear cscs - like iteration method * + _ let @xmath52 be a non - hermitian toeplitz matrix ; @xmath14 and @xmath15 are the circulant and skew - circulant parts of @xmath6 given in ( [ ku4x ] ) and ( [ ku4y ] ) and they are both positive definite . given an initial guess @xmath53 and compute @xmath44 for @xmath54 , using the following iteration scheme until @xmath55 satisfies the following stopping criterion : @xmath82 where @xmath29 is a given positive constant . _ define @xmath83 and @xmath84 then the nonlinear cscs - like iterative scheme can be equivalently expressed as @xmath85 the ostrowski theorem , i.e. , theorem 10.1.3 in @xcite , gives a local convergence theory about a one - step stationary nonlinear iteration . based on this , zhu and zhang established the local convergence theory for the nonlinear cscs - like iteration method in @xcite . however , these convergence theory has a strict requirement that @xmath86 is @xmath87-differentiable at a point @xmath88 such that @xmath89 . obviously , the absolute value function @xmath1 is non - differentiable . leveraging the smoothing approximate function introduced in @xcite , we can establish the following local convergence theory for nonlinear cscs - like iterative method . but firstly , we must review this smoothing approximation and its properties , which will be used in the next section . define @xmath90 by @xmath91 it is clear that @xmath92 is a smoothing function of @xmath1 , now we give some properties of @xmath93 , which will be used in the next section . ( @xcite ) @xmath92 is a uniformly smoothing approximation function of @xmath1 , i.e. , @xmath94 ( @xcite ) for any @xmath95 , the jacobian of @xmath92 at @xmath96 is @xmath97 [ lemma3 ] assume that @xmath98 is @xmath87-differentiable at a point @xmath99 such that @xmath100 . suppose that @xmath14 and @xmath15 are the circulant and the skew - circulant parts of the matrix @xmath101 given in ( [ ku4x ] ) and ( [ ku4y ] ) , and @xmath14 and @xmath15 both are positive definite matrices . denote by @xmath102 and @xmath103 @xmath104 then @xmath105 holds ; in other word , @xmath106 is a point of attraction of the nonlinear cscs - like iteration , provided @xmath107 . leveraging the smoothing approximate function @xmath92 in ( [ ku9x ] ) , we define @xmath108 and @xmath109 then the nonlinear cscs - like iterative scheme can be equivalently expressed as @xmath110 [ the1 ] assume that the condition of lemma [ lemma3 ] are satisfied , @xmath14 and @xmath15 be circulant and skew - circulant parts of the toeplitz matrix @xmath6 , respectively . for any initial guess @xmath111 , the iteration sequence @xmath112 produced by the nonlinear cscs - like iteration method can be instead approximately by that produced by its smoothed nonlinear cscs - like iterative scheme ( [ ku12x ] ) , i.e. , @xmath113 provided @xmath114 * proof*. first , we give the well - known inequality @xmath115 and the result @xmath116 achieved in @xcite . then based on iterative scheme ( [ ku9x ] ) and ( [ ku12x ] ) , we obtain @xmath117 for @xmath118 holds , provided @xmath114 this completes the proof . @xmath119 assume that the conditions of theorem [ the1 ] are satisfied . denoted by @xmath120 and @xmath121 then the spectral radius @xmath122 of the matrix @xmath123 is less than 1 , where @xmath124 and @xmath125 is the the jacobian of @xmath92 at @xmath126 defined in ( [ ku11 ] ) , provided that @xmath127 that is to say , for any initial guess @xmath128 , the iteration sequence @xmath112 produced by the nonlinear cscs - like iteration method converges to @xmath64 , or @xmath64 is a point of attraction of the ninlinear cscs - like iteration , provided the condition ( [ ku16 ] ) . * proof*. for @xmath129 , we only need to prove @xmath130 where @xmath131 is defined in ( [ ku9x ] ) and @xmath132 is defined in ( [ ku12x ] ) . via using the theorem [ the1 ] , the former part @xmath133 holds for @xmath134 , provided @xmath114 as the uniformly smoothing approximation function @xmath92 of @xmath1 is @xmath87-differentiable at a point @xmath135 such that @xmath136 , according lemma [ lemma3 ] , @xmath64 is a point of attraction of the nonlinear cscs - like iteration , that is the second part in ( [ ku17 ] ) @xmath137 holds for @xmath138 , provided @xmath105 . next we prove @xmath105 . via straightforward computations we have @xmath139 where @xmath125 is the jacobian of the smoothing approximation function @xmath92 at @xmath64 , also since @xmath140 we obtain @xmath141 now , under the condition @xmath142 , we easily obtain @xmath143.@xmath119 * remark 1 . * an attractive feature of the nonlinear cscs - like iterative method is that it avoids the use of the differentiable in actual iterative scheme , although we employe it in the convergence analysis . thus , the smoothing approximate function @xmath92 in ( [ ku9y ] ) is not necessary in actual implementation . at the end of this subsection , we remark that the main steps in nonlinear cscs - like iteration method can be alternatively reformulated into residual - updating form similar to those in the picard - cscs iterative method as follows . * algorithm 5 ( the nonlinear cscs - like iteration method * ( residual - updating variant ) + _ let @xmath52 be a non - hermitian toeplitz matrix ; @xmath14 and @xmath15 are the circulant and skew - circulant parts of @xmath6 given in ( [ ku4x ] ) and ( [ ku4y ] ) and they are both positive definite . given an initial guess @xmath53 and compute @xmath44 for @xmath54 , using the following iteration scheme until @xmath55 satisfies the following stopping criterion : @xmath144 where @xmath29 is a given positive constant . in this section , the numerical properties of the picard - cscs and the nonlinear cscs - like methods are examined and compared experimentally by a suit of test problems . all the tests are performed in matlab r2014a on intel(r ) core(tm ) i5 - 3470 cpu @ 3.2 ghz and 8.00 gb of ram , with machine precision @xmath145 , and terminated when the current residual satisfies @xmath146 where @xmath25 is the computed solution by each of the methods at iteration step @xmath51 , and a maximum number of the iterations 200 is used . in addition , the stopping criterion for the inner iterations of the picard - cscs method are set to be @xmath147 where @xmath148 is the number of the inner iteration steps and @xmath149 is the prescribed tolerance for controlling the accuracy of the inner iterations at the @xmath51-th outer iteration step . if @xmath149 is fixed for all @xmath51 , then it is simply denoted by @xmath150 . in our numerical experiments , we use the zero vector as the initial guess , the accuracy of the inner iterations @xmath149 for both picard - cscs and picard - hss iterative methods is fixed and set to @xmath151 , a maximum number of iterations 15 ( @xmath152 ) for inner iterations , and the right - hand side vector @xmath153 of aves ( 1 ) is taken in such a way that the vector @xmath154 with @xmath155 be the exact solution . the two sub - systems of linear equations involved are solved in the way if @xmath156 , then @xmath157 . moreover , if the two sub - systems of linear equations involved in the picard - cscs and the nonlinear cscs - like iteration methods are solved by making use of the method presented in @xcite and using parallel computing , the numerical results of the picard - cscs and the nonlinear cscs - like iteration methods must be better . in practical implementations , the optimal parameter @xmath158 recommended in @xcite is employed for the picard - hss and the nonlinear hss - like methods , where @xmath159 and @xmath160 are the minimum and the maximum eigenvalues of the hermitian part @xmath70 of the coefficient matrix @xmath6 . similarly , we adopt the optimal parameters @xmath161 given in ( * ? ? ? * theorem 2 ) for the picard - cscs and the nonlinear cscs - like methods . at the same time , it is remarkable that they only minimize the bound of the convergence factor of the iteration matrix , but not the spectral radius of the iteration matrix . admittedly , the optimal parameters are crucial for guaranteeing fast convergence speeds of these parameter - dependent iteration methods , but they are generally very difficult to be determined , refer to e.g. @xcite for a discussion of these issues . to show that the proposed iteration methods can also be efficiently applied to deal with complex system of aves ( [ ku1 ] ) , we construct and test the following example , which is a toeplitz system of aves with complex matrix . * example 1*. we consider that @xmath7 is a complex non - hermitian , sparse and positive definite toeplitz matrix with the following form @xmath162 where @xmath163 . it means that the matrices @xmath6 in the target aves are defined as eq . ( [ matrix2 ] ) . according to the performances of hss - based methods , see @xcite , compared with other early established methods , we compare the proposed cscs - based methods with hss - based methods in example 1 . then we will choose different parameters @xmath164 and @xmath165 and present the corresponding numerical results in tables [ tab2]-[tab3 ] . .the optimal parameters @xmath166 for example 1 . [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] [ tab7 ] based on the numerical results in tables [ tab5]-[tab7 ] , it is notable that these four iterative solvers , i.e. , the picard - cscs and the nonlinear cscs - like , can successfully obtain approximate solutions to the system of aves for all different matrix dimensions ; whereas both the gn - gmres(5 ) and the gn - tfqmr iterative methods fully fail to converge . it should be because the newton - like iterative methods are usually sensitive to the initial guess and the accuracy of solving inner linear systems . when the matrix dimension @xmath167 is increasing , the number of outer iteration steps are almost fixed or or decreasing slightly for all iteration methods , whereas the number of inner iteration steps show the contrary phenomena for the cases with @xmath168 and @xmath169 . meanwhile , the total cpu times and total iteration steps for both the picard - cscs and the nonlinear cscs - like iterative methods are increasing quickly except the cases of @xmath170 with @xmath171 and @xmath172 . on the other hand , from tables [ tab5]-[tab7 ] , we also observe that both the nonlinear cscs - like method is almost more competitive than the picard - cscs iteration methods in terms of the number of iteration steps and the cpu elapsed time for solving the system of aves . in particular , it is remarkable that the nonlinear cscs - like method can use slightly less number of iteration steps to converge than the picard - cscs iterative solver , but the picard - cscs iterative solver can save a little elapsed cpu time with compared to the nonlinear cscs - like iterative method in our implementations . however , it still concludes that the nonlinear cscs - like iterative method is the first choice for solving the aves concerning in example 2 . at the same time , the picard - cscs iterative method can be viewed as a good alternative . in this paper , we have proposed two cscs - based methods for solving the system of aves with non - hermitian toeplitz structure . two cscs - based iterative methods are based on separable property of the linear term @xmath173 and nonlinear term @xmath42 as well as the circulant and skew - circulant splitting ( cscs ) of involved non - hermitian definite toeplitz matrix @xmath6 . by leveraging the smoothing approximate function , the locally convergence have been analysed . further numerical experiments have shown that the picard - cscs and the nonlinear cscs - like iteration methods are feasible and efficient nonlinear solvers for the ave . moreover , in particular , the nonlinear cscs - like method often does better than the picard - cscs method to solve ave is that the smoothing approximate function is introduced in the convergence analysis although is avoid in implement algorithm . hence , to find a better theoretical proof for cscs - like will be a topics and suitable accelerated techniques @xcite in the future research . 99 s .- l . hu , z .- h . huang , a note on absolute value equations , optim . lett . , 4 ( 2010 ) , pp . 417 - 424 . o. prokopyev , on equivalent reformulations for absolute value equations . appl . , 44 ( 2009 ) , pp . 363 - 372 . mangasarian , absolute value equation solution via concave minimization , optim . lett . , 1 ( 2007 ) , pp . 3 - 8 . j. rohn , v. hooshyarbakhsh , r. farhadsefat , an iterative method for solving absolute value equations and sufficient conditions for unique solvability , optim . lett . , 8 ( 2014 ) , 35 - 44 . bai , g.h . golub , c .- k . li , covergence properties of preconditioned hermitian and skew - hermitian splitting methods for non - hermitian positive semidefinite matrices , math . comput . , 76 ( 2007 ) , pp . 287 - 298 . ortega , w.c . rheinboldt , iterative solution of nonlinear equations in several variables , siam , philadelphia , usa , 2000 . l. yong , particle swarm optimization for absolute value equations , j. comput . systems , 6 ( 7 ) ( 2010 ) , pp . 2359 - 2366 . huang , a practical formula for computing optimal parameters in the hss iteration methods , j. comput . appl . , 255 ( 2014 ) , pp . 142 - 149 . choi , s.k . chung , y.j . lee , numerical solutions for space fractional dispersion equations with nonlinear source terms , bull . korean math . soc . , 47 ( 2010 ) , pp . 1225 - 1234 . gu , t .- z . huang , h .- b . li , l. li , w .- h . luo , on @xmath51-step cscs - based polynomial preconditioners for toeplitz linear systems with application to fractional diffusion equations , appl . lett . , 42 ( 2015 ) , pp .

recently , two kinds of hss - based iteration methods are constructed for coping with the absolute value equation ( ave ) , which is a family of non - differentiable np - hard problem . in present paper , we focus on developing the cscs - based methods for solving the absolute value equation ( ave ) involving the toeplitz matrix , and propose the picard - cscs method and the nonlinear cscs - like iterative method . with the help of introducing a smoothing approximate function , we give some theoretical analyses for the convergence of the cscs - based iteration methods for the ave . the advantage of these methods is that they do not require storage of coefficient matrix , and the linear sub - systems can be solved efficiently via the fast fourier transform ( fft ) . therefore , computational cost and computer storage may be saved in actual implementations . extensive numerical experiments involving the numerical solutions of fractional diffusion equations are employed to demonstrate the robustness and effectiveness of the proposed methods and to compare with the recent methods . * key words * : absolute value equation ; cscs - based iteration ; toeplitz matrix ; convergence analysis ; smoothing approximate function ; fast fourier transform * ams classification * : 65f12 ; 65l05 ; 65n22 .

introduction the cscs method two cscs-based methods for aves numerical results conclusions acknowledgments

arxiv

surface - attached polymers are an ubiquitous phenomenon in living matter . on the cellular level , prominent examples are the cytoskeletal actin network beneath the cell s plasma membrane @xcite and the extracellular matrix @xcite . while the semiflexible actin polymers interact at multiple sites with the intracellular face of the cell s membrane , the dominant component of the extracellular matrix , the hyaluronic acid ( ha ) , rather is a flexible polymer , end - grafted to the extracellular face of the membrane and extending to the cell s environment . to provide an efficient barrier that is capable of protecting the cell , extracellular ha is modified by rigid aggrecan combs that are linked to the flexible ha backbone . as a consequence , the cell is protected by a jungle of ha - aggrecan complexes . we have previously shown that attaching a single rigid side chain to a flexible , end - grafted polymer ( resembling the ha - aggrecan complex ) can lead to a considerable stiffening of the backbone @xcite , hinting at a mechanism by which cells can tune the rigidity of their protective matrix . in this study , we had used a dynamic monte carlo algorithm on a regular cubic lattice to study the steady - state properties of a flexible , end - grafted backbone polymer with a rigid side chain . extending this approach to a system of many interacting backbones , however , is computationally challenging due to the massive increase in the system s autocorrelation time . an alternative approach to a brute - force monte carlo sampling is a further coarse graining of the system that reduces the problem to its basic observables , namely the height of the backbones and the orientation of the rigid side chains . here , a combination of the potts model ( describing the height levels ) and the two - dimensional xy - model ( describing the orientation of the side chains ) appears as a promising candidate . using this approach , however , one has to deal with an intricate problem of statistical physics the prediction of the system s phase behavior and phase transition . the potts model ( with @xmath0 height levels ) undergoes a second order phase transition @xcite while the two - dimensional xy - model shows a kosterlitz - thouless transition @xcite . it is far from evident which phase transition will be encountered when coupling these two systems . aiming at modeling the behavior of the extracellular matrix in terms of such a combined model thus requires to elucidate the model s phase behavior in the first place . inspired by the potential value of the combined potts - xy - model in describing features of the extracellular matrix , we have investigated the phase transition of the model . in particular , we have studied the phase transition in a model that couples the @xmath1 potts model with the xy - model by means of extensive monte carlo simulations . we find that the phase transition is dominated by the potts model , i.e. a second order phase transition is observed . the ordered state at low temperatures shows domains of parallel aligned xy - spins with the potts levels collectively exploring all states in a stochastic fashion . in contrast , for high temperatures the xy - spins are disordered with an average potts level @xmath2 that only varies to a minor extent . relating these findings to the original biological problem , we speculate that changing the interaction strength between individual ha - aggrecan complexes , i.e. driving the system through the phase transition , may be one way for the cell to build up a stochastically fluctuating protective barrier . as described previously @xcite , the basic unit of the extracellular matrix may be reduced to a single flexible , end - grafted backbone with a rigid side chain ( fig . [ fig : fig01 ] ) . while a simulation of this unit yields valuable insights into the stiffening of the individual backbone due to the attached rigid side chain , many of the details may not be necessary to understand the generic behavior of the multi - complex system . the basic quantities describing the interaction of many individual units are the height @xmath3 of each side chain above the substrate and its orientation angle @xmath4 with respect to the @xmath5-axis . thus , the individual backbone side - chain complexes are interpreted as mutually interacting rotors aligned on a regular two - dimensional lattice . of the side chain above the substrate and its orientation angle @xmath4 , are retained . a system of mutually interacting complexes is obtained by using a two - dimensional lattice of such rotors ( right ) where the dynamics is determined by eq . ( [ eq : hybrid_model ] ) . , width=264 ] we assume here that the sites at which the backbones are attached to the substrate are sufficiently far aparat from each other so that any interaction between the complexes is mediated by the side chains only . this also means that side chains only interact when they have the same height @xmath3 . this aspect can be modeled by a potts - type hamiltonian @xcite : @xmath6 here , @xmath7 denotes the height of side chain @xmath8 above the substrate , @xmath9 is the total number of allowed heights , and @xmath10 is the coupling constant ( in units of @xmath11 ) . the interaction between side chains with the same @xmath3 will be considered as a nearest - neighbor ferromagnetic interaction leading to a prefered parallel alignment of the side chains . this kind of interaction seems plausible as a first approach since it mimics the repulsive forces between side chains that may be due to steric or short - ranged electrostatic potentials . consequently , we use for this part of the interaction the two - dimensional xy - model hamiltonian : @xmath12 with the coupling constant @xmath13 and @xmath14 denoting a summation over nearest neighbors . for simplicity , the length of the rotors ( = side chains ) @xmath15 is set to unity while the angle @xmath16 denotes the relative orientation of the side chains @xmath8 and @xmath17 . combining the above models eq . ( [ eq : potts_model ] ) and eq . ( [ eq : xy_model ] ) yields the hybrid model that is used to describe the surface - attached polymer complexes of the extracellular matrix : @xmath18 the two - dimensional potts model is known to undergo a second order phase transition for @xmath0 @xcite while the xy - model shows a kosterlitz - thouless transition @xcite . for @xmath1 our model eq . ( [ eq : hybrid_model ] ) can be interpreted as a 3-state potts model with a randomly varying coupling strength @xmath19 between neighbouring potts spins @xmath8 and @xmath17 . as this local coupling may become very small and even zero , the nature of the phase transition of the hybrid model may be influenced by _ vacancies _ that are known to drive the transition of the ( @xmath0)-state potts model to a first order behavior @xcite . given this variety of phase behaviors , it appears difficult to predict the system s behavior _ a priori_. per rotor shows a sigmoidal increase with temperature @xmath20 for the pure xy - model [ eq . ( [ eq : xy_model ] ) ] , the pure @xmath21-state potts model [ eq . ( [ eq : potts_model ] ) ] , and the hybrid model [ eq . ( [ eq : hybrid_model ] ) ] . the transition temperature @xmath22 for the latter appears to be shifted towards lower values while the slope in the transition region is strongly increased . full lines are guides to the eye.,width=264 ] for simulations of the hybrid model eq . ( [ eq : hybrid_model ] ) we relied on the monte carlo ( mc ) method ( see , e.g. , @xcite for an introduction ) . far away from the critical temperature , we employed the metropolis mc algorithm @xcite . in each step , a rotation of each rotor ( side chain ) by some angle @xmath23 and a change of its potts level ( height ) were proposed and this combined move was accepted or rejected for each rotor according to the metropolis criterion . close to the critical temperature , the standard metropolis mc suffers from critical slowing down , i.e. it becomes difficult to generate enough statistically independent configurations for a reliable statistics . to overcome this problem , we used a cluster mc approach @xcite , i.e. a modification of wolff s cluster algorithm @xcite . here , one deals with two degrees of freedom , one discrete ( potts level @xmath3 ) and one continuous ( orientation angle @xmath4 ) . thus , two different kinds of clusters are grown and flipped in each step one with respect to @xmath3 ( embedded in an ensemble of fixed @xmath24 s ) and another one with respect to @xmath4 ( embedded in an ensemble of fixed @xmath7 s ) . the treatment of the potts degree of freedom has to be carried out carefully as ferromagnetic and anti - ferromagnetic bonds can occur between neighboring rotors . in the beginning , a pair of values for @xmath3 is chosen randomly , i.e. ( 0,1 ) , ( 0,2 ) , or ( 1,2 ) , respectively . rotors in the remaining potts state are kept unchanged , i.e.,the identity operator is applied , and thus do not contribute to the ongoing cluster growing step . the effective ( local ) coupling between rotors @xmath8 and @xmath17 is @xmath25 . for @xmath26 a ferromagnetic bond is established with probability @xmath27 between rotors on the same potts level . if @xmath28 , an anti - ferromagnetic bond is formed with probability@xmath29 between rotors on different potts levels . when the cluster is grown , all potts levels of the involved rotors are switched according to the initially chosen pair of @xmath3-values ( e.g. 0 @xmath30 1 ) . following wolff s embedding trick for growing and flipping clusters in the xy - model @xcite the rotors are projected onto the @xmath31-axis . this results in an ising model of the @xmath31-components @xmath32 with random couplings . these effective spins are connected by a bond with probability @xmath33 \right).$ ] please note that the potts states of the rotors are accounted for by @xmath34 meaning that bonds are only formed between rotors on the same potts level . finally , when the cluster is grown , all @xmath31-components of the constituing rotors are inverted . to study the thermodynamic properties of a system of interacting rotors modelled by the hamiltonian eq . ( [ eq : hybrid_model ] ) with @xmath1 we conducted extensive mc simulations using the method described in the preceeding section . the linear system size @xmath35 ( i.e. using @xmath36 rotors ) was varied in the range @xmath37 in order to apply a finite - size scaling analysis . we have concentrated here in particular on the inner ( potential ) energy @xmath38 per rotor and the transition temperature @xmath22 . below and above the critical temperature @xmath22 shows only a single maximum for a system with @xmath39 , while for @xmath40 a slight double - peak structure is observed . ( b ) indeed , this double - peak structure at @xmath40 is even better visible for small systems and seems to subside for larger system sizes.,width=264 ] metropolis mc simulations of small systems revealed significant differences in the behavior between the hybrid model and the two constituting models ( potts and xy ) . fig . [ fig : fig02 ] shows the variation of @xmath38 with temperature for all three models . obviously , the transition point @xmath22 of the hybrid model is much lower than those of the constituting models . furthermore , the transition region is much steeper , suggesting a first - order transition . at this point , however , it can not be determined if the slope really diverges at @xmath22 which would be a criterion for a discontinuous phase transition . this behavior would be qualitatively different from that shown by the constituting models , yet would agree with the random potts model @xcite . please note that for @xmath41 the energy @xmath38 of the hybrid model resembles essentially that of the corresponding pure xy - model . thus , the potts degree of freedom seems to be frozen in this regime while all excitations occur with respect to the xy - degree of freedom . $ ] of the minimum between the double peak in @xmath42 appears to vanish with system size @xmath35 , thus indicating a second - order phase transition . ( b ) in agreement with this , the phase transition temperature shows a scaling @xmath43 . ( c ) the tendency of the 4@xmath44-order cumulant towards a value @xmath45 is also in support of a second - order phase transition.,width=264 ] for a more thorough investigation , we used the cluster algorithm described in the preceeding section . in contrast to the standard metropolis mc method it allowed for extensive simulations of large systems close to @xmath22 due to a reduced dynamic exponent . we focussed on the inner energy @xmath38 as we found no proper order parameter to follow the phase behavior of the hybrid model . as can be seen from fig . [ fig : fig03]a , the distribution @xmath42 shows pronounced single peaks for @xmath46 and @xmath47 , while for @xmath40 a double - peak seems to emerge . in fig . [ fig : fig03]b , the distribution @xmath42 is shown for three different system sizes at @xmath48 . the distributions were obtained by _ reweighting _ @xcite of distributions close to @xmath22 . our numerical data indeed highlight a double - peak structure in @xmath42 , indicating the coexistence of two phases and thus a first - order phase transition . to extrapolate if this finding persists in the thermodynamic limit , we first considered the logarithm of the relative gap depth @xmath49 $ ] at @xmath48 for growing system sizes @xmath35 . for a first - order transition , this quantity is supposed to grow with @xmath35 . in the hybrid model studied here , however , @xmath50 for @xmath51 ( fig . [ fig : fig04]a ) . thus , the energy distribution assumes a gauss - like shape in the thermodynamic limit , consistent with a second - order phase transition with only a single phase existing in the system at @xmath22 . considering the finite - size scaling of the transition temperature @xmath22 with the inverse system size @xmath52 supports this notion since we find @xmath53 as anticipated for a second - order transition ( fig . [ fig : fig04]b ) . further evidence for a second order phase transition comes from the 4@xmath44 order cumulant of the energy distribution @xcite @xmath54 for a gaussian distribution , this quantity takes on the value @xmath55 while distributions with a double - peak structure are characterized by @xmath56 . while a small dip in @xmath57 is observed for small system sizes ( fig . [ fig : fig04]c ) , it subsides for large @xmath35 and only very small deviations from @xmath45 are seen . in the thermodynamic limit , one thus expects @xmath55 , i.e. a gauss(-like ) distribution for all temperatures . the phase behavior of the system above and below the critical temperature is shown in fig . [ fig : fig05 ] . while for @xmath41 the rotors ( = rigid side chains ) are ordered ( fig . [ fig : fig05]a ) , they assume random orientations for @xmath58 . the mean potts level ( = the average height of the rigid side chains ) in both cases is @xmath2 ( fig . [ fig : fig05]c ) , yet for @xmath41 collective fluctuations of the rotors to the three degenerate potts levels are observed . given the small transition probabilities for these height fluctuations for fixed temperatures ( which we circumvented via the mc cluster algorithm ) the dwell time at an individual potts levels may be very large . and ( b ) @xmath58 . rotor orientations are indicated by arrows , potts levels @xmath59 are labeled in black , dark grey and light grey , respectively . ( c ) probability distribution of the mean potts level in the system @xmath60 . in the ordered state at @xmath41 ( dashed ) , all potts states @xmath59 are populated with similar probabilities while for @xmath40 ( dash - dotted ) and @xmath58 ( full ) the state @xmath61 is clearly favored.,width=264 ] in this work , we have proposed a model for a system of surface - attached polymers with rigid side chains that is found , e.g. , in the extracellular matrix of mammalian cells . to reduce the computational effort when simulating large systems of end - grafted flexible polymers with rigid side chains , a coarse grained approach has been chosen combining two well - known models of statistical physics , namely the potts and the xy - model . the thermodynamics of the hybrid model , given by the hamiltonian eq . ( [ eq : hybrid_model ] ) , has been studied in monte carlo simulations and a second - order phase transition was found . given our results , it seems reasonable to call the hybrid model rather a modified potts model than a modified xy - model as the phase behavior of the system is dominated by the 3-state potts model : at low temperatures , the rotors essentially condensate into a single potts state . this corresponds to the ( three - fold degenerate ) ground state of the potts model showing long - range order with respect to @xmath3 . excitations in this regime arise due to perturbations in the ordered state of the xy - model . at high temperatures , the influence of the xy degree of freedom becomes manifest as a randomized coupling of the potts spins which includes the existence of vacancies that may drive the transition of the 3-state potts model from second to first order @xcite . indeed , at first glance , the transition of the hybrid model appears to change character ( cf . [ fig : fig02 ] and fig . [ fig : fig03 ] ) . only more thorough investigations revealed the transition to be still continuous , i.e. to remain second order . thus , the vacancies evoked by the random coupling drive the transition towards first order , but do not suffice to fully change its character . coming back to the original problem that inspired the model , i.e. the dynamics of the extracellular matrix , the observed ordering of the system for @xmath41 may actually be used by a cell to change the thickness of its protective layer . a cell may use the transition to the regime @xmath41 to freeze the extracellular matrix in a desired homogenous ordered state / height that is not necessarily thicker than in the disordered regime . thus , speaking in terms of the original problem , the cell may tune the thickness of its extracellular matrix by tuning the interactions of the ( polyelectrolytic ) aggrecan side chains , thereby driving the system through the phase transition . it will be interesting to experimentally manipulate the aggrecan interaction by changing ion conditions for the extracellular matrix and thereby to induce the above described phase transition . this work was supported by the institute for modeling and simulation in the biosciences ( bioms ) in heidelberg . mh acknowledges financial support by the helmholtz alliance _ sbcancer_. yd thanks the alexander von humboldt foundation . , , , * * , ( ) , http://eutils.ncbi.nlm.nih.gov / entrez / eutils / elink.fcgi?cmd = p% rlinks&dbfrom = pubmed&retmode = ref&id=17930056[http://eutils.ncbi.nlm.nih.gov / entrez / eutils / elink.fcgi?cmd = p% rlinks&dbfrom = pubmed&retmode = ref&id=17930056 ] .

we present a simple yet generic model for the behavior of a system of many surface - attached flexible polymers with rigid side chains . beyond its potential application in describing the dynamics of the extracellular matrix of mammalian cells , the model itself shows an interesting phase transition behavior since the underlying models ( a two - dimensional potts model and a xy - model ) undergo different phase transitions .

introduction model simulation method results conclusions

arxiv

quantum annealing ( qa ) aims to solve computational problems by using a guided quantum drive . the dynamics is generated by a time - dependent hamiltonian along a trajectory that ends at a final target hamiltonian whose ground state contains the solution of the problem @xcite . qa is based on the adiabatic theorem , which guarantees that if the hamiltonian is changed sufficiently slowly , transitions to excited states are suppressed during the adiabatic evolution , thus preparing states that are close to the target ground state . unfortunately , the adiabatic condition that ensures that the system remains in the instantaneous ground state leads to long time scales for the solution of the hard computational problem . the adiabatic trajectory is not the only path to reaching the ground state of a final hamiltonian encoding the solution of the computational problem . more generally , one could imagine many other paths , including those where the hamiltonian is varied rapidly , that land at the desired state or , of practical interest , reach low energy states . in fact , it has been already found that for certain hard instances of problems , fast , nonadiabatic paths can sometimes prevent the system from getting stuck at local minima , thus improve the search results @xcite . the variational quantum algorithm ( vqa ) is an example where one searches for such possible paths , using optimization of the outcome via the variation of a fixed number of parameters . a hybrid machine , combining classical optimization and quantum evolution , optimizes the variational parameters . such hybrid variational approaches have proven useful in the context of quantum state preparation @xcite . recently , ref . @xcite introduced a variational quantum eigensolver ( vqe ) for applications in quantum chemistry . this idea was further explored in @xcite and experimentally tested in @xcite . in a related approach @xcite , farhi _ et al . _ introduced a quantum approximate optimization algorithm ( qaoa ) for combinatorial optimization problems based on a parameterized square - pulse ansatz for dynamical evolution of the solver . in this paper , we make a connection between vqa and optimal control theory @xcite . vqa is essentially an adaptive feedback control @xcite of a quantum system with the objective function encoding the solution of a computational problem , see fig . [ fig : vqa](a ) . it utilizes a hybrid system comprised of a classical computer that searches for the optimal variational protocol using measurements done on a quantum machine , which generates the final states for variational protocols , via a closed - loop learning method @xcite . using pontryagin s minimum principle of optimal control , we show that the optimal protocol for vqa has a `` bang - bang '' form . our results put the bang - bang ansatz of qaoa on a rigorous ground in contrast to vqa with a continuous - time evolution . a comparison of the performance of the optimal nonadiabatic bang - bang protocol with quantum adiabatic algorithm ( qaa ) demonstrates that the former reduces error in the final state by orders of magnitude . furthermore , we perform a quantitative analysis of the characteristics of these optimal protocols . we numerically find a system - size independent distribution function for the duration of individual pulses , which may facilitate the development of effective algorithms for the classical optimizer . consider a computational optimization problem such as finding a sequence of @xmath0 bits that minimizes a certain function of all of the bits . to solve this problem with vqa , we consider a system of @xmath0 qubits with a parameterized hamiltonian @xmath1 generically , we can cast the problem into generating a state @xmath2 that minimizes a certain cost function such as the expectation value of an operator @xmath3 with respect to @xmath2 . a common example is finding the ground state of a disorderd classical ising hamiltonian @xcite , where the operator @xmath3 is a hamiltonian diagonal in the computational basis . in the context of quantum chemistry , vqe considers the operator @xmath3 to be the hamiltonian of a molecule @xcite . the essence of vqa , as depicted in fig . [ fig : vqa ] , is finding the time - dependent parameters @xmath4 over a time period @xmath5 such that @xmath6 minimizes a cost function @xmath7 . generically , the controls @xmath4 belong to a permissible set determined by the experimental setup . a common such set is given by simple bounds as seen in eq . the ideal solution could be a unique state @xmath8 [ as depicted in fig.[fig : vqa](b ) ] that is the ground state of the target hamiltonian or more generally a set of states in the hilbert space with an optimal figure of merit . one can either fix the initial state @xmath9 or add it to the list of the variational parameters ( here we fix it motivated by experimental constraints ) . generally , the longer the total time @xmath5 , the closer we can get to an ideal solution . one way to view this is to consider the reachable set of all the final states one can reach by using one of the infinite number of permissible controls . the reachable set , naturally , grows with @xmath5 ( in fact , if @xmath10 is allowed , the reachable set for @xmath11 is strictly a subset of the reachable set for @xmath12 ) . as seen in fig . [ fig : vqa](b ) , there could be a critical time beyond which the reachable set includes the target state and an exact solution is possible . there is no advantage in increasing @xmath5 beyond this critical time . generically , for smaller @xmath5 , where the reachable set does not include the target state , the optimal protocols are highly constrained as they should prepare the closest point(s ) of the reachable set to the target . for times longer than the critical time mentioned above , we expect an infinite number of protocols to produce the target and the evolution has extra time to meander in the hilbert space . our strategy is to fix @xmath5 and find the best variational protocol @xmath4 . if the solution is not acceptable , we can increase @xmath5 . next we present how pontryagin s minimum principle from optimal control theory determines the form of optimal @xmath4 functions . the parameters in hamiltonian are generically constrained by their range : @xmath13 during the evolution @xmath14 . . implies that , by assumption , each @xmath15 can be tunned in the above range independently of the values of the other control parameters . for fixed initial state @xmath9 , the coupling constants @xmath4 uniquely determine the final wave function . consequently the cost function , which we take as an arbitrary function of the final state , is a functional of @xmath4 @xmath16={\cal f}(|\psi(t)\rangle).\ ] ] the pontryagin s minimum principle @xcite is directly applicable here . briefly , this theorem states that for a set of dynamical variables @xmath17 evolving from given initial values @xmath18 with the equations of motions @xmath19 , where @xmath20 are a set of control functions , the control functions @xmath21 that minimize an arbitrary function @xmath22 $ ] of the final values of the dynamical variable satisfy @xmath23 at any point in time and for each of the control functions . the optimal - control hamiltonian is defined as @xmath24 for conjugate variables @xmath25 that evolve as @xmath26 with boundary conditions @xmath27 $ ] . here the `` @xmath28 '' superscript indicates the optimal solution corresponding to @xmath21 . an important consequence of eq . is that if the equations of motion for @xmath17 , and consequently the optimal - control hamiltonian @xmath29 , are linear in @xmath30 , generically , the optimal protocol is bang - bang , i.e. , at any time during the evolution we have @xmath31 or @xmath32 . this follows from the fact that at any point in time we need to choose @xmath15 to minimize @xmath33 . if the sign of the coefficient of @xmath15 in the optimal - control hamiltonian is positive ( negative ) , we should then choose the smallest ( largest ) @xmath15 from the permissible range . in other words , the optimal protocol for each control function involves a sequence of sudden jumps between its minimum and maximum permissible values . the only caveat for the above argument is the possibility that the coefficient of a particular @xmath15 in @xmath33 vanishes over a finite interval ( since the sign of this coefficient determines whether we should choose the minimum or maximum value ) . in the quantum mechanical context , if the physical hamiltonian is linear in the controls , the equations of motion and consequently the optimal - control hamiltonian will also be linear , giving rise to bang - bang protocols as verified in several recent studies on optimal topological quantum computing @xcite . to find the protocol @xmath20 that minimizes the cost function in our case , we expand the wave function in a complete orthonormal basis , e.g. , the computational basis @xmath34 as @xmath35 and treat the real and imaginary parts of the amplitudes @xmath36 as dynamical variables , which evolve according to the schrdinger equation @xmath37,\\ \dot{a}^i_z&=&{1\over 2}\sum_{\alpha , z ' } g_\alpha \left[-\left(h_\alpha^{zz'}+h_\alpha^{z'z}\right)a^r_{z ' } -i\left(h_\alpha^{zz'}-h_\alpha^{z'z}\right)a^i_{z ' } \right],\end{aligned}\ ] ] where @xmath38 and @xmath39 . clearly , these equations of motion are linear in the control functions @xmath15 and the cost function is a function of only the final values of the dynamical variables . thus , the argument above holds and the optimal protocol is generically bang - bang regardless of the number of variational parameters . from practical point of view , it is important to assess the validity of the closed system findings in the presence of decoherence . again , a straightforward application of the pontryagin principle extends the above results for a closed system evolution to an open quantum system with markovian dynamics described by a lindblad equation @xmath40+\sum_\beta f_\beta(t)\left(2f_\beta \ \rho \ f^\dagger_\beta-\left\{f^\dagger_\beta f_\beta,\ \rho\right\}\right ) \label{eqlindblad}\ ] ] where the optimal protocol @xmath41 , if controllable , are of type bang - bang . this is due to the linearity of the dynamical equation ( [ eqlindblad ] ) . a decoherence operator @xmath42 can represent either noise in the hamiltonian parameters ( in which case , @xmath42 is hermitian @xcite ) or an engineered bath @xcite . in the former case , @xmath43 s are constant rates of noise processes and in the latter , @xmath44 s are control knobs that the pontryagin s minimum principle says should vary in bang - bang form for an optimal protocol . in the rest of the paper , we only focus on closed systems schrdinger dynamics . we now focus on a canonical problem in combinatorial optimization , namely the sherrington - kirkpatrick ( sk ) ising spin glass @xcite with the energy function @xmath45 where @xmath46 and @xmath47 are independent gaussian random variables with zero mean and variance @xmath48 , and each @xmath49 spin can take the values @xmath50 . the goal is to minimize @xmath51 over all the @xmath52 spin configurations . a multitude of practical combinatorial optimization problems map to this model . the computational cost of finding the minimum with classical algorithms is exponential in @xmath53 . in analogy with the simple instances of quantum annealing , we focus on the case with only one control function @xmath54 and use the following parameterized hamiltonian : @xmath55b,\ ] ] with the operator @xmath56 representing a transverse field , which generates quantum fluctuations . for the initial state , we choose the ground states of @xmath57 . it is easy to prepare product state @xmath58 commonly used in other schemes such as the qaa . here @xmath59 and @xmath60 . we would like to minimize the cost function @xmath61 . in the adiabatic scheme , a smooth ramp such as @xmath62 is applied for @xmath14 and we can generate large overlap with the ground state of @xmath51 in the limit of large @xmath5 . here , we allow for arbitrary time dependence of the control function in the fixed range @xmath63 . according to the general argument above , the optimal solution is bang - bang . as discussed in the introduction , in vqa a classical optimization algorithm commands a quantum system to find the optimal protocol variationally from measurement of the cost function for many protocols . this requires many repetitions and it is to our advantage to use the shortest possible time @xmath5 for which the final state has an acceptable overlap with the ground state of @xmath51 ( projective measurement is ultimately used in generating the ground state ) . in the adiabatic scheme , we only need one shot but there are important restrictions from the small energy gaps along the adiabatic trajectory ( vqa is insensitive to instantaneous ground states and spectral gaps ) , which can lead to exceedingly long time scales , over which quantum coherence can not be even approximately sustained . furthermore , the presence of noise or modulation in the control fields places important limitations on adiabatic schemes due to the emergence of the recently proposed noise - induced anti - adiabaticity in the long - time limit @xcite . given the limitations of adiabatic scheme , a quantum approximate algorithm has been introduced for solving combinatorial optimization problems @xcite in the spirit of vqa . the algorithm of ref . @xcite uses an ansatz @xmath64 where the evolution operators are given by @xmath65 and @xmath66 . the integer @xmath67 is a parameter characterizing a variational ansatz . for a given @xmath67 , the goal of the algorithm is to find a set of variational parameters that minimizes the expectation value of @xmath68 , which ensures that the state @xmath69 approaches the ground state of @xmath51 . physically , the ansatz describes time evolution for a total time @xmath70 , and a sequence of sudden switching between the hamiltonians @xmath57 and @xmath51 . while this ansatz with a finite @xmath67 is an intelligent guess , the result that we derived using pontryagin s minimum principle implies that given bounded independent control of hamiltonian terms , the ansatz ( [ eq : ansatz ] ) is the true choice for a vqa approach to optimization . so far we have not discussed how large @xmath67 is or even if it is finite . this requires an analysis of the characteristic time scales of the pulses , which we carry out in what follows . we start by verifying for small system sizes and short annealing times that the optimal annealing protocol is indeed bang - bang , by using a metropolis monte carlo ( mc ) algorithm , which makes no assumptions about the nature of the protocol . we divide the total time @xmath5 into @xmath71 slices and use a piece - wise constant protocol with @xmath71 pieces of duration @xmath72 . the method approaches an unbiased optimization , i.e. , it explores all permissible controls and chooses the optimal one , if the protocols obtained converge upon increasing @xmath71 . we then proceed by carrying out a mc simulation starting from random initial protocols , without any assumption regarding the bang - bang nature of the protocol . in each step , we slightly change the control parameter @xmath73 in a randomly chosen discretized time interval . if the cost function gets smaller , we accept the attempt ; if the cost function gets larger , we accept the attempt with probability @xmath74 , where @xmath75 is a fictitious temperature that is gradually reduced to zero . in fig . [ fig : protocol](a ) , we show the optimal protocol obtained from such mc simulation for a fixed instance of hamiltonian ( [ eq1 ] ) with @xmath76 spins and total time @xmath77 . ) with @xmath76 spins and total annealing time @xmath77 . different colors represent different initial protocols . the plots are for @xmath78 , but the optimal protocol does not change upon increasing @xmath71 . ( b ) a typical protocol obtained for a given instance of hamiltonian ( [ eq1 ] ) with @xmath76 spins and @xmath79 , using a classical optimization solver . we start from a uniform initial protocol with @xmath71 slices such that @xmath80.,width=328 ] indeed , the mc simulation converge to a bang - bang protocol for different initial protocols in agreement with the pontryagin s minimum principle . despite the convergence for short total time , the mc simulations often fail to converge for longer times and larger systems , signaling the difficulty of implementing vqa without any a priori knowledge about the form of the optimal protocol . however , based on the mathematical proof of the bang - bang nature of the optimal protocols , we can parameterize the protocol similar to qaoa @xcite and use the durations of the pulses as variational parameters to be optimized with the interior - point minimization method ( ipmm ) , increasing @xmath67 to achieve convergence . we have checked that ipmm results are indeed in agreement with mc results , e.g. , fig . [ fig : protocol](a ) ( it also runs much faster ) . in fig.[fig : protocol](b ) , we show a typical optimized protocol obtained with ipmm for a certain instance of hamiltonian ( [ eq1 ] ) with @xmath76 spins and @xmath79 . guided by mc results , we choose around @xmath81 variational parameters , which proved to be adequate ( we converged to a smaller number of bangs than we allowed in the ansatz ) . we now turn to the critical question of the time scales of the pulses . we observe numerically and then argue analytically that the typical time scale of each bang is independent of the system size , and is only determined by some characteristic energy scale of hamiltonian ( [ eq1 ] ) . therefore , from a complexity theory perspective , this result implies that the hardness of the optimization problem should translate into the number of pulses and/or the hardness of the protocol optimization . in fig . [ fig : distribution ] we plot the distribution of the time scales of each bang @xmath82 for system sizes @xmath83 and 7 . for each system size , we fix the total annealing time to be @xmath79 , and average over 50 instances of the hamiltonian ( [ eq1 ] ) . and 7 . the total annealing time is fixed to be @xmath79 , leading to an average success rate around @xmath84 , depending on the system sizes . each curve is averaged over 50 instances of hamiltonian ( [ eq1]).,width=302 ] we find that the distributions for the bang time collapse for different system sizes , and peak at almost the same value . this observation suggests a universal average distribution of the bang times for the near optimal protocols , and a typical time scale ( peak or average value ) that is independent of the system size . although we have only considered a few system sizes , the dependence on @xmath53 is extremely weak and we expect our results to extrapolate to large @xmath53 . as shown in the appendix , the typical time scale of the interval @xmath85 is given by the first root for @xmath86 of the following function : @xmath87 \label{eqn20}\ ] ] where @xmath88 is the energy of configuration @xmath89 , @xmath90 is the configuration @xmath89 with the @xmath91-th spin flipped , @xmath92 is the conjugate momentum of @xmath93 , and @xmath94 is the beginning of a bang with @xmath95 . while we can not derive analytically the average first root for @xmath86 from eq . , one can see that the only time dependence in eq . ( [ eqn20 ] ) within the current interval is in @xmath96 . thus , the energy difference @xmath97 , which has zero mean ( as both @xmath88 and @xmath98 clearly have zero mean ) and variance @xmath99 , sets the characteristic time scale proportional to @xmath100 observed in fig . [ fig : distribution ] . importantly , this time scale is finite , distinguishing our bang - bang type optimal protocol from the trotterization of generic protocols , where the duration of individual pulses must be taken to zero . while the above scaling argument suggests system size independence , an analytical derivation of the distribution in fig . [ fig : distribution ] has remained elusive . finally we comment on the performance of our protocols . the cost function we minimize is the expectation value @xmath101 . minimizing the energy expectation value results in larger overlap with the ground state of @xmath51 . as expected , the time scales for our protocols are significantly shorter than those of the adiabatic algorithm with similar success rate . a comparison between the optimal bang - bang protocols and linear ramps @xmath102 is shown in fig . [ fig : error ] . the energy @xmath101 and the errors @xmath103 are averaged over the 20 instances ( out of 50 generated realizations ) with the highest success rates , for the optimal bang - bang protocol and linear qaa ramp respectively . remarkably , we find that for the same amount of time , the nonadiabatic bang - bang protocol on average beats the linear ramp ( commonly used in qaa ) by orders of magnitude . of course , in practice one needs to include the overhead of searching for the optimal solution , and understand how it scales as function of system size . we have shown that the optimal vqa with bounded linear control parameters is of the bang - bang form . we verified this prediction by finding numerically the optimal protocol that minimizes the energy of a sk spin glass . moreover , we find that the optimal nonadiabatic bang - bang protocol reduces error by orders of magnitude when compared to qaa within the same running time . we argue that the characteristic time scale between bangs is fixed by the energy scales in the problem and is independent of system size , which we confirmed numerically . the finding that the time scale for each pulse is size - independent would constrain the number of variational parameters to scale with the running time allotted for the protocol . our results , that the bang - bang protocols are optimal and the bang time is size - independent , inform the search for effective hybrid classical - quantum machine learning schemes for solving combinatorial optimization problems . here we show that the pontryagin s minimum principle can tell about not only the form of optimal solution for vqa but also when the pulses should be switched on and off . we analyze the dynamical equations for the sk model studied in the main text . using the computational basis @xmath104 , we represent the wave function as @xmath105 . the initial state with all the spins in the @xmath106 direction corresponds to @xmath107 , and the schrdinger equation reads @xmath108\sum_{k=1}^n a_{\bar { z}(k)}(t),\ ] ] with @xmath109 , where @xmath110 represents a flipped spin with respect to @xmath111 and @xmath88 is the energy function we would like to minimize . in terms of the real and imaginary parts of @xmath112 , we can then write @xmath113\sum_{k=1}^n i_{\bar { z}(k)}(t),\\ \partial_t i_z(t)&=&-g(t)c_zr_z(t)-[1-g(t)]\sum_{k=1}^n r_{\bar { z}(k)}(t).\end{aligned}\ ] ] introducing conjugate momenta @xmath114 and @xmath115 respectively for the real and imaginary parts of @xmath36 , the explicit form of the optimal - control hamiltonian is given by @xmath116\\ & + [ 1-g(t)]\sum_{k=1}^n \left[p_z(t)i_{\bar { z}(k)}(t)-q_z(t)r_{\bar { z}(k)}(t)\right]\bigg\}. \end{split}\ ] ] the equations of motion for the conjugate momenta are @xmath117 and @xmath118 , which can be written explicitly as @xmath119\sum_{k=1}^n q_{\bar { z}(k)}(t),\\ \partial_t q_z(t)&=&-g(t)c_zp_z(t)-[1-g(t)]\sum_{k=1}^n p_{\bar { z}(k)}(t),\end{aligned}\ ] ] where we have used the relationships @xmath120 and @xmath121 . the cost function @xmath122=\sum_z|a_z(t)|^2c_z\ ] ] leads to the following boundary conditions at @xmath123 for the conjugate momenta : @xmath124 clearly @xmath54 uniquely determines @xmath36 . from @xmath125 and the expression above , we can find @xmath126 and @xmath127 , solve the equations of motion backward in time and determine the conjugate momenta as a function of time . therefore , @xmath54 also uniquely determines @xmath114 and @xmath115 . the potryagin s minimum principle states that the optimal protocol @xmath128 satisfies @xmath129 where @xmath130 are the corresponding optimal solution . as argued in sec . [ sec : pontryagin ] , @xmath128 is bang - bang and can only take two values of @xmath131 and @xmath132 . which value will depend on the sign of @xmath133 given by the expression @xmath134\\ & -\sum_{k=1}^n \left[p^*_z(t)i^*_{\bar { z}(k)}(t)-q^*_z(t)r^*_{\bar { z}(k)}(t)\right]\bigg\}. \end{split } \label{eq : coeff}\ ] ] let us first combine @xmath135 and @xmath136 into one complex momentum @xmath137 as we argued above , the optimal protocol is bang - bang with @xmath138 . in any interval with @xmath85 , we can write @xmath139 where @xmath94 is the beginning of the current bang @xmath95 . the above equation allows for an estimation of the typical time scale of the interval @xmath85 . suppose at some @xmath94 , @xmath54 switches from 0 to 1 , i.e. a bang starts . from the discussion above , we know that at @xmath142 we must have @xmath143 . the time when the bang stops corresponds to the next @xmath86 when @xmath144 . more explicitly , in the interval @xmath85 , eq . ( [ eq : coeff ] ) can be written as : @xmath87,\ ] ] which is exactly eq . in sec . [ sec : numerical ] . p. omalley , r. babbush , i. kivlichan , j. romero , j. mcclean , r. barends , j. kelly , p. roushan , a. tranter , n. ding , b. campbell , y. chen , z. chen , b. chiaro , a. dunsworth , a. fowler , e. jeffrey , a. megrant , j. mutus , c. neill , c. quintana , d. sank , a. vainsencher , j. wenner , t. white , p. coveney , p. love , h. neven , a. aspuru - guzik , j. martinis . rev . a * 6 * , 031007 ( 2016 ) .

we use pontryagin s minimum principle to optimize variational quantum algorithms . we show that for a fixed computation time , the optimal evolution has a bang - bang ( square pulse ) form , both for closed and open quantum systems with markovian decoherence . our findings support the choice of evolution ansatz in the recently proposed quantum approximate optimization algorithm . focusing on the sherrington - kirkpatrick spin glass as an example , we find a system - size independent distribution of the duration of pulses , with characteristic time scale set by the inverse of the coupling constants in the hamiltonian . we numerically demonstrate that our optimal nonadiabatic bang - bang protocols can outperform quantum annealing by orders of magnitude within the same running time . the optimality of the bang - bang protocols and the characteristic time scale of the pulses inform the search for effective hybrid ( classical and quantum ) machine learning schemes for tackling combinatorial optimization problems .

introduction variational quantum algorithm pontryagins minimum principle applied to variational quantum algorithm spin-glass model numerical studies summary pulse duration from the pontryagins minimum principle

arxiv

large graphs and networks are natural mathematical models of interacting objects such as computers on the internet or articles in citation networks . numerous examples can be found in the biomedical context from metabolic pathways and gene regulatory networks to neural networks @xcite . the present work is dedicated to one type of such biomedical network , namely epidemic networks @xcite : such a network models the transmission of a directly transmitted infectious disease by recording individuals and their contacts , other individuals to whom they can pass infection . understanding the dynamic of the transmission of diseases on real world networks can lead to major improvements in public health by enabling effective disease control thanks to better information about risky behavior , targeted vaccination campaigns , etc . while transmissions can be studied on artificial networks , e.g. , some specific types of random networks @xcite , such networks fail to exhibit all the characteristics observed in real social networks ( see e.g.@xcite ) . it is therefore important to get access to and to analyze large and complex real world epidemic networks . as pointed out in @xcite , the actual definition of the social network on which the propagation takes place is difficult , especially for airborne pathogens , as the probability of disease transmission depends strongly on the type of interaction between persons . this explains partially why sexually transmitted diseases ( std ) epidemic networks have been studied more frequently than other networks @xcite . we study in this paper a large hiv epidemic network that has some unique characteristics : it records almost 5400 hiv / aids cases in cuba from 1986 to 2004 ; roughly 2400 persons fall into a single connected component of the infection network . std networks studied in the literature are generally smaller and/or do not exhibit such a large connected component and/or contain a very small number of infected persons . for instance , the manitoba study ( in canada , @xcite ) covers 4544 individuals with some std , but the largest connected component covers only 82 persons . the older colorado springs study @xcite covers around 2200 persons among which 965 falls in connected component ( the full network is larger but mixes sexual contacts and social ones ; additionally , the sexual networks contains only a very small number of hiv positive persons ) . while the large size and coverage of the studied network is promising , it has also a main negative consequence : manual analysis and direct visual exploration , as done in e.g. @xcite , is not possible . we propose therefore to analyze the network with state - of - the - art graph visualization methods @xcite . we first describe the epidemic network in section [ sec : cuban - hiva - datab ] and give an example of the limited possibilities of macroscopic analysis on this dataset . then section [ sec : visual - mining ] recalls briefly the visual mining technique introduced in @xcite and shows how it leads to the discovery of two non obvious sub - networks with distinctive features . the present work studies an anonymized national dataset which lists 5389 cuban residents with hiv / aids , detected between 1986 and 2004 . each patient is described by several variables including gender , sexual orientation , age at hiv / aids detection , etc . ( see @xcite for details . ) the cuban hiv / aids program produces this global monitoring using several sources that range from systematic testing of pregnant women and all blood donations to general practitioner testing recommendations . in addition , the program conducts an extended form of infection tracing that leads to the epidemic network studied in this work . indeed , each new infected patient is interviewed by health workers and invited to list his / her sexual partners from the last two years . the primary use of this approach is to discover potentially infected persons and to offer them hiv testing . an indirect result is the construction of a network of infected patients . sexual partnerships are indeed recorded in the database for all infected persons . additionally , a probable infection date and a transmission direction are inferred from other medical information , leading to a partially oriented infection network . while this methodology is not contact tracing _ stricto sensu _ as non infected patients are not included in the database ( contrarily to e.g. @xcite ) , the program records the total number of sexual partners declared for the two years period as well as a few other details , leading to an extended form of infection tracing . ( see @xcite for differences between contact and infection tracing . ) the 5389 patients are linked by 4073 declared sexual relations among which 2287 are oriented by transmission direction . a significant fraction of the patients ( 44 % ) belong to a giant connected component with 2386 members . the rest of the patients are either isolated ( 1627 cases ) or members of very small components ( the second largest connected component contains only 17 members ) . as the sexual behavior has a strong influence on hiv transmission , it seems important to study the relations between the network structure and sexual orientation of the patients . in the database , female hiv / aids patients are all considered to be heterosexual as almost no hiv transmission between female has been confirmed @xcite . male patients are categorized into heterosexual man and `` man having sex with men '' ( msm ) ; the latter being men with at least one male sexual partner identified during their interview . the distributions of genders and of sexual orientations is given in table [ tab : gender : so ] : the giant component contains proportionally more msm than the full population ; this seems logical because of the higher probability of hiv transmission between men @xcite . .gender and sexual orientation distributions in the whole network and in the giant component [ cols= " > , > , > , > , > " , ] -1.2em this analysis shows that the two groups made of atypical clusters are far from each other compared to their internal distances . this is confirmed by the detection date analysis displayed on figure [ fig : groups : year ] . it appears that the epidemic in the giant component has two separated components . one mostly male homosexual component tends to dominate the recent cases ( note that even typical clusters contain at least 57 % of msm ) , while a mixed component with a large percentage of female patients was dominating the early epidemic , but tends to diminish recently . it should also be noted that this mix component is dominated by the growth of the homosexual component , but seems to decay only slightly in absolute terms . in other words , the reduction should be seen as an inability to control the growth homosexual epidemic rather than as a success in eradicating the heterosexual epidemic . the proposed visual mining method for graphs has been shown to provide valuable insights on the epidemic network . it is based on links between modularity and visualization and leverages recent computationally efficient modularity maximizing methods . future works include the integration of the proposed methods in graph mining tools such as @xcite and its validation on other aspects of epidemic networks analysis . clmenon , s. , de arazoza , h. , rossi , f. , tran , v.c . : hierarchical clustering for graph visualization . in : proceedings of xviiith european symposium on artificial neural networks ( esann 2011 ) . bruges , belgique ( april 2011 ) , to be published kwakwa , h.a . , ghobrial , m.w . : female - to - female transmission of human immunodeficiency virus . clinical infectious diseases : an official publication of the infectious diseases society of america 36(3 ) ( february 2003 ) noack , a. , rotta , r. : multi - level algorithms for modularity clustering . in : sea 09 : proceedings of the 8th international symposium on experimental algorithms . . 257268 . springer - verlag , berlin , heidelberg ( 2009 ) rothenberg , r.b . , woodhouse , d.e . , potterat , j.j . , muth , s.q . , darrow , w.w . , klovdahl , a.s . : social networks in disease transmission : the colorado springs study . in : needle , r.h . , coyle , s.l . , genser , s.g . , trotter ii , r.t . social networks , drug abuse , and hiv transmission , pp . 318 . no . 151 in research monographs , national institute on drug abuse ( 1995 ) varghese , b. , maher , j. , peterman , t. , branson , b. , steketee , r. : reducing the risk of sexual hiv transmission : quantifying the per - act risk for hiv on the basis of choice of partner , sex act , and condom use . sexually transmitted diseases 29(1 ) , 3843 ( january 2002 )

we show how an interactive graph visualization method based on maximal modularity clustering can be used to explore a large epidemic network . the visual representation is used to display statistical tests results that expose the relations between the propagation of hiv in a sexual contact network and the sexual orientation of the patients .

introduction cuban hiv/aids database conclusion

arxiv

many theoretical models exist to describe the valence parton distribution amplitude ( da ) of the @xmath1 , the @xmath0 ( both longitudinally and transversally polarized ) , and other mesons . in particular the pion , the lightest bound state within quantum chromodynamics ( qcd ) , provides a suitable `` laboratory '' for testing new ideas and techniques to catch the main ingredients of the underlying quark - gluon dynamics entering exclusive qcd processes ( see @xcite for reviews ) . in this paper we will present an amplification of the recent analysis by one of us in @xcite , continued in @xcite , for the pion and consider its extension to the longitudinally polarized @xmath0 meson ( @xmath2 ) . the new mode of thought in @xcite is based on the ubiquitous phenomenon of synchronization ( sync for short ) in complex systems and we will expand the status of this subject towards a deeper understanding of the meson das . in this way , we will redetermine the @xmath2 da using qcd sum rules with nonlocal condensates ( nlc ) within the approach developed in @xcite . our primary findings to be discussed later can be summarized as follows : first , we provide more details about the structure of the sync - inspired shorttailed ( i.e. , endpoint - suppressed ) platykurtic ( pk ) @xmath1 da , proposed in @xcite , and quantify the uncertainties of its expansion coefficients . this da is a kind of chimera state because it mimics within the nlc - based approach characteristics pertaining to the dynamical chiral symmetry breaking ( dcsb ) described in terms of dyson schwinger equations ( dse ) @xcite . in particular it conserves nlc - generated endpoint suppression of the @xmath1 da combining it with a broad downward concave shape in the central region . second , using similar mathematical techniques , we obtain within the reliability range of the nlc approach a regime of das for the @xmath2 meson characterized by a shorttailed platykurtic profile . third , we employ statistical measures , like the kurtosis , to classify meson das with respect to their peakedness relative to the tail flatness and heaviness . the rest of the paper is organized as follows . the next two sections ( sec . [ sec : da - qcd ] and sec . [ sec : gegen - repres ] ) discuss the theoretical basis for the description of meson das within qcd . section [ sec : nlc - sr ] sketches the derivation of the @xmath1 and @xmath2 das from qcd sum rules with nlcs . we will then proceed to investigate how meson das can be analyzed in terms of synchronization concepts ( sec . [ sec : sync ] ) . synthetic meson das will be considered in sec . [ sec : synthetic - da ] , where also the important chimera das with a shorttailed platykurtic profile will be presented . finally , sec . [ sec : concl ] will be reserved for the summary of our main results and conclusions . let us consider the pion da , starting with its definition . applying collinear factorization in qcd , the @xmath1 da of leading - twist two , @xmath3 , encodes the distribution of the longitudinal momentum of the pion between its two valence constituents : the quark and the antiquark , with corresponding longitudinal - momentum fractions @xmath4 and @xmath5 , respectively . its momentum - scale dependence stems from the renormalization of the current operator in @xmath6 q(0 ) | \pi(p ) \rangle|_{z^{2}=0 } & & \!\!\ ! \ ! \ ! = if_\pi p_\mu \int_{0}^{1 } dx e^{i x ( z\cdot p ) } \nonumber \\ & & \times \varphi_{\pi}^{(2 ) } \left(x,\mu^2\right ) \ , , \label{eq : pion - da}\end{aligned}\ ] ] where the gauge link @xmath7 = \mathcal{p}\exp [ -ig \int_{0}^{z } dw_{\mu}a_{a}^{\mu}(w)t_{a } ] = 1 $ ] is set equal to unity on account of the lightcone gauge @xmath8 ( @xmath9 ) . the pion da is related to the bethe salpeter wave function @xmath10 by integrating over the transverse parton momentum @xmath11 , i.e. , @xmath12 because the dependence on the momentum scale of any meson da is controlled by the efremov - radyushkin brodsky - lepage ( erbl ) @xcite evolution equation , each meson da can be expressed in terms of the one - loop eigenfunctions of this equation , @xmath13 , with the asymptotic ( asy ) da being given by @xmath14 , and @xmath15 denoting the gegenbauer polynomials of order @xmath16 within the complete and orthonormal basis on @xmath17 $ ] with respect to the weight @xmath18 . thus , one has the ( scale - dependent ) conformal expansion @xmath19 in terms of the nonperturbative coefficients @xmath20 . by virtue of the normalization condition @xmath21 , @xmath22 at any scale @xmath23 . gev@xmath24 , defind in sec . [ sec : gegen - repres ] . the family of the @xmath1 platykurtic das is shown as a narrow shaded strip in red color , obtained with @xmath25 gev@xmath24 at the edge of the nlc regime . the solid lines within both regions denote , respectively , the bms model from @xcite and the pk da discussed in the text and in @xcite . the broken lines show the unimodal dse - db da ( dashed ) , the dse - rl da ( dashed - dotted ) ( both from @xcite ) , and the asymptotic da ( dashed - dotted - dotted ) . the lower panel illustrates various @xmath26 das . the larger blue shaded area contains the family of das obtained in @xcite using qcd sum rules with nlcs and @xmath27 gev@xmath24 , while the narrower strip in its interior indicates the platykurtic regime of these das . the solid lines within each band denote , respectively , the bimodal da from @xcite ( lowest ( blue ) solid line ) and the platykurtic da ( upper solid line ) derived in this work . the lower dashed line represents the da obtained from the dse approach @xcite , whereas the dashed - dotted - dotted line displays again the asymptotic da . all das in both panels refer to the scale @xmath28 gev@xmath24 after two - loop erbl evolution , provided the initial proprietary scale was lower than this . [ fig : pi - rho - das],title="fig:",scaledwidth=40.0% ] gev@xmath24 , defind in sec . [ sec : gegen - repres ] . the family of the @xmath1 platykurtic das is shown as a narrow shaded strip in red color , obtained with @xmath25 gev@xmath24 at the edge of the nlc regime . the solid lines within both regions denote , respectively , the bms model from @xcite and the pk da discussed in the text and in @xcite . the broken lines show the unimodal dse - db da ( dashed ) , the dse - rl da ( dashed - dotted ) ( both from @xcite ) , and the asymptotic da ( dashed - dotted - dotted ) . the lower panel illustrates various @xmath26 das . the larger blue shaded area contains the family of das obtained in @xcite using qcd sum rules with nlcs and @xmath27 gev@xmath24 , while the narrower strip in its interior indicates the platykurtic regime of these das . the solid lines within each band denote , respectively , the bimodal da from @xcite ( lowest ( blue ) solid line ) and the platykurtic da ( upper solid line ) derived in this work . the lower dashed line represents the da obtained from the dse approach @xcite , whereas the dashed - dotted - dotted line displays again the asymptotic da . all das in both panels refer to the scale @xmath28 gev@xmath24 after two - loop erbl evolution , provided the initial proprietary scale was lower than this . [ fig : pi - rho - das],title="fig:",scaledwidth=40.0% ] in our approach @xcite , based on qcd sum rules with nonlocal condensates @xcite , we calculated the moments @xmath29 up to @xmath30 together with their intrinsic theoretical uncertainties at the typical hadronic scale @xmath31 gev@xmath24 @xcite . detailed estimates of the moment uncertainties can be found in @xcite ( see also table [ tab : parameters ] ) ) . this scale represents the average value of the borel parameter @xmath32 in the stability window of the sum rule , notably , @xmath33 $ ] , where @xmath34 is the continuum threshold and @xmath35 is the physical mass of the @xmath0 meson . note that the moments @xmath36 coincide by construction with the central moments @xmath37 = \int_{0}^{1 } dx ( x-\mu[\varphi_\pi])^n \varphi_\pi(x ) = 2^{-n}\langle \xi^n\rangle_\pi \label{eq : central - moments}\ ] ] of @xmath38 , where @xmath39 = \int_{0}^{1 } dx x \varphi_\pi(x ) = \frac{1}{2 } \label{eq : mean}\ ] ] is the mean of the da . using standard techniques ( see , for example , @xcite ) , we extracted from the moments the corresponding conformal coefficients @xmath40 entering eq . ( [ eq : gegen - exp ] ) . these quantities contain nonperturbative information and implicitly depend on the finite virtuality of the vacuum quarks , the latter expressed by means of the nonlocality parameter @xmath41 $ ] gev@xmath42 . it was found that the first two coefficients ( see appendix c in @xcite ) @xmath43 dominate . their values have been calculated with controlled accuracy in @xcite . they read @xmath44 and @xmath45 , whereas the next higher coefficients were computed in the same work as well and found to be much smaller , viz . , @xmath46 , but bearing large uncertainties . their inclusion can add refinements to the method , as we have discussed in @xcite . this apparent hierarchy , with each subsequent coefficient becoming smaller with the order of the conformal expansion , is not following from general principles ; it is an inherent element of our specific approach . indeed , one can even have an inverse ordering of the coefficients @xmath40 see @xcite for such das . in the final analysis , the pion da at the scale @xmath47 gev@xmath24 can be written in the form ( @xmath48 ) @xmath49 \ , , \label{eq : truncated}\ ] ] where the label means bakulev , mikhailov , stefanis @xcite . this simple model probably offers a biased picture of the pion structure but its chief predictions are in good agreement with measurements and various lattice simulations , as detailed in a series of papers @xcite . the reason is that physical observables , like the pion - photon transition or the pion s electromagnetic form factor are given in terms of integrals of the das with smooth coefficient functions . because the leading - order anomalous dimensions of the involved operators in the matrix elements between the meson state and the vacuum are positive ( except @xmath50 ) , the higher coefficients are logarithmically suppressed at large scales , so that only @xmath51 survives this is particularly visible in the inverse moment of the pion da , cf . ( [ eq : inv - mom ] ) , in which the oscillating terms are washed out and strongly suppressed as the momentum increases . the profiles of the meson das , considered in this work , are shown in the upper ( @xmath1 ) and the lower ( @xmath26 ) panel of fig . [ fig : pi - rho - das ] , respectively . to leading logarithmic accuracy , the conformal coefficients @xmath40 are multiplicatively renormalizable and the anomalous dimensions are known in closed form , see , for example , @xcite . at the next - to - leading - order ( nlo ) level , the momentum - scale dependence of the conformal coefficients ( or moments ) is more complicated owing to the mixing of the operators under renormalization @xcite . the diagonal elements of the corresponding two - loop anomalous - dimension matrix coincide with the flavor nonsinglet anomalous dimensions known from deeply inelastic scattering . they have been computed in @xcite , corrected in @xcite , and verified in @xcite . the nlo erbl kernel was calculated in @xcite , while the analytic expressions for the mixing coefficients to obtain the corresponding eigenfunctions were given in @xcite . bear in mind that the next - to - leading - order corrections under a change of scale using a running coupling appear as a two - loop contribution of the eigenvalues and as an @xmath52 correction to the eigenfunctions . a detailed exposition of the erbl evolution of the pion da at the two - loop level , as used in the present work , is provided in appendix d in @xcite . it is convenient to employ another gegenbauer representation ( `` gegenbauer-@xmath53 '' ) proposed in @xcite , notably , @xmath54 where @xmath55 and @xmath56 , @xmath57 , where @xmath58 is the euler beta function . the gegenbauer polynomials @xmath59 form an orthonormal set over @xmath17 $ ] with respect to the weight @xmath60^{\alpha_{-}}$ ] . the difference to the conformal expansion in eq . ( [ eq : gegen - exp ] ) is that the order of the gegenbauer polynomials is not a priori fixed to the value @xmath16 , but is allowed to vary in order to accelerate the reconstruction procedure of meson das on @xmath17 $ ] . however , expansion ( 10 ) is not directly amenable to erbl evolution because the functions @xmath61 are not its eigenfunctions . to evolve @xmath62 , expressed via ( [ eq : da - gen - gegen ] ) , to another scale @xmath63 , one has to project it first onto the conformal basis @xmath64 and then determine @xmath65 and @xmath66 at the new scale . the authors of the works in refs . @xcite find that it is sufficient to include only one coefficient in this expansion , namely , @xmath67 , so that eq . ( [ eq : da - gen - gegen ] ) reduces to @xmath68 \ , . \label{eq : dse - da}\ ] ] below , we will use for our analysis both representations in parallel and present the results in the form @xmath69 and @xmath70 . it is worth bearing in mind that broad das of the form @xmath71 are not well represented by eq . ( [ eq : truncated ] ) which only employs the first two conformal coefficients . to approximate such das with sufficient accuracy , one would have to include a large number of coefficients of order 50 or more , see , for example , @xcite . in this section , we discuss the derivation of the @xmath26 da using qcd sum rules with nonlocal condensates within the scheme developed in @xcite . the main conceptual idea is to apply the sum - rule method in combination with vacuum averages of nonlocal operators @xcite . following this rationale , one can determine the moments @xmath72 of a meson da using a sum rule with nonlocal condensates . in addition , due to the absence of endpoint singularities in the nlc approach , one is able to calculate the inverse moment @xmath73 via an independent sum rule at the same low renormalization scale @xmath74 gev . this moment is an integral characteristic of the pion da and encodes information on the maximum possible weight of the higher - order conformal coefficients . moreover , it is particularly relevant for phenomenological applications because @xmath75 where @xmath76 is the leading - order ( lo ) expression of the pion - photon transition form factor , which has been measured in several experiments from a few gev@xmath24 up to 40 gev@xmath24 @xcite . here the ellipsis represents corrections due to higher eigenfunctions and evolution of the gegenbauer coefficients @xmath40 to the considered scale @xmath77 is assumed ( see @xcite for more details ) . thus , experimental evidence can be used to validate or reject particular pion das . for instance , we know ( see , e.g. , @xcite ) that all existing data demand @xmath78 , implying that the @xmath1 da has to be broader than @xmath79 at accessible momentum values . to obtain the @xmath1 da , we employ the following qcd sum rule ( @xmath80 , @xmath81 gev ) @xmath82 whereas for the @xmath26 da we have @xmath83 with @xmath84 gev and @xmath85 gev . the effective @xmath86-meson state with the decay constant @xmath87 gev and the mass @xmath88 gev comprises the @xmath89 and the @xmath90 mesons and is described by the da @xmath91 . the perturbative contribution to the sum rules is expressed via the perturbative spectral density for which we use the corrected expression published as eq . ( 1 ) in the erratum to @xcite , viz . , @xmath92 \ , , \label{eq : spec - dens - nlo}\ ] ] where @xmath93 for @xmath94 . radiative correction entering the perturbative contribution to the sum rule considered in @xcite . ] the nonperturbative content of the sum rule is contained in the expression @xmath95 where @xmath32 denotes the borel parameter and @xmath34 marks in each case the threshold value which separates the lowest resonance state from higher states . to saturate the sum rules for the first @xmath30 moments of @xmath96 @xcite , we use @xmath97 gev@xmath24 . the various contributions , forming @xmath98 , pertain to the following terms : ( i ) @xmath99 ( four - quark condensate ) ; ( ii ) @xmath100 ( quark - gluon - antiquark condensate ) ( iii ) @xmath101 ( vector quark condensate ) ; ( iv ) @xmath102 ( gluon condensate ) . their explicit expressions can be found in appendix a of ref . the basic assumption here is that higher - order correlations are less important than two - particle correlations ( vacuum - dominance hypothesis @xcite ) . this assumption is employed in order to reduce the four - quark condensate to the product of two - quark condensates ignoring corresponding uncertainties . one notes that the four - quark contribution enters the sum rule for @xmath26 in ( [ eq : srrholterms ] ) with the opposite sign with respect to @xmath1 . as a result , it reduces the relative weight of this condensate in the sum rule entailing smaller values of the da moments in contrast to the pion case . in fact , we found in @xcite ( see table 1 there ) the following relation : @xmath103 with @xmath104 . similarly to the pion da , the da of the longitudinally polarized @xmath0 is defined by the matrix element @xmath105 while the definition of the transversal ( @xmath106 ) @xmath0 da reads @xmath107 where we have again employed the gauge @xmath108 . the @xmath109 da will not be considered in this work . llllllllll model da & @xmath110 & @xmath111 & @xmath112 & @xmath65 & @xmath67 & @xmath113 & @xmath114 & @xmath115 & @xmath116 + asy & @xmath117 & @xmath117 & @xmath118 & @xmath119 & @xmath117 & @xmath120 & @xmath121 & @xmath122 & @xmath123 + @xmath26 @xcite @xmath124 & @xmath125 & @xmath126 & @xmath127 & @xmath128 & @xmath129 & @xmath130 & 0.22 & 0.093 & @xmath131 + @xmath26 @xcite & 0.032(46 ) & -0.038(81 ) & @xmath132 & @xmath133 & @xmath134 & @xmath135 & 0.211(16 ) & 0.088(7 ) & @xmath136 + @xmath137 ( here ) & @xmath138 & @xmath139 & @xmath140 & @xmath141 & @xmath142 & @xmath143 & 0.206(8 ) & 0.087(6 ) & @xmath144 + @xmath26 @xcite & @xmath145 & @xmath146 & @xmath147 & @xmath148 & @xmath149 & @xmath150 & 0.197 & 0.081 & @xmath151 + @xmath26 @xcite & @xmath152 & @xmath153 & @xmath154 & @xmath155 & @xmath156 & @xmath157 & 0.204 & 0.088 & @xmath158 + @xmath26 @xcite & @xmath159 & @xmath117 & @xmath118 & @xmath119 & @xmath159 & @xmath160 & 0.234 & 0.109 & @xmath161 + @xmath26 @xcite @xmath162 & @xmath163 & @xmath164 & @xmath165 & @xmath166 & @xmath117 & @xmath167 & 0.232 & 0.110 & @xmath168 + @xmath26 @xcite @xmath169 & @xmath170 & @xmath171 & @xmath172 & @xmath173 & @xmath174 & @xmath167 & 0.220 & 0.099 & @xmath175 + @xmath176 @xcite & @xmath177 & @xmath178 & 20.49 & @xmath179 & @xmath180 & @xmath181 & @xmath182 & @xmath183 & @xmath184 + @xmath185 @xcite & @xmath186 & @xmath187 & 7.78 & @xmath188 & @xmath189 & @xmath190 & @xmath191 & @xmath192 & @xmath193 + @xmath194 @xcite & @xmath195 & @xmath196 & @xmath197 & @xmath198 & @xmath199 & @xmath200 & 0.251 & 0.128 & @xmath201 + @xmath202 @xcite @xmath203 & @xmath204 & @xmath205 & @xmath206 & @xmath207 & @xmath208 & @xmath209 & 0.280 & 0.151 & @xmath210 + @xmath211 @xcite @xmath212 & @xmath213 & @xmath214 & @xmath215 & @xmath216 & @xmath117 & @xmath217 & 0.237 & 0.114 & @xmath218 ) and is therefore not a precise estimate . ] + @xmath219 @xcite & @xmath220 & @xmath117 & @xmath221 & @xmath222 & @xmath220 & @xmath223 & 0.343 & 0.181 & @xmath224 + @xmath225 @xcite & @xmath226 & & & & & & @xmath227 & & + to construct @xmath228 , we compute the moments @xmath229 up to @xmath30 from the qcd sum rule ( [ eq : srrholconden ] ) and determine from them the corresponding conformal coefficients @xmath40 with @xmath230 . because the moments with @xmath231 turn out to have values close to the asymptotic ones , we can safely set the associated conformal coefficients equal to zero and express @xmath228 in the form of eq . ( [ eq : truncated ] ) . in addition , we use the gegenbauer-@xmath53 expansion , given by eq . ( [ eq : da - gen - gegen ] ) , via the parameters @xmath232 . the accessible regions of these parameters for the longitudinal @xmath0 da , determined within qcd sum rules with nonlocal condensates , are displayed as shaded blue areas closer to the @xmath233 axis in fig . [ fig : rhol ] . the left panel displays the results in the @xmath69 plane , while the right panel contains the analogous results in the @xmath232 plane . we have also depicted in both panels the parameter regions referring to the pion case for @xmath234 gev@xmath24 ( larger green slanted rectangles further to the right ) and for @xmath235 gev@xmath24 ( transparent rectangles within dashed boundaries ) . the graphics in fig . [ fig : rhol ] include the platykurtic regimes of both das , determined in our present nlc - based analysis to be outlined later . those areas referring to @xmath26 have red color and are located closer to the ordinate ( left panel ) and further to the top ( right panel ) . the analogous areas for the pion appear in light green color and are adjacent to the previous ones . in addition , we incorporate several other @xmath1 and @xmath26 das with individual designations explained in the figure caption for the readers s convenience . the uncertainties of the presented pion and rho - meson das , obtained with our nlc qcd sr approach , include only those stemming from the srs themselves and are related to the variation of the borel parameter within the stability window of the srs @xcite . experimental uncertainties in the input physical parameters have little influence on the results and have been ignored . the values of all parameters of the displayed models are listed in table [ tab : parameters ] at the reference scale @xmath28 gev@xmath24 . this scale is employed in lattice calculations because it naturally arises by the matching of the bare ( lattice ) operators at @xmath236 ( @xmath237 being the lattice spacing ) to those in the @xmath238 scheme in continuum qcd . it is also used in various works based on the dse approach . in this work , we obtained our own results at the initial scale @xmath47 gev@xmath24 and evolved them to this higher scale using erbl evolution at the next - to - leading order level . the conformal coefficients @xmath110 and @xmath111 , and the moments @xmath239 and @xmath240 along with the inverse moment @xmath241 in table [ tab : parameters ] have for each da their own original values at @xmath28 gev@xmath24 . in cases where the original scale was lower , we have evolved these quantities to the scale @xmath28 gev@xmath24 using two - loop erbl evolution . we have included in the table the ads / qcd model of the pion da derived in @xcite within holographic qcd . this model reads @xmath242 and approaches at @xmath243 the asymptotic da @xmath244 , while it has a very different @xmath245 behavior at finite @xmath77 @xcite . the conformal coefficients of this da have been computed in the arxiv version of @xcite at the initial scale @xmath246 gev@xmath24 to obtain @xmath247 and @xmath248 from which the first two conformal coefficients @xmath249 and @xmath250 were determined . using nlo scaling relations , these coefficients have been evolved to the reference scale @xmath28 gev@xmath24 and are given in table [ tab : parameters ] . the value of the second moment was later computed in @xcite for @xmath251 gev and @xmath252 gev using lo evolution and found to be almost the same , notably , @xmath253 . note that a broad pion da @xmath254 was considered before in @xcite using qcd sum rules with nlcs . an even broader da was discussed earlier in @xcite , giving @xmath255}$ ] . to complete this discussion , we mention that the coefficient @xmath110 ( and other parameters ) for the pion da proposed by chernyak and zhitnitsky @xcite has been included in table [ tab : parameters ] using nlo evolution to the scale @xmath28 gev@xmath24 , but we have not displayed it in fig . [ fig : rhol ] because its value is outside the range of the graphs . the most important observations from fig . [ fig : rhol ] are the following : ( i ) there is no overlap of the @xmath256 , or , equivalently , @xmath232 , regions for the pion and the @xmath26 meson computed with nlc sum rules in @xcite and @xcite . ( ii ) while the platykurtic regime for the @xmath26 da is entirely enclosed by the region determined with @xmath234 gev@xmath24 , the platykurtic regime for the pion da appears as an exclave . this is , because in order to obtain a platykurtic pion da one has to use the slightly larger value @xmath235 gev@xmath24 . ( iii ) the dse - based das for the pion and the @xmath26 meson are in both panels far away from our nlc estimates . but keep in mind that the locations of the dse das in the @xmath256 plane are only indicative because , as we have already mentioned , their parametrization by means of eq . ( [ eq : truncated ] ) is a very crude approximation . ( iv ) we observe in the @xmath256 plane an intriguing alignment of the unimodal dse - based das ( @xmath1 and @xmath2 ) along an upward pointing `` diagonal '' and another branch of bimodal nlc - based das steered downwards along @xmath257 . this second `` orbit '' of das roughly follows the values of @xmath258 evolved to the scale 4 gev@xmath24 @xcite . crucially , there is a small region where both branches overlap allowing the combination of endpoint suppression with unimodality the chimera regime of our interest . ( v ) on the other hand , all das , blended together according to eq . ( [ eq : hybrid ] ) , lie on the straight line with @xmath259 and stray away from the nlc @xmath256 region with decreasing values of the mixing parameter @xmath237 . ultimately , i.e. , for @xmath243 , all das with the correct @xmath245 asymptotics will evolve either along the upper `` diagonal '' ( if they are unimodal ) or along the lower one ( if they are bimodal ) toward the asymptotic da ( ) with @xmath260 , as predicted by perturbative qcd . issues ( iv ) and ( v ) will be addressed later in full detail . the meson da at a fixed scale @xmath23 is a distribution of @xmath245 values in the interval @xmath261 $ ] . while a unimodal da profile may seem more `` natural '' in appearance for the ground state of the pion , the interpretation of a bimodal structure causes discomfort and gives rise to debates . so there is a desire for an explanation and rationalization of this issue . recently , it was argued by one of us @xcite that in order to better understand the patterns of the pion and other meson das it is useful to develop some ideas which are drawn from the subject of synchronization of nonlinear oscillators in the theory of complex systems natural and engineered ( see @xcite for reviews ) . the nub of the idea , as stefanis put it in ref . @xcite , is to represent the @xmath245 values , accessible to the meson das in the interval @xmath261 $ ] , in terms of the phases of a large number ( @xmath262 ) of interacting oscillators . the dynamics of this kind of systems is describable in terms of the kuramoto model @xcite and its descendants , albeit its specifics is not relevant for the present analysis . what is more important is that the synchronization of the oscillator phases , alias the longitudinal momentum fractions carried by the valence quark vs. that of the antiquark , gives rise to the formation of particular patterns of the @xmath245 distribution . these patterns emerge from the `` organization '' of the phase spectrum ( i.e , the @xmath245 distribution ) and reflect the specific approach used to describe the partonic interactions in the pion bound state described by the da . in other words , each particular da profile is latent in the underlying theoretical method and pertains to a patterned arrangement of synchronized coupled oscillators in the kuramoto context . these methods can be qcd sum rules with nonlocal condensates , as employed in this work and in @xcite ( see also @xcite ) , local condensates @xcite , lightcone sum rules ( lcsr)s @xcite , instanton models @xcite , approaches based on dyson schwinger - equations @xcite ( reviewed in @xcite ) , light - front models , e.g. , @xcite , holographic qcd @xcite see @xcite for a recent review , etc . examples of @xmath1 and @xmath2 da profiles are depicted in fig . [ fig : pi - rho - das ] . thus , the sync concept provides a universal canvas to study the characteristics of very different meson das without taking recourse to a specific calculational scheme . in particular , it puts a theoretical basis beneath the interpretation of the bimodality of meson das , as we will show next . indeed , it was pointed out in @xcite that at scales of the order of @xmath263 gev , nonlocal condensates , which are used to parameterize the vacuum nonlocality in terms of a nonvanishing quark virtuality @xmath264 cause the distribution over @xmath245 to flock into two clusters , giving rise within a broad range of uncertainties in the midregion of @xmath245 to bimodal das for the pion @xcite and the @xmath26 meson @xcite . the corresponding families of das are shown in the form of the larger shaded bands in fig . [ fig : pi - rho - das ] . the upper panel refers to the @xmath1 and the lower one to the @xmath2 meson , both at the scale @xmath28 gev@xmath24 after nlo evolution from the initial scale @xmath265 gev@xmath24 . the bimodality strength of the bms - type of das is controlled by the nonlocality parameter @xmath264 , which endows vacuum fluctuations with a characteristic correlation length @xmath266 . lower values of @xmath264 tend to increase the bimodality character of the da and reduce the value of @xmath267 , while larger values enhance the midregion of @xmath245 driving @xmath268 closer to a unimodal distributional shape . this behavior is deeply rooted in the combined effect of the perturbative part and the power - behaved terms in the qcd sum rule for the moments @xmath72 considered in @xcite and in @xcite . for @xmath269 , one recovers the qcd sum rules of chernyak - zhitnitsky in ref . @xcite with an infinite correlation length of the vacuum fluctuations . the numerically most important term is the scalar - condensate contribution @xcite , encountered in ( [ eq : srrholterms ] ) , ) pertains to a gaussian model for the quark condensate . ] @xmath270 \ ! + \ ! \left(\bar x\rightarrow x\right ) \ ! + \ ! \theta(1>\delta ) \right . \nonumber \\ & & \!\!\times \left . \theta\left(\delta > x>\bar\delta\right ) \ ! \left[\bar\delta + \left(\delta-2\bar xx\right ) \right . \nonumber \\ & & \!\!\times \ ! \left . \left . \ln(\delta)\right ] \right\ } \ , , \label{phi - scalar}\end{aligned}\ ] ] where @xmath271 , with the four - quark contribution being given by @xmath272 gev@xmath273 and @xmath274 gev@xmath275 @xcite . here , @xmath276 , @xmath277 , and @xmath278 gev@xmath24 is the borel parameter @xmath279 $ ] . the sum rule should not be sensitive to the choice of this parameter . the procedure for minimizing the dependence on @xmath32 has been described in @xcite and is applied here . larger values of @xmath280 shift the balance in the sum rule in favor of the perturbative contribution which has a single mode at @xmath281 , thus entailing a reduction of the two peaks at @xmath282 and @xmath283 until they ultimately collapse into a single more rounded peak at the center . but despite this shift towards the central region of @xmath245 , the tails of the bms das remain suppressed within only a small range of theoretical uncertainties , as one also observes from fig . [ fig : pi - rho - das ] . moreover , it was shown in @xcite that the endpoint behavior of the pion da can be related to the `` decay rate '' of the correlation length of the scalar condensate . it is worth noting in this context that in the local version of the condensate model , i.e. , for @xmath269 , all nonperturbative contributions are concentrated just at the endpoints because @xmath284/(m^2)^2 $ ] . viewed from the kuramoto prism , the two peaks of the bms das correspond to two distinct groups of rather strongly synchronized oscillators with characteristic `` frequencies '' located in the lower and upper quartiles of the @xmath245 distribution , respectively . on the other hand , the tails correspond to tiny cohorts of oscillators with natural `` frequencies '' close to the rare values @xmath285 and @xmath286 with almost nil phase - locking , while a partly synchronized arrangement of oscillators with values around @xmath281 connects the two clusters across the dip in the central region . consider now the implementation of dcsb to meson das . it was stated in @xcite that the dcsb and the concomitant mass generation of quarks and gluons within the dse - based framework @xcite tend to enhance , both the central region of @xmath245 values but also the tails of the meson das down to the kinematic endpoints @xmath287 , leading to a homogenization of the @xmath245 values of the valence @xmath288 pair and to broad unimodal das for all considered mesons @xcite . these das have downward concave profiles in the whole interval @xmath17 $ ] . there are two variants of pion das derived from the dse - based approach @xcite . they were computed via a large number of moments @xmath289 @xmath290 and were then expressed by means of eq . ( [ eq : dse - da ] ) using two different dse truncations at the scale @xmath28 gev@xmath24 . the associated values of the parameters @xmath70 and other relevant metrics are provided in table [ tab : parameters ] . one @xmath1 da , dubbed dse - rl , was obtained using the rainbow - ladder ( rl ) approximation of the bethe salpeter kernel in the dses , while the other , termed dse - db , was derived with a dcsb - improved kernel ( abbreviated by db ) , which includes nonperturbative dcsb - generated effects that were not taken into account in the rl truncated version . both das are much broader relative to @xmath291 , with the dse - rl da being flatter and broader than the dse - db da . the profiles of these das are given in the upper panel of fig . [ fig : pi - rho - das ] : dse - rl dashed - dotted line ; dse - db dashed line . also the @xmath2 da obtained with the dse computational scheme @xcite , is a relatively broad everywhere downward concave curve ( though less pronounced than both pion dse das ) , as one can see from the lower panel of fig . [ fig : pi - rho - das ] ( lower dashed line ) . the broadening of the dse das is a direct consequence of the nonperturbative dcsb interactions which give rise to the dressed quark s selfenergy see @xcite for a detailed review of the method and further explanations . also note that the dse das cross the nlc - based ones twice on each side of the mean ( fig . [ fig : pi - rho - das ] ) . thus , these das show a pattern of higher - lower - higher on each side , related to their heavier tails . this behavior can be quantified by employing the kurtosis statistic , defined by @xmath292 = \frac{e(x-\mu[\varphi])^4}{(e(x-\mu[\varphi])^2)^2 } = \frac{\mu_4[\varphi]}{\sigma^4[\varphi ] } = \frac{\langle \xi^4 \rangle_\pi}{\left(\langle \xi^2 \rangle_\pi\right)^2 } \ , , \label{eq : kurtosis}\ ] ] where @xmath293 = \int_{0}^{1 } dx ( x-\mu[\varphi])^2 \varphi(x ) = \frac{1}{4}\langle \xi^2\rangle_\pi \label{eq : variance}\ ] ] is the variance of the distribution @xmath294 . together with the skewness ( vanishing here but being relevant for mesons composed of light and heavy quarks like the kaon ) @xmath295 = \frac{e(x-\mu[\varphi])^3}{\sigma^3[\varphi ] } \ , , \label{eq : skewness}\ ] ] it describes the central tendency , variability , and shape of a distribution . in particular the kurtosis serves to measure the peakedness in the central region of a distribution against the flatness of its tails . as one sees from table [ tab : parameters ] , the unimodal , downward concave dse - based das can be ordered as follows : @xmath296 . this result confirms a similar qualitative statement in @xcite . such broad da morphologies describe a loosely synchronized assortment of oscillators spread over the entire range of their native `` frequencies '' in @xmath297 $ ] . the enhancement of the generic midregion , which corresponds from a physical perspective to `` egalitarian '' partonic configurations in the pion in which the valence quark and the valence antiquark carry comparable fractions of longitudinal momentum , may be welcome . but at the same time dse das also overestimate the weight of `` aristocratic '' configurations with low probability occurrence in which a single valence parton takes the lion s share of the momentum with @xmath298 or @xmath299 to go far - off shell . because these are more specific and rare configurations of the dispersion of the valence parton s longitudinal momentum far away from the typical values around the mean @xmath300 , tail enhancement is at odds with our understanding of the qcd description of the pion bound state based on off - shell gluon exchanges ( see @xcite and @xcite for explanations ) and , as we have seen , leads to more variation in the oscillator `` frequencies '' , thus entailing less synchronization . note that the broader and flatter a unimodal downward concave @xmath245 distribution is , the closer to random the oscillator phases will be . comparison in earlier works , e.g. , @xcite , of predictions for the pion - photon transition form factor obtained within the lcsr framework with all existing data , indicates that strict qcd scaling behavior at high @xmath77 is very sensitive to the endpoint - behavior of the pion da . this behavior is intimately related to the inverse moment @xmath301 . recalling eq . ( [ eq : diagonal ] ) , this implies that the conformal coefficients have to balance each other in such a way so that the excess over @xmath302 is not too large . otherwise , the form factor would overshoot the data . and in fact , using the dse das as nonperturbative input in a lcsr - framework , we have shown @xcite that the predictions obtained herewith exceed the asymptotic limit even at the highest momenta around 40 gev@xmath24 probed in current experiments @xcite . this is also obvious from table [ tab : parameters ] . the inverse moment @xmath303 obtained with both pion dse das , has very large values to be compatible with the data . we mention in this context that a recent analysis @xcite of the meson structure in lightfront holographic qcd , which employs a broad unimodal @xmath1 da , finds predictions for the pion - photon transition form factor that are in quite poor agreement with all experimental data ( see fig . 18 there ) . but notice that a subsequent dse - based computation @xcite of the tff , in which the role of the inverse moment is not so crucial , finds good agreement with the cello , cleo , and belle data in the entire domain of spacelike momenta . in the sync analogy , the nlc within our approach causes a generic clustering of the da into two clusters liaised with massive endpoint suppression within the range of intrinsic theoretical uncertainties and only a moderate reduction of the da in the central region , which is controlled by the strength of the nonlocality parameter as discussed above in connection with eq . ( [ phi - scalar ] ) . thus , configurations with a highly asymmetric dispersion of the valence parton s longitudinal momentum are suppressed . it is highly unlikely that the pion can remain intact and rebound as a whole for such partonic configurations . in the sync analogy they would play the role of very idiosyncratic oscillators incapable to synchronize , being either too slow ( with `` frequency '' values close to 0 ) or too fast with `` frequencies '' tending to unity . in this sense , the nlc acts like a negative feedback opposing the excessive dcsb - induced enhancement in the endpoint regions by turning off the corresponding oscillators . a figurative explanation of these issues is provided in fig . [ fig : oxymoron ] in which the two main antithetic effects in the @xmath245 behavior of the valence quark in the pion da are illustrated . this is how stefanis envisioned and mapped out the sync properties of the bms - like and dse - like pion das in @xcite . so there are , it seems , two very distinct da patterns with telltale signatures that include , but are not limited to , the pion case : one is related to nlcs , which encode particle correlations in the range @xmath304 fermi , while the other implements dcsb which causes the dynamical generation of quark masses and entails dressing of the quark propagator describing the confined quark in the pion . both effects are manifestations of confinement and neither exists in isolation see @xcite for a recent review of strong - interaction dynamics . unfortunately , they can not be described simultaneously within a single analytic approach at present . thus , for the time being , there are two physical paradigms with their own computational techniques , each applying only within its own sphere of acceptance and validity . $ ] ( mirror graph in the interval @xmath305 $ ] not shown ) . the lower ( green ) line denotes the bms da , with the range of such das being indicated by the green vertical strip ( cf . green shaded band in the upper panel of fig . [ fig : pi - rho - das ] ) . the upper ( blue ) line marks the dse - db da in the central @xmath245 region . the unimodal dse - db da shows enhancement of the tails and the midregion around @xmath306 , while the bms @xmath1 das are characterized by suppression of the endpoint region @xmath286 . the corresponding main trends are indicated by vertical arrows : dcsb enhancement ; nlc suppression . [ fig : oxymoron],scaledwidth=45.0% ] there are basically two options : ( i ) either the @xmath245 distribution of the pion da is described by a single da over the whole range of values in the interval @xmath297 $ ] , or ( ii ) the `` true '' pion da is rather a mixture of two different das , one better applicable to the central region and the other controlling the tails . the first option was discussed above and at length in the literature . following the second scenario , stefanis proposed in @xcite a synthetic da of the form @xmath307 where @xmath237 is a mixing parameter with values within the interval @xmath261 $ ] . mixtures of the form of eq . ( [ eq : hybrid ] ) are quite common in statistics when a single distribution , like the gaussian distribution function , the poisson distribution , etc . , has to be combined with a distribution with a different type of mathematical behavior in the tails , e.g. , the generalized pareto distribution . in fact , hybrid - like das have been constructed and profoundly studied by bergmann and stefanis long ago for the nucleon @xcite and also for the @xmath308 resonance @xcite ( termed `` heterotic '' das ) , although they were motivated by other concerns . a comprehensive and detailed review of such baryon das has been given in @xcite . in the present case , the coexistence of distinct domains of oscillators , some coherent and phase - locked ( bms peaks ) , and others which describe unsynchronized oscillators ( heavy tails of the dse da ) , would give rise to a so - called chimera state @xcite . it was argued in @xcite that for values of the mixing coefficient @xmath237 close to 1 , the synthetic da would still belong to the family of pion bms das shown in terms of the wide shaded band in fig . [ fig : pi - rho - das ] . more generally , the synthesized da is supposed to encapsulate both manifestations of qcd confinement , reflecting the perpetual balance which arises from the appropriate combination of the two basic effects , one associated with nlc formation , via qcd sum rules ( bms - da @xcite ) , the other being the result of dcsb , expressed in terms of a da computed within the dse - based approach @xcite . a proper combination has to balance the enhancement impact of dcsb against the suppression due to nlc , as exposed in fig . [ fig : oxymoron ] . under erbl evolution the synthetic da would develop at @xmath243 to the asymptotic da which represents the most synchronized configuration of the valence @xmath288 pair within the pion being still a bound state @xcite after all quark - gluon interactions have died out . this is also evident from table [ tab : parameters ] from which we see that the asymptotic da has the largest kurtosis , i.e. , @xmath79 is the most leptokurtic meson da . in this paper , we work out eq . ( [ eq : hybrid ] ) in more certain terms and exploit the whole range of possible values of @xmath237 . from the synchronization point of view , a synthetic da represents an attempt of combining ensembles of synchronized and unsynchronized ( but otherwise identical ) oscillators in order to enhance or suppress particular frequency values amounting to a chimera state . the question is whether the simple one - parametric design of eq . ( [ eq : hybrid ] ) is indeed capable of providing the desired properties addressed above in the adjunct discussion of fig . [ fig : oxymoron ] . ) at the scale @xmath28 gev@xmath24 for various values of the mixing parameter @xmath309 $ ] . the shorttailed platykurtic ( pk ) da is shown by the upper red solid line for comparison ; it does not belong to this class of das ( see text ) . [ fig : da - mixture],scaledwidth=45.0% ] figure [ fig : da - mixture ] shows various samples of synthesized das obtained with different values of the mixing parameter between 0 and 1 . as one clearly sees from these plots , when @xmath237 is larger than , say , 0.75 , the main characteristics of the bms das persist . this is also obvious from the location of this da , denoted by the symbol ( ) , in the plane @xmath256 in the left panel of fig . [ fig : rhol ] . however , for small @xmath237 values close to 0.25 and below , the obtained da profiles show no endpoint suppression , as desired , but exhibit instead tail enhancement , inherited to them by the dse da , while the profile is still bimodal , see ( @xmath310 ) and ( @xmath311 ) in the same figure . this means that the bimodality of the da prevails from large down to quite small values of the mixing parameter while at the same time for these small values of @xmath237 the tails of the da get strongly enhanced . hence , eq . ( [ eq : hybrid ] ) can not supply a pion da which combines unimodality in the central region with suppression of the tails , despite the fact that it generates das with the same value of @xmath110 , see , fig . [ fig : rhol ] , left panel . nor can this be realized via erbl evolution.$ ] @xcite which obviously affects the endpoint behavior of the meson da independently of its shape albeit endpoint - enhanced das receive larger nlo corrections @xcite . however , given that at scales @xmath312 gev@xmath24 the running coupling is already sufficiently small , this effect can be safely neglected , see @xcite . ] in order to embody endpoint suppression of the pion da , one has to build it in right from the start , resorting to qcd sum rules with nlcs and looking for das which would mimic the characteristics of the dse das in the central region , while preserving suppression of the tails in compliance with fig . [ fig : oxymoron ] . to achieve this goal we have to keep in mind that if mass is moved from the shoulders to the center of a distribution , then one has to compensate the accompanying movement of mass to the tails , leaving the variance almost unchanged but increasing the kurtosis . the existence of such a chimera da for the pion , which binds these diverse aspects of coherence and incoherence into a single da , was first discussed in @xcite and predictions for the pion - photon transition form factor @xmath313 were presented which are fully compliant with all experimental data compatible with qcd scaling . the overall quality of these predictions resembles that of the bms - type das @xcite . this can be traced back to the fact that they both lead to an inverse moment with just the appropriate size in order to agree with the @xmath314 data . in fact , a brand - new simultaneous fit to the cleo @xcite and belle @xcite data in @xcite favors a profile of the pion da which is very close to the platykurtic one ( see fig . [ fig : pi - rho - das ] and table [ tab : parameters ] ) . on the other hand , das with downward concave shapes in the entire interval @xmath17 $ ] , will tend to overestimate most data of the belle collaboration @xcite , the reason being that they give rise to large inverse - moment values ( table [ tab : parameters ] ) and @xcite ( see also @xcite ) . [ the opposite behavior was found in @xcite , as already mentioned . ] in this work , we determine a whole domain of such chimera @xmath1 das employing our nlc technology and allowing for a slightly larger value of the quark virtuality , viz . , @xmath25 gev@xmath24 . the core area of these das is illustrated in the upper panel of fig . [ fig : pi - rho - das ] in the form of a narrow strip in red color . the corresponding parameters and the range of theoretical uncertainties for both used gegenbauer parametrizations are given at the reference scale @xmath28 gev@xmath24 in table [ tab : parameters ] . this table also includes the brand - new lattice estimates for the second moment @xmath239 from @xcite at the same scale using the @xmath238 scheme . note that this @xmath110 value was not calculated from the second moment @xmath315 via eq . ( [ eq : conf - coeff - a2 ] ) at finite lattice spacing , taking subsequently the continuum limit . instead , the value of @xmath110 was calculated directly on the lattice and was then extrapolated to the continuum limit via @xmath316 . this implies that eq . ( [ eq : conf - coeff - a2 ] ) is broken by lattice artifacts . a small variation in the lattice spacing around @xmath317 may result in an increase of @xmath110 of the order of @xmath318 @xcite . the final result at @xmath28 gev@xmath24 reads @xmath319 , while the reported value of the second moment is @xmath320 . the first error is statistical and originates from the chiral expansion , whereas the second one pertains to the uncertainties of the renormalization factors . it agrees within errors with @xmath321 @xcite and also with @xmath322 ( see table [ tab : parameters ] ) . in contrast , while the lattice estimate @xmath323 agrees with the bms coefficient @xmath324 , determined in the year 2001 @xcite ( see table [ tab : parameters ] ) , it turns out to be larger than @xmath325 . but one should be cautious . extracting @xmath110 via @xmath326 , one would obtain @xmath327 , which is indeed compatible with the range of the platykurtic @xmath110 values . thus , one can not exclude the influence of significant discretization effects that would require simulations at smaller lattice spacings of the order of @xmath328 fm @xcite . be that as it may , one should recall that the second moment @xmath329 is related to the variance of the da given by eq . ( [ eq : variance ] ) . this statistic is not sufficient to draw any conclusions about the shape of the distribution in the central region . indeed , as one observes from table [ tab : parameters ] , the unimodal dse - db pion da yields a conformal coefficient @xmath330 which fully agrees with the new lattice result but also with @xmath331 . we note that this is valid for the second moment as well , which has the value @xmath332 , ( cf . ( [ eq : conf - coeff - a2 ] ) ) and thus almost coincides with the second moment of the pion bms da given above , being also close to @xmath333 . on the other hand , the fourth moment @xmath240 and the conformal coefficient @xmath111 of the dse and the bms ( pk ) das are different in value and sign , respectively see table [ tab : parameters ] . what is far more significant is the fact that , as it is evident from the left panel of fig . [ fig : rhol ] , _ all _ das lying on the straight vertical line at @xmath334 agree equally well with the new lattice estimate for @xmath110 . this makes it apparent that a single lattice constraint can not fix the profile of the pion da uniquely , however precise it may be . the chimera das have shorttailed platykurtic profiles and overlap with the dse - db da in the midregion of @xmath245 but descend at the endpoints at low angle to zero , similar to a typical bms da . as one observes from fig . [ fig : rhol ] ( both panels ) , there is an imbrication of the platykurtic regimes ( small rectangles in light - green color ) with the domains of the bimodal pion das obtained with nlc sum rules for the quark virtuality @xmath235 gev@xmath24 ( transparent rectangles bounded by a dashed line ) . for this value the conformal coefficients for @xmath335 at @xmath28 gev@xmath24 read @xmath336 and @xmath337 while the inverse moment is @xmath338 , a value which agrees well with @xmath339 in table [ tab : parameters ] . the prediction for @xmath314 obtained with the shorttailed platykurtic @xmath1 da appears in line with all data of the belle @xcite and the babar collaboration @xcite compatible with qcd scaling @xcite . these unique features of the pk pion da look indeed very attractive . but is it more than mere coincidence or can it provide a general mode of accessing meson das and offer a deeper perspective on meson das in general ? to this end , we turn our attention to the @xmath0 meson case and attempt to determine a platykurtic regime for the @xmath2 da using as a selector the behavior illustrated in fig . [ fig : oxymoron ] . evaluating the sum rule in eq . ( [ eq : srrholconden ] ) , we compute the reliability range of the conformal coefficients up to the order @xmath30 by first determining the central moments @xmath340 of the same order . their values at the initial scale @xmath341 gev@xmath24 can be found in @xcite . also the corresponding values of the @xmath342 meson are given there together with the conformal coefficients . we will not repeat these details here . we concentrate instead on our primary goal to extract a platykurtic domain of these parameters . it turns out that this is possible even for the somewhat smaller value of the quark virtuality @xmath234 gev@xmath24 used originally for the extraction of the pion da in @xcite . the extracted domains are shown in fig . [ fig : rhol ] in the form of the red slanted rectangles surrounded by the larger blue bands of coefficient values computed with the nlc sum rules . the left panel displays the results for the first two conformal coefficients @xmath110 and @xmath111 , whereas the right panel provides the areas of the coefficients @xmath65 and @xmath67 . in both graphics the platykurtic model for the @xmath2 da is denoted by the symbol . the values of all these parameters , accompanied by their intrinsic errors , are compiled in table [ tab : parameters ] , while the platykurtic @xmath2 da profiles are displayed in the form of a narrow red strip in the lower panel of fig . [ fig : pi - rho - das ] . for the sake of direct comparison with the dse results , all graphics and the values in the table are given at the reference scale @xmath28 gev@xmath24 after two - loop evolution . one immediately observes from this figure that , similar to the pion case , the platykurtic @xmath2 da has a single rounded central peak bearing endpoint suppression relative to the @xmath2 da obtained within the dse - based approach @xcite . in comparison to the platykurtic da of the pion , it features a slightly narrower profile with the kurtosis value @xmath343 . tangible consequences of the platykurtic @xmath2 da will be studied elsewhere . we have performed an intensive study of the pion and the @xmath2 das within qcd , fortified with the knowledge of synchronization concepts used in the description of complex systems . these concepts provide a unifying rationale of _ how _ the various da profiles emerge instead of asking _ they should have a particular shape , thus avoiding descriptive comparisons of das obtained with unrelated theoretical frameworks . furthermore , guided by these concepts , we have used controlled theory tools to obtain a new kind of chimera das for the pion and the @xmath2 meson using qcd sum rules with nlcs . these das are capable of mingling in situ the best of both worlds endpoint suppression via nlc and unimodality due to dcsb , giving rise to shorttailed platykurtic profiles and realizing the scenario illustrated in fig . [ fig : oxymoron ] , while preserving the asymptotic @xmath245 behavior predicted by perturbative qcd . in the sync picture , they correspond to a vast number of phase - locked oscillators between the lower and the upper quartile of the @xmath245 distribution , whereas oscillators with extremely high or low `` frequencies '' , located close to the tails @xmath287 , are in limbo . while the characteristics of these new das in the central @xmath245 region resemble the gross behavior of dse - based das , their suppressed tails are following the same trend as the bms das . in mathematical terms , the bms - like das and the platykurtic ones are very different as regards their profiles ( fig . [ fig : pi - rho - das ] ) and gegenbauer coefficients ( fig . [ fig : rhol ] and table [ tab : parameters ] ) . but from the nlc point of view , the bimodal bms @xmath1 das and the bimodal @xmath2 da of @xcite , which in the sync analogy correspond to two clusters of synchronized oscillators , are on the same theoretical footing as the unimodal shorttailed platykurtic das for these mesons , which unite the phase - locked oscillators in a single group . moreover , in the pion case they yield coinciding predictions for the pion - photon transition form factor which agree well with all available experimental data compatible with qcd scaling above @xmath344 gev@xmath24 . given all these results , we do nt want to stretch the importance of unimodality too far . too broad das with downward concave profiles encompassing the tails , as those derived for mesons with the aid of dses @xcite , imply that there is no particular @xmath245 value standing out because even the remote regions close to the endpoints @xmath287 have a significant weight almost comparable to that of the central region especially the dse - rl pion da . the extreme case of a flat - top da , like @xmath345^{-1 } ( x\bar{x})^{\alpha } $ ] with @xmath346 @xcite , translates into a population of oscillators with a very strong variation of native `` frequencies '' so that these oscillators can hardly synchronize and as a result phase locking diminishes . physically , this kind of @xmath245 distribution comprises extremely asymmetric partonic configurations that can spoil scale locality and thus collinear factorization . on the experimental side , flat - top das yield predictions for the scaled pion - photon transition form factor which have a tendency to increase with @xmath77 at least in the domain of currently accessible momentum values in the range 10 - 40 gev@xmath24 where one would expect scaling to be visible @xcite . the high-@xmath77 data of the babar collaboration @xcite indicate such a trend , but are not supported by the belle data @xcite in the same region . the next - generation experiments to measure the pion - photon transition form factor with the belle ii detector at the upgraded kekb accelerator ( superkekb ) in japan and more precise data on the electromagnetic pion form factor expected at the jefferson laboratory ( jlab ) after its upgrade will provide extraordinary tools to test our predictions and assertions . we thank sergey mikhailov for collaboration on various aspects related to the present work and for useful discussions and comments . thanks pengming zhang for the warm hospitality and support at the institute of modern physics of the chinese academy of sciences , where the final stage of this work was carried out . this work was partially supported by the heisenberg landau program ( grant 2015 ) , the russian foundation for fundamental research under grants no . 14 - 01 - 00647 and no . 15 - 52 - 04023 , the jinr - belrffr grant f14d-007 , the major state basic research development program in china ( no . 2015cb856903 ) , and the national natural science foundation of china ( grant no . 11575254 and 11175215 ) . 99 v. l. chernyak and a. r. zhitnitsky , phys . * 112 * ( 1984 ) 173 . s. j. brodsky and g. p. lepage , adv . high energy phys . * 5 * ( 1989 ) 93 . n. g. stefanis , eur . j. direct c * 7 * ( 1999 ) 1 , hep - ph/9911375 . n. g. stefanis , phys . b * 738 * ( 2014 ) 483 , arxiv:1405.0959 . n. g. stefanis , s. v. mikhailov , and a. v. pimikov , few body syst . * 56 * ( 2015 ) 295 , arxiv:1411.0528 . a. p. bakulev , s. v. mikhailov , and n. g. stefanis , phys . b * 508 * ( 2001 ) 279 , hep - ph/0103119 ; 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using qcd sum rules with nonlocal condensates , we show that the distribution amplitude of the longitudinally polarized @xmath0-meson may have a shorttailed platykurtic profile in close analogy to our recently proposed platykurtic distribution amplitude for the pion . such a chimera distribution de facto amalgamates the broad unimodal profile of the distribution amplitude , obtained with a dyson schwinger equations - based computational scheme , with the suppressed tails characterizing the bimodal distribution amplitudes derived from qcd sum rules with nonlocal condensates . we argue that pattern formation , emerging from the collective synchronization of coupled oscillators , can provide a single theoretical scaffolding to study unimodal and bimodal distribution amplitudes of light mesons without recourse to particular computational schemes and the reasons for them .

introduction meson da in qcd gegenbauer da representations @xmath1 and @xmath2 da from nlc qcd sum rules meson da as patterns of synchronization synthetic meson da conclusions

arxiv

the ability to achieve inversion and lasing in atomic and solid state systems is a topic of continuing interest @xcite . in the domain of cavity - qed , regimes of `` single atom lasing '' and non - classical light emission have been studied @xcite . according to the common wisdom , any scheme to achieve population inversion must exploit a multi - level manifold of material states ; steady - state inversion is not possible for a driven two - level system . this wisdom is restricted , however , to a simple two - level system ( tls ) coupled to classical fields in a non - engineered environment . when interacting with a _ quantized _ radiation mode , a tls may be inverted by two - photon excitation to higher lying levels of the quantized _ matter plus field _ @xcite . also , when in interaction with an environment near a photonic bandgap , a tls may be inverted by way of an asymmetry of the mollow triplet ; although , very high ( many orders of magnitude ) contrast in the sideband decay rates is required @xcite . in contrast to atoms , the unique features of semiconductor quantum dots ( qds ) include their large optical dipole moment and engineerable emission wavelength . other attractive features are their fixed position and potential for integration with cavities and waveguides using developed semiconductor fabrication techniques @xcite . although semiconductor microcavity lasers are today quite common @xcite , there is an ongoing push to realize more exotic regimes , such as `` single qd lasing , '' with the aim of producing a non - classical light source in a solid state nanostructure . single dot lasing _ per se _ is not addressed in this work . rather , we focus on the fundamental mechanisms available to achieve population inversion in a semiconductor qd - cavity system . semiconductor qds differ from atoms at the level of fundamental physics , and this calls for extra care when describing the light - matter interaction . typically , qds are embedded in a solid state lattice where electron - phonon interactions , though sometimes ignored in quantum optical studies , are known to impact optical properties ; they affect photoluminescence lineshapes @xcite , coherent rabi oscillation @xcite , and the mollow triplet spectrum of resonance fluorescence ; phonon - mediated scattering can cause excitation - induced dephasing @xcite , which is detrimental to the exploitation of quantum optical interactions . it is interesting , then , to ask what impact this scattering has when one tries to coherently drive an excitonic transition coupled to a quantized radiation mode into a regime of population inversion @xcite . in answer we find a surprising result : contrary to the expectation that it might be detrimental , acoustic - phonon - mediated scattering can be exploited to achieve significant exciton inversion via a mechanism entirely different to those previously reported to invert a simple tls . the mechanism is richer and significantly more efficient . it relies on the natural asymmetry of phonon emission and absorption at low bath temperatures , and for cavity driving , exploits a one - way coupling ( phonon - assisted scattering ) between the exciton and the cavity mode when the latter is blue detuned with respect to the zero phonon line ( zpl ) . population inversion in a qd in the presence of phonons without cavity coupling has been reported in previous theoretical studies . _ @xcite employed path integral and correlation expansion techniques to explore the long - time behaviour of a coherently driven qd with phonon coupling and noted that away from resonance large population inversions can be realized . the authors compare their results with a simple model based on thermalization in dressed states and find good agreement for the parameters considered . one limitation of this work is the omission of zpl decay processes ; these are generally needed for the quantitative modeling of experiments . et al._@xcite also studied the driven exciton - phonon system , again with no cavity coupling , and report population inversion in a _ unstructured _ phonon bath . they consider a coherently driven double - dot system in the large pump limit in a dressed - state approach . phonon coupling is included as a perturbation and exciton inversion reported for a phonon bath with subquadratic spectral density . both of these works exploit phonons to achieve tls inversion with continuous wave ( cw ) drive . with the qd embedded in a cavity , it is not known how these processes change , nor is it known what differences , if any , arise between the two methods of driving now made available : direct driving of the exciton or driving through the cavity . it is not known how zpl decay processes effect the achievable population inversion . in this work we investigate several mechanisms realizing pump - induced exciton inversion in a qd - cavity system with cw driving . we propose and characterize the phonon - assisted scheme using a semiconductor quantum - optics formalism that includes phonon interactions by way of an effective master equation ( me ) . the me uses a polaron transform to take electron - acoustic - phonon interactions into account at a microscopic level , and includes cavity and qd decay in a lindblad description . it yields an intuitive and transparent modeling of the qd - cavity dynamics through a combination of photon and phonon scattering . we consider a qd embedded within a small high @xmath2 dielectric cavity , as shown schematically in figs . [ fig : schematic1](a ) and [ fig : schematic1](b ) . either the exciton or cavity mode is subject to coherent cw driving ( @xmath3 and @xmath4 in the figure ) . a possible advantage of cavity over exciton driving is that it might mitigate problems with excitation - induced dephasing , a process that accompanies excitation via the exciton - phonon reservoir ; it also allows for pumping through a waveguide input channel @xcite [ cf . [ fig : schematic1](b ) ] , allowing chip - based quantum optics using semiconductor fabrication techniques . the paper is organized as follows . in sec . [ introduction ] we review the me theory and present a simple analytical solution for the exciton density in the absence of cavity coupling . we also discuss our numerical scheme for treating the full ladder of cavity photons in the presence of either exciton or cavity driving . our results for the exciton inversion in a qd - cavity system are presented in sec . [ results ] , where we explore the effect of cavity - exciton detuning and study the difference between exciton versus cavity driving , demonstrating the role of phonon - induced scattering in each case . the path to inversion is found to be qualitatively different for exciton versus cavity driving , with inversion occurring in the latter case only with the cavity suitably blue - shifted with respect to the exciton resonance . we introduce a simple analytical model to explain our results . conclusions are offered in sec . [ conclusions ] . we work in a frame rotating at the laser pump frequency @xmath5 . the simplest hamiltonian for a single qd interacting with a cavity mode and phonons , excluding qd and cavity decay , is @xmath6 where @xmath7 and @xmath8 are creation and annihilation operators for mode @xmath9 of the phonon bath , @xmath10 is the ( real ) exciton - phonon coupling , @xmath11 and @xmath12 are photon creation and annihilation operators for the cavity mode , and @xmath13 and @xmath14 are pauli raising and lowering operators for the exicton ; @xmath15 ( @xmath16 ) designates the detuning of the exciton ( frequency @xmath17 ) and cavity ( frequency @xmath18 ) from the laser drive , and @xmath19 is the drive hamiltonian . since we deal with quasi - resonant coherent excitation , higher lying exciton states and continuum levels in the qd material are neglected . we allow for driving of the exciton and the cavity but consider the two cases separately i.e . , we adopt either non - zero @xmath3 or non - zero @xmath4 . in moving from hamiltonian ( [ sec1eq1 ] ) to the effective me , one first transforms to a polaron frame , which formally recovers the independent boson model @xcite in the appropriate limit ; the independent boson model is known to capture the characteristic spectrum of an exciton coupled to a phonon bath @xcite . the derived me treats the coherent electron - phonon interaction nonperturbatively through a mean phonon displacement , @xmath20 = \exp\left[-\frac{1}{2}\int^{\infty}_{0}d\omega\frac{j(\omega)}{\omega^{2}}\coth\left ( \frac{\hbar\omega } { 2k_bt } \right ) \right],\ ] ] where @xmath21 is the mean phonon number ( bose - einstein distribution at temperature @xmath22 ) , and @xmath23 is the phonon spectral function . the incoherent interaction ( scattering ) is treated within the second - order born approximation ( see refs . for details ) , though it is important to note that higher - order contributions are included by the polaron transform . adding qd and cavity decay , the time - convolutionless me for the reduced density operator is @xmath24+{\cal l}_{\rm ph}\rho+{\cal l}\rho,\ ] ] with polaron - transformed hamiltonian @xmath25 polaron shift @xmath26 ( absorbed by @xmath27 below ) , and phonon scattering term @xmath28 + { \rm h.c . } , \label{eq : fullrates}\ ] ] where @xmath29 , with @xmath30 , @xmath31 , @xmath32 . the rescaling @xmath33 was pointed out some time ago by wilson - rae and imamolu @xcite . it is important to note that @xmath34 and @xmath35 are temperature dependent ; although this dependence is often ignored when fitting experiments , and compensated for by changing other parameters in an attempt to improve the fit . the response functions @xmath36 - 1 $ ] and @xmath37 $ ] are polaron green functions @xcite , defined by the phonon phase term @xmath38 \\ & = \int^{\infty}_{0}d\omega\frac{j(\omega)}{\omega^{2}}\left[\coth\left(\frac{\hbar\omega}{2k_bt}\right)\cos(\omega t)-i\sin(\omega t)\right],\end{aligned}\ ] ] which clearly includes contributions from multi - phonon scattering . the last term in eq . ( [ me ] ) is a sum of three lindblads : @xmath39+(\gamma/2){\cal l}[\sigma^-]+(\gamma^\prime/2){\cal l}[\sigma_{ee}]$ ] , with @xmath40\rho=2\xi\rho\xi^\dagger-\xi^\dagger\xi\rho-\rho\xi^\dagger\xi$ ] and @xmath41 . it accounts for cavity decay at rate @xmath42 , exciton decay at rate @xmath43 , and pure dephasing of the exciton at rate @xmath44 . these processes broaden the zpl , an essential effect not captured by the independent boson model . we adopt the established phonon spectral function , @xmath45 , which describes the electron - acoustic - phonon interaction via a deformation potential , the dominant source of phonon scattering for inas and gasas qds @xcite . appropriate numbers are obtained by fitting experimental measurements made on inas qds @xcite . our model as outlined yields an involved solution scheme and little physical insight . we turn therefore to an effective lindblad form of the phonon scattering term , which is shown by roy and hughes to be in very good agreement with the predictions of eq . ( [ me ] ) . for cavity excitation it makes the replacement : @xmath46\rho + \frac{\gamma_{\rm ph}^{a^{\dagger}\sigma^{-}}}{2}{\cal l}[{a}^{\dagger}{\sigma}^{-}]\rho,\ ] ] with scattering rates @xmath47,\ ] ] where @xmath48 is the cavity - exciton detuning . the replacement follows by making the approximation @xmath49 , with @xmath50 , in eq . ( [ eq : fullrates ] ) . this approximation is good when @xmath51 and @xmath52 are much smaller than the phonon correlation time ( or when the detunings are larger than @xmath3 and @xmath53 ) . with it we capture the dependence of the phonon scattering rates on detuning for detunings that are large enough to have a significant impact on the integral in eq . ( [ eq : phononrates ] ) . in this prescription phonon scattering amounts to a one - way coupling between the driven cavity mode and qd exciton expressed through quantum jumps . there are jumps in two directions photon creation accompanied by exciton deexitation ( rate @xmath54 ) and photon annihilation accompanied by excitation of the exciton ( rate @xmath55)though eqs . ( [ eq : phase ] ) and ( [ eq : phononrates ] ) yield an asymmetry of rates . the asymmetry of rates allows for phonon - mediated inversion . for exciton excitation , the lindblad form of phonon scattering term has two additional contributions : @xmath56\rho + \frac{\gamma_{\rm ph}^{a^{\dagger}\sigma^{-}}}{2}{\cal l}[{a}^{\dagger}{\sigma}^{-}]\rho + \frac{\gamma_{\rm ph}^{\sigma^{+}}}{2}{\cal l}[{\sigma}^{+}]\rho + \frac{\gamma_{\rm ph}^{\sigma^{-}}}{2}{\cal l}[{\sigma}^{-}]\rho,\end{aligned}\ ] ] with additional scattering rates @xmath57,\ ] ] where @xmath58 ( up scattering ) results in incoherent excitation and @xmath59 ( down scattering ) causes enhanced decay . further details appear in ref . . the asymmetry in these rates also allows for phonon - mediated inversion . in the absence of cavity coupling , it is straightforward to solve for the steady - state exciton population analytically from the above me model . the solution is @xmath60,\end{aligned}\ ] ] with polarization decay rate @xmath61 . this formula was recently presented to model photoluminescence experiments on single in(ga)as qds ( without any connection to inversion ) and found to be in excellent agreement with the experiments @xcite . as we show below , the process of phonon - mediated incoherent excitation [ @xmath58 process , shown schematically in fig . [ fig : schematic2](a ) ] can lead to population inversion . @xcite numerically solve for the exciton population in the absence of a cavity , neglecting zpl decays ( neglecting @xmath43 and @xmath44 ) . they find good agreement with an assumed thermal occupation of the dot - photon dressed states , which yields @xmath62 , \end{aligned}\ ] ] where @xmath63 . below we show that , in comparison to eq . ( [ nx : analytic ] ) , this formula is only a reasonable approximation for very large pump fields , and even then fails in general by omitting the important influence of the zpl decays . the full cavity - qd results which follow are obtained by solving the me , eq . ( [ me ] ) , numerically with electron - phonon scattering treated in the lindblad approximation [ eq . ( [ eq : lph ] ) or eq . ( [ eq : lph2 ] ) ] . as depicted in fig . [ fig : schematic2](b)-(d ) , we adopt a basis of truncated photon states , @xmath64 , @xmath65 , and the exciton states @xmath66 and @xmath67 . we find that truncation at @xmath68 is required for convergence at the chosen pump levels , especially in the presence of electron - phonon scattering . for a simplest picture of the inversion mechanism with cavity excitation , fig . [ fig : schematic2](d ) shows the two harmonic oscillator ladders , coordinated with @xmath66 and @xmath67 , that result when the jaynes - cummings interaction term @xmath69 is dropped from the polaron - transformed hamiltonian . with this simplification , if the cavity is detuned to the blue of the zero - photon line , as shown below ( fig . [ fig:2 ] ) , @xmath55 may dominate over @xmath54 and pump the exciton into its excited state . for exciton excitation , a similar argument holds based on the asymmetry of @xmath58 and @xmath59 ; it is unclear , however , what the cavity will do , as two quanta excitation can also efficiently load the exciton in the absence of phonon scattering [ cf . fig . [ fig : schematic2](c ) ] @xcite . the polaron me is expected to be valid when the drive rates and exciton - cavity coupling rates are smaller than the phonon coupling cut - off frequency @xmath70 @xcite , the case of interest in this work . when the drive strength is much larger than @xmath70 , one can adopt an elegant variational me approach @xcite , or resort to numerical path integral or correlation expansion techniques @xcite . a major advantage of a me approach is the clear connection to the underlying scattering processes provided and its ability to account for the various decay processes that must typically be included to explain experiments . we choose material parameters suitable for inas qds , with @xmath71 and @xmath72 , and consider a qd - cavity system in the strong coupling regime of cavity qed , with parameters @xmath73 , @xmath74 , @xmath75 , and @xmath76 . these numbers are consistent with various qd - cavity experiments and show good agreement with experiments on in(ga)as qds , both with and without @xcite cavity coupling . to gain essential insight into the relevant phonon - induced scattering rates , we plot two of them as a function of cavity - exciton detuning , at two different bath temperatures , in fig . [ fig:2](a ) . the figure highlights the asymmetry between @xmath77 and @xmath78 , which is more pronounced at the lower temperature of @xmath1 than at @xmath79 ( when phonon absorption is less likely ) . note also that the scattering rates at @xmath79 are larger than at @xmath1 , even though @xmath80 is lower than @xmath81 . we plot the phonon correlation function , @xmath82 - 1 $ ] , in fig . [ fig:2](b ) . the plot shows that higher temperatures are more heavily damped , and thus the peak in the scattering rates shifts to lower frequencies at higher temperature . the trend for @xmath58 and @xmath59 is exactly the same , only with respect to a @xmath83 detuning dependence . ( solid ) and @xmath54 ( dashed ) as a function of cavity detuning from the zero phonon line ; for an effective cavity - exciton coupling @xmath84 , spectral parameters @xmath71 , @xmath72 , and bath temperatures @xmath85 ( blue , thicker curves ) and @xmath86 ( red , thinner curves ) . the rates @xmath87 and @xmath88 corresponding to exciton driving ( not shown ) have the same functional dependence with @xmath83 replacing @xmath89 ( @xmath90 replaces @xmath91 as the drive ) . ( b ) corresponding phonon correlation function , @xmath92 - 1 $ ] , showing both the real ( solid ) and imaginary part ( dashed ) . ] k. steady state exciton populations are plotted for two different drive strengths , @xmath93 ( blue dashed ) and @xmath94mev ( red solid ) . ( a ) without phonon - induced scattering there is no inversion . ( b ) with phonon - induced scattering an inversion ( @xmath95 ) is produced by the larger drive ; the mechanism is incoherent excitation through the phonon bath . we also plot results for the thermal occupation model [ eq . ( [ nx : analytic ] ) ] in ( b ) ; green dashed curve for @xmath93 and green solid curve for @xmath96 . ] and a bath temperature of @xmath97k . steady state exciton and cavity populations are plotted for detunings @xmath98 ( a , b ) and @xmath99 ( c , d ) ; solid red curves with phonon scattering and blue dashed curves without phonon scattering . the bare cavity resonance is indicated by a vertical black line . the narrow inversion peak appearing without phonon scattering is due to two - photon excitation of the state @xmath100 @xcite . ] mev and a bath temperature of @xmath97k . steady - state exciton populations are plotted as a function of laser frequency for cavity - exciton detunnings @xmath101 , @xmath102 , @xmath103 , @xmath104 , @xmath105 , and @xmath106 ( lower to upper ) . blue dashed curves show the population without phonon scattering while the red solid curves include phonon scattering . the bare cavity resonance is indicated by the vertical black line . the grey solid curve ( near @xmath99 ) shows results obtained with the jaynes - cummings term omitted from the polaron - transformed hamiltonian [ coupled harmonic ladders , cf . [ fig : schematic2](d ) ] . ] with these scattering rates as input , we first explore the steady - state exciton population in the absence of a cavity , i.e. , the predictions of the analytical expression eq . ( [ nx : analytic ] ) . we adopt the lower bath temperature , which yields the larger asymmetry of rates . in fig . [ fig:3n ] we plot the exciton population versus detuning , without [ frame ( a ) ] and with [ frame ( b ) ] phonon scattering , and for two different strengths of the drive . the larger drive produces substantial power broadening and population inversion when phonon scattering is included . the prediction of inversion is consistent with previously reported results @xcite . we note that it is obtained here from an analytical expression with a simple physical interpretation , and including zpl broadening mechanisms that have been shown to be necessary for good agreement with experiments @xcite . to connect with the results of ref . @xcite , we also plot ( green curves ) the prediction of the thermal occupation model , eq . ( [ nx : analytic ] ) . it clearly overestimates the populations , and , moreover , fails entirely for lower pump strengths as might be expected . next , we add in cavity coupling while keeping the exciton drive . it was recently shown that coupling to an off - resonant cavity in this configuration can invert a simple tls via two - photon pumping through the state @xmath100 [ cf . [ fig : schematic2](c ) ] @xcite . it is of interest how phonon scattering affects this result . we again adopt the lower bath temperature , and we choose the larger value , @xmath107 , for the drive . figures [ fig:4n](a ) and [ fig:4n](c ) show results for cavities red and blue shifted , respectively , by @xmath108 ; corresponding mean cavity photon numbers appear in fig . [ fig:4n](b , d ) . the chosen cavity - exciton detuning realizes inversion via two - photon resonance while also having a large asymmetry of the phonon rates . in the absence of phonon scattering ( blue dashed lines ) , we confirm the results of ref . @xcite : an inversion peak appears approximately midway between the cavity and exciton resonances . when phonon scattering is included , the peak is either suppressed ( red shifted cavity ) or subsumed by an enhanced domain of inversion associated with the @xmath58 process ( blue shifted cavity ) . as might be anticipated from eq . ( [ nx : analytic ] ) and ref . @xcite , both cavity detunings show substantial inversion due to the @xmath58 process on the blue side of the exciton ; although , details of the @xmath109 and @xmath110 lineshapes depend on cavity detuning , with the @xmath110 profiles , in particular , retaining clear signatures of the two - photon resonance . it remains to explore cavity coupling with cavity excitation . we again compare exciton populations with phonon scattering against those with the phonons turned off ( note that @xmath53 is replaced by @xmath34 in the latter case ) . the comparison is made in fig . [ fig:6 ] , which shows the exciton population passing through a peak as a function of @xmath89 , more or less in line with the peak in the scattering rates . most notably , large population inversions are obtained when the cavity is blue - shifted , the configuration that allows phonon scattering between ground- and excited - state branches of the photon ladder of states to excite the exciton [ cf . [ fig : schematic2](d ) ] . the reverse process , _ cavity feeding _ , has been identified in semiconductor cavity - qed studies . here we see that the phonon - mediated @xmath55 process can create large population inversions . we stress that the mechanism is quite different to the one reported in refs . @xcite and @xcite and is considerably more efficient [ cf . [ fig:4n](a , c ) , dashed blue lines ] . not only do we see pronounced inversion in the presence of phonon scattering , with @xmath111 at @xmath99 in fig . [ fig:6 ] , but also significant inversion over a broad detuning range . we see , however , that the role of the cavity resonance is much more pronounced than with the exciton drive . for cavity excitation the @xmath55 process dominates , while for exciton excitation the @xmath58 process dominates . although this inverse of cavity - feeding is to be expected , it is frequently omitted from qd me theories and , to the best of our knowledge , has not been noticed in any qd cavity - qed experiments to date . phonon - mediated population inversion is unique to the solid state environment where it may allow for cavity - pumped single exciton lasing . it is important to note that all these results are reproduced by the full non - lindblad me . at the optimal cavity - exciton detuning , near @xmath99 , the exciton population is greater than @xmath0 , and even higher numbers result at higher values of the drive . as indicated above [ cf . [ fig : schematic2](d ) ] , the mechanism underlying this cavity - excited inversion is seen , in a simplest model , to be electron - phonon scattering between a pair of photon ladders , one coordinated with the ground state of the exciton and the other with the excited state . this simplest model neglects the jaynes - cummings interaction term @xmath69 in the polaron - transformed hamiltonian [ eq . ( [ eq : h_s ] ) ] . to test it , we also carried out calculations with the jaynes - cummings term omitted . results for @xmath99 appear as the solid grey curve in fig . [ fig:6 ] . they confirm the qualitative correctness of the model ; although , as one would expect , the exciton resonance is completely missed . the quantum trajectory simulations @xcite presented in fig . [ fig:7 ] provide insight into the role of the neglected jaynes - cummings term . for a detuning of @xmath112mev , the phonon scattering dynamic at @xmath1 and @xmath79 is displayed and compared . short magenta lines signal cavity photon jumps , while green circles and black squares identify phonon scattering jumps photon annihilation accompanied by exciton exitation ( green circles ) and photon creation accompanied by deexcitation ( black squares ) . phonon scattering excites the exciton within a cavity lifetime . the excitation is maintained over a relatively long time at @xmath1 , and eventually lost primarily through radiative decay ( the @xmath43-jump is not seen in the figure ) . reverse phonon scattering is the main source of deexcitation at @xmath79 , resulting in a reduced population ( from @xmath0 to @xmath104 ) . the jaynes cummings term produces the oscillation between photon scattering events . it hardly alters the pattern of quantum jumps but reduces the mean population a little . and ( b ) @xmath79 , for optimal detuning , @xmath99 , and other parameters as in fig . [ fig:2 ] . orange ( light ) lines display the photon number expectation @xmath113 ( @xmath68 ) and blue ( dark ) lines the exciton expectation @xmath114 . the expectations are _ conditioned _ upon the record of quantum jumps : either photon decay ( magenta lines ) or phonon - mediated scattering [ green circles ( black squares ) for scattering at rate @xmath55 ( @xmath54 ) ] . ] , for detunings @xmath115mev . the numerical solution of the me ( red solid curves ) is compared with eq . ( [ nx : analytic2 ] ) ( green dashed curves ) . to improve the fit the decay rate enhancement has been set at @xmath116 . ] finally , it is instructive to develop a formula similar to eq . ( [ nx : analytic ] ) to help explain the qualitative difference in the response for cavity versus exciton driving . to this end , we first ignore the exciton - cavity coupling , so the cavity simply fills up with coherent light . the steady - state density operator factorizes as @xmath117 , with coherent state amplitude @xmath118 we then substitute the _ ansatz _ into the me , take the trace over the cavity mode , and make the approximation @xmath119 , i.e. , write @xmath120 . this yields a me with effective exciton drive @xmath121 . the solution for the exciton population takes the same form as eq . ( [ nx : analytic ] ) : @xmath122,\end{aligned}\ ] ] with @xmath123 , @xmath124 , @xmath125^{-1/2}$ ] , and @xmath126 . in place of @xmath127 and @xmath43 , enhanced rates @xmath128 and @xmath129 have been introduced . the fitting parameter @xmath130 aims to account for absorption ( enhancing @xmath127 ) and the purcell effect ( enhancing @xmath43 ) , which are overlooked when the exciton - cavity coupling is ignored . figure [ fig:6extra ] compares the full numerical solution ( red solid curve ) with eq . ( [ nx : analytic2 ] ) ( green dashed curves ) for two different values of cavity - exciton detuning , and for the choice @xmath116 . the agreement is surprisingly good for both positive [ fig . [ fig:6extra](a ) ] and negative [ fig . [ fig:6extra](b ) ] detunings , and , in contrast to the grey curve in fig . [ fig:6 ] , the model captures the exciton peak as well as the peak around the cavity resonance . the model demonstrates the effect of cavity filtering on the phonon - mediated scattering rates @xmath131 and @xmath132 , which are both proportional to @xmath133 . this filtering is important in the cavity - driven system , as it is the primary control over when the inversion turns off as the drive is detuned . the situation is quite different with direct driving of the exciton and no cavity . in that case the inversion process turns off because of the @xmath83 factor inside the integral , eq . ( [ eq : phononrates2 ] ) , defining the phonon scattering rates . we have reported on a number of different ways to create exciton inversion in a semiconductor quantum - dot cavity system with cw drive . the underlying mechanism exploits a highly efficient phonon - mediated scattering process to create excitons via direct incoherent excitation or through the annihilation of a cavity photon ; the reverse process takes place at a much reduced rate in a low temperature bath . at a phonon bath temperature of @xmath1 , exciton populations greater than 0.9 are easily achievable . direct exciton excitation and excitation through a coupled cavity are found to be qualitatively different . in particular , in the latter case , the effect of cavity filtering is important and depends upon the sign of the cavity - exciton detuning . this work was supported by the national sciences and engineering research council of canada and the marsden fund of the royal society of new zealand . the authors thank p. michler , s. ulrich , a. ulhaq , and s. weiler for useful discussions . 99 see , e.g. , k. hennessy , a. badolato , m. winger , d. gerace , m. atatre , s. gulde , s. flt , e. l. hu and a. imamolu , nature 445 , 896 ( 2007 ) ; m. nomura , n. kumagai , s. iwamoto , y. ota , and y. arakawa , nature phys . * 6 * , 279 ( 2010 ) . l. besombes , k. kheng , l. marsal , and h. mariette , phys . b * 63 * , 155307 ( 2001 ) . a. j. ramsay , a. v. gopal , e. m. gauger , a. nazir , b. w. lovett , a. m. fox , and m. s. skolnick , phys . lett . * 104 * , 017402 ( 2010 ) . s. m. ulrich , s. ates , s. reitzenstein , a. lffler , a. forchel and p. michler , phys . lett . * 106 * , 247402 ( 2011 ) . c. roy and s. hughes , phys . rev . lett . * 106 * , 247403 ( 2011 ) . j. frstner , c. weber , j. danckwerts , and a. knorr , phys . rev . lett . * 91 * , 127401 , 2003 . c. roy and s. hughes , phys . x * 1 * , 021009 ( 2011 ) . s. hughes , p. yao , f. milde , a. knorr , d. dalacu , k. mnaymneh , v. sazonova , p. j. poole , g. c. aers , j. lapointe , r. cheriton and r. l. williams , phys . b * 83 * , 165313 ( 2011 ) . t. takagahara , phys . b * 60 * , 2638 ( 1999 ) .

we investigate pump - induced exciton inversion in a quantum - dot cavity system with continuous wave drive . using a polaron - based master equation , we demonstrate excited - state populations above @xmath0 for an inas dot at a phonon bath temperature of @xmath1 . in an exciton - driven system , the dominant mechanism is incoherent excitation from the phonon bath . for cavity driving , the mechanism is phonon - mediated switching between ground- and excited - state branches of the ladder of photon states , as quantum trajectory simulations clearly show . the exciton inversion as a function of detuning is found to be qualitatively different for exciton and cavity driving , primarily due to cavity filtering . the master equation approach allows us to include important radiative and non - radiative decay processes on the zero phonon line , provides a clear underlying dynamic in terms of photon and phonon scattering , and admits simple analytical approximations that help to explain the physics .

introduction theory results conclusions acknowledgments references

arxiv

directed polymers interacting with a one dimensional defect line are quite rich in physical and biological applications , and lately have started to attract much attention also in the mathematical literature @xcite , @xcite@xcite . in particular , they are an ideal framework to model @xmath2dimensional interface wetting phenomena @xcite , the problem of depinning of flux lines from columnar defects in type ii superconductors @xcite and the denaturation transition of dna in the poland scheraga approximation @xcite . in many situations , the polymer defect interaction is neither homogeneous nor periodic along the line . this corresponds for instance to the presence of impurities on the wall in the case of the wetting problem , and to the non periodic arrangement of base pairs a t , g c along dna sequences . therefore , one resorts very naturally to quenched disordered models . the interplay between the ( _ energetic _ ) pinning effect , which tends to keep the polymer close to the defect line , and the ( _ entropic _ ) depinning one , favoring configurations which wander away from the line , is responsible for a non trivial pinning / depinning ( or localization / delocalization ) phase transition . the depinned and pinned phases are characterized by a different behavior of the order parameter , the _ contact fraction _ , which is essentially the density of polymer defect contacts along the line . in the pinned phase , the contact fraction stays positive in the thermodynamic limit , while it vanishes in the interior of the depinned phase ( finite size estimates of the latter statement can be found in @xcite ) . a very interesting problem is to understand what happens at the _ critical line _ separating the two phases . recently , with g. giacomin we proved that , as soon as disorder is present , the contact fraction vanishes continuously when the critical line is approached from the pinned region @xcite . this is in striking contrast with the situation in pure ( i.e. , non disordered ) pinning models , where the transition can be either of first or of higher order , depending for instance on the space dimension . given this result , it is very natural to investigate how fast the contact fraction vanishes with system size , _ at _ the critical line or in a small critical window around it . this question is addressed in theorem [ th : fss ] of the present paper , where it is shown for instance that , in the disordered situation , the contact fraction is at most of order @xmath3 at criticality . inside the localized region , the length of the maximal excursion of the polymer ( i.e. , of the longest portion of the polymer without contacts with the defect line ) is @xmath4 @xcite , where @xmath1 is a certain annealed exponent ( cf . section [ sec : versus ] for its definition ) and @xmath5 is the total length of the polymer . when the critical line is approached @xmath1 tends to zero , as well as the free energy @xmath0 . in theorem [ th : bounds ] we prove an inequality which essentially relates the critical exponents which govern the vanishing of @xmath1 and @xmath0 at the critical line . this inequality is interesting also because , in the particular case of a @xmath2dimensional wetting model , we prove in theorem [ th : correlazioni ] that @xmath6 and @xmath7 coincide with the the typical and disorder averaged correlation lengths of the system , respectively . as we discuss briefly in section [ sec : gen ] , the finite size estimates of theorem [ th : fss ] and the bounds of theorem [ th : bounds ] have a very natural generalization to the case of random copolymers at a selective interface between two solvents @xcite , which also show a localization / delocalization transition . in this case , the relevant order parameter is not the contact fraction but the fraction of monomers in the unfavorable solvent . let @xmath8 be a time homogeneous process with law @xmath9 , taking values in some set @xmath10 and such that @xmath11 . we will be especially interested in the returns to zero of @xmath12 : we let @xmath13 and , for @xmath14 , @xmath15 . if @xmath16 , then by convention @xmath17 . the only assumptions we make on @xmath9 is that @xmath18 is a sequence of iid random variables taking values in @xmath19 and that , defining @xmath20 , there exists @xmath21 such that @xmath22 and @xmath23 if @xmath24 , for some @xmath25 and a function @xmath26 varying slowly at infinity , i.e. , a positive function such that @xmath27 for every @xmath28 @xcite . an example of slowly varying function is @xmath29 , for @xmath30 , but also @xmath31 , for @xmath32 , as well as any positive function for which @xmath33 . on the _ defect line _ @xmath34 are placed random charges @xmath35 which we assume to be iid _ bounded _ random variables with law @xmath36 . we will assume that @xmath37=0 $ ] and @xmath38=1 $ ] ( which , as will be clear from below , implies no loss of generality ) . the hamiltonian describing the interaction between the polymer and the defect line depends on two parameters , @xmath39 ( playing the role of the strength of the disorder ) and @xmath40 ( where @xmath41 represents the average energetic gain of a polymer line contact ) : @xmath42 the corresponding boltzmann distribution is @xmath43 and , of course , the partition function is given by @xmath44 here and in the following , we assume that @xmath45 , even when not explicitly stated . as equation shows , the polymer tends to touch the defect line at points where @xmath46 and to avoid it in the opposite situation . note that there is a competition between an energetic effect ( trying to touch as many favorable points as possible along the line ) and an entropic one ( trajectories which stay close to the line are much less numerous than those which wander away ) . therefore , it is quite intuitive ( and actually well known ) that a ( de)localization transition takes place when the strength of the polymer line interaction is varied . this will be discussed below . we restrict to bounded disorder variables @xmath47 just for simplicity of exposition . the results below can be extended to more general situations but we will not pursue this line . let us just mention that all the results of this paper hold also in the gaussian case @xmath48 . in more general cases of continuous , unbounded disorder variables , a sufficient condition for the results to hold is that the sub gaussian concentration inequality is satisfied by @xmath36 and that a certain condition on the smoothness of the density of @xmath49 with respect to the lebesgue measure on @xmath50 holds ( cf . @xcite , condition * c2 * ) . a discussion of the relevance of concentration of measure inequalities in pinning and copolymer models can be found in @xcite . note that only the model with endpoint @xmath51 pinned to zero is being considered , cf . this is just for simplicity of exposition , since this way one has for @xmath52 @xmath53 ( @xmath54 is the left shift : @xmath55 ) , a property we will use several times in the proofs of section [ sec : proofs ] . by the way , note that implies that the sequence @xmath56 is super additive in @xmath5 . one could also leave the endpoint free : in this case , in the r.h.s . of eq . error terms of order @xmath57 would appear ( cf . e.g. ( * ? ? ? * remark 1.1 ) ) . as a consequence , in the proof of the theorems one would have to keep track of harmless but annoying logarithmic error terms . to make condition more explicit note that , for instance , if @xmath58 is the srw ( simple random walk ) on @xmath59 , then holds with @xmath60 and @xmath61 for @xmath62 and @xmath63 for @xmath64 . the poland scheraga model of dna denaturation also fits into our framework ; in this case , the physically relevant value of @xmath65 is around @xmath66 @xcite . for the poland scheraga model , the contact fraction defined in eq . below corresponds to the fraction of bound base pairs . as it is well known the infinite volume free energy , i.e. the limit @xmath67 exists , is almost surely independent of @xmath68 and satisfies @xmath69 ( cf . e.g. @xcite , @xcite , but proofs of these facts have appeared several times in the literature . the non negativity of @xmath0 is proven by simply restricting the average in to the configurations which do not touch zero between sites @xmath70 and @xmath5 , and using eq . . ) one decomposes the phase diagram @xmath71 into depinned ( or delocalized ) and pinned ( or localized ) phases , @xmath72 and @xmath73 , defined as @xmath74 and @xmath75 , separated by a critical line @xmath76 . various properties of the critical curve are known @xcite @xcite : in particular , under our assumptions one has that , for every @xmath77 , @xmath78 note that @xmath79 , and @xmath80 iff @xmath12 is transient . moreover , @xmath81 is a convex increasing function , as follows easily from the convexity of @xmath0 with respect to its arguments and from . the order parameter associated to the ( de)localization transition is the _ contact fraction _ , defined as @xmath82 since @xmath0 is clearly convex as a function of @xmath83 , and since it is differentiable in @xmath83 for every @xmath84 ( as was proven in @xcite ) , from the definitions of @xmath85 it follows that , @xmath86a.s . , @xmath87 while @xmath88 however , much more than is true : indeed , in @xcite it was proven that , for @xmath89 , @xmath90 if @xmath91 , for some constants @xmath92 . in other words , the number of contacts with the defect line grows , typically , linearly with @xmath5 for @xmath84 and at most logarithmically in @xmath5 for @xmath89 . finally , in @xcite@xcite it was proven that @xmath93 vanishes continuously for @xmath94 if @xmath95 , which implies that , @xmath86a.s . , @xmath96 in view of these facts , it is very natural to ask what is the typical size of the contact fraction for finite @xmath5 , _ at _ the critical point or very close to it . this question will be addressed in the next section . since we are interested in the finite size scaling behavior of the system in a window around the critical point , shrinking to zero with the system size , we allow in general @xmath83 to depend on @xmath5 , and write explicitly @xmath97 . [ th : fss ] let @xmath95 and @xmath25 . assume that @xmath98 for some @xmath99 . then , 1 . if @xmath100 , then for @xmath101 sufficiently large @xmath102 2 . if @xmath103 and @xmath104 , then for @xmath101 sufficiently large @xmath105 3 . if @xmath103 and @xmath106 , then for @xmath101 sufficiently large @xmath107 it is understood that the constant @xmath101 above can depend on @xmath108 , @xmath65 and @xmath109 . note that , for @xmath110 and @xmath104 , one finds back the known estimates on the contact fraction valid in the interior of @xmath72 @xcite . the estimates of theorem [ th : fss ] need not be optimal , in general . indeed , as will be clear in section [ sec : proofs ] , our proof is based on the fact that @xmath0 vanishes at least quadratically when the critical line is approached from the localized region and @xmath95 @xcite : @xmath111 for some constant @xmath112 , if @xmath84 . on the other hand it is quite reasonable , and actually expected in the physics literature , that the transition is smoother in various situations , for instance if @xmath113 and @xmath108 small . following the proof of theorem [ th : fss ] in section [ sec : proofs ] it is not difficult to realize ( cf . remark [ rem : ipo ] below ) that , if one assumes @xmath114 for every @xmath84 then , for instance , @xmath115 for @xmath101 sufficiently large . if @xmath116 , this would clearly improve the upper bound on the contact fraction at the critical point given by theorem [ th : fss ] . estimates could also be similarly improved for all values of @xmath117 and @xmath109 . unfortunately , up to now there are no known cases where one can prove an estimate like , with @xmath116 , for non zero values of @xmath108 . in refs . @xcite and @xcite , the quantity @xmath119\end{aligned}\ ] ] was introduced . as it was proved there , in the localized phase @xmath1 is strictly positive and related to maximal excursions of the polymer from the defect line : indeed , for the polymer of length @xmath5 the maximal distance between two successive returns to zero of @xmath12 is typically @xmath120 . when @xmath83 approaches @xmath121 from below , @xmath1 tends to zero and therefore the length of the maximal excursion diverges , on the scale @xmath57 . more precisely , the following bounds were proven in @xcite : for every @xmath95 there exists @xmath122 such that @xmath123 where the lower bound holds , say , for @xmath124 . our next result significantly improves the lower bound in : [ th : bounds ] for every @xmath95 there exists @xmath125 such that @xmath126 if @xmath124 . in order to give a more readable form to these bounds assume that , for @xmath95 and @xmath127 , @xmath128 and @xmath129 for some functions @xmath130 slowly varying in @xmath131 for @xmath132 ( of course @xmath133 , as a consequence of eq . and of the upper bound in ; in principle , @xmath134 can depend on @xmath108 ) . then , recalling the definition of slow variation and the fact that @xmath0 is convex in @xmath83 , one realizes that eq . implies @xmath135 while from follows that @xmath136 beyond giving informations about the divergence of the longest excursion close to ( but below ) the critical line , bounds like involving @xmath1 and @xmath0 are of interest because it is rather natural to expect that @xmath7 ( respectively @xmath6 ) has the same divergence , for @xmath83 approaching @xmath121 from the localized phase , as the average ( respectively typical ) correlation length of the system . our next result , theorem [ th : correlazioni ] , makes this conjecture precise at least in a specific model of @xmath2dimensional wetting . recall that in ( * ? ? ? * theorem 2.2 ) it was proven that , for every bounded local observable @xmath137 ( i.e. , bounded function which depends on @xmath138 only for @xmath139 in a finite subset of @xmath140 ) , the infinite volume limit @xmath141 exists @xmath86almost surely , if @xmath142 . moreover , in @xmath73 truncated correlation functions decay exponentially fast with distance . in fact , for every bounded local observables @xmath143 define the local observable @xmath144 as @xmath145 , where @xmath54 is the left shift , @xmath146 . then , there exist a constant @xmath147 , an almost surely finite random variable @xmath148 and a constant @xmath149 such that @xcite , in @xmath73 , @xmath150 and @xmath151 however , in @xcite the @xmath71 dependence of the constant @xmath152 was not tracked , and lower bounds complementary to eqs . , were not obtained . it turns out that this gap can be filled , at least in the case of a rather natural @xmath2dimensional wetting model we define now . this model still belongs to the class described by the boltzmann distribution but , in addition to the basic assumptions of section [ sec : model ] , we require that the state space of the process @xmath12 is @xmath153 ( i.e. , there is an impenetrable wall which prevents @xmath154 ) and that actually @xmath12 is the srw with increments @xmath155 , conditioned to be non negative ( the condition @xmath156 could be somewhat relaxed in the theorem below , at the price of some further technical work . we will not pursue this line ) . note that in this case holds with @xmath61 and @xmath60 . this model has a natural interpretation as a _ @xmath2dimensional wetting model of a disordered substrate _ the defect line represents a wall with impurities , and @xmath12 the interface between two coexisting phases ( say , liquid below the interface and vapor above ) . when @xmath157 the underlying homogeneous substrate repels the liquid phase , and vice versa for @xmath158 . @xmath73 corresponds then to the _ dry phase _ ( microscopic liquid layer at the wall ) and @xmath72 to the _ wet phase _ ( macroscopic layer ) . then , one has : [ th : correlazioni ] for the wetting model just introduced , the following holds : for every @xmath39 and @xmath84 , @xmath159 and , @xmath86a.s . , @xmath160 here it is understood that @xmath161 , due to the periodicity of the simple random walk . it would be extremely interesting , especially in view of theorem [ th : correlazioni ] , to fill the gap between the upper bound in and the lower bound ( or equivalently , between and ) . in the case of the @xmath2dimensional wetting model with @xmath162 increments , this would answer the question whether typical and average correlation lengths have the same critical behavior close to the depinning transition , or if their divergence is governed by different critical exponents , as it happens for instance in the disordered ising spin chain with random transverse field of ref . @xcite . in this section we sketch briefly how theorems [ th : fss ] and [ th : bounds ] can be extended to the model of _ random copolymer at a selective interface _ @xcite . we refer for instance to @xcite for physical motivations of this model . in this case , the state space of @xmath12 is @xmath163 and , in addition to time homogeneity of @xmath12 and to the iid property of the sequence @xmath164 , one assumes that @xmath165 and that @xmath9 is invariant under the transformation @xmath166 . the hamiltonian is replaced by @xmath167 where , without loss of generality in view of the symmetry of @xmath9 , we can assume that @xmath168 . the variables @xmath169 are iid centered and satisfy the same boundedness assumption as in section [ sec : model ] . the boltzmann distribution and the partition function @xmath170 are defined as in eqs . , , provided that @xmath171 is replaced by @xmath172 . one should imagine the model as describing a polymer @xmath12 in proximity of the interface ( @xmath34 ) between two solvents a and b , placed in the half planes @xmath173 and @xmath174 , respectively . note that @xmath175 has the tendency to be in a whenever @xmath176 and in b if @xmath177 . note also that , if @xmath158 , for a typical disorder realization the polymer has a net preference to be in a , which will be called the _ favorable solvent_. again , it is known @xcite that the infinite volume free energy @xmath178 exists , is almost surely independent of @xmath68 and non negative , so that one can define the localized and delocalized phases , @xmath179 and @xmath180 , is analogy to section [ sec : model ] . upper @xcite and lower @xcite bounds are known for the critical curve @xmath181 but , on the basis of careful numerical simulations plus concentration of measure considerations , none of them is believed to be optimal in general @xcite . in contrast with the case of the pinning models of section [ sec : model ] , for the copolymer the order parameter associated to the localization / delocalization transition is the _ fraction of monomers in the unfavorable solvent _ : @xmath182 this is rather intuitive since , comparing definitions and , one notices that the role of @xmath183 is now played by @xmath184 . like for the contact fraction in pinning models , various estimates on the order parameter are known : @xmath185 is of order @xmath186 in @xmath179 , at most of order @xmath187 in the interior of @xmath180 @xcite and @xmath188 for @xmath189 at the critical line @xcite . the methods we introduce in the present paper allow to make the last statement sharper : indeed , theorem [ th : fss ] holds unchanged also for the copolymer model , provided that @xmath190 is replaced by @xmath191 . in particular , therefore , @xmath192 is at most of order @xmath3 at the critical point . theorem [ th : bounds ] also admits a natural extension to the copolymer case : if @xmath193 is defined as in , with @xmath194 replaced by @xmath170 , then again eq . holds with @xmath195 replaced by @xmath196 . in order to avoid a useless duplication of the proofs of theorems [ th : fss ] and [ th : bounds ] , in section [ sec : proofs ] we will consider only the case of pinning models and we will not give details for the copolymer case : as it was also the case in refs . @xcite@xcite , it is easy to realize that the two models can be treated analogously , if the correct order parameter is used in each case . just to give an example , eq . below holds also for the copolymer , if @xmath190 is replaced by @xmath197 , as was proven in ( * ? ? ? * lemma 2.1 ) . given a set @xmath198 of polymer configurations , measurable with respect to @xmath9 , it is convenient to set @xmath199 our basic technical tool is the following classical concentration inequality @xcite : if @xmath200 is a sequence of iid bounded random variables with law @xmath36 , there exist constants @xmath201 such that , for every convex lipschitz function @xmath202 , one has : @xmath203 for every @xmath204 , where @xmath205 is the lipschitz norm of @xmath206 with respect to the euclidean norm in @xmath207 , i.e. , the smallest @xmath208 such that @xmath209^{1/2}}\le m.\ ] ] the way we will employ this inequality is by noting that @xmath210 , considered as a function of @xmath211 , is convex and has a lipschitz constant at most @xmath212 . more generally , one has the following ( * ? ? ? * lemma 2.1 ) : let @xmath213 be a set of polymer trajectories such that @xmath214 for every @xmath215 . then , @xmath216 this is simply proven by noting that @xmath217 has a lipschitz constant at most @xmath218 . for @xmath219 , consider the restricted partition function @xmath220 where the number of contacts with the line , @xmath221 , is constrained to @xmath222 . thanks to the fact that the differences @xmath223 between successive return times to zero of @xmath12 are independent under the law @xmath9 , one has @xmath224 where @xmath225\right).\end{aligned}\ ] ] the limits in exist for monotonicity reasons . in @xcite it was proven that , under some assumptions on @xmath36 ( assumptions which are satisfied , in particular , in the case of bounded random variables we are considering here ) , one has for @xmath95 @xmath226 for some constant @xmath227 depending only on the law @xmath36 . [ rem : ipo ] equation follows simply from eq . and from the fact that , as was proven in @xcite , @xmath0 is related to the function @xmath228 of eq . via the legendre transformation @xmath229}\left(\phi(\beta , x)-hx\right).\end{aligned}\ ] ] ( actually , in @xcite the reverse path was followed : first was proven , and then was deduced ) . if one could prove eq . with @xmath116 , would be immediately improved into @xmath230 for some @xmath231 . equation , together with , implies that for every @xmath45 , @xmath219 @xmath232 let us consider first the case @xmath104 , @xmath103 . then , for @xmath5 sufficiently large one has , uniformly in @xmath222 , @xmath233 we let @xmath234 be the event @xmath235 to estimate the probability of @xmath234 , we employ eq . and we find @xmath236\le c_1\sum_{m\ge c n^{2t}\log n } e^{-c_2\frac { b^2 m n^{-2t}}{16\beta^2}}\end{aligned}\ ] ] which decays to zero for @xmath189 , if @xmath101 is large enough . on the complementary of the event @xmath234 , one the other hand , one has @xmath237 which also decays to zero . in we used the obvious bound @xmath238 cf . eq . and the definition of slowly varying function . equations and together imply . next , consider the case @xmath239 . it is immediate to check that , for @xmath5 sufficiently large and @xmath240 , the r.h.s . of eq . is smaller than @xmath241 then , one defines @xmath242 and notes that , in analogy with eqs . and , @xmath243\le c_1\sum_{m\ge c n^{2/3}\log n } e^{-c_2\frac { c_4(\beta)^2 m^3 } { 16\alpha^2n^2\beta^2}}\end{aligned}\ ] ] while , on the complementary of the event @xmath244 , @xmath245 which together imply , for @xmath101 large . finally , the case @xmath106 and @xmath246 . one realizes easily that , for @xmath247 sufficiently large and @xmath248 , the r.h.s . of eq . is smaller than @xmath249 then , one defines @xmath250 and notes that @xmath251\le c_1\sum_{m\ge c n^{1-t } } e^{-c_2 \frac { b^2}{4\beta^2}n^{-2t}m } \ ] ] which decays to zero for @xmath189 since @xmath246 while , on the complementary of the event @xmath252 , @xmath253 equation follows as in the previous cases . @xmath254 define preliminarily , for every @xmath255 $ ] and @xmath256 , @xmath257\right)<\frac{{\textsc{f}}(\beta , h)}2 \right\}.\end{aligned}\ ] ] then , @xmath258 \ , & \le \ , \exp\left(-n{\textsc{f}}(\beta , h)/2\right)+ { { \ensuremath{\mathbb e } } } \left[\frac{{\mathbf{1}}_{\{e_{n , x,\varepsilon}\}}}{z_{n,{\omega}}^{\beta , h}}\right ] \\ & \le\exp\left(-n{\textsc{f}}(\beta , h)/2\right)+ c_5 n^{2\alpha}{{\ensuremath{\mathbb p } } } [ e_{n , x,\varepsilon } ] . \end{split}\ ] ] thanks to the legendre transformation relation and from the infinite differentiability of the free energy for @xmath84 @xcite , it follows that the value @xmath259 , which realizes the supremum in eq . , is unique , smooth as a function of @xmath83 and satisfies @xmath260 . moreover , since @xmath261 , one has immediately @xmath262\right)={\textsc{f}}(\beta , h).\end{aligned}\ ] ] thanks to eq . , one has then for @xmath263 sufficiently small @xmath264 \le c_1e^{-c_2 \frac{n { \textsc{f}}(\beta , h)^2 } { 8\beta^2(-\partial_h{\textsc{f}}(\beta , h ) ) } } \end{aligned}\ ] ] for @xmath5 sufficiently large . therefore , recalling eq . , always for @xmath5 large one finds @xmath265 \ , & \le \ , \exp\left(-n{\textsc{f}}(\beta , h)/2\right)+ c_1 e^{-c_2 \frac{n { \textsc{f}}(\beta , h)^2 } { 16\beta^2(-\partial_h{\textsc{f}}(\beta , h ) ) } } \end{split}\ ] ] which immediately implies eq . for @xmath266 sufficiently small . indeed , since @xmath267 is a convex function and @xmath268 , one has @xmath269 which implies that , for @xmath270 small , the second term in the r.h.s . of eq . is the larger one . @xmath271 recall that here @xmath161 . we start with the upper bounds on the correlation lengths , which are somewhat easier . observe first that @xmath272 , \end{split}\end{aligned}\ ] ] where @xmath273 is the product gibbs measure for two independent , identical copies @xmath274 of the polymer and @xmath275 is the event @xmath276 indeed , the expectation in vanishes if conditioned on the complementary of @xmath275 , as is immediately realized via a symmetry argument based on the markov property of the srw conditioned to be non negative . an analogous trick was used in the proof of ( * ? ? ? * theorem 2.2 ) . then , it follows that @xmath277 where in the second and third steps we used the fact that , since the polymer trajectories have increments of unit length , @xmath278 and @xmath279 can not cross without touching . at this point , let us condition on the last return to zero of @xmath12 before @xmath280 , which we call @xmath222 , and on its first return @xmath281 after @xmath282 , and observe that @xmath283 where , we recall , @xmath54 is the left shift : @xmath55 . from one obtains @xmath284 recalling the definition of @xmath1 and the fact that @xmath285 converges to @xmath286 @xmath86a.s . for @xmath287 , one obtains for every @xmath288 @xmath289 and @xmath290 where @xmath291 is @xmath86almost surely finite . here and in the following we omit the possible dependence on @xmath292 and @xmath293 of the constants , in order to keep notations lighter note however that @xmath294 can be chosen independent of @xmath293 . since neither @xmath294 nor @xmath291 depend on @xmath5 , the @xmath189 limit can be taken in the l.h.s . of eqs . , . as for the lower bound , we start by observing that , by eq . , one has the identity @xmath295 indeed , thanks to the constraint @xmath296 , it can not happen that @xmath297 , otherwise also @xmath298 , since @xmath299 and @xmath300 . similarly , it can not happen that @xmath301 . for this reason , the first term in the last line of gives the r.h.s . of . in view of analogous considerations , the second term is identically zero , since there are no polymer configurations belonging to @xmath275 , i.e. , not crossing each other , and satisfying @xmath302 . on the other hand , thanks to ( * ? ? ? * lemma a.1 ) , one can bound @xmath303 for some @xmath304 independent of @xmath305 , provided that @xmath306 . indeed , lemma a.1 of @xcite states that there exists an @xmath68independent constant @xmath307 such that for every @xmath308 , @xmath309 and every @xmath68 we have @xmath310 from which inequality easily follows . in order to keep notations in the following formulas simple , let us introduce some useful sets of polymer trajectories ( see figure [ fig : figura ] ) : @xmath311 of course , @xmath312 is non empty only for @xmath313 sufficiently large so that @xmath314 . = 14 cm [ c]@xmath293 [ c]@xmath315 [ c]@xmath316 [ c]@xmath317 [ c]@xmath318 [ c]@xmath319 [ c]@xmath320 [ c ] [ c]@xmath278 [ c]@xmath279 [ c]@xmath321 [ c]@xmath322 [ c](a ) [ c](b ) if @xmath323 is a @xmath324measurable set of trajectories of @xmath274 we define , in analogy with , @xmath325 then , one has the obvious lower bound @xmath326 and , thanks to eq . , @xmath327 the numerator in can be bounded below requiring that @xmath328 and @xmath329 . at this point the constraint @xmath330 becomes superfluous , since it is automatically satisfied if @xmath328 and @xmath329 , and one obtains @xmath331 note that the trajectories belonging to @xmath332 never touch the defect line in the interval @xmath333 . therefore , in @xmath334 the pinning hamiltonian gives no contribution except at the boundary point @xmath313 , and one is left with a srw computation . an easy counting of allowed trajectories gives , for large @xmath313 , @xmath335 uniformly in @xmath68 . secondly , applying repeatedly ( * ? ? ? * lemma a.1 ) one obtains @xmath336 plugging the lower bounds , into and taking the @xmath189 limit one finally finds @xmath337 the conclusions @xmath338 and @xmath339 are obtained , for every @xmath288 , by recalling the definition of @xmath340 and the fact that @xmath341 converges to @xmath286 almost surely . together with eqs . , , these imply the desired results , . @xmath342 it is interesting to compare the strategy leading to the upper bounds , with the coupling method introduced in ref . @xcite to estimate the speed of convergence to equilibrium of some special renewal sequences . the connection between polymer measures and renewal equations is not casual : for instance , a moment of reflection ( or a look at appendix a of @xcite ) shows that , in the homogeneous case @xmath343 , the polymer measure can be rewritten exactly in terms of the renewal process where the probability that the time elapsed between two successive renewals is @xmath344 is given by @xmath345 . i am extremely grateful to giambattista giacomin for countless conversations on these topics , and in particular about the contents of section [ sec : xi ] . most of this work was written during a stay at the isaac newton institute in cambridge , in the framework of the programme `` principles of the dynamics of non - equilibrium systems '' . this work was partially supported by the gip anr project jc05_42461 ( _ polintbio _ ) .

we consider models of directed random polymers interacting with a defect line , which are known to undergo a pinning / depinning ( or localization / delocalization ) phase transition . we are interested in critical properties and we prove , in particular , finite size upper bounds on the order parameter ( the _ contact fraction _ ) in a window around the critical point , shrinking with the system size . moreover , we derive a new inequality relating the free energy @xmath0 and an annealed exponent @xmath1 which describes extreme fluctuations of the polymer in the localized region . for the particular case of a @xmath2dimensional interface wetting model , we show that this implies an inequality between the critical exponents which govern the divergence of the disorder averaged correlation length and of the typical one . our results are based on on the recently proven smoothness property of the depinning transition in presence of quenched disorder and on concentration of measure ideas . + + 2000 _ mathematics subject classification : 82b27 , 82b44 , 82b41 _ + + _ keywords : directed polymers , pinning and wetting models , copolymers , depinning transition , finite size estimates , concentration of measure , typical and average correlation lengths . _

introduction random pinning models main results generalization to copolymers at a selective interface proof of the results acknowledgments

arxiv

neutrino oscillation experiments involving neutrinos and antineutrinos coming from astrophysical and terrestrial sources have found compelling evidence that neutrinos have finite but small masses . to accommodate this observation , the minimal standard model ( sm ) must be extended . generating neutrino masses through the seesaw mechanism @xcite is among the most attractive ones . it explains the smallness of neutrino mass by supplying a suppression factor of the ratio of electroweak scale to a new physics scale . there are different ways to realize seesaw mechanism . they can be categorized as type i , type ii and type iii seesaw mechanisms . the main ingredients of these models are as the followings . * type i * @xcite : introducing singlet right - handed neutrinos @xmath6 which transform as : @xmath7 under sm @xmath8 gauge group . it is clear that @xmath6 does not have sm gauge interactions . the neutrino masses @xmath9 are given by @xmath10 , where @xmath11 is the vacuum expectation value ( vev ) of the higgs doublet in the sm , @xmath12 is the yukawa coupling and @xmath13 is the right - handed neutrino mass , which sets the new physics scale @xmath14 . if @xmath15 , to obtain the light neutrino mass of order an ev or smaller , @xmath13 is required to be of order @xmath16 gev . this makes it impossible to directly detect @xmath6 at laboratory experiment . however , the yukawa coupling @xmath17 does not need to be of order one . if it turns out to be similar to or smaller than the yukawa coupling for electron , @xmath13 can be as low as a tev . * type ii * @xcite : introducing a triplet higgs representation @xmath18 transforming as : @xmath19 . in this type of models , the neutrino masses are given by : @xmath20 , where @xmath21 is the vev of the neutral component of the triplet and @xmath22 is the yukawa coupling . with a doublet and triplet mixing via a dimensionful parameter @xmath23 , the electroweak symmetry breaking ( ewsb ) leads to a relation @xmath24 , where @xmath25 is the mass of the triplet . in this case the scale @xmath14 is replaced by @xmath26 . with @xmath27 and @xmath28 , the scale @xmath14 is also @xmath16 gev . again a lower value of order a tev for @xmath29 is possible . * type iii * @xcite : introducing triplet lepton representations @xmath30 with @xmath31 sm quantum numbers . the resulting mass matrix for neutrinos has the same form as that in type i seesaw . the high scale @xmath14 is replaced by the mass of the leptons in the @xmath0 triplet representation which can also be as low as a tev . in the absence of more experimental data , it is impossible to tell which , if any , of the mechanisms is actually correct . different models should be studied using available data or future ones . the most direct way of verifying the seesaw mechanism is , of course , to produce the heavy degrees of freedom in the models if they are light enough , and study their properties . the large hadron collider ( lhc ) at cern with the unprecedented high energy and luminosity is the best place to carry out such a test . major discoveries of exciting new physics at the terascale at the lhc are highly anticipated . test of seesaw mechanism at lhc has received a lot of attentions recently @xcite . however , it is believed that any signal of @xmath6 would indicate a more subtle mechanism beyond the simple type i seesaw due to the otherwise naturally small mixing @xmath32 between the heavy neutrinos and the sm leptons . some of the ways to evade such a situation are to have some new gauge interactions @xcite or to find solutions where @xmath33 are large which can happen in inverse seesaw models @xcite . the possibility of testing the type ii seesaw mechanism at the lhc has been considered by several groups @xcite . recently one group including one of us systematically explored the parameter space in this model @xcite . using preferred parameters from experimental data , they found that in the optimistic scenarios , by identifying the flavor structure of the lepton number violating decays of the charged higgs bosons , one can establish the neutrino mass pattern of the normal hierarchy ( nh ) , inverted hierarchy ( ih ) or quasi - degenerate ( qd ) . many other signatures of type ii seesaw at the lhc have been studied @xcite . there have also been studies to test type iii seesaw at the lhc @xcite . due to the fact that the @xmath0 triplet @xmath34 has gauge interactions , the production of the heavy triplet particles can have a much larger cross section compared with that in type i seesaw . the type iii seesaw can be tested in a more comprehensive way up to the tev range . in this paper we further study some features of the type iii seesaw at lhc . to detect the signals of the heavy triplet leptons , one needs to understand their decays to sm particles which depend on how light and heavy leptons mix with each other . similar to type - i seesaw , in this model it is also possible to have small and large mixing @xmath33 between light and heavy leptons @xcite . the usual solutions with light and heavy lepton mixing of order the square root of the ratio of light and heavy masses , @xmath1 could lead to a visible displaced vertex in the detector at the lhc @xcite . this fact can be used to distinguish small mixing and large mixing between light and heavy leptons . the latter does not lead to a displaced vertex . it has long been realized that it is possible to have large light and heavy neutrino mixing originated from the so - called inverse seesaw @xcite . this possibility has also received a lot of attentions recently @xcite . with a large mixing between light and heavy leptons , one can also study single heavy lepton production @xcite . this can also be used to distinguish model parameter spaces . we will concentrate on the usual small light and heavy mixing solutions . the analysis carried out in this work , in many ways , is similar to that in ref . @xcite since in both cases the productions of heavy lepton pairs are through gauge boson mediation , and also the light and heavy lepton mixing comes from seesaw mechanism . the main differences are that in this model the heavy leptons have electroweak interactions and the mediating gauge bosons in productions are @xmath35 and @xmath36 , while in the model discussed in ref . @xcite , the heavy neutrinos do not have electroweak interactions and the mediating particle is the new neutral gauge boson @xmath37 . our analysis also has overlaps with that in ref . @xcite where type i+iii seesaw was studied , but detailed correlations are different since the model in ref . @xcite has both heavy neutrinos from type i which do not have electroweak interactions and also the triplet heavy leptons from type iii we are considering . we have checked that when applicable , our results agree with those obtained in ref . @xcite . we find that there is a relation between the low energy neutrino oscillation and mass parameters , and the heavy triplet lepton decay parameters which has not been considered before in this model . we first derive this relation , and then make concrete predictions of the heavy triplet lepton signals using this relation for the small mixing solutions . we consider two ideal production channels , 1 ) @xmath2 and 2 ) @xmath3 in detail . we also include @xmath4 events reconstruction in the analysis which turns out to give some interesting additional information . with judicious cuts at the lhc , the discovery of the heavy triplet leptons as high as a tev can be achieved with 100@xmath5 integrated luminosity . with 300@xmath5 integrated luminosity , the reach of the scale for heavy triplet leptons can be higher . the paper is arranged as the following . in sec . ii we summarize some basic features of type iii seesaw model , paying particular attention to the heavy triplet lepton couplings to sm bosons and light leptons , and display relations between the low energy neutrino oscillation and mass parameters . in sec . iii we study constraints on the relevant parameters in the model , taking full advantage of the relations obtained in sec . ii . in sec . iv we study the heavy triplet lepton decays . in sec . v we study production of heavy triplet leptons and the detection signals at the lhc . finally in sec . vi we summarize our main results . we also include two appendices , appendix a and appendix b , to provide more details on the derivation of the relation displayed in sec . ii and the general expressions for the heavy triplet lepton decay parameters . the type iii seesaw model consists , in addition to the sm particles , left - handed triplet leptons with zero hypercharge , @xmath38 under @xmath39 @xcite . we write the component fields as @xmath40 the charge conjugated form is @xmath41 note that @xmath42 is right - handed . the renormalizable lagrangian involving @xmath43 is given by @xmath44-{1\over 2}{\rm tr}[\overline{\sigma_l^c}m_\sigma \sigma_l+\overline{\sigma_l}m_\sigma^\ast \sigma_l^c]-\overline{l_l}\sqrt{2}y_\sigma^\dagger \sigma_l^c \tilde{h}-\tilde{h}^\dagger \overline{\sigma_l^c}\sqrt{2}y_\sigma l_l\;.\end{aligned}\ ] ] here we have defined that @xmath45 with @xmath46 . in the above @xmath47 is the left - handed doublet lepton field , and @xmath48 is the higgs doublet filed . with a non - zero vacuum expectation value @xmath49 for the higgs field , the doublet leptons receive masses , and also mix the doublet and triplet leptons . the relevant terms in the lagrangian for mass matrices are given by @xmath50 the second line above gives the seesaw mass matrix for neutrinos . there are many different features for type iii seesaw compared with the other types . unlike type i seesaw model , in this model the doublet charged leptons mix with the triplet charged leptons leading to tree level flavor changing neutral current involving changed leptons @xcite . the fact that the heavy triplet leptons in type iii seesaw have gauge interaction also leads to other different phenomenology @xcite . different extensions of the simplest model can also achieve different goals @xcite . for detailed studies , one needs to understand the mass matrices in eq . [ mass - matrix ] and their diagonalization further . the diagonalization of the mass matrices can be achieved by making unitary transformations on the triplet , the charged and neutral , leptons defined in the following @xmath51 where @xmath52 and @xmath53 are @xmath54 unitary matrices , for 3 light doublet and 3 heavy triplet lepton fields , which we decompose into @xmath55 block matrices as @xmath56 for our studies we need to know gauge and higgs boson couplings to leptonic fields . in the weak interaction basis , they can be written as @xmath57 where @xmath58 . in the mass eigen - state basis , the photon couplings to fermions are diagonal , but @xmath36 couplings are more complicated . we have @xmath59 where @xmath60,\nonumber\\ \mathcal{l}_{ncz}^b&=&-{g\over 2c_w}\overline{\sigma_{ml}^0}v^l_{z\sigma\sigma}\gamma^\mu p_l \sigma_{m'l}^0z^0_\mu\;,\nonumber\\ \mathcal{l}_{ncz}^c&=&{g\over 2c_w}\overline{\nu_m}v^l_{z\nu\sigma}\gamma^\mu p_l \sigma_{m'l}^0 z_\mu^0,\nonumber\\ \mathcal{l}_{ncz}^d&=&{g\over \sqrt{2}c_w}[\overline{l_m}v_{zl\psi}^l\gamma^\mu p_l \psi_{m'}z_\mu^0+\overline{l_m}v_{zl\psi}^r\gamma^\mu p_r \psi_{m'}z_\mu^0],\nonumber\\ \mathcal{l}_{ncz}^e&=&-{g\over 2c_w}\overline{\nu_m}v^l_{z\nu\nu}\gamma^\mu p_l \nu_{m'}z^0_\mu,\\ \mathcal{l}_{ncz}^f&= & -{g\over c_w}[\overline{l_m}v_{zll}^l\gamma^\mu p_l l_{m'}z_\mu^0+\overline{l_m}v_{zll}^r\gamma^\mu p_r l_{m'}z_\mu^0],\nonumber\end{aligned}\ ] ] and @xmath61 for the charged current interactions , we have @xmath62 where @xmath63,\nonumber\\ \mathcal{l}_{cc}^b&=&-{g\over \sqrt{2}}[\overline{l_m}v^l_{l\sigma}\gamma^\mu p_l \sigma_{m'l}^0 w^-_\mu+\overline{l_m}v^r_{l\sigma}\gamma^\mu p_r \sigma_{m'l}^{0c}w^-_\mu],\nonumber\\ \mathcal{l}_{cc}^c&=&-{g\over \sqrt{2}}[\overline{\psi_m}v_{\psi \nu}^{l}\gamma^\mu p_l \nu_{m'l}w_\mu^-+\overline{\psi_m}v_{\psi \nu}^{r}\gamma^\mu p_r \nu_{m'l}^{c}w_\mu^-],\\ \mathcal{l}_{cc}^d&=&-{g\over \sqrt{2}}[\overline{l_m}v^l_{l\nu}\gamma^\mu p_l \nu_{m'l } w^-_\mu+\overline{l_m}v^r_{l\nu}\gamma^\mu p_r \nu_{m'l}^{c}w^-_\mu]\;,\nonumber\end{aligned}\ ] ] and @xmath64 in the above we have made the approximation @xmath65 . strictly speaking @xmath66 is not unitary as the usual definition of the unitary @xmath55 @xmath67 matrix . the correction is at the order of @xmath68 . it is a good approximation since we are working with the small light and heavy neutrino mixing scenario . one finds an interesting relation @xmath69 the detailed derivation is given in appendix a. a similar relation without the last two terms on the right in the above equation for type i seesaw has been derived in ref . @xcite . the physical higgs @xmath70 interactions with leptonic fields , in the mass eigen - state basis , are given by @xmath71 where @xmath72h^0\;,\end{aligned}\ ] ] and @xmath73/\sqrt{2},\nonumber\\ & & v_{sl\psi}^l = u^\dagger_{r\psi l}y_\sigma u_{ll\psi } + { 1\over \sqrt{2 } v } u^\dagger_{lll } m_l u_{rl\psi } , \ \ v_{sl\psi}^r = u^\dagger_{lll}y_\sigma^\dagger u_{r\psi\psi } + { 1\over \sqrt{2 } v } u^\dagger_{rll } m_l u_{ll\psi}\;.\end{aligned}\ ] ] in principle , the matrices @xmath52 and @xmath74 can be expressed in terms of @xmath75 , @xmath76 and @xmath77 . since for seesaw mechanism to work , @xmath78 should be small , one can expand @xmath52 and @xmath74 in powers of @xmath79 to keep track of the leading order contributions . for this purpose , it is convenient to write the leading order expressions up to @xmath80 in the basis where @xmath76 and @xmath77 are already diagonalized , without loss of generality . the following results have been obtained in the literature @xcite @xmath81 to leading order in @xmath78 , we have interaction terms involving heavy triplet leptons as @xmath82z_\mu^0\;,\nonumber\\ & & \mathcal{l}_{cc}=-g [ \overline{e}\gamma^\mu n + { 1\over \sqrt{2}}\overline{l}v_{ln}\gamma^\mu p_ln _ { } + \overline{e}v_{ln}^tv_{pmns}^\ast\gamma^\mu p_r\nu_{}]w^-_\mu+h.c.\;,\label{sss}\\ & & \mathcal{l}_{s}={g\over 2m_w}[\overline{\nu}(v_{pmns}^\dagger v_{ln}m_{n}^{diag}p_r+v_{pmns}^tv^{\ast}_{ln}m_{n}^{diag}p_l)n_{}+ \sqrt{2}\overline{l}v_{ln}m_{e}^{diag}p_re_{}]h^0+h.c.\;,\nonumber\end{aligned}\ ] ] with @xmath83 . in the above , all fields are in mass eigen - states . the @xmath84 , @xmath85 and @xmath86 are mass eigen - states of @xmath87 , @xmath34 , and the eigen - mass matrices , respectively . note that the interactions involving light neutrinos in the above have the additional @xmath67 factor compared with those involving light charged leptons . to the same order , we also have @xmath88 this equation plays an important role in constraining the elements in the coupling matrix @xmath89 . in the study of decay of @xmath84 and @xmath85 into sm particles , the interaction matrix @xmath89 plays an important role . knowledge about it is crucial . in this section we study constraints on @xmath89 and the decay branching ratios of @xmath84 and @xmath85 . [ vl ] provides very important constraints on @xmath89 . as have been mentioned before that there are two classes of solutions , the cases with small and large mixing between light and heavy leptons . the small mixing case is characterized by the fact that in the limit , @xmath90 goes to zero , the elements in @xmath91 also go to zero , and the elements in @xmath91 are of order @xmath1 . but with more than one generations it is possible to have non - trivial solutions for eq . [ vl ] which have large mixing between light and heavy leptons , as have been shown in refs . @xcite and @xcite . the cases with small and large mixing have very different experimental signatures . the small mixing solution case will lead to a visible displaced vertex in the detector at the lhc . while for the large mixing case , one can also study single heavy lepton production @xcite . the aim of this paper is to study the correlations of heavy lepton productions and decays with low energy neutrino oscillation parameters and masses . therefore in this section we will discuss constraints on the physical parameters for small mixing solutions . on the right - handed side of eq . [ vl ] , the parameters involved are in principle measurable parameters , the neutrino masses and mixing angles . therefore in order to understand the constraints we need to know as much as these parameters . as has been mentioned before that in our case the @xmath67 is , in general , not unitary . however , since the deviation is of order @xmath92 , to a good approximation , we can neglect these corrections and use a unitary matrix to represent it which can be written as @xmath93 where @xmath94 , @xmath95 , @xmath96 and @xmath97 . the phase @xmath98 is the dirac cp phase , and @xmath99 are the majorana phases . the experimental constraints on the neutrino masses and mixing parameters , at @xmath100 level @xcite , are @xmath101 and no constraints on the phases . the neutrino masses are bounded by @xmath102 @xcite . for a complete discussion of these constraints see reference @xcite . following the convention , we denote the case @xmath103 as the normal hierarchy ( nh ) and otherwise the inverted hierarchy ( ih ) . in our later discussions unless specified for the input values of relevant parameters , when scanning the parameters space we will always allow @xmath104 to run the above allowed ranges , and the lightest neutrino mass for nh and ih cases to run the range @xmath105 ev . [ vl ] relates @xmath89 to low energy measurable quantities , but the elements in @xmath89 can not be fully determined . certain assumptions or new parameters need to be introduced to describe the ranges for the elements in @xmath89 . in the following we consider in details for the size of @xmath89 with the majorana phases set to zero first , and then comment on the effects of non - zero majorana phases . we start with a simple but interesting case where the heavy triplet leptons are degenerate . in this case [ vl ] becomes simple on the left hand side with @xmath106 . here the superscript `` @xmath107 '' runs over the three light generation leptons and `` @xmath108 '' runs over the three heavy triplet leptons . @xmath109 is the heavy triplet mass . we have a simple expression from eq . [ vl ] @xmath110 where @xmath111 . explicitly we have @xmath112 if the phases in @xmath91 are all zero , the right - handed sides in the above equations are all real . we can formally write latexmath:[\[\begin{aligned } m_n \sum_{j=1,2,3 } ( v_{ln}^{ij*})^{2 } = m_n \sum_{j=1,2,3 } particular case as case i. if indeed the three heavy triplet leptons are degenerate or almost degenerate , experimentally when they are produced , one would not be able to distinguish them and therefore must sum over the heavy ones . the above equation allows one to fix the couplings completely in terms of low energy parameters . we emphases that this is true only for the case that all phases in @xmath89 are zero . the experimental information on neutrino masses and mixing indicates that the neutrino mass matrix @xmath114 presents the following patterns @xmath115 more detailed discussions can be found in ref . we plot the allowed values for the normalized couplings , @xmath116 , as a function of the lightest neutrino mass for both the nh ( left panels ) and the ih ( right panels ) cases . we see two distinctive regions in terms of the lightest neutrino mass . in the case @xmath117 ev , @xmath118 for nh and @xmath119 for ih . on the other hand , for @xmath120 ev , we have the quasi - degenerate spectrum @xmath121 as expected . [ cols="^,^ " , ] we have studied the properties of heavy @xmath0 triplet lepton in type iii seesaw model and also their signatures at the lhc for small mixing solution between light and heavy leptons . the small mixing solution is characterized by the fact that in the limit that the light neutrino masses go to zero , the mixing also goes to zero . the smallness of light neutrino masses then leads to the fact that the total decay widths of heavy leptons are small . with such small decay widths , although not considered as long - lived for large triplet mass , the heavy lepton decays could lead to a visible displaced vertex in the detector at the lhc . this displaced vertex can be observed through @xmath84 and @xmath85 reconstructions . we summarize our main results with small mixing in the following : * to a good approximation , the couplings of light charged lepton and heavy triplet leptons @xmath89 to @xmath36 , @xmath35 and @xmath70 bosons in eq . [ sss ] can be expressed with measurable neutrino mass and mixing through eq . [ vl ] with three unknown complex parameters @xmath122 in a @xmath55 orthogonal matrix given in eq . this allowed us to study the correlation between the decays of heavy triplet leptons , light neutrino masses and mixing and model parameters . with real @xmath122 , the mixing between light and heavy is small which leads to displaced vertex at the lhc for heavy leptons decays if produced . * using the relation in eq . [ vl ] , we have tried to study possible correlation in neutrino mass hierarchy and heavy lepton decays with real @xmath122 . we find that only in certain limited cases , for example case i studied in this paper , the correlation is strong . the study of heavy lepton productions and decays at the lhc may help to determine the neutrino mass hierarchy . for the more general situation case ii when heavy neutrinos are not degenerate , no such information can be extracted because the correlation is weak . however , even for this case interesting information about the model can still be extracted . if in the future the neutrino mass pattern is determined from other experiments and the sizes of elements in @xmath89 from analysis of heavy lepton productions and decays at the lhc , one may be able to obtain more information about the model parameters such as the angles @xmath122 and the majorana phases @xmath99 . * we have studied production and detection of heavy triplet leptons at lhc with judicious cuts to reduce sm background to see how large the seesaw scale can be reach at the lhc . the associated production @xmath123 is crucial to identify the quantum numbers of the triplet leptons and to distinguish between the neutrino mass hierarchies . even with only the cleanest channels @xmath124 , the signal observability can reach about @xmath125 tev for @xmath126 luminosity and @xmath127 tev for @xmath128 luminosity . * although the rate of pair production @xmath129 is smaller than @xmath123 , we demonstrated that besides the clean 4-lepton channels from @xmath130 , the @xmath4 final state can be effectively reconstructed as well . even with only the cleanest channels @xmath131 , the signal observability can reach @xmath132 tev for @xmath126 luminosity and @xmath133 tev for @xmath128 luminosity . if nature does use low scale , as low as 1 tev , to facilitate seesaw mechanism , there will be a lot of surprises to come soon after lhc will be in full operation . we urge our experimentalists to carry out searches for low scale seesaw effects . xgh was supported in part by the nsc and ncts . we acknowledge tao han for providing his fortran codes hanlib for our calculations . t. li would like to thank tao han and pavel fileviez prez for helpful discussions . to derive eq . [ vvl ] , we need to have some detailed relation of the block matrices in the unitary matrices @xmath74 and @xmath52 . for @xmath53 , we have @xmath134 from neutrino mass matrix diagonalization , we have @xmath135 and @xmath136 for @xmath52 , we have @xmath137 from charged lepton mass matrix diagonalization , we have @xmath138 and @xmath139 combining the above relations and the definition of @xmath140 , and using the approximation @xmath141 , we obtain @xmath142 which leads to @xmath143 then we have @xmath144 from the definition of @xmath145 we can also get @xmath146 which leads to @xmath147 combining eqs . [ ee ] and [ eee ] , we finally obtain @xmath148 from eq . 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we study lhc signatures of type iii seesaw in which @xmath0 triplet leptons are introduced to supply the heavy seesaw masses . to detect the signals of these heavy triplet leptons , one needs to understand their decays to standard model particles which depend on how light and heavy leptons mix with each other . we concentrate on the usual solutions with small light and heavy lepton mixing of order the square root of the ratio of light and heavy masses , @xmath1 . this class of solutions can lead to a visible displaced vertex detectable at the lhc which can be used to distinguish small mixing and large mixing between light and heavy leptons . we show that , in this case , the couplings of light and heavy triplet leptons to gauge and higgs bosons , which determine the decay widths and branching ratios , can be expressed in terms of light neutrino masses and their mixing . using these relations , we study heavy triplet lepton decay patterns and production cross section at the lhc . if these heavy triplet leptons are below a tev or so , they can be easily produced at the lhc due to their gauge interactions from being non - trivial representations of @xmath0 . we consider two ideal production channels , 1 ) @xmath2 and 2 ) @xmath3 in detail . for case 1 ) , we find that with one or two of the light leptons being @xmath4 it can also be effectively studied . with judicious cuts at the lhc , the discovery of the heavy triplet leptons as high as a tev can be achieved with 100@xmath5 integrated luminosity .

introduction the type iii seesaw model constraints on the physical parameters summary derivation of equation explicit expressions of @xmath89 for case ii

arxiv

there are several automated tools for inferring the properties of the stellar populations contributing to the integrated galaxy spectra . the list includes moped @xcite , starlight @xcite , steckmap @xcite , vespa @xcite , or ulyss , as well as the use of line indices like the lick indices @xcite . similarly , there are semi - automatic procedures to deduce the properties of the gas ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , including the so - called strong - line ratio methods . these tools are ( and will be ) fundamental for understanding the galaxy formation and evolution , but the blind use of the codes results quite unsatisfactory from a physical stand point . one obtains a precise quantitative description of the stellar populations contributing to the integrated spectra , but ignores the reason why the code has chosen them rather than other potential alternatives . the educated eye of an astronomer is often far more telling from a physical point of view . unfortunately , the know - how of qualitatively interpreting a spectrum is learned after a long experience of working in the field . the information on which particular spectral feature informs of which particular physical property is scattered among a large number of technical publications , difficult to identify and to deal with for a newcomer . this paper aims at providing a step - by - step guide to qualitative interpretation of galaxy spectra . moreover , it will be compared with up - to - date numerical techniques to show that both qualitative and quantitative results are in excellent agreement . the work was originally planned as a mere academic exercise to understand the nature of the classes resulting from the k - means classification of all the galaxy spectra in the sloan digital sky survey data release 7 ( sdss - dr7 , * ? ? ? we wanted to translate the spectral shapes into physical units like stellar ages and metallicities , so that this information can be used to tailor class - based searches , or when interpreting spectra ( e.g. , * ? ? ? * ) . however , the exercise is of interest beyond the original scope . the simple decision tree we use is suitable to characterize any galaxy spectrum . we know of its generality because it allows to separate and characterize the 28 automated spectroscopic k - means - based ( ask ) classes @xcite which , by construction , are proxies that condense the properties of the some one - million sdss spectra @xcite . the ask class characterization represents a significant part of the paper , that are discussed in detail as an illustration of the procedure . as we stress above , our qualitative analysis may have several other applications , e.g. , ( 1 ) to gain physical insight when interpreting quantitative star formation histories ( sfhs ) derived from modern automated tools , ( 2 ) for quick - look galaxy classification ( not only in the local universe , but also at moderate - high redshifts , since the hubble expansion shifts the uv - visible spectrum to the near ir ) , ( 3 ) for interpreting noisy spectra where eyeball inspection is often better than detailed inversion , ( 4 ) as reference for identifying unusual galaxies , or ( 5 ) for educational purposes to develop physical intuition . the paper is organized as follows . section [ ask_class ] introduces the ask spectral classification of galaxy spectra whose templates serve as reference point . section [ list_features ] lists and discusses spectral features commonly used when interpreting galaxy spectra . they are employed to set up the recipe introduced in sect . [ decision_tree ] , which is abridged in a schematic shown in fig . [ decision ] . the recipe ( or algorithm ) is used in sect . [ qualitative_classes ] to disclose the physical properties of all the ask classes . the results of such qualitative analysis are compared with state - of - the - art quantitative analyses in sects . [ starlight ] and [ quantitative_lines ] sect . [ starlight ] deals with the comparison of stellar components , whereas sect . [ quantitative_lines ] refers to the gas components . section [ additional_results ] discusses several additional properties of the ask templates , whereas sect . [ conclusions ] summarizes the proposed qualitative analysis . @xcite classified all the galaxies with spectra in sdss - dr7 into only 28 ask classes . the original sdss covers one - fourth of the sky and contains the spectra of all the galaxies above an apparent magnitude threshold ( sdss @xmath0 ) . therefore , the some one million sdss spectra can be regarded as representative of the galaxies of the local universe , and so do the ask classes inferred from them . the ask classification is detailed in @xcite , with additional properties of the classes discussed elsewhere . for the sake of comprehensiveness , however , we summarize here the main properties . all galaxies with redshift smaller than 0.25 were transformed to a common rest - frame wavelength scale , and then re - normalized to the integrated flux in the sdss @xmath1-filter . these two are the only manipulations the spectra underwent before classification . we wanted the classification to be driven only by the shape of the visible spectrum ( from 400 to 770 nm ) , and these two corrections remove obvious undesired dependencies of the observed spectra on redshift and galaxy apparent magnitude . we deliberately avoided correcting for other effects requiring modeling and assumptions ( e.g. , dust extinction , seeing , or aperture effects ) . the employed classification algorithm , k - means , is a robust workhorse that allows the simultaneous classification of the full data set ( @xmath212 gb ) . it is commonly employed in data mining , machine learning , and artificial intelligence ( e.g. , * ? ? ? * ; * ? ? ? * ) , and it guarantees that similar rest - frame spectra belong to the same class . most galaxies ( 99% ) were assigned to only 17 major classes , with 11 additional minor classes including the remaining 1% . it is unclear whether the ask classes represent genuine clusters in the 1637-dimensional classification space , or if they slice a continuous distribution probably the two kinds of classes are present ( see * ? ? ? * ; * ? ? ? all the galaxies in a class have very similar spectra , which are also similar to the class template spectrum formed as the average of all the spectra of the galaxies in the class . these template spectra are the ones analyzed in the paper . the averaging is slightly different from the one in @xcite , and the novelty allows us to reach the near uv of the spectrum . the sdss spectrograph detects from 3800 to 9200(e.g . * ) , however , the templates cover from 3000 to 9200 . the uv extension is recovered because the classified galaxies have redshifts up to 0.25 , which moves the rest - frame @xmath33000 within the observed range . rather than averaging the spectral range common to all galaxies , the new templates consider the full range of available rest - frame wavelengths . given a wavelength bin , it includes the spectra of all the galaxies in the class that have been observed at that particular rest - frame wavelength . consequently , the template spectra ( i.e. , the average spectra ) include wavelengths down to 3000 . the templates thus obtained vary smoothly and continuously . they are labeled according to the @xmath4 color , from the reddest , ask 0 , to the bluest , ask 27 . the use of numbers to label the classes does not implicitly assumes the spectra to follow a one dimensional family . the numbers only name the classes . the sorting ( and , so , the naming ) would have been slightly different using other bandpasses to define colors . in general , however , the smaller the ask class number the redder the spectrum . the ask classification of all galaxies with spectra in sdss - dr7 is publicly available , templates includedftp://ask:[email protected]/ + http://sdc.cab.inta-csic.es/ask/index.jsp in the spanish virtual observatory . ] . wavelengths of sdss spectra ( and of ask templates ) are vacuum wavelengths . however , all the spectra shown in this paper are transformed to air wavelengths according to the equations by @xcite . the sdss spectra used for classification , and so the templates shown along the paper , are given as flux per unit wavelength . galaxies have composite spectra . they integrate contributions from different stars of different stellar populations , from hii regions , from active galactic nuclei ( agns ) , as well as from other possible components . our qualitative analysis builds on this fact , and tries to separate each spectrum into a minimum number of components . we consider the ( ionized ) gas and the stars separately , that is to say , the emission and absorption lines separately . each one of these two components is assumed to have one or two sub - components . the details on the characterization are summarized as a decision tree in sect . [ decision_tree ] . it is based on the analysis of a set of general spectral features , listed in the next sub - section . the main spectral features that can be considered are listed in the section , ordered from the more obvious to the subtle details . each item names the particular feature , and then outlines its main properties and interest . the actual features are illustrated using the appropriate ask templates . 1 . the shape of the continuum and the presence or not of emission and absorption lines must be considered . the emission lines trace the ionized gas and its excitation mechanism . the absorption lines trace the stellar populations , their ages and metallicities . the overall continuum shape is modulated by the gas , the stars , as well as by the presence of dust . figure [ dust_effect ] shows the prototype red galaxy with passively evolving stellar populations ( ask 0 ) . although red , the continuum is rather flat from 6000 on . spectra even redder must be shaped by dust extinction ( see ask 1 in fig . [ dust_effect ] ) . the so - called 4000 _ break _ is produced by the absorption of metallic lines of a variety of elements in various states of ionization , including caii h and k ( @xmath53969 and 3934 ) and high - order lines of the balmer series ( h@xmath63970 , h@xmath73889 , h@xmath83835 , ) ( see * ? ? ? * and also fig . [ uvbreaks ] ) . the opacity suddenly increases for photons bluer than this wavelength , which produces an intensity drop . it is enhanced in old stellar populations ( ask 0 ) , which tend to be metal rich , but it is also present in younger galaxies ( ask 19 in fig . [ uvbreaks ] ) . the balmer lines become deeper and broader with time from the starburst , with a characteristic time - scale of the order of one gyr ( e.g. , * ? ? ? the limit of the balmer series and the blending of the high - order balmer lines also produces a notable discontinuity of the spectrum blueward of 3650 . it is the _ balmer break _ see fig . [ uvbreaks ] . ( photons bluer than this limit ionize the excited hydrogen , thus h becomes an important source of continuum opacity . ) it is present in young and old stellar populations , but it is more important in the young populations where h is a major constituent of the opacity ( especially in the balmer continuum beyond the discontinuity ) . the break amplitude and position is a proxy for the age of the stellar population ( e.g. , * ? ? ? 4 . the caii h and k lines ( @xmath5 3969 and 3934 , respectively ) are typical of old metal - rich stars . caii h is blended with h@xmath9 which , as the rest of the balmer series , appears in absorption in young stars ( say , a stars ) . in case of mixed populations of old and young stars , the relative intensities of caii h and caii k ( actually , of caii k and caii h+h@xmath9 ) is a proxy for the relative importance of the young and old populations . when caii k is larger than caii h , then the old population dominates the spectrum ( ask 2 in fig . [ cahk ] ) . as the young population becomes more important then caii h becomes stronger than caii k ( ask 9 in fig [ cahk ] ) . the relative growth reverts when the hii regions accompanying the young stellar populations produce enough h@xmath9 emission , which fills the caii h+h@xmath9 absorption profile ( ask 14 in fig [ cahk ] ) . + . in case of mixed young and old stellar populations , the relative importance of caii k and the blended caii h informs on the relative importance of the two populations . caii k dominates in old populations ( ask 2 ) , and caii h dominates when the young stars are more important ( ask 9 ) , but the relationship saturates when h@xmath9 starts to show up in emission ( ask 14 ) . ] the uv continuum flux is also an age indicator for very young stellar populations . it increases with decreasing age when the ages are only a few myr see fig . [ uvcontinuum ] and , e.g. , . a symptom of extreme youth is the balmer continuum showing up in emission ( @xmath10 ) , as it happens with ask 25 in fig . [ uvcontinuum ] . + 6 . the ratio between the fluxes of h@xmath11 and [ nii]@xmath126583 is an indicator of whether the nebula is ionized by a starburst ( h@xmath13}\lambda6853 $ ] ) , or by a source of harder uv flux like an agn or low mass evolved stars ( @xmath14}\lambda6853 $ ] ) . figure [ star_agn ] shows examples of the two possibilities : a starburst ( ask 9 ) and a liner - like excitation ( ask 0 ) . once the ratio [ nii]@xmath126583 to h@xmath11 is observed to be outside the starburst regime , seyferts and liners can be distinguished according to the ratio of fluxes between [ oiii]@xmath125007 and h@xmath15 ( fig . [ bpt2nd ] ) the high ionization source is a seyfert if [ oiii]@xmath16 ( ask 6 ) or liner - like if [ oiii]@xmath125007 @xmath17 h@xmath15 ( ask 5 ) . this recipe is a qualitative rendering of the so - called bpt diagram @xcite and its updates @xcite . + and [ nii]@xmath126583 indicates whether the nebula is ionized by a starburst , or by other type of source with higher ionization power ( e.g. , agns ) . h@xmath18}\lambda6853 $ ] indicates starburst ( ask 9 ) whereas @xmath19}\lambda6853 \ge { \rm h}\alpha$ ] is a symptom of higher ionization ( ask 0 ) . ] + and [ oiii]@xmath125007 indicates whether the ionization is powered by a strong agn ( [ oiii]@xmath125007 @xmath20 h@xmath15 ask 6 ) or by a liner - like source ( [ oiii]@xmath125007 @xmath17 h@xmath15 ask 5 ) . ] [ bptitem ] the ratio between the fluxes of [ nii]@xmath126583 and h@xmath11 also provides an estimate of gas metallicity in star - forming galaxies @xcite . the sensitivity is high : the ratio goes from 1/3 to 1/300 when the metallicity ranges from solar to one tenth solar @xcite . this high contrast makes it simple to distinguish between solar and sub - solar metallicities . the ratio is not suitable to diagnose super - solar metallicities . in this case one can use supplementary line ratios like the so - called r23 or 03n2 . [ line4363 ] the presence of [ oiii]@xmath124363 is also an indicator of low metallicity . the line is used to compute electron temperatures in hii regions , and it weakens with increasing metallicity to disappear at around @xmath21 ( e.g. , * ? ? ? tio bands at approximately 7150 , 7600 , and 8500 are characteristic of m stars ( dwarf , giant and super - giants ; e.g. , * ? ? ? * ) , and reveal the presence of evolved stellar populations . when the stellar population is young , massive stars outshine the contribution of m stars , making these spectral features invisible . ( red super - giants are young massive stars showing tio bands , but they are outnumbered by the associated blue super - giants that overshadow their contribution to the integrated galaxy spectrum e.g. , ; @xcite @xcite . ) figure [ tiobands ] shows how the tio bands appear in almost all ask classes , except for the bluest ones , and how the strength of the bands decrease as the spectrum becomes bluer . as we mention in sect . [ qualitative_classes ] , the bands are hardly noticeable at ask 16 , and they are absent at ask 20 and bluer types . + 8498 , 8542 , and 8662 ) . ] 10 . the ir caii triplet at @xmath58498 , 8542 , and 8662is an indicator of metallicity and gravity . in stars , its equivalent width ( ew ) increases with increasing metallicity until 2/3 of the solar metallicity @xcite . above this metallicity it depends only on gravity , with the ew increasing with decreasing gravity from dwarfs to super - giants @xcite . the combined effect on galaxy spectra must be modeled , but the existence of a caii absorption with significant strength is always a sign of high metallicity and of the presence of giant stars . contrarily , absence of the triplet indicates low metallicity . we find it in all ask spectra except for the bluest classes ( see fig . [ tiobands ] ) . as we mention in sect . [ qualitative_classes ] , the lines are almost absent in ask 20 and bluer classes . 11 . the so - called mg@xmath22 and h@xmath15 lick indices are in the same spectral region ( h@xmath15 from 4848 to 4877 , and mg@xmath22 from 5154 to 5197 ; see fig . [ hbetamg ] ) , and they were designed ( and are used ) to determine simultaneously age and metallicity in galaxies with old stellar populations @xcite . one can generally say that h@xmath15 mostly depends on age , and to less extent on metallicity , and the opposite happens with mg@xmath22 ( e.g. , * ? ? ? * ; * ? ? ? however , their quantitative application require modeling ( e.g. , they may depend on the relative abundance of the metals , rather than on a single global metallicity ) . + and mg@xmath22 ( h@xmath15 index from 4848 to 4877 , and mg@xmath22 from 5154 to 5197 ) . both indices combined allow us to set mean age and mean metallicity in galaxies with old stellar populations . the labels mark the h@xmath15 line and the position of the three mg i lines contributing to mg@xmath22 . ] 12 . the interstellar medium ( ism ) that reddens the spectra also produces absorption in the nai d line ( @xmath55891,5896 , e.g. , * ? ? ? * ; * ? ? ? therefore , one would expect that the strength of the ism nai d line sorts galaxies according to extinction @xcite . the example in fig . [ nadline ] corresponds to the two spectra in fig . [ dust_effect ] , where ask 1 is known to present a substantial dust extinction . its nai d is stronger than that for the class without extinction , ask 0 , being the rest of the spectrum similar . + ) . ] the mere presence of high excitation lines like [ nev]@xmath123426 , [ fevii]@xmath126087 , or [ fex]@xmath126375 , tells us that the galaxy hosts an agn ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? the example in fig . [ agn_excitation ] shows spectra in the range of [ fevii]@xmath126087 , whose emission is clear in seyferts ( ask 7 and 8) , but is non - existing in starbursts ( ask 20 ) as well as in passively evolving red galaxies with liner - like emission ( ask 0 ; see sect . [ qualitative_classes ] ) . [ heii]@xmath124686 is also indicative of agn , though it is sometimes found in star forming galaxies . + @xmath126087 , clear in seyferts ( ask 7 and 8) but non - existing otherwise . ( ask 20 is a starbursts , whereas ask 0 represents a passively evolving red galaxy . ) ] considering the spectral features described in the previous section , we have set up a simple decision tree ( a questionary ) that leads to classifying a galaxy spectrum by replying to a few questions ( fig . [ decision ] ) . emission and absorption lines are analyzed separately , therefore , the natural outcome would be galaxy types with two components , one for the stars and other for the gas . one should begin the questionary from top to bottom , to end up with the characteristics of both the gas and the stars . the decision tree in fig . [ decision ] is self - explanatory , although a few clarifications on the terminology are required . the symbol g stands for galaxy . _ broad spectral lines _ means lines in excess of 2000 kms@xmath23 , and they separate seyfert 1 and quasars from the other kinds of agns . such broad lines are not illustrated in sect . [ list_features ] since the ask classes lack seyfert 1 and quasars , that were excluded from the list of galaxy targets directly at the sdss distribution ( see * ? ? ? when we mention young , old and a mixture of old - and - young stellar populations , we loosely speaking refer to stellar ages @xmath24yr ( young ) , @xmath25yr ( old ) , and the intermediate range in between , @xmath26@xmath27yr . when metal poor gas is mentioned , we mean clearly sub - solar ( say , less than 1/3 solar ) . bl lac objects are also included to complete the questionary , so that it considers the possibility that neither emission nor absorption lines are present in the spectrum . several criteria in fig . [ decision ] compare emission lines such comparison refers to the fluxes of the lines . one may think of this section as analogous to the section on _ individual objects _ , common in many papers , except that the targets are ask templates representing objects too numerous to be described one by one . we use the criteria put forward in the previous section to determine the properties of all ask classes individually . the thread of the argumentation follows the decision tree in fig . [ decision ] . a summary with the properties of all classes is also given in table [ table_summary ] . at the end of the section , we define a stellar age index that sorts the ask classes according to their mean stellar age . similarly , we define an index to sort the emission line spectra by metallicity . both are relative quantities , devised to compare our qualitative analysis with quantitative estimates of ages and metallicities . clllccccc 0&agn or liner - like&old metal - rich & h@xmath11 [email protected]&0&&&[email protected]&[email protected] + 1&liner - like & old metal - rich & dust reddened , edge on disks?&0&&&[email protected]&[email protected] + 2&agn or liner - like & old metal - rich & [ siii]@xmath129069 emission&1&&&[email protected]&[email protected] + 3&liner - like & old metal - rich & continuum bluer than ask 0 and 2&2&& & [email protected]&[email protected] + 4&liner - like?&old & young&edge - on disks?&3&& & [email protected]&[email protected] + 5&liner - like?&old & young&green valley galaxies&3&& & [email protected]&[email protected] + 6&seyfert 2&old & young&[fevii]@xmath126087 emission&3&& & [email protected]&[email protected] + 7&seyfert 2&old & young & younger than 6 , [ fevii]@xmath126087 emission&4&& & [email protected]&[email protected] + 8&seyfert 2&old & young & younger than 7 , [ fevii]@xmath126087 emission&5&& & [email protected]&[email protected] + 9 & liner - like&old & young & metal - rich starburst?&3&&&[email protected]&[email protected] + 10&metal - rich starburst&old & young&liner - like?&3&-0.35&&[email protected]&[email protected] + 11&metal - rich starburst&old & young&liner - like ? , stars younger than 9 and 10&4&-0.36& & [email protected]&[email protected] + 12&metal - rich starburst&old & young&starburst prototype , stars younger than 11&6&-0.43&[email protected] & [email protected]&[email protected] + 13&metal - rich starburst&old & young&stars similar to 12&6&-0.46& & [email protected]&[email protected] + 14&metal - rich starburst&old & young&starburst prototype , stars younger than 12&7&-0.46&[email protected]&[email protected]&[email protected] + 15&metal - poor starburst&no absorption & hii g , youngest ask&&-1.67&[email protected]&& + 16&metal - poor starburst&old & young&&7&-0.63&[email protected] & [email protected]&[email protected] + 17&metal - poor starburst & young & hii g , stars older than 15&13&-1.58&[email protected] & [email protected]&[email protected] + 18&metal - poor starburst & young & stars younger than 16&8&-0.54&[email protected]&[email protected]&[email protected] + 19&metal - poor starburst & young & stars as in 18&8&-0.75&[email protected] & [email protected]&[email protected] + 20 & metal - poor starburst&young & hii g , stars younger than 18 , older than 17&13&-1.42&[email protected]&[email protected]&[email protected] + 21 & metal - poor starburst&young & hii g , like 20 , gas slightly metal - poorer&13&-1.45&[email protected] & [email protected]&[email protected] + 22 & metal - poor starburst&young & like 19 , stars younger , gas metal - richer&8&-0.93&[email protected] & [email protected]&[email protected] + 23 & metal - poor starburst&young & like 19 and 22 , stars younger&9&-0.79&[email protected] & [email protected]&[email protected] + 24 & metal - poor starburst&young & like 23 , stars younger&10&-1.07&[email protected] & [email protected]&[email protected] + 25 & metal - poor starburst&young & like 20 and 21 , stars older , gas metal - richer&12&-1.27&[email protected] & [email protected]&[email protected] + 26 & metal - poor starburst&young & like 25 , stars older , gas metal - richer&11&-1.09&[email protected] & [email protected]&[email protected] + 27 & metal - poor starburst&young & like 25&12&-1.18&[email protected] & [email protected]&[email protected] [ table_summary ] @xmath30 ask 0 has an absorption line spectrum with very weak emission lines ( fig . [ dust_effect ] ) . it is not a starburst since @xmath19}\lambda6853 > { \rm h}\alpha$ ] , but [ oiii]@xmath125007 and h@xmath15 are too weak to decide whether the excitation is seyfert - like or liner - like . note , however , that the ew of h@xmath11 is very small ( fig . [ star_agn ] ) , which according to @xcite indicates that the ionization is produced by hot - low mass stars . the absorption line spectrum does not show the balmer break ( fig . [ uvbreaks ] ) but the 4000 break is conspicuous , consequently , the absorption spectrum is produced by an old metal - rich stellar population . @xmath30 ask 1 also has an absorption line spectrum with weak emission lines ( fig . [ dust_effect ] ) . @xmath19}\lambda6853 \simeq { \rm h}\alpha$ ] , and therefore it is not a starburst . h@xmath15 is smaller than [ oiii]@xmath125007 , which may naively indicate agn excitation . however , the lines are so weak that the underlying h@xmath15 absorption is important and , therefore the corrected h@xmath15 emission is similar to that of [ oiii]@xmath125007 . consequently , the emission line spectrum is probably in the liner region of the bpt diagram . the absorption line spectrum is also very similar to ask 0 , which was assigned to an old metal - rich stellar population . the main difference with respect to ask 0 is the continuum , which steepens redward of 6000 ( fig . [ dust_effect ] ) , and is a signature of dust reddening . additional independent arguments also corroborate that ask 1 owes much of its red colors to reddening . ask 1 galaxies tend to have very elongated morphologies , a fact difficult to interpret unless they are edge - on disks @xcite , which are known to be significantly dust reddened with respect to their face - on counterparts ( e.g. , * ? ? ? * ; * ? ? ? @xmath30 ask 2 is very similar to ask 0 , so does our assignation emission consistent with agn or liner - like excitation plus absorption corresponding to old metal - poor stars . the difference is in the continuum , which is somewhat redder in ask 0 , and also in the emission line [ siii]@xmath129069 , which shows up in ask 2 but not in ask 0 . @xmath30 ask 3 also shows an absorption line spectrum with weak emission . emissions and absorptions are similar to those of ask 0 , therefore , the associated stellar population is old . as it happens with ask 0 , [ oiii]@xmath125007 and h@xmath15 are too weak to decipher whether the emission is seyfert or liner - like . the difference with ask 0 , 1 and 2 is the continuum , which is bluest in ask 3 ( see fig . [ early_types ] ) . except for ask 1 , such variation reflects differences in the stellar populations , ask 3 being the youngest . ask 0 , 2 , and 3 were used to select a clean sample of red ellipticals by . @xmath30 ask 4 spectrum has absorption lines and significant emission lines . the continuum is fairly red , similar to that of ask 0 in fig . [ dust_effect ] . @xmath31}\lambda6853 $ ] and h@xmath32 [ oiii]@xmath125007 , therefore , according to the decision tree ( fig . [ decision ] ) , it should be a liner - like galaxy . however , it is in the region of the bpt diagram where agn activity and star - formation are difficult to disentangle ( see fig . 13 in * ? the absorption line spectrum shows both the balmer break and the 4000 break , which correspond to a mixture of old and young stellar populations . the region around the break is shown in fig . [ breaks ] . @xmath30 ask 5 has absorption and emission lines . the continuum is significantly bluer than that for ask 4 . @xmath31}\lambda6853 $ ] and h@xmath32 [ oiii]@xmath125007 , therefore , according to the decision tree ( fig . [ decision ] ) , it should be a liner - like galaxy ( with the caveat issued for ask 4 still applying ) . the absorption line spectrum shows both the balmer break at 3650and the 4000 break , which correspond to a mixture of old and young stellar populations ( like ask 4 in fig . [ breaks ] ) . @xmath30 ask 6 has intense emission lines on top of an absorption spectrum . emission lines are broad ( fig . [ broad_lines ] ) , but not broad enough to be a seyfert 1 galaxy ( larger than 2000kms@xmath23 ) . it appears in the seyfert region of the bpt diagram , therefore it is a seyfert 2 . [ fevii]@xmath126087 and [ nev]@xmath123426 show up in emission confirming the agn nature of the emission . it shows the balmer break at 3650and the 4000 break , which correspond to a mixture of old and young stellar populations . the breaks are extremely similar to that of ask 4 in fig . [ breaks ] . @xmath30 ask 7 is similar to ask 6 . the emission lines are broad ( fig . [ broad_lines ] ) , and it is also classified as seyfert 2 according to the decision tree . [ fevii]@xmath126087 is detected , confirming the agn nature of the emission ( fig . [ agn_excitation ] ) . the absorption spectrum has a clear 4000 break , and the balmer break is present but less pronounced than in the case of ask 6 because the balmer continuum rises blueward of the balmer break . the absorption spectrum is also produced by a mixture of old and young stars , but probably younger than for ask 6 . @xmath30 ask 8 is similar to ask 6 and 7 , but the emission lines are even broader . it is also a seyfert 2 . [ fevii]@xmath126087 is detected , confirming the agn nature of the emission ( fig . [ agn_excitation ] ) . following the trend from ask 6 to ask 7 , the absorption spectrum shows the two breaks ( balmer and 4000 ) , but the balmer continuum ( blueward of 3650 ) is more intense . the spectrum is also produced by a mixture of old and young stars , but probably younger than for ask 7 . @xmath30 ask 9 has absorption and emission lines . the lines are narrow , and @xmath31}\lambda6853 $ ] with h@xmath32 [ oiii]@xmath125007 . according with the decision tree it has liner - like emission , although it is close to being classified as a metal - rich starburst . the absorption spectrum presents well defined balmer and 4000 breaks , therefore , it is produced by a mixture of old and young stellar populations . the region of the breaks is similar to that of ask 5 . @xmath30 ask 10 has absorption and emission lines . the continuum is similar to that of ask 9 except that it becomes redder beyond 7000 . @xmath33}\lambda6853 $ ] with h@xmath34}\lambda$]5007 . according with the decision tree it corresponds to a metal - rich starburst , although it is close to the divide with liner - like emission . the absorption spectrum is almost identical to the spectrum of ask 9 in the region of the balmer and 4000 breaks , and it is produced by a mixture of old and young stellar populations . @xmath30 ask 11 has both absorption and emission , but the emission lines are very intense . the red continuum is redder than in ask 9 and 10 , but the emission lines of ask 11 are stronger . @xmath35}\lambda6853 $ ] with h@xmath36 [ oiii]@xmath125007 . it corresponds to a metal - rich starburst , although it is close to the border to present liner - like emission . the absorption spectrum is almost identical to the spectrum of ask 9 in the region of the balmer and 4000 breaks , except that the contribution of the balmer lines is more important . it is produced by a mixture of old and young stellar populations , but the young population is more important than in the case of ask 9 and 10 . @xmath30 ask 12 spectrum has both absorption and emission lines . the continuum is bluer than that of ask 10 and 11 , but the emission lines are weaker . @xmath33}\lambda6853 $ ] with h@xmath37}\lambda$]5007 . it represents a typical metal - rich starburst it is right on the head of the _ seagull _ of the local bpt diagram corresponding to prototypical starbursts . the absorption line spectrum has the balmer and 4000 breaks , but the 4000 break is less pronounced than that in ask 10 and 11 , and the balmer series more intense . the spectrum corresponds to mixed old and young stellar populations , but the young population is more important than in the case of ask 9 , 10 , and 11 . @xmath30 ask 13 spectrum has both absorption and emission lines . the continuum is bluer than that of ask 11 and 12 , but the emission lines are weaker . @xmath35}\lambda6853 $ ] with h@xmath36 [ oiii]@xmath125007 . it is a starburst . the absorption line spectrum has the balmer and 4000 breaks , and they are almost identical to those for ask 12 . the spectrum corresponds to mixed old and young stellar populations similar to ask 12 . @xmath30 ask 14 spectrum has both absorption and emission lines . the continuum is bluer than that of ask 12 and 13 , and the emission lines more pronounced . @xmath35}\lambda6853 $ ] with h@xmath37}\lambda$]5007 . it corresponds to a typical metal - rich starburst . the absorption line spectrum has the balmer and 4000 breaks , but the 4000 break is barely noticeable . the spectrum corresponds to mixed old and young stellar populations , but the young population is more important than in the case of ask 12 , and 13 . @xmath30 ask 15 is a pure emission line spectrum . the ew of h@xmath15 is of the order 200 therefore , according the decision tree , it is an hii galaxy . @xmath38}\lambda6853 $ ] with h@xmath39}\lambda$]5007 , which corresponds to a low - metallicty starburst . the spectrum shows neither the balmer break nor the 4000 break ( even more extreme than ask 17 in fig . [ hiigalax ] ) . there are no metallic lines , and even the balmer series shows no trace of absorption . this spectrum corresponds to the youngest stellar populations of the ask series . ask 15 has only 68 members @xcite , most of which look compact galaxies , like those described by @xcite and @xcite , but a few of them are hii regions in resolved galaxies . @xmath30 ask 16 spectrum has both absorption and emission lines . the continuum is bluer than that of ask 13 and 14 including the upturn at the uv . $ ] with h@xmath41 [ oiii]@xmath125007 , which corresponds to a metal - poor starburst . the absorption line spectrum does show the balmer break , but the 4000 break is barely noticeable . consequently , the absorption line spectrum corresponds to a young stellar population , with hints of an old component . the tio bands are hardly noticeable . @xmath30 ask 17 spectrum has only emission lines . ( one can barely notice the absorption of some of the balmer lines ; see fig . [ hiigalax ] ) . since the ew of h@xmath42 150 , according to the decision tree ask 17 is an hii galaxy . the continuum is as blue as that of ask 15 and includes the uv upturn ( fig . [ hiigalax ] ) . @xmath40}\lambda6853 $ ] with h@xmath43 [ oiii]@xmath125007 , which corresponds to a metal - poor starburst . even though absorption lines are not obvious , the spectrum shows the balmer break ( see fig . [ hiigalax ] ) . this two features correspond to extremely young stellar populations ( although not as young as those involved in ask 15 ) . @xmath30 ask 18 spectrum has absorption and strong emission lines . the continuum is similar to that of ask 16 but the upturn of the uv continuum is more pronounced . the emission lines are also stronger than those of ask 16 . @xmath40}\lambda6853 $ ] with h@xmath44 [ oiii]@xmath125007 , which corresponds to a metal - poor starburst . the absorption line spectrum does show the balmer break , but it does not have a 4000 break ( similar to ask 22 in fig . [ breaks ] ) . the absorption line spectrum corresponds to a young stellar population . @xmath30 ask 19 spectrum presents absorption and strong emission lines . the continuum is bluer than that of ask 18 but the emission lines are weaker . @xmath40}\lambda6853 $ ] with h@xmath43 [ oiii]@xmath125007 , which corresponds to a metal - poor starburst . the absorption line spectrum shows the balmer break , but it does not have a 4000 break , and it is similar to ask 18 . it corresponds to a young stellar population . @xmath30 ask 20 spectrum is dominated by strong emission lines but it also shows weak absorptions in the balmer lines . the weak continuum is as blue as that of ask 15 , 17 or 19 , and includes an uv upturn . we identify ask 20 as an hii galaxy . @xmath40}\lambda6853 $ ] with h@xmath45}\lambda$]5007 , which corresponds to a metal - poor starburst . the absorption line spectrum does show the balmer break , but it does not have a 4000 break . it also contains metallic lines ( caii handk ) . the absorption line spectrum corresponds to a young stellar population . stars are older than those in ask 17 , but younger than those in ask 19 . the tio bands are absent , and the ir ca triplet is almost gone with a hint of showing up in emission . @xmath30 ask 21 is very similar to ask 20 , except that the lines are somewhat weaker . probably the gas - phase metallicity is a bit higher in ask 21 as judged from the ratio between @xmath46 and @xmath19}\lambda6853 $ ] . in any case , the starburst is metal - poor . @xmath30 ask 22 spectrum presents absorption and strong emission lines . the continuum is similar to ask 19 , but bluer . the emission lines are somewhat stronger than those in ask 19 . @xmath40}\lambda6853 $ ] with h@xmath45}\lambda$]5007 , which corresponds to a metal - poor starburst . the gas metallicity is a bit higher in ask 19 , as judged from the ratio between @xmath46 and @xmath19}\lambda6853 $ ] . the absorption line spectrum shows the balmer break , but it does not have a 4000 break , and it is similar to ask 19 . the absorption line spectrum corresponds to a young stellar population , probably younger than that in ask 19 . @xmath30 ask 23 spectrum is very similar to that of ask 22 and ask 19 , except for having larger emission lines . the continuum is also a bit bluer . it corresponds to a metal - poor starburst with a young stellar population , presumedly younger than that for ask 19 and 22 . @xmath30 ask 24 has a spectrum similar to ask 23 ( and so to ask 22 and 19 ) , with stronger emission lines . the continuum is bluer than in ask 23 . as judged from the ratio between @xmath46 and @xmath19}\lambda6853 $ ] , the gas metallicity of ask 23 is higher . @xmath30 ask 25 spectrum is similar to ask 20 and 21 , with the continuum a bit redder , and the lines weaker . as judged from the ratio between @xmath46 and @xmath19}\lambda6853 $ ] , the gas metallicity of ask 20 and 21 are smaller . the balmer continuum shows up in emission ( fig . [ hiigalax ] ) . @xmath30 ask 26 spectrum is similar to ask 25 , with the continuum a bit redder , and the lines weaker . as judged from the ratio between @xmath46 and @xmath19}\lambda6853 $ ] , the gas metallicity of ask 25 is smaller . @xmath30 ask 27 spectrum is very similar to that of ask 25 . in order to carry out the comparison of this qualitative analysis with the quantitative analysis in sect . [ starlight ] , we define a _ stellar age index _ that sorts the ask classes according to the age of their stellar populations . the stellar age index ( sai ) is defined as follows . based on the absorption lines in the region containing the uv breaks and the uv continuum , we order the ask classes according to their relative stellar ages . for instance , having stronger broader balmer lines implies being older . since the ordering is rough , we allow for several classes to share the same age bin . once the order has been set , we assign a sequential number to this order from the older stellar population sai=0 ( ask 0 ) to the youngest stellar population sai=13 ( ask 20 ) . the sai thus defined qualitatively orders the stellar populations from the oldest to the youngest , although it does not assign specific ages to the ask classes . sais are included in table [ table_summary ] . similarly , a gas metallicity index ( gmi ) is defined to sort the classes according to their gas - phase metallicities . in this case we use the n2 index , i.e. , @xmath47\lambda6583/{\rm h}\alpha)$ ] , which is a well known proxy for gas metallicity ( see item # [ bptitem ] in sect . [ list_features ] ) . again , the index ( i.e. , n2 renamed as gmi ) is listed among the properties of the classes in table [ table_summary ] . gmi is used for comparison with the quantitative analysis described in sect . [ quantitative_lines ] . we use the star formation histories ( sfh ) derived using the code starlight @xcite to cross - check the qualitative analysis carried out in the previous sections . starlight decomposes the observed absorption spectrum in terms of a sum of single stellar populations ( ssp ) , i.e. , coeval starbursts with an assumed initial mass function , a common metallicity ( _ the metallicity _ ) , and observed with a time - lag with respect to the burst ( _ the age _ ) . each ssp produces a known spectrum , and a linear combination of these spectra is fitted to the observed spectrum , being the amplitudes applied to each ssp the free parameters of the fit . extinction is modeled as a foreground dust screen , with its wavelength dependence given by the extinction law of @xcite , and then scaled during the fitting process using a single degree of freedom . ( other extinction laws were tried and yield equivalent results ; see @xcite @xcite . ) the amplitudes of the ssps represent the measured sfh . starlight uses the metropolis scheme to carry out the @xmath48 minimization ( for a full description of the code , see * ? ? ? the amplitudes thus derived are proportional to the fraction of the galaxy mass produced by each individual ssp burst , and they are the parameters used in our study ( after suitable normalization to 100 ) . in our particular rendering , starlight employs 150 ssps from @xcite , combined according to the padova 1994 evolutionary tracks . the ssps cover a grid of 6 metallicities ( from 0.005 to 2.5 times solar ) and 25 ages ( from 1 myr to 18 gyr ) . further details are given in sect . 2.1 of @xcite . under these assumptions , we computed the sfh of each galaxy with a spectrum in sdss - dr7 . figure [ sfh_ask ] shows mean sfhs for a number of representative ask classes . the average considers all the sdss - dr7 galaxies in each ask class . the classes have been chosen so that they cover the full range of possibilities , from the oldest reddest stellar populations ( ask 0 ) to the youngest bluest ones ( ask 20 ) . ask 5 , 14 and 18 illustrate intermediate cases . figure [ sfh_ask_light ] is equivalent to fig . [ sfh_ask ] except that , rather than mass , it shows the percentage of present light ( at 4020 ) produced by each one of the ssps . note how light is strongly biased towards young populations , as compared to mass which is held by old populations . the dotted lines in the figure represent @xmath29 one standard deviation considering all the galaxies in sdss - dr7 corresponding to a given ask class . these are the histograms used to compute the luminosity - weighted averages and dispersions discussed in the next paragraph . figure [ age_vs_age]a shows the relationship between the mean luminosity - weighted age as derived from starlight and the estimate of relative age carried out in sect . [ qualitative_classes ] ( sai ) . the error bars give the rms fluctuations among the ages of the ssps that contribute to each class . the correlation age - sai is extremely good , implying that our quick qualitative estimate is consistent with the detailed up - to - date modeling . moreover , the existence of an almost one - to - one correlation provides specific timescales to our qualitative dating . sai between 0 and 2 correspond to a single old metal rich population , with ages between 11.2 and 6.7gyr ( see the sfh for ask 0 in fig [ sfh_ask ] ) . sai between 3 and 7 has two stellar populations assigned , one old and one young ( table [ table_summary ] ) . they have mean ages between @xmath49gyr and 1.2gyr . finally , from sai 8 onwards , we qualitatively find young populations , and their mean starlight ages go from 250myr to 5myr . figure [ age_vs_age]b displays the mean stellar metallicity corresponding to each sai . the metallicity is high ( slightly super - solar ) when sai @xmath50 , i.e. , in the classes our qualitative analysis catalogued as having old stellar populations . in this case the scatter is fairly small ( see the error bars in the figure ) , meaning that all their old stars are metal - rich . the scatter increases and the mean metallicity decreases for younger populations . we interpret this result as an increase of the number of stellar populations that contribute to the galaxy spectra , which is corroborated by the sfhs of ask 5 , 14 and 18 in fig . [ sfh_ask ] ( with sai 3 , 7 and 8 , respectively ) . the stellar metallicity grows slightly for spectra corresponding to even younger stellar populations , and it becomes slightly sub - solar for the youngest ask classes . the scatter remains large , also reflecting the significant number of stellar components in these galaxies . derived from starlight vs sai . the error bars give the rms fluctuations among the ages and metallicities of the ssps included in each class . the scatter of metallicities is large except for sai@xmath51 as usual , @xmath52 stands for the solar metallicity . , scaledwidth=50.0% ] one of the most sophisticated techniques of analysis of ionized nebulae involves measuring emission - line fluxes of many atomic species to derive their relative abundances . adding up all the ionization states of an element provides its abundance . this approach is the so - called direct method or temperature - based method.the fluxes depend on atomic parameters as well as on the physical conditions of the plasma . once the atomic parameters are known ( or assumed ) , one can use the observed lines to retrieve , simultaneously , the elemental abundances and the physical conditions of the nebula . for instance , using collisional excited lines of the same species having different excitation potentials , one can determine the electron temperature ( e.g. , [ oiii]@xmath124363 and [ oiii]@xmath125007 ) . similarly , lines of the same species with the same excitation potential but different collisional de - excitation rates , provide diagnostics for the electron density ( e.g. , [ sii]@xmath126731 to [ sii]@xmath126717 ) . we have applied this technique to determine the oxygen abundance characteristic of the emission lines of the ask classes that are starbursts . the actual recipe is described by @xcite and @xcite , and it has been widely used ( e.g. , * ? ? ? * ; * ? ? ? * ) .we refer to the original references for details on the technique and atomic parameters . whenever possible , the electron temperature was inferred from [ oiii]@xmath124363 . this line weakens with increasing metallicity , therefore , it can not be used with the classes of large metallicities ( see item # [ line4363 ] in sect . [ list_features ] ) . the problem was bypassed in these cases using [ siii]@xmath126312 and [ siii]@xmath129069 to derive the sulphur electron temperature , which was then used for oxygen after scaling @xcite . ask classes 17 , 20 , 21 , 22 24 , 25 , 26 , and 27 have [ oiii]@xmath124363 intense enough to determine electron temperatures . the line is not detectable in classes 12 , 14 , 16 , 18 , 19 and 23 , however they show [ siii]@xmath126312 , which we used for deriving electron temperatures . finally , classes ask 10 , 11 and 13 do not allow us to measure either [ oiii]@xmath124363 or [ siii]@xmath126312 , and so we could not assign an oxygen abundance using the direct method . classes 20 , 21 , 22 , 24 , 25 , 26 , and 27 allow to determine electron temperatures from both [ oiii]@xmath124363 and [ siii]@xmath126312 . the oxygen abundances obtained using the two ways of estimating temperature agree within @xmath290.02 dex . all the abundances thus obtained are listed in table [ table_summary ] . as we explain in sect . [ qualitative_classes ] , the metallicity of the gaseous component of the template spectra was judged based on the ratio between [ nii]@xmath126583 and h@xmath11 . this ratio is therefore our qualitative metallicity index ( table [ table_summary ] ) , which is compared with the direct oxygen abundance in fig . [ pplike ] . the correlation is extremely good , at least from solar metallicity ( log[o / h]@xmath53 ; ) to one tenth the solar value . the fluctuations of the actual data with respect to a linear fit are just 0.06dex , which is significantly smaller than the same correlation obtained from individual galaxies e.g. , @xcite claim 0.2dex . @xmath126583 to h@xmath11 , which is the proxy used to estimate the metallicity in our qualitative scheme . error bars are computed in a monte - carlo simulation to be consistent with the errors assigned to the observed fluxes . the straight lines correspond to various estimates of the relationship from individual galaxies and hii regions by ( * ? ? ? * the dotted - dashed line ) and by ( * ? ? ? * the dashed line ) , and from our ask templates ( the solid line ) . our linear fit excludes the three rightmost points , and it reads , @xmath54}\lambda6583/{\rm h}\alpha ) + ( 9.31\pm 0.07)$ ] . ] from the very good correlation between oxygen abundance and [ nii]@xmath126583/h@xmath11 we conclude that the qualitative analysis of nebular metallicities is consistent with the quantitative estimate using the best techniques available . figure [ scatterplot]a shows the index used to determine the gas metallicity , @xmath55}\lambda6583/{\rm h}\alpha$ ] , vs the index used to characterize the age of the stellar populations , sai . it is clear that the two indices are correlated , indicating that the templates with the lowest oxygen content also have the youngest stellar populations . this is explicitly shown in fig . [ scatterplot]b , which presents the same kind of relationship but using quantitative determinations of ages and gas - phase metallicities . ( the two last points deviating from the linear relationship will be ignored since the trend they represent is not present in fig . [ scatterplot]a , and they have particularly weak [ oiii]@xmath124363 lines , with the uncertainties that this entails see item # [ line4363 ] in sect . [ list_features ] . ) the correlation is similar to that found by @xcite . the physical origin of the relationship is unclear . it may be a side - effect of the galaxy mass ( a phenomenon often referred to as downsizing ; see , e.g. , * ? ? ? first , the mass - metallicity relationship implies that low - mass galaxies are less metallic ( e.g. , * ? ? ? second , the mass - age relationship ( e.g. , * ? ? ? * ) implies that low - mass galaxies also have younger stellar populations . finally , the bluest ask classes contain more dwarf galaxies @xcite , therefore , they are less metallic and with younger stars , giving as a side - effect the observed correlation . even though this explanation is feasible , the relationship between gas - metallicity and stellar - age shown in figs . [ scatterplot ] is so clean that it looks fundamental rather than derived from the combined effect of two other relationships . this conjecture is supported by the scatter plots in fig . [ mass_age ] , that include the two variables involved in fig . [ scatterplot]b plus the galaxy mass . assigning masses to the ask templates is not without ambiguity , since the spectra of the individual galaxies were normalized before averaging ( sect . [ ask_class ] ) . however , we computed the mean and standard deviation among the masses of all the galaxies in each ask class , and those are the masses assigned to the classes in figs . [ mass_age ] . one can see that the templates follow a mass - metallicity relationship ( fig . [ mass_age]b ) and a mass - age relationship ( fig . [ mass_age]a ) , but both are less tight than the metallicity - age relationship in fig . [ scatterplot]b , which seems to be the primary relationship . } \lambda6583/{\rm h}\alpha$ ] , vs the index used to characterize the age of the stellar populations , sai . they are correlated . ( b ) same representation as ( a ) but using quantitative determinations of gas metallicity and age . the age of the point corresponding to ask 15 ( i.e. , the youngest class with the lowest oxygen content ) is just an upper limit , which we include to show that the relationship continues to the youngest targets . ] b. ( b ) scatter plot of the mass of the galaxies in a class versus the gas - phase metallicity . the error bars have the same meaning as in panel ( a ) . ] in short , the properties of the gas and the stars are not independent but tightly correlated in real galaxies . galaxy mass does not seem to be the only factor driving such correlation . galaxy spectra seem to follow a 1d sequence , with a secondary branch for agns @xcite . in other words , an independent parameter ( affine parameter ) characterizes most properties of galaxy spectra , from the red passive ones to those actively forming stars . the actual nature of the affine parameter is unknown , but the results in this paper suggest it to be the mean age of the stellar population . the ask templates can be naturally ordered by mean stellar age ( or by sai , in our parlance ) , and the order thus obtained turns out to be extremely similar to the one obtained using minimal spanning trees by @xcite . the latter represents a non - trivial exercise to find the location of the templates in the 1637-dimensional space where the ask classification was carried out ( i.e. , a space where each galaxy is a point , and the 1637 coordinates represent the flux at particular wavelengths ) . they are organized in a 1d sequence with the same order given by the luminosity - weighted mean stellar age . we take the agreement between the two orderings as a strong suggestion that stellar age is the affine parameter . note that the emission line spectrum is prominent in blue galaxies and so it plays a major role in shaping the galaxy spectra . that fact that the spectrum of a galaxy is ( mostly ) dictated by the age of the stellar population implies that the emission lines and the absorption lines are not independent . this is indeed the conclusion reached in the previous paragraph through a totally different argument . we argued in sect . [ ask_class ] that the ask classes are representative of all local galaxies since they condense the properties of some one million galaxies of the local universe . even though we endorse the statement , it must be clarified . ask templates are representative of the most common galaxies , however , some important but uncommon galaxies are not included . in particular , the most massive galaxies that dominate the centers of galaxy clusters ( brightest cluster galaxies and cd galaxies ) are not properly described . these massive red galaxies have old stellar populations so they are classified as ask 0 and ask 2 .however , they represent a small fraction of all the galaxies in these classes , so that their contribution to the average ( template ) spectra of ask 0 and 2 is negligible . the same may happen with other kinds of rare objects like bl lac , objects with extreme star formation rates ( e.g. , * ? ? ? * ) , extremely metal poor galaxies ( e.g. , * ? ? ? * ) , and others . the fact that some objects may escape the simplified schematic in sect . [ decision_tree ] do not invalidate the analysis it will be useful to indicate that these objects are unusual . the comparison between figs . [ sfh_ask ] and [ sfh_ask_light ] evidences a fact that is well documented in the literature , but which still results somewhat surprising . most galaxies formed a significant fraction of their stellar mass long ago when the universe was just a few gyr old , even those forming stars today ( e.g. , * ? ? ? ? * ; * ? ? ? this fact is obviously true for ask classes representing passively evolving red galaxies ( see ask 0 in fig . [ sfh_ask ] ) , but it also holds true for young ask classes see the important contribution of old stellar populations to the ask 20 sfh in fig . [ sfh_ask ] , even though its luminosity - weighted mean age is just 4.5 myr ( table [ table_summary ] ) . when the mass contribution is transformed to light contribution ( fig . [ sfh_ask_light ] ) , it becomes clear how newborn stars outshine the older populations , that are heavily underrepresented in the composite galaxy spectrum . if a galaxy happens to undergo a significant starburst , spectrum - wise it looks young . there is a conspicuous difference between the old stellar populations present in passively evolving galaxies and in star - forming galaxies . the metallicity of the old stars is high in passive galaxies and very low in starbursts ( compare the sfhs of ask 0 and ask 20 in the first column of fig . [ sfh_ask ] ) . the dominance of old metal rich stellar populations in red galaxies is well known , so does the fact that the old stars in dwarf galaxies of the local group have extremely low metallicity ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? as argued in the introduction , we have sophisticated computer codes for inferring the properties of the stellar populations contributing to the observed galaxy spectra . similarly , tools are available for qualitative diagnostics of the physical properties of the galaxy gas . they have been developed by specialist groups , and then kindly offered to a much broader community . technicalities often complicate the interpretation of the results , therefore , there is a natural tendency to apply these sophisticated tools in black - box fashion , which turns out to be quite unsatisfactory for a physical stand point . one obtains a detailed description of the stars and gas producing the observed galaxy spectra , but overlooks the reasons why the computer code has preferred them rather than other alternatives . we provide a simple step - by - step guide to qualitative interpretation of galaxy spectra . it is not precise , and has not been planed as an alternative to the existing tools . however , it allows a quick - look that yields the main properties of the spectra in a intuitive fashion . this may be of interest in various applications , e.g. , to provide physical insight when using sophisticated tools , or to interpret noisy spectra . moreover , the results of the qualitative analysis agree with those inferred using up - to - date computer codes . the step - by - step guide is described in sect . [ decision_tree ] , and it has been summarized as a simple questionary in fig . [ decision ] . emission and absorption lines are analyzed separately , which give rise to a classification with one entry for the gas and another for the stars . ( in real galaxies , however , the properties of gas and stars are tightly correlated ; see sect . [ additional_results ] . ) the analysis has been systematically applied to the set of ask template spectra that resulted from the classification of all galaxy spectra in sdss - dr7 ( see sect . [ ask_class ] ) . their physical properties are summarized in table [ table_summary ] . with the caveats pointed out in sect . [ additional_results ] , the ask classes represent a comprehensive set of galaxy spectra , that go all the way from passively evolving red galaxies ( e.g. , ask 0 ) to hii galaxies , dominated by massive newborn stars having no absorption lines ( e.g. , ask 15 ) . since it works for this set , the analysis should work for most galaxies . the qualitative analysis is found to be in excellent agreement with quantitative numerical codes . we show how the index for stellar - age ( sai ) follows an almost one - to - one correlation with the mean stellar age assigned by the code starlight ( fig . [ age_vs_age ] ) . similarly , we found how the proxy for gas metallicity is in good agreement with the ( oxygen ) metallicity inferred by applying the direct method to the emission lines of the ask templates ( fig . [ pplike ] ) . the ask templates are freely available ( see footnote # [ my_foot ] ) and , together with their physical properties listed in table [ table_summary ] , they can be used as benchmarks so that any other galaxy spectrum can be analyzed by reference to them . thanks are due to c. ramos almeida and e. prez - montero for discussions and help with references . this work has been funded by the spanish micin project _ estallidos _ , aya 2010 - 21887-c04 - 04 . et and rt acknowledge also financial support by the mexican research council ( conacyt ) , through grants cb-2005 - 01 - 49847 , 2007 - 01 - 84746 and 2008 - 103365-f . we are members of the consolider - ingenio 2010 program , grant micinn csd2006 - 00070 : first science with gtc . funding for the sdss and sdss - ii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , the u.s . department of energy , the national aeronautics and space administration , the japanese monbukagakusho , the max planck society , and the higher education funding council for england . the sdss is managed by the astrophysical research consortium for the participating institutions ( for details , see the sdss web site at http://www.sdss.org/ ) . the starlight project is supported by the brazilian agencies cnpq , capes and fapesp and by the france - brazil capes / cofecub program .

we describe a simple step - by - step guide to qualitative interpretation of galaxy spectra ( fig . [ decision ] ) . rather than an alternative to existing automated tools , it is put forward as an instrument for quick - look analysis , and for gaining physical insight when interpreting the outputs provided by automated tools . though the recipe is of general application , it was developed for understanding the nature of the automatic spectroscopic k - means based ( ask ) template spectra . they resulted from the classification of all the galaxy spectra in the sloan digital sky survey data release 7 ( sdss - dr7 ) , thus being a comprehensive representation of the galaxy spectra in the local universe . using the recipe , we give a description of the properties of the gas and the stars that characterize the ask classes , from those corresponding to passively evolving galaxies , to hii galaxies undergoing a galaxy - wide starburst . the qualitative analysis is found to be in excellent agreement with quantitative analyses of the same spectra . we compare the mean ages of the stellar populations with those inferred using the code starlight.we also examine the estimated gas - phase metallicity with the metallicities obtained using electron - temperature based methods . a number of byproducts follow from the analysis . there is a tight correlation between the age of the stellar population and the metallicity of the gas , which is stronger than the correlations between galaxy mass and stellar age , and galaxy mass and gas metallicity . the galaxy spectra are known to follow a 1-dimensional sequence , and we identify the luminosity - weighted mean stellar age as the affine parameter that describes the sequence . all ask classes happen to have a significant fraction of old stars , although spectrum - wise they are outshined by the youngest populations . old stars are metal rich or metal poor depending on whether they reside in passive galaxies or in star - forming galaxies .

introduction ask classification recipe for qualitative interpretation of galaxy spectra qualitative analysis of the ask classes quantitative analysis of the stellar populations using starlight quantitative analyses of the emission line spectra additional results and discussions conclusions

arxiv

the atlas collaboration has carried out a detailed study to detect the susy signature in the framework of one of the most popular model , sugra @xcite,@xcite . it has been shown @xcite that if susy exists at the electro - weak scale , it should be discovered by atlas and a general method has been given to determine approximately the mass scale of the susy particles . in subsequent papers @xcite@xcite it was shown in five representative points of the parameter space that some of the susy particles can be reconstructed and using the obtained characteristics ( masses , branching ratios ) the model parameters can be precisely determined @xcite . all these studies have been carried out assuming that @xmath2 parity is conserved . in this note we consider that @xmath2 parity is broken in such a way that the lepton number @xmath1 is violated through @xmath3-type couplings . the present experimental limits @xcite can not completely exclude such a scenario . in this case one of the prominent signatures of susy , the missing energy is considerably weakened because the lightest susy particle ( lsp ) is allowed to decay . due to this decay the lepton and/or jet multiplicity increases considerably and some efficient cuts ( e.g. lepton veto against the @xmath8 background ) can not be applied . on the other hand , the decay products of the lsp in some cases allow its direct reconstruction . therefore the event topology and the search strategies are different of the case when @xmath2 is conserved . this has motivated us to revisit the feasability to detect susy and to determine the parameters of the sugra model using the atlas detector . in section 2 we give a brief description of the phenomenology of the @xmath2 parity violation and the event generator used . section 3 deals with the atlas detector and with the fast simulation of its response . in section 4 we present the domain of the parameter space where a susy signal can be expected by atlas . in the subsequent three sections the reconstruction of the susy particles and the determination of the model parameters are described in the lhc points 1,3 and 5 which represent a heavy , light and medium susy mass scale . we summarize the obtained results in the concluding section . 0.5 cm 0.5 cm @xmath2-parity has been introduced @xcite in order to avoid fast nucleon decay and flavor changing neutral currents ( fcnc ) . if the multiplicative quantum number @xmath9 is conserved it guarantees automatically baryon number ( @xmath10 ) and lepton number ( @xmath1 ) conservation . @xmath2 is + 1 for standard model ( sm ) particles and its value is -1 for their superpartners . the most important experimental consequences of the conservation of @xmath2 are that super partners should be produced in pairs and the lightest superpartner ( lsp ) should be stable . the lsp interacts weakly , therefore the prominent signature of susy in case of @xmath2 parity conservation is a considerable amount of missing ( transverse ) energy ( @xmath11 ) . although no violation of @xmath10 or @xmath1 has been observed yet , there is no firm theoretical argument which would require exact conservation of them and that of the @xmath2 parity . in fact the following term in the superpotential @xmath12 which violates explicitely @xmath10 , @xmath1 and @xmath2 parity , can not be ruled out experimentally . here @xmath1 and @xmath13 are isodublet and isosinglet lepton , @xmath14 and @xmath15 are isodublet and isosinglet quark superfields , the indices @xmath16 , @xmath17 and @xmath18 run for the three lepton and quark families . the suffix @xmath19 denotes charge conjugate . the first two terms violate explicitely @xmath1 whilst the last one violates @xmath10 . present limits on the proton lifetime suggests that either the @xmath1 or the @xmath10 violating terms ( i.e. the corresponding @xmath20 couplings ) should vanish for the first family . other experimental limits e.g. on lepton number violation : double @xmath6 decay , or on @xmath21 oscillation , etc . indicate that the couplings in equ . ( [ eq : lag ] ) should nt be expected to exceed a few percent , and usually are much smaller than the gauge couplings . even so , if @xmath2 parity is violated the topology of the expected susy signal changes substantially . since the lsp is no more stable , the missing energy is considerably reduced on the other hand the decay products of the lsp increase the average number of jets and/or leptons in an event . in general , the event topology depends crucially on the size of the couplings . if , e.g. the couplings are of the order of @xmath22 or larger , the mass spectra , branching ratios , etc . will be different in the two cases where @xmath2 is conserved or violated . if however the couplings are smaller than the above value , the dominant effect of the @xmath2 parity violation is that the lsp becomes unstable . an estimation of the lsp lifetime as a function of the couplings show @xcite that we can distinguish four subcases giving rise to different detection strategies : + @xmath23 @xmath24 + @xmath25 @xmath26 + @xmath27 @xmath28 + @xmath29 @xmath30 + in case @xmath23 only the event topology changes w.r.t . the case of @xmath2 conservation . in case @xmath25 one can observe a displaced vertex at the lhc energies . in case @xmath27 the lsp decays outside of a typical lhc detector , however it can be catched by special purpose detectors @xcite . finally the case @xmath29 can not be distinguished experimentally from the case of @xmath2 parity conservation . in this study we have deliberately chosen to study case @xmath23 and compare the result with the case of @xmath2 parity conservation , because it represents a more difficult experimental situation than case @xmath25 where a displaced vertex could disentangle the lsp from the rest of the event . moreover we have assumed that @xmath31 and only one of the @xmath32 coupling is different from zero in equ . ( [ eq : lag ] ) . nonzero @xmath33 and @xmath34 are subject of other reports inside atlas @xcite . the hierarchical structure observed for the yukawa couplings in the sm motivates our hypothesis above . the lagrangian corresponding to the superpotential of equ . ( [ eq : lag ] ) can be written in terms of particle fields for our case as : @xmath35 where @xmath36 and @xmath37 are lepton fields , the tilde denotes the field of the superpartner , the @xmath19 stands for charge conjugation , @xmath38 for complex conjugation and @xmath39 are the flavor indices . as stated before , if the @xmath3 couplings are smaller than @xmath40 , which is our case , the sparticle mass spectrum practically does nt change and the main consequence of the @xmath2 parity violation is the decay of the lsp . this process is depicted in fig.[lspdecay1 ] where we assume that the lsp is the lightest neutralino ( @xmath41 ) . the decay proceeds through an @xmath2 conserving and an @xmath2 violating vertex and in the final state there are always three leptons out of them at least two of _ different _ flavours , one neutral and the other two of opposite charges , since the lsp is supposed to be neutral . the prominent signature of this type of susy event is the spectacular increase of `` stable '' leptons : electrons and muons in the final state . the neutral leptons , neutrinos , give rise to some missing transverse energy , but its magnitude is much less than that if @xmath2 parity is conserved . the flavour of the lepton in the final state depends on the values of the indices @xmath39 . since @xmath32 is antisymmetric in @xmath16 and @xmath17 there are only 9 independent couplings which we choose as : @xmath42 . the first two families , 1 and 2 give rise always to `` stable '' leptons , electrons and muons , ( and the corresponding neutrinos ) in the final state . if an index 3 appears , the lepton is not stable if it is a @xmath43 , and its decay products are most of the time different from electrons or muons . since @xmath44 in equ . ( [ eq : lag ] ) is an isosinglet , @xmath45 is a @xmath43 . the number of the stable leptons is the most prominent for @xmath46 and @xmath47 . it is less spectacular , if an index 3 appears at the second place , and even less if the 3 appears in the third place . finally , if two indices have value of 3 one has the least number of stable leptons . on the other hand , the number of the neutrinos , and with that the magnitude of the missing energy increases in the order of the above mentioned cases . the expected extra number of stable charged leptons and neutrinos per each lsp decay for the different @xmath3 couplings calculated by the program @xcite are given in table [ tb : brs ] . it is clear that the average number of the leptons for different flavours gives a strong hint on the coupling which is realized . e.g. the coupling @xmath47 gives rise predominantly to muons whilst the coupling @xmath48 results in equal number of electrons and muons , etc . -0.5 cm -10.0 cm 0.5 cm 0.5 cm [ cols="^,^,^,^,^",options="header " , ] [ tb : slope_ptanb ] 0.5 cm -0.3 cm & @xmath47 & @xmath47 & @xmath48 & @xmath47 & @xmath48 + @xmath49 & & [email protected] & [email protected]@xmath51 & & + @xmath52 & 932@xmath5020 & & & 662@xmath5012 & + @xmath53 & & & & 685@xmath5020 & 686@xmath5012 + @xmath54 & & & & 504@xmath5020 & + @xmath55 & & [email protected] & [email protected]@xmath51 & & + @xmath56 & & & & [email protected] & + @xmath57 & [email protected] & & & [email protected] & + @xmath58 & [email protected] & [email protected] & & [email protected] & [email protected] + @xmath59 & 169.8@xmath60 & 44.8@xmath61 & & 122.6@xmath62 & + @xmath63 & [email protected] & [email protected]@xmath64 & [email protected]@xmath64 & [email protected] & [email protected] + @xmath65 & & & & & [email protected] + @xmath66 & & [email protected] & 52.9@xmath67 @xmath51 & & + + + + + + + [ tb : values ] 0.5 cm 0.5 cm relative errors on the & & & + sugra parameters & & & + & @xmath47 & @xmath47 & @xmath48 & @xmath68 & @xmath69 & @xmath47 & @xmath48 + @xmath70 ( % ) & 12 & 4.4 & 7.3 & 4 & 5.8 & 2.9 & 9.7 + @xmath71 ( % ) & 0.3 & 0.3 & 0.6 & 0.2 & 0.4 & 0.5 & 1.4 + @xmath72 ( % ) & 5 & 3.3 & 3.3 & 1.8 & 1.8 & 6 & 6.2 + + + + [ tb : sugrafitp1 ] we have studied the feasibility to detect a susy signal by atlas in the framework of the sugra model and to determine its parameters in the case when @xmath2 parity is broken in conjunction with lepton number violation : @xmath73 . for this purpose we have chosen three representativ points in the sugra parameter space and two different type of couplings , both having a value @xmath74 , small enough to concentrate the effect in the lsp decay but large enough not to see displaced vertex in this decay . \3 . in the case of couplings with absence of a @xmath43 among the decay products of the @xmath41 ( e.g. @xmath47 ) one can reconstruct the susy particles and this reconstruction can be used for a precision determination of the model parameters . the achieved precision turns out to be better than it was the case with conserved @xmath2 parity . this is because one can reconstruct the lsp from its decay products . at the low energy point where the chargino or second lightest neutralino produces additional leptons this determination is slightly handicapped by the combinatorial background and the most complex structure . \4 . in the case of lsp decay with a @xmath43 particle in the final state the full reconstruction of the lsp , i.e. the determination of its four momentum , is not always possible , however , one can still estimate its mass ( except at point 3 - @xmath48 ) . it allows ones to determine the parameters of the sugra model in spite of the large combinatorial background due to the leptonic decay of the lsp . this determination in most of the cases is better or at least comparable in precision with that when @xmath2 parity is conserved . 1.0 cm * aknowledgements * + + we would like to thank daniel froidevaux for suggesting us to carry out this study , his constant help and encouragement . we had many useful discussions within the atlas susy working group and would like to express our gratitude to its members , especially to ian hinchliffe , frank paige and giacomo polesello . 1.0 cm l.alvarez-gaume , j.polchinski , and m.b.wise , nucl . b221 ( 1983 ) 495 , + l.iba@xmath78ez , phys . lett . 118b ( 1982 ) 73 , + j.ellis , d.v.nanopoulos and k.tamvakis , phys . lett . 121b ( 1983 ) 123 , + k.inoue _ et al . _ prog . theor . phys . 68 ( 1982 ) 927 , + a.h.chamseddine , r.arnowitt and p.nath , phys . 49 ( 1982 ) 970 . d. froidevaux , precision susy measurements with atlas , lhcc susy workshop , cern , 30/10/1996 , unpublished + g.polesello , overview of sugra studies in atlas , atlas physics workshop , grenoble , 3/04/1998 , unpublished f.paige and s.protopopescu , in it supercollider physics , p. 41 d.soper ( world scientific , 1986 ) + h.bauer , f.paige , s.protopopescu and x.tata , in it proceedings of the workshop of physics at current accelerators and supercolliders , ed . j.hewett , a.white and d.zeppenfeld , ( argonne national laboratory , 1993 )

simulating @xmath0 collisions at lhc energies in the framework of the sugra model and the detection of the produced leptons and jets by atlas we demonstrate that a clean signature of susy can be obtained over a large domain of the parameter space in the case of @xmath1-violating @xmath2 parity breaking ( @xmath3 couplings ) . the obtained signal allows the reconstruction of the susy particles and thereby the precise determination of the model parameters @xmath4 , @xmath5 , tan@xmath6 and sign@xmath7 . 0.5 cm

introduction basic phenomenology conclusions

arxiv

a _ partial chord diagram _ , is a special kind of graph , which can be specified as follows . the graph consists of a number of line segments ( which we will also call backbones ) arranged along the real line ( hence they come with an ordering ) with a number of vertices on each . a number of semi - circles ( called chords ) arranged in the upper half plan are attached at a subset of the vertices of the line segments , in such a way that no two chords have endpoints on the line segments in common . the vertices which are not attached to chord ends are called the marked points . chord diagram _ is by definition a partial chord diagram with no marked points . partial chord diagrams occur in many branches of mathematics , including topology @xcite , geometry @xcite and representation theory @xcite . furthermore , they play a very prominent role in macro molecular biology . please see the introduction of @xcite for a short review of these applications . as documented in @xcite , the notion of a _ fatgraph _ @xcite is a useful concept when studying partial chord diagrams . a fatgraph is a graph together with a cyclic ordering on each collection of half - edges incident on a common vertex . a partial linear chord diagram @xmath0 has a natural fatgraph structure induced from its presentation in the plane . the fatgraph @xmath0 has canonically a two dimensional surface with boundary @xmath1 associated to it ( e.g. see figure [ partial_chord ] ) . the partial chord diagram @xmath0 and the surface @xmath1 associated to the fatgraph with marked points . this partial chord diagram has the type @xmath2 . the boundary length - point spectra are @xmath3 . , width=453 ] we now recall the basic definitions from @xcite for a partial chord diagram @xmath0 . * the number of chords , the number of marked points , and the number of backbones of @xmath0 are denoted @xmath4 , @xmath5 , and @xmath6 respectively . * the euler characteristic and the genus of @xmath1 , are denoted @xmath7 and @xmath8 respectively . if @xmath9 is the number of boundary components of @xmath1 , we have that @xmath10 and @xmath8 obeys euler s relation @xmath11 * the _ backbone spectrum _ @xmath12 are assigned to @xmath0 , if it has @xmath13 backbones with precisely @xmath14 vertices ( of degree either two or three ) ; + * the _ boundary point spectrum _ @xmath15 is assigned to @xmath0 , if its boundary contains @xmath16 connected components with @xmath17 marked points ; + * the _ boundary length spectrum _ @xmath18 is assigned to @xmath0 , if the boundary cycles of the diagram consist of @xmath19 edge - paths of length @xmath20 , where the _ length _ of a boundary cycle is the number of chords it traverses counted with multiplicity ( as usual on the graph obtained from the diagram by collapsing each backbone to a distinct point ) _ plus _ the number of backbone undersides it traverses ( or in other words , the number of traversed connected components obtained by removing all the chord endpoints from all the backbones ) . + we now introduce the combination of the boundary length spectrum and the boundary point spectrum , namely our new boundary length and point spectrum . * the _ boundary length and point spectrum _ @xmath21 is assigned to @xmath0 , if its boundary contains @xmath22 connected components of length @xmath23 with marked point spectrum @xmath24 , meaning that there cyclically around the boundary components are @xmath25 marked points , then a chord or a backbone underside , then @xmath26 marked points , then a chord or a backbone underside , and so on all the way around the boundary component . in fact we will not need to distinguish which way around the boundary we go . hence it is only the cyclic ordered tuple of the numbers @xmath27 , which we need and which we denote as @xmath28 . we remark that some of the @xmath29 ( @xmath30 ) might be zero . + we have the following relations @xmath31 where @xmath32 . for all @xmath23 and @xmath17 , we also have that @xmath33 we define @xmath34 to be the number of connected partial chord diagrams of type @xmath35 taken to be zero if there is none of the specified type . in @xcite , @xmath36 is defined as the number of distinct connected partial chord diagrams of type @xmath37 . we find the relation between these numbers by the following formula @xmath38 where @xmath39 in particular , the numbers @xmath40 and @xmath41 are given by @xmath42 for the index @xmath43 , we consider the variable @xmath44 and denote @xmath45 and for the index @xmath46 , we consider the variable @xmath47 and denote @xmath48 for any @xmath49 . we define the orientable , multi - backbone , boundary length and point spectrum generating function @xmath50 , where @xmath51 for an element @xmath52 , where each @xmath53 , we write @xmath54 where @xmath55 contains all the positive entries and @xmath56 the absolute value of all the negative ones , which we assume to both be finite . we define the differential operator @xmath57 we now define @xmath58 , @xmath59 and @xmath60 to be strings like @xmath61 given by the following formulae @xmath62 where @xmath63 denotes the sequence @xmath64 where the component @xmath65 appears only at the entry indexed by @xmath66 . we further define the index @xmath67 by the formula @xmath68 which is identical to the index on the last term of the above assignments . [ thm1 ] .1 in define the first and second order linear differential operators @xmath69 and the quadratic differential operator @xmath70 then the following partial differential equations hold @xmath71 together with the initial conditions @xmath72 they determine the functions @xmath73 and @xmath74 uniquely . in this article , we also consider the non - oriented analogue of partial chord diagrams . the generalization of the above analysis is straightforward , as we will now explain . non - oriented _ partial chord diagrams , is a partial chord diagram together with a decoration of a binary variable at each chord , which indicates if the chord is _ twisted _ or not . when associating the surface @xmath75 , to a non - oriented partial chord diagram , a twisted band is associated along twisted chords as indicated in figure [ partial_chord_non ] . by this construction , @xmath76 orientable and non - orientable surfaces are obtained from one partial chord diagram with @xmath4 chords , when we vary over all assignments of twisting or not to the @xmath4 chords . in the non - oriented case , we have the following definition of the euler characteristic . * euler characteristic @xmath7 . + the euler characteristic of the two dimensional surface @xmath1 is defined by the formula @xmath77 where @xmath78 is the number of cross - caps and we have euler s relation @xmath79 with this set - up , the enumeration of the non - oriented partial chord diagrams is considered in parallel to the oriented case discussed above with a small change for the boundary length and point spectrum @xmath80 . in this non - oriented case , there are now induced orientation on the boundaries of @xmath1 and hence for an index @xmath28 corresponding some boundary component of @xmath1 , we not only need to consider this tuple up to cyclic permutation of the tuple , but also reversal of the order @xmath81 the non - oriented surface constructed out of untwisted and twisted chords.,width=453 ] let @xmath82 be the number of non - oriented partial chord diagrams of type @xmath83 . in @xcite , @xmath84 is defined as the number of non - oriented connected partial chord diagrams of type @xmath85 . these numbers are related by the following formula @xmath86 and the numbers @xmath87 and @xmath88 are given by @xmath89 we define the non - oriented generating function @xmath90 to be given by @xmath91 we define @xmath92 , @xmath93 and @xmath94 to be by @xmath95 and we also define indices @xmath96 by the formula @xmath97 which again , we note is identical to the index on the last term of the above assignments . [ enumeration of non - oriented partial chord diagrams filtered by their boundary length and point spectrum ] [ thm2 ] .1 in define the first and second order linear differential operators @xmath98 and the quadratic differential operator @xmath99 then the following partial differential equations hold @xmath100 together with the following initial conditions @xmath101 determines @xmath102 and @xmath103 uniquely . this paper is organized as follows . section [ sec2 ] contains basic combinatorial results on the boundary length and point spectra of partial chord diagrams and derives the recursion relation of the number of diagrams ( proposition [ prop1 ] ) , by the cut - and - join method . this cut - and - join equation is rewritten as a second order , non - linear , algebraic partial differential equation for generating function of the number of partial chord diagrams filtered by the boundary length and point spectrum ( proposition [ prop2 ] ) . section [ sec3 ] extends these results to include the non - oriented analogues of the partial chord diagrams . the cut - and - join equation is extended to provide a recursion on the number of non - oriented partial chord diagrams ( proposition [ prop3 ] ) , and is also rewritten as partial differential equation ( proposition [ prop4 ] ) . in this section , we devote to prove theorem [ thm1 ] . the partial differential equation ( [ pde_ori ] ) is equivalent to the following recursion relation for the numbers of connected partial chord diagrams . [ prop1 ] the numbers @xmath104 enumerating connected partial chord diagrams of type @xmath105 obey the following recursion relation @xmath106 \nonumber \\ & + \frac{1}{2}\sum_{k\ge 1}\sum_{l\ge 1}\sum_{{\pmb{d}}_k}\sum_{{\pmb{f}}_l } ( m_{{\pmb{d}}_k}+1)(m_{{\pmb{f}}_l}+1-\delta_{{\pmb{d}}_k,{\pmb{f}}_l } ) \nonumber \\ & \quad\times\sum_{i=1}^k\sum_{j=1}^l\sum_{\ell=0}^{d_i-1}\sum_{m=0}^{f_j-1 } { \mathcal m}_{g-1,k-1,l+2}\left({\pmb{b}},{\pmb{m}}+q_{i , j,\ell , m}({\pmb{d}}_k,{\pmb{f}}_l)\right ) \nonumber \\ & + \frac{1}{2}\sum_{k\ge 1}\sum_{l\ge 1}\sum_{{\pmb{d}}_k}\sum_{{\pmb{f}}_l } \sum_{g_1+g_2=g}\;\;\sum_{k_1+k_2=k-1}\;\ ; \sum_{b^{(1)}+b^{(2)}=b } \nonumber \\ & \quad\times \sum_{i=1}^k\sum_{j=1}^l\sum_{\ell=0}^{d_i-1}\sum_{m=0}^{f_j-1 } \sum _ { \substack { { \pmb{m}}^{(1)}+{\pmb{m}}^{(2 ) } \\ = { \pmb{m}}+q_{i , j,\ell , m}({\pmb{d}}_k,{\pmb{f}}_l ) } } \nonumber \\ & \quad\quad\times m_{{\pmb{d}}_k}^{(1)}m_{{\pmb{f}}_l}^{(2 ) } \frac{b!}{b^{(1)}!b^{(2 ) } ! } { \mathcal m}_{g_1,k_1,l_1}\bigl({\pmb{b}}^{(1)},{\pmb{m}}^{(1)}\bigr ) { \mathcal m}_{g_2,k_2,l_2}\bigl({\pmb{b}}^{(2)},{\pmb{m}}^{(2)}\bigr ) . \label{cut_join}\end{aligned}\ ] ] this recursion relation is referred to as the _ cut - and - join equation _ , since it follows from a cut - and - join argument , which we shall now provide . when one removes one chord from a partial chord diagram , there are essentially three distinct possible outcomes . first of all the diagram can stay connected and then there are two cases to consider . in the first one , the chord that is removed is adjacent to two different boundary components and in the second one it is adjacent to just one . the third case is when the chord diagram becomes disconnected . in the first case , the genus of the partial chord diagram is not changed , but two boundary components join into one component . on the other hand , in the second case , the genus decreases by one , and one boundary component splits into two components . removal of a chord in case one . the chord is depicted as a band . after the removal of this chord , two boundary components join into one component . left : the clusters of marked points @xmath107 and @xmath108 join into two clusters @xmath29 and @xmath109 right : the clusters of marked points @xmath110 and @xmath111 join into one cluster @xmath29.,width=453 ] in the first case , and let us say that after removing this chord , the two adjacent boundary components join into one component with the marked point spectrum @xmath112 . ( see figure [ comb1 ] . ) under this elimination , the numbers @xmath4 and @xmath9 change to @xmath113 and @xmath114 , the genus @xmath8 is not changed ( c.f . euler s relation @xmath115 ) . the number of marked points @xmath5 changes to @xmath116 , because the chord ends of the chord which is removed become new marked points . there are two distinct possible sub cases , namely either the chord ends belong to two distinct clusters of marked points @xmath29 and @xmath109 in the resulting chord diagram , or chord ends belong to the same cluster of marked points @xmath29 . we will consider the former kind of chord , and assume @xmath117 without loss of generality . before we remove the chord , the two boundaries adjacent to the chord needs to have the following two marked point spectra @xmath118 when removing the chord , we connect the clusters of marked points @xmath119 and @xmath120 . if the original partial chord diagram has the boundary length - point spectrum @xmath80 , the resulting diagram has @xmath121 for the latter kind , we must have two boundary components with the marked point spectra @xmath122 and removing the chord connects the clusters of marked points @xmath110 and @xmath123 . this manipulation changes the boundary length and point spectrum @xmath80 into @xmath124 for both of these two kinds of removal , there are @xmath125 possibilities to choose the boundary components in the partial chord diagram . therefore , the number of possibilities for the first way of removal is @xmath126 . \label{p1}\end{aligned}\ ] ] in the second case ( see figure [ comb2 ] ) , the removal changes the numbers @xmath4 and @xmath9 to @xmath113 and @xmath127 and the genus of the partial chord diagram decreases by one . for partial chord diagram with a boundary with marked point spectrum @xmath128 we remove the chord which connects the two clusters @xmath129 and @xmath107 of marked points . the boundary component then splits into two boundary components with marked point spectra @xmath112 and @xmath130 . if the original partial chord diagram has the boundary length and point spectrum @xmath80 , after removal of this chord , we find that @xmath131 the number of possibilities of this removal is @xmath132 for @xmath133 . if @xmath134 , the number of possibilities becomes @xmath135 . in total , the number of possibilities for the second way of elimination is @xmath136 the factor @xmath137 in front of the sum takes care of the over counting in the cases @xmath133 . the second and third way of elimination of a chord . after the elimination of this chord , a boundary component split into two different boundary components . , width=453 ] in the third case , the partial chord diagram split into two connected components . we consider the case that the original diagram has the type @xmath105 and the resulting two connected components have types @xmath138 and @xmath139 . these types are related such that @xmath140 since a boundary component also split into two components , the boundary length and point spectrum changes in the same manner as in the second case . there are @xmath142 ways to choose the boundary components which are to be fused under the inverse operation of chord removal . and the number of different ordered splittings of a @xmath6-backbone diagram is @xmath143 where @xmath144 ( @xmath145 ) . therefore , the total number of possibilities of this case is @xmath146 the factor @xmath137 corrects for the over counting due to the ordering of the two connected components . the sum of the contributions ( [ p1 ] ) , ( [ p2 ] ) , and ( [ p3 ] ) from the three different cases of chord removals equals @xmath147 , because there are @xmath4 possibilities for the choice of the chord to be removed . this gives the cut - and - join equation ( [ cut_join ] ) . [ prop2 ] the generating function @xmath148 is uniquely determined by the differential equation @xmath149 where @xmath150 . the generating function @xmath151 $ ] of the number of connected and disconnected partial chord diagrams satisfies @xmath152 and is as such determined by the initial conditions @xmath153 it is straightforward to check that the differential equation @xmath154 is equivalent to the cut - and - join equation ( [ cut_join ] ) . the actions in the quadratic differential @xmath155 on @xmath74 can be rewritten by following relation @xmath156 the derivatives on the right hand side are contained in @xmath157 , and the differential equation @xmath158 follows from that of @xmath74 . on the initial condition , every partial chord diagram of type @xmath159 can be obtained from the disjoint collection of type @xmath160 with multiplicity @xmath13 by connecting them with @xmath4 chords . this implies @xmath161 . since this is the first order differential equation of @xmath162 , the coefficient of @xmath163 is determined uniquely using this initial condition . in this section , we will prove theorem [ thm2 ] . we first establish the following proposition . [ prop3 ] the number @xmath164 of connected non - oriented partial chord diagrams of type @xmath105 obeys the following recursion relation @xmath165 \nonumber \\ & + \frac{1}{2}\sum_{k\ge 1}\sum_{l\ge 1}\sum_{{\pmb{d}}_k}\sum_{{\pmb{f}}_l } ( m_{{\pmb{d}}_k}+1)(m_{{\pmb{f}}_l}+1-\delta_{{\pmb{d}}_k,{\pmb{f}}_l } ) \nonumber \\ & \ ; \times\sum_{i=1}^k\sum_{j=1}^l\sum_{\ell=0}^{d_i-1}\sum_{m=0}^{f_j-1 } \bigl\ { \widetilde{\mathcal m}_{h-2,k-1,l+2}\left({\pmb{b}},{\pmb{m}}+q_{i , j,\ell , m}({\pmb{d}}_k,{\pmb{f}}_l)\right ) \nonumber \\ & \quad\ ; \hspace*{3 cm } + \widetilde{\mathcal m}_{h-2,k-1,l+2}\bigl({\pmb{b}},{\pmb{m}}+q^{\times}_{i , j,\ell , m}({\pmb{d}}_k,{\pmb{f}}_l)\bigr ) \bigr\ } \nonumber \\ & + \frac{1}{2}\sum_{k\ge 1}\sum_{l\ge 1}\sum_{{\pmb{d}}_k}\sum_{{\pmb{f}}_l } \sum_{h_1+h_2=h}\sum_{k_1+k_2=k-1 } \sum_{b^{(1)}+b^{(2)}=b } \nonumber \\ & \quad\times \sum_{i=1}^k\sum_{j=1}^l\sum_{\ell=0}^{d_i-1}\sum_{m=0}^{f_j-1 } \vspace{0.3 cm } \left ( \sum _ { \substack { { \pmb{m}}^{(1)}+{\pmb{m}}^{(2 ) } \\ = { \pmb{m}}+q_{i , j,\ell , m}({\pmb{d}}_k,{\pmb{f}}_l ) } } + \sum _ { \substack { { \pmb{m}}^{(1)}+{\pmb{m}}^{(2 ) } \\ = { \pmb{m}}+q^{\times}_{i , j,\ell , m}({\pmb{d}}_k,{\pmb{f}}_l ) } } \right ) m_{{\pmb{d}}_k}^{(1)}m_{{\pmb{f}}_l}^{(2 ) } \nonumber \\ & \quad\times \frac{b!}{b^{(1)}!b^{(2 ) } ! } \widetilde{\mathcal m}_{h_1,k_1,l_1}\bigl({\pmb{b}}^{(1)},{\pmb{m}}^{(1)}\bigr ) \widetilde{\mathcal m}_{h_2,k_2,l_2}\bigl({\pmb{b}}^{(2)},{\pmb{m}}^{(2)}\bigr ) . \label{cut_join_non}\end{aligned}\ ] ] if we remove a non - twisted chord , then we find the same recursive structure as for the numbers ( [ p1 ] ) , ( [ p2 ] ) , and ( [ p3 ] ) for @xmath166 in the oriented case . as we did in the proof of proposition [ prop1 ] , we also consider three cases , organised the same way , when removing a twisted chord . in the first case ( see figure [ comb1_non ] ) , there are again two possibilities , namely the twisted chord ends belong to two different or the same clusters of marked points on the boundary component in the resulting diagram after removal . contrary to the case of non - twisted chords , the boundary cycle does not split , but the marked point spectrum changes due to the recombination of the boundary component . for both of these two cases , the numbers @xmath4 and @xmath9 change to @xmath113 and @xmath9 , and the cross - cap number @xmath78 decreases by one under this elimination ( c.f . euler s relation @xmath167 ) . the chord ends become marked points and @xmath5 changes to @xmath116 . removal of a twisted chord from a non - oriented partial chord diagram . the chord is depicted as a twisted band . after the elimination of this chord , the boundary component is reconnected into one component with different marked point spectrum . left : the clusters of marked points @xmath107 and @xmath108 join into two clusters @xmath29 and @xmath109 . right : the clusters of marked points @xmath110 and @xmath111 join into one cluster @xmath29.,width=453 ] in the former situation , we must have a boundary component with the marked point spectrum @xmath168 from which we remove one twisted chord with one end between the two clusters @xmath110 and the other between @xmath169 . then the removal will result in a boundary component with the marked point spectrum @xmath66 and the boundary length and point spectrum @xmath80 is changed as follows @xmath170 the possible number of choices for this kind of removal is @xmath125 , and the total number of diagrams which can be obtained in this way is @xmath171 for the removal of the latter kind of twisted chords , we must start with a diagram with a boundary component with the marked point spectrum @xmath172 from which we remove one twisted chords with one end between the two clusters @xmath173 and the other one between the two clusters @xmath174 . after removal , we obtain a boundary component with the marked point spectrum @xmath66 . thus , the boundary length and point spectrum @xmath80 is changed to @xmath175 the number of such chords to be removed is @xmath125 , and the total number of partial chord diagrams obtained in this way is @xmath176 next , we consider the second case ( see figure [ comb2_non ] ) , where we must start with a non - oriented partial chord diagram with a boundary component with the marked point spectrum @xmath177 from which we remove a twisted chord with one end between the two clusters @xmath129 and the other end between the two clusters @xmath178 . after removal of this chord , the boundary component has been split into two components with spectra @xmath66 and @xmath179 , and the cross - cap number @xmath78 decreases by two . then , the boundary length and point spectrum @xmath80 changes to @xmath180 the number of choices for the chord to be removed is @xmath132 for @xmath181 and @xmath135 for @xmath182 , and the total number of partial chord diagrams obtained this way is @xmath183 in case three partial chord diagram split into two connected components when we remove the chord . assume that the original diagram has the type @xmath184 and the resulting two connected components have types @xmath185 and @xmath186 . then these types are related by @xmath187 the marked point spectrum changes in the same way as the second case @xmath188 the total number of resulting diagrams is @xmath189 therefore , in total , the number of possible partial chord diagrams obtained by removing a twisted or a non - twisted chord is the sum of ( [ p1_non ] ) ( [ p4_non ] ) and of ( [ p1 ] ) ( [ p3 ] ) for @xmath166 . this number gives the right hand side of equation ( [ cut_join_non ] ) , which we have just argued also gives the left side of equation ( [ cut_join_non ] ) . [ prop4 ] the generating function @xmath190 is uniquely determined by the differential equation @xmath191 where @xmath192 and @xmath193 . the generating function @xmath194 $ ] of the number of connected and disconnected partial chord diagrams filtered by the boundary length and point spectrum satisfies @xmath195 as such they are uniquely determined by the initial conditions @xmath196 n. v. alexeev , j. e. andersen , r. c. penner , and p. zograf , _ enumeration of chord diagrams on many intervals and their non - orientable analogs , _ adv . math . * 289 * ( 2016 ) 10561081 , arxiv:1307.0967 [ math.co ] . j. e. andersen , l. o. chekhov , r. c. penner , c. m. reidys , and p. sulkowski , _ topological recursion for chord diagrams , rna complexes , and cells in moduli spaces , _ nucl . phys . * b866 * ( 2013 ) 414 - 443 , arxiv:1205.0658 [ hep - th ] . j. e. andersen , l. o. chekhov , r. c. penner , c. m. reidys , and p. sulkowski , _ enumeration of rna complexes via random matrix theory , _ biochem . * 41 * ( 2013 ) 652 - 655 , arxiv:1303.1326 [ q-bio.qm ] . r. c. penner , _ cell decomposition and compactification of riemann s moduli space in decorated teichmller theory , _ in tongring , n. and penner , r.c . ( eds ) woods hole mathematics - perspectives in math and physics , world scientific , singapore , arxiv : math/0306190 [ math.gt ] .

we introduce the boundary length and point spectrum , as a joint generalization of the boundary length spectrum and boundary point spectrum in @xcite . we establish by cut - and - join methods that the number of partial chord diagrams filtered by the boundary length and point spectrum satisfies a recursion relation , which combined with an initial condition determines these numbers uniquely . this recursion relation is equivalent to a second order , non - linear , algebraic partial differential equation for the generating function of the numbers of partial chord diagrams filtered by the boundary length and point spectrum . , grant - in - aid for scientific research(c ) [ # 26400079 ] , and grant - in - aid for scientific research(b ) [ # 16h03927 ] from the japan ministry of education , culture , sports , science and technology . ]

introduction combinatorial proof of the cut-and-join equation non-oriented analogue of the cut-and-join equation

arxiv

apart from their technological importance , emulsions have served as model systems accessible to rigorous theoretical modeling . the study of emulsions consisting of droplets of a liquid dispersed in another liquid has thus contributed substantially to our understanding of the rheology of complex fluids . however , although major theoretical achievements date back to the beginning of the @xmath0 century , further progress turned out to be difficult . the macroscopic rheological properties of emulsions are determined by the reaction of the individual drops to the flow field , which in turn is modified by the presence of other drops . the mutual hydrodynamic interactions of drops complicates substantially the mathematical description . moreover , depending on system parameters and flow type , droplets may break under steady flow conditions if a certain critical strain rate is exceeded . rigorous calculations of the constitutive equation have therefore concentrated on very dilute emulsions and on conditions where drops are only weakly deformed . sometimes , however , it is desirable to have an approximate expression , which though not rigorous can serve for practical purposes as a quantitative description in the parameter region beyond the ideal limits . as far as the dependence of the viscoelastic properties of an emulsion on the volume fraction @xmath1 of the dispersed phase is concerned , such an approximation has been given by @xcite . it is not rigorous beyond first order in @xmath1 but serves well some practical purposes even at rather high volume fractions . it seems not reasonable to look for a comparably simple approximation for the dependence of shear viscosity @xmath2 on shear rate @xmath3 that covers the whole range of parameters , where all kinds of difficult break up scenarios are known to occur . in the next section , we propose instead a less predictive expression which contains an average drop size @xmath4 ( that may change with shear rate ) as a phenomenological parameter . the latter has to be determined independently either from theory or experiment . it turns out , however , that for a substantial range of viscosity ratios and shear rates , the expression for @xmath5 is to a large extent independent of morphology . for conditions , where drops do not break outside this region , we point out a similarity relation between this expression and the frequency dependent viscoelastic moduli @xmath6 , @xmath7 , similar to the _ rule in polymer physics . more precisely , we show that in the limit of small drop deformation , the constitutive equation of an emulsion composed of newtonian constituents of equal density can be obtained from the frequency dependent linear response to leading order in capillary number @xmath8 and ( reciprocal ) viscosity ratio @xmath9 . and we argue that the identification is likely to represent a good approximation beyond this limit in a larger part of the @xmath10 parameter plane . experimentally , this similarity relation can be tested directly , without interference of theoretical modeling , by comparing two independent sets of rheological data . in summary , our theoretical discussion provokes two major empirical questions . ( 1 ) if drops break : does the expression for the viscosity derived in eq.([eq : cm_miracle ] ) describe the data with @xmath4 the average drop size at a given shear rate ? ( 2 ) if drops do not break below a certain characteristic capillary number @xmath11 : does the proposed similarity rule hold ? to what extent does it generalize to non newtonian constituents ? in section [ sec : experiment ] we address mainly the second question by experiments with a quasi static droplet phase of a mixture of moderately viscoelastic polymer solutions . a common way to characterize the rheological properties of complex fluids such as emulsions , suspensions , and polymer solutions , is by means of a _ constitutive equation _ or an equation of state that relates the components @xmath12 of the _ stress tensor _ to the _ rate of strain tensor _ @xmath13 . this relation can account for all the internal heterogeneity and the complexity and interactions of the constituents if only the system may be represented as a homogeneous fluid on macroscopic scales . the form of possible constitutive equations is restricted by general _ symmetry arguments _ , which provide guidelines for the construction of phenomenological expressions @xcite . on the other hand , for special model systems the constitutive equations may be calculated directly at least for some restricted range of parameters . an early example for a direct computation of the constitutive equation of a complex fluid is einstein s formula @xmath14 for the shear viscosity @xmath15 of a dilute suspension ( particle volume fraction @xmath16 ) . it is obtained by solving stokes equation for an infinite homogeneous fluid of viscosity @xmath17 containing a single solid sphere . for a sufficiently dilute suspension , the contributions of different particles to the overall viscosity @xmath2 can be added independently , giving an effect proportional to @xmath1 . in close analogy @xcite calculated @xmath2 for a steadily sheared dilute suspension of droplets of an incompressible liquid of viscosity @xmath18 in another incompressible liquid of viscosity @xmath17 . for weakly deformed drops he obtained @xmath19 which we abbreviate by @xmath20 in the following . this expression includes einstein s result as the limiting case of a highly viscous droplet , @xmath21 . as in einstein s calculation , interactions of the drops are neglected . the result is independent of surface tension @xmath22 , shear rate @xmath3 , and drop radius @xmath4 ; i.e. , it is a mere consequence of the presence of a certain amount @xmath1 of dispersed drops , regardless of drop size and deformation ( as long as the latter is small ) . moreover , the dynamic ( frequency dependent ) linear response of an emulsion has been calculated by @xcite . his results are quoted in section [ sec : cox - merz ] below . under steady flow conditions , drop deformation itself is proportional to the magnitude of the rate of strain tensor @xmath13 . more precisely , for simple shear flow with constant shear rate @xmath23 , the characteristic measure of drop deformation for given @xmath24 is the _ capillary number _ @xmath25 also introduced by @xcite . it appears as dimensionless expansion parameter in a perturbation series of the drop shape under shear . to derive taylor s eq.([eq : taylor ] ) it is sufficient to represent the drops by their spherical equilibrium shape . aiming to improve the constitutive equation , @xcite took into account deformations of drops to first order in @xmath26 . the refined analysis did not affect the off diagonal elements of the constitutive equation , i.e. taylor s eq.([eq : taylor ] ) for the viscosity , but it gave the ( unequal ) normal stresses to order @xmath27 . another limit , where exact results can be obtained , is the limit of large viscosity ratios @xmath21 @xcite . to clarify the physical significance of the different limits we want to give a brief qualitative description of the behavior of a suspended drop under shear , based on work by @xcite and @xcite . in a quiescent matrix fluid of viscosity @xmath17 , a single weakly deformed drop relaxes exponentially into its spherical equilibrium shape ; i.e. , defining dimensionless deformation by @xmath28 with @xmath29 and @xmath30 the major and minor axis of the elongated drop , one has for a small initial deformation @xmath31 , @xmath32 the characteristic _ relaxation time _ @xcite @xmath33 also characterizes the macroscopic _ stress relaxation _ in an unstrained region of a dilute emulsion . at @xmath34 one observes the characteristic relaxation mode in the frequency dependent moduli . the relaxation time diverges for @xmath35 since it takes longer for a weak surface tension to drive a viscous drop back to equilibrium . what happens if the matrix is steadily sheared at shear rate @xmath36 ? for @xmath37 , the flow induced in the drop by the external driving is weak compared to the internal relaxation dynamics and the equilibrium state is only slightly disturbed , i.e. , the drop is only weakly deformed . similarly , for large viscosity ratio @xmath24 , the elongation of the drop becomes very slow compared to vorticity , and hence again very small in the steady state , even if @xmath38 is not small . technically , this is due to the asymptotic proportionality to @xmath9 of the shear rate within the drop . in both limits of weak deformation , the time @xmath39 also controls the orientation of the major axis of the drop with respect to the flow according to @xmath40 eq.([eq : d_relax ] ) and eq.([eq : angle ] ) both can be used to determine the surface tension @xmath22 from observations of single drops under a microscope . in passing , we note that the classical method based on the result obtained by @xcite for the steady state deformation , can only be used if @xmath24 is not too large , whereas eq.([eq : d_relax ] ) and eq.([eq : angle ] ) are more general . the exact calculations mentioned so far became feasible because ( and are applicable if ) deviations of the drop from its spherical equilibrium shape are small . on the other hand , if neither the capillary number ( or @xmath38 ) is small nor the viscosity ratio is large , i.e. , @xmath41 drops can be strongly deformed by the symmetric part of the flow field . experiments with single drops by @xcite and others have shown that this eventually leads to drop break up if @xmath24 is smaller than some critical viscosity ratio . no general rigorous result for the viscosity of an emulsion is known in this regime , where arbitrary drop deformations and break up may occur . below we will also be interested in such cases , where the conditions required for the rigorous calculations are not fulfilled . for the following discussion we introduce some additional notation . the rate of strain tensor @xmath13 and the vorticity tensor @xmath42 are defined as symmetric and antisymmetric parts of the velocity gradient @xmath43 . in particular , for a steady simple shear flow @xmath44 , and @xmath45 the components @xmath46 of the stress tensor in the shear plane ( @xmath47 ) as obtained for finite @xmath24 by @xcite read @xmath48 as usual , the material derivative has been defined by @xmath49 where summation over repeated indices is implied , and the first two terms vanish for steady shear flow . note that since @xmath50 is diagonal , eq.([eq : schowalter ] ) implies @xmath51 , and hence eq.([eq : taylor ] ) remains valid to first order in @xmath8 as we mentioned already . ) , ( [ eq : cm_miracle ] ) for the viscosity @xmath2 of an emulsion give rigorous results . in the shaded region they predict @xmath2 to be practically independent of capillary number . the frame of the box is meant to comprise the whole @xmath26-@xmath24 parameter plane from zero to infinity ( small reynolds number understood ) . the curved solid line is a sketch of the break up curve for steadily sheared isolated newtonian drops according to @xcite . dashed lines indicate schematically our viscosity measurements ( see section [ sec : experiment ] ) . ] can we extrapolate the exact second order result eq.([eq : schowalter ] ) for the stress tensor to arbitrary @xmath8 and @xmath24 by using the constraints provided by general invariance arguments ? for example , since the shear stress has to change sign if the direction of the shear strain is inverted whereas the normal stresses do not , the shear stress and the normal stresses have to be odd / even functions of @xmath3 , respectively . from this observation we could have foreseen that eq.([eq : taylor ] ) can not be improved by calculating the next order in @xmath3 , i.e. , by considering droplet deformation to lowest order . more important are galilean invariance and invariance under transformations to rotating coordinate frames , which give rise to the material derivative introduced above . applying the operator @xmath52 to eq.([eq : schowalter ] ) adds to the right hand side of the equation a term @xmath53 plus a term of order @xmath54 , so that one obtains ( in the shear plane ) @xmath55 as another short hand notation we have introduced a second characteristic time @xmath56 , which to the present level of accuracy in @xmath1 is given by @xmath57 it sets the time scale for _ strain relaxation _ in an unstressed region and was named _ retardation _ time by @xcite . @xcite realized that up to the partly unknown terms of order @xmath58 on the right hand side , eq.([eq : acrivos ] ) belongs to a class of possible viscoelastic equations of state already discussed by @xcite . hence , setting @xmath59 on the right hand side of eq.([eq : acrivos ] ) , we can define a ( minimal ) model viscoelastic fluid that behaves identical to the emulsion described by eq.([eq : schowalter ] ) for small shear rates . in contrast to eq.([eq : acrivos ] ) , the truncated formula for the viscosity @xmath60\ ] ] thus obtained has a manifestly non perturbative form . however , no phenomenological parameters had to be introduced . note that eq.([eq : miracle ] ) comprises both exactly known limits : @xmath61 for fixed @xmath24 , and @xmath21 for arbitrary @xmath8 . obviously , @xcite have forgotten a term @xmath62 in their eq.(3.6 ) for @xmath46 in the limit @xmath21 for fixed @xmath26 . if the latter is included , eq.([eq : miracle ] ) is also in accord with their @xmath63analysis . moreover , eq.([eq : miracle ] ) has the proper limiting behavior for @xmath64 , @xmath65 , i.e. @xmath66 , which is an extreme case of eq.([eq : restriction ] ) . since we assume equal densities for the two phases , the two phase fluid actually reduces to a one phase fluid in this degenerate case , and the viscosity is simply @xmath17 , independent of morphology . for illustration , the rigorous limits of eq.([eq : miracle ] ) in the @xmath10 plane are depicted graphically in fig . [ fig : limits ] . following @xcite , a qualitative break up curve for single drops under steady shear is also sketched . in summary , eq.([eq : miracle ] ) is correct for arbitrary @xmath24 if @xmath67 , and for arbitrary @xmath8 if @xmath21 , and for small and large @xmath8 if @xmath64 . therefore , one can expect that eq.([eq : miracle ] ) works reasonably well within a large parameter range ( small reynolds number understood ) . this is further supported by the observation that the error made in going from eq.([eq : schowalter ] ) to eq.([eq : miracle ] ) rather concerns the _ shape _ of the droplet than its _ extension _ ( it consists in truncating a perturbation series in shape parametrisation ) . the final result , though sensitive to the latter , is probably less sensitive to the former . nevertheless , one would not be surprised to see deviations from eq.([eq : miracle ] ) when drops become extremely elongated . finally , due to changes in morphology by break up and coalescence , the avarage drop size @xmath4 may change . observe , however , that for most viscosity ratios ( @xmath24 not close to unity ) , eq.([eq : miracle ] ) is practically independent of capillary number ( and thus of @xmath4 ) when break up might be expected according @xcite and others . as an analytic function that is physically known to be bounded from above and from below ( the latter at least by the viscosity of a stratified two phase fluid depicted in fig . [ fig : mixing ] ) , @xmath68 has to have vanishing slope in the @xmath69direction for large @xmath8 . according to eq.([eq : miracle ] ) , @xmath68 is almost independent of @xmath8 for @xmath70 where @xmath71 is the turning point in the dilute limit , determined by @xmath72 . for finite volume fractions @xmath73 from eq.([eq : tau_1_oldroyd ] ) replaces @xmath39 . hence , for @xmath74 , eq.([eq : miracle ] ) and its extension to higher volume fractions derived below in eq.([eq : cm_miracle ] ) are practically independent of drop deformation and morphology . for finite volume fractions a rough interpolation for @xmath71 derived from eq.([eq : cm_miracle ] ) is given by @xmath75 . in a large part of parameter space we thus expect eqs.([eq : miracle ] ) , ( [ eq : cm_miracle ] ) to be applicable to monotonic shear histories with @xmath4 given by the average initial radius of the droplets . for non monotonic shear histories , there can of course be hysteresis effects in @xmath76 that result from morphological changes for @xmath74 . these can only be avoided by substituting for @xmath4 the radius corresponding to the actual average drop size at the applied shear rate @xmath36 . of the continuous phase / the viscosity @xmath2 of the whole emulsion , within / outside the `` free volume sphere '' of radius @xmath77 . ] a general limitation of the equations discussed so far , is the restriction to small volume fractions . above , we have implicitly assumed that second order effects from drop interactions are small compared to second order effects from drop deformation . any direct ( coalescence ) and indirect ( hydrodynamic ) interactions of droplets have been neglected in the derivation . hydrodynamic interactions can approximately be taken into account by various types of cell models . recently @xcite proposed a self consistent method analogous to the clausius mossotti or lorentz sphere method of electrostatics . for the case of a disordered spatial distribution of drops his results reduce to those already obtained by @xcite . oldroyd artificially divides the volume around a droplet of viscosity @xmath78 into an interior `` free volume '' with a viscosity @xmath17 of the bare continuous phase and an exterior part with the viscosity @xmath2 of the whole emulsion ( see fig . [ fig : mf - trick ] ) . according to this scheme , an improved version of eq.([eq : taylor ] ) should be @xcite @xmath79 this equation predicts a larger viscosity than its truncation to first order in @xmath1 , eq.([eq : taylor ] ) . both are shown as dot dashed lines in fig . [ fig : mixing ] . eq.([eq : cm_taylor ] ) is _ qualitatively _ superior to eq.([eq : taylor ] ) . we note , however , that the limit @xmath80 deviates in second order in @xmath1 from the result obtained for suspensions by @xcite . eq.([eq : cm_taylor ] ) and likewise all of the following equations containing quantities @xmath81 , @xmath73 , @xmath82 are only rigorous to first order in @xmath1 . the same reasoning as to the viscosity applies to the characteristic times @xmath39 and @xmath56 which now read @xcite @xmath83[2\lambda+3 - 2\phi(\lambda-1 ) ] } { 40(\lambda+1)-8\phi(5\lambda+2 ) } \;,\ ] ] @xmath84[2\lambda+3 + 3\phi(\lambda-1 ) ] } { 40(\lambda+1)+12\phi(5\lambda+2 ) } \;.\ ] ] their ratio @xmath85 is still given by eq.([eq : tau_2 ] ) . finally , eq.([eq : miracle ] ) becomes @xmath86 which to our knowledge has not been given before , and is one of our main results . ( for a graphical representation see fig . [ fig : graph ] . ) in the limit @xmath87 it reduces to eq.([eq : cm_taylor ] ) , whereas for @xmath88 only the second term in parentheses contributes and the _ curved _ dashed lines in fig . [ fig : mixing ] are obtained . from the foregoing discussion one should expect eq.([eq : cm_miracle ] ) to be applicable within a large range of shear rates , viscosities , and volume fractions . finally , we note that more cumbersome expressions for @xmath89 , @xmath90 and @xmath82 have been derived within a different cell model by @xcite . here we only quote their expression for @xmath89 , @xmath91 } { 4(\lambda+1)-5(5\lambda+2)\phi+42\lambda\phi^{5/3}-5(5\lambda-2)\phi^{7/3 } + 4(\lambda-1)\phi^{10/3}}\ ] ] which is also represented graphically by the dotted lines in fig . [ fig : mixing ] . since our data favor eq.([eq : cm_taylor ] ) over eq.([eq : choi ] ) , and similar observations have been made by others before ( see section [ sec : experiment ] ) , we will not pursue this alternative approach further in the present contribution . ( chosen arbitrarily ) . the dot dashed and dashed _ straight _ lines pertain to dilute emulsions described by the extrapolation formula eq.([eq : miracle ] ) , which reduces to taylor s formula eq.([eq : taylor ] ) for small capillary numbers . the corresponding _ curved _ lines are obtained from eq.([eq : cm_miracle ] ) where interactions of the droplets are taken into account in a mean field approximation . the curved _ dotted _ lines are the predictions of the cell model by @xcite for small shear rates , eq.([eq : choi ] ) . the curved _ solid _ line represents the viscosity @xmath92 $ ] of a two phase stratified fluid and is a lower bound for any viscosity mixing rule . ] it is interesting to observe that if morphology is conserved ( drop size @xmath4 independent of shear rate ) for @xmath93 , our eq.([eq : cm_miracle ] ) for the nonlinear shear viscosity is closely related to the expressions for the frequency dependent complex viscosity @xmath94 of an emulsion of two incompressible newtonian liquids as derived by @xcite , @xmath95 for convenience , we also give the corresponding viscoelastic shear modulus @xmath96 , @xmath97 obviously , the shear rate dependent viscosity of eq.([eq : cm_miracle ] ) is obtained from the real part of the complex frequency dependent viscosity @xmath98 by substituting @xmath36 for @xmath99 , @xmath100 in the same way , the first normal stress difference @xmath101 from eq.([eq : schowalter ] ) is obtained to leading order in @xmath1 and @xmath36 from @xmath102 , i.e. , @xmath103 reverting the line of reasoning pursued so far , we can conclude that to leading order in @xmath8 and/or @xmath9 the weak deformation limit of the constitutive equation of emulsions is obtained from the linear viscoelastic spectra @xmath104 , @xmath105 . further , this identification can possibly be extended ( at least approximately ) into regions of the @xmath10 plane where the critical capillary number for drop breakup is somewhat larger than @xmath71 of eq.([eq : region ] ) . in hindsight , it is not surprising that in the case of weakly deformed drops the frequency dependent viscosity and the steady shear viscosity are related . note that under steady shear , drops undergo oscillatory deformations at a frequencey @xmath106 if observed from a co rotating frame turning with vorticity at a frequency @xmath107 . if we take eq.([eq : first_normal_stress ] ) seriously beyond the rigorously known limit , we obtain an interesting prediction for the first normal stress difference . in contrast to eq.([eq : fnsd_schowalter ] ) , eq.([eq : first_normal_stress ] ) implies that the first normal stress difference saturates at a finite value @xmath108 ^ 2 $ ] for high shear rates . thus , although the initial slope of the first normal stress difference with @xmath3 increases with @xmath24 , its limit for large @xmath3 decreases with @xmath24 . finally , we remark that based on qualitative theoretical arguments , the similarity relation contained in eq.([eq : modified_cox - merz ] ) and eq.([eq : first_normal_stress ] ) has recently been proposed also for polymer melts @xcite . usually , in polymer physics a slightly different relation is considered ; namely a similarity between @xmath109 and @xmath110 , also known as _ cox merz rule _ @xcite . in our case , since @xmath111 , we can write @xmath112 under the conditions mentioned at the beginning of this section , the usual cox merz rule is fulfilled to first order in @xmath1 for an emulsion . eqs.([eq : modified_cox - merz ] ) , ( [ eq : first_normal_stress ] ) are interesting from the theoretical point of view , because they suggest a similarity of two _ a priory _ rather different quantities . the results of this section also can be of practical use , since they suggest that two different methods may be applied to measure a quantity of interest . ) normalized to @xmath17 as a function of viscosity ratio @xmath24 and capillary number @xmath8 . the volume fraction of the dispersed phase is chosen to be @xmath113 . ] generalization of the above theoretical discussion to the case of non newtonian constituents is not straightforward . indeed , as @xcite already knew , his linear response results quoted in eq.([eq : eta_w_oldroyd ] ) and eq.([eq : g_w_oldroyd ] ) are readily generalized to viscoelastic constituents by replacing the viscosities @xmath114 in the expression for @xmath115 ( or @xmath116 ) by complex viscosities @xmath117 @xcite . as a consequence , the decompositions of @xmath115 and @xmath116 in real and imaginary parts are no longer those of eqs.([eq : eta_w_oldroyd ] ) and ( [ eq : g_w_oldroyd ] ) , and @xmath118 , @xmath119 , @xmath120 , @xmath121 are given by more cumbersome expressions . for the steady state viscosity , on the other hand , one has to deal with a non homogeneous viscosity even within homogeneous regions of the emulsion , since the strain rate itself is non homogeneous and the viscosities are strain rate dependent . we do not attempt to solve this problem here , nor do we try to account for elasticity in the nonlinear case . yet , it is an intriguing question , whether the similarity rule eq.([eq : modified_cox - merz ] ) can be generalized to the case of non - newtonian constituents if the constituents themselves obey the cox merz rule ( what many polymer melts and solutions do ) . if both constituents have similar phase angles @xmath122 the generalized viscosity ratio @xmath123 that enters the expressions for @xmath115 and @xmath116 , transforms approximately to @xmath124 by applying the cox merz rule . therefore , in this particular example , eq.([eq : cm_miracle ] ) supports the expectation that the generalization may work at least approximately . if , on the other hand , the phase angles of the constituents behave very differently , the answer is less obvious . this problem has been investigated experimentally and is further discussed in section [ sec : experiment ] . in any case , the generalization can only work if the representation of the emulsion by a simple shear rate dependent viscosity ratio @xmath125 , with @xmath36 the external shear rate , is justified . in the remainder of this section we construct an argument that allows us to estimate the effective shear rate dependent viscosity ratio that should replace @xmath24 in eq.([eq : cm_miracle ] ) . we take into account the deviation of the strain rate from the externally imposed flow only within the drops , because outside the drops the discrepancy is always small . inside a drop , the strain rate can be small even for high external shear rates if the viscosity ratio @xmath126 is large . since we are looking for an effective viscosity @xmath127 for the whole drop to replace the viscosity @xmath128 at small shear rate , we replace the non newtonian drop of non homogeneous viscosity by an effective pseudo newtonian drop of homogeneous but shear rate dependent viscosity . a possible ansatz for @xmath129 is obtained by requiring that the total energy dissipated within the drop remains constant upon this substitution . hence , we have @xmath130 where @xmath131 and @xmath132 denote the rate of strain fields in the real drop and in the corresponding model drop of effective viscosity @xmath129 , respectively . they both depend on the position within the drop , whereas the average strain rate @xmath133 that enters the right hand side of eq.([eq : dissipation ] ) does not . since the strain field @xmath132 within a drop of homogeneous viscosity is unique for a given system in a given flow field , @xmath129 itself can be expressed as a function @xmath134 of the average strain rate . here , we approximate this functional dependence by the strain rate dependence @xmath135 of the viscosity of the dispersed fluid . further , neglecting drop deformation we calculate @xmath136 from the velocity field within a spherical drop @xcite and obtain @xmath137 a different prefactor ( @xmath138 in place of @xmath139 ) in the expression for the effective strain rate @xmath140 was obtained by @xcite using instead of the average in eq.([eq : avg_strain ] ) the maximum norm of @xmath132 . to obtain the correction to eq.([eq : cm_miracle ] ) due to eq.([eq : eta_eff ] ) in the case of non newtonian constituents , the implicit equation for @xmath141 has to be solved for given functions @xmath142 and @xmath135 . for shear thinning constituents , eq.([eq : eta_eff ] ) implies a tendency of eq.([eq : cm_miracle ] ) to overestimate @xmath143 if @xmath36 and @xmath24 are large . in the actual case of interest , for the constituents that were used in the experiments discussed in section [ sec : experiment ] , the viscosity ratio @xmath24 ( @xmath128 and @xmath17 taken at the external shear rate @xmath3 ) varies almost by a factor of 10 . however , the corrections discussed in this section only become important for high shear rates , where the constituents are shear thinning . in this regime , the viscosity ratio ( viscosities taken at the external shear rate ) only varies between @xmath144 and @xmath145 , and hence the corrections expected from eq.([eq : eta_eff ] ) are at best marginally significant at the level of accuracy of both eq.([eq : eta_eff ] ) and the present measurements . therefore , a representation of the drops by pseudo newtonian drops of homogeneous but shear rate dependent viscosity is most probably not a problem for the measurements presented in the following section . the question as to a generalization of the similarity rule eq.([eq : modified_cox - merz ] ) to non newtonian constituents seems well defined . the experimental investigation deals with a phase separated aqueous solution containing a polysaccharide ( alginate ) and a protein ( caseinate ) . this type of solutions are currently used in the food industry . the methods for characterizing the individual polymers in solution are in general known , especially when dealing with non gelling solutions where composition and temperature are the only relevant parameters . the polymers are water soluble . when the two biopolymers in solution are mixed , a miscibility region appears in the low concentrations range and phase separation at higher concentrations . the binodal and the tie lines of the phase diagram can then be established by measuring the composition of each phase at a fixed temperature . in general , the rheological behavior of phase separated systems is difficult to investigate , and a suitable procedure is not fully established . in some cases , two phase solutions macroscopically separate by gravity within a short period of time , but in some other cases ( such as ours ) they remain stable for hours or days without appearance of any visible interface . these `` emulsion type '' solutions have no added surfactant . following approaches developed for immiscible blends , one may try to characterize the partially separated solution as an effective emulsion if the coarsening is slow enough . in order to establish a comparison between phase separated solutions and emulsions , it is necessary to know * the volume fraction of the phases , * their shear rate dependent viscosities ( flow curves ) , * their viscoelastic spectra , * the interfacial tension @xmath22 between the phases , * the average radius of @xmath4 the drops a difficulty when working with phase separated solutions , as opposed to immiscible polymer melts , arises from the fact that each phase is itself a mixture ( and not a pure liquid ) and therefore the rheology of the phase depends on its particular composition . if one wishes to minimize the number of parameters , it is important to keep the composition of the phases constant upon changing the volume fractions . this can be achieved by working along a tie line of the phase diagram . and this is precisely the procedure that we followed . the polymers were first dissolved , then a large quantity of the ternary mixture was prepared ( 350 ml ) and was centrifuged . the two phases were then collected separately . both pure phases were found to be viscoelastic and to exhibit shear thinning behavior , which is especially pronounced for the alginate rich phase with @xmath147 , while the caseinate rich phase is almost newtonian below @xmath148 s@xmath149 . the viscosity of the alginate rich phase is higher than that of the caseinate rich phase for shear rates below @xmath150 s@xmath149 and lower for higher shear rates . we checked that both phases obey the usual cox merz rule in the whole range of applied shear rates . by mixing various amounts of each phase , the volume fraction of the dispersed phase was varied between @xmath151 and @xmath152 while the composition of each phase was kept constant . in particular , the temperature was kept constant and equal to the centrifugation temperature in order to avoid redissolution of the constituents . to prepare the emulsion , the required quantities of each phase were mixed in a vial and gently shaken . then the mixture was poured on the plate of the rheometer ( ar 1000 from ta instruments fitted with a cone and plate geometry 6 cm/@xmath153 ) and a constant shear rate was applied . the apparent viscosity for a particular shear rate was then recorded versus time until it reached a stable value . by shearing at a fixed shear rate , one may expect to create a steady size distribution of droplets , with a shear rate dependent average size . after each shear experiment a complete dynamic spectrum was performed . in this way , shear rates ranging between @xmath154 s@xmath149 and @xmath155 s@xmath149 were applied . the analysis of each spectrum according to @xcite allowed us to derive by curve fitting the average drop radius @xmath4 at the corresponding shear rate . more technical details about the experimental investigation along with more experimental results will be presented elsewhere . here , we concentrate on the analysis of those aspects of the rheological measurements pertinent to the theoretical discussion in section [ sec : theory ] . in this section we present our experimental observations and address the questions posed at the end of the introduction . before we present our own data we want to comment briefly on related data recently obtained for polymer melts by @xcite . these authors measured the shear rate dependent viscosity of binary polymer melts and compared them to the low volume fraction limit of eq.([eq : cm_miracle ] ) , i.e. eq.([eq : miracle ] ) , and to ( a truncated form of ) results of @xcite . they reported much better agreement with eq.([eq : miracle ] ) than with the truncated series from @xcite . comparison with the full expressions of @xcite would have made the disagreement even worse ( cf . fig [ fig : mixing ] ) . the average radius @xmath4 that enters the equation , was determined independently for each shear rate applied . the constituents where moderately non newtonian polymer melts , the viscosity ratio varying between @xmath156 over the range of shear rates applied . hence , these experiments , are located in the interesting parameter range , where eqs.([eq : cm_miracle ] ) and eq.([eq : miracle ] ) for @xmath157 are expected to be sensitive to drop deformation and break up . surprisingly , the results show that they describe the data very well over the whole range of shear rates although one would not necessarily expect average drop deformation to be very small . unfortunately , drop sizes have not been reported by the authors , so conclusions concerning the location in the @xmath10 parameter plane and the validity of the similarity rule eq.([eq : modified_cox - merz ] ) can not be drawn . also the question , whether eq.([eq : cm_miracle ] ) holds for small viscosity ratios @xmath158 , can not be answered . ( opaque squares ) and the real part @xmath159 of the dynamic viscosity @xmath160 ( lines ) of a droplet phase of a mixture of weakly viscoelastic polymer solutions ( alginate / caseinate ) . also shown is eq.([eq : cm_miracle ] ) for the viscosity of an emulsion of newtonian constituents evaluated for the actually non newtonian viscosities of the constituting phases with the drop size obtained from the spectra ( open triangles ) . due to the scatter in the dynamic viscosity at low frequencies there is an uncertainty in the average drop size , resulting in corresponding error bars ( multiple points ) for eq.([eq : cm_miracle ] ) . ] our own measurements were located in about the same @xmath161range . as we noted in the preceding section , only the ratio @xmath146 enters rheological equations and knowledge of either @xmath4 or @xmath22 allows the other quantity to be inferred from rheological measurements . @xcite determined the interfacial tension of the alginate / caseinate system used in our experiments by observing drop relaxation under a microscope and analyzing the data according to section [ sec : theory ] . they found @xmath162 n / m . using this , we obtained an average drop size @xmath163 m from the measured spectra @xmath104 , @xmath164 according to @xcite for the experiments reported in fig . [ fig : cox - merz ] . by the method based on @xcite , we could not detect a decrease in drop size with shear rate as expected from the phenomenological phase diagram for single newtonian drops under shear as established by @xcite and others . thus , the limit of high capillary numbers and moderate viscosity ratios ( the region above the break up curve ) in fig . [ fig : limits ] has been accessed experimentally . corresponding locations have been indicated qualitatively in the figure by dashed lines . with the drop size being constant , one can try to test the proposed similarity rule eq.([eq : modified_cox - merz ] ) . by identifying the axis for frequency @xmath99 and shear rate @xmath36 , data for the real part @xmath159 of the frequency dependent dynamic viscosity @xmath165 are compared to data for the shear rate dependent viscosity @xmath157 in fig . [ fig : cox - merz ] . the emulsion containing @xmath152 of the alginate rich phase and @xmath166 of the caseinate rich phase has been prepared at room temperature as described in the preceding section . steady shear rates ranging between @xmath154 s@xmath149 and @xmath155 s@xmath149 corresponding to capillary numbers @xmath167 have been applied . the shear viscosity ( opaque squares ) is reported in the figure for each of these individual measurements . the multiple data sets for @xmath118 ( lines ) taken each between two successive steady shear measurements , superimpose fairly well ; i.e. the spectra appear to be remarkably independent of the preceding steady shear rate . the good coincidence of @xmath168 and @xmath157 in fig . [ fig : cox - merz ] show that the data obey the proposed similarity rule eq.([eq : modified_cox - merz ] ) over a large range of shear rates . the agreement near @xmath169 s@xmath149 is a consequence of the proximity to the trivial limit @xmath64 , @xmath170 . nevertheless , the data provide strong evidence that eq.([eq : modified_cox - merz ] ) is an excellent approximation for a large range of viscosity ratios and capillary numbers . similar results ( not shown ) have been obtained for other volume fractions . comparison with eq.([eq : cm_miracle ] ) represented by the open triangles in fig . [ fig : cox - merz ] , on the other hand , is less successful at large shear rates , although it is still not too far off for a theoretical curve without any adjustable parameter . a discrepancy had to be expected as a consequence of the non newtonian character of the constituents at large shear rates , which is definitely not taken into account in eq.([eq : cm_miracle ] ) . for the plot of eq.([eq : cm_miracle ] ) in fig . [ fig : cox - merz ] we merely substituted @xmath171 taken at the external shear rate @xmath36 for @xmath24 . the average drop size @xmath4 was obtained from fitting the viscoelastic moduli . the scatter in the dynamic viscosity data gives rise to an uncertainty in @xmath4 , which is reflected by the multiple open triangles at low shear rates . it seems that the similarity rule eq.([eq : modified_cox - merz ] ) is more general than eq.([eq : cm_miracle ] ) , i.e. , it still holds for rather viscoelastic constituents ( that obey the usual cox merz rule ) , where the latter fails . this relation certainly deserves further investigation with different materials and methods . in summary , we have succeeded in establishing an analogy first between a partially phase separated polymer solution and an emulsion , and further between the viscoelastic spectrum of the system and its nonlinear shear viscosity even in the case of ( moderately ) non newtonian constituents . this work was supported by the european community under contract no fair / ct97 - 3022 . we thank p. ding and a. w. pacek ( university of birmingham ) for measuring the surface tension and s. costeux and g. haagh for helpful discussions and suggestions . 1998 the morphology - dependent rheological behavior of an immiscible model polymer blend . in _ proceedings of the @xmath173 european rheology conference _ i. emri & r. cvelbar ) , _ progress and trends in rheology _ , vol . 5 , p. 80 . darmstadt : steinkopff .

combining direct computations with invariance arguments , taylor s constitutive equation for an emulsion can be extrapolated to high shear rates . we show that the resulting expression is consistent with the rigorous limits of small drop deformation and that it bears a strong similarity to an _ a priori _ unrelated rheological quantity , namely the dynamic ( frequency dependent ) linear shear response . more precisely , within a large parameter region the nonlinear steady state shear viscosity is obtained from the real part of the complex dynamic viscosity , while the first normal stress difference is obtained from its imaginary part . our experiments with a droplet phase of a binary polymer solution ( alginate / caseinate ) can be interpreted by an emulsion analogy . they indicate that the predicted similarity rule generalizes to the case of moderately viscoelastic constituents that obey the cox merz rule .

introduction theory experiment

arxiv

turbulent flow occurs in the natural environmental and in technological applications with such frequency that it could be considered the `` natural '' state of fluid flows to be found around us . traditionally , fluid flows have been observed and studied in the eulerian perspective where a fixed position is observed for a definite interval of time . the other perspective , the lagrangian , follows the motion of the fluid and thus is better suited to study aspects of fluid flow such as material transport or the deformation of fluid material in a given state of motion . the use of stretching quantifiers such as the lyapunov exponents , which measure the relative separation between particles @xcite , has broadly improved the lagrangian study of fluid flows . on the one hand lyapunov methods provide information on time scales for dispersion processes , with its relevance for mixing and stirring of fluids @xcite . on the other , they are useful to detect the so - called lagrangian coherent structures ( lcs ) . lcss @xcite are templates for particle advection in complex flows , separating regions with different dynamical behavior and acting as barriers and avenues to transport , fronts or eddy boundaries @xcite . relationships of lcss with lyapunov fields have been established for the case of finite - time lyapunov exponents ( ftles ) @xcite . these relationships state that lcs can be identified with the ridges ( generalized maxima ) of the ftle field . furthermore they state that the flux through the lcs is inversely proportional to the strength of the ridge and to the integration time of the ftle field calculation . this flux is shown to be small and the lcs extracted as the ridges of ftle fields are considered to be almost material - like surfaces . this identification has become widely used in the field although it should be mentioned that there are other more precise definitions of lcs @xcite , that consider lcs to be exact material surfaces admitting zero flux across them . in our work , we use instead finite - size lyapunov exponents ( fsles ) , which quantify the separation rate of fluid particles between two given distance thresholds @xcite . they turn out to be convenient for the case of bounded flows in which characteristic spatial scales are more direct to identify than temporal ones and have been shown to be robust with respect to noisy or poorly resolved velocity fields @xcite . although a rigorous connection between the fsle and lcss has not been established yet , previous work @xcite has shown that the ridges of the fsle behave in a similar fashion as the ridges of the ftle field . following these works we assume that lcss can be computed as ridges of fsles , and that they are transported by the flow as almost material surfaces / lines , with negligible flux of particles through them . observations presented here are consistent with those assumptions . despite its relevance in real flows , the full three - dimensional ( 3d ) structure of lcss is still an open subject . in 3d flows , lcs were explored in atmospheric contexts @xcite , and in a turbulent channel flow at @xmath0 in @xcite . a kinematic abc flow was studied in @xcite . in the ocean , where it is widely recognized that filamental structures , eddies , and in general oceanic meso- and submeso - scale structures have a great influence on marine ecosystems @xcite , the identification of lcss and the study of their role in the transport of biogeochemical tracers has primarily been restricted to two - dimensional ( 2d ) layers @xcite . there are two concurrent reasons for this : a ) because of stratification and rotation , vertical motions in the ocean are usually very small when compared to horizontal displacements ; b ) synoptic measurements ( e.g. from satellites ) of relevant quantities are restricted to the surface . a few previous results for lagrangian eddies in 3d were obtained in refs . @xcite , by applying the methodology of lobe dynamics and the turnstile mechanism . also , @xcite used 3d fsle fields to identify lcs in oceanic flows . in particular , a mesoscale eddy in the southern atlantic was studied in @xcite , and it was shown that oceanic lcss presented a vertical curtain - like shape , i.e. they look mostly like vertical sheets , and that material transport into and out of the mesoscale eddy occurred through filamentary deformation of such structures . in this paper , we use 3d fields of fsle to identify lcss in a turbulent channel flow and in an oceanic flow . observations of the similarities and differences between the two systems , both in their computation and their physical meaning , helps to appreciate the power and scope of this lagrangian technique in the analysis of fluid flows . in section [ sec : methods ] we describe the methodology used to identify lcss in 3d turbulent flows . sections [ sec : channel_flow ] and [ sec : ocean_flow ] are devoted to the turbulent channel flow and the oceanic flow , respectively , and section [ sec : conclus ] presents our conclusions and directions for future work . in order to study non - asymptotic dispersion processes such as stretching at finite scales and bounded domains , the finite size lyapunov exponent was introduced @xcite . it is defined as : @xmath1 where @xmath2 is the time it takes for the separation between two particles , initially @xmath3 , to reach a value @xmath4 . in addition to the dependence on the values of @xmath5 and @xmath6 , the fsle depends also on the initial position of the particles and on the time of deployment . locations ( i.e. initial positions ) leading to high values of this lyapunov field identify regions of strong separation between particles , i.e. , regions that will exhibit strong stretching during evolution , that can be identified with the lcs @xcite . in principle , to compute fsle in 3d , the method of @xcite can be extended to include the third dimension , by computing the time it takes for particles initially separated by @xmath7^{1/2}$ ] to reach a final distance of @xmath8^{1/2}$ ] . we will proceed this way for the turbulent channel , but , as indicated in @xcite , vertical displacements are much smaller than horizontal ones in ocean flows . therefore , the displacement in the @xmath9 direction does not contribute significatively to the calculation of @xmath6 in the ocean , which prompt us to implement a quasi-3d computation of fsles : we use the full 3d velocity field for particle advection but particles are initialized in 2d horizontal ocean layers and the contribution @xmath10 is not considered when computing @xmath6 ( see more details in @xcite ) . in any case , since we allow the full 3d trajectories of particles , we take into account the vertical dynamics of the oceanic flows . . , scaledwidth=50.0% ] concerning the turbulent channel , where we can implement a fully 3d computation of the fsle , we proceed as follows . a grid of initial locations @xmath11 is set up at time @xmath12 , fixing the spatial resolution of the fsle field ( figure [ fig1 ] ) . particles are released from each grid point and their three - dimensional trajectories are calculated . the distances of each neighbor particle with respect to the central one ( initially @xmath5 ) is monitored until one of the separations reaches a value @xmath4 . in both systems considered , we obtain two different types of fsle maps by integrating the three - dimensional particle trajectories backward and forward in time : the attracting lcss ( for the backward ) , and the repelling lcss ( forward ) @xcite . we obtain in this way fsle fields with a spatial resolution given by @xmath5 . when a particle leaves the velocity field domain or reaches a no - slip boundary , the fsle value at its initial position and initial time is set to zero . if the interparticle separation remains smaller than @xmath4 past a maximum integration time @xmath13 , then the fsle for that location is also set to zero . the identification of lcs calculated from lyapunov fields in 2d flows is straightforward since they practically coincide with ( finite - time ) stable and unstable manifolds of relevant hyperbolic structures in the flow @xcite ( but see @xcite ) . the structure of these manifolds in 3d is generally much more complex than in 2d @xcite , and they can be locally either lines or surfaces . differently than 2d , where lcs can be visually identified as the maxima of the fsle field , in 3d they are hidden within the volume data and one needs to explicitly compute and extract them , using the definition of lcss as the ridges of the fsle field . a ridge @xmath14 is a co - dimension 1 orientable , differentiable manifold ( which means that for a 3d domain @xmath15 , ridges are surfaces ) satisfying the following conditions @xcite : 1 . the field @xmath16 attains a local extremum at @xmath14 . the direction perpendicular to the ridge is the direction of fastest descent of @xmath16 at @xmath14 . the method used to extract the ridges from the scalar field @xmath17 is from @xcite . it uses an earlier @xcite definition of ridge in the context of image analysis , as a generalized local maxima of scalar fields . for a scalar field @xmath18 with gradient @xmath19 and hessian @xmath20 , a _ d_-dimensional height ridge is given by the conditions @xmath21 where @xmath22 , are the eigenvalues of @xmath20 , ordered such that @xmath23 , and @xmath24 is the eigenvector of @xmath20 associated with @xmath25 . for @xmath26 , eq . ( [ ridg1 ] ) becomes @xmath27 in other words , in @xmath28 the @xmath29 eigenvectors point locally along the ridge and the @xmath30 eigenvector is orthogonal to it , so the ridge maximizes the scalar field in the normal direction to it and in this direction the field is more convex than in any other direction , since the eigenvector associated with the most negative eigenvalue is oriented along the direction of maximum negative curvature of the scalar field . the extraction process progresses by calculating the points where the ridge conditions are verified and the ridge strength @xmath31 is higher than a predefined threshold @xmath32 so that ridge points whose value of @xmath33 is lower ( in absolute value ) than @xmath32 are discarded from the extraction process . since the ridges are constructed by triangulations of the set of extracted ridge points , the strength threshold greatly determines the size and shape of the extracted ridge , by filtering out regions of the ridge that have low strength . the reader is referred to @xcite for details about the ridge extraction method . the height ridge definition has been used to extract lcs from ftle fields in several works ( see , among others , @xcite ) . since the @xmath16 value of a point on the ridge and the ridge strength @xmath33 are only related through the expressions ( [ ridg1 ] ) and ( [ ridg2 ] ) , the relationship between the two quantities is not direct , which makes difficult to choose the appropriate strength threshold @xmath32 . a too small value of @xmath32 will result in the extraction of very small lcss that appear to have little influence on the dynamics , while a large value will result in only a partial rendering of the larger and more significant lcs , limiting the possibility of observing their real impact on the flow . the ridges extracted from the backward fsle map approximate the attracting lcss , and the ridges extracted from the forward fsle map approximate the repelling lcss . the attracting ones are the more interesting from a physical point of view @xcite , since particles ( or any passive scalar driven by the flow ) typically approach them and spread along them , so that they are good candidates to be identified with the typical filamentary structures observed in tracer advection . turbulent channel flow is a turbulent flow between two stationary , parallel walls separated by a distance @xmath34 . it has been studied extensively due to its geometrical simplicity and its wall - bounded nature , which makes it a suitable platform to study phenomena appearing in more complex turbulent wall - bounded flows of great technological interest . the coordinates of the flow are : @xmath35 for the streamwise direction , @xmath36 for the cross - stream coordinate that separates the two plates , and @xmath9 for the spanwise direction . the flow is maintained by a downstream pressure gradient @xmath37 acting against the wall shear stress . the laminar flow solution @xmath38 is a cross - stream parabolic profile given by @xmath39 where @xmath40 is the dynamic viscosity . following the reynolds averaging method @xcite , the turbulent flow velocity @xmath41 is decomposed in a mean @xmath42 and a fluctuating component @xmath43 . the mean turbulent velocity profile @xmath44 differs from the laminar one , @xmath45 , by a lower centerline velocity @xmath46 and increased near - wall velocity giving it a flatter shape . due to the increase in mean velocity near the wall , the shear stress near the wall is higher for the turbulent case . the total shear stress @xmath2 appearing in the averaged reynolds equations gets contributions from both the viscous stress and the reynolds stress @xmath47 associated to the velocity fluctuations : @xmath48 @xmath49 is the kinematic viscosity . the symmetries of the domain and the reynolds equations imply that @xmath2 depends only on the cross - stream coordinate @xmath36 , and the dependence is linear , so that it can be written as @xmath50 the shear velocity @xmath51 gives the velocity scale of the turbulent velocity fluctuations . the formula @xcite : @xmath52 allows to compute @xmath53 from measurements of the mean velocity profile from the simulations . a length scale can be formed by combining @xmath53 with @xmath54 : the wall scale @xmath55 . the wall distance can now be expressed as @xmath56 , and the same normalization could be done for the rest of coordinates . the viscous reynolds number @xmath57 is simply the ratio between the two relevant length scales . the existence of coherent structures in turbulent wall - bounded flows has been known for several decades from investigations on intermittency in the interface between turbulent and potential flow regions , on the large eddy motions in the outer regions of the boundary layer , and on coherent features in the near - wall region ( @xcite and references therein ) . since then , through experimental and numerical investigations , a picture of the organization of these coherent structures in the turbulent boundary layer has emerged , which has become rather complete from the eulerian point of view @xcite . our approach is a contribution to the lagrangian exploration of these coherent structures , as in @xcite and @xcite . the longitudinal velocity field in the inner region of the channel ( the viscous sublayer adjacent to the wall and the intermediate buffer region ) is organized into alternating streamwise streaks of high and low speed fluid . turbulence production occurs mainly in the buffer region in association with intermittent and violent outward ejections of low - speed fluid and inrushes of high - speed fluid towards the wall . the outer region is characterized by the existence of three - dimensional @xmath58-scale bulges that form on the turbulent / potential flows interface . irrotational valleys appear at the edges of the bulges , entraining high - speed fluid into the turbulent inner region . a central element in the structure of the turbulent boundary layer is the hairpin vortex , mainly because it is a structure with the capability of transporting mass and momentum across the mean velocity gradient and because it provides a paradigm with which to explain several observations of wall turbulence @xcite . the data used to extract the lcs come from a direct numerical simulation ( dns ) of turbulent channel flow at a viscous reynolds number @xmath0 . the setup of the simulation follows that of @xcite and is summarized in table [ tab1 ] . the simulations were conducted using the cfd solver ` channelflow.org ` @xcite . the ` channelflow.org ` code solves the incompressible navier - stokes equations in a rectangular box with dimensions @xmath59 , with periodic boundary conditions in the @xmath35 ( so that fluid leaving the computational domain in the direction of the mean flow at @xmath60 reenters it at @xmath61 ) and in the spanwise @xmath9 direction . no - slip conditions are imposed on @xmath62 . the unsteady velocity field @xmath63 is represented as a combination of fourier modes in the @xmath35 and @xmath9 directions and of chebyshev polynomials in the wall - normal direction . the pressure gradient necessary to balance the friction at the walls was chosen as to maintain a constant bulk velocity of @xmath64 . time stepping is a 3rd - order semi - implicit backward differentiation . note that in our computations @xmath65 so that in wall units @xmath66 . the flow was integrated from an initial base - flow with parabolic profile and a small disturbance that evolved into a fully developed turbulent flow . the total integration time was @xmath67 time units that in dimensionless form @xmath68 gives @xmath69 . after an initial transient of about @xmath70 time units the simulations reached a statistically stationary state from which statistics was accumulated . the mean quantities and first order statistics of our simulations where compared to those of @xcite and the agreement is quite good . the profile of the mean velocity in wall units is shown in figure [ fig2 ] . the profile for the reynolds stress @xmath47 shows that the maximum ( in absolute value ) is located at approximately @xmath71 , in the outer limit of the buffer layer ( see figure [ fig3 ] ) . . solid line : our simulations ; squares : @xcite ; dashed line : logarithmic profile @xmath72 . , scaledwidth=70.0% ] profile at @xmath0 . solid line : our simulations ; squares : @xcite ( given up to the channel centerline ) . , scaledwidth=70.0% ] @ll|ll|ll @xmath73 channel center & @xmath74 & @xmath75 nominal & @xmath76 & @xmath75 actual & @xmath77 + @xmath78 & @xmath79 & @xmath58 & @xmath80 & @xmath81 & @xmath82 + @xmath83 & @xmath84 & @xmath85 & @xmath86 & @xmath87 & @xmath88 + @xmath89 & 128 & @xmath90 & 129 & @xmath91 & 128 + @xmath92 & @xmath93 & @xmath94 & @xmath95 & @xmath96 & @xmath97 + the lcs were extracted from the turbulent velocity field data described in the previous section . a calculation of fsle field in the entire turbulent channel was conducted in order to understand the statistical properties of the fsle field in this class of turbulent flows . a subsequent calculation in a subdomain of the channel was used to extract the lcs in that subdomain for a sequence of time instants . the setup of both calculations is shown in table [ tab2 ] . @lllll calculation & @xmath5 & @xmath98 & @xmath13 & @xmath99 + complete channel & @xmath100 & @xmath101 & @xmath101 & @xmath102 + lcs subdomain & @xmath103 & @xmath104 & @xmath105 & @xmath102 + the 3d backward fsle field for the entire channel was calculated at a single time instant in the statistically steady state . the initial and final distances @xmath5 and @xmath6 were chosen as a balance between encompassing the widest possible range of scales of motion ( measured by the ratio @xmath106 ) , and adequate resolution and computational cost . the initial distance is of the order of @xmath107 and the final distance of the order of @xmath108 a typical scale of coherent structures found in the turbulent channel flow so that the ratio of scales , @xmath98 , is approximately @xmath109 . shown on a streamwise / wall - normal plane in the turbulent channel . walls are at the top and bottom of the figure . mean velocity is in the streamwise direction from left to right . , height=264 ] , as a function of the cross - stream normalized coordinate @xmath110 . only half of the channel is shown since the profile is quasi - symmetric about the channel centerline . , scaledwidth=70.0% ] figure [ fig4 ] shows an instantaneous configuration of the fsle values in a streamwise / wall - normal plane . the maxima of the fsle appear to be located close to the walls with ocasional sloping structures extending to the midchannel region . the channel center is devoid of high fsle values but coherent patches of low fsle values can still be observed . these structures are not distributed uniformly along the length of the channel but appear to be organized in packets . this organization bears resemblance to the widely accepted picture of organized structures in wall turbulence where the outer region is dominated by packets of sloping hairpin vortices and the inner region by near wall vortices ( the hairpin vortices legs ) and shear layers @xcite . a cross - stream fsle profile is obtained by averaging the 3d field over the periodic directions @xmath35 and @xmath9 . it is shown in figure [ fig5 ] . the profile is symmetric about the channel centerline and shows a maximum at approximately @xmath111 , inside the viscous sublayer ( this location corresponds to the first grid point off the wall ) . because of the periodic boundary conditions in the @xmath35 and @xmath9 directions the average profiles along these directions are rather unstructured , and we resort to two - point correlation functions to quantify the statistical structure properties . for each plane parallel to the walls , i.e. for each value of @xmath110 , we compute the fluctuations of the fsle values around the average in that plane : @xmath112 . from this quantity we define the streamwise @xmath113 correlation function as : @xmath114 and the spanwise @xmath115 correlation function @xmath116 in the above equations the averages are over the periodic directions @xmath117 and @xmath118 . the correlations are shown in figs . [ fig6 ] and [ fig7 ] at different distances from the walls : one smaller , one larger , and one approximately coincident with the location of the maximum reynolds stress . these functions reveal sizes and organization of the different structures in the lagrangian fsle field , to be contrasted with eulerian correlation functions in the same system @xcite . as a function of the streamwise separation @xmath119 , at four distances from the lower wall : continuous line : @xmath120 ; dashed line @xmath121 ; dash - dot line @xmath122 ; dotted line : @xmath123 . , scaledwidth=70.0% ] as a function of the spanwise separation @xmath124 , at four distances from the lower wall : continuous line : @xmath120 ; dashed line @xmath121 ; dash - dot line @xmath122 ; dotted line : @xmath123.,scaledwidth=70.0% ] close to the wall ( @xmath120 and @xmath121 ) , viscous effects dominate . the correlations show that the fsle field is organized in streamwise structures of length scale approximately @xmath125 wall units . in the transverse direction @xmath118 the oscillations seen in @xmath126 for @xmath120 indicate an approximately periodic arrangement of the streaks @xcite , with a spacing @xmath127 wall units . this pattern of organization is similar to what is seen in eulerian descriptions @xcite . at planes further away from the wall ( @xmath122 and @xmath123 in figs . [ fig6 ] and [ fig7 ] ) , correlation functions in both directions become shorter ranged , and periodic features are progressively lost . this corresponds to a rather disordered distribution of structures , each with a typical size related to the width of the correlation functions , i.e. of the order of 50 wall units , as also seen in figure [ fig4 ] . an instantaneous near - wall fsle field is shown in figure [ fig8 ] , where the high fsle values appear in slender and elongated structures with length and width corresponding to the streamwise and spanwise correlation lengths discussed above . it is unclear whether the correlation lengths result from a single streamwise structure or from the overlaping of shorter structures ( a feature of the near wall coherent structure arrangement @xcite ) . these are the highest fsle values that are to be found in the channel as the plot in figure [ fig5 ] shows . the mechanism for the formation of these structures could be the lifting of low speed fluid close to the wall by the action of counter rotating vortex pairs located above the viscous sublayer ( see figure [ fig9 ] ) . this mechanism is widely known in the eulerian view of coherent structures of turbulent wall bounded flows ( _ ejections _ or _ bursting _ , @xcite ) . . the time is the same as in figure [ fig4 ] , scaledwidth=90.0%,height=264 ] the near wall fluid is advected away from the wall by the action of these vortices . this mechanism could be responsible for very fast particle separation in particle pairs where one particle is lifted away and the other remains in the low speed zone close to the wall . we note that the particle separation would increase not only by the wall normal distance between particles but also because the ejected particle would move to a region with higher streamwise velocity . shear layers near the wall is another possible way to produce large particle dispersion . these mechanisms would explain the fact that the maximum average fsle is located so close to the wall and not on the buffer region where turbulence production is larger . to conclude , we note that these high fsle regions near the wall seem to extend to the midchannel region in an inclined fashion . it is not clear whether this pattern signals the existence of a hairpin vortex with streamwise legs and inclined head or if there are two separate structures : the streamwise vortices _ and _ the hairpin arch or head @xcite . also , we note that the interpretation of the high fsle regions near the wall do not require the existence of a counter rotation pair of vortices , as only one vortex would suffice . to illustrate these mechanisms , a map of the fsle field in a spanwise / wall normal plane for the lcs domain calculation is shown in figure [ fig10 ] , together with a set of passive particles initially located in a rectangular region close to the wall and released some instants before the time of the fsle map . in order to focus just on the above mentioned ejection mechanism involving only the vertical motion of the particles , the trajectory integration was made in a 2d fashion by setting the longitudinal component of the particles velocity to zero . plane located at @xmath128 ( @xmath129 ) . the time of the map is @xmath130 . together with a set of particles initially located in rectangular region @xmath131 $ ] and @xmath132 $ ] . the particles were released at @xmath133 . particle trajectories were integrated using only the spanwise and wall normal velocity components . the mean flow is moving out of the page . , scaledwidth=90.0% ] the particles seem to have been lifted from wall by a streamwise vortex located to the left of the particle plume , with center at @xmath134 . we note that the structures are moving with the mean flow and that the continuous motion of the particles away from the wall is due to the passage of a streamwise structure that imparts this sustained motion to the particles for long enough time . to compare the eulerian and lagrangian coherent structures , figure [ fig11 ] shows the turbulent velocity components in the same plane at the nearest time available in the turbulent dataset . the signature of the streamwise vortex discussed above can be seen in the eulerian map at the same location . it is embedded in a patch of negative streamwise velocity fluctuation @xmath135 . to the right , close to @xmath136 , a vertical shear layer appears dividing the negative and positive patches of @xmath135 . the lagrangian signature of this vertical shear layer is not very strong and appears in figure [ fig10 ] as quasi - vertical line of moderate fsle extending from @xmath137 to @xmath138 . on the lower right of the map , there is a set of high fsle lines almost parallel to wall , signalling the existence of high particle dispersion . in the eulerian map ( figure [ fig11 ] ) , it can be seen that there is a shear layer parallel to the wall at the same location ( @xmath139 and @xmath140 ) . the fact that this shear layer has a much stronger lagrangian signature than the vertical shear layer could be because it has the same orientation and sign of the mean shear and therefore acts together with the latter to disperse neighboring particles across the wall normal direction . the high fsle line seen at the middle of the map in figure [ fig10 ] , separating the two convoluted features can be seen to be related to the existence of two counter - rotating vortices , one with center located at @xmath141 and the other at @xmath142 . the line of high fsle line is seen to be located at the boundary between both vortices . in section [ 3d_lcs_cf ] , we present a 3d view of these structures and their evolution in time . ( @xmath129 ) and @xmath143 . velocity vectors correspond to the inplane velocity components @xmath144 , together with contours of streamwise turbulent velocity @xmath135 . dashed contours are negative @xmath135 ( into the paper ) and continuous countours are positive @xmath135 ( from the paper ) . , scaledwidth=90.0% ] in turbulent channel flow the velocity perturbations propagate in the streamwise direction aproximately with the velocity of the mean flow@xcite . in the case of lyapunov exponents , @xcite measured the ftle field in an 2d turbulent boundary layer velocity field obtained by time - resolved piv measurements . the ftle maxima were found to move with the mean flow velocity . we measured the propagation velocity of the fsle field perturbation using a space - time correlation of the form : @xmath145 where @xmath119 and @xmath146 are the delays in the streamwise direction and time . the time delay is fixed and the propagation velocity is defined as @xmath147 where @xmath148 is the streamwise lag for which @xmath149 is maximum . the choice of the time delay is related to the time scale of the fsle field . a first rule is to choose a time delay that gives reasonable peaks in the correlation . if there are several time scales present , several @xmath146 will result in correlations exhibiting peaks . the calculation of ( [ vprop ] ) was made for a full length and height spanwise section of the channel . a time series of fsle fields with time step of @xmath150 and time length @xmath151 was calculated for this section to offset the effects of the limited spanwise extent of the section . the final time lag used in ( [ vprop ] ) was equal to @xmath152 . all larger delays produced correlations with no significant peak . a reason for this could be the fact that by setting the fsle final distance the length scales of turbulence retained in the fsle field is fixed , and then there will be only one time delay producing a peak in the correlation ( [ ruu ] ) , specifically that corresponding to @xmath153 . ) and mean flow ( @xmath154 ) . , scaledwidth=70.0% ] the profile of the propagation velocity is shown in figure [ fig12 ] . the propagation velocity is very close to the mean flow velocity . the result shows that the maxima of the fsle field , that produce high values of @xmath149 and where we expect to find the ridges of the fsle field , move with the flow . hence , one may conclude , as expected , that the fsle ridges also move with the flow approximately as material surfaces . the previous description summarized the statistical properties of the different structures appearing in an instantaneous fsle field . to make further progress we now extract three - dimensional attracting lcss in a region of the channel at a series of time instants . the extraction domain had dimensions @xmath155 . the initial separation @xmath5 and distance ratio @xmath156 were increased from the previous calculation to improve the resolution and extract smoother structures , but sacrificing a complete view of 3d lcs in the turbulent channel . the extraction threshold was set to @xmath157 , a compromise value between speed and cost of extraction and continuity of the extracted surfaces . the fsle fields were calculated for an interval of @xmath158 time units with a time step of @xmath159 units . ( @xmath129 ) . time goes from top to bottom , at intervals of 0.1 time units . the flow direction is in the positive @xmath35 direction in each panel , and a wall is at the bottom . the sequence shows how one of the flow structures is advected and passes through the @xmath128 plane . , scaledwidth=90.0%,height=755 ] the 3d lcss are rendered in figure [ fig13 ] , in a sequence of time instants , as they pass through the calculation domain . they have a clearly 3d shape and move with the flow . the lcs seem to create a boundary between the inner turbulent region and the outer region that is practically devoid of fsle . the highest lcs have @xmath58-scale heights above the wall , and have a distinct mushroom shape enclosing the regions of the channel closer to the wall , where high fsle values can be found . near the wall , the lcs adopt the shape of sheets parallel to it , which reflects the high rates of shear that occur in that region . these sheets form the base of the mushroom - shaped excursions up to the channel center . contrarily to the turbulent flow of the previous section , large scale oceanic flows , naturally turbulent , can be considered as almost 2d due to rotation and stratification effects . this fact makes the theory of 2d turbulence a very important tool to understand the ocean processes that occur at large scales . the main characteristic of 2d turbulence is the existence of an inverse energy cascade , from the small to the large scales and a direct enstrophy cascade . these cascades manifests themselves by the creation of large coherent vortices , and by the process of filamentation by which strain regions in the boundaries of the vortices produce lines of vorticity that are continuously stretched and deformed by the flow , concentrating the vorticity gradient in the small scales . this behavior is often observed in oceanic flows thereby confirming the importance of the 2d turbulent processes . the results presented in this section were obtained in the benguela ocean region , situated off the west coast of southern africa . it is characterized by a substantial mesoscale activity in the form of eddies and filaments , and also by the northward drift of agulhas eddies . the velocity data set comes from a regional ocean model ( roms ) simulation of the benguela region @xcite . additional details on this work can be found in @xcite . the three - dimensional fsle fields were calculated for a @xmath160 day period starting september 17 of year 9 , with snapshots taken every @xmath161 days . the fields were calculated for an area of the benguela ocean region between latitudes 20s and 30s and longitudes 8e to 16e . the calculation domain extended vertically from @xmath101 up to @xmath162 m of depth . both backward and forward calculations were made in order to extract the attracting and repelling lcs . in the left panel of figure [ fig14 ] a snapshot of the attracting lcss for day 1 of the calculation period the structures appear as thin vertical curtains , most of them extending throughout the whole depth of the calculation domain . the horizontal slices of the fsle field in figure [ fig14 ] ( left panel ) show that the attracting lcs fall on the maximum fsle field lines , as in the case of the turbulent channel flow ( figure [ fig13 ] ) . the fsle fields themselves exhibit a variation in intensity that decreases with depth , altough a local maximum is found at @xmath163 m ( not shown ) . the ridges also seem to be weaker as the depth increases since for the same strength threshold , the extracted portions of the ridges become less extent and eventually vanish . the atracting and repelling lcs ( figure [ fig14 ] , right panel ) populate the calculation region , testifying the enhanced mixing activity that is known to occur in that particular ocean region . the quite entangled `` web '' in which attracting and repelling lcss intersect mutually provides the skeleton for the barriers and pathways controlling transport @xcite . , scaledwidth=90.0% ] at this point , it may help to stress the differences between the eulerian and lagrangian detection of coherent structures . this can be seen in figure [ fig15 ] where the boundaries of a mesoscale eddy are shown using the q - criterion and the attracting and repelling lcs . the q - criterion @xcite uses the second invariant of @xmath164 : @xmath165 where @xmath166 , @xmath167 , and @xmath168 , @xmath169 are the antisymmetric and symmetric components of @xmath164 , to identify regions where rotation dominates strain ( @xmath170 ) , commonly identified with coherent vortices , and strain dominated regions ( @xmath171 ) . we refer the reader to @xcite and @xcite for reviews and criticism of several eulerian criteria . eulerian and lagrangian measures limit approximately the same region , but are substantially different . the q - criterion is related to the instantaneous configuration of the second invariant of @xmath164 and therefore conveys only local information about fluid flow processes . the lagrangian perspective , on the other hand , provides an integration of the temporal evolution of material properties of the flow , e.g. material transport , and thus should give more meaningful information about the processes that rely on ocean material transport . this issue can be further explored by looking at a filamentation event ( described more extensively in @xcite ) . a set of particles were released inside the eddy at day 1 at a depth of 50 m. at day 11 of the calculation period ( see figure [ fig15 ] ) , they have formed a filament that is expelled from the eddy , so that particles clearly cross the q - criterion isosurface . this shows that the eulerian criteria is inadequate as an indicator of regions of material transport in the flow . on the contrary , it can be observed that the lagrangian description of the eddy boundaries does bear relation with material transport into and out of the eddy , since the particle filament leaves the enclosed region that we associate with the mesoscale eddy by following one of the identified lagrangian boundaries . ( red ) . the particles ( black dots ) were released inside the eddy at day 1 at a depth of 50 m and are leaving now the eddy as a filament along the upper part of the attracting lcs . ] lyapunov exponents are useful to identify lagrangian coherent structures in turbulent flows . these constitute the pattern determining the pathways of particle transport in the flow and thus strongly influence the transport and mixing properties in the fluid . in this paper we have used a particular type of lyapunov exponents , the so - called finite - size lyapunov exponents , to identify lcs in 3d flows . the finite size lyapunov exponent was used to measure the rate of streching of initially nearby fluid particles in the flow domain and the lagrangian coherent structures where identified as the the ridges of the fsle field . these ridges were filtered in order to retain only the strongest attracting or repelling structures . in a turbulent channel flow , the fsle field is organized into longitudinal structures close to the wall that develop into sloping ones away from the wall . correlations in the streamwise and spanwise direction show the typical dimensions of these structures . they were found to be similar to the eulerian coherent structures that are known to exist in this same regions of the turbulent channel . specially , elongated streamwise vortices that move low speed fluid away from the wall into the channel core . in 3d , the lcss appear as mushroom - shaped excursions of near - wall sheet - like structures of a scale comparable to the channel width . they separate the channel into an interior region , where the fsle attains high values , and an exterior region , showing low fsle values . the distribution of lcs in the turbulent channel resembles the commonly accepted picture where upward excursions of near wall fluid coexist with inward rushes of mid - channel irrotational flow . further work is necessary to elucidate the relations between lcs and fluid transport in these type of flows , not least because the visualization of 3d structures and transport in turbulence is a complex and time - consuming subject . in a quasi-2d mesoscale oceanic flow , the lcss appear as quasi - vertical surfaces highlighting the fact that dispersion in this case is mainly horizontal . the high mixing activity can be deduced from the proliferation of lcs in the flow domain and their mutual intersection . these lcs were seen to provide barriers and pathways to transport in the case of a mesoscale eddy , contrary to eulerian measures that failed to provide indicative locations or directions of major transport events . the main difference between these two 3d turbulent flows with respect to the lcss seems to be the fact that in the case of oceanic flow , turbulence was limited to the horizontal plane wheras in the channel flow case , turbulent fluctuations in all three space directions had similar magnitude , thereby producing much more complex 3d shapes in this latter case . in the oceanic flow , vertical motions have a tendency to be supressed by the combined effects of the earth s rotation and the stratification of the ocean . this results in the aforementioned dominance of horizontal dispersion . the quasi - horizontal character of oceanic flows results in a phenomenology of turbulence similar to that of 2d turbulence rather than to 3d turbulence . we note that there are fundamental differences between the lagrangian and eulerian coherent structures , although they can actually have a common interpretation as vortices or shear layers . lagrangian coherent structures have a clear impact in particle trajectories whereas eulerian coherent structures are related to space / time coherency in , e.g. , velocity signals and do not necessarily affect particles . in the above comparison , only the strongest fsle features had a clear connection to the features in the eulerian distribution , which indicates that , inversely , only the eulerian features that live long enough or are strong enough to affect particles in a discernible fashion will appear in the lagrangian point of view of coherent structures . the results shown in this paper highlight the usefulness of lyapunov analysis and dynamical systems theory as a tool to study transport and mixing in fluid flows , through the concept of lagrangian coherent structures . this work was supported by ministerio de economa y competitividad ( spain ) and fondo europeo de desarrollo regional through project fisicos ( fis2007 - 60327 ) . jhb acknowledges financial support of the portuguese fct ( foundation for science and technology ) and fundo social europeu ( fse / qren / poph ) through the predoctoral grant sfrh / bd/63840/2009 . 10 url # 1#1urlprefix[2][]#2 artale v , boffetta g , celani a , cencini m and vulpiani a 1997 _ phys . fluids _ * 9 * 31623171 bakun a 1996 _ patterns in the ocean . ocean processes and marine population dynamics _ ( california sea grant college system , noaa and centro de investigaciones biolgicas del noroeste , la paz , bcs mxico )

in this paper we use the finite size lyapunov exponent ( fsle ) to characterize lagrangian coherent structures in three - dimensional ( 3d ) turbulent flows . lagrangian coherent structures act as the organizers of transport in fluid flows and are crucial to understand their stirring and mixing properties . generalized maxima ( ridges ) of the fsle fields are used to locate these coherent structures . three - dimensional fsle fields are calculated in two phenomenologically distinct turbulent flows : a wall - bounded flow ( channel flow ) and a regional oceanic flow obtained by numerical solution of the primitive equations where two - dimensional turbulence dominates . in the channel flow , autocorrelations of the fsle field show that the structure is substantially different from the near wall to the mid - channel region and relates well to the more widely studied eulerian coherent structure of the turbulent channel flow . the ridges of the fsle field have complex shapes due to the 3d character of the turbulent fluctuations . in the oceanic flow , strong horizontal stirring is present and the flow regime is similar to that of 2d turbulence where the domain is populated by coherent eddies that interact strongly . this in turn results in the presence of high fsle lines throughout the domain leading to strong non - local mixing . the ridges of the fsle field are quasi - vertical surfaces , indicating that the horizontal dynamics dominates the flow . indeed , due to rotation and stratification , vertical motions in the ocean are much less intense than horizontal ones . this suppression is absent in the channel flow , as the 3d character of the fsle ridges shows .

introduction methods turbulent channel flow oceanic flow conclusions acknowledgements references

arxiv

one of the hallmark distinctions between qso absorption systems containing strong c iv lines and those which do not has been , since early in the history of their study , the difference in clustering between the two populations as seen along single lines of sight . the ly @xmath4 forest has always been shown to be weakly clustered compared to c iv absorbers ( webb 1987 , hu et al . 1995 , chernomordik 1995 , cristiani et al . 1995 ) , and perhaps not detectably clustered at all ( sargent et al . 1980 , rauch et al . 1992 , lu et al . 1996 , kirkman & tytler 1997 ) . in contrast , even early studies showed that c iv absorbers cluster significant in velocity along single sightlines ( young et al . 1982 ) , with perhaps the best evidence coming from a large compilation of qso spectra at approximately 100 km s@xmath2 resolution ( sargent et al . 1988 , hereafter ssb ) . this is usually described by the two - point correlation function @xmath5 which equals the excess number of absorbers over random expectation found at a certain location with respect to a given absorber , usually quantified as a spatial or velocity separation between the two locations ( and with @xmath5 normalized by dividing by the random expectation ) . depending on the sample selected , for line - of - sight velocity differences @xmath6 of 200 - 600 km s@xmath2 , values of @xmath5 for c iv absorbers were found with @xmath7 or larger . similar behavior is found in large c iv samples at spectral resolution higher than that of ssb e.g. within the largest clustering sample , 10 sightlines at 18 to 40 km s@xmath2 fwhm resolution , analyzed by petitjean and bergeron ( 1994 - pb ) . ( other recent works at even higher resolution are based on even fewer sightlines : songaila & cowie 1996 , fernndez - soto et al . 1996 , rauch et al . 1996 . ) pb , like ssb , find an average @xmath7 for @xmath8 - 600 km s@xmath2 , dominated by a broad , slowly declining component . ( specifically , they fit @xmath5 with two components , of widths @xmath9 and 525 km s@xmath2 , with the broader component containing 71% of the pair count over 30 - 1000 km s@xmath2 and 93% over 200 - 600 km s@xmath2 . ) in contrast , even in those papers which found some clustering in the ly @xmath4 forest , also on approximately these @xmath6 scales or slightly smaller , the signal rarely exceeded @xmath10 . this is seen as clear evidence for a difference between these population , possibly implying distinct origins for the two . there are caveats to this interpretation which suggest caution in comparing the single - sightline @xmath5 values of the ly @xmath4 forest and c iv absorber populations . first , it is possible that single - sightline velocity splittings might arise from internal motions within absorbers , in which case the differences between the two populations single - sightline @xmath5 functions are not clearly tied to their spatial clustering behavior . ssb argue that , for c iv absorption arising in galaxy haloes , these velocity splittings can not be due to gravitational orbits within these haloes , and clustering still contributes the dominant portion of the observed @xmath5 on scales larger than @xmath11 km s@xmath2 . indeed , many papers have modeled qso absorption - line clustering in terms of spatial separations indicated by their relative velocities , whereas for highly over - dense structures , large differences between velocity clustering and spatial clustering become apparent e.g. kaiser ( 1997 ) . models have been proposed , however , where non - gravitational acceleration might lead to splittings with large @xmath6 ( york et al . 1986 ) . second , the intrinsic width of ly @xmath4 lines , up to @xmath12 km s@xmath2 , is much higher than for c iv because of thermal broadening , and significant on the scale of the clustering in @xmath6 being discussed . line - of - sight blending of ly @xmath4 lines does seem to obscure some of the clustering power seen in their corresponding c iv lines fernndez - soto et al . ( 1996 ) . one way to assess the importance of such effects is to study the clustering of absorbers in adjacent sightlines close enough together so that the angular separation between them is less than or comparable to the velocity scales where clustering is seen or sought in single sightlines , here assuming that a hubble expansion law at high redshift can be used to relate @xmath6 and transverse separation . this addresses all of the above concerns . first , purely internal velocity splittings can not shift absorbers to a different sightline , and , second , blending can not eliminate all of the small @xmath6 absorber pairs that would otherwise exist between sightlines . even if blending decreases or splittings increase the number of close absorbers pairs between sightlines , there is much less effect on @xmath5 because these effects also change the total numbers of pairs used to normalize @xmath5 . if all absorbers in a population are equally likely to cluster i.e. if all within a population cluster in a way described purely by the same @xmath5 , the effects of blending or line splitting on close pairs and distant pairs cancel . the 1623 + 27 qso triplet discovered by sramek and weedman ( 1978 ) and has been used to measure the spatial two - point correlation function ( here also denoted by @xmath5 ) of ly @xmath4 absorbers ( crotts 1989 , crotts & fang 1996 , with some members of the triplet also observed by sargent et al . 1982 , and ssb ) . we have obtained keck hires spectra of these three qsos , yielding a sample of c iv absorbers large enough and unambiguous enough that a useful comparison of spatial c iv clustering can be made to ly @xmath4 clustering and single - sightline c iv clustering . these c iv clustering results are rather different from prior results from single sightlines alone , which changes our understanding of clustering at high redshift . on the night of 20 may 1996 , we used the hires spectrograph ( vogt 1994 ) on the keck-1 10 m telescope to obtain spectra of the qso triplet q1623 + 27 . each of the qsos was observed with the same setup , providing wavelength coverage from 3872 to 6299 . the observations were performed sequentially over a four hour period , and the spectrograph was not moved between observations . we exposed for 5400s on both q1623.7 + 268a ( kp 76 , @xmath13 , @xmath14 ) and q1623.9 + 268 ( kp 78 , @xmath15 , @xmath16 ) , and 3000s on q1623.7 + 268b ( kp 77 , @xmath17 , @xmath18 ) . the exposures were taken with a 1.14@xmath19 slit , which gave a resolution of 8 km s@xmath2 and adequate sky coverage . the images were processed and the spectra were optimally extracted using an automated routine specifically designed by t. a. barlow for hires spectra . the routine performs baseline subtraction , bias and flat - field corrections , and utilizes a bright standard star to trace the echelle orders and define the apertures for extraction . thorium - argon lamp images were obtained immediately after the observations to provide wavelength calibrations in each echelle order . the root - mean - square residuals in the wavelength calibration for each echelle order was less than 0.3 km s@xmath2 . all wavelengths are vacuum values in the heliocentric frame . each echelle order was continuum fit with a legendre polynomial to normalize the unabsorbed qso flux level to unity . as an example of these data , we present figure 1 , which shows a particularly complex c iv doublet from the faintest qso , kp 78 , fit by 10 components . the positions of components and best fit flux from vpfit ( carswell et al . 1992 ) were determined ; seen in figure 1 for the @xmath20 system , along with the continuum fit . the spectrum does not have useful snr in the blue , where the correcponding lya lines lie . the redshifts of c iv doublets found in these data are listed in table 1 . we include only those redward of the ly @xmath4 forest , and list them according to the 200 km s@xmath2 `` blended '' sample treated below . in comparison , the crotts and fang ( 1997 ) @xmath21 kpno 4 m sample contains within this redshift interval the two stronger c iv doublets in kp 76 , all of the kp 77 sample ( with 1.878027 and 1.880660 blended together ) , and all of the kp 78 sample except 2.115063 , with 2.061465 listed as uncertain . their uncertain system at @xmath22 kp 77 appears to be a misinterpretation of the confused region of c iv and mg ii doublets near 5280 . sightline cross - correlating pairs for all c iv systems results in 36 in the first bin ( @xmath23 km s@xmath2 ) , compared to the random expectation from a linear fit over the first 20000 km s@xmath2 of @xmath24 for the first 500 km s@xmath2 bin . these counts are obviously highly non - poisson , so we do not assign an error estimate to the resulting two point function of @xmath25 km s@xmath26 . it is more reasonable to consider merging c iv redshifts close to each other in the same sightline , since it seems likely that these are multiple representatives of the same absorber . if we over - correct for this effect , we do not damage the cross - sightline @xmath5 , since blending together systems does not reduce the fraction of pairs between sightlines due to close cross - sightline @xmath6 values , compared to the total number of cross pairs . we choose to blend together all systems in the same sightline within 200 km s@xmath2 of each other , starting with the smallest splitting first this is the same criterion adopted by ssb , so it leads to a direct comparison . the cross correlation of this sample ( and samples defined by further criteria ) are shown in figure 2 . in correspondence with ssb , we reduce the sample to only those systems which would likely have been detected by their survey . this is a heterogeneous selection in terms of rest equivalent width @xmath27 , and corresponds roughly to @xmath28 for their `` sample a2 '' ( which also excludes all absorbers within 5000 km s@xmath2 of the emission - line redshift . a homogeneous sample in @xmath27 requires a cut at 0.15 ( their sample `` a4 , '' also with @xmath295000 km s@xmath2 ) . the randomly expected number of pairs in the first 500 km s@xmath2 bin for each of these subsamples ( `` blended , '' `` a2 , '' and `` a4 '' ) are @xmath30 , @xmath31 , and @xmath32 , respectively , whereas the actually observed number of pairs in the first bin for each sample is 4 , 1 and 0 , respectively , leading to @xmath5 values of @xmath33 , @xmath34 , and @xmath35 , respectively . ( these are 68% confidence intervals , corresponding to @xmath36 , assuming poisson errors in pair counts , which is close to correct . ) at @xmath37 ( and for @xmath38 ) the transverse separations between the three qsos correspond to velocities in the hubble flow of 286 to 399 km s@xmath2 . ( for @xmath39 they are 0.95 times smaller . ) therefore , structure dominated by the hubble flow over 200 to 600 km s@xmath2 would contribute to clustering on these scales , and correspond to proper separations of 0.36 to 1.07 @xmath40 mpc . since separations between the sightlines are smaller than this ( 0.51 to 0.71 @xmath40 mpc ) , one must add a perpendicular ( line of sight ) velocity component up to about 500 km s@xmath2 ( although more typically about 200 km s@xmath2 ) . correlational activity from such a signal should be restricted to the first 500 km s@xmath2 bin in figure 2 . nevertheless , the single sightlines over 200 - 600 km s@xmath2 and the multiple sightlines for @xmath41 km s@xmath2 probe slightly different volumes around each absorber . we can evaluate the importance of the different sampling regions by considering the analytic fit by pb to the number of pairs in these velocity intervals . they find that the number of pairs is well - approximated by the sum of two gaussians , with standard - deviation velocity widths of 109 and 525 km s@xmath2 , and with the wider gaussian contributing 30% of the number of pairs to the sum of the gaussians at their peak at zero velocity . when we integrate this distribution over the ssb sampling volume covering 200 - 600 km s@xmath2 , we find 1.15 times as many pairs as when we integrate over the triple - sightline sampling volume . this is a relatively minor effect which we neglect hereafter , but one which tends to lower slightly the discrepancy we discuss below . the actual value seen by ssb for the a2 sample is @xmath42 km s@xmath43 , which should be compared to our @xmath44 , and for a4 @xmath42 km s@xmath45 , which is even more directly comparable to our @xmath46 . the a2 result is inconsistent at about the @xmath47 level , while the a4 result is discrepant by about @xmath48 , both in the sense that the absorbers are less clustered in adjacent sightlines than is predicted by the single - sightlines result assuming pure hubble flow . we confirmed that the line - of - sight clustering in our three spectra are consistent with those in the ssb sample of 55 qsos . we constructed the c iv redshift auto - correlation function for @xmath49 along each of the three sightlines taken individually , then summed together . for all systems , one finds @xmath50 , and @xmath51 , for all c iv systems , blended systems , `` a2 '' and a4 samples , respectively , in the first 600 km s@xmath2 bin . these are constructed using the average number of pairs in 600 km s@xmath2 bins with 600 km s@xmath52 km s@xmath2 . these measurements are consistent with their corresponding ssb values , albeit at much poorer @xmath53 due to the smaller number of qso sightlines . the inconsistency of the two - point correlation function derived from figure 2 with that of ssb implies that single sightline correlation functions can not be used to study the _ spatial _ clustering of absorbers on velocity scales of several hundred km s@xmath2 . this may be due either to internal velocities within absorbers that are caused by non - gravitational processes , or by motion within gravitational potentials that have separated from the hubble flow . these structures , either individual absorbers or clusters of absorbers , must be small enough to add little clustering power on scales of 0.5 to 1.1 @xmath40 mpc . in either case , most of the line - of - sight correlational power is due to behavior not described by the absorber positions alone , but some peculiar motion . line - of - sight absorber correlation functions should not be compared directly to galaxy correlation functions usually expressed as @xmath54 in terms of a radial separation vector @xmath55 in space . this spatial clustering of c iv absorbers is much weaker than would be expected for galaxies at @xmath56 . a direct comparison involves averaging the galaxy correlation function @xmath57 , where we assume @xmath58 and @xmath59 mpc ( derived from park et al . 1994 , although there are lower @xmath60 values for different samples e.g. fisher et al . this is averaged over a line segment extending from the tangent point at closest approach ( 0.512 , 0.593 , and 0.714 @xmath40 mpc for each of the sightline pairs ) and extending to the point at @xmath61 km s@xmath2 ( 1.031 , 1.073 , and 1.144 @xmath40 mpc , respectively ) . averaged over all three sightlines , @xmath62 . assumably , this can be back - evolved to @xmath37 with stable hierarchical clustering formalism ( davis & peebles 1977 ) if @xmath63 , according to @xmath64 . ( formally , this assumes @xmath38 , but remaining non - linear over all relevant @xmath65 , is a close approximation for other cosmilogical densities . ) note that high-@xmath65 clustering ( hudon & lilly 1996 ) measured at @xmath66 corresponds to @xmath67 , and extrapolates to @xmath68 at @xmath37 assuming stable clustering . these result is consistent with any of the comparable values obtained above for c iv absorbers . even though there are large differences between the line - of - sight clustering of the ly @xmath4 forest and c iv systems , their clustering power between different lines of sight is more similar . the strength of c iv clustering is consistent with that of the ly @xmath4 forest at similar redshifts . crotts & fang ( 1996 ) show , for these same sightlines at nearly the same redshift @xmath69 , that @xmath70 for ly @xmath71 lines with @xmath72 and @xmath73 km s@xmath2 . for @xmath23 km s@xmath2 , there are 51 pairs observed versus 19 expected , implying @xmath74 ( @xmath75 ) . measured in this way , it is unclear that c iv clustering is stronger than ly @xmath4 clustering . however , ly @xmath4 spatial clustering is less than the expectation for galaxies assuming stable hierarchical clustering ( @xmath76 ) , but not necessarily weaker than when we start from the hudon & lilly result . one remaining question is whether the structure we probe on 0.5 mpc scales might actually probe the scale of individual absorbers . we are fairly confident that this analysis of @xmath5 addresses more the clustering of absorbers than some measure of their characteristic size . the size of c iv absorbers is indicated by the transverse separation at which absorbers in one sightline have high probability of appearing in the adjacent sightline . for gravitationally - lensed qsos ( steidel & sargent 1991 ) and for distinct qso pairs ( crotts et al . 1994 ) , strong correspondence between adjacent absorption lines indicates c iv absorber sizes of only a few tens of kiloparsecs . on scales smaller than this , absorbers are presumed to merge . on scales larger than this , up to the 0.5 @xmath40 mpc sampled by the qso triplet , it is still possible that motion within objects that have collapsed out of the hubble flow might still be responsible for much of the clustering signal for 200 km s@xmath77 km s@xmath2 reported by ssb . calculating @xmath78 for the galaxy two - point correlation function at @xmath37 , assuming stable hierarchical clustering development , one finds @xmath79 for separations ( along a sightline ) of 40 @xmath40 kpc to 0.5 @xmath40 mpc , still larger than @xmath5 found for the a2 or a4 samples of ssb . the ssb 200 - 600 km s@xmath2 clustering signal might plausibly be explained as clusters of absorbers smaller than 0.5 @xmath40 mpc with internal velocities of a few hundred km s@xmath2 . indeed , high resolution simulations of fragments collapsing ultimately into galaxies at @xmath80 show that these fragments at @xmath81 subtend such spatial scales e.g. rauch , haehnelt & steinmetz ( 1997 ) . these conclusions are based on a single group of sightlines , and such close groupings of reasonably bright , sufficiently high @xmath65 qsos are extremely rare . nonetheless , a larger sample to check and refine these conclusions is desired . this research was supported in part by a david and lucile packard foundation fellowship to a.c . and nsf grant ast-9420443 and nasa grant nagw-4497 to d.t . we thank p.j.e . peebles and c. cress for helpful discussion . we thank tom barlow and bob carswell for software which made the data reduction and analysis possible . figure 2 shows the cross - correlation between pairs of the three sightlines , for three restricted samples : all systems after blending within 200 km s@xmath2 ( dashed curve ) , blended systems with rest equivalent width @xmath82 for c iv @xmath831548 , in close analogy to ssb sample a2 ( horizontally - striped bars ) , and blended systems with rest equivalent width @xmath84 , as in ssb sample a4 . cccl & 1.845177 & @xmath85 & 1.845177 & 2.112378 & @xmath86 & 2.111746 , 2.112022 , 2.112872 & 2.156867 & @xmath87 & 2.156484 , 2.157249 & 2.245817 & @xmath88 & 2.245372 , 2.246084 , 2.246438*kp 77 * & 1.878027 & @xmath89 & 1.878027 & 1.880660 & @xmath90 & 1.880084 , 1.881235 & 1.972602 & @xmath91 & 1.972276 , 1.972929 & 2.050746 & @xmath92 & 2.049659 , 2.049868 , 2.050201 , 2.051020 , 2.051807 , 2.052194 & 2.052938 & @xmath93 & 2.052644 , 2.052866 , 2.053120 & 2.161619 & @xmath94 & 2.161104 , 2.161317 , 2.161332@xmath95 , 2.162024 & 2.244602 & @xmath96 & 2.244602 & 2.400640 & @xmath94 & 2.399910 , 2.400782 , 2.401035 , 2.401195 , 2.401791 & 2.444659 & @xmath97 & 2.443576 , 2.444170 , 2.445444 & 2.528857 & @xmath98 & 2.528504 , 2.529211*kp 78 * & 1.985477 & @xmath99 & 1.985368 , 1.985587 & 2.042732 & @xmath100 & 2.042464@xmath101 , 2.043000 & 2.061465 & @xmath102 & 2.061347 , 2.061583 & 2.094603 & @xmath103 & 2.093318 , 2.094057 , 2.094360 , 2.095019 , 2.095867 & 2.097187 & @xmath104 & 2.097187 & 2.115063 & @xmath105 & 2.115063 & 2.240122 & @xmath106 & 2.238328 , 2.238913 , 2.239351 , 2.239765 , 2.240104 , 2.240775 , & & & 2.241570 , 2.242268 , 2.242837 , 2.243204 & 2.550918 & @xmath107 & 2.550744 , 2.551092

we observe with keck / hires the @xmath0 qso triplet 1623 + 27 in order to explore on the scale of a megaparsec the spatial clustering of c iv absorbers between adjacent sightlines . we find this signal to be significantly weaker than the clustering in velocity on corresponding scales along single sightlines , assuming that the relative velocity of absorbers is dominated by the hubble flow . this indicates that small - scale clustering ( 200 km s@xmath1 km s@xmath2 ) of the c iv absorbers can not be interpreted in terms of the positions of the absorbers in space , but must be considered as internal motions within individual absorbers , or within clusters of absorbers whose internal velocities dominate over hubble expansion across the cluster scale . if the single - sightline signal is due to spatial clustering , it is caused by absorber clusters smaller than would be implied by their velocities if a hubble flow is assumed . the spatial clustering of c iv absorbers at @xmath3 is consistent with data on ly @xmath4 forest clustering measured in the same way at the same redshifts . however , present - day galaxy clustering , evolved back to @xmath3 , is consistent with c iv spatial clustering but perhaps not with that of the ly @xmath4 forest . even so , one can not as yet distinguish the two absorber populations on the basis of spatial clustering on these small scales .

introduction observations and analysis results discussion and conclusions

arxiv

pedestrian flow modelling has attracted the interest of a large number of scientists from different research fields , planners and designers . while planning the architecture of buildings one might be interested in how people move around their intended design so that shops , entrances , corridors , emergency exits and seating can be placed in useful locations . pedestrian models are helpful in improving efficiency and safety in public places such as airport terminals , train stations , theatres and shopping malls . they are not only used as a tool for understanding pedestrian dynamics at public places , but also support transportation planners or managers to design timetables . a large number of models for pedestrian flow have appeared on different levels of description in recent years . the microscopic ( individual - based ) level models based on newton type equations as well as vision - based models or cellular automata models have been developed , see refs . , and hydrodynamic pedestrian flow equations involving equation for density and mean velocity of the flow are derived in refs modeling of pedestrian flow with scalar conservation laws coupled to the solution of the eikonal equation has been presented and investigated in refs . @xcite . the modelling of pedestrian behaviour in a real - world environment is a complex problem , mainly due to the unpredictable nature of human decision making . literature @xcite shows for example , that , 70 @xmath0 of people in a crowd are moving in groups and social interactions which can greatly influence crowd behaviour . most of the models mentioned above treat pedestrians as individual agents and neglect the group dynamics among them . the influence of group dynamics on the behaviour of pedestrians , the differences between the behaviour of pedestrians walking with groups and those who walk alone , and the interpretation of passenger s social characteristics and group behaviour using a simulation model has been presented in several recent works . we refer to @xcite . in this work , we closely follow a procedure for interacting particle systems used , for example , in the description of coherent motion of animal groups such as schools of fish , flocks of birds or swarms of insects , see ref . it has been applied to pedestrian flow modelling in ref . @xcite . there , a classical microscopic social force model for pedestrians has been extended with an optimal path computation as for example in ref . @xcite . the main objective of the present paper is to include multi - group behaviour and the impact of group dynamics , addressing in particular larger groups in a pedestrian crowd , into the set - up developped in @xcite . we extend the model developed there to the description of multi - group pedestrian flows using a multi - phase approach . for the numerical simulations we use , as in ref . @xcite , particle methods on the microscopic and macroscopic level of the model hierarchy . these methods are straightforward for microscopic equations . in case of the macroscopic equations particle methods are based on a lagrangian formulation of these equations . the dependence of the solutions on the level of attraction between the group members and the relaxation time towards the desired optimal path field is investigated and discussed . as a general result we observe the increase in evacuation time either by increasing the attraction between the group members or by decreasing the relaxation time towards the velocity given by the eikonal equation . the paper is organized in the following way : in section 2 the hierarchy of multi - group pedestrian models is presented . section 3 contains a description of the particle methods used in the simulations . section 4 contains the numerical results . we consider an evacuation problem . a comparison of the solutions of the macroscopic equations is presented for different parameters together with a comparison of the assicated evacuation times . finally , section 5 concludes the work . in this section , we start with a multi - group microscopic model for pedestrian flow using a microscopic social force model and a hughes - type model including the solution of the eikonal equation . we proceed by deriving multi - group hydrodynamic and scalar models from the microscopic model . we consider a microscopic social force model for pedestrian flow including an optimal path computation . for references , see for example refs . . for @xmath1 pedestrians divided into @xmath2 groups , we obtain a two - dimensional interacting particle system with locations @xmath3 , and velocity @xmath4 . here , the index @xmath5 is used to number all pedestrians , the index @xmath6 denotes the group to which the pedestrian belongs . @xmath7 denotes the set of all @xmath8 which are in group @xmath9 and @xmath10 denotes the number of pedestrians in group @xmath9 with @xmath11 . the equations of motion are @xmath12 where @xmath13 is an interaction potential denoting the interaction between members of groups @xmath9 and @xmath14 . a common choice is the morse potential @xmath15 here , @xmath16 , @xmath17 are attractive and repulsive strengths and @xmath18 , @xmath19 are their respective length scales . these constants depend on the groups @xmath9 and @xmath14 under consideration . an attractive interaction force acts only between members of the same group . the repulsive force acts between all pedestrians . the acceleration towards the desired direction is given by @xmath20 moreover , @xmath21 is given by @xmath22 where @xmath23 is a smoothed version of the @xmath24-distribution with @xmath25 finally , @xmath26 is given by the solution of the eikonal equation @xmath27 @xmath28 is a density - dependent velocity function , @xmath29 , @xmath30 denotes a reaction time . moreover , we use the notation @xmath31 such that @xmath32 . in @xcite an attractive interaction of the members of the group with the center of mass is postulated . this gives an additional term @xmath33 we define this attraction using the attractive part of a morse potential @xmath34 again , the constants might depend on the group @xmath9 . following @xcite one rescales the interaction potential of equation ( [ eq:2.1 ] ) with a factor @xmath35 and derives a kinetic mean field equation . this procedure is adapted to the multi - group case in the following . our scaled microscopic model states @xmath36 the empirical measures @xmath37 of the stochastic processes @xmath38 are given by @xmath39 where @xmath24 denotes the usual dirac distribution and @xmath40 . the mean field limit describes the convergence for @xmath41 of the empirical measure @xmath37 towards the deterministic distribution @xmath42 of the stochastic process @xmath43 governed by the so - called nonlinear mckean vlasov equation @xmath44 where @xmath45 and @xmath46 . the corresponding differential equation for the evolution of the distribution functions @xmath47 on state space , which is determined using it s formula , is called the mean field equation . it is given by @xmath48 with force term @xmath49 we note that due to our definitions we have @xmath50 for the following we define the momentum by @xmath51 for @xmath52 . the additional term @xmath53 gives in the limit @xmath54 with @xmath55 hydrodynamic limits for interacting particle systems have been derived for example in @xcite . we consider the mean field equation ( [ eq:2.4 ] ) and integrate the kinetic equation against @xmath56 and @xmath57 . using a mono - kinetic distribution function to close the resulting balance equations means that the velocity distribution function is assumed to be concentrated in the direction of the mean velocity , @xmath58 integrating the mean field equation wrto @xmath56 one obtains the continuity equation for group @xmath9 @xmath59 integrating wrto @xmath60 yields the second balance equation for group @xmath9 @xmath61 using now the mono - kinetic closure function we obtain @xmath62 and @xmath63 thus , equation ( [ bal2 ] ) becomes the momentum equation @xmath64 with @xmath65 for @xmath52 and this is coupled to @xmath66 in this section , we reduce the hydrodynamic description deriving scalar models . we assume again an interaction potential depending only on @xmath67 . starting from the hydrodynamic momentum equation derived from the standard maxwellian closure we neglect time changes in this equation and obtain an equation for @xmath68 as @xmath69 using equation ( [ eq:2.7 ] ) , we get @xmath70 thus , the resulting scalar equation for @xmath71 is @xmath72 = 0,\ ] ] for @xmath52 . a further simplification is obtained approximating the potential @xmath73 by a @xmath74 distribution , i.e. , @xmath75 with the constant @xmath76 given by @xmath77 this yields straightforwardly @xmath78 hence , equation ( [ eq:2.8 ] ) becomes a multi - group version of the hughes equations @xmath79 where @xmath52 . this is again combined with the eikonal equation @xmath80 [ discussion ] looking at equation ( [ eq:2.9 ] ) one observes that the influence of the diffusive term can be decreased in different ways . for example , adding or increasing the attraction interaction will decrease the value of @xmath81 . decreasing the value of the relaxation time @xmath30 will also decrease the diffusion . moreover , from considering ( [ eq:2.9 ] ) together with a monotone decaying function @xmath28 one would expect an increase of the diffusion to lead to a decay of the maximal values of the density and then to a faster transport in the direction of the eikonal field . in this section , we discuss the numerical methods for the two multi - group models ( [ eq:2.5 ] ) , ( [ eq:2.6 ] ) and ( [ eq:2.8 ] ) . to solve the hydrodynamic limit equations numerically we use a particle method , see , for example @xcite . mesh - less or particle methods are an appropriate way to solve pedestrian flow problems due to the appearance of situations with complicated geometries , free and moving boundaries and potentially large deformations of the domain of computation , i.e. the region where the density of pedestrians is non - zero . the particle method is based on a lagrangian formulation of hydrodynamic equations ( [ eq:2.5 ] ) and ( [ eq:2.6 ] ) . we consider @xmath82 where @xmath83 and @xmath84 . one evaluates these quantities at the particle locations and approximates the spatial derivatives of @xmath85 by a difference approximation . the simplest way to evaluate the integral over the interaction potential is to use a straightforward first order integration rule using an approximation of the local area around a particle determined by nearest neighbour search . this works fine for a well resolved situation with a sufficiently large number of gridpoints . in case the situation is underresolved we refer to @xcite for a thorough discussion of this issue . the resulting system of odes is then solved by a suitable time discretization method . diffusive terms can be included as well in a straightforward way . obviously , the above considerations show that the actual macroscopic computations are very similar to the microscopic ones if the number of macroscopic grid points is equal to the number of microscopic particles . however , in the macroscopic situation the particles are not physical particles as in the microscopic case . they play the role of discretization points . in particular , if the number of real particles is very large , that does not mean that the number of macroscopic particles in the particle method has to be increased in the same way . the number of macroscopic particles is only chosen according to accuracy considerations . in this sense , the numerical methodology for the hydrodynamic equations ranges from a `` nearly '' microscopic `` solver to a purely macroscopic solver depending on the number of grid - particles involved in the computation compared to the ' ' real " number of physical particles . we note that boundary conditions are realized by using fixed boundary particles with a suitable interaction potential . the scalar equation is solved with a similar particle method . in this case the so called diffusion velocity methods is used , i.e. , the equation ( [ eq:2.8 ] ) is written as a pure transport problem @xmath86 with @xmath87 and then solved in a lagrangian way . the approximation of the convolution term and the realization of the boundary conditions is done as for the hydrodynamic models . in all cases the solution of the eikonal equation is coupled to the flow simulation . the eikonal equation is solved by a fast marching method , see ref . it is solved on the same point cloud as the fluid dynamic equations . we use a similar methodology as described in ref . @xcite to solve the eikonal equation on an unstructured fixed grid . we update the eikonal solution in every tenth time step in order to save computational time . in this section , we present a series of numerical experiments for different parameters in the multi - group hydrodynamic ( [ eq:2.5 ] ) , ( [ eq:2.6 ] ) and scalar ( [ eq:2.8 ] ) equations . we investigate the models numerically for a configuration defined in ref . consider a railway platform of @xmath88 length and @xmath89 width with a square obstacle in the middle . initially , pedestrians are concentrated in the left boundary . they can leave at either of the two exits of @xmath90 width on the right boundary . we use 570 grid points or grid particles to describe the initial configuration . the initial density in this region is @xmath91 . s , title="fig : " ] we consider two situations : a single and a multi - group pedestrian flow as shown in figure [ fig:1 ] . for the single group pedestrian model the interaction between the pedestrians is given by a purely repulsive interaction potential . the multi - group pedestrian model contains two kinds of pedestrian : 75 @xmath92 are individual pedestrians . they interact with a purely repulsive interaction term as in the single group model . the second group consists of the remaining 25 @xmath92 of the pedestrians . between the members of this group the interaction potential contains an additional attraction term . see the right picture in [ fig:1 ] for the location of the two groups . we choose the inflow velocity as @xmath93 , @xmath94 , where the speed - density relation , @xmath95 if @xmath96 and @xmath97 otherwise . we have chosen @xmath98 . moreover , we choose @xmath99 and @xmath100 . we are choosing different kind of attractive strength @xmath101 , repulsive strength @xmath102 , attractive interaction length @xmath103 , and repulsive length @xmath104 . we use an explicit time integration for solving the hydrodynamic and scalar models with the constant time step @xmath105 for all experiments . in the following we investigate three issues . first single- and multi - group models are compared to each other . second , the parameter dependence of the multigroup model is investigated for the hydrodynamic case . in particular , we consider situations with increasing attraction between the members of the group . third , the hydrodynamic and scalar multi - group models are compared for different parameters . here , an increase of the relaxation time @xmath30 is investigated . as discussed in remark [ discussion ] increasing the attraction and decreasing the relaxation time have similar effects . in both cases diffusion is reduced and the overall evacuation time is increased . for the numerical simulation of the hydrodynamic models , we use the parameters : @xmath106 , @xmath107 and @xmath108 for single group pedestrian and @xmath109 , @xmath107 , @xmath110 , and @xmath108 for multi - group pedestrian . + + + figures [ fig:2 ] show the time evolution of the grid particles in single and multi - group hydrodynamic models for time t = 10.5s , t = 21s , t = 42s and t = 59s , respectively . single group pedestrians are faster then the multi - group pedestrians . in the multi - group model , grouped pedestrians walk slower compared to individual pedestrians and some individual pedestrians become slower since the grouped pedestrians play the role of obstacles for them . figure [ fig:2b ] shows the corresponding density plots for the time @xmath111 for single and multi - group case . . , title="fig : " ] + . , title="fig : " ] figure [ fig:3 ] shows the percentage of grid particles being in the computational domain for single and multi - group hydrodynamic models with respect to time . as expected one observes , that the evacuation time is larger in the case of grouped pedestrians . for the numerical comparison displayed in figure [ fig:4 ] and [ fig:5 ] , we use two different sets of interaction parameters : case 1 ( weak attraction ) : @xmath112 , @xmath113 , @xmath114 , and @xmath115 . case 2 ( strong attraction ) : @xmath116 , @xmath113 , @xmath114 , and @xmath115 . figures [ fig:4 ] show the time evolution of the grid particles with multi - group hydrodynamic models for the parameters case 1 and case 2 for time @xmath117 , and @xmath118 , respectively . in case 2 , the grouped pedestrians join into a single group which walks together , but slower compared to case 1 as shown in figures [ fig:4 ] and [ fig:5 ] . in general , at least for larger values of the attraction constant @xmath16 , one obtains a monotonic behaviour . the evacuation times increase with increasing attraction . this corresponds to the discussion in remark [ discussion ] . + + + + for the numerical simulation of the both models , we use the parameters : @xmath109 , @xmath107 , @xmath110 , and @xmath108 . figure [ fig:6 ] shows the time evolution of the number of grid particles in the computational domain for hydrodynamic and scalar equations for @xmath119 , @xmath120 and @xmath121 . the results coincide well for small values of @xmath30 . in this case the hydrodynamic model is near to a scalar model with vanishing interaction term , i.e. the original hughes equation . in contrast they differ for larger values of the relaxation time . for both models one observes that the increase of @xmath30 leads to reduced evacuation times . this effect is more pronounced for the hydrodynamic model . this corresponds again to the discussion in remark [ discussion ] . and @xmath121.,title="fig : " ] + and @xmath121.,title="fig : " ] + and @xmath121.,title="fig : " ] for the numerical simulation , we use interaction parameters as @xmath122 , @xmath113 , @xmath114 , and @xmath115 . figure [ fig:9 ] shows the number of grid particles in the computational domain for the case with center of mass interaction . in figure [ fig:9 ] we display 3 results . case 1 is the case without center of mass attraction . case 2 uses @xmath123 and case 3 uses @xmath124 . obviously , the center of mass attraction has a similiar influence as the reciprocal interaction in subsection [ attraction ] . one obtains , at least for larger values of @xmath125 a monotone behaviour . we have presented a multi - group microscopic model combining a social force model and an optimal path computation for pedestrians flows . hydrodynamic and scalar models are derived from the microscopic model . a meshfree particle method to solve the governing equations is presented and used for the computation of several numerical example analysing single- , and multi - group hydrodynamic and scalar models for different parameters such as interaction coefficients and relaxation time . the dependence of the solutions on these parameters is investigated and discussed . as a general result we observe the increase in evacuation time as the diffusive flow decreases , either by increasing the attraction between the group members or by decreasing the relaxation time towards the velocity given by the eikonal equation . future research topics are the consideration of real data , the determination of suitable parameters and the development of simulations for more than two groups of pedestrians . this work is supported by the german research foundation , dfg grant kl 1105/27 - 1 , by rtg grk 1932 `` stochastic models for innovations in the engineering sciences '' , project area p1 and by the daad phd program mic `` mathematics in industry and commerce '' . l. cheng , v. reddy , c. fookes , and p. k. d. v. yarlagadda , _ impact of passenger group dynamics on an airport evacuation process using an agent - based model _ , international conference on computational science and computational intelligence , las vegas , nevada , usa ( 2014 ) . + retrieved from http://eprints.qut.edu.au/69071/ p. degond , c. appert - rolland , m. moussaid , j. pettre and g. theraulaz , _ a hierarchy of heuristic - based models of crowd dynamics _ , j. stat . phys . 152 ( 2013 ) 1033 - 1068 . d. helbing , i.j . farkas , p. molnar , and t. vicsek , _ simulation of pedestrian crowds in normal and evacuation situations , in : m. schreckenberg , s.d . sharma(eds . ) _ , pedestrian and evacuation dynamics , springer - verlag , berlin , 2002 , pp . 21 - 58 . hughes , _ the flow of human crowds _ , ann . fluid mech . 35 ( 2003 ) 169 - 182 . a. klar , s. tiwari , a multi - scale meshfree particle method for macroscopic mean field interacting particle models , siam multiscale mod . 12 , 3 h. ling , s.c . wong , m. zhang , c.h . shu , and w.h.k . lam , _ revisiting hughes dynamics continuum model for pedestrian flow and the development of an efficient solution algorithm _ , transp . part b : methodological , 43 ( 1 ) ( 2009 ) , pp . 127 - 141 . b. maury , a. roudneff - chupin and f. santambrogio , _ a macroscopic crowd motion model of the gradient - flow type _ models methods appl . ( 2010 ) 1787 - 1921 . m. moussaid , n. perozo , s. garnier , d. helbing , and g. theraulaz , _ walking behaviour of pedestrian social groups and its impact on crowd dynamics _ , [ doi:10.1371/journal.pone.0010047 ] , plos one , 5(4 ) , e10047(2010 ) . + retrieved from http://dx.doi.org/10.1371%2fjournal.pone.0010047 . h. singha , r. arterb , l. doddb , p. langstonb , e. lesterb , j. druryc modelling subgroup behaviour in crowd dynamics dem simulation applied mathematical modelling 33 ( 12 ) , 44084423 , 2009 s. tiwari , and j. kuhnert , _ modelling of two - phase flow with surface tension by finite pointset method(fpm ) _ , j. comp . math , 203 ( 2007 ) , pp . 376 - 386 .

we consider a multi - group microscopic model for pedestrian flow describing the behaviour of large groups . it is based on an interacting particle system coupled to an eikonal equation . hydrodynamic multi - group models are derived from the underlying particle system as well as scalar multi - group models . the eikonal equation is used to compute optimal paths for the pedestrians . particle methods are used to solve the macroscopic equations . numerical test cases are investigated and the models and , in particular , the resulting evacuation times are compared for a wide range of different parameters . keywords : interacting particle system ; multi - group equations ; mean field equation ; eikonal equation ; macroscopic limits ; particle methods .

introduction multi-group pedestrian flow models numerical methods numerical results concluding remarks acknowledgment

arxiv

random matrix theory ( rmt ) has been applied in physics as well as in various other scientific disciplines @xcite . in physics , the most notable applications of rmt are found in statistical nuclear physics , quantum chaotic systems , and mesoscopic and disordered systems . rmt facilitates a theoretical understanding of the spectral correlations of a physical or a model complex system . an important aspect of rmt is the universality of spectral correlations , making the results of the classical rmt ensembles , viz . the gaussian ensembles , useful in many fields . parameter - dependent rmt models @xcite also yield universal results @xcite . these models are applicable to complex systems in which spectral statistics is governed by an external parameter . in these models one may also consider spectral cross correlations at different parameter values @xcite . such spectral correlations are associated with the level motion with respect to the parameter , and they are referred to as parametric level correlations . these correlations have been studied extensively , and their universality has been tested in diverse systems , e. g. , a hydrogen atom in a uniform magnetic field with the strength of the field as the parameter , resonances in quartz blocks at a uniform temperature where the temperature is an external parameter , and chaotic billiards where aharonov - bohm flux or the background potential or boundary parameters are treated as an external parameter @xcite . in these studies , analytic results for the density - density correlation function @xcite are of fundamental importance , and they have been obtained using the supersymmetric nonlinear @xmath0-model for disordered systems . recent studies of the parametric spectral cross - form factor and the fidelity @xcite reemphasized the importance of parametric correlations . in addition to the above references we also mention ref . @xcite , which made important contributions to the study of parametric correlations . to estimate how long the correlations are sustained in a parameter - driven complex system , it is suggestive to study integrated measures such as a number variance . in this context such a measure appears in the literature @xcite . however , in our opinion , it is a non - local measure as it involves variance of the staircase function from the ground state . this motivated us to introduce the number covariance as a local measure to study the parametric correlations . it is defined as the covariance of the number of energy levels in intervals of fixed length between spectra for two values of the parameter . by definition it is local , and thus it fulfills the basic requirement for applying the rmt . the number covariance can be calculated numerically from the above - mentioned density - density correlation function , which is known in the form of a multiple integral , or from the spectral cross - form factor which is somewhat simpler . it is surprising that while many measures have been used in this context , the number covariance , which is a natural quantity to use for comparison with numerical data , has not been investigated . in this paper we consider the binary correlation method @xcite to derive compact expressions for the number covariance . it turns out to be very close to the results obtained from numerical integrations of the exact correlation functions . we show that our results agree extremely well with the number covariance calculated for the spectra of quantum kicked rotors introduced in @xcite . we also consider a local version of the measure proposed in @xcite . we consider the three invariant gaussian ensembles ( ges ) of hermitian matrices @xmath1 of dimension @xmath2 , viz . the gaussian orthogonal ensemble ( goe ) , the gaussian unitary ensemble ( gue ) , and the gaussian symplectic ensemble ( gse ) . we use the dyson index @xmath3 , where @xmath4 and @xmath5 respectively for these ensembles @xcite . the joint probability density of matrix elements is given by @xmath6 . here the @xmath7 are the variances for @xmath3 distinct classes of the off - diagonal matrix elements . parametric variations in the ges are described with respect to a parameter @xmath8 by the ensembles of matrices , @xmath9 , defined as @xmath10 . here both @xmath11 and @xmath12 belong to the same invariance class of the gaussian ensembles , and they are independently distributed . it is worth pointing out that similar models are defined for the crossover ensembles @xcite with the matrices corresponding to different symmetry classes . variance @xmath7 is the same for @xmath13 , and @xmath12 . thus @xmath9 and @xmath11 are identically distributed gaussian ensembles with correlation coefficient @xmath14 between @xmath15 and @xmath16 for all @xmath17 . the scale of the spectral statistics is supplied by @xmath7 , which we fix by @xmath18 @xcite . in the limit of large @xmath2 , the ensembles - averaged spectral density , @xmath19 , is described by wigner s semicircle law @xcite , @xmath20 , where @xmath21 . notice that the same density is valid for all @xmath8 and @xmath3 . we introduce the number covariance , @xmath22 , which is defined as the covariance of the number of levels in the interval @xmath23 $ ] for @xmath9 and @xmath11 . in the limit of large @xmath2 , the number covariance becomes a function of @xmath24 and @xmath25 , where @xmath26 is the average number of eigenvalues in @xmath23 $ ] , and @xmath25 is the rescaled parameter defined by @xcite @xmath27 where @xmath28 is the average level spacing . we remark that @xmath25 depends on @xmath29 since @xmath30 depends on @xmath29 . it has been shown in transition studies @xcite that , for @xmath31 and @xmath32 , the transition in the two - point correlation is abrupt as a function of @xmath8 but smooth with respect to @xmath25 . in terms of @xmath24 and @xmath25 , the number covariance , @xmath33 , is given by @xmath34 where _ overbar _ denotes ensemble averaging . @xmath35 is the number of eigenvalues in the interval @xmath23 $ ] at parameter value @xmath25 . notice that the number variance is given by @xmath36 . we also introduce the parametric number variance ( pnv ) , as @xmath37 this is the local equivalent of pnv introduced in @xcite . note that @xmath38 in eq . ( [ pnv - ncov ] ) whereas @xmath39 in @xcite . for @xmath40 , @xmath41 becomes @xmath42 , whereas in the latter case , it diverges as @xmath43 @xcite , confirming thereby the nonlocality . we are interested in local quantities defined in the large-@xmath2 limit . for instance the unfolded cluster function , @xmath66 , and the spectral cross - form factor , @xmath67 , are defined by @xmath68 the cross - form factor has been useful in the semi - classical study @xcite , in calculating the current correlator @xcite , and also in the fidelity analysis @xcite . note that for @xmath69 , @xmath70 and @xmath71 give respectively the unfolded cluster function and the spectral form factor of the corresponding ges , viz . , @xmath72 , and @xmath73 , as defined in @xcite . to obtain @xmath70 from eqs . ( [ srho - assym ] , [ y11 ] ) , we replace the summation by an integral , using @xmath74 . ignoring the rapidly oscillating part of the @xmath75\cos[\zeta\psi(y)]$ ] term in eq . ( [ srho - assym ] ) we finally get @xmath76 note that @xmath77 is the small @xmath78 expansion of @xmath79 . as in @xcite , to improve the approximation we can replace this term by @xmath80 . comparison of the resulting equation with eq . ( [ y11 ] ) yields , @xmath81 see also @xcite , where similar results have been given for the crossover ensembles . in an alternative method used in @xcite , the summation in eq . ( [ srho - assym ] ) can be evaluated using an exponential cut - off factor . we introduce @xmath82 , replacing @xmath64 by @xmath83 as @xmath84 $ ] in eq . ( [ srho - assym ] ) , to obtain @xmath85.\ ] ] it follows from the stationarity of @xmath86 that the number covariance , which is a double integral as in ( [ sig - srho ] ) , can be written as @xmath87dk.\end{aligned}\ ] ] using eq . ( [ bin - ep - kbeta ] ) in the second equality of eq . ( [ sig11_k ] ) we get the compact answer , @xmath88.\ ] ] the cut - off term has to be fixed with respect to the @xmath69 result , i.e. @xmath89 . since this term has small variation with respect to @xmath24 , we fix its value at which @xmath90 in ( [ rncov - unf ] ) fits the exact @xmath89 for large @xmath24 . we find @xmath91 , and @xmath92 , respectively for @xmath93 and @xmath5 . for these values of @xmath82 we find that both of our approximations ( [ bin_kbeta ] , [ bin - ep - kbeta ] ) are close to each other for small @xmath78 . finally we remark that the above result is valid for @xmath94 . pnv can be calculated from eq . ( [ pnv - ncov ] ) along with eq . ( [ sig11_k ] ) for finite @xmath24 . for @xmath95 and finite @xmath25 , we find @xmath96 using eqs . ( [ pnv - ncov],[rncov - unf ] ) we obtain , @xmath97 we remark that the result given in @xcite is half of our result in eq . ( [ bc - pnv ] ) because their interval @xmath23 $ ] , used in eq . ( [ pnv - ncov ] ) , starts from the ground state . moreover , they have used the approximation ( [ bin_kbeta ] ) instead of ( [ bin - ep - kbeta ] ) . the exact results for the density - density correlation function , @xmath98 , and the cross - form factor , @xmath71 , are known @xcite . note that for @xmath69 , @xmath99 gives @xmath100 , where @xmath101 is the usual two - level correlation function @xcite . these can be used to obtain exact numerical results for @xmath102 . the density - density correlation functions in terms of our above parameter @xmath25 are given by @xmath103\nonumber \\ & \times & \exp\big[\dfrac{\pi^{2}\lambda}{2 } ( x^{2 } + y^{2 } + z^{2 } -2x^{2}y^{2 } -1 ) \big ] , \\ \label{para-20 } \mathsf{r}^{(2)}_{11}(r;\lambda ) & = & 1 + \int_{0}^{1 } dx \int_{1}^{\infty } dy \cos(\pi x r ) \cos(\pi y r ) \exp\left[{\pi^{2}\lambda(x^{2}-y^{2})}\right ] , \\ \label{para-21 } { \mathsf r^{(4)}_{11}}(r;\lambda ) & = & 1 + \re \int_{-1}^{1 } dx \int_0^{1}dy \int_{1}^{\infty } dz \frac{(xy - z)^{2 } ( z^{2}-1 ) } { ( x^{2 } + y^{2 } + z^{2 } -2xyz -1 ) ^{2 } } \exp\left [ - \ii(2\pi r+\ii\delta)(xy - z)\right ] \nonumber \\ & \times & \exp\left[-4\pi^{2}\lambda ( x^{2 } + y^{2 } + z^{2 } -2x^{2}y^{2}-1 ) \right],\end{aligned}\ ] ] where @xmath104 . using eqs . ( [ para-20 ] ) in ( [ y11 ] ) one obtains a compact result for @xmath105 : @xmath106 on the other hand , using ( [ para-19],[para-21 ] ) in ( [ y11 ] ) , for @xmath107 and @xmath5 , we get @xmath67 as double - integrals of the variables @xmath108 and @xmath109 : @xmath110 one can verify ( [ bin_kbeta ] ) for small @xmath78 , but otherwise the exact results are difficult to deal with analytically . we evaluate @xmath111 and @xmath112 by solving the double integrals numerically . next , we use @xmath67 in eq . ( [ sig11_k ] ) and evaluate @xmath33 numerically . it is worth pointing out that our approximate results eqs . ( [ bin_kbeta],[bin - ep - kbeta ] ) work well for small @xmath78 . however , both approximations yield @xmath102 close to the exact ones for @xmath94 . it comes about because of the @xmath113 term which suppresses the contribution of @xmath114 for large @xmath78 in @xmath33 . vs @xmath115 , for @xmath107 ( top ) and @xmath116 ( bottom ) , at four different values of the parameter @xmath25 , viz . @xmath117 and @xmath118 shown respectively in _ black _ ( upper ) , _ brown _ ( mid upper ) , _ magenta _ ( mid lower ) , and _ turquoise _ ( lower ) . solid lines represent the exact results and wriggled curves are obtained using eq . ( [ kbeta_kick ] ) for eigenangle spectra calculated at @xmath119 ( top ) and @xmath120 ( bottom ) . we have used local averaging in the range @xmath121 to reduce the statistical fluctuations . , scaledwidth=45.0% ] for the ge models , we have considered a @xmath122-member gaussian ensemble of @xmath123-dimensional @xmath9 matrices for all three @xmath3 at different values of @xmath8 . the variance is fixed such that the semicircle has radius @xmath116 . since @xmath25 depends on @xmath19 , we choose only @xmath124 middle levels from each spectrum to ensure that for a given @xmath8 , the density @xmath30 and therefore @xmath25 do not vary appreciably . vs @xmath25 , for @xmath93 , and @xmath5 from top to bottom , at three different values of @xmath24 , viz . @xmath125 and @xmath126 shown respectively in _ black _ ( lower ) , _ orange _ ( mid ) , and _ brown _ ( upper ) . solid lines represent the exact results obtained from the numerical integration and dashed lines represent approximate results ( [ rncov - unf ] ) . circles represent the rmt data for all three @xmath3 and crosses represent the kicked rotor data for @xmath107 and @xmath116 . , scaledwidth=45.0% ] the quantum kicked rotor is a prototypical example of quantum chaotic systems . we consider the eigenangle spectra of quantum kicked rotors @xcite . the quantum map is generated in an @xmath2-dimensional hilbert space by the time - evolution operator @xmath127 of a kicked rotor with torus boundary conditions . the standard case is that of a singly - kicked rotor with @xmath128 where @xmath129 $ ] and @xmath130 $ ] with @xmath131 and @xmath132 being the position and momentum operators . here , @xmath133 is the kicking parameter , @xmath134 is the parity - breaking parameter , and @xmath135 is the time - reversal - breaking parameter @xmath136.we consider parametric correlations arising from small variations @xmath137 in the kicking strength @xmath133 . parametric correlation can also be studied with variations in @xmath138 or @xmath135 @xcite . in the position representation , @xmath139\delta_{mn}$ ] and @xmath140 $ ] , for @xmath141 , @xmath142 , @xmath143 and we set @xmath144 . we choose the parameter @xmath145 for parity breaking . for @xmath119 , it corresponds to the @xmath107 symmetry class , and otherwise it rapidly approaches @xmath146 . eigenangle density for the system is constant . the @xmath25 parameter is given by @xcite @xmath147 , where @xmath137 is variation in the initial @xmath133 . this can be proved from the first equality of eq . ( [ alphalambda ] ) by making the correspondence @xmath148 , @xmath149 , and @xmath150/\beta n^{2}=1/2\beta n$ ] . the spectral cross - form factor is calculated as @xmath151 where @xmath152 is an integer and @xmath153 . in numerics we consider @xmath154 dimensional matrices @xmath127 with @xmath155 and @xmath119 and @xmath156 respectively for @xmath107 and @xmath116 . initially @xmath133 is @xmath157 and then varied in small steps of @xmath158 . this represents one member of the ensemble at different @xmath133 values . the other independent members of the ensemble are obtained by increasing the initial value of @xmath133 in steps of @xmath157 . finally , we consider @xmath159 such members of the ensembles . vs @xmath24 , for @xmath107 ( top ) and @xmath116 ( bottom ) , at three different @xmath25 , viz . @xmath160 , and @xmath161 for @xmath107 , and @xmath162 , and @xmath163 for @xmath146 , shown respectively in _ black _ ( lower ) , _ orange _ ( mid ) , and _ brown _ ( upper ) . as in fig . [ fig - ncov ] , we use lines and symbols for the theory and data respectively.,scaledwidth=45.0% ] in fig . [ fig - kbeta ] we illustrate @xmath67 vs @xmath115 for the kicked rotor data evaluated at @xmath164 and @xmath118 . in fig . [ fig - ncov ] , we show @xmath165 , @xmath166 , and @xmath167 as a function of @xmath25 at three values of @xmath24 , viz . @xmath168 , @xmath169 , and @xmath126 . difference in results obtained from the approximations ( [ bin_kbeta ] ) and ( [ bin - ep - kbeta ] ) is nominal and therefore the former approximation is not shown . in fig . [ fig - pnv ] we illustrate @xmath170 and @xmath171 as a function of @xmath24 at several values of @xmath25 . in this figure we consider @xmath24 upto @xmath172 . for @xmath173 , @xmath174 becomes almost independent of @xmath24 . it is evident from these figures that exact results are in excellent agreement with the kicked rotor data . also , our binary correlation results yield a very good approximation to the exact results . in conclusion , we have defined the parametric number covariance to study parametric correlations in quantum chaotic spectra . we have shown that the local spectral fluctuations become rapidly independent as the parameter @xmath8 of the system is varied . smooth statistical variations are found as a function of a rescaled parameter @xmath175 . for spectra with @xmath176 , we find @xmath177 when @xmath178 . for such small values of @xmath8 the global correlations between the spectra are close to 1 . we have dealt with the three @xmath3 cases and derived the number covariance for the gaussian ensembles , using the binary correlation method , which is close to the results obtained form numerical integration of the exact formula . we have shown its universality in the quantum kicked rotor spectra for time - reversal invariant and time - reversal non - invariant systems . o. bohigas , m. j. giannoni , in _ mathematical and computational methods in nuclear physics _ , edited by j. s. dehesa , j. m. g. gomez , and a. polls , in lecture notes in physics , vol . 209 ( springer , berlin , 1984 ) , p. 1 . f. haake _ quantum signatures of chaos _ ( springer - verlag , berlin , 2001 ) . c. w. j. beenakker , rev . phys . * 69 * , 731 ( 1997 ) . , edited by g. akemann , j. baik , and p. di francesco ( oxford university press , oxford , 2011 ) . a. pandey and m. l. mehta , commun . 87 * , 449 ( 1983 ) ; m. l. mehta and a. pandey , j. phys . a : math . gen . * 16 * , 2655 ( 1983 ) ; a. pandey and p. shukla , j. phys . a : math . gen . * 24 * , 3907 ( 1991 ) ; vinayak and a. pandey , j. phys . a : math . theor . * 42 * , 315101 ( 2009 ) ; s. kumar and a. pandey , ann . phys . * 326 * , 1877 ( 2011 ) . a. szafer and b. l. altshuler , phys lett . * 70 * , 587(1993 ) ; b. d. simons and b. l. altshuler , phys . * 70 * , 4063 ( 1993 ) ; b. d. simons , p. a. lee , and b. l. altshuler , phys . b * 48 * , 11450 ( r ) ( 1993 ) ; c. w. j. beenakker , phys . rev . lett . * 70 * , 4126 ( 1993 ) ; c. w. j. beenakker and b. rejaei , physica a * 203 * , 61 ( 1994 ) ; e. brezin and a. zee , phys . e * 49 * , 2588 ( 1994 ) ; e. r. mucciolo , b. s. shastry , b. d. simons , and b. l. altshuler , phys b * 49 * , 15197 ( 1994 ) . b. d. simons , a. hashimoto , m. courtney , d. kleppner , and b. l. altshuler , phys . rev . lett . * 71 * , 2899 ( 1993 ) ; m. barth , u. kuhl , and h .- j . stckmann , phys . lett . * 82 * , 2026 ( 1999 ) ; p. bertelsen , c. ellegaard , t. guhr , m. oxborrow , and k. schaadt , phys . * 83 * , 2171 ( 1999 ) . r. schfer , t. gorin , t. h. seligman , and h. j. stckmann , new j. phys . * 7 * , 152 ( 2005 ) ; h .- j . stckmann and r. schfer , new j. phys . * 6 * , 199 ( 2004 ) ; h .- j . stckmann and r. schfer , phys . lett . * 94 * , 244101 ( 2005 ) ; t. gorin , t. prosen , t. h. seligman , and m. nidari , phys . rep . * 435 * , 33 ( 2006 ) ; h. kohler , t. nagao , and h .- j . stckmann , phys . e * 84 * , 061133 ( 2011 ) . t. gorin and p. c. lopez vzquez , phys . e * 88 * , 012906 ( 2013 ) ; b. gutkin , d. waltner , m. gutierrez , j. kuipers , and k. richter , phys . e * 81 * , 036222 ( 2010 ) ; b. kber , u. kuhl , h .- j . stckmann , t. gorin , d. v. savin , and t. h. seligman , phys . e * 82 * , 036207 ( 2010 ) . j. zakrzewski and d. delande , phys . e * 47 * , 1650 ( 1993 ) ; f. von oppen , phys . lett . * 73 * , 798 ( 1994 ) ; y. v. fyodorov and h .- j . sommers , phys . e * 51 * , r2719 ( 1995 ) ; y. v. fyodorov and h - j . sommers , z. phys . b * 99 * , 123 ( 1995 ) ; n. taniguchi , a. v. andreev , and b. l. altshuler , euro . phys * 29 * , 515 ( 1995 ) ; y. alhassid and h. attias , phys . rev . lett . * 74 * , 4635 ( 1995 ) ; b. dietz , m. lombardi , t. h. seligman , phys . a * 215 * , 181 ( 1996 ) ; a. m. s. maedo , europhys . * 26 * , 641 ( 1994 ) ; n. savytskyy , a. kohler , sz . bauch , r. bmel , and l. sirko , phys . e * 64 * , 036211(2001 ) ; m. v. berry and j. p. keating , j. phys . gen * 27 * 6167 ( 1994 ) .

we study spectral parametric correlations in quantum chaotic systems and introduce the number covariance as a measure of such correlations . we derive analytic results for the classical random matrix ensembles using the binary correlation method and obtain compact expressions for the covariance . we illustrate the universality of this measure by presenting the spectral analysis of the quantum kicked rotors for the time - reversal invariant and time - reversal non - invariant cases . a local version of the parametric number variance introduced earlier is also investigated .

introduction parametric gaussian ensembles and the number covariance the binary correlation method for the number covariance the parametric number variance exact results numerics of the gaussian ensembles the quantum kicked rotor numerical results conclusion

arxiv

during the last three decades the investigation of phenomena related with strongly interacting electrons has developed to a central field of condensed matter physics . in this context , high - temperature superconductivity and heavy - fermion behavior are maybe the most important examples . it has been clearly turned out that such systems require true many - body approaches that properly take into account the dominant strong electronic correlations . in the past , many powerful numerical methods like exact diagonalization @xcite , numerical renormalization group @xcite , quantum monte - carlo @xcite , the density - matrix renormalization group @xcite , or the dynamical mean - field theory @xcite have been developed to study strongly correlated electronic systems . in contrast , only very few analytical approaches are available to tackle such systems . in this regard , renormalization schemes for hamiltonians developed in the nineties of the last century @xcite represent an important new direction in the field where renormalization schemes are implemented in the liouville space ( that is built up by all operators of the hilbert space ) . thus , these approaches can be considered as further developments of common renormalization group theory @xcite that is based on a renormalization within the hilbert space . in this review we want to discuss the projector - based renormalization method [ prm , ref . ] that shares some basic concepts with the renormalization schemes for hamiltonians mentioned above @xcite . all these approaches including the prm generate effective hamiltonians by applying a sequence of unitary transformations to the initial hamiltonian of the physical system . however , there is one distinct difference between these methods : both similarity renormalization @xcite and wegner s flow equation method @xcite start from a continuous formulation of the unitary transformation by means of a differential form . in contrast , the prm is based on discrete transformations so that a direct link to perturbation theory can be provided . this review is organized as follows : in the next section we discuss the basic concepts of the prm : we introduce projection operators in the liouville space that allow the definition of an effective hamiltonian . if these ingredients are combined with unitary transformations one can derive a new kind of perturbation theory that is not restricted to the ground - state but also allows to investigate excitations . ( to illustrate this point we briefly discuss the triplet dispersion relation of a dimerized and frustrated spin chain in the appendix . ) however , this perturbation theory is not the focus of this review and can be considered as an interesting side - product of the development of the prm , a renormalization scheme based on the same ingredients . to illustrate the method in some detail , the exactly solvable fano - anderson model is considered . improving our previous publications on the prm , we show here the relation of the prm to wegner s flow equation method @xcite for the first time . it turns out the latter method can be understood within the framework of the prm by choosing a complementary unitary transformations to generate the effective hamiltonian . for demonstration , the fano - anderson model is solved with this approach , too . as a more physical example , the electron - phonon interaction is studied in sec . in particular , the prm is compared in some detail with the flow equation method @xcite and the similarity transformation @xcite . furthermore , we introduce a possible modification of the prm that allows to derive block - diagonal hamiltonians , and we discuss in some detail the freedom in choosing the generator of the unitary transformation the prm is based on . finally , we show how phase transitions can be studied within the prm by adding symmetry breaking fields to the hamiltonian . in sec . [ pam ] the prm is applied to the periodic anderson model to describe heavy - fermion behavior . whereas the famous slave - boson mean - field theory @xcite obtains an effectively free system consisting of two non - interacting fermionic quasi - particles , here the periodic anderson model is mapped onto an effective model that still takes into account electronic correlations . thus , in principle both mixed and integral valence solution can be found . however , here we restrict ourself to an analytical solution of the renormalization equations that is limited to the mixed valence case . as third application of the prm the one - dimensional holstein model of spinless fermions is discussed . it is well known that the system undergoes a quantum phase transition from a metallic to a peierls distorted state if the electron - phonon coupling exceeds a critical value . first , for the metallic state we discuss the crossover behavior between the adiabatic and anti - adiabatic case in sec . [ holstein_cross ] . all physical properties are shown to strongly depend on the ratio of phonon and hopping energy in the system . in sec . [ holstein_qp ] , a unified description of the quantum - phase transition is given for the one - dimensional model in the adiabatic case . finally , as a second example for a quantum phase transition , we discuss in sec . [ holstein_sc ] the competition of charge ordering and superconductivity in the two - dimensional holstein model . based on the prm both charge density wave and superconductivity are studied within one theoretical framework . we summarize in sec . [ summary ] . in this section we introduce the concepts of the prm @xcite where we particularly pay attention to a general notation that is used throughout the review for all applications of the approach . we define projection operators of the liouville space and define an effective hamiltonian where , in contrast to common approaches , _ excitations _ instead of _ states _ are integrated out . in this way , not only a perturbation theory is derived but also and more important a renormalization scheme ( that we call prm in the following ) is established which allows to diagonalize or at least to quasi - diagonalize many - particle hamiltonians . as an illustrative example , the exactly solvable fano - anderson model is discussed . the prm is based on a sequence of _ finite _ unitary transformations whereas wegner s flow equations start from a continuous formulation of unitary transformations by means of a differential form . it turns out that such a continuous transformation can also be understood in the framework of the prm if a complementary choice for the generator of the unitary transformation is used and infinitely small transformation steps are considered . to discuss the differences between the two formulations of the prm in more detail , we also solve the fano - anderson model using the developed continuous approach . the projector - based renormalization method ( prm ) @xcite starts from the usual decomposition of a given many - particle hamiltonian , @xmath0 where the perturbation @xmath1 should not contain any terms that commute with the unperturbed part @xmath2 . thus , the interaction @xmath1 consists of the transitions between eigenstates of @xmath2 with corresponding _ non - zero _ transition energies . the presence of @xmath1 usually prevents an exact solution of the eigenvalue problem of the full hamiltonian @xmath3 so that suited approximations are necessary . the aim is to construct an effective hamiltonian @xmath4 with a renormalized unperturbed part @xmath5 and a remaining perturbation @xmath6 @xmath7 with the following properties : 1 . the eigenvalue problem of the renormalized hamiltonian @xmath8 is diagonal @xmath9 with @xmath10-dependent eigenvalues @xmath11 and eigenvectors @xmath12 . the effective hamiltonian @xmath4 is constructed in such a way so that ( measured with respect to @xmath8 ) all non - diagonal contributions with transition energies larger than some cutoff energy @xmath10 vanish . 3 . @xmath4 has the same eigenvalues as the original hamiltonian @xmath13 . the eigenvalue problem of @xmath5 is crucial for the construction of @xmath14 because it can be used to define projection operators , @xmath15 note that neither @xmath12 nor @xmath16 need to be low- or high - energy eigenstates of @xmath8 . @xmath17 and @xmath18 are super - operators acting on operators @xmath19 of the hilbert space of the system . thus , @xmath17 and @xmath18 can be interpreted as projection operators of the liouville space that is built up by all operators of the hilbert space . @xmath17 projects on those parts of an operator @xmath19 which only consist of transition operators @xmath20 with energy differences @xmath21 less than a given cutoff @xmath10 , whereas @xmath18 projects onto the high - energy transitions of @xmath19 . in terms of the projection operators @xmath22 and @xmath23 the property of @xmath14 to allow no transitions between the eigenstates of @xmath5 with energies larger than @xmath10 reads @xmath24 for an actual construction of the effective hamiltonian we now assume that the effective hamiltonian @xmath14 can be obtained from the original hamiltonian by a unitary transformation , @xmath25 which shall automatically guarantee that condition ( iii ) above is fulfilled . in the following the evaluation of the effective hamiltonian is done in two ways : at first a perturbative treatment is derived . after that we develop a much more sophisticated renormalization where we interprete the unitary transformation of eq . as a sequence of small transformations . the projector - based perturbation theory discussed in the next subsection is important for the understanding of the renormalization scheme derived later . however , the main focus of this review is the prm . in the following we evaluate the effective hamiltonian @xmath4 in perturbation theory . for this purpose the effective hamiltonian @xmath4 from eqs . and is simplified in a crucial point : the projection operators are now defined with respect to the eigenvalue problem of the unperturbed part of the _ original _ hamiltonian @xmath26 , @xmath27 thus , these projection operators differ from the formerly defined projectors @xmath17 and @xmath18 and can be written as follows @xmath28 & & \quad\times\theta(\lambda -|e_n^{}-e_m^{}| ) , \nonumber \\ \label{b2b } \mathbf{\bar{q}}_{\lambda } & = & \mathbf{1 } - \mathbf{\bar{p}}_{\lambda}.\end{aligned}\ ] ] the renormalized hamiltonian @xmath14 is now obtained from the unitary transformation , @xmath29 where @xmath30 is the generator of this transformation . to find @xmath31 , we employ the modified condition : all matrix elements of @xmath4 for transitions with energies larger than @xmath10 vanish , _ i.e. _ @xmath32 first we expand @xmath14 with respect to @xmath31 , @xmath33 + \frac{1}{2 ! } \left [ x_{\lambda } , \left [ x_{\lambda } , \mathcal{h } \right ] \right ] \\ & & + \ , \frac{1}{3 ! } \left [ x_{\lambda } , \left [ x_{\lambda } , \left [ x_{\lambda } , \mathcal{h } \right ] \right ] \right ] + \dots \ ; . \nonumber\end{aligned}\ ] ] and assume that the generator @xmath31 can be written as a power series in the interaction @xmath1 , @xmath34 thus inserting in eq . , the effective hamiltonian @xmath4 can be rewritten as a power series in the interaction @xmath1 @xmath35 + \left [ x_{\lambda}^{(1 ) } , \mathcal{h}_{1 } \right ] \\ & & + \ , \left [ x_{\lambda}^{(2 ) } , \mathcal{h}_{0 } \right ] + \frac{1}{2 ! } \left [ x_{\lambda}^{(1 ) } , \left [ x_{\lambda}^{(1 ) } , \mathcal{h}_{0 } \right ] \right ] + { \cal o}(\mathcal{h}_{1}^{3 } ) . \nonumber\end{aligned}\ ] ] the contributions @xmath36 to the generator of the unitary transformation can successively be determined by employing eq . . one finds @xmath37 \phantom{aaa}\\ & & -\ , \frac{1}{{\bf l}_{0 } } \mathbf{\bar{q}}_{\lambda}^ { } \left [ ( \mathbf{\bar{p}}_{\lambda}^{}\mathcal{h}_{1 } ) , \frac{1 } { \mathbf{l}_{0 } } ( \mathbf{\bar{q}}_{\lambda}^{}\mathcal{h}_{1 } ) \right ] . \nonumber\end{aligned}\ ] ] here , @xmath38 is the liouville operator of the unperturbed hamiltonian @xmath2 which is defined by @xmath39 $ ] for any operator variable @xmath40 . as one can see from and , no information about the low - energy part @xmath41 of the generator @xmath31 can be deduced from . therefore , we set for simplicity @xmath42 inserting eqs . , , and into the power series for @xmath4 , the desired perturbation theory is found , @xmath43 \nonumber\\ \label{b11 } & & -\ , \mathbf{\bar{p}}_{\lambda}^ { } \left [ ( \mathbf{\bar{p}}_{\lambda}^{}{\cal h}_{1 } ) , \frac{1 } { \mathbf{l}_{0 } } ( \mathbf{\bar{q}}_{\lambda}^{}{\cal h}_{1 } ) \right ] + \mathcal{o}(\mathcal{h}_{1}^{3}),\end{aligned}\ ] ] which can easily be extended to higher order terms . note that the correct size dependence of the hamiltonian is automatically guaranteed by the commutators in eq . . the limit @xmath44 is of particular interest because in this case the complete interaction @xmath1 is integrated out . usual perturbation theory derives effective hamiltonians that are only valid for a certain range of the system s hilbert space . in contrast , @xmath45 , as derived above , has no limitations with respect to the hilbert space so that it can also be used to study excited states . to illustrate this important aspect of our projector - based perturbation theory , we discuss the dimerized and frustrated spin chain in the appendix . at this point we would like to note that eq . can also be derived in a different way . it turns out that @xmath46 is only needed to fulfill the requirement @xmath47 if we restrict ourselves to second order perturbation theory . thus , in this case @xmath46 can be set to @xmath48 if the projector @xmath49 is applied to the right hand side of eq . , @xmath50 \\ & + & \mathbf{\bar{p}}_{\lambda}^ { } \left [ x_{\lambda}^{(1 ) } , \mathcal{h}_{1 } \right ] + \frac{1}{2 ! } \mathbf{\bar{p}}_{\lambda}^ { } \left [ x_{\lambda}^{(1 ) } , \left [ x_{\lambda}^{(1 ) } , \mathcal{h}_{0 } \right ] \right ] + \cdots . \nonumber\end{aligned}\ ] ] it is easy to proof that eq . again leads to the result eq . if and is used . in appendix [ spin_chain ] , the developed perturbation theory is applied to the dimerized and frustrated spin chain where ground - state energy and triplet dispersion relation have been calculated . a perturbation theory based on wegner s flow equations @xcite , that also allows a description of the complete hilbert space , has been derived in refs . and . however , this approach requires an equidistant spectrum of the unperturbed hamiltonian @xmath2 . in contrast , the perturbation theory presented here can be applied to systems with arbitrary hilbert space , and has similarities to a cumulant approach to effective hamiltonians @xcite . in the previous subsection the effective hamiltonian @xmath4 as defined by eqs . and has been evaluated within a new kind of perturbation theory . however , if the unitary transformation is interpreted as a sequence of unitary transformation a renormalization scheme can be developed based on the same definition of the effective hamiltonian . because again the projection operators @xmath17 and @xmath18 play a key role we call the derived method @xcite projector - based renormalization method ( prm ) . let us start from a renormalized hamiltonian @xmath51 that has been obtained after all transitions with energy differences larger than @xmath10 have already been integrated out . of course , @xmath5 and @xmath6 will differ from the original @xmath26 and @xmath52 . furthermore , we assume @xmath4 has the properties ( i)-(iii ) proposed in subsection [ basic_concepts ] . now we want to eliminate all excitations within the energy range between @xmath10 and a smaller new energy cutoff @xmath53 . thereby we use a unitary transformation , @xmath54 so that the effective hamiltonian @xmath55 has the same eigenspectrum as the hamiltonian @xmath56 . note that the generator @xmath57 needs to be chosen anti - hermitian , @xmath58 , to ensure that @xmath55 is hermitian when @xmath14 was hermitian before . to find an appropriate generator @xmath57 of the unitary transformation , we employ the condition that @xmath4 has ( with respect to @xmath8 ) only vanishing matrix elements for transitions with energies larger than @xmath10 , _ i.e. _ @xmath59 . similarly , also @xmath60 must be fulfilled , where @xmath61 is now defined with respect to the excitations of @xmath62 . in principle , there are two strategies to evaluate eqs . and : the first uses perturbation theory as derived in subsection [ b_perturbation ] . in this case @xmath63 can be written as @xmath64 \nonumber\\ & & + \ , \mathbf{p}_{(\lambda-\delta\lambda ) } \left [ x_{\lambda,\delta\lambda } , \mathcal{h}_{1,\lambda } \right ] \nonumber\\ & & + \ , \frac{1}{2 } \mathbf{p}_{(\lambda-\delta\lambda ) } \left [ x_{\lambda,\delta\lambda } , \left [ x_{\lambda,\delta\lambda } , \mathcal{h}_{0,\lambda } \right ] \right ] + { \cal o}(\mathcal{h}_{1,\lambda}^{3 } ) . \nonumber\end{aligned}\ ] ] the generator @xmath65 has to be chosen corresponding to eq . , @xmath66 + \cdots \hspace*{0.2cm}.\end{aligned}\ ] ] for details of the derivation we refer to subsection [ b_perturbation ] . this approach has been successfully applied to the electron - phonon interaction to describe superconductivity @xcite . alternatively , one can also start from an appropriate ansatz for the generator in order to calculate @xmath63 in a non - perturbative manner @xcite . an ansatz for the generator with the same operator structure as eq . is often a very good choice . this approach has been applied to the periodic anderson model to describe heavy - fermion behavior @xcite . it turns out that the second strategy has the great advantage to successfully prevent diverging renormalization contributions . however , in both cases , eqs . and describe a renormalization step that lowers the energy cutoff of the effective hamiltonian from @xmath10 to @xmath53 . consequently , difference equations for the hamiltonian @xmath4 can be derived , and the resulting equations for the @xmath10 dependence of the parameters of the hamiltonian are called renormalization equations . by starting from the original model @xmath67 the hamiltonian is renormalized by reducing the cutoff @xmath10 in steps @xmath68 . the limit @xmath44 provides the desired effective hamiltonian @xmath69 without any interaction . note that the results strongly depend on the parameters of the original hamiltonian @xmath3 . it turns out that the generator @xmath65 of the unitary transformation is not yet completely determined by eqs . and . instead , the low - energetic excitations included in @xmath65 , namely the part @xmath70 , can be chosen arbitrarily . the result of the renormalization scheme should not depend on the particular choice of @xmath70 as long as all renormalization steps are performed without approximations . however , approximations will be necessary for practically all interacting systems of interest so the choice @xmath70 becomes relevant . if @xmath71 is chosen the minimal transformation is performed to match the requirement . such an approach of `` minimal '' transformations avoid errors caused by approximations necessary for every renormalization step as much as possible . note that in order to derive the expression this choice of @xmath70 was used . however , in particular cases a non - zero choice for @xmath70 might help to circumvent problems in the evaluation of the renormalization equations . in general , new interaction terms can be generated in every renormalization step . this might allow the investigation of competing interactions which naturally emerge within the renormalization procedure . however , actual calculations require a closed set of renormalization equations . thus , often a factorization approximation has to be performed in order to trace back complicated operators to terms already appearing in the renormalization ansatz . consequently , derived effective hamiltonians might be limited in their possible applications if important operators have not been appropriately included in the renormalization scheme . if a factorization approximation needs to be performed the obtained renormalization equations will contain expectation values that must be calculated separately . in principle , these expectation values are defined with respect to @xmath4 because the factorization approximation was employed for the renormalization step that transformed @xmath4 to @xmath63 . however , @xmath4 still contains interactions that prevent a straight evaluation of required expectation values . the easiest way to circumvent this difficulty is to neglect the interactions and to use the diagonal unperturbed part @xmath8 instead of @xmath4 for the calculation of the expectation values . this approach has been successfully applied to the holstein model to investigate single - particle excitations and phonon softening @xcite . however , it turns out that often the interaction term in @xmath4 is crucial for a proper calculation of the required expectation values . thus , usually a more involved approximation has been used that neglects the @xmath10 dependence of the expectation values but includes interaction effects by calculating the expectation values with respect to the full hamiltonian @xmath3 instead of @xmath4 . in this case , the renormalization equations need to be solved in a self - consistent manner because they depend on expectation values defined with respect to the full hamiltonian @xmath3 which are not known from the very beginning but can be determined from the fully renormalized ( and diagonal ) hamiltonian @xmath72 . there exist two ways to calculate expectation values of the full hamiltonian from the renormalized hamiltonian . the first one is based on the free energy that can be calculated either from the original model @xmath3 or the renormalized hamiltonian @xmath73 , @xmath74 because @xmath73 is obtained from @xmath3 by unitary transformations . the desired expectation values can then be determined from the free energy by functional derivatives . this approach has advantages as long as the derivatives can be evaluated analytically as , for example , in refs . and . the second way to calculate expectation values of the full hamiltonian employs unitarity for any operator variable @xmath40 , @xmath75 where we defined @xmath76 . thus , additional renormalization equations need to be derived for the required operator variables @xmath77 where the same sequence of unitary transformations has to be applied to the operator variable @xmath40 as to the hamiltonian @xmath3 . in this subsection we want to illustrate the prm discussed above by considering an exactly solvable model , namely the fano - anderson model @xcite , @xmath78 the hamiltonian describes dispersion - less @xmath79 electrons interacting with conduction electrons where all correlation effects are neglected . @xmath80 denotes the wave vector , and the one - particle energies are measured with respect to the chemical potential . both types of electrons are assumed to have the same orbital index @xmath81 with values @xmath82 . the model is easily diagonalized , @xmath83 where @xmath84 and @xmath85 are given by linear combinations of the original fermionic operators @xmath86 and @xmath87 , @xmath88 @xmath89 here , we defined @xmath90 , and the eigenvalues of @xmath3 are given by @xmath91 in the following , we want to apply the prm as introduced above to the fano - anderson model where we mainly use the formulation of ref . @xcite . the goal is to integrate out the hybridization term @xmath1 so that we finally obtain an effectively free model . therefore , having in mind the exact solution of the model , we make the following renormalization ansatz : @xmath92 note that @xmath93 includes a cutoff function in order to ensure that the requirement @xmath94 is fulfilled . in the next step we want to eliminate excitations with energies within the energy shell between @xmath10 and @xmath95 by means of an unitary transformation similar to . by inspecting the perturbation expansion corresponding to subsection [ b_perturbation ] , the generator of the unitary transformation must have the following form : @xmath96 \label{b24}\end{aligned}\ ] ] where the parameters @xmath97 need to be properly determined so that eq . is fulfilled . to evaluate the transformation , we now consider the transformations of the operators appearing in the renormalization ansatz . for example , we obtain @xmath98 - 1 \right\ } \left ( c^{\dagger}_{\mathbf{k}m}c_{\mathbf{k}m } - f^{\dagger}_{\mathbf{k}m}f_{\mathbf{k}m } \right ) \\ & & + \ , \sin\left [ 2 a_{\mathbf{k}}(\lambda,\delta\lambda)\right ] \left ( f^{\dagger}_{\mathbf{k}m}c_{\mathbf{k}m } + c^{\dagger}_{\mathbf{k}m}f_{\mathbf{k}m } \right).\end{aligned}\ ] ] here it is important to notice that due to the fermionic anti - commutator relations the different @xmath80 are not coupled with each other . very similar transformations can also be found for @xmath99 and @xmath100 . inserting these transformations into leads to the following renormalization equations : @xmath101 - 1 \right\ } \left ( \varepsilon^{c}_{\mathbf{k},\lambda } - \varepsilon^{f}_{\mathbf{k},\lambda } \right ) \nonumber \\ & & + \ , v_{\mathbf{k},\lambda } \sin\left[2 a_{\mathbf{k } } ( \lambda,\delta\lambda)\right ] , \nonumber\end{aligned}\ ] ] @xmath102 now we need to determine the parameters @xmath97 . for this purpose we employ the condition : first , from @xmath103 we conclude @xmath104 , where we have defined @xmath105 . moreover , from @xmath106 we find @xmath107 \,=\ , } & & \\ & = & \left[1 - \theta_{\mathbf{k}(\lambda- \delta \lambda ) } \right ] \theta_{\mathbf{k}\lambda } \ , \frac{2v_{\mathbf{k},\lambda } } { \varepsilon^{f}_{\mathbf{k},\lambda } - \varepsilon^{c}_{\mathbf{k},\lambda } } \nonumber\end{aligned}\ ] ] which shows that also @xmath108 contains the cutoff factor @xmath109 . note that in the expression the low excitation - energy part of the generator was chosen to be zero @xmath71 . as one can see from eqs . - , the renormalization of the parameters of a given @xmath80 is _ not _ affected by other @xmath80 values . furthermore , it is important to notice that @xmath110 . consequently , each @xmath80 value is renormalized only once during the renormalization procedure eliminating excitations from large to small @xmath10 values . such a steplike renormalization allows an easy solution of the renormalization equations - where @xmath10 is replaced by the cutoff @xmath111 of the original model and we set @xmath112 . here , one needs to consider that the parameter @xmath113 changes its sign if the difference @xmath114 changes its sign . thus , we find the following renormalized hamiltonian @xmath115 \label{b28}\end{aligned}\ ] ] where the renormalized energies are given by @xmath116 % \label{b30 } \tilde{\varepsilon}_{\mathbf{k}}^{c } & = & \frac{\varepsilon_{f } + \varepsilon_{\mathbf{k}}}{2 } - \frac { \mathrm{sgn } ( \varepsilon_{f } -\varepsilon_{\mathbf{k } } ) } { 2 } w_{\mathbf{k}}.\end{aligned}\ ] ] the results of the renormalization and the diagonalization are completely comparable for physical accessible quantities like quasiparticle energies [ compare with eqs . and ] or expectation values . however , there is also an important difference between the two approaches : whereas the eigenmodes @xmath117 and @xmath118 of the diagonalized hamiltonian change there character as function of the wave vector @xmath80 [ compare and ] , the operators @xmath119 and @xmath120 of @xmath73 remain @xmath79-like and @xmath121-like for all @xmath80 values . in return , the quasi - particle energies @xmath122 and @xmath123 show a steplike behavior as function of @xmath80 at @xmath124 so that the deviations from the original one - particle energies @xmath125 and @xmath126 remain relatively small for all @xmath80 values . as already mentioned in subsection [ genut ] , the low - energetic excitations included in the generator @xmath127 of the unitary transformation can be chosen arbitrarily , i.e. @xmath128 is not determined by the condition . in the previous subsection an approach of `` minimal '' transformations has been applied to the fano - anderson model where @xmath128 is set to zero . however , in the following we want to demonstrate that it is also possible to take advantage of this freedom to choose the generator @xmath127 and to derive a continuous version of the prm . as it will turn out in sec . [ ep_flow ] the prm can also be connected to wegner s flow equation method @xcite . by allowing a nonzero part @xmath129 the generator @xmath127 of the unitary transformation can be written as follows @xmath130 here the part @xmath131 ensures that eq . , @xmath132 , is fulfilled . note however , one may also choose the remaining part @xmath128 in such a way that it almost completely integrates out all the interactions _ before _ the cutoff energy @xmath10 approaches their corresponding transition energies . as it will be discussed in sec . [ ep ] in more detail , the flow equation method @xcite and the prm ( in its minimal form ) take advantage of the freedom to chose the generator of the unitary transformation in a very different way . in the prm , the low transition - energy projection part of the generator , @xmath133 , is set to zero for convenience . the flow equation approach instead uses exactly this part to eliminate the interaction . even though the prm resembles the similarity transformation @xcite and wegner s flow equation method @xcite in some aspects there is an important difference : the latter two methods start from _ continuous _ transformations in differential form . this has the advantage that one can use available computer subroutines to solve the differential flow equations . in contrast , the prm is based on _ discrete _ transformations which lead to coupled difference equations . the advantage of the prm is to provides a direct link to perturbation theory ( as already discussed in subsection [ b_perturbation ] ) . moreover , the stepwise renormalization of the prm allows a unified treatment on both sides of a quantum phase transition ( see for example sec . [ holstein_qp ] ) which seems not to be possible in the flow equation method . however , as we show in the following the idea of continuous unitary transformations can also be implemented in the framework of the prm . now we want to demonstrate that the freedom in choosing the generator of the unitary transformation can be employed in order to derive a continuous renormalization scheme within the framework of the prm . as an example we again discuss the fano - anderson model . as already discussed , the part @xmath134 of the generator @xmath127 of the unitary transformation is not fixed by the prm . in the former treatment of the fano - anderson model in subsection [ b_fano - anderson ] we had chosen @xmath135 for simplicity . in the following we want to take advantage of this freedom in a different way . according to eq . , the generator of the fano - anderson model is given by @xmath136 where the most general form of @xmath97 can be written as @xmath137 \nonumber \\ & & + a''_{\mathbf{k}}(\lambda,\delta\lambda ) \ , \theta_{\mathbf{k } , \lambda } \theta_{\mathbf{k } , \lambda - \delta\lambda}.\end{aligned}\ ] ] here , the renormalization contributions related with @xmath134 and @xmath138 are described by the parameters @xmath139 and @xmath140 , respectively . a possible choice for @xmath139 is @xmath141^{2 } } \ , \delta\lambda . \end{aligned}\ ] ] of course , there is no derivation for eq . but it will turn out that this is indeed a reasonable choice . in particular we will show that in the limit of small @xmath142 a rapid decay for the hybridization @xmath93 is obtained in this way . thus , the part @xmath143 of the generator is not important anymore for the renormalization procedure and can be neglected in the following . in eq . , @xmath144 denotes an energy constant to ensure a dimensionless @xmath139 . note that @xmath139 is chosen proportional to @xmath142 to reduce the impact of the actual value of @xmath142 on the final results of the renormalization . in order to derive continuous renormalization equations note that the parameter @xmath145 is approximately proportional to @xmath68 . by neglecting the part @xmath143 of the generator one can rewrite eqs . and in the limit @xmath146 @xmath147 where higher order terms have been neglected . furthermore , we defined @xmath148^{2 } } \nonumber .\end{aligned}\ ] ] a similar equation can also be derived for @xmath149 , @xmath150 to solve these equations we rewrite , @xmath151 and insert into . using @xmath152 we obtain @xmath153 eq . is easily integrated and leads to a quadratic equation for @xmath154 which corresponds to the former result . moreover , @xmath155 is found from @xmath156 . according to and the @xmath10-dependence of @xmath149 is governed by @xmath157 ^ 2 } \ , \theta(\lambda -|\varepsilon^f_{{\mathbf k},\lambda } -\varepsilon^c_{{\mathbf k},\lambda } | ) \nonumber \\ \label{b30e2 } & & \end{aligned}\ ] ] as one can easily see from eq . , 1 . the interaction @xmath158 is always renormalized to smaller values when the cutoff energy @xmath10 is lowered , 2 . and at @xmath159 the renormalized coupling @xmath158 vanishes , i.e. it has completely integrated out by the present choice of the generator @xmath160 . the classical bcs - theory @xcite is essentially based on attractive electron - electron interactions @xcite . it is well - known that such an interaction can be mediated via phonons coupled to the electronic system @xcite . in this section we want to revisit this problem because it has been studied @xcite by wegner s flow equation method @xcite , by a similarity transformation proposed by gazek and wilson @xcite , and by the prm @xcite . therefore , the electron - phonon interaction is a perfectly suited test case to discuss differences and similarities of the three methods . in this section we consider the following hamiltonian @xmath161 \nonumber\end{aligned}\ ] ] which describes electrons @xmath162 and phonons @xmath163 that interact with each other . in the following we apply a slightly modified version of the prm to the electron - phonon problem in order to derive an effective electron - electron interaction . it turns out that frhlich s transformation @xcite is re - examined in this way . in [ ep_flow ] the approach is modified in the spirit of the ideas developed in subsections [ b_generalize_g ] and [ fa_rev ] . thus , allowing a more continuous renormalization of the electron - phonon interaction we derive the result of ref . obtained by the flow equation method . in subsection [ ep_bcs ] a much more sophisticated scheme is introduced by adding a symmetry breaking field to the hamiltonian so that a gap equation can be derived . the effective electron - electron interaction is then obtained by comparing with the famous bcs - gap equation . the strategy to introduce symmetry breaking fields turns out to be of general importance for the investigation of phase transitions within the prm . finally , the different results for the electron - phonon interaction are discussed in subsection [ ep_disc ] . in this subsection we want to apply the prm to the electron - phonon problem in order to derive an effective electron - electron interaction . here , we start from the renormalization ansatz , @xmath164 % \mathcal{h}_{0 } & = & \sum_{\mathbf{k},\sigma } \varepsilon_{\mathbf{k } } \ , c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k}\sigma } + \sum_{\mathbf{q } } \omega_{\mathbf{q } } \ , b_{\mathbf{q}}^{\dagger}b_{\mathbf{q } } , \nonumber\end{aligned}\ ] ] @xmath165 % \mathcal{h}_{1,\lambda}^{\mathrm{el , ph } } & = & \sum_{\mathbf{k } , \mathbf{q } , \sigma}\left [ g_{\mathbf{k } , \mathbf{q},\lambda } \ , b_{-\mathbf{q}}^{\dagger } \right . \nonumber\\ & & \qquad \left . \phantom{b_{-\mathbf{q}}^{\dagger}}+\ , g_{\mathbf{k}+\mathbf{q } , -\mathbf{q } , \lambda } \ , b_{\mathbf{q } } \right ] c_{(\mathbf{k}+\mathbf{q})\sigma}^{\dagger}c_{\mathbf{k}\sigma } , \nonumber \\[1ex ] % \mathcal{h}_{1,\lambda}^{\mathrm{el , el } } & = & \sum_{\mathbf{k } , \sigma , \mathbf{k ' } , \sigma ' , \mathbf{q } } v_{\mathbf{k } , \mathbf{k ' } , \mathbf{q } , \lambda } \ , c_{(\mathbf{k}+\mathbf{q})\sigma}^{\dagger } c_{(\mathbf{k'}-\mathbf{q})\sigma'}^{\dagger } c_{\mathbf{k'}\sigma'}c_{\mathbf{k}\sigma } , \nonumber\end{aligned}\ ] ] that was also used in ref . where the flow equation method was applied to the same system note that the parameters of @xmath166 contain a cutoff function in order to ensure that only transitions with energies smaller than @xmath10 are included . the parameters of @xmath4 depend on the energy cutoff @xmath10 because all transitions with energies larger than @xmath10 have already been integrated out . however , we shall restrict ourselves to the second order renormalization contributions to @xmath166 . therefore , @xmath2 is assumed to be @xmath10 independent . in the following we want to integrate out all transitions which create or annihilate phonons , however keeping all electronic transitions . therefore , the present calculation differs from the previous ones where all parts of the unperturbed hamiltonian @xmath167 were subject to the renormalization procedure . as it turns out , the electron - phonon coupling will be replaced by an effective electron - electron interaction . however , the final hamiltonian containing the electron - electron interaction is not diagonal any more as required for the standard prm . instead , we want to derive a block - diagonal hamiltonian so that the renormalization approach has to be modified . for this purpose , we define projection operators @xmath168 and @xmath169 that are defined with respect to the phonon part of the unperturbed hamiltonian @xmath2 . these new projectors now replace those of the full unperturbed hamiltonian . thus , from @xmath170 we conclude @xmath171 , where we have defined @xmath172 . moreover , following ref . , the generated electron - electron interaction @xmath173 is not considered in determining the generator of the unitary transformation . thus , the generator can be written as @xmath174 \nonumber\end{aligned}\ ] ] where the parameter @xmath175 needs to be properly determined in the following : corresponding to , @xmath176 must be fulfilled . as already discussed , the part @xmath177 of the generator of the unitary transformation is not fixed by the prm . thus , the parameters @xmath178 have the following general form @xmath179 \nonumber \\ & & + \ , a''_{\mathbf{k},\mathbf{q}}(\lambda,\delta\lambda ) \ , \theta_{\mathbf{q } , \lambda } \theta_{\mathbf{q } , \lambda - \delta\lambda}.\end{aligned}\ ] ] note that both parts of @xmath180 include the factor @xmath181 . however , in the following @xmath128 and @xmath182 are set to zero for simplicity . note that a different choices for @xmath183 will be used in the subsequent subsection . we restrict ourselves to second order renormalization contributions so that the unitary transformation can easily be evaluated where operator terms are only kept if they are included in the ansatz . thus , we directly obtain difference equation for the electron - phonon coupling , @xmath184 \ , a_{\mathbf{k}+\mathbf{q},-\mathbf{q}}(\lambda , \delta\lambda ) , \nonumber \end{aligned}\ ] ] and for the effective electron - electron interaction , @xmath185 because we have set @xmath186 , renormalization contributions only appear if the phonon energy @xmath187 is in the energy shell between @xmath188 and @xmath10 . consequently , we find a step - like renormalization of the electron - phonon coupling @xmath189 and the generated electron - electron interaction @xmath190 . the parameter @xmath175 defined in has to be chosen in such a way that @xmath191 . from equation we obtain @xmath192 . \nonumber \\ & & \end{aligned}\ ] ] as one can see by inserting eq . into , the electron - phonon coupling has no @xmath193-dependence in the present approximation , i.e. @xmath194 . now we insert eq . into the renormalization equation and consider the limit @xmath195 , @xmath196 where we exactly find frhlich s result @xcite . wegner s flow equation method @xcite was applied to the electron - phonon system in ref . where a renormalization ansatz similar to was used . however , a less singular expression for the effective electron - electron interaction could be derived in this way . in the following we want to analyze how this different result can be understood in the framework of the prm . in order to derive continuous renormalization equations the part @xmath177 of the generator of the unitary transformation is chosen to be non - zero so that now @xmath183 needs to be considered in eq . . furthermore , @xmath197 can be neglected if @xmath183 leads to a rapid decay of the interaction terms . thus , neglecting @xmath197 and employing the limit @xmath198 we obtain from eqs . and @xmath199 \ , { \alpha}_{\mathbf{k},\mathbf{q},\lambda},\\ % % \label{ep13 } \frac{\mathrm{d}}{\mathrm{d}\lambda } v_{\mathbf{k } , \mathbf{k ' } , \mathbf{q } , \lambda } & = & g_{\mathbf{k}+\mathbf{q},-\mathbf{q},\lambda } \ , { \alpha}_{\mathbf{k'},-\mathbf{q},\lambda } \\ & & + \ , g_{\mathbf{k},\mathbf{q},\lambda } \ , { \alpha}_{\mathbf{k ' } + \mathbf{q},\mathbf{q},\lambda}. \nonumber\end{aligned}\ ] ] here , we introduced @xmath200 . again the parameter @xmath201 is chosen proportional to @xmath142 so that the third and the fourth term on the right side of eq . can be neglected in the limit @xmath198 . the commonly used generator of the flow equation method is chosen in such a way that the matrix elements of the interaction , which shall be integrated out , show an exponential decay with respect to the flow parameter . consequently , _ all _ matrix elements change continuously during the renormalization procedure . we adapt the idea of such a continuous renormalization and assume an exponential decay for the electron - phonon interaction , @xmath202 where @xmath144 is just a constant to ensure a dimensionless exponent . note that ansatz is inspired by the results of ref . . of course , eq . is only useful as long as the considered renormalization contributions are restricted to second order in the original electron - phonon interaction . note also that ansatz meets the basic requirement of the prm , @xmath203 . now we need to determine the parameter @xmath204 of the unitary transformation . for this purpose , eq . is divided by @xmath189 and integrated between the cutoff @xmath205 and @xmath206 by using eq . . we find @xmath207 } { \kappa \left ( \lambda - \omega_{\mathbf{q}}\right)^{2}}. \end{aligned}\ ] ] note that this result is equivalent to the choice for @xmath183 used for the fano - anderson model in [ fa_rev ] [ compare with equations and ] . using this solution and the ansatz for the electron - phonon coupling @xmath189 , eq . is easily integrated where the constant @xmath144 is canceled . thus , the renormalized values @xmath208 can be obtained and reads @xmath209 this is the final version of the effective electron - electron interaction after eliminating the electron - phonon interaction . obviously , differs from frhlich s result @xcite that had been derived above . however , eq . coincides with the result of ref . that had been obtained by wegner s flow equation method @xcite . at this point it is important to notice that the approaches of [ ep_froehlich ] and [ ep_flow ] are based on the same renormalization ansatz . therefore , the different results are only caused by different choices for the generator . due to the continuous renormalization , the electron - phonon coupling becomes dependent on the electronic one - particle energies @xmath126 so that the approach of [ ep_flow ] involves more degrees of freedom . the main goal of this subsection was to demonstrate that wegner s flow equation method @xcite can be understood within the prm@xcite , as already for the case of the fano - anderson model in the previous section . however , the idea of a continuous renormalization , as implemented here , can also be very useful for other applications . in this regards , the discussion line needs to be changed : one starts from an ansatz for the generator @xmath65 of the unitary transformation similar to eqs . , , and demonstrates _ afterwards _ that the interaction decays as function of @xmath10 as required . so far the discussion of the electron - phonon problem was focused on the phonon - induced electron - electron interaction . thus , we derived block - diagonal hamiltonians with constant phonon occupation numbers within each block . however , in the following we want to tackle the electron - phonon problem in a different way because an effective phonon mediated electron - electron interaction is mainly discussed with respect to superconductivity . the idea is to obtain the superconducting properties directly from the electron - phonon system . the goal is again to decouple the electron and the phonon system but now we want to derive a truly diagonal renormalized hamiltonian . for this purpose the prm shall be applied to the electron - phonon system in conjunction with a bogoliubov transformation @xcite as it was done in ref . . whereas the hamiltonian is gauge invariant , a bcs - like hamiltonian breaks this symmetry @xcite . therefore , in order to describe superconducting properties , the renormalized hamiltonian should contain a symmetry breaking field as well so that the renormalization ansatz reads @xmath210 % % \mathcal{h}_{0,\lambda } & = & \sum_{\mathbf{k},\sigma } \varepsilon_{\mathbf{k } } \ , c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k}\sigma } + \sum_{\mathbf{q } } \omega_{\mathbf{q } } \ , b_{\mathbf{q}}^{\dagger}b_{\mathbf{q } } \nonumber\\ & & -\ , \sum_{\bf k } \left ( \delta_{{\bf k},\lambda } \ , c_{{\bf k}\uparrow}^{\dagger } c_{-{\bf k}\downarrow}^{\dagger } + \delta_{{\bf k},\lambda}^ { * } \ , c_{-{\bf k}\downarrow } c_{{\bf k}\uparrow } \right ) + c_{\lambda } , \nonumber\\[1ex ] % % { \cal h}_{1,\lambda } & = & { \bf p}_{\lambda } \sum_{{\bf k},{\bf q},\sigma } g_{\bf q } \left [ c_{{\bf k}\sigma}^{\dagger } c_{({\bf k}+{\bf q})\sigma } b_{\bf q}^{\dagger } + c_{({\bf k}+{\bf q})\sigma}^{\dagger } c_{{\bf k}\sigma } b_{\bf q } \right ] . \nonumber\end{aligned}\ ] ] here , the fields @xmath211 and @xmath212 break the gauge invariance and can be interpreted as the superconducting gap function . the initial values for @xmath211 and the energy shift @xmath213 are given by those of the original model , @xmath214 , @xmath215 . note that in the following the projectors @xmath17 and @xmath18 are defined as usual with respect to @xmath8 and not only to the phonon part . furthermore , renormalization contributions to electronic and phononic one - particle energies and to the electron - phonon coupling will be neglected for simplicity . at this point it is important to realize that the introduction of symmetry breaking fields is a general concept to study phase transitions within the prm . the same approach has also been successfully applied to the holstein model and its quantum phase transition @xcite ; this model will be discussed in sec . [ holstein_qp ] . to perform our renormalization scheme as introduced in section [ prm1 ] we need to solve the eigenvalue problem of @xmath8 . for this purpose we utilize the well - known bogoliubov transformation @xcite and introduce new @xmath10 dependent fermionic operators , @xmath216 where the coefficients read @xmath217 hence , @xmath8 can be rewritten in diagonal form , @xmath218 where the fermionic excitation energies are given by @xmath219 . in the following , we restrict ourselves to second order renormalization contributions so that the first order of the generator @xmath65 of the unitary transformation is sufficient [ see eq . and the discussion in [ b_perturbation ] ] . thus , @xmath65 can be written as , @xmath220 \nonumber\end{aligned}\ ] ] where @xmath221 % \theta_{\mathbf{k},\mathbf{q}}(\lambda , \delta\lambda ) & = & \left [ 1 - \theta\left ( \lambda - \delta\lambda - \left| \varepsilon_{\mathbf{k } } - \varepsilon_{\mathbf{k}+\mathbf{q } } + \omega_{\mathbf{q } } \right| \right ) \right ] \nonumber \\ & & \times \theta\left ( \lambda - \left| \varepsilon_{\mathbf{k } } - \varepsilon_{\mathbf{k}+\mathbf{q } } + \omega_{\mathbf{q } } \right| \right ) . \nonumber\end{aligned}\ ] ] note that the generator @xmath222 as defined in eq . almost completely agrees with the one used to re - examine frhlich s transformation in subsection [ ep_froehlich ] [ see eqs . and ] . however , now the @xmath223 functions do not only refer to the phonon energies @xmath187 but also to the electronic one - particle energies @xmath126 because of the different definitions of the @xmath17 projection operators . to perform the renormalization step reducing the cutoff from @xmath10 to @xmath53 , one would need to express the electronic creation and annihilation operators by the quasi - particle operators . after considering the renormalization contributions , the quasi - particle operators have to be transformed back to the original electron operators . however , this involved procedure is only necessary if we are interested in renormalization contributions beyond second order perturbation theory . therefore , here the symmetry breaking fields @xmath224 and @xmath225 are only generated by the renormalization scheme but not considered in the evaluation of energy denominators or projection operators . taking into account all simplifications related with second order perturbation theory , the unitary transformation is easily evaluated where generated operator terms are only kept if their mean - field approximations renormalize the symmetry breaking fields , @xmath224 and @xmath225 , or the energy shift , @xmath213 . thus , for sufficiently small steps @xmath142 we obtain the following renormalization equations @xmath226 \nonumber\\ & & \times \ , \left\ { 1 - \theta\left [ \lambda - \delta\lambda - \left| \varepsilon_{\bf k } - \varepsilon_{({\bf k}+{\bf q } ) } \right| + \omega_{\bf q } \right ] \right\ } \nonumber \\ & & \times \ , \frac { \left| g_{\bf q } \right|^{2 } \theta\left [ \omega_{\bf q } - \left| \varepsilon_{\bf k } - \varepsilon_{({\bf k}+{\bf q } ) } \right| \right ] } { \left| \varepsilon_{\bf k } - \varepsilon_{({\bf k}+{\bf q } ) } \right| + \omega_{\bf q } } \left\langle c_{-({\bf k}+{\bf q}),\downarrow } c_{({\bf k}+{\bf q}),\uparrow } \right\rangle,\nonumber % % \end{aligned}\ ] ] @xmath227 . \nonumber \\[-1ex ] \label{ep26 } & & \end{aligned}\ ] ] by summing up all difference equations between the cutoff @xmath111 of the original model and the lower cutoff @xmath195 , one easily finds @xmath228 } { \left| \varepsilon_{\bf k } - \varepsilon_{({\bf k}+{\bf q } ) } \right| + \omega_{\bf q } } \nonumber \\ \label{ep27 } & & \qquad\times\ , \left\langle c_{-({\bf k}+{\bf q}),\downarrow } c_{({\bf k}+{\bf q}),\uparrow } \right\rangle , \\[2ex ] % \label{ep28 } \tilde c & = & c_{\lambda } + \sum_{\bf k } \left\langle c_{{\bf k},\uparrow}^{\dagger } c_{-{\bf k},\downarrow}^{\dagger } \right\rangle \left ( \tilde\delta_{\bf k } - \delta_{{\bf k},\lambda } \right).\end{aligned}\ ] ] here we defined @xmath229 , @xmath230 . the final hamiltonian @xmath231 can easily be diagonalized by a bogoliubov transformation and reads according @xmath232 where @xmath233 , @xmath234 , and @xmath235 . its parameters depend on the original system , on the initial conditions , @xmath214 , @xmath215 , and on expectation values @xmath236 that need to be determined self - consistently . following the approach of ref . , we consider the free energy which can be calculated either from @xmath3 or from the renormalized hamiltonian @xmath73 . thus , the required expectation values are easily found by functional derivatives , @xmath237 , so that eq . can be rewritten as @xmath238 } { \left| \varepsilon_{\bf k } - \varepsilon_{({\bf k}+{\bf q } ) } \right| + \omega_{\bf q } } \right\ } \\ & & \qquad \times\ , \frac { \tilde{\delta}_{{\bf k}+{\bf q}}^ { * } \left [ 1 - 2f ( \tilde{e}_{{\bf k}+{\bf q } } ) \right ] } { 2\sqrt { \varepsilon_{{\bf k}+{\bf q}}^{2 } + \left| \tilde \delta_{{\bf k}+{\bf q } } \right|^{2 } } } \nonumber\end{aligned}\ ] ] where the initial condition @xmath214 has been used . eq . has the form of the famous bcs - gap equation so that the term inside the braces @xmath239 can be interpreted as parameter of the effective phonon induced electron - electron interaction , @xmath240 } { \left| \varepsilon_{\bf k } - \varepsilon_{({\bf k}+{\bf q } ) } \right| + \omega_{\bf q } } \end{aligned}\ ] ] which is responsible for the formation of cooper pairs . even though we have here derived an effective electron - electron interaction as well there is a significant difference to the approaches of [ ep_froehlich ] and [ ep_flow ] : in the present formalism both the attractive electron - electron interaction and the superconducting gap function were derived in _ one step _ by applying the prm to the electron - phonon system with additional symmetry breaking fields . in the following we want to discuss the different approaches to the phonon - induced effective electron - electron interaction in more detail . at first we summarize the results derived above where we focus on the interaction between electrons of a cooper pair . frhlich s classical result [ see ref . and eq . ] reads @xmath241 however , there is an important problem related with eq . : it diverges at @xmath242 . thus , a cutoff function is introduced by hand in the classical bcs - theory to suppress repulsive contributions to the effective electron - electron interaction . in contrast to the frhlich interaction , the results obtained by wegner s flow equation method @xcite , by similarity transformation @xcite , and by the prm @xcite are less singular , @xmath243 % % v_{\mathbf{k } , -\mathbf{k } , \mathbf{q } , \lambda}^{\mbox{\tiny mielke } } & = & -\ , \frac { \left| g_{\mathbf{q } } \right|^{2 } \theta\left ( \left| \varepsilon_{\mathbf{k}+\mathbf{q } } - \varepsilon_{\mathbf{k } } \right| + \omega_{\mathbf{q } } - \lambda \right ) } { \left| \varepsilon_{\mathbf{k}+\mathbf{q } } - \varepsilon_{\mathbf{k } } \right| + \omega_{\mathbf{q } } } , \nonumber\\[-1ex ] \label{ep34 } & & \\[1ex ] % % \label{ep35 } v_{\mathbf{k},-\mathbf{k},\mathbf{q},\lambda}^ { \mbox{\tiny h\"{u}bsch / becker } } & = & -\ , \frac { \left| g_{\mathbf{q } } \right|^{2 } \theta\left ( \omega_{\mathbf{q } } - \left| \varepsilon_{\mathbf{k}+\mathbf{q } } - \varepsilon_{\mathbf{k } } \right| \right ) } { \left| \varepsilon_{\mathbf{k}+\mathbf{q } } - \varepsilon_{\mathbf{k } } \right| + \omega_{\mathbf{q } } } .\end{aligned}\ ] ] ( note that eqs . and have already been derived above , compare with and . the @xmath10 dependence of the electronic and phononic one - particle energies are suppressed in for simplicity . ) all three results for the effective phonon - mediated electron - electron interaction are never repulsive as long as @xmath244 is fulfilled . at first we want to discuss mielke s result @xcite , an effective electron - electron interaction that depends on the energy cutoff @xmath10 . as wegner s flow equation method @xcite , the used similarity transformation @xcite is based on continuous unitary transformations and leads to differential equations for the parameters of the hamiltonian . however , like the prm , the similarity transformation leads to a band - diagonal structure of the renormalized hamiltonian with respect to the eigenenergies of the unperturbed hamiltonian whereas the flow equation method generates block - diagonal hamiltonians . mielke derived the phonon - mediated electron - electron interaction by eliminating excitations with energies larger than @xmath10 where excitation energies are measured with respect to the unperturbed hamiltonian consisting of both electronic and bosonic degrees of freedom . the obtained effective interaction becomes @xmath10 independent for the einstein model ( of dispersion - less phonons ) if @xmath10 is chosen smaller than the phonon frequency @xmath245 . for this case mielke s result is very similar to ours obtained by the prm with symmetry - breaking fields . however , in contrast to our result , the cutoff function @xmath246 is absent in . this difference might be related with different choices for the generator of the unitary transformation in the two methods but could also be caused by a systematic problem in mielke s approach : setting @xmath247 , the final renormalized hamiltonian contains non - diagonal terms with respect to the used unperturbed hamiltonian . this seems to contradict a basic premise of the similarity transformation . lenz and wegner @xcite applied the flow equation method to the electron - phonon problem as discussed here and obtained an effective electron - electron interaction as shown in eq . . as one can see in fig . [ fig_bcs ] , their result is quite similar to ours derived using the prm as long as @xmath248 is fulfilled . however , in contrast to our result , the interaction remains finite even for @xmath249 . probably , this difference is caused by the different choices for the generator of the unitary transformation that also require different approximations in order to obtain closed sets of renormalization equations . the periodic anderson model ( pam ) is considered to be the basic microscopic model for the theoretical investigation of heavy - fermion ( hf ) systems @xcite . it describes localized , strongly correlated @xmath79 electrons interacting with itinerant conduction electrons . here we focus on the limit of infinitely large coulomb repulsion on @xmath79 sites so that the hamiltonian of the pam can be written as @xmath250 @xmath251 the one - particle energies @xmath125 and @xmath252 , and , as a simplification , both types of electrons have the same angular momentum index @xmath253 . the hubbard operators , @xmath254 take into account the infinitely large local coulomb repulsion and only allow either empty or singly occupied @xmath79 sites . the prm has already been applied to the pam in ref . where approximations have been employed that allow to map the renormalization equations of the pam onto those of the uncorrelated fano - anderson model ( see subsection [ b_fano - anderson ] ) . thus , hf behavior and a possible valence transition between mixed and integral valent states could be studied . however , the approach of refs . has a significant disadvantage : the renormalization of the one - particle energies show as function the cutoff @xmath10 a steplike behavior that leads to serious problems in the ( numerical ) evaluation . therefore , a constant renormalized @xmath79 energy had to be chosen for all values of the energy cutoff @xmath10 to ensure a continuous behavior of the one - particle energies as required for physical reasons . in the following we modify the approach of refs . to ensure a more continuous renormalization of all parameters of the hamiltonian . for this purpose , the ideas of [ fa_rev ] and [ ep_flow ] are transferred to the pam . however , to explore all features of this continuous approach is beyond the scope of this review , we re - derive the analytical solution of ref . instead . much of the physics of the pam can be understood in terms of an effective uncorrelated model that consists of two non - interacting fermionic quasi - particle bands . various theoretical approaches have been used to generate such effective hamiltonians ; the most popular among them is the slave - boson mean - field ( sb ) theory @xcite . however , as discussed in ref . , such approaches _ do not prevent _ from unphysical multiple occupation of @xmath79 sites and are therefore restricted to heavy - fermion like solutions . [ the sb solutions break down if the original @xmath79 level @xmath125 is located too far below the fermi level or if the hybridization between @xmath79 and conduction electrons becomes too weak @xcite . ] to reliably prevent the system from unphysical states with multiple occupations of @xmath79 sites we here follow ref . and start from a renormalization ansatz that keeps the hubbard operators during the whole renormalization procedure , @xmath255 % \mathcal{h}_{0,\lambda } & = & e_{f,\lambda } \sum_{\mathbf{k},m } \hat{f}^{\dagger}_{\mathbf{k}m } \hat{f}_{\mathbf{k}m } + \sum_{\mathbf{k},m } \delta_{\mathbf{k},\lambda } \left ( \hat{f}^{\dagger } _ { \mathbf{k}m } \hat{f}_{\mathbf{k}m } \right)_{\mathrm{nl } } \nonumber \\ & & + \sum _ { { \bf k},m } \varepsilon_{{\bf k},\lambda } \ c^{\dagger}_{{\bf k}m } c_{{\bf k}m } + e_{\lambda } , \nonumber \\[1ex ] % \mathcal{h}_{1,\lambda } & = & \mathbf{p}_{\lambda } \mathcal{h}_{1,\lambda } \,=\ , \sum_{\mathbf{k},m } v_{\mathbf{k},\lambda } \ \left ( \hat{f}^{\dagger}_{\mathbf{k}m } c_{{\bf k}m } + \mathrm{h.c . } \right ) . \nonumber\end{aligned}\ ] ] eq . is obtained after all excitations between eigenstates of @xmath5 with transition energies larger than the cutoff @xmath10 have been eliminated , i.e. @xmath256 holds . furthermore , we introduced fourier transformed hubbard operators , @xmath257 the @xmath10 dependencies of the parameters are caused by the renormalization procedure . note that @xmath93 includes a cutoff function in order to ensure that the requirement @xmath94 is fulfilled . furthermore , an additional energy shift @xmath258 and direct hopping between @xmath79 sites , @xmath259 have been generated . finally , we need the initial parameter values of the original model ( with cutoff @xmath111 ) to fully determine the renormalization , @xmath260 to implement our prm scheme we also need the commutator of the unperturbed part @xmath8 of the @xmath10 dependent hamiltonian @xmath4 with the interaction @xmath166 ( in the present case the hybridization between @xmath79 and conduction electrons ) . to shorten the notation we here introduce the ( unperturbed ) liouville operator @xmath261 that is defined as @xmath262 $ ] for any operator @xmath40 . because of the correlations included in the hubbard operators @xmath263 , the required commutator relation can not be calculated exactly and additional approximations are necessary . here , the one - particle operators @xmath263 and @xmath120 are considered as approximative eigenoperators of @xmath261 so that we obtain @xmath264 \label{pam4 } & & \end{aligned}\ ] ] here we introduced the local @xmath79 energy , @xmath265 the averaged @xmath79 dispersion , @xmath266 , and defined @xmath267 . note that the factors @xmath268 in eqs . and are caused by the hubbard operators @xmath263 where a factorization approximation has been employed . to ensure that @xmath269 is fulfilled by , the hybridization matrix elements must include an additional @xmath223-function , @xmath270 , where we have defined @xmath271 in order to derive the renormalization equations for the parameters of @xmath4 we have to consider the unitary transformation to eliminate excitations within the energy shell between @xmath53 and @xmath10 . corresponding to eq . , such a unitary transformation is determined by its generator @xmath65 . as in ref . we use an ansatz that is motivated by perturbation theory [ see eq . ] , @xmath272 \label{pam6 } & & \end{aligned}\ ] ] the parameter @xmath97 of the generator @xmath273 needs to be chosen in such a way that eq . , @xmath274 , is fulfilled . however , as already discussed before , this requirement only determines the part @xmath138 of the generator of the unitary transformation whereas @xmath134 can be chosen arbitrarily . thus , @xmath275 is usually chosen to perform the minimal transformation to match the requirement . in this way , the impact of approximations necessary for every renormalization step can be minimized . on the other hand , the approach of `` minimal '' approximations can also lead to some problems if a steplike renormalization behavior for the parameter of the hamiltonian is found . this is the case for the prm approach of refs . where a constant renormalized @xmath79 energy @xmath276 have been used for all cutoff values @xmath10 to ensure a continuous behavior of the one - particle energies as required for physical reasons . therefore , in the following @xmath134 shall again be chosen non - zero in order to ensure a more continuous renormalization of all parameters of the hamiltonian . in close analogy to subsection [ fa_rev ] , we choose a proper generator @xmath277 , not yet specified , which almost completely integrates out interactions _ before _ the cutoff energy @xmath10 approaches their corresponding transition energies . in the limit of small @xmath142 , we again expect an exponential decay for the hybridization @xmath93 in this way . in comparison to the approach of refs . , the derivation of the renormalization equation is simplified : having in mind @xmath278 , where @xmath68 is a small quantity , we can restrict ourselves to first order renormalization contributions and neglect the @xmath140 part of @xmath127 altogether . thus , eliminating excitations within the energy shell between @xmath53 and @xmath10 , the renormalized hamiltonian @xmath63 can be calculated based on eq . . to derive the renormalization equations for the parameters of the hamiltonian , we compare the coefficients of the different operator terms in the renormalization ansatz at cutoff @xmath53 and in the explicitly evaluated eq . . thus , based on similar approximations as the approach of refs . and , we obtain the following equations : @xmath279,\end{aligned}\ ] ] @xmath280 \nonumber \\ & & \qquad \times \ , \left\ { 1 + \left ( \nu_{f } - 1 \right ) \left\langle c_{\mathbf{k}m}^{\dagger } c_{\mathbf{k}m } \right\rangle \right\ } \nonumber \\ & & + \ , \frac{\nu_{f } - 1}{n } \sum_{k } \theta\left ( \mathbf{k } , \lambda - \delta\lambda \right ) \ , a''_{\mathbf{k}}(\lambda,\delta\lambda ) \ , \left ( \delta_{\mathbf{k},\lambda } - \bar\delta_{\lambda } \right ) \nonumber \\ & & \qquad \times \ , \left\langle \hat{f}_{\mathbf{k}m}^{\dagger } c_{\mathbf{k}m } + \mathrm{h.c . } \right\rangle , \nonumber\end{aligned}\ ] ] @xmath281 \nonumber\end{aligned}\ ] ] @xmath282 \\ & & - \ , \frac{\langle \hat{n}_{i}^{f } \rangle}{d } \sum_{\mathbf{k } } \left [ \varepsilon_{\mathbf{k},\lambda-\delta\lambda } - \varepsilon_{\mathbf{k},\lambda } \right ] . \nonumber\end{aligned}\ ] ] here , the condition @xmath283 has to be fulfilled . note that higher order terms in these equations have been evaluated in refs . and for the case that the generator @xmath127 was fixed by @xmath284 . in deriving the renormalization equations - a factorization approximation has been employed in order to trace back all terms to operators appearing in the renormalization ansatz . thus , the renormalization equations still depend on expectation values which have to be determined simultaneously . following the approach of ref . , we neglect the @xmath10 dependency of all expectation values and calculate them with respect to the full hamiltonian @xmath3 . as discussed in subsection [ genut ] , there are two strategies to obtain such expectation values : the first one is based on the free energy which we will use later for the analytical solution in [ pam_analytical ] . however , the evaluation of the free energy is complicated as long as the renormalized hamiltonian contains hubbard operators @xmath285 . thus , here it would be more convenient to use the second strategy to calculate expectation values and to derive renormalization equations for additional operator expressions ( see refs . and for more details ) . however , such involved approach is only needed in case of a numerical treatment of the renormalization equations which will be discussed below . the further calculations can be simplified by considering the limit @xmath198 and to transform the difference equations - into differential equations . for this purpose we define @xmath286 so that we obtain @xmath287 % % \label{pam14 } \frac{\mathrm{d}\delta_{\mathbf{k},\lambda}}{\mathrm{d}\lambda } & = & - \ , \frac{1}{d } \frac{\mathrm{d}\varepsilon_{\mathbf{k},\lambda}}{\mathrm{d}\lambda } \\[1ex ] % % \frac{\mathrm{d}e_{f,\lambda}}{\mathrm{d}\lambda } & = & -\ , \frac{1}{d}\frac{1}{n}\sum_{\mathbf{k } } \left\ { 1 + \left ( \nu_{f } - 1 \right ) \left\langle c_{\mathbf{k}m}^{\dagger } c_{\mathbf{k}m } \right\rangle \right\ } \frac{\mathrm{d}\varepsilon_{\mathbf{k},\lambda}}{\mathrm{d}\lambda } , \nonumber \\ & & -\ , \frac{\nu_{f } - 1}{n } \sum_{k } \theta ( \mathbf{k } , \lambda ) \ , \alpha_{\mathbf{k}}(\lambda ) \ , \left ( \delta_{\mathbf{k},\lambda } - \bar\delta_{\lambda } \right ) \nonumber \\ \label{pam15 } & & \qquad \times \ , \left\langle \hat{f}_{\mathbf{k}m}^{\dagger } c_{\mathbf{k}m } + \mathrm{h.c . } \right\rangle , \\[1ex ] % % \label{pam16 } \frac{\mathrm{d}v_{\mathbf{k},\lambda}}{\mathrm{d}\lambda } & = & \left [ e_{f,\lambda } + d\left ( \delta_{\mathbf{k},\lambda } - \bar{\delta}_{\lambda } \right ) - \varepsilon_{\mathbf{k},\lambda } \right ] \ , \alpha_{\mathbf{k}}(\lambda ) , \\[1ex ] % % \label{pam17 } \frac{\mathrm{d}e_{\lambda}}{\mathrm{d}\lambda } & = & -\ , n \langle \hat{n}_{i}^{f } \rangle \frac{\mathrm{d}e_{f,\lambda}}{\mathrm{d}\lambda } - \frac{\langle \hat{n}_{i}^{f } \rangle}{d } \sum_{\mathbf{k } } \frac{\mathrm{d}\varepsilon_{\mathbf{k},\lambda}}{\mathrm{d}\lambda } .\end{aligned}\ ] ] in the following , we concentrate on an analytical solution of the renormalization equations - by assuming a @xmath10 independent energy of the @xmath79 electrons . the aim is to demonstrate that the analytical solution of ref . can also be derived from the renormalization equations - or likewise - obtained here . in particular , we want to derive an analytical solution that describes hf behavior . as in ref . , we use the following approximations : 1 . all expectation values ( which appear due to the employed factorization approximation ) are considered as independent from the renormalization parameter @xmath10 and are calculated with respect to the full hamiltonian @xmath3 . 2 . as mentioned , the @xmath10 dependence of the renormalized @xmath79 level is neglected and we approximate @xmath288 to decouple the renormalization of the different @xmath80 values . note that such a renormalized @xmath79 energy is also used from the very beginning in the sb theory . 3 . to obtain the analytical solution of ref . we set @xmath289 for further simplification . the hubbard operators are replaced by usual fermionic operators where we employ @xmath290 thus , on a mean - field level , the system is prevented from generating unphysical states but a multiple occupation of @xmath79 sites is _ not _ completely suppressed by this approximation . therefore , we can only obtain useful results as long as only very few @xmath79 type states below the fermi level are occupied . it turns out that the analytical solution of ref . is obtained if the approximations ( i)-(iii ) are applied to the renormalization equations - . employing approximation ( iv ) , the desired renormalized hamiltonian @xmath291 is a free system consisting of two non - interacting fermionic quasi - particle bands , @xmath292 eqs . and can be easily integrated between @xmath247 and the cutoff @xmath111 of the original model , @xmath293 , \\[1ex ] % \tilde{e } & = & -\ , n \langle \hat{n}_{i}^{f } \rangle \left [ \tilde{\varepsilon}_{f } - \varepsilon_{f } \right ] + \frac{d-1}{d } \langle \hat{n}_{i}^{f } \rangle \sum_{\mathbf{k } } \left [ \tilde{\varepsilon}_{\mathbf{k } } - \varepsilon_{\mathbf{k } } \right ] \nonumber\\[-1ex ] \label{pam20 } & \approx & -\ , n \langle \hat{n}_{i}^{f } \rangle \left [ \tilde{\varepsilon}_{f } - \varepsilon_{f } \right],\end{aligned}\ ] ] where approximation ( iii ) has been used . the equation can also be solved if the renormalizations of the different @xmath294 values are decoupled from each other by approximations ( i ) and ( ii ) . thus , eq . can be rewritten as @xmath295 and inserted into so that we obtain @xmath296 eq . can easily be integrated and a quadratic equation for @xmath297 is obtained our recent work on the pam @xcite has shown that the quasi - particles in the final hamiltonian @xmath73 do not change their ( @xmath121 or @xmath79 ) character as function of the wave vector @xmath80 . therefore , @xmath298 jumps between the two solutions of the obtained quadratic equation in order to minimize its deviations from the original @xmath126 , @xmath299 the second quasi - particle band is given by @xmath300 \label{pam24 } & & \end{aligned}\ ] ] thus , we have obtained the same effective hamiltonian and the same quasi - particle energies and as found in ref . . finally , we need to determine the renormalized @xmath79 energy @xmath276 and the expectation values . because the final renormalized hamiltonian consists of non - interacting fermionic quasi - particles , it is straightforward to calculate all desired quantities from the free energy as it was done in ref . . because the effective model @xmath73 is connected with the original hamiltonian @xmath3 by an unitary transformation the free energy can also be calculated from @xmath73 , @xmath301 the expectation value of the @xmath79 occupation is found from the free energy by functional derivative , @xmath302 thus , we finally obtain a relation of the following structure @xmath303 in the cases of mixed valence and heavy fermion behavior the derivatives in eq . are non - zero so that both brace expressions can be set equal to zero to find equations of self - consistency for the renormalized @xmath79 level and the averaged @xmath79 occupation number , @xmath304 % & & + \ , \frac{\nu_{f}}{n } \sum_{\mathbf{k } } f(\tilde{\omega}_{\mathbf{k } } ) \left\ { \frac{1}{2 } + \mathrm{sgn}(\varepsilon_{\mathbf{k } } - \tilde{\varepsilon}_{f } ) \frac{\varepsilon_{\mathbf{k}}-\tilde{\varepsilon}_{f}}{2w_{\mathbf{k } } } \ , \right\ } , \,\phantom{a } \nonumber\end{aligned}\ ] ] @xmath305 % & & + \ , \frac{\nu_{f}-1}{n } \sum_{\mathbf{k } } \mathrm{sgn}(\varepsilon_{\mathbf{k } } - \tilde{\varepsilon}_{f } ) \ , f(\tilde{\omega}_{\mathbf{k } } ) \frac{|v_{\mathrm{k}}|^{2}}{w_{\mathbf{k } } } \nonumber.\end{aligned}\ ] ] these equations are quite similar to the results of the sb theory @xcite . in particular , the limit @xmath306 of eqs . and leads to the sb equations . note that expectation values @xmath307 and @xmath308 can be calculated similar to eq . , see ref . for details . note that for the analytical solution in the preceeding subsection an explicit expression for the generator @xmath309 , was not needed . the reason was that a @xmath10 independent @xmath79 electron energy @xmath310 was assumed in close analogy to what is done in the well known slave boson mean field approach for the periodic anderson model . for an improved treatment an explicit expression for @xmath311 should be used . following the discussion in subsection [ fa_rev ] we make the following ansatz for @xmath139 @xmath312^{2 } } \ , \delta\lambda \nonumber .\end{aligned}\ ] ] in the limit of small @xmath142 , we again expect an exponential decay for the hybridization @xmath93 in this way . in eq . , @xmath144 denotes an energy constant to ensure a dimensionless @xmath139 . note that @xmath139 is chosen proportional to @xmath142 to reduce the impact of the actual value of @xmath142 on the final results of the renormalization . using and the basic renormalization equations - was solved numerically in ref . . [ fig_n_f ] shows the @xmath79 occupation @xmath313 as function of the bare @xmath79 energy @xmath314 at degeneracy @xmath315 for two cases , ( i ) for fixed total particle occupation @xmath316 ( in red ) and ( ii ) for fixed chemical potential @xmath317 ( in green ) . here , @xmath318 is the conduction electron occupation . for the first case the result from the prm approach shows a rather smooth decay from the integer valence region with @xmath319 , when @xmath314 is located far below the fermi level , to an empty state with no @xmath79 electrons @xmath320 , when @xmath314 is far above the fermi level ( black line ) . note that this analytical prm result almost completely agrees with the result from recent dmrg calculations from ref . for the same parameter values . for comparison , the figure also contains a curve obtained from the prm approach when the chemical potential @xmath317 instead of @xmath321 was fixed in the calculation ( red curve ) . note that in this case @xmath322 as function of @xmath314 shows an abrupt change from an completely filled to an empty @xmath79 state . obviously the latter behavior can easily be understood as change of the @xmath79 charge when @xmath314 crosses the fixed chemical potential . in contrast , for fixed total occupation @xmath321 the fermi level is shifted upwards , when the @xmath79 level is partially depleted when @xmath314 comes closer to the fermi level . for details we refer to ref .. in this section we discuss the one - dimensional holstein model . as is well known , this model shows a quantum phase transition between a metallic and a charge ordered state as function of the electron - phonon coupling . in the present section we restrict ourselves to the metallic state . let us start with the hamiltonian of the one - dimensional holstein model of spinless fermions ( hm ) which reads , @xmath323 this model is perhaps the simplest realization of an electron - phonon ( ep ) system and describes the interaction between the local electron density @xmath324 and dispersion - less phonons with frequency @xmath325 . here , the @xmath326 ( @xmath327 ) denote creation operators of electrons ( phonons ) , and the summation @xmath328 runs over all pairs of neighboring lattice sites . with increasing ep coupling @xmath329 , the hm undergoes the quantum - phase transition from a metallic to a charge - ordered insulating state . at half - filling , the insulating state of the hm is a dimerized peierls phase . because the hm is not exactly solvable , a number of different analytical and numerical methods have been applied : strong coupling expansions @xcite , monte carlo simulations @xcite , variational @xcite and renormalization group @xcite approaches , exact diagonalization ( ed ) techniques @xcite , density matrix renormalization group @xcite and dynamical mean - field theory ( dmft ) @xcite . however , most of these approaches are restricted in their application , and the infinite phononic hilbert space ( even for finite systems ) demands the application of truncation schemes in numerical methods or involved reduction procedures . the prm represents an alternative analytical approach . in the following the prm is applied to the hm where we mainly follow refs . , and . here we focus on the investigation of the change of physical properties by passing from the adiabatic to the anti - adiabatic limit . furthermore , we discuss electronic and phononic quasi - particle energies as well as the impact of the system filling . for the metallic phase of the hm a very simple renormalization scheme is sufficient where only the electronic and phononic one - particle energies are renormalized . following refs . and , we make the following ansatz for the renormalized hamiltonian @xmath330 % % \mathcal{h}_{0,\lambda } & = & \sum_{k } \varepsilon_{k,\lambda } c^{\dagger}_{k}c_{k } + \sum_{q } \omega_{q,\lambda } b^{\dagger}_{q } b_{q } + e_{\lambda } , \nonumber\\[1ex ] % \mathcal{h}_{1,\lambda } & = & \frac{g}{\sqrt{n}}\sum_{k , q } \theta_{k , q,\lambda } \ , \left ( b^{\dagger}_{q } c^{\dagger}_{k}c_{k+q } + b_{q } c^{\dagger}_{k+q}c_{k } \right ) \nonumber\end{aligned}\ ] ] here , all excitations with energies larger than a given cutoff @xmath10 are thought to be integrated out . moreover , we have defined @xmath331 . note that fourier - transformed one - particle operators have been used for convenience . next , all transitions within the energy shell between @xmath53 and @xmath10 will be removed by use of a unitary transformation ( eq . ) , @xmath332 where the following ansatz is made for the generator @xmath65 of the transformation @xmath333 \label{hm4 } & & \end{aligned}\ ] ] the part @xmath70 has been set equal to zero . therefore @xmath334 reads @xmath335.\end{aligned}\ ] ] as before , the ansatz is suggested by the form of the first order expression of the generator @xmath65 . later , the coefficients @xmath336 will be fixed in a way that @xmath106 is fulfilled , so that @xmath63 contains no transitions larger than the new cutoff @xmath53 . by evaluating , terms with four fermionic and bosonic one - particle operators and higher order terms are generated . in order to restrict the renormalization scheme to the terms included in the ansatz , a factorization approximation has to be employed , @xmath337 % b^{\dagger}_{q } b_{q } c^{\dagger}_{k}c_{k } & \approx & b^{\dagger}_{q } b_{q } \langle c^{\dagger}_{k}c_{k } \rangle + \langle b^{\dagger}_{q } b_{q } \rangle c^{\dagger}_{k}c_{k } - \langle b^{\dagger}_{q } b_{q } \rangle \langle c^{\dagger}_{k}c_{k } \rangle .\end{aligned}\ ] ] in this way , it is possible to sum up the series expansion from transformation . the parameters @xmath336 as well as the renormalization equations for @xmath338 , @xmath339 , @xmath340 , and @xmath258 can be found by comparing the final result obtained from the explicit evaluation of the unitary transformation with the renormalization ansatz , where @xmath10 is replaced by @xmath341 . the result is given in ref . . it can be further simplified in the thermodynamic limit @xmath342 . by expanding the renormalization equations from ref . in powers of @xmath329 , one finds that only terms of quadratic or linear order in @xmath329 survive . the final equations read @xmath343 @xmath344 where @xmath345 , @xmath346 , and @xmath347 $ ] . note that the renormalization equations still depend on unknown expectation values @xmath348 and @xmath349 which follow from the factorization approximation . following ref . , they are best evaluated with respect to the full hamiltonian @xmath3 . exploiting @xmath350 , we derive additional renormalization equations for the fermionic and bosonic one - particle operators , @xmath351 and @xmath352 . they have the following form according to refs . and , @xmath353 \label{hm5 } & & \\ % % \label{hm6 } b_{q,\lambda}^{\dagger } & = & \phi_{q,\lambda } \ , b_{q}^{\dagger } + \eta_{q,\lambda } \ , b_{-q } + \sum_{k } \psi_{k , q,\lambda } \ , c_{k+q}^{\dagger } c_{k } .\end{aligned}\ ] ] the set of renormalization equations has to be solved self - consistently : one chooses some values for the expectation values . with these values , the numerical evaluation starts from the cutoff @xmath111 of the original model @xmath3 and proceeds step by step to @xmath354 . for @xmath354 , the hamiltonian and the one - particle operators are fully renormalized . the case @xmath354 allows the re - calculation of all expectation values , and the renormalization procedure starts again with the improved expectation values by reducing again the cutoff from @xmath111 to @xmath247 . after a sufficient number of such cycles , the expectation values are converged and the renormalization equations are solved self - consistently . thus , we finally obtain an effectively free model , @xmath355 where we have introduced the renormalized dispersion relations @xmath356 and @xmath357 , and the energy shift @xmath358 . for the numerical evaluation of the renormalization equations we choose a lattice size of @xmath359 sites the temperature is fixed to @xmath360 . at first , let us discuss our results for the so - called adiabatic case @xmath361 . they are shown in panel ( a ) of figs . [ fig_phonon_dispersion_2 ] , [ fig_phonon_expect ] , [ fig_electron_dispersion ] , and in panels ( a ) and ( b ) of fig . [ fig_phonon_dispersion ] . first , according to fig . [ fig_phonon_dispersion_2]a the phononic quasi - particle energies @xmath362 ( half - filling ) are found to gain dispersion due to the coupling between electronic and phononic degrees of freedom in particular around @xmath363 . furthermore , if the coupling exceeds a critical value @xmath364 non - physical negative energies at @xmath363 occur . this feature signals the break - down of the present description for the metallic phase at the quantum - phase transition to the insulating peierls state . whereas at half - filling the phonon softening occurs at the brillouin - zone boundary , soft phonon modes are found at @xmath365 and at @xmath366 for filling @xmath367 and @xmath368 , respectively . this can be seen in fig . [ fig_phonon_dispersion ] . since the phonon softening can be considered as a precursor effect of the metal - insulator transition , the type of the broken symmetry in the insulating phase _ strongly _ depends on the filling of the electronic band . note that the critical ep coupling @xmath364 of the phase transition may be determined from the vanishing of the phonon mode ( see ref . ) . at half - filling and for @xmath369 , a value of @xmath370 is found , which is somewhat larger than the dmrg result of @xmath371 of refs . and . in subsection [ hm_uniform ] the determination of the critical coupling @xmath372 within our prm approach will be discussed in more detail . [ fig_phonon_expect]a shows the phonon distribution @xmath373 for the same parameter values as in fig . [ fig_phonon_dispersion_2]a . there are two pronounced maxima found at wave numbers @xmath374 and @xmath375 . the peak at @xmath376 is directly connected to the softening of @xmath377 at the zone boundary and can therefore be considered as a precursor of the transition to a dimerized state . for the critical ep coupling @xmath378 a divergency of @xmath379 should appear at @xmath374 . the second peak around @xmath380 follows from renormalization contributions which become strong for small @xmath381 for the adiabatic case @xmath382 . this will be explained in more detail in the discussion part below . finally , in fig . [ fig_electron_dispersion]a the renormalized fermionic one - particle energy @xmath383 is shown in relation to the original dispersion @xmath384 for the same parameter values as in fig . [ fig_phonon_dispersion_2]a . though the absolute changes are quite small , the difference between @xmath383 and @xmath385 is strongest in the vicinity of @xmath386 and @xmath387 . in particular , we find @xmath388 for @xmath386 and @xmath389 for @xmath387 , so that the renormalized bandwidth becomes larger than 4 t , i.e. larger than the original bandwidth . next , let us discuss the results for phonon frequencies @xmath325 of the order of the hopping matrix element @xmath390 ( intermediate case ) . the results are found in the panels ( b ) of figs . [ fig_phonon_dispersion_2 ] , [ fig_phonon_expect ] , [ fig_electron_dispersion ] . in contrast to the adiabatic case , the renormalized phonon energy @xmath377 ( fig . [ fig_phonon_dispersion_2]b ) now shows a noticeable kink at an intermediate wave vector ( for @xmath391 ) . this particular @xmath381 value , which will be called @xmath392 in the following strongly depends on the initial phonon energy @xmath325 . the appearance of such a kink at @xmath393 is a specific feature of the intermediate case . the wave number @xmath392 is characterized by a strong renormalization of the phonon energy in a small @xmath381-range around @xmath392 , where @xmath394 for @xmath395 and @xmath396 for @xmath397 holds . the origin of these features will be discussed in more detail below . similar to @xmath377 , also the phonon distribution @xmath379 in fig . [ fig_phonon_expect]b shows a pronounced structure of considerable weight around @xmath392 . finally , in fig . [ fig_electron_dispersion]b the difference of the fermionic one - particle energies @xmath398 is shown . again a remarkable structure is found , though the absolute changes are small for the present @xmath329-values . finally , let us discuss the results for the anti - adiabatic case @xmath399 . in panels ( c ) of figs . [ fig_phonon_dispersion_2 ] , [ fig_phonon_expect ] , [ fig_electron_dispersion ] a value of @xmath400 was used . as most important feature a stiffening of the renormalized phonon frequency @xmath377 ( fig . [ fig_phonon_dispersion_2]c ) is found instead of a softening as in the adiabatic case . in particular , for large values of the ep coupling no softening of the phonon modes is found at @xmath363 . moreover , no large renormalization contributions occur in any limited @xmath381-space regime which would lead to peak - like structures . instead an overall smooth behavior is found in the entire brillouin zone . also the phonon distribution @xmath379 ( fig . [ fig_phonon_expect]c ) shows a smooth behavior with a maximum at @xmath363 . the lack of strong peak - like structures in @xmath381 space indicates that there is no phonon mode that gives a dominant contribution to the renormalization processes . if one compares the renormalized electronic bandwidth for the anti - adiabatic case ( fig . [ fig_electron_dispersion]c ) with that of the adiabatic case ( fig . [ fig_electron_dispersion]a ) , one observes a relatively strong reduction of the bandwidth . this indicates the tendency to localization in the anti - adiabatic case . it also indicates that the metal - insulator transition in the anti - adiabatic limit can be understood as the formation of small immobile polarons with electrons surrounded by clouds of phonon excitations . in the present prm approach , a renormalized one - particle excitation like @xmath383 corresponds to a quasiparticle of the coupled many - particle system . therefore , a completely flat @xmath193 dependence of @xmath383 would be expected to be found in the insulating regime . it may be worthwhile to demonstrate that the prm approach has the advantage that all features of the results for @xmath377 and @xmath379 or @xmath383 can easily be understood on the basis of the former renormalization equations . for simplicity , we shall restrict ourselves to the case of half - filling and to the renormalization of the phonon energies @xmath362 . the basic equation is the renormalization equation . due to the @xmath223-functions @xmath401 in all equations a renormalization approximately occurs when the energy difference @xmath402 lies within a small energy shell between @xmath10 and @xmath341 . as one can see from the most dominant renormalization processes take place for small values of the cutoff @xmath10 . therefore , the largest renormalization contributions come from @xmath193 and @xmath381 values that fulfill the condition @xmath403 from directly follows a second condition for the renormalization contributions to @xmath339 . due to the expectation values @xmath404 in the renormalization of @xmath339 is caused from the coupling to particle - hole excitations . therefore , the energies @xmath338 and @xmath405 have to be either below or above the fermi level , i.e. @xmath406 and @xmath407 or @xmath408 and @xmath409 . let us first discuss the adiabatic case @xmath382 . the most dominant contributions to the renormalization are expected when both conditions are simultaneously fulfilled . this is the case for @xmath410 or partially also for @xmath411 . note that for @xmath363 practically all @xmath193-values can contribute to the renormalization of , which is not the case for @xmath381-values different from @xmath412 . for instance , for @xmath375 only few @xmath193 points from the sum in can contribute which are located in a small region around the fermi momentum @xmath413 . on the other hand , for @xmath380 , the energy denominator is almost zero so that still some noticeable renormalization structures are found in fig . [ fig_phonon_dispersion_2]a . moreover , for the adiabatic case , where @xmath339 is small , the energy denominator of can be replaced by @xmath414 . therefore , almost all particle - hole contributions to @xmath339 are negative because @xmath404 and @xmath414 have always different signs . one concludes that in the adiabatic case @xmath339 will be renormalized to smaller values where the renormalization at @xmath374 should be dominant . the behavior of @xmath377 for the case of intermediate phonon frequencies ( @xmath391 in fig . [ fig_phonon_dispersion_2]b and fig . [ fig_phonon_expect]b ) can again be understood on the basis of the renormalization equations and condition . as was already discussed , particle - hole excitations lead to the renormalization of @xmath339 . therefore , from the sum over @xmath193 in eq . only @xmath193 terms contribute where either @xmath406 and @xmath407 or @xmath408 and @xmath409 . for the latter case always @xmath415 is valid so that can not be fulfilled . therefore , we can restrict ourselves to contributions @xmath416 und @xmath407 , for which always @xmath417 and @xmath418 holds . the largest renormalization should result from a small @xmath381 region around some @xmath381 vector @xmath392 for which @xmath419 is approximately fulfilled . since @xmath325 is of the order of @xmath390 , @xmath392 is located somewhere in the middle of the brillouin zone and depends strongly on @xmath325 . from eq . also follows that renormalization contributions to @xmath377 change their sign at @xmath392 due to the sign change in the energy denominator . finally , from equation one may point out also the stiffening of the phonon modes in the anti - adiabatic case @xmath400 . in this case the phonon energy @xmath325 is much larger than the electronic bandwidth . therefore , for all @xmath10 a positive energy denominator @xmath420 is obtained . nevertheless , for half - filling in the @xmath193 sum on the right hand side of there are as many negative as positive terms due to the factor @xmath404 . since from @xmath421 always follows @xmath422 , the negative terms have larger energy denominators and are always smaller than the positive terms . the resulting renormalization of @xmath339 is therefore positive for all @xmath381 values and largest for @xmath363 due to the smallest energy denominator . in this section we want to demonstrate the ability of the prm approach to describe also quantum phase transitions . in particular , we shall investigate the transition from the metallic to the insulating charge ordered phase when the electron - phonon coupling @xmath329 exceeds a critical value . in the following we present a uniform description that covers the metallic as well as the insulating phase of the hm in the adiabatic case . we mainly follow the approach of ref . where we have discussed methodological aspects in more detail . as already mentioned above , the simple approach of subsection [ hm_metal ] breaks down for ep couplings @xmath329 larger than some critical value @xmath372 where a long - range charge density wave occurs and the ions are shifted away from their symmetric positions . an adequate theoretical description needs to take into account a broken symmetry field . for this purpose , the underlying idea of subsection [ ep_bcs ] to take such a term into account in the renormalization ansatz will be transferred to the present case . as one can see from fig . [ fig_phonon_dispersion ] , the order parameter of the insulating phase strongly depends on the filling of the electronic band . therefore , in the following we restrict ourselves to the case of half - filling . here , the unit cell is doubled and a dimerization occurs in the insulating phase . following ref . , the hamiltonian in the reduced brillouin zone including symmetry breaking fields reads @xmath423 % % \mathcal{h}_{0,\lambda } & = & \sum_{k>0,\alpha } \varepsilon_{\alpha , k,\lambda } c_{\alpha , k}^{\dag } c_{\alpha , k } + \sum_{q>0 , \gamma } \omega_{\gamma , q,\lambda } b_{\gamma , q}^{\dag } b_{\gamma , q } \nonumber \\ & & + \ , e_{\lambda } + \sum_{k } \delta_{k,\lambda}^{\mathrm{c } } \left ( c_{0,k}^{\dag } c_{1,k } + \mathrm{h.c . } \right ) \nonumber \\ & & + \ , \sqrt{n } \delta_{\lambda}^{b } \left ( b_{1,q}^{\dag } + \mathrm{h.c . } \right ) , \nonumber \\[1ex ] % % \mathcal{h}_{1,\lambda } & = & \frac{1}{\sqrt{n } } \sum _ { \genfrac{}{}{0pt}{1 } { \genfrac{}{}{0pt}{1}{k , q>0 } { \alpha,\beta,\gamma } } { } } g_{k , q,\lambda}^{\alpha,\beta,\gamma } \left\ { \delta(b_{\gamma , q}^{\dag } ) \delta(c_{\alpha , k}^{\dag } c_{\beta , k+q } ) + \mathrm{h.c . } \right\ } . \nonumber\end{aligned}\ ] ] where @xmath424 and @xmath425 are the appropriate order parameters for the electronic and the phononic symmetry breaking fields . note that the reduced brillouin zone leads to additional band indices @xmath426 of both electronic and phononic one - particle operators . furthermore , we defined @xmath427 and @xmath428 . the ansatz is restricted to the one - dimensional case at half - filling . to extend the approach to higher dimensions one would need to take into account all @xmath429 wave vectors of the brillouin zone boundary . before we can proceed we need to diagonalize @xmath8 . for this purpose a rotation in the fermionic subspace and a translation to new ionic equilibrium positions are performed in order to diagonalize @xmath8 @xmath430 with new fermionic and bosonic creation an annihilation operators , @xmath431 and @xmath432 , and we rewrite @xmath166 in terms of the new operators , @xmath431 and @xmath432 . finally , we have to transform @xmath14 to @xmath433 according to to derive the renormalization equations for the parameters of @xmath4 . here the ansatz @xmath434 is used . the coefficients @xmath435 have to be fixed in such a way so that only excitations with energies smaller than @xmath188 contribute to @xmath436 . the renormalization equations for the parameters @xmath437 , and @xmath438 are finally obtained by comparison with after the creation and annihilation operators @xmath439 have been transformed back to the original operators @xmath440 . the actual calculations are done in close analogy to subsection [ hm_metal ] . note that again a factorization approximation was used and only operators of the same structure as in are kept . therefore , the final renormalization equations still depend on unknown expectation values , which are evaluated with the full hamiltonian @xmath3 . note that in order to evaluate the expectation values @xmath441 additional renormalization equations have also to be found for the fermionic and bosonic one - particle operators , @xmath442 and @xmath443 . by using the same approximations as for the hamiltonian a resulting set of renormalization equations is derived . it is solved numerically where the equations for the expectation values are taken into account in a self - consistency loop . by eliminating all excitations in steps @xmath68 we finally arrive at cutoff @xmath247 which again provides an effectively free model @xmath444 . it reads @xmath445 where it was defined @xmath446 , @xmath447 , @xmath448 , and @xmath449 . note that all excitations from @xmath6 were used up to renormalize the parameters of @xmath450 . the expectation values are also calculated in the limit @xmath195 . because @xmath73 is a free model they can easily be determined from @xmath451 in the following , we first demonstrate that the prm can be used to investigate the peierls transition of the one - dimensional spinless holstein model at half - filling . the phonon energy is fixed to @xmath369 . in particular , our analytical approach provides a simultaneous theoretical description for both the metallic and the insulating phase . finally , we compare our results with recent dmrg calculations . first , let us consider the critical electron - phonon coupling @xmath364 . for that purpose , in fig . [ fig_hm_delta ] a characteristic electronic excitation gap @xmath452 for infinite system size is plotted as function of the ep coupling @xmath329 , where @xmath452 was determined from the opening of a gap in the quasi - particle energy @xmath383 ( see text below ) . a closer inspection of the data shows that an insulating phase with a finite excitation gap is obtained for @xmath329 values larger than the critical ep coupling @xmath453 . a comparison with the critical value @xmath454 obtained from dmrg calculations shows that the critical values from the prm approach might be somewhat too small . however , this difference can be attributed to the exploited factorization approximation in the prm which suppresses fluctuations so that the ordered insulating phase is stabilized . note that in order to determine @xmath364 a careful finite - size scaling was performed as shown for some @xmath329 values in the inset of fig . [ fig_hm_delta ] . a linear regression was applied to extrapolate our results to infinite system size . note that the finite size scaling may be affected by two different effects : suppression of long - range fluctuations by the finite cluster size and by the used factorization approximation so that a rather unusual dependence on the system size is found . in contrast to other methods , the prm directly provides the quasi - particle energies : after the renormalization equations were solved self - consistently the electronic and phononic quasi - particle energies of the system , @xmath455 and @xmath362 , respectively , are given by the limit @xmath195 of the parameters @xmath456 and @xmath457 of the diagonal hamiltonian @xmath458 of . in fig . [ fig_4 ] the renormalized one - particle energies @xmath459 and @xmath460 as quasi - particle of the full system are shown for different values of the ep coupling @xmath329 . the upper panel shows that the electronic one - particle energies depend only slightly on @xmath329 as long as @xmath329 is smaller than the critical value @xmath453 . if the ep coupling @xmath329 is further increased a gap @xmath452 opens at the fermi energy so that the system becomes an insulator . remember that the gap @xmath452 has been used as order parameter to determine the critical ep coupling @xmath364 of the metal - insulator transition ( see fig . [ fig_hm_delta ] ) . the lower panel of fig . [ fig_4 ] shows the results for the phononic one - particle energy @xmath362 . one can see that @xmath362 gains dispersion due to the coupling @xmath329 between the electronic and phononic degrees of freedom . in particular , the phonon mode at momentum @xmath461 , i.e. at the brillouin - zone boundary becomes soft if the ep coupling is increased up to @xmath453 . however , in contrast to the metallic solution of subsection [ hm_metal ] @xmath377 at @xmath461 always remains positive though it is very small . note that for @xmath329 values larger than @xmath372 the energy @xmath377 increases again . this phonon softening at the phase transition has to be interpreted as a lattice instability which leads to the formation of the insulating peierls state for @xmath462 . the phase transition is associated with a shift of the ionic equilibrium positions . a lattice stiffening occurs if @xmath329 is further increased to values much larger than the critical value @xmath453 . note also that the critical coupling @xmath453 obtained from the opening of the gap in @xmath383 is significantly smaller than the @xmath364 value of @xmath463 which was found from the vanishing of the phonon mode at the brillouin zone boundary in the metallic solution of subsection [ hm_metal ] . instead , one would expect that both the gap in @xmath455 and the vanishing of @xmath362 should occur at the same @xmath372 value . this inconsistency can again be understood from the factorization approximation in the prm : as discussed above , the inclusion of additional fluctuations leads to a less stable insulating phase so that a @xmath372 value larger than @xmath464 would follow . on the other hand , the dispersion of @xmath377 due to renormalization processes would be enhanced by taking additional fluctuations into account . thus , a @xmath372 value smaller than @xmath465 would follow . in this way , both ways to determine @xmath372 would be consistent with each other and could lead to a common result for @xmath372 in between @xmath464 and @xmath466 . this would be in agreement with the dmrg value of @xmath467 ( ) . as a second example for a quantum phase transition , we now study the competition of charge - density waves ( cdw ) and superconductivity ( sc ) for the two - dimensional half - filled holstein model by use of the projector - based renormalization method . in one dimension the coupling of electrons to phonons gives rise to a metal - insulator transition . in two dimensions the electron - phonon interaction may also be responsible for the formation of cooper pairs . in the following , the competing influence of superconductivity and charge order will be discussed for two dimensions . the prm not only allows to study sc and cdw correlation functions but gives direct access to the order parameters . the discussion closely follows the approach of ref . + the relationship between a possible superconducting and an insulating peierls - cdw phase in the 2d - holstein model has been subject to a number of studies in the literature ( for details we refer to ref . ) . in general , it is believed that the onset of strong sc correlations suppresses the development of cdw correlations and vice versa . thus close to the phase transition , both types of correlations must be taken into account . to find a uniform description of both the superconducting ( sc ) and the insulating cdw phase , two fields , which break the translation and the gauge symmetry should be added to the hamiltonian . thus , the model on a square lattice is given by @xmath468 where @xmath294 is the wave vector on the reciprocal lattice and @xmath469 is the characteristic wave vector of the cdw phase @xmath470 . assuming an electron hopping between nearest - neighbor sites , the electronic dispersion is given by @xmath471 , where @xmath317 is the chemical potential . moreover , @xmath325 is the dispersionless phonon energy , and @xmath329 denotes the coupling strength between the electrons and phonons . at the beginning of the renormalization the two symmetry breaking fields @xmath472 and @xmath473 , as well as @xmath474 , are assumed to be infinitesimally small ( @xmath475 , @xmath476 ) . the unperturbed hamiltonian @xmath26 can be diagonalized , since its electronic part is quadratic in the fermionic operators . note that due to the doubling of the unit cell in the insulating phase , in @xmath26 the creation operator @xmath477 is coupled to @xmath478 . in addition the coupling of @xmath479 to @xmath480 is caused by superconductivity . therefore , the eigenmodes of @xmath26 can be represented as a linear combination of the following four operators @xmath481 in the renormalization procedure , all transitions with energies larger than @xmath10 will be integrated out . as can be seen , the renormalized hamiltonian can again be divided into @xmath482 . if one denotes by @xmath483 ( @xmath484 ) the @xmath10 dependent eigenmodes of @xmath5 the electronic part of the renormalized hamiltonian @xmath5can be written as @xmath485 where the eigenenergies are given by @xmath486 for @xmath487 , whereas for @xmath488 the @xmath489-signs have to be reversed . note that in the sum of the two order parameters squared enter the energies @xmath490 of . in order to derive the renormalization equations , the unitary transformation has to be evaluated explicitly . thereby , also the interaction @xmath6 has to be expressed in terms of the eigenmodes @xmath491 of @xmath5 . moreover , an ansatz for @xmath127 has to be made in analogy to what was done in the previous sections . the explicit calculation is found in ref . . for the numerical evaluation of the renormalization equations , we consider a square lattice with @xmath492 sites . the temperature is set equal to @xmath360 , and a small value of @xmath493 is chosen . for simplicity , we also restrict ourselves to @xmath494-wave - like superconducting solutions . the results are shown in fig . [ fig_1/2 ] , where the @xmath294-dependent symmetry breaking fields @xmath495 ( black ) and @xmath496 ( red ) for @xmath497 are plotted as function of the electron - phonon coupling @xmath329 . the coupling @xmath329 is restricted to small values @xmath498 . as can be seen from fig . [ fig_1/2 ] , for small values of @xmath499 the system is in a pure superconducting state , i.e. no charge order is present . for small @xmath329 , the superconducting gap increases roughly proportional to @xmath500 . in the intermediate @xmath329 range , @xmath501 , a coexistence of both phases is found . the system is in a combined superconducting - charge ordered phase . here , the @xmath329 dependence of @xmath502 is no longer quadratic as in the small @xmath329 regime . instead , @xmath502 reaches a maximum value and drops down to zero with increasing @xmath329 . finally , for @xmath503 the superconducting phase is completely suppressed and the system is in a pure charge ordered state . the aim of this contribution was to discuss the basic ideas of a new theoretical approach for many - particle systems which is called projector - based renormalization method ( prm ) and its application to a number of nontrivial physical problems . instead of eliminating high - energy states as in usual renormalization group methods in the prm high - energy transitions are successively eliminated . thereby , a unitary transformation is used where all states of the unitary space of the interacting system are kept . in that respect , the prm is closely related to the similarity transformation introduced by wilson and glazek and to wegner s flow equation method though both approaches start from a continuous formulation of the unitary transformation . the prm starts from a hamiltonian which can be decomposed into a solvable unperturbed part and a perturbation , @xmath504 , where the latter part induces transitions between the eigenstates of @xmath26 . suppose a renormalized hamiltonian @xmath14 has been constructed which only contains transitions with transition energies smaller than some given cutoff energy @xmath10 . the hamiltonian @xmath14 can be further renormalized by eliminating all transitions from , roughly speaking , the energy shell between the cutoff @xmath10 and a reduced cutoff @xmath505 , and so on . this is done by a unitary transformation @xmath506 which guarantees that the eigenspectrum is not changed . the generator of the unitary transformation @xmath127 is specified by the condition @xmath507 where @xmath508 is the projector on all transitions with energy differences larger than @xmath505 . the latter condition implies that all transitions from the shell between @xmath10 and @xmath509 are eliminated and lead to a renormalization of @xmath433 . note that only the equivalent part @xmath510 of @xmath127 is fixed whereas the orthogonal part @xmath511 can be chosen arbitrarily . note that this additional freedom can be used in a different way . whereas in the original version of the prm the remaining part @xmath511 of @xmath127 was set equal to zero for simplicity this part was used in wegner s flow equation method as the only relevant part when the transformation was performed continuously . in this case , the interaction parameters were chosen to decay exponentially . by proceeding the renormalization up to the final cutoff @xmath512 all transitions induced by @xmath6 are eliminated . the final renormalized hamiltonian @xmath513 is diagonal and allows to evaluate in principle any correlation function of physical interest . in particular the one - particle excitations of @xmath514 can be considered as quasi - particles of the coupled many - particle system since the eigenspectrum of the original interacting hamiltonian @xmath13 and of @xmath514 are in principle the same since both are connected by a unitary transformation . note that the present approach has the advantage of formulating the renormalization quite universally . by specifying the unitary transformation of the many - particle system both the prm and wegner s flow equation method can be derived from the same basic ideas . however , the stepwise transformation of the prm has its own merits . firstly , as was shown in sec . [ ep_bcs ] , sec . [ holstein_qp ] , and sec . [ holstein_sc ] the physical behavior on both sides of a quantum critical point can be described within the same prm scheme . this seems not the case for the flow equation approach . in particular , by allowing symmetry breaking terms in the unperturbed part @xmath5 , the transformation of eigenmodes of the liouville operator @xmath515 can be followed in each renormalization step . this makes the description of quantum critical points possible . secondly , in sec . [ b_perturbation ] a perturbation theory for @xmath45 was given . this allows to evaluate physical properties in perturbation theory . in contrast to a recent perturbation approach on the basis of the flow equation method , in the prm no equidistant spectrum of @xmath26 is required . we would like to acknowledge stimulating and enlightening discussions with a. mai and j. schne . this work was supported by the dfg through the research program sfb 463 . in this appendix we are going to investigate ground - state properties of a dimerized and frustrated spin chain . we apply the projector - based perturbation theory and use expression for @xmath14 and chose @xmath44 right from the beginning . in this case , the interaction @xmath52 is completely integrated out in one step . the starting hamiltonian reads @xmath516 % \mathcal{h}_{0 } & = & j \sum_{i } \mathbf{s}_{2i } \mathbf{s}_{2i+1 } , \nonumber \\ % \mathcal{h}_{1 } & = & j \sum_{i } [ \alpha \mathbf{s}_{2i } \mathbf{s}_{2i-1 } + \beta\left ( \mathbf{s}_{2i } \mathbf{s}_{2i-2 } + \mathbf{s}_{2i-1 } \mathbf{s}_{2i+1 } \right ) ] , \nonumber\end{aligned}\ ] ] the model itself is of some physical interest because it can be used to describe some spin - peierls compounds like cugeo@xmath517 or ttfcubdt @xcite . in the following we are interested in the limit of strong dimerization of the model so that we start from isolated dimers as described by @xmath2 . every dimer can be in the singlet state or in one of the three degenerated triplet states . here , the dimer states are energetically separated by the singlet - triplet splitting , @xmath518 . thus , triplets can be considered as the basic excitations of the system . following the ideas of refs . and , the contributions to the perturbation @xmath1 can be classified according to the number of created or annihilated local triplets , @xmath519\nonumber\end{aligned}\ ] ] the introduced excitation operators @xmath520 only act on the local dimers with indices @xmath521 and @xmath522 , and create @xmath81 local triplets . the @xmath520 are eigenoperators of the liouville operator @xmath523 and the corresponding eigenvalues are @xmath524 . the actual contributions to @xmath525 , @xmath526 , and @xmath527 are summarized in table [ tab_dimer ] , and @xmath528 and @xmath529 are given by the relation @xmath530^{\dagger}$ ] . in the limit of strong dimerization , hilbert space sectors with different numbers of triplets in the system are energetically separated because the unperturbed part @xmath2 of the hamiltonian does not change the number of triplets in the system , and the interaction @xmath1 only leads to modest corrections . consequently , the evaluation of the effective hamiltonian can be simplified if one concentrates on a hilbert space sector with a given fixed number of triplets . in the following , actual calculations are presented for the two energetically lowest sectors where the system contains no or only one triplet . the subspace without triplets consists of a single state , i.e. the singlet product state , @xmath531 . because the effective hamiltonian @xmath532 is obtained from the original hamiltonian @xmath3 by means of a unitary transformation , the ground - state energy can be calculated from @xmath533 here @xmath534 is given by where @xmath535 and @xmath536 is the remaining part of . using the notation of ref . , one easily finds @xmath537 this result agrees with findings of refs . and . note that higher order terms can easily be calculated by implementing a computer based evaluation algorithm as discussed in ref . where a cumulant method @xcite was applied to the same model . the case of a single triplet in the system is more complex because a triplet can easily move along the chain . consequently , it is advantageous to introduce momentum dependent states , @xmath538 and the eigenvalues of this hilbert space sector can be calculated by @xmath539 . we again employ the useful notation of ref . and obtain @xmath540 note that the energy gap of the system can easily be determined from eq . by considering the case @xmath386 . the @xmath193 dependence of @xmath541 describes the triplet dispersion relation . furthermore , the calculation can easily be extended to higher orders . the same model was also studied @xcite based on wegner s flow equation method @xcite where both ground - state energy and triplet dispersion relation were calculated in high orders . however , for this purpose a set of coupled differential equations had to be integrated so that this approach is restricted to systems with an equidistant eigenvalue spectrum of the unperturbed part @xmath2 of the hamiltonian . see , for example , h.q . lin and j.e . gubernatis , comput . * 7 * , 400 ( 1993 ) , and references therein . wilson , rev . mod . phys . * 47 * , 773 ( 1975 ) ; for a recent review see r. bulla , t. costi , and t. pruschke , cond - mat/0701105 . w. von der linden , physics rep . * 220 * , 53 ( 1992 ) . s. white , phys . 69 * , 2863 ( 1992 ) ; for a recent review see p. schollwck , rev . phys . * 77 * , 259 ( 2005 ) . w. metzner and d. vollhardt , phys . * 62 * , 324 ( 1989 ) ; g. kotliar and d. vollhardt , physics today , march 2004 , p. 53 ; for a review see a. georges , g. kotliar , w. krauth , and m.j . rozenberg , rev . phys , * 68 * , 13 ( 1996 ) gazek and k.g . wilson , phys . d * 48 * , 5863 ( 1993 ) . gazek and k.g . wilson , phys . d * 49 * , 4214 ( 1994 ) . f. wegner , ann . ( leipzig ) * 3 * , 77 ( 1994 ) ; see also s. kehrein , _ the flow equation approach to many - particle systems _ , springer tracts in modern physics , springer - verlag gmbh , 2006 . see , for example , j. zinn - justin , _ quantum field theory and critical phenomena _ , oxford , clarendon press 2002 . becker , a. hbsch , and t. sommer , phys . b * 66 * , 235115 ( 2002 ) . p. coleman , phys . b * 29 * , 3035 ( 1984 ) . for a review see , for example , p. fulde , j. keller , and g. zwicknagl , in _ solid state physics _ , edited by h. ehrenreich and d. turnbull ( academic , san diego , 1988 ) , vol . 41 , p. 1 . j. stein , j. stat . phys . * 88 * , 487 ( 1997 ) . c. knetter and g.s . uhrig , eur . j. b * 13 * , 209 ( 2000 ) . a. hbsch , m. vojta , and k.w . becker , j. phys . : condens . matter * 11 * , 8523 ( 1999 ) . j. riera and a. dobry , phys . b * 51 * , 16098 ( 1995 ) . g. castilla , s. chakravarty , and v. j. emery , phys . rev . lett . * 75 * , 1823 ( 1995 ) . bray , l.v . interante , i.c . jacobs , j.c . bonner , in _ extended linear chain compounds _ , edited by j.s . miller ( plenum press , new york , 1983 ) , vol . 3 , p. 353 . s. sykora , a. hbsch , and k.w . becker , phys . b * 70 * , 054408 ( 2004 ) . a. hbsch and k.w . becker , eur . j. b * 33 * , 391 ( 2003 ) . a. hbsch and k.w . becker , phys . b * 71 * , 155116 ( 2005 ) . a. hbsch and k.w . becker , eur . j. b * 52 * , 345 ( 2006 ) . s. sykora , a. hbsch , k.w . becker , g. wellein , and h. fehske , phys . b * 71 * , 045112 ( 2005 ) anderson , phys . rev . * 124 * , 41 ( 1961 ) . u. fano , phys . rev . * 124 * , 1866 ( 1961 ) . j. bardeen , l.n . cooper , and j.r . schrieffer , phys . rev . * 108 * , 1175 ( 1957 ) . cooper , phys . rev . * 104 * , 1189 ( 1956 ) . h. frhlich , proc . london a * 215 * , 291 ( 1952 ) . p. lenz and f. wegner , nucl . b * 482 * , 693 ( 1996 ) . a. mielke , ann . physik ( leipzig ) * 6 * , 215 ( 1997 ) . bogoliubov , nuovo cim . * 7 * , 794 ( 1958 ) . s. sykora , a. hbsch , and k.w . becker , eur . j. b * 51 * , 181 ( 2006 ) . s. sykora , a. hbsch , and k.w . becker , europhys . lett . * 76 * , 644 ( 2006 ) . lee , t.m . rice , j.w . serene , l.j . sham , and j.w . wilkins , comments condens . matter phys . * 12 * , 99 ( 1986 ) . r. franco , m.s . figueira , m.e . foglio , phys . b * 66 * , 045112 ( 2002 ) . a. mai , p.v . nham , a. hbsch , and k.w . becker , unpublished . myake j.e . hirsch and e. fradkin , phys . b * 27 * , 4302 ( 1983 ) . mckenzie , c.j . hamer , and d.w . murray , phys . b * 53 * , 9676 ( 1996 ) . h. zheng , d. feinberg , and m. avignon , phys . b * 39 * , 9405 ( 1989 ) . caron and c. bourbonnais , phys . b * 29 * , 4230 ( 1984 ) ; g. benfatto , g. gallovotti , and j.l . lebowitz , helv . acta * 68 * , 312 ( 1995 ) . a. weie and h. fehske , phys . b * 58 * , 13526 ( 1998 ) ; h. fehske , m. holicki , and a. weie , advances in solid state physics * 40 * , 235 ( 2000 ) . bursill , r.h . mckenzie , and c.j . hamer , phys . * 80 * , 5607 ( 1998 ) . e. jeckelmann , c. zhang , and s.r . white , phys . b 60 , 7950 - 7955 ( 1999 ) . h. fehske , g. wellein , g. hager , a. weie , k.w . becker , and a.r . bishop , physica b * 359 - 361 * , 699 ( 2005 ) . d. meyer , a.c . hewson , and r. bulla , phys . lett . * 89 * , 196401 ( 2002 ) . scalettar , n.e . bickers , and d.j . scalapino , phys . b * 40 * , 197 ( 1989 ) . f. marsiglio , phys . b * 42 * , 2416 ( 1990 ) . m. vekic , r.m . noack , and s.r . white , phys . b * 46 * , 271 ( 1992 ) . f. marsiglio , j.e . hirsch , phys . b * 49 * , 1366 ( 1994 ) . e. berger , p. valasek , and w. von der linden , phys . b * 52 * , 4806 ( 1995 ) . s. sykora , a. hbsch , and k.w . becker , to be published .

despite the advances in the development of numerical methods analytical approaches play a key role on the way towards a deeper understanding of strongly interacting systems . in this regards , renormalization schemes for hamiltonians represent an important new direction in the field . among these renormalization schemes the projector - based renormalization method ( prm ) reviewed here might be the approach with the widest range of possible applications : as demonstrated in this review , continuous unitary transformations , perturbation theory , non - perturbative phenomena , and quantum - phase transitions can be understood within the same theoretical framework . this review starts from the definition of an effective hamiltonian by means of projection operators that allows the evaluation within perturbation theory as well as the formulation of a renormalization scheme . the developed approach is then applied to three different many - particle systems : at first , we study the electron - phonon problem to discuss several modifications of the method and to demonstrate how phase transitions can be described within the prm . secondly , to show that non - perturbative phenomena are accessible by the prm , the periodic anderson is investigated to describe heavy - fermion behavior . finally , we discuss the quantum - phase transition in the one - dimensional holstein model of spinless fermions where both metallic and insulating phase are described within the same theoretical framework .

introduction projector-based renormalization method (prm) renormalization of the electron-phonon interaction heavy-fermion behavior in the periodic anderson model crossover behavior in the metallic one-dimensional holstein model quantum phase transition in the one-dimensional holstein model charge ordering and superconductivity in the two-dimensional holstein model summary acknowledgments example: dimerized and frustrated spin chain

arxiv

weakly bound few - body systems are being studied since a long time back and have achieved revived interest recently as the physics of such weakly bound systems can be investigated experimentally in ultracold atomic gases @xcite . utilizing the feshbach resonance , the effective inter - atomic interaction can be changed essentially to any desired values @xcite . the recent experiments on cold atoms also provide evidence of the existence of large weakly bound clusters . thus our present study is motivated by the recent experiments on ultracold bose gas . we treat the three - dimensional bosonic cluster with maximum up to @xmath4 rb atoms interacting through two - body van der waals potential . alkali atoms , specially rb atoms , are good candidates for laser manipulation and to observe bose - einstein condensate @xcite . at ultracold temperature the interatomic interaction is fairly well represented by a single parameter @xmath5 , the @xmath6wave scattering length . for our present system we keep @xmath7 @xmath8 which corresponds to the jila experiment @xcite . thus the system is weakly interacting , and diffuse as the average size of the cluster increases with cluster size . the binding of such @xmath9body cluster is provided by the two - body van der waals potential having a short range repulsive core below a cutoff radius and a @xmath10 tail which represents the long range attractive interaction . the stability of such @xmath9body clusters , their energetics and various structural properties are recently studied @xcite . we propose the use of two - body basis function to describe various properties of bosonic clusters . with more than three particles the system becomes more complex as the number of degrees of freedom increases . we have investigated correlations between energies of the @xmath11 and @xmath12 systems and observe the generalized tjon line @xcite for large cluster . now we consider the spectral statistics and spectral correlation of the atomic clusters of different sizes as these contain rich physics and also plays an pivotal role to establish the universal properties of quantum systems . berry and tabor conjectured that the fluctuation property of energy levels of a quantum system whose classical analog is regular , is characterised by poisson statistics @xcite . whereas , the fluctuation property of energy levels of a quantum system whose corresponding classical dynamical system is fully chaotic obeys the bohigas - giannoni - schmit ( bgs ) conjecture @xcite . this tells that gaussian orthogonal ensemble ( goe ) or gaussian unitary ensemble ( gue ) or gaussian sympletic ensemble ( gse ) statistics of random matrix theory , depending on time reversal symmetry and rotational symmetry of the system , will describe the fluctuation properties . however this conjecture is often interpreted in another way and the observation of level repulsion in the spectrum is treated as an indication of the non - integrability of the system . the poisson distribution implies complete randomness in the relative positions of energy levels as they are completely uncorrelated . on the other hand wigner distribution implies strong correlation among the energy levels . earlier the spectral properties of many different quantum systems like atoms , atomic nuclei , quantum billiards have been studied @xcite . also some attempts have been made for non - interacting many - bosons and interacting bosonic system @xcite . recently we have reported the level spacing distribution of ultra - cold interacting bosons trapped in a harmonic potential @xcite . we found intriguing effect of both the interatomic interaction and the trap and observed deviation from the bgs cojecture . in this paper we are interested in similar type of calculation in the van der waals bosonic clusters . unlike the bose - einstein condensate where the external trapping provides the stability of the condensate , the van der waals clusters are bound due to the van der waals interaction . in the very dilute condition one may treat it as a uniform bose gas . apart from the experimental interest , this kind of systems are also challenging for the following reasons . first , solving the many - body schrdinger equation itself is a challenging numerical task due to many degrees of freedom and the obvious question is what kind of approximation is to be valid for the description of such clusters . secondly for large cluster size when the system becomes very much correlated , one may expect wigner type spectral distribution . however it needs an exhaustive study as level repulsion in the energy spectrum may not always lead to wigner distribution which signifies chaos . it indicates that one may need to use some deformed goe type of distribution for the correct description of nonintegrable but non - chaotic system . we propose to study several measures of spectral fluctuations and spectral correlation to determine the degree of influence of the interatomic interaction . this kind of study is also relevant as the statistical fluctuation can be directly observed experimentally in the context of ultracold bose gases . we calculate nearest neighbour level spacing distribution ( nnsd ) @xmath0 , the level number variance @xmath1 and the dyson - mehta @xmath13-statistics @xcite for various cluster sizes . however all these measures require unfolding of the spectrum to remove variation in the density of energy levels in different parts of the spectrum . we can either unfold the spectrum of each member of the ensemble separately and form ensemble averaged nnsd or a single unfolding function can be used for all the members of the ensemble . depending on the unfolding procedure , the final outcome of nnsd may vary . moreover suitable unfolding function is not always known a priori and generally is approximated by higher order polynomials . therefore to verify the outcome of the nnsd , we further analyze the distribution of quotients of successive spacings @xmath3 which does not require any unfolding and is independent of the energy level density . the paper is organised as follows . in section ii , we introduce the many - body potential harmonic expansion method . section iii discusses the numerical results and section iv concludes with the summary of our work . to study the spectral statistics and different spectral correlations we need to calculate a large number of energy levels of the diffuse rb cluster . we approximately solve the full many - body schrdinger equation by our recently developed potential harmonic expansion method . we have earlier applied it successfully to study different properties of bec @xcite and atomic clusters @xcite . the methodology has already been described in detail in our earlier works @xcite . hence here we describe it briefly for interested readers . we consider a system of @xmath14 rb atoms , each of mass @xmath15 and interacting via two - body potential . the time - independent quantum many - body schrdinger equation is given by @xmath16\psi(\vec{r}_{1}, ... ,\vec{r}_{n})=0 , \ ] ] where @xmath17 is the total energy of the system , @xmath18 is the two - body potential and @xmath19 is the position vector of the @xmath20th particle . it is usual practice to decompose the motion of a many - body system into the motion of the center of mass and the relative motion of the particles in center of mass frame . in absence of any confinig potential the center of mass behaves as a free particle in laboratory frame and we set its energy as zero . hence , after elimination of the center of mass motion and using standard jacobi coordinates , defined as @xcite @xmath21 we obtain the equation for the relative motion of the atoms @xmath22\psi(\vec{\zeta}_{1 } , ... , \vec{\zeta}_{\mathcal{n } } ) = 0\hspace*{.1cm},\ ] ] @xmath23 is the sum of all pair - wise interactions . now it is to be noted that hyperspherical harmonic expansion method is an _ ab - initio _ tool to solve the many - body schrdinger equation where the total wave function is expanded in the complete set of hyperspherical basis @xcite . although hyperspherical harmonic expansion method is a complete many - body approach and includes all possible correlations , it is highly restricted to @xmath24 only . but for a diffuse cluster like rb - cluster , only two - body correlation and pairwise interaction are important . therefore we can decompose the total wave function @xmath25 into two - body faddeev component for the interacting @xmath26 pair as @xmath27 it is important to note that @xmath28 is a function of two - body separation ( @xmath29 ) only and the global hyperradius @xmath30 , which is defined as @xmath31 . thus the effect of two - body correlation comes through the two - body interaction in the expansion basis . @xmath28 is symmetric under the exchange operator @xmath32 for bosonic atoms and satisfy the faddeev equation @xmath33\phi_{ij } = -v(\vec{r}_{ij})\sum_{kl > k}^{n}\phi_{kl}\ ] ] where @xmath34 is the total kinetic energy operator . in this approach , we assume that when ( @xmath35 ) pair interacts , the rest of the bosons are inert spectators . thus the total hyperangular momentum quantum number as also the orbital angular momentum of the whole system is contributed by the interacting pair only . next the @xmath26th faddeev component is expanded in the set of potential harmonics ( ph ) ( which is a subset of hyperspherical harmonic ( hh ) basis and sufficient for the expansion of @xmath36 ) appropriate for the ( @xmath35 ) partition as @xmath37 @xmath38 denotes the full set of hyperangles in the @xmath39-dimensional space corresponding to the @xmath26 interacting pair and @xmath40 is called the ph . it has an analytic expression : @xmath41 @xmath42 is the hh of order zero in the @xmath43 dimensional space spanned by @xmath44 jacobi vectors ; @xmath45 is the hyperangle between the @xmath46-th jacobi vector @xmath47 and the hyperradius @xmath30 and is given by @xmath48 = @xmath49 . for the remaining @xmath50 noninteracting bosons we define hyperradius as @xmath51 such that @xmath52 . the set of @xmath53 quantum numbers of hh is now reduced to _ only _ @xmath54 as for the @xmath50 non - interacting pair @xmath55 and for the interacting pair @xmath56 , @xmath57 and @xmath58 . thus the @xmath39 dimensional schrdinger equation reduces effectively to a four dimensional equation with the relevant set of quantum numbers : energy @xmath17 , orbital angular momentum quantum number @xmath59 , azimuthal quantum number @xmath15 and grand orbital quantum number @xmath60 for any @xmath11 . substituting in eq(4 ) and projecting on a particular ph , a set of coupled differential equation for the partial wave @xmath61 is obtained @xmath62u_{kl}(r ) + \displaystyle{\sum_{k^{\prime}}}f_{kl}v_{kk^{\prime}}(r ) f_{k^{\prime}l } u_{k^{\prime}l}(r ) = 0&\\ \hspace*{.1 cm } , \end{array } \label{eq.cde}\ ] ] + where @xmath63 , @xmath64 , @xmath65 and @xmath66 . + @xmath67 is a constant and represents the overlap of the ph for interacting partition with the sum of phs corresponding to all partitions @xcite . the potential matrix element @xmath68 is given by @xmath69 here we would like to point out that we did not require the additional short - range correlation function @xmath70 for rb clusters as was necessary for dilute bec . a bec is designed to be very dilute and hence confined by a harmonic oscillator potential of low frequency ( @xmath71 hz ) . the average interatomic separation is thus very large ( @xmath72 ) compared with the range of atom - atom interaction ( @xmath73 ) . moreover the kinetic energy of the atoms is extremely small . hence the effective interaction for large @xmath74 is controlled by the @xmath75-wave scattering length ( @xmath5 ) @xcite . this is achieved by the inclusion of the correlation function @xcite . on the other hand , diffuse van der waals clusters are weakly bound by the actual interatomic van der waals potential ( of range @xmath76 ) , without any confinement . hence no correlation function is needed . the average inter - particle separation is large enough , so that only two - body correlations are expected to be adequate , at least for light clusters . as pointed earlier we choose the van der waals potential with a hard core of radius @xmath77 as the interaction potential , @xmath78= @xmath79 for @xmath80 and = @xmath81 for @xmath82 . for rb atoms , the value of @xmath83 is 2803 ev @xmath84 @xcite . the unmanipulated scattering length corresponding to rb - dimer is @xmath85 @xmath8 . we obtain @xmath5 by solving the two - body schrdinger equation for zero - energy @xcite . we adjust the hard core radius in the two - body equation to obtain the dimer scattering length . in the fig . 1 of ref . @xcite , we see the value of @xmath5 changes from negative to positive passing through an infinite discontinuity as @xmath77 decreases . each discontinuity corresponds to one extra two - body bound state . we observe that tiny change in @xmath77 across the infinite discontinuity causes @xmath5 to jump from very large positive value to very large negative value . for our present calculation , we tune @xmath77 such that it corresponds to single bound state of the dimer . thus calculated @xmath77 is @xmath86 for dimer scattering length of rb atoms . with this set of values of @xmath83 and @xmath77 , we next solve the coupled differential equation [ [ eq.cde ] ] by hyperspherical adiabatic approximation @xcite . in hyperspherical adiabatic approximation , the hyperradial motion is assumed slow compared to hyperangular motion . for the hyperangular motion for a fixed value of @xmath30 , we diagonalize the potential matrix together with the hypercentrifugal term . thus the effective potential for the hyperradial motion is obtained as a parametric function of @xmath30 . for the ground state of the system we choose the lowest eigenpotential @xmath87 [ corresponding eigen column vector being @xmath88 as the effective potential . we plot the effective potential @xmath87 as a function of hyperradius @xmath30 , at the dimer scattering length and for various cluster size @xmath893 , 5 and 40 in fig . [ fig.pot ] . [ cols="^,^ " , ] [ table1 ] study of energy level statistics plays an important role in elucidating the universal properties of quantum systems . berry and tabor conjectured that the eigenenergy levels of a quantum system whose classical dynamics shows integrability , must exhibit the fluctuation property as determined by the uncorrelated poisson statistics . this is in sharp contrast with the bgs conjecture which asserts that the fluctuation property of energy levels of a quantum system whose classical dynamics should exhibit goe ( or gue or gse ) statistics . however , complicated quantum many - body systems often lie between these two contrasting conjectures . thus the purpose of present paper is to consider a relatively complex quantum system whose experimental realization is possible . the van der waals bosonic cluster is such a quantum system which starts to be more and more complex with increase in cluster size . the above mentioned contrasting conjectures have been examined thoroughly by using various statistical observables like nnsd , level number variance @xmath1 and the spectral rigidity @xmath90 . these observables highlight the short and long range correlation , level repulsion , level clustering and how the features of the above observables crucially depend on the cluster size are also focussed . our detailed numerical analysis reveals that for smaller cluster when the system is very close to integrability , berry and tabor conjecture is followed . for large cluster although we observe similar to bgs conjecture , however deviation occurs . for large clusters the system becomes strongly correlated but does not exhibit true chaos . however our present study reveals that the deformed goe type of distribution may be suitable for future investigation . this work is supported by the department of atomic energy ( dae ) , government of india , through grant no . skh acknowledges the council of scientific and industrial reaserch ( csir ) , india for a senior research fellowship through net ( grant no : 08/561(0001)/2010-emr-1 ) . ndc acknowledges financial support from the university grants commission ( ugc ) , india [ grant no : f.40 - 425/2011 ( sr ) ] . skh also acknowledges hospitality of the maharaja sayajirao university of baroda , vadodara , india during a recent visit for this work . o. bohigas and m. -j . giannoni , in mathematical and computational methods in nuclear physics , vol . 209 of lecture notes in physics , edited by j. s. dehesa , j. m. g. gomez , and a. polls ( springer , new york , 1984 ) .

we present a statistical analysis of eigenenergies and discuss several measures of spectral fluctuations and spectral correlations for the van der waals clusters of different sizes . we show that the clusters become more and more complex with increase in cluster size . we study nearest - neighbour level spacing distribution @xmath0 , the level number variance @xmath1 , and the dyson - mehta @xmath2statistics for various cluster sizes . for large clusters we find that although the bohigas - giannoni - schmit ( bgs ) conjecture seems to be valid , it does not exhibit true signatures of quantum chaos . however contrasting conjecture of berry and tabor is observed with smaller cluster size . for small number of bosons , we observe the existence of large number of quasi - degenerate states in low - lying excitation which exhibits the shnirelman peak in @xmath0 distribution . we also find a narrow region of intermediate spectrum which can be described by semi - poisson statistics whereas the higher levels are regular and exhibit poisson statistics . these observations are further supported by the analysis of the distribution of the ratio of consecutive level spacings @xmath3 which is independent of unfolding procedure and thereby provides a tool for more transparent comparison with experimental findings than @xmath0 . thus our detail numerical study clearly shows that the van der waals clusters become more correlated with the increase in cluster size .

introduction methodology:many-body calculation with potential harmonic basis results conclusions

arxiv

pilot systems play an important role in negative discharge development process . in metre - scale spark pilots appear as bipolar formations from a point in space called `` space stem '' and grow in both directions away from and towards the high - voltage electrode . in longer gaps pilots can transform into space leaders @xcite . such space leaders also appear in front of a lightning leader channel , grow in both directions and finally attach to the main channel . the stepped propagation is a common feature of both negative long laboratory sparks and lightning leaders . lightning leader steps are also closely associated with discrete intense bursts of x - ray radiation @xcite and may even be responsible for terrestrial gamma - ray flashes ( tgfs ) , as was previously suggested in @xcite and recently modeled in @xcite . similarly , as shown in @xcite , pilots are involved in x - ray burst generation in long laboratory sparks . thus , experimental study of pilot system formation and development can provide more information about lightning leader x - rays and tgfs . however , our understanding of pilots is mostly based on streak photographs obtained in the last century . the existence of bipolar structures in long laboratory discharges was first shown in 1960 s @xcite . in 1981 the les renardires group performed a fundamental study on negative discharges using various electrodes , gap distances and voltage rise times @xcite . this study provided the first systematic description of the various phases of negative discharge development . phenomena such as negative leader , space stem , pilot system , and space leader were photographed , identified and described . theoretical efforts and models to explain long atmospheric discharges were presented by gallimberti ( @xcite and citations therein ) but these do not explain how pilot systems form and develop @xcite . it is assumed that pilots appear from a `` space stem '' ahead of the leader tip in virgin air . in 2003 vernon cooray @xcite formulated the situation as follows : _ `` the pilot system consists of a bright spot called the space stem of short duration , from which streamers of both polarity develop in opposite directions''_. this is consistent with our observations . we will demonstrate below that pilots are preceded by negative streamer heads and often contain several bright spots . + in this work we first show the pilot system development in the laboratory with high spatial and temporal resolution from the very beginning till attachment to the hv electrode . it is demonstrated how a single negative streamer creates such a complex bipolar structure . then we propose a 1d model of the collective ionization front evolution that is capable to capture most important observed details , such as glowing beads , bipolarity and ionization front collisions . striking similarity between pilots and high - altitude sprites will be discussed at the end . the setup was available at the high - voltage laboratory in eindhoven university of technology from 2008 till 2014 but it is currently dismantled . a 2.4 mv haefely marx generator was used to create metre - long sparks . the voltage was set at 1 mv with 1.2/50 @xmath0s rise / fall time when not loaded . the generator was connected to a spark gap between two conical electrodes . the setup and all measuring equipment was exhaustively described in series of publications @xcite and only its optical part will be repeated here for consistency . to obtain images of the pre - breakdown phenomena between two electrodes , a ns - fast 4 picos camera @xcite was located at 4 m distance from the gap , perpendicular to developing discharge . the camera contains a charge coupled image sensor preceded by a fast switched image intensifier ( iccd ) . the image intensifier is a micro - channel plate that allows adjustment of the camera sensitivity by varying the applied voltage between 600 and 1000 v. the ccd is read out with 12-bit and 780x580 pixels resolution . lenses were either nikon 35 mm f2.8 fixed focus or sigma 70 - 300 mm f/45.6 zoom . the field of view of the camera covers the region below the hv electrode . the camera has a black and white ccd and is not calibrated . the applied color scheme is linked to the light intensity and intended to increase visual perception , but it does not represent the actual plasma temperature . the camera was placed inside an emc cabinet . appropriate shielding protected the camera and its communication cables against electromagnetic interference . more emc aspects of the setup have been discussed in @xcite . in 2009 cooray et al . proposed a mechanism of x - ray generation in long laboratory sparks @xcite . it was suggested that the x - ray bursts are caused by encounters of negative and positive streamer fronts . although the negative discharge development process was simplified in this model , the main idea of streamer encounter as the emission source has recently been experimentally supported . with positive high - voltage pulse , x - rays indeed appear at the moment when positive corona from the hv electrode merge with negative corona from the grounded electrode @xcite . many encounters between individual streamers of opposite polarity occur at this moment . the development of negative discharges is more complex . we photographed the x - ray source region with ns - fast camera , and showed that positive streamers appear in the vicinity of the negative hv electrode @xcite . the positive streamers originate from bipolar pilot systems and encounter nearby negative streamers . for more details and properties of the x - ray emission we refer to @xcite ; a statistical analysis is given in @xcite . it is assumed that the x - rays are generated by high energy electrons in bremsstrahlung process . the first attempt to measure such electrons directly has recently been published in @xcite . here it should be noted that a recent simulation of the x - ray emission questioned the streamer encounter mechanism @xcite . while it was confirmed that such encounters dramatically increase the electric field between two streamer fronts to values much higher than the cold runaway breakdown , the field persists only for several picoseconds due to rapid rise of electron density . the ionization quickly collapses the field , giving no time for the electrons to accumulate high energy . it is possible , however , that the action of electric field on electron acceleration was underestimated in that work , as discussed in subsection [ ssec : discuss_xray ] . nevertheless , in measurements the x - rays bursts and pilots coincide in space and time , so the precise role of the latter requires further investigation . for the sake of consistency , we reproduce here the pilot development as reported in @xcite , and discuss additional features . figure [ fig : collage ] shows the pilot system development process . every image was exposed for 50 ns and represents a single individual discharge at the moment of the most intense x - ray emission . the time delay between two consecutive discharges was at least 10 s. the depicted area is located below the hv electrode . the hv electrode tip is only visible in images ( b ) and ( f ) at the top in the middle . the electrical signals of the full discharge are shown in figure [ fig : plot ] , the voltage @xmath1 over the gap , the currents @xmath2 and @xmath3 through both electrodes and the x - ray signals . two labr@xmath4 scintillation x - ray detectors were placed next to each other . both simultaneously registered a 400 kev signal . all images of figure [ fig : collage ] are taken with 50 ns shutter time that fell within the x - ray time window , or between @xmath5s . the pilot systems occur at an advanced stage of the discharge development . returning to figure [ fig : collage ] , we observe the negative streamers ( 1 ) that originate from the hv electrode ( image ( a ) ) . some of them leave isolated beads ( 2 ) behind during the propagation . we will call them `` streamer beads '' or just beads ; the reader should not confuse these with bead lightning @xcite . in 2d images the beads appear at @xmath6 cm intervals . some beads become branching points of the negative streamer . eventually the channels and heads of the branched streamers form the negative streamer corona . the streamer propagation velocity is measured by two different techniques : ( i ) by measuring the streamer trace length in an image with known exposure time and ( ii ) by measuring the displacement of a streamer head comparing two consecutive short exposure images taken by two cameras . the velocities are measured for many different streamers in different discharges ; we further refer to @xcite for details on negative streamer and corona development . we found that the negative streamer heads propagate at @xmath710@xmath8 m / s , which is in agreement with previously reported in @xcite for 2 m gaps . since the projection into the camera plane always reduces distances , the actual distance and velocity is likely to be closer to the highest measured value than to the average . at the same moment , positive streamers start at the beads , either as single intense streak or as branches . the branches first move perpendicularly ( image ( c ) ) to the streamer channel driven by the streamer electric field , and then they start to propagate towards the hv electrode ( images ( b ) - ( f ) ) . this gives the branches the appearance of a stack of the greek letters @xmath9 . to substantiate the movement towards the electrode , we used two high - speed cameras triggered in sequence ; the results have earlier been published in @xcite and are reproduced here for consistency . figure [ fig : two_cameras ] shows two subsequent images made with 3 ns exposure time . the first image was placed on the red layer of an rgb picture , the second image was delayed by 10 ns with respect to the first one , and placed on the blue layer of the same picture . the arrows indicate the displacement from the red to the blue images . clearly , many streamers move towards the cathode . the positive streamer velocity is difficult to measure correctly due to the limited extension in space and apparent dependence on other factors , such as the proximity of the hv electrode . we estimated velocity of the positive streamer head at @xmath1010@xmath8 m / s , or half the speed of the negative ones . as a result , positive streamers appear brighter in the images than negative assuming equal intrinsic brightness . the entire structure - the negative corona , streamer beads with @xmath9s , is called a pilot system ( 4 ) in @xcite . pilots are encircled in images ( c ) - ( e ) by dashed ellipses . in our setup the last bead appears at @xmath11 cm distance from the tip . we can use the streamer velocity data of figure 6 in @xcite to go back to the moment that the negative streamer head was at this point ( 0.5 @xmath0s in figure [ fig : plot ] ) . then the applied voltage @xmath1 was 500 kv , or the local electric field @xmath12 kv / cm assuming a @xmath13-dependence for @xmath14 . this field is close to the so - called `` stability field '' @xmath15 @xcite . it also indicates that the streamer experiences a shortage in charge and electric field which hinders smooth propagation . the blockade may lead to beading . as a support for this suggestion we refer to @xcite where it was demonstrated that a shorter voltage rise time , i.e. larger @xmath16 , leads to smoother discharge development . the upper parts of figure [ fig : collage](c ) and ( d ) show many @xmath9s that are about to collide with negative streamers or the cathode . such collisions provide the kick - off for the few electrons that become run - away in the electric field and produce x - ray by bremsstrahlung a few nanosecond later @xcite . we observed high frequency cathode current oscillations @xcite simultaneously with the x - rays , and also attribute the oscillation to the collisions . figure [ fig : sketch ] summarizes the pilot system development . it starts with a negative streamer that leaves one or more beads behind , that may grow into @xmath9s , and that finally transforms into the bipolar structure . an attempt to combine this history with the streak photographs of @xcite is hindered by the space - time convolution inherent to streak images . to this adds the larger gaps of 2 and 7 m versus ours of 1 up to 1.5 m and the longer voltage rise time of 6 @xmath0s versus 1.2 @xmath0s . figure 6.1.5(b ) in @xcite shows at least three stationary beads at 10 times larger separation than ours that last for about 0.5 @xmath0s ; the beads seem to appear out of the blue in virgin air . our images demonstrate that a precursor streamer initiates the beads . though it is not apparent from the photographs presented here , previous observations on longer spark gaps show that the pilot system becomes a hot space leader if given sufficient time @xcite . figure 6.3.2 in @xcite shows that the instantaneous velocity of the space stem ranges from @xmath17 to @xmath18 m / s ; the higher velocity goes with the shorter voltage rise time . such space stem behavior can be explained by subsequent launching of positive streamers starting from the first bead to the last . naturally , this will appear as a moving space stem on streak photographs . two types of positive corona emanate from the beads . the first is a single positive streamer , for instance image(a ) in figure [ fig : positive_part ] ; the second , our stack of @xmath9s in image ( b ) , has many streamers . the single positive streamer is significantly brighter and wider than all other streamers . the @xmath9s are shown in detail in figure [ fig : positive_part ] image ( b ) , and also in figure [ fig : collage ] images ( c ) - ( f ) . the streamers follow the local electric field lines towards the hv electrode . there are no visible beads anymore , which indicates that they fade away quickly , in fact about 10 times quicker than those in figure 6.1.5(b ) of @xcite . the faint speckle trace is visible in images ( a ) and ( b ) and indicated by small arrows . in image ( a ) it runs from the hv electrode tip down in the middle of the picture . in image ( b ) it comes from the left upper corner and goes through the structure down . it is a camera artifact . the camera s electronic shutter is switched off during the final breakdown , but some light can still leak through it and appear in the images as a trace . it helps to identify the streamer that grows into the final spark . + figure [ fig : reconnection ] shows an example of the pilot reconnection . both types of pilot systems , as described above , are clearly visible . a reconnection between positive streamers in stp ambient air was previously shown in @xcite and possible mechanism proposed in @xcite . it is shown here that the reconnection also occurs between negative streamers , in this case interconnecting two pilot systems . the characteristic curvature of the streamer path and termination on the edge of another streamer channel indicates that they merged . numerical modeling of streamer development in air has currently reached a rather advanced stage . the streamers discharge has been modelled in full 3d space and based solely on microscopic physical mechanisms , e.g. @xcite . however , obtaining numerically in this way a fully developed branching fractal streamer pattern is computationally difficult and still under development . to study the streamer structures with limited computational resources , one can introduce macroscopic physics , i.e. , assumptions about the details of streamers which are not modelled microscopically @xcite . the `` pilots '' occur only at an advanced stage of a streamer discharge , by which we mean that the streamer corona have been fully developed and the individual streamers may have undergone possibly multiple branching . in order to understand the pilots , we are thus forced to follow the route of macroscopic ( simplified ) modeling . so that we can simplify the modeling , we make a rather crude assumption of spherical symmetry in the developed structured streamer discharge , with physical values being functions only of a single ( radial ) coordinate . the individual streamer branches are thus not considered but are treated collectively , and we represent the collective streamer effects in transverse ( angular ) direction in terms of the effective curvature , which is thus different from the curvature of individual streamer heads . the overall schematic representation of the model used is shown in figure [ fig : model_cartoon ] . we shall model the development of a discharge in quasi - electrostatic approximation , i.e. , neglecting the effects of electromagnetic waves . this can be done since the ratios of typical length and time scales , including the typical streamer velocities @xmath19 m / s ( given above in this paper ) , are much less than the speed of light . in principle , the relatively large @xmath20 m size of the electrode gap can give importance to electromagnetic effects of very fast phenomena happening at time scales @xmath211 ns , which could occur , e.g. , during the streamer collisions ; this is a topic for our future research . the quasi - electrostatic equations may be represented as : @xmath22 in these equations , @xmath23 , @xmath24 , @xmath25 are electric field , potential , and charge density , respectively , @xmath26 is the electric conductivity , @xmath27 is the electron density , @xmath28 and @xmath29 are effective ionization and attachment rates which describe propagation of streamers , which are different from the physical ( microscopic ) ionization and attachment rates ( denoted here by @xmath30 ) , and @xmath31 is the photoionization source . this model includes a row of simplifying assumptions . first , the electron mobility @xmath32 is assumed constant because it varies very little in a large range of electric fields , e.g. , from @xmath33 m@xmath34s@xmath35v@xmath35 at @xmath36 to @xmath37 m@xmath34s@xmath35v@xmath35 at @xmath38 , where @xmath39 v / m is the electric breakdown field @xcite . we may therefore justify our assumption of constant electron mobility by arguing that the discharge - related processes occurring at @xmath40 are much less important than at fields @xmath41 . second , we neglect ion conductivity due to the fact that ion mobility is at least by a factor of @xmath42 smaller @xcite . third , we neglect electron advection effects , since the velocity of electron drift is much smaller than the streamer velocity . this , in particular , leads to having no difference in propagation of positive and negative ionization fronts in our model , while the observed velocities differed by a factor of @xmath432 in the same background field . fourth , we neglected electron diffusion @xmath44 , valued at @xmath45@xmath46 m@xmath34/s in the range of electric fields of interest @xcite , because the characteristic kolmogorov - petrovskii - piskunov velocity of the ionization @xmath47 m / s @xcite is also much smaller than the observed streamer velocities . the electric breakdown occurs above @xmath48 , where the ionization prevails over attachment . however , in a 1d situation the propagation of negative streamers occurs above the negative streamer sustainment field @xmath49 , which is lower than @xmath48 . thus , we choose the functional dependence of the effective 1d rates @xmath50 in such a way that the ionization occurs at @xmath51 v / m @xcite . we approximate these rates with power functions : @xmath52 where @xmath53 ns is the typical ionization time . we note that the empirical data for the physical values of @xmath30 @xcite may also be fitted with power - law functions ( [ eq : nu_powerlaw ] ) , but we of course must use @xmath48 instead of @xmath49 in these formulas . in the case of the physical values @xmath30 , the best - fitting coefficients are @xmath54 and @xmath55 . another source of ionization growth is @xmath31 , the photoionization , which is one of the mechanisms responsible for ionization front propagation ( the other mechanisms being the neglected electron advection and electron diffusion ) . the photoionization is a non - local source @xmath56 where @xmath57 is the `` local '' ionization rate . we model it as the `` exponential profile '' model @xcite : @xmath58 where @xmath59 and @xmath60 may be understood as the characteristic `` length '' and the `` strength '' of photoionization , respectively . this model is chosen for computational efficiency , because we can find @xmath31 as the solution of helmholtz equation : @xmath61 the electrode gap is modelled as 1-dimensional interval @xmath62 $ ] , where the emitting electrode ( the cathode in the case of the negative discharge study considered here ) is at @xmath63 , while the grounded electrode is at @xmath64 m. the system is assumed to be symmetric in the transverse direction . voltage @xmath65 is applied at @xmath63 , while @xmath66 is held at @xmath67 . the discharge is started by small initial ionization at @xmath68 , @xmath63 . the curvature of ionization front @xmath69 is included through the expression for divergence of vectors @xmath70 : @xmath71 for example , a spherically - symmetric system may be modelled by taking @xmath72 . for the results presented here , we take a constant value @xmath73 for simplicity , which is equivalent to the transverse area of the discharge growing exponentially with distance . this may be justified by the streamers branching repeatedly with a fixed interval , and the transverse area being proportional to the number of streamers . in the process of the nonlinear development of ionization growth , the ionization developed into multiple persistent peaks ( seen in figure [ fig : reverse_streamers ] ) with complex dynamics ( i.e. , moving in both directions ) which may be interpreted as various luminous features of streamers . in particular , the first peak in an ionization wave may be interpreted as the streamer head , while the consequent peaks , which move in the same direction or become stationary , may be interpreted as beads . under some conditions ( specified in the next paragraph ) we observed a particular class of such peaks , which exhibited the following stages of evolution : ( 1 ) an ionization peak appears at the most advanced point of the ionization wave ; ( 2 ) the ionization is extinguished in the part of the gap between this peak and the emitting electrode ; ( 3 ) as the voltage at the emitting electrode increases further , a reverse ionization wave separates from the peak and moves towards the emitting electrode , while a direct ionization waves moves towards it , until they collide ; ( 4 ) after even further voltage increase , the ionization peak continued its movement toward the grounded electrode . in view of the observations reported above , these ionization peaks may be interpreted as the pilots which exhibit similar behavior , namely that they launch ionization waves in both directions : positive streamers towards the emitting electrode ( cathode ) and the negative streamers towards the grounded electrode ( anode ) . figure [ fig : reverse_streamers ] presents a snapshot of the process described above ( at stage 3 ) . note that we used a slower - growing shape of ionization function ( @xmath74 instead of 5.5 ) . the higher values of power coefficient @xmath75 create steeper ionization edges . the simulations with other values of @xmath75 were also performed ( e.g. , @xmath76 , @xmath77 ) and they also produce similar results ( i.e. , formation of the `` reverse streamers '' and the hv electrode current pulsations , see below ) . the chosen lower value of @xmath75 in a 1d case creates a smoother front , and thus simulates the uncertainty in the position of the individual streamer heads ( they may be distributed around some average position in @xmath78 ) ; unfortunately , the exact value of @xmath75 which is best suited for this is not known to us at the current stage of research . we also chose the value of the photoionization length @xmath59 so that the simulation produces approximately the observed streamer velocities , while the value of @xmath79 is about the same as can be obtained from experimental data @xcite . ) $ ] kv , @xmath80 m , @xmath81 , @xmath82 , @xmath74 , @xmath83 . arrows show the apparent movement of ionization enhancements . ] as we see , the variations in the ionization rate functional dependencies on @xmath84 did not change qualitatively the result of having `` islands '' of ionization and reverse ionization waves ( stages 23 described above ) . by varying other parameters we preliminary conclude that this stages only appear when ( 1 ) the voltage has a stage when it gradually increases with time and ( 2 ) the photoionization effect is rather large . although the propagation of streamers does need a large photoionization , and the `` islands '' of stage 2 appeared even at small values of @xmath79 , the reverse ionization waves ( stage 3 ) only appeared when @xmath79 exceeded a certain threshold . beside pilots and reverse streamers , another interesting outcome of the presented 1d model were the quasi - periodic current pulses at the high - voltage electrode ( cathode ) , shown in figure [ fig : current_pulses ] , which reproduce , at least qualitatively , the experimentally observed pulses shown in figure [ fig : plot ] . the current was calculated taking into account both conductivity and displacement currents : @xmath85 where the area of the cathode @xmath86 m@xmath34 is chosen so that the current is of the same order as those in figure [ fig : plot ] . , except @xmath87 m was taken in order to capture more pulses . ] in the past , the pilots and stepping mechanism were studied numerically in long negative sparks ( leaders ) by 1d modeling @xcite . however , these models treated separately the stages of the discharge which lead to the development of the pilots and stepping and therefore also take into account transition from the streamer corona to a leader discharge . in contrast to these models , we demonstrated that the leader development is not necessary for the pilot formation , and the similar stepping stages appear automatically from the nonlinear nature of the system without artificially subdividing the process . the colliding individual streamers were considered by @xcite in the context of x - ray production by high electric field , but the streamer configuration was taken as an input on the basis of previous results and only @xmath84 field was calculated . the modeling of a streamer collision , taking into account the microscopic physical processes , was also performed by @xcite . we emphasize that , in contrast to the presented model , the last two works describe collision of two pre - existing streamers and do not deal with the development of the global streamer discharge system . in most existing theoretical treatments ( see chapter 12.3 in @xcite ) the streamer channel is understood as a linear structure along which the ionization ( and therefore the electric field and current ) vary monotonously . for example , the ionization is highest at the head of the streamer channel and gradually decreases in the backward direction . however , the high - speed videos of sprites @xcite reveal luminous `` beads '' which move along each individual streamer channel . we may therefore propose a hypothesis that a streamer channel is not monotonous but has peaks of enhanced conductivity and/or field which manifest themselves as `` beads '' . the presented simulation results supports this hypothesis . namely , there are multi - peak structures in figure [ fig : reverse_streamers ] even when the peaks are moving in the same direction ( so that they are not parts of disconnected streamers ) . however , another interpretation of multi - peak structures in the simulation results is also possible . namely , since we modelled a whole group of streamers , the peaks may be just heads of separate monotonous streamer channels or groups thereof . the beads and periodic structures in ionization waves were attributed to the attachment instability by @xcite , who presented a linear - wave analysis of this phenomenon in their appendix a. the mechanisms included in the two models are slightly different ( e.g. , electron advection in @xcite vs. the effective curvature in the present work ) , so this similarity requires further investigation . pilots strikingly resemble high - altitude discharges known as sprites @xcite . such peculiar features as glowing beads , streamer branching on beads , counter - propagating streamers originated from the beads , difference in brightness of the top and bottom parts and finally reconnection , characterise both phenomena . for visual comparison we refer to figure 16 in @xcite . two known types of sprites , carrot and column , can be directly compared to two types of pilots , as shown in section [ sec : positive_part ] . similarity is clear , despite the fact that in the laboratory pilots are pointed towards the sharp electrode tip , while sprites in nature originate from a dispersed charge region and appear vertically in 2d images . the last concern of scientific community regarding different polarity of pilots and sprites has recently been eliminated . it has been known since 1999 that sprites are not uniquely associated with positive cloud - to - ground ( @xmath88cg ) lightnings , but can also be triggered by negative @xmath89cg flashes @xcite . however , the community continued to be skeptical in accepting sprite polarity asymmetry and existence of `` negative sprites '' [ private conversations at agu / egu meetings ] . five more evidences of sprites following a negative @xmath89cg discharge were published in 2016 @xcite . with simple considerations put forward in the next subsection , we can not yet explain the existence of pilots with opposite polarity . in this case , new experimental studies of positive laboratory discharges with different voltage rise times are highly desirable . as it was mentioned in the introduction , the pilots may transform into space leaders @xcite . we may also speculate how pilots determine the polarity asymmetry between the modes of propagation of negative and positive leaders . let us consider the conditions for formation of a system of forward and reverse streamers moving towards each other , such as one that appeared during stage 3 of the simulation described in section [ ssec : model_results ] , and consider the differences for two different leader polarities . in a negative leader discharge , the electrostatic field in front of the leader converges towards its tip . the negative forward streamers , being closer to the electrode , experience a higher field than the positive reverse streamers that are initiated from a position further away from the leader tip and are on their way to encounter the negative forward streamers . this is consistent with the fact that negative streamers need a higher field to support their propagation than the positive streamers . thus , both forward and reverse streamers may exist at the same time . on the other hand , in a positive leader discharge , the forward streamers are positive while the reverse streamers ( if they appeared ) would be negative . we see that the lower field , which the reverse negative streamers would experience , can not support their propagation . thus , we do not expect formation of reverse streamers in the positive - leader case , the ionization gap ( if it appeared ) would be filled only by forward positive streamers . the role of the decreasing field in the differences between positive and negative leader propagation was also discussed by @xcite , where they suggested that in the negative leader corona the forward - moving electrons attach in the lower field , which interrupts the current and leads to stepping . another implication of experiments and modelling is that the positive and negative streamers , which travel towards each other , will collide at a certain moment of the discharge . this may lead to a increased electric field in the gap between them , which subsequently leads to generation of high - energy electrons @xcite . it has been proposed as the possible mechanism of x - ray production in laboratory spark discharges @xcite . this mechanism has been modelled by @xcite who found that the number of x - ray photons produced may not be sufficient to explain observations . however , we think that the electric field in @xcite has been underestimated . the boundary conditions with fixed potential , represent a perfectly conducting boundary and lead to image charges , whose field reduces the field in the modelling domain . thus , the actual field may be higher and also occupy a bigger volume . on the other hand , a higher field between the two colliding streamers would also lead to an increased streamer velocity , which would reduce the time of the existence of the region with high field . since there are two counteracting effects , it is not clear without repeating the full simulation whether corrected boundary conditions would lead to increase of x - ray production . here , we may note that the observations confirmed the coincidence of occurrence of x - rays in laboratory sparks with colliding positive and negative streamers @xcite , even if the exact mechanism of electron acceleration is still open for discussion . in this work we first tracked a single pilot system development in the laboratory between two conical electrodes under 1 mv applied voltage . it was demonstrated for the first time that pilots do not develop from `` nowhere '' , as was thought before @xcite , but from isolated streamer beads , created in the wake of the negative streamer head . the beads , in principle , can be called a `` space stem '' for consistency with the previous studies , but it is important to highlight that they do not appear in virgin air , but behind a negative streamer . the 1d model of the ionization front evolution demonstrated that such beads and reverse ionization waves can appear with certain photoionization parameters . however , we have not yet demonstrated the differences in the discharge polarity , as the electron drift was neglected ; this is a subject of a future work . taking into account many similarities between pilot systems and sprites , not only in appearance but also in progression , we conclude that these are two manifestations of the same phenomenon . it is very desirable to investigate the pilot system development under different conditions , i.e. temperature , pressure , voltage rise time etc . in addition , the main modelling parameters as electron and photon density , their energy spectrum , the local e - field and potential with respect to ground remain hidden in all measurements . some specially designed probes would solve some ambiguities . this work was supported by the european research council under the european union s seventh framework programme ( fp7/2007 - 2013)/erc grant agreement n. 320839 and the research council of norway under contracts 208028/f50 and 223252/f50 ( coe ) . pavlo kochkin acknowledges financial support by stw - project 10757 , where stichting technische wetenschappen ( stw ) is part of the netherlands organization for scientific research nwo . van deursen a and kochkin p 2015 some emc aspects of a 2 mv marx generator with sensitive diagnostic equipment in the immedeate vicinity _ 2015 ieee international symposium on electromagnetic compatibility ( emc ) _ ( ieee ) pp 13881391

the pilot system development in metre - scale negative laboratory discharges is studied with ns - fast photography . the systems appear as bipolar structures in the vicinity of the negative high - voltage electrode . they appear as a result of a single negative streamer propagation and determine further discharge development . such systems possess features like glowing beads , bipolarity , different brightness of the top and bottom parts , and mutual reconnection . a 1d model of the ionization evolution in the spark gap is proposed . in the process of the nonlinear development of ionization growth , the model shows features similar to those observed . the visual similarities between high - altitude sprites and laboratory pilots are striking and may indicate that they are two manifestations of the same natural phenomenon . version of +

introduction experimental setup the pilot system and its features 1d model of the ionization evolution in the electrode gap discussion conclusions acknowledgement

arxiv

a topological space is said to be _ punctiform _ if it does not contain a compact connected set with more than one point . the most common examples of connected punctiform spaces are connected graphs of hereditarily discontinuous functions and connected spaces with dispersion points . a function @xmath0 is _ hereditarily discontinuous _ if @xmath1 is dense in @xmath2 . if @xmath3 satisfies the conclusion of the intermediate value theorem , sometimes called the _ darboux property _ , and @xmath4 is a countable dense subset of @xmath5 , then the graph of @xmath6 , denoted @xmath7 , is a completely metrizable example of the first type . for instance , let @xmath8 for @xmath9 and put @xmath10 . enumerate the rationals @xmath11 . the function defined by @xmath12 has the darboux property and @xmath13 , therefore @xmath7 is a punctiform connected @xmath14 set in the plane . this example is due to kuratowski and sierpinski @xcite . for examples with similar properties , see mazurkiewicz @xcite and jones @xcite . knaster and kuratowski @xcite gave the first example of a connected space with a dispersion point ( a point @xmath15 in a connected space @xmath2 is a _ dispersion point _ if @xmath16 has no connected subset with more than one point ) . the knaster - kuratowski fan fails to be completely metrizable , but there are connected @xmath14 sets in the plane with dispersion points . see @xcite and @xcite . a connected space is _ widely - connected _ if each of its nondegenerate connected subsets is dense ( p.m. swingle @xcite ) , and _ biconnected _ if it is not the union of two disjoint connected sets each containing more than one point . connected spaces ] widely - connected and biconnected hausdorff spaces are punctiform as a result of the boundary bumping theorem , lemma 6.1.25 in @xcite . none of examples above are widely - connected , though dispersion point spaces are clearly biconnected . widely - connected metric spaces do exist , but it seems that all examples up to this point fail to be completely metrizable . specifically , the planar examples in @xcite and @xcite are not complete ( section 6 of this paper ) . p. erds ( g3 in @xcite and @xcite ) and h. cook ( problem 123 in @xcite ) asked : is there a completely metrizable widely - connected space ? in this paper we show that the answer is yes by constructing a widely - connected not biconnected @xmath14 set in the plane . we do not know if there is a completely metrizable biconnected set without a dispersion point . also , j. mioduszewski ( 23(c)(2 - 3 ) in @xcite ) asked : does every separable metric widely - connected @xmath17 have a metric compactification @xmath18 such that for every composant @xmath19 of @xmath18 , @xmath20 is finite ? we show that the answer is no , even if ` finite ' is replaced with ` countable ' . the question remains open if ` finite ' is replaced with ` hereditarily disconnected ' . let @xmath2 be a space . a subset of @xmath2 is said to be _ clopen _ if it is both closed and open in the topology of @xmath2 . @xmath2 is _ connected _ if it is not the union of two nonempty and disjoint clopen subsets . the _ quasicomponent _ of a point @xmath21 is the intersection of all clopen subsets of @xmath2 which contain @xmath15 . the _ component _ of @xmath15 is the union of all connected subsets of @xmath2 which contain @xmath15 . @xmath2 is _ totally disconnected _ if all of its quasicomponents are singletons , and _ hereditarily disconnected _ if all of its components are singletons . a _ continuum _ is a connected compact hausdorff space . if @xmath2 is a continuum , then @xmath19 is a _ composant _ of @xmath2 if there exists @xmath21 such that @xmath19 is the union of all proper subcontinua of @xmath2 which contain @xmath15 . each composant of @xmath2 is connected and dense in @xmath2this is a standard result in continuum theory . a _ compactification _ of @xmath2 is a compact hausdorff space which contains a dense copy of @xmath2 . @xmath2 is _ completely metrizable _ if its topology is induced by a complete metric , equivalently , if it is metrizable and @xmath14 in one ( each ) of its compactifications . @xmath2 is _ baire _ if the intersection of any countable collection of dense open subsets of @xmath2 is dense in @xmath2 , equivalently , if @xmath2 is not the union of countably many closed nowhere dense subsets . the baire category theorem says that completely metrizable spaces and ( locally ) compact hausdorff spaces are baire . let @xmath19 denote the middle - thirds cantor set in the interval @xmath22 $ ] . if @xmath2 is a subset of @xmath23 , then @xmath2 is _ connectible _ if @xmath24 whenever @xmath25 is clopen in @xmath26 and @xmath27 . block _ is a dense subset of @xmath23 that is connectible and hereditarily disconnected . note that @xmath2 is connectible if and only if @xmath28 is connected , so a connected subset of the cantor fan with a dispersion point becomes a connectible set when the dispersion point is removed . this is the central idea behind two of our examples . let @xmath29 be the knaster bucket - handle continuum in @xmath30 ^ 2 $ ] ; @xmath29 is _ indecomposable _ , meaning that every proper subcontinuum of @xmath29 has empty interior . indeed , every proper subcontinuum of @xmath29 is either a single point or a nowhere dense arc . another useful property of @xmath29 is that @xmath31 is locally homeomorphic to the cantor set times the unit interval . let @xmath32\}$ ] be the diagonal cantor set in @xmath29 . then @xmath33 is the union of @xmath34-many copies of @xmath23 , to wit , @xmath35 . ] ] suppose that @xmath2 is a block . for each @xmath36 let @xmath37 be the simple bending homeomorphism , and let @xmath38=\delta\cup \bigcup_{i\in\omega } \varphi_i [ x].\ ] ] [ one]@xmath39 $ ] is widely - connected . @xmath40 $ ] is connected : suppose that @xmath25 and @xmath41 are nonempty disjoint closed subsets of @xmath17 and @xmath42 . then @xmath43 because @xmath17 is dense in @xmath29 . there exists @xmath44 because @xmath29 is connected . @xmath45 because @xmath46 . there exists @xmath47 such that @xmath48 . thinking of @xmath49 as @xmath50 $ ] , with @xmath51 , give coordinates @xmath52 to @xmath15 ( so @xmath53 ) . assume that @xmath54 . there is a sequence of points @xmath55 in @xmath56 converging to @xmath15 . then @xmath57 for each @xmath36 because @xmath2 is connectible . this sequence converges to @xmath58 . since @xmath41 is closed in @xmath17 , @xmath59 . we have @xmath60 , a contradiction . @xmath17 is _ widely_-connected : let @xmath25 be a non - dense connected subset of @xmath17 , and suppose that @xmath61 . then @xmath62 is a non - degenerate proper subcontinuum of @xmath29 , and is therefore an arc . let @xmath63\hookrightarrow k$ ] be a homeomorphic embedding such that @xmath64)=\overline a$ ] . let @xmath65 with @xmath66 , and let @xmath67 $ ] such that @xmath68 and @xmath69 . since @xmath25 is connected , @xmath70 $ ] is connected , and so @xmath71\subseteq e^{-1}[a]$ ] . clearly @xmath72)\not \subseteq \delta$ ] , so there exists @xmath36 such that @xmath72)\cap k_i\neq\varnothing$ ] . then @xmath73\cap [ r , s]$ ] is a nonempty open subset of @xmath71 $ ] and thus contains a non - degenerate interval @xmath74 . then @xmath75 $ ] is a non - degenerate connected subset of @xmath76 , contrary to the fact that @xmath77 $ ] is hereditarily disconnected . a widely - connected set can not contain an interval of reals , so every widely - connected subset of @xmath29 is dense in @xmath29 . moreover , if @xmath19 is a composant of @xmath29 , then @xmath19 is the union of countably many arcs , each of which is hereditarily disconnected when intersected with @xmath17 . it follows that @xmath20 is a countable union of closed zero dimensional subsets , and therefore has dimension zero . let @xmath78 be the set of all endpoints of intervals removed from @xmath30 $ ] in the process of constructing @xmath19 , and let @xmath79 . let @xmath80 where @xmath81 and @xmath82 are the rationals and irrationals , respectively . clearly @xmath83 is hereditarily disconnected and dense in @xmath23 . the reader may recognize @xmath83 as the knaster - kuratowski fan @xcite minus its dispersion point . proving that @xmath83 is connectible is essentially the same as proving that the knaster - kuratowski fan is connected , so some details have been omitted from the proof below . @xmath83 is connectible . let @xmath25 be a nonempty clopen subset of @xmath84 and let @xmath85 . suppose for a contradiction that @xmath86 . there is an open @xmath87 such that @xmath88 and @xmath89 for some open @xmath90 $ ] . enumerate @xmath91=\{q_i : i\in\omega\}$ ] . for each @xmath36 let @xmath92 each @xmath93 is closed and nowhere dense . by the baire category theorem , @xmath94 is dense , so there exists @xmath95 . @xmath62 and @xmath96 form a nontrivial clopen partition of the connected set @xmath97 , a contradiction . if @xmath18 is a metric compactification of @xmath98 $ ] , then there is a composant @xmath19 of @xmath18 such that @xmath99 . by lavrentieff s theorem , there is a homeomorphism between @xmath14 sets @xmath100 and @xmath101 which extends an automorphism of @xmath83 . note that @xmath102\ ] ] is a countable intersection of dense @xmath14 s in @xmath19 ( see 1.4.c(c ) in @xcite ) . so there exists @xmath103 such that @xmath104\subseteq z$ ] . the corresponding arc in @xmath105 is a proper subcontinuum of @xmath18 , and its intersection with @xmath17 is a copy of the irrationals . the ` visible ' composant of @xmath29 is the one - to - one continuous image of the half line @xmath106 , while every other composant of @xmath29 is a one - to - one continuous image of the real line @xmath107 . @xmath108 $ ] is essentially the subspace of @xmath29 consisting of the rational points in the visible composant and the irrational points in every other composant . define @xmath109 by @xmath110 the hilbert space @xmath111 is the set @xmath112 with the norm topology generated by @xmath113 . the subspace @xmath114 of @xmath111 is called the _ complete erds space_. , while @xmath115 is reserved for the set of rational points in the hilbert space ( the _ erds space _ ) . in the absence of the latter space we have chosen to use the symbol @xmath115 . it is closed in the complete metric space @xmath111 , hence complete . " ] let @xmath116 $ ] be the function with the following graph . ] the cantor set is the unique zero - dimensional compact metric space without isolated points , therefore @xmath117 . we will think of these spaces as being equal . define @xmath118 note that @xmath119 is continuous , as the inclusion @xmath120 is continuous and @xmath6 and @xmath113 are both continuous . let @xmath121.\ ] ] if instead @xmath116 $ ] is defined by @xmath122 , then @xmath119 is a homeomorphic embedding ( see @xcite and @xcite 6.3.24 ) . by adding @xmath123 to @xmath124 $ ] and then contracting it to a point , one obtains a completely metrizable dispersion point space . however , in this case @xmath124 $ ] is nowhere dense in @xmath23 , and thus fails to be a block . we will see that the sinusoidal @xmath6 makes @xmath119 a dense embedding , but destroys completeness . @xmath125 is dense in @xmath23 . let @xmath126 be a nonempty open subset of @xmath23 . we think of @xmath127 as a basic open subset of @xmath128 ; @xmath129 and @xmath130 . for each @xmath131 choose @xmath132 . let @xmath133 . there exists @xmath134 such that @xmath135 . let @xmath136 . claim : there exists @xmath137 such that @xmath138 . observe that there is an increasing sequence @xmath139 of nonnegative rationals converging to @xmath140 , with @xmath141 . each @xmath142 , @xmath47 , is nonnegative rational and can thus be written as a finite sum of terms in @xmath143 . as @xmath144 , we can replace each positive term with a finite sum of terms @xmath145 . summing all terms gives @xmath140 , establishing the claim . now consider @xmath146 . we have @xmath147 and @xmath148 , so that @xmath149 . so @xmath150 . the following is due to erds @xcite . [ bound]@xmath151 is unbounded whenever @xmath25 is nonempty and clopen in @xmath115 . let @xmath25 be a nonempty subset of @xmath115 such that @xmath151 is bounded ; we show @xmath25 has nonempty boundary . let @xmath152 such that @xmath153 for each @xmath154 , and let @xmath155 . define @xmath156 as follows . there is a least @xmath157 $ ] such that @xmath158 ( replacing @xmath159 with @xmath160 for each @xmath161 ) . let @xmath162 if @xmath163 and @xmath164 otherwise . then @xmath165 and @xmath166 . let @xmath167 and @xmath168 . suppose @xmath169 and @xmath170 and increasing integers @xmath171 have been defined , @xmath172 , such that @xmath173 and @xmath174 for @xmath175 . there exists @xmath176 such that @xmath177 whenever @xmath178 . there is a least @xmath179 $ ] such that @xmath180 let @xmath181 . define @xmath182 by letting @xmath183 if @xmath184 and @xmath185 otherwise . finally , define @xmath186 by setting it equal to @xmath182 on @xmath187 $ ] , @xmath188 . @xmath189 , otherwise there is a finite sum @xmath190 greater than @xmath191 , but then @xmath192 if @xmath193 . thus @xmath194 , and so @xmath195 as @xmath196 . so @xmath197 as @xmath198 . therefore @xmath186 is a limit point of @xmath25 . also , @xmath199 , where @xmath200 is equal to @xmath182 with @xmath201 increased to @xmath202 ( so @xmath203 ) . by construction @xmath204 , so it follows that @xmath205 as @xmath198 . therefore @xmath186 is also a limit point of @xmath206 . @xmath125 is connectible . suppose that @xmath25 is a clopen subset of @xmath124\cup c\times \{0\}$ ] , @xmath207 , and @xmath208\setminus a$ ] . there is a clopen @xmath87 such that @xmath209\cap u\times \{0\}\subseteq \xi[\mathfrak e]\setminus a,\ ] ] so that @xmath210 \setminus a$ ] . let @xmath211 be an even integer greater than @xmath212 . then @xmath213\big]\cap \{x\in \mathfrak e:\|x\|<n\}=\xi^{-1}\big[a\cap u\times [ 0,1]\big]\cap \{x\in \mathfrak e:\|x\|\leq n\}\ ] ] is a nonempty ( it contains @xmath186 ) clopen subset of @xmath115 . its set of norms is bounded above ( by @xmath211 ) , contradicting proposition [ bound ] . [ comp]@xmath39 $ ] is completely metrizable if and only if the same is true of @xmath2 . suppose that @xmath2 is completely metrizable . then each @xmath214 $ ] is @xmath14 in @xmath29 . further , the sets @xmath215 $ ] lie in pairwise disjoint open regions of @xmath29 , and so their union is @xmath14 in @xmath29 . thus @xmath39 $ ] is the union of two @xmath14 s , @xmath216 and @xmath217 $ ] , which is again a @xmath14 ( in @xmath29 ) . the converse is true because @xmath39 $ ] contains an open copy of @xmath2 . @xmath83 is not completely metrizable because it contains closed copies of the rationals , e.g. in the vertical lines above points in @xmath78 . therefore @xmath108 $ ] is not complete . let @xmath2 and @xmath105 be baire spaces , with @xmath105 second countable . if @xmath218 is a dense @xmath14 in @xmath219 , then @xmath220 is a dense @xmath14 in @xmath2 . let @xmath221 be a collection of open sets in @xmath219 such that @xmath222 , and let @xmath223 be a countable basis for @xmath105 consisting of nonempty sets . for each @xmath224 and @xmath225 , define @xmath226 @xmath227 : let @xmath228 and @xmath229 . then @xmath230 by density of @xmath231 in @xmath232 . as @xmath233 , we have @xmath234 , so that @xmath235 . thus @xmath236 . now suppose that @xmath237 . fix @xmath36 . as @xmath238 for each @xmath239 , we have that @xmath240 is dense in @xmath241 . thus @xmath242 is a countable intersection of dense open subsets of @xmath241 . by the baire property of @xmath105 , @xmath243 is a dense @xmath14 in @xmath241 , whence @xmath244 . each @xmath245 is closed and nowhere dense in @xmath2 : fix @xmath229 and let @xmath246 . @xmath247 is closed in @xmath2 : let @xmath248 . there exists @xmath249 such that @xmath250 . there is an open set @xmath251 with @xmath252 and @xmath253 . then @xmath254 . so @xmath255 is open . @xmath247 is nowhere dense in @xmath2 : if @xmath256 is nonempty and open , then by density of @xmath218 there exists @xmath257 . then @xmath258 witnesses that @xmath259 . it now follows that from the baire property of @xmath2 that @xmath260 is a dense @xmath14 in @xmath2 . it follows that every dense @xmath14 in @xmath50 $ ] has uncountable intersection with some arc @xmath104 $ ] . @xmath261 $ ] is not complete because @xmath125 has at most one point from each arc . moreover , we find that every dense @xmath14 in @xmath29 has an uncountable intersection with some composant . the original widely - connected space by swingle @xcite has only one point from each composant , while miller s biconnected set @xcite , also widely - connected , has only countably many points in any given composant . both spaces are of course dense in @xmath29 , and so they fail to be complete . [ pr]let @xmath262 be a punctiform connected space . if @xmath263 is a compactification of @xmath262 in which @xmath262 is a @xmath14 set , then @xmath264 contains a compact non - disconnecting set that meets every composant of @xmath263 . let @xmath265 be a collection of compact sets such that @xmath266 . each @xmath267 is non - disconnecting , as @xmath262 is a dense connected subset of @xmath268 . let @xmath127 and @xmath269 be nonempty open subsets of @xmath263 with @xmath270 . for each @xmath271 , let @xmath272 be the component of @xmath15 in @xmath273 . let @xmath274 with the quotient topology and @xmath275 . each member of @xmath276 is a continuum with more than one point . since @xmath262 is punctiform @xmath277=\mathscr c$ ] , where @xmath278 is the canonical epimorphism . claim : @xmath19 is closed in @xmath273 . let @xmath279 . the components and quasicomponents of @xmath273 coincide , so there is a collection @xmath280 of clopen subsets of @xmath273 such that @xmath281 , the component of @xmath282 in @xmath273 , equals @xmath283 . since @xmath284 and @xmath285 is compact , there is a finite @xmath286 such that @xmath287 . then @xmath288 is clopen in @xmath273 . it contains @xmath282 and misses @xmath19 . by similar arguments , @xmath276 is hausdorff , so that @xmath276 is compact hausdorff . by the baire category theorem there exists @xmath152 and an open @xmath289 such that @xmath290 $ ] . then @xmath291 has a point from each component of @xmath292 $ ] , each component of @xmath292 $ ] is contained in a composant of @xmath263 , and @xmath292 $ ] meets every composant because it has nonempty interior in @xmath263 . therefore @xmath291 meets each composant . if @xmath17 is a completely metrizable widely - connected subset of @xmath29 , then there is a closed non - disconnecting @xmath293 such that @xmath294 and @xmath247 intersects each composant of @xmath29 . @xmath17 is punctiform and dense in @xmath29 by the remark following proposition [ one ] . apply theorem [ pr ] with @xmath295 and @xmath296 . the main result of @xcite is that @xmath29 contains a closed non - disconnecting set @xmath247 that meets each composant . considered as a subset of @xmath50 $ ] , @xmath247 is the closure of the graph of a certain hereditarily discontinuous function . specifically , let @xmath297 be a countable dense subset of the non - endpoints in @xmath19 , let @xmath298 be a sequence of positive real numbers such that @xmath299 , and define @xmath300 $ ] by @xmath301 then @xmath302 . [ line]if @xmath127 is open in @xmath19 and @xmath303 is an open interval in @xmath5 , then every quasicomponent of @xmath304 is of the form @xmath305 . because @xmath19 is zero dimensional , each @xmath305 contains a quasicomponent of @xmath304 . now we prove the reverse inclusion . for each @xmath47 , let @xmath306 and @xmath307 . since @xmath6 is a non - decreasing function with set of ( jump ) discontinuities @xmath262 , @xmath247 meets a vertical line @xmath104 $ ] in exactly one point if @xmath308 , and in exactly two points @xmath309 and @xmath310 if @xmath311 . fix @xmath47 , and suppose that @xmath312 . then @xmath313 is equal to @xmath314 depending on whether neither , one , or both of @xmath315 and @xmath316 are in @xmath269 . because @xmath317 is a non - endpoint of @xmath19 , it is the limit of increasing and decreasing sequences of points in @xmath19 . suppose that @xmath318 . then the intervals @xmath319 , @xmath320 , are well - defined subsets of @xmath321 . they limit to points vertically above and below @xmath309 , so that @xmath322 is contained in a quasicomponent of @xmath321 . similarly , if @xmath323 then in @xmath321 the intervals @xmath324 , @xmath325 , bridge the gap in @xmath326 . thus @xmath327 is contained in quasicomponent of @xmath321 . since @xmath262 is dense in @xmath127 and @xmath327 is dense in @xmath328 for each @xmath47 , @xmath329 is contained in a quasicomponent of @xmath321 whenever @xmath330 . when @xmath312 and @xmath331 enumerate the rationals @xmath332 , and for each @xmath47 let @xmath333 . let @xmath334 clearly @xmath105 is @xmath14 in @xmath335 . each @xmath336 is a real line minus one or two shifted copies of @xmath81 , and is therefore hereditarily disconnected and dense in @xmath337 . therefore @xmath105 is hereditarily disconnected and dense in @xmath335 . [ ll]every quasicomponent of @xmath105 is of the form @xmath336 . we need to show each @xmath336 is contained in a quasicomponent of @xmath105 . suppose not . then there exists @xmath103 and a partition of @xmath105 into disjoint clopen sets @xmath25 and @xmath41 such that @xmath338 and @xmath339 . density of @xmath105 in @xmath335 implies that @xmath340 . in particular , @xmath341 , so @xmath342 by connectedness of @xmath337 . now apply the baire category theorem in @xmath343 . as @xmath344 , there exists @xmath152 , an open @xmath87 , and an open interval @xmath345 such that @xmath346 notice that @xmath347 so @xmath348 and @xmath349 are disjoint open sets whose union is @xmath350 . by proposition [ line ] , @xmath351 is contained in either @xmath62 or @xmath96 whenever @xmath330 . so @xmath352\text { and } u_2:=\pi\big[\overline b \cap ( u\times v\setminus f_n)\big]\ ] ] are disjoint open sets , @xmath353 being the first coordinate projection ( @xmath353 is open ) . further , @xmath354 because @xmath355 for each @xmath103 . hence @xmath356 and @xmath357 form a clopen partition of @xmath358 . as @xmath359 and @xmath360 , we have @xmath361 , a contradiction . let @xmath362 be a homeomorphism . then @xmath363 by @xmath364 is a dense homeomorphic embedding that preserves the form of quasicomponents . let @xmath365 $ ] . [ oo]@xmath366 is connectible . suppose that @xmath25 is clopen in @xmath367 and @xmath27 . then @xmath368 by proposition [ ll ] . since @xmath366 is dense in @xmath104 $ ] , we have @xmath24 . in summary , @xmath366 is a completely metrizable block . by propositions [ one ] and [ comp ] , @xmath369 $ ] is a completely metrizable widely - connected space . [ 10]@xmath369 $ ] is not biconnected . claim : every disconnecting subset of @xmath370 $ ] is uncountable . let @xmath371 be a countable subset of @xmath17 . note that proposition [ line ] holds if @xmath247 is replaced by @xmath372 , for any @xmath373 . by the proofs of propositions [ ll ] and [ oo ] , it follows that each @xmath374 is connectible , where @xmath265 is the collection of closed sets removed from @xmath23 during the construction of @xmath366 . therefore @xmath375 is connected . let @xmath376 . we want to show that @xmath377 is connected . suppose to the contrary that @xmath378 is a nontrivial clopen partition of @xmath377 . . there exists @xmath380 . there exists @xmath239 such that @xmath381 . there is a @xmath29-neighborhood of @xmath15 that identifies with @xmath382 $ ] , so that @xmath383\subseteq \overline{k_0}$ ] , @xmath50\subseteq \overline{k_j}$ ] , and @xmath384 . note that @xmath385 $ ] is dense in @xmath386 $ ] for each @xmath103 . if @xmath15 is the limit of sequences of points in @xmath25 and @xmath41 that are in @xmath123 , then since @xmath387 and @xmath388 are connectible , points in @xmath389 $ ] would be limit points of lines in @xmath25 and @xmath41 . this can not happen . therefore there is an open @xmath87 such that @xmath390 , without loss of generality . there exists @xmath391 such that @xmath392 . there is an open @xmath393 with @xmath394 . since @xmath395 is dense in @xmath216 , there exists @xmath396 . we have a contradiction : the line @xmath397 $ ] meets both @xmath25 and @xmath41 . therefore @xmath377 is connected , establishing the claim . since @xmath17 is complete , each of its closed disconnecting subsets has size @xmath398 . let @xmath399 be an enumeration of the closed disconnecting subsets of @xmath17 . let @xmath400 and @xmath401 be distinct points in @xmath402 . if @xmath403 and @xmath404 and @xmath405 have been defined for @xmath406 , then there are two distinct points @xmath407 . then @xmath408 and @xmath409 are disjoint dense connected subsets of @xmath17 . so @xmath410 and @xmath411 are disjoint connected sets whose union is @xmath17 . is there a completely metrizable biconnected set without a dispersion point ? additional set - theoretic assumptions may be required to construct an example , as all currently known metric examples of biconnected spaces without dispersion points , due to e.w . miller @xcite and m.e . rudin @xcite , were constructed under ch or ma . by the proof of proposition [ 10 ] , a separable example would have to contain a countable closed disconnecting set . miller s biconnected set has many such subsets , but , as previously shown , is not complete . we conclude by discussing some interesting open problems . widely - connected spaces are usually constructed as dense subsets of indecomposable continua , and therefore have indecomposable compactifications . this is true of all examples in this paper . the obvious question is whether every ( completely regular ) widely - connected space has an indecomposable compactification . equivalently : is the stone - ech compactification of a ( completely regular ) widely - connected space necessarily an indecomposable continuum ? due to the technique in @xcite , an indecomposable continuum contains a dense widely - connected set provided that it has at least as many composants as its number of closed disconnecting subsets . every metric indecomposable continuum has a dense widely - connected subset because it has the same number ( @xmath398 ) of composants as closed subsets . the technique does not apply in general , as there are ( non - metric ) indecomposable continua with very few composants . the stone - ech remainder of @xmath106 is an indecomposable continuum with only one composant if the axiom ncf is assumed , and bellamy constructed indecomposable continua with one and two composants in zfc . is there a dense widely - connected subset of some indecomposable continuum that has only one composant ? we now return to the variation of mioduszewski s question . if @xmath17 is separable and metrizable , then does @xmath17 have a metric compactification @xmath18 such that for every composant @xmath19 of @xmath18 , @xmath20 is hereditarily disconnected ? we do not even know if every separable metric @xmath17 has a compactification with more than one composant . if true , this would imply a positive answer to question 2 in the case of separable metric spaces . also of interest now are questions 2 and 4 in the context of completely metrizable spaces . the author would like to thank his advisor , michel smith , for his thoughtful input on this topic . rudin , lectures on set theoretic topology , expository lectures from the cbms regional conference held at the university of wyoming , laramie , wyo . , august 12 - 16 , 1974 , conference board of the mathematical sciences regional conference series in mathematics , no . 23 , providence , r.i . ,

a widely - connected set is a connected set that is irreducible between every two of its points , meaning that no proper closed connected subset has more than one point . the object of this paper is the construction of widely - connected subsets of the plane . we give a completely metrizable example that answers a question of paul erds and howard cook . a similar example answers a question of jerzy mioduszewski .

introduction terminology assembly of blocks solution to mioduszewskis problem a totally disconnected block necessary conditions for completeness solution to the erds-cook problem conclusion acknowledgements

arxiv

over the past decade numerous surveys of young stars in star - forming regions have found evidence that the overall frequency of binary and multiple systems is consistently higher than in the general field ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , although the enhancement may vary somewhat from region to region ( see , e.g. , * ? ? ? in general those surveys have indicated roughly twice as many binaries as in the classical study by @xcite , which focussed on solar - type stars in the solar neighborhood that are typically much older . the results for the younger binaries refer mostly to relatively wide systems ( @xmath7 au ) , spatially resolved by high - resolution imaging techniques such as infrared speckle interferometry or adaptive optics ( see , e.g. , * ? ? ? * ) . studies of binaries with smaller separations ( spectroscopic binaries ) in star - forming regions have not been as systematic , but may indicate a similar trend ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? partial lists of known spectroscopic binaries among pre main sequence ( pms ) stars have been presented by @xcite , @xcite , and others , and new binaries are slowly but continuously being added . more than two dozen of the systems with known orbits are double - lined spectroscopic binaries , which provide the most information on the individual masses of the components ( compared to single - lined spectroscopic binaries ) . the young star hd 34700 has recently been reported to be a double - lined spectroscopic binary by @xcite , who also provided evidence of velocity variations . hd 34700 was originally identified as a young object from its strong infrared excess as observed by iras @xcite , a characteristic usually interpreted in terms of a circumstellar disk in other similar vega - excess " stars . it is located in orion , although it is not clear whether it is associated with any of the star - forming complexes in the general vicinity . the optical , infrared , and millimeter - wave properties of hd 34700 have been modeled extensively by @xcite and @xcite . they inferred that the disk has inner and outer radii of roughly 2550 au and 550 au , respectively , depending on the model , and that the circumstellar material is composed mostly of relatively small dust grains ( size @xmath8 m ) . sub - millimeter observations have led to the detection of @xmath9co and @xmath10co emission , from which a total mass of @xmath61 m@xmath11 in dust particles has been derived @xcite . other estimates have varied between 0.2 m@xmath11 and 3.7 m@xmath11 @xcite , depending also on the assumed distance . molecular hydrogen in the disk has been searched for , but not seen @xcite . low - level optical linear polarization has been detected by @xcite and @xcite , consistent with a non - spherically symmetric distribution of dust ( a disk ) . the infrared luminosity of the star is a considerable fraction of its bolometric luminosity ( @xmath6835% ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? as in other young objects , the @xmath46708 absorption line is quite prominent in hd 34700 , and the h@xmath5 line is seen in emission with highly variable profiles @xcite . it is also a strong x - ray source . the spectral type of the star has been listed as g0 @xcite , or more recently as ge @xcite . given the interest in the object and the evidence that it may be a spectroscopic binary , the main motivation for this paper is to present high - resolution spectroscopic observations that indeed confirm its double - lined nature . from these observations we derive an accurate double - lined orbital solution . we discuss also the available indicators of youth . hd 34700 ( also hip 24855 , sao 112630 , iras 05170 + 0535 , @xmath12 , @xmath13 , j2000 , @xmath14 ) was originally placed on our observing program at the harvard - smithsonian center for astrophysics ( cfa ) in 1996 as part of a project to monitor the radial velocities of several dozen nearby stars believed to be young from a variety of spectroscopic and photometric indicators ( infrared excess , h@xmath5 emission , strong @xmath46708 absorption , etc . ) . observations were obtained mostly with an echelle spectrograph on the 1.5-m wyeth reflector at the oak ridge observatory ( harvard , massachusetts ) , and occasionally also with a nearly identical instrument on the 1.5-m tillinghast reflector at the f. l. whipple observatory ( mt . hopkins , arizona ) . a single echelle order centered at 5187 was recorded using intensified photon - counting reticon detectors , giving a spectral window of 45 . the resolving power of these observations is @xmath15 . the signal - to - noise ( s / n ) ratios range mostly from about 20 to 30 per resolution element of 8.5 , with the exception of our first exploratory exposure . that observation in march of 1996 clearly revealed the double - lined nature of the system from the double peaks in the cross - correlation function , despite being a very weak spectrum with a s / n ratio of only 7 . the relative strengths of the peaks were similar , indicating stars of nearly equal luminosity . once we realized this , exposure times were increased accordingly . subsequent observations confirmed the double peaks , and we continued to monitor the star for another four years , obtaining a total of 35 spectra . radial velocities were derived using todcor @xcite , a two - dimensional cross - correlation algorithm well suited to our relatively low s / n spectra . todcor uses two templates , one for each component of the binary , and combines one - dimensional correlation functions into a two - dimensional function that avoids the common blending problems of the standard procedures . the one - dimensional correlation functions were computed using the iraf task xcsao @xcite . the templates were selected from a large library of synthetic spectra based on model atmospheres by r. l.kurucz , computed for us by jon morse ( see also * ? ? ? * ; * ? ? ? * ) . these calculated spectra are available for a wide range of effective temperatures ( @xmath16 ) , projected rotational velocities ( @xmath3 ) , surface gravities ( @xmath17 ) and metallicities . experience has shown that radial velocities are largely insensitive to the surface gravity and metallicity adopted for the templates . consequently , the optimum template for each star was determined from grids of cross - correlations over broad ranges in temperature and rotational velocity ( since these are the parameters that affect the radial velocities the most ) , for an adopted surface gravity of @xmath18 ( based on the system s probable pre main sequence status ) and solar composition . the values obtained are @xmath19 k and @xmath20 for the primary star , and @xmath21 k and @xmath22 for the secondary , with estimated uncertainties of @xmath6150 k and 1 , respectively . these temperatures are consistent with the reported spectral type g0 . we see no evidence in our spectra of the phenomenon of veiling , which is common in many other young stars and was reported by @xcite to be present in hd 34700 . however , veiling can be variable . table [ tab : rvs ] lists the radial velocities for both components , referred to the heliocentric frame . typical uncertainties are given below . the stability of the zero - point of our velocity system was monitored by means of exposures of the dusk and dawn sky , and small systematic run - to - run corrections were applied in the manner described by @xcite . the accuracy of the cfa velocity system , which is within about 0.14 of the reference frame defined by minor planets in the solar system , is documented in the previous citation and also by @xcite and @xcite . following @xcite , we have also determined the light ratio between the secondary and the primary of hd 34700 at the mean wavelength of our spectroscopic observations ( 5187 ) , which is close to the @xmath23 band : @xmath24 . @xcite measured the equivalent widths of 105 relatively unblended lines in both components in one of their high resolution spectra , over the wavelength range 42606770 . the average ratio they found between the lines of the secondary and those of the primary is @xmath25 . since the stars are of very similar temperature , the line - strength ratio should be close to the light ratio between the stars . the @xcite value is somewhat higher than our own . we adopt in the following the straight average , @xmath26 . a double - lined orbital solution was easily obtained from our radial velocities , showing a period of 23.5 days and a significant eccentricity . this orbit is shown graphically in fig . [ fig : orbit ] , and the elements are listed in table [ tab : elem ] , where the symbols have their usual meaning . rms residuals for the primary and secondary are 1.70 and 1.34 , respectively . the slightly larger errors for the primary are explained by its higher value of @xmath27 . velocity residuals from our observations are listed in table [ tab : rvs ] . the heliocentric center - of - mass velocity we derive is @xmath28 . this is in excellent agreement with the radial velocity for the co material in the circumstellar disk measured by @xcite , which is @xmath2921 ( with a fwhm of 4.6 for the @xmath302 transition of co from which the velocity was measured ) . we note also that this radial velocity is not far from typical values in the orion star - forming region , which are about @xmath2925 ( although with a spread of several ; see , e.g. , * ? ? ? * ) . in reporting double lines for hd 34700 , @xcite also presented their radial velocity measurements for both components based on the three high - resolution spectra they obtained . those measurements are in fair agreement with our orbit , as shown in fig . [ fig : orbit ] smaller ) . if the smaller errors are adopted , then the agreement with our orbit for two of their velocities is considerably worse . ] . as described in [ sec : intro ] , there are fairly compelling indications that hd 34700 is a young object from a number of independent studies based on a wide variety of observational techniques . @xcite considered the age to be @xmath610 myr or less , although based only on circumstantial evidence . the fundamental difficulty is that the distance to hd 34700 is essentially unknown , and therefore it can not be placed on the h - r diagram and compared to model isochrones in order to estimate the age . the star was measured by the hipparcos mission @xcite , but the published parallax is rather uncertain ( @xmath31 mas ) , and corresponds to a formal distance of 1160 pc . assumptions on the distance to hd 34700 in the literature have ranged from 55 pc @xcite to 90 pc @xcite , derived by adopting absolute magnitudes and intrinsic colors from the spectral type , and accounting roughly for extinction . @xcite adopted a lower limit of 180 pc , and @xcite relied on the hipparcos determination . based on the light ratio of @xmath26 ( [ sec : orbit ] ) and the apparent system magnitude of @xmath14 @xcite , we infer individual magnitudes of @xmath32 and @xmath33 assuming there is no extinction . we may then use theoretical isochrones such as those by @xcite ( see also * ? ? ? * ) along with our effective temperatures to compute the distance to each star assuming they are located on the zero - age main sequence ( zams ) , and adopting the solar value for the metallicity . this exercise results in distances of 122 pc and 125 pc for the primary and secondary , respectively . if the system were any closer , the stars would fall below the zams in the h - r diagram . an extinction value of @xmath34 changes these minimum distances slightly to 111 pc and 114 pc . thus we conclude that the distance to hd 34700 is unlikely to be less than about 100 pc , and therefore that the smaller values adopted to infer properties of the system in most of the previous studies ( which did not account for the binarity of the object , since it was not known at the time ) are probably unrealistic . a parallax such as that corresponding to 100 pc ( 10 mas ) is large enough that hipparcos would most likely have been able to measure it accurately . a distance of 250 pc ( @xmath35 mas ) , on the other hand , is only about 2@xmath36 from the nominal value reported in the catalog , and could have been mismeasured . at this distance the age inferred from the above models would be about 9 myr for both stars , in the absence of extinction . hd 34700 is located in orion ( @xmath68 northwest of the belt ) , though it is not particularly near any of the more conspicuous star - forming regions in that area . perhaps the closest connection that can be found is the similarity between its center - of - mass velocity and the typical radial velocities of other young stars in orion , mentioned earlier . nevertheless , if we assume the star s distance to be the same as the main star - forming complexes , or @xmath6460 pc , the age inferred from the brightness would be as young as 3 myr according to the model isochrones used above . despite the quite extensive studies of hd 34700 over the past 15 years , it is rather surprising that a measurement of the lithium abundance one of the key indicators of youth has not been made until very recently . @xcite measured more than 100 spectral lines for both components of the binary , including @xmath46708 . the results were not mentioned explicitly in the text of their paper , but were reported in their table 2 ( available only in electronic form ) . the li equivalent width for the primary ( redward component in their spectra , according to our fig . [ sec : orbit ] ) is 0.0881 , and that for the secondary is 0.0790 . examination of their figure 2 suggests , however , that the li line for the blueward component ( secondary ) may be blended with the @xmath46705.1 line of the redward component , in which case their li equivalent width may be overestimated for the secondary . we adopt the measurements at face value . correction for binarity using @xmath26 leads to final values of 0.17 for both components . equivalent widths such as these are not large compared to typical values in t tauri stars , but hd 34700 is considerably hotter than most t tauri stars . lithium depletion is a strong function of temperature , and possibly also of rotational velocity . comparison with diagrams of li equivalent width vs. temperature such as those by @xcite ( their fig . 3 ) , @xcite ( fig . 2 ) , or @xcite ( fig . 1 ) , and others , indicate that the stars in hd 34700 lie essentially on the upper envelope of the pleiades distribution ( age @xmath6120 myr ) , and that the li strength is not quite as strong as in stars in the younger cluster ic 2602 ( age @xmath635 myr ) , which have equivalent widths of up to about 0.25 at these temperatures . we note , however , that if veiling were significant , as discussed by @xcite , the measured equivalent widths for hd 34700 could be underestimated because the lines may be filled in by featureless continuum radiation . therefore , the age of roughly 100 myr or so is only an upper limit . nevertheless , on the basis of the li strength alone it is quite possible that the age is several tens of myr instead of a few myr , and therefore that it is not as young as some of the other evidence presented above would seem to suggest . as proposed by @xcite , hd 34700 could be a young g star near the end of the t tauri phase . from fig . 5 in @xcite and using the effective temperatures we determined for each star , we estimate the abundance of li as @xmath37(li ) @xmath38 3.13.2 on a scale where @xmath37(h ) @xmath39 . hd 34700 shows h@xmath5 in emission , and the line profile changes significantly on very short timescales ( @xmath61 day ) , as illustrated by @xcite . the equivalent width they measured is only 0.6 , which places it in the weak - line t tauri class as opposed to the classical t tauri group emission ( @xmath60.2 ) can be derived from fig . 2 of @xcite . profile variations in other lines have also been reported by @xcite . such changes are not uncommon in young stars . the object is also a strong x - ray source , with a ratio of x - ray to optical flux of @xmath40 . it is listed in the rosat all - sky survey bright source catalogue @xcite under the designation 1rxs j051945.3@xmath29053509 , and its x - ray properties ( flux , hardness ratios , etc . ) are consistent with those of other young stars ( see , e.g. , * ? ? ? the projected rotational velocity measurements we reported earlier ( 28 and 22 for the primary and secondary , respectively ; [ sec : obs ] ) are quite close to those given by @xcite ( 25 and 23 ) , although very different from the @xmath3 of @xmath41 determined by @xcite . the latter value is based on two high - resolution spectra ( @xmath42 , slightly higher than the resolution of our own observations ) taken at similar orbital phases of 0.67 and 0.72 ( see fig . [ fig : orbit ] ) . by chance one of our exposures was obtained on the same night as the first of the mora et al . spectra . fig . [ fig : badvsini ] shows that the lines in that spectrum are severely blended due in part to their intrinsic broadening , which explains the large @xmath3 measurement by @xcite . the velocity separation at this phase is @xmath640 . at other phases the lines are clearly separated , as seen in the figure . the @xmath3 values for the components of hd 34700 are relatively large , and one may ask whether they could be the result of tidal locking of the rotation with the orbital motion of the binary , which often goes together with orbital circularization in closer systems . the significant eccentricity of the orbit and relatively long period of 23.5 days seem to argue against this , although as a rule orbital circularization occurs on much longer timescales than rotational synchronization ( see , e.g. , * ? ? ? if the rotation period @xmath43 of each star were the same as the orbital period , the measured @xmath3 values and the relation @xmath44 lead to values for the projected radii of @xmath45 r@xmath46 and @xmath47 r@xmath46 . these seem much too large for pms stars of this temperature . even if the rotational period corresponded to the orbital motion at periastron ( pseudo - synchronization " ) , eq.(42 ) of @xcite predicts a correction factor for @xmath43 of 0.72 at the measured eccentricity of hd 34700 . the projected radii would then be @xmath48 r@xmath46 and @xmath49 r@xmath46 , still too large in absolute terms by the time the increase due to the projection factor is accounted for . for comparison , at the distance of orion ( and an age of 3 myr ) the same theoretical isochrones used above predict sizes for the stars of about 4.4 r@xmath46 . thus , the components in this system are spinning much more rapidly than synchronization would imply ( at least twice as fast ) . this may perhaps be considered one more indication of youth , since main sequence stars of this spectral type do not usually rotate at more than a few ( e.g. , * ? ? ? based on the pms models by @xcite and assuming that hd 34700 is at a representative distance of 200 pc , we estimate that the absolute masses of the components are roughly 1.11.2 m@xmath46 , similar to what is expected for main sequence stars of the same spectral type . the minimum masses from our orbital solution ( table [ tab : elem ] ) , in turn , imply an inclination angle for the orbit of about 50 . the semimajor axis is then 0.21 au , which is well within the inner radius of the circumstellar disk as modeled by @xcite and @xcite . at the same distance of 200 pc the angular semimajor axis would be 1 mas , difficult to resolve with current interferometers , although the eccentricity of the orbit would make the actual separation some 25% larger at certain phases . if the system is at the distance of orion , the masses inferred from the evolutionary tracks are about twice as large ( and the orbital inclination angle @xmath639 ) , the linear semimajor axis only slightly larger than before ( 0.26 au ) , and the angular semimajor axis would be about 0.6 mas . detecting the motion of the _ photocenter _ of the pair is far beyond the capabilities of the hipparcos mission ( initially we had considered the unmodeled photocentric motion as a possible explanation for the large error in the parallax ) . from the mass ratio and light ratio determined here , we estimate the semimajor axis of the photocenter to be only 2.3% of the angular semimajor axis of the relative orbit . this is mostly because the stars are of similar brightness , so their center of light does not move much . at the distance of orion , the 14 @xmath50as signal would be challenging to measure even for nasa s space interferometry mission . finally , we note that the hipparcos epoch photometry ( @xmath51 band ) shows no sign of photometric variability at the few milli - magnitude level over the duration of the satellite s 3-year mission , which is somewhat unexpected if the star is very young , but perhaps not so much so if it is already near the zams . similar evidence in the @xmath52 bands was presented by @xcite . we have confirmed the young star hd 34700 to be a double - lined spectroscopic binary , and presented an accurate orbital solution with a period of 23.4877 days and a significant eccentricity ( @xmath53 ) . the components are of very nearly equal mass , temperature , and luminosity . the measured projected rotational velocities indicate super - synchronous rotation in both stars . \(f ) rapid rotation of the components ( 28 and 22 , @xmath610 times faster than main - sequence stars of similar spectral type ) in an orbit that is wide enough that the stars are not tidally synchronized . it is difficult to estimate a precise age for the system due to the lack of an accurate parallax . if it is at the distance of orion , theoretical isochrones indicate it is only a few myr old . however , it could be closer and therefore be approaching the zams , in which case the age could be several tens of myr . the @xmath46708 absorption is not quite as strong as seen in other very young stars , but the li measurements for hd 34700 may perhaps be affected by veiling . this research has made use of the simbad database , operated at cds , strasbourg , france , and of nasa s astrophysics data system abstract service . we are grateful to a. arellano - ferro for providing the electronic version of their table 2 in advance of publication , to r.neuhuser for pointers to the x - ray activity , and to the referee for helpful comments . partial support for this work from nasa s massif sim key project ( jpl grant 1240033 ) is also acknowledged . eiroa , c. , garzn , f. , alberdi , a. , de winter , d. , ferlet , r. , grady , c. a. , cameron , a. , davies , j. k. , deeg , h. j. , harris , a.w . , horne , k. , mern , b. , miranda , l. f. , montesinos , b. , mora , a. , oudmaijer , r. d. , palacios , j. , penny , a. , quirrenbach , a. , rauer , h. , schneider , j. , solano , e. , tsapras , y. , & wesselius , p.r . 2001 , , 365 , 110 mora , a. , mern , b. , solano , e. , montesinos , b. , de winter , d. , eiroa , c. , ferlet , r. , grady , c. a. , davies , j. k. , miranda , l. f. , oudmaijer , r. d. , palacios , j. , quirrenbach , a. , harris , a. w. , rauer , h. , cameron , a. , deeg , h. j. , garzn , f. , penny , a. , schneider , j. , tsapras , y. , & wesselius , p. r. 2001 , , 378 , 116 oudmaijer , r. d. , palacios , j. , eiroa , c. , davies , j. k. , de winter , d. , ferlet , r. , garzn , f. , grady , c. a. , cameron , a. , deeg , h. j. , harris , a. w. , horne , k. , mern , b. , miranda , l.f . , montesinos , b. , mora , a. , penny , a. , quirrenbach , a. , rauer , h. , schneider , j. , solano , e. , tsapras , y. , & wesselius , p. r. 2001 , , 379 , 564 voges , w. , aschenbach , b. , boller , th . , br " auninger , h. , briel , u. , burkert , w. , dennerl , k. , englhauser , j. , gruber , r. , haberl , f. , hartner , g. , hasinger , g. , k " urster , m. , pfeffermann , e. , pietsch , w. , predehl , p. , rosso , c. , schmitt , j. h. m. m. , tr " umper , j. , & zimmermann , h. u. 1999 , , 349 , 389 crrrrc 50154.6196 & @xmath2959.90 & @xmath5511.87 & @xmath294.30 & @xmath292.11 & 0.918 + 50382.7060 & @xmath553.56 & @xmath2942.73 & @xmath290.06 & @xmath553.27 & 0.629 + 50411.9077 & @xmath2939.65 & @xmath550.94 & @xmath552.34 & @xmath550.74 & 0.872 + 50448.7009 & @xmath556.86 & @xmath2949.23 & @xmath550.20 & @xmath290.15 & 0.439 + 50710.8038 & @xmath555.50 & @xmath2948.42 & @xmath290.17 & @xmath290.35 & 0.598 + 50729.8705 & @xmath557.53 & @xmath2946.76 & @xmath552.33 & @xmath550.83 & 0.410 + 50752.8929 & @xmath553.27 & @xmath2946.66 & @xmath290.61 & @xmath290.39 & 0.390 + 50781.7730 & @xmath554.28 & @xmath2946.47 & @xmath290.04 & @xmath550.24 & 0.620 + 50804.7755 & @xmath555.30 & @xmath2946.88 & @xmath290.32 & @xmath551.14 & 0.599 + 50836.6429 & @xmath2963.33 & @xmath5523.95 & @xmath551.40 & @xmath550.72 & 0.956 + 50845.6472 & @xmath293.74 & @xmath2943.21 & @xmath293.00 & @xmath291.63 & 0.339 + 50854.5896 & @xmath296.32 & @xmath2935.56 & @xmath550.28 & @xmath550.09 & 0.720 + 50872.6278 & @xmath5510.28 & @xmath2952.14 & @xmath552.31 & @xmath291.74 & 0.488 + 50900.5368 & @xmath291.28 & @xmath2942.12 & @xmath290.47 & @xmath290.61 & 0.676 + 51061.8857 & @xmath558.74 & @xmath2948.45 & @xmath551.06 & @xmath551.67 & 0.546 + 51080.8535 & @xmath290.67 & @xmath2943.00 & @xmath291.39 & @xmath550.06 & 0.353 + 51102.8486 & @xmath298.04 & @xmath2936.35 & @xmath290.86 & @xmath291.29 & 0.290 + 51111.8753 & @xmath291.39 & @xmath2942.33 & @xmath290.83 & @xmath290.57 & 0.674 + 51126.7643 & @xmath296.68 & @xmath2937.16 & @xmath292.11 & @xmath550.55 & 0.308 + 51154.8194 & @xmath556.11 & @xmath2951.72 & @xmath291.97 & @xmath291.20 & 0.502 + 51174.7082 & @xmath551.56 & @xmath2940.03 & @xmath551.25 & @xmath552.62 & 0.349 + 51198.6539 & @xmath551.54 & @xmath2946.46 & @xmath290.62 & @xmath291.94 & 0.369 + 51240.5926 & @xmath2935.28 & @xmath292.74 & @xmath551.44 & @xmath552.40 & 0.154 + 51467.8701 & @xmath2928.09 & @xmath2912.33 & @xmath551.86 & @xmath290.34 & 0.831 + 51522.7615 & @xmath2931.40 & @xmath297.55 & @xmath551.58 & @xmath551.37 & 0.168 + 51542.5413 & @xmath2970.80 & @xmath5528.28 & @xmath291.34 & @xmath550.26 & 0.010 + 51544.6439 & @xmath2952.37 & @xmath5510.82 & @xmath550.28 & @xmath290.18 & 0.099 + 51566.5918 & @xmath2967.81 & @xmath5524.01 & @xmath290.33 & @xmath292.01 & 0.034 + 51576.5787 & @xmath555.35 & @xmath2949.46 & @xmath292.02 & @xmath550.34 & 0.459 + 51577.6052 & @xmath556.70 & @xmath2951.04 & @xmath291.38 & @xmath290.52 & 0.503 + 51590.6213 & @xmath2963.32 & @xmath5522.72 & @xmath550.16 & @xmath550.75 & 0.057 + 51610.4871 & @xmath2952.42 & @xmath5510.79 & @xmath291.36 & @xmath551.40 & 0.903 + 51612.5918 & @xmath2969.90 & @xmath5527.96 & @xmath290.61 & @xmath550.10 & 0.992 + 51623.5376 & @xmath5510.50 & @xmath2950.26 & @xmath553.15 & @xmath290.48 & 0.458 + 51626.5283 & @xmath555.03 & @xmath2948.35 & @xmath291.28 & @xmath550.37 & 0.586 + lccc@cc + @xmath56 ( days ) & 23.4877 @xmath57 0.0013 + @xmath58 ( ) & @xmath59 @xmath57 0.18 + @xmath60 ( ) & 38.83 @xmath57 0.42 + @xmath61 ( ) & 39.33 @xmath57 0.34 + @xmath62 & 0.2501 @xmath57 0.0068 + @xmath63 ( deg ) & 358.1 @xmath57 1.6 + @xmath64 ( hjd@xmath552,400,000 ) & 51072.558 @xmath57 0.093 + + @xmath65 ( m@xmath46 ) & 0.531 @xmath57 0.011 + @xmath66 ( m@xmath46 ) & 0.524 @xmath57 0.012 + @xmath67 & 0.987 @xmath57 0.014 + @xmath68 ( 10@xmath69 km ) & 12.14 @xmath57 0.13 + @xmath70 ( 10@xmath69 km ) & 12.30 @xmath57 0.11 + @xmath71 ( r@xmath46 ) & 35.12 @xmath57 0.25 + + @xmath72 & 35 + time span ( days ) & 1472 + @xmath73 ( ) & 1.70 + @xmath74 ( ) & 1.34 +

we report high - resolution spectroscopic observations of the young star hd 34700 , which confirm it to be a double - lined spectroscopic binary . we derive an accurate orbital solution with a period of @xmath0 days and an eccentricity of @xmath1 . the stars are found to be of similar mass ( @xmath2 ) and luminosity . we derive also the effective temperatures ( 5900 k and 5800 k ) and projected rotational velocities ( 28 and 22 ) of the components . these values of @xmath3 are much higher than expected for main - sequence stars of similar spectral type ( g0 ) , and are not due to tidal synchronization . we discuss also the indicators of youth available for the object . although there is considerable evidence that the system is young strong infrared excess , x - ray emission , @xmath46708 absorption ( 0.17 equivalent width ) , h@xmath5 emission ( @xmath60.6 ) , rapid rotation the precise age can not yet be established because the distance is unknown .

introduction observations radial velocities and orbital solution discussion concluding remarks

arxiv

there is controversy in the literature about the existence of a bimodality in the distribution of radio - to - optical flux ratio , @xmath0 , for quasars ( the so - called quasar radio dichotomy ) . for example , white _ et al . _ ( 2000 ) suggested that previous detections of radio dichotomy were caused by selection effects . on the other hand , ivezi et al . ( 2002 , hereafter i02 ) claimed that a sample of quasars detected by the sdss and first surveys supports the existence of a radio dichotomy . the latter result was recently questioned on statistical grounds by cirasuolo et al . ( 2003 , hereafter c03 ) . i02 determined the distribution of @xmath3 for narrow regions in the @xmath4 ( radio ab magnitude ) vs. @xmath5 ( optical magnitude ) plane that were oriented perpendicular to the @xmath6=const . lines ( see top left panel in figure 1 ) . in other words , the quasar density in the @xmath4 vs. @xmath5 plane , @xmath7 , was found to be a separable function @xmath8 . the @xmath6 distribution , @xmath9 , determined this way has a strong maximum at @xmath10 , and declines towards smaller @xmath6 ( bottom left panel in fig . 1 ) . since a large majority ( @xmath11 ) of quasars undetected by first form another peak at @xmath12 , the local minimum at @xmath13 01 implies the existence of a radio - dichotomy . c03 claimed that a more meaningful quantity is the conditional probability distribution @xmath14 , that is , the @xmath6 distribution for a given ( narrow range of ) @xmath5 , with @xmath15 . here @xmath16 is the differential @xmath5 distribution ( `` optical counts '' ) . for comparison with their work , in this contribution we analyze the behavior of @xmath14 . in the top right panel in figure 1 , we show the @xmath6 vs. @xmath5 distribution for @xmath210,000 quasar candidates detected by both sdss and first ( for more details see york 2000 , i02 , schneider et al . 2003 , and references therein ) . the corresponding @xmath14 displayed in the bottom right panel does not decrease smoothly with @xmath6 ; rather , it suggests a possible local minimum around @xmath17 , and a local maximum around @xmath18 . this distribution is consistent with the c03 best - fit shown by the dashed line in the lower right panel ( the latter is in fact a bimodal function ) . note that , given the first flux limit shown as the diagonal dot - dashed line in the top right panel , only quasars _ brighter _ than @xmath19 can be used to directly constrain the position of the local minimum in @xmath14 , and thus a large area optical survey such as sdss is required ( as opposed to a deeper survey of a smaller area ) . -0.25 in when analyzing the @xmath6 distribution , it is important to realize that the scatter due to k - corrections and quasar variability is much larger than the measurement errors . the uncertainty in @xmath6 ( @xmath20.2 - 0.3 ) is mostly due to optical and radio flux k - corrections , and optical variability . even if the intrinsic @xmath6 were the same for all quasars ( i.e. a @xmath20-function ) , its observed distribution would still have a finite width because of this uncertainty . in practice , this effect smears any features in @xmath6 distribution and would reduce any bimodality , if not taken into account . the need to account for the k - correction can be inferred from the improved agreement between different @xmath6 histograms when the sample is divided into redshift bins . we compared the @xmath14 distributions in different redshift bins using the _ uncorrected _ @xmath6 , and found them to be systematically different . furthermore , the differences between the @xmath14 distributions for different @xmath5 bins in a given narrow redshift bin are smaller than when the whole redshift range is considered . this systematic behavior disappears when a proper k - correction for @xmath6 is applied . we determined the k - correction for @xmath6 , such that @xmath21 , as @xmath22 where @xmath23 and @xmath24 are radio and optical spectral slopes , respectively ( @xmath25 ) . we use the difference between @xmath26 color for a particular source and the median @xmath26 color at the redshift of that source to estimate the optical spectral slope ( richards et al . 2003 ) . for radio spectral slope we assume @xmath23 = -0.5 , which is the median value of radio spectral index for a sample of @xmath2400 quasars with sdss , gb6 , first , nvss and wenss detections ( ivezi et al . , in prep . ) . -0.25 in figure 2 compares the distribution of sdss - first quasars with redshifts in the range 0.52.5 in the @xmath6 vs. @xmath5 plane when @xmath6 is * not * k - corrected ( left ) , and when @xmath6 * is * k - corrected using eq . 1 ( right ) . as evident in the bottom panels , accounting for k - correction increases the significance of the detected bimodality . it is important to use an estimate of the optical spectral slope on an _ object - by - object _ basis it is insufficient to use a mean slope as obtained from e.g. a composite quasar spectrum . the dashed line in the bottom right panel in figure 2 is the best - fit using the same functional form proposed by c03 ( a double gaussian ) . it has a local minimum at @xmath17 and a local maximum at @xmath27 , with the maximum - to - minimum ratio of @xmath22 . as reported by ivezi et al . ( 2002 ) , the fraction of sources with @xmath28 is 8@xmath291 % . the remaining 92% of quasars , most of which are not detected by first , are responsible for the steep rise of @xmath14 for @xmath30 . we conclude that accurate optical and radio measurements for a large and homogeneous sample of radio quasars obtained by sdss and first provide conclusive evidence for the existence of the quasar radio - dichotomy . funding for the creation and distribution of the sdss archive has been provided by the alfred p. sloan foundation , the participating institutions , the national aeronautics and space administration , the national science foundation , the u.s . department of energy , the japanese monbukagakusho , and the max planck society . the sdss web site is http://www.sdss.org/. -0.4 in -0.3 in cirasuolo , m. , et al . 2003 , astro - ph/0306415 ivezi , , et al . 2002 , aj , 124 , 2364 richards , g.t . 2003 , aj , 126 , 1131 schneider , d.p . , 2003 , aj , in press , astro - ph/0308443 white , r.l . , et al . 2000 , apjs , 126 , 133 york , d.g . , et al . 2000 , aj , 120 , 1579

recent claims by ivezi et al . ( 2002 ) that the distribution of the radio - to - optical flux ratio , @xmath0 , for quasars is bimodal ( the so - called quasar radio dichotomy ) were questioned on statistical grounds by cirasuolo et al . ( 2003 ) . we apply the approach suggested by cirasuolo et al . to a sample of @xmath1 objects detected by sdss and first , and find support for the quasar radio dichotomy . the discrepancy between the claims by cirasuolo et al . and the results presented here is most likely because 1 ) the @xmath2100 times larger sample based on two homogeneous surveys that is used here allows a direct determination of the @xmath0 distribution , rather than relying on indirect inferences based on monte carlo simulations of several heterogeneous surveys 2 ) the accurate sdss colors and redshift information allow robust determination of the k - correction for @xmath0 , which , if unaccounted for , introduces significant scatter that masks the intrinsic properties of the quasar @xmath0 distribution . # 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in -0.5 in

what statistics to use? to k-correct, or not? evidence for quasar radio dichotomy

arxiv

the baryon - baryon interactions are basic issues in nuclear physics . in the past years , there has been a lot of work on the nucleon - nucleon interactions ( @xmath4 ) , for there exists much experimental information from nucleon - nucleon scattering . however , the interactions of hyperon - nucleon ( @xmath5 ) are much less known than these of @xmath4 due to the difficulties in performing scattering experiments with the unstable hyperons . in these @xmath5 interactions , only the @xmath6 interaction is known for us by studying the hyper - nuclei . the hyperon - hyperon ( @xmath7 ) interactions are the least known ones in baryon - baryon interactions for the very scarce information in experiments . more studies of @xmath5 , @xmath7 interactions are needed not only by the development of the nuclear physics , but also by the application in the other fields , such as in astrophysics . there have been some typical models for the study of @xmath5 and @xmath7 interactions . such as the @xmath8 quark model @xcite , the @xmath9 chiral quark model @xcite , the chiral effective field theory @xcite , the lattice qcd @xcite , the chiral unitary approach @xcite , the meson - exchange model @xcite and the qcdsr @xcite . in this work , we will study the @xmath1 interactions with finite - density qcdsr , which had been developed in the series papers @xcite . with this approach , the properties of nucleons , @xmath10- and @xmath0-hyperons in nucleonic nuclear matter have been reasonably described . based on the sum rules for @xmath11 interactions @xcite , the @xmath1 interactions can be described by substituting the in - medium condensates in nucleonic nuclear matter with pure @xmath0 matter , this approach has been adopted in our previous work , in which the @xmath10@xmath10 interactions were discussed @xcite . the finite - density qcdsr approach focuses on a correlation function of interpolating fields , made up of quark fields , which carry the quantum numbers of the hadron of interest . unlike usual ones , the correlation function is evaluated in the ground state of the nuclear matter rather than in vacuum . the correlation function can represent in a simple phenomenological ansatz for these spectral densities on the one hand . on the other hand , the correlation function can be evaluated at large space - like momenta using an operator product expansion ( ope ) . finally , one can deduce the sum - rules by equating these two different representations using appropriately weighted integrals . the baryon self - energies in medium matter can be related to qcd lagrangian parameters and finite - density condensates . for simplicity , only the leading order of the in - medium condensates are taken into account in this work , which is a reasonable approximation at low nuclear densities @xcite . in the ope for @xmath0 correlation function , we consider all condensates up to dimension 4 , and the terms up to the first order in the strange quark mass @xmath12 . in addition , the contributions from the dimension-6 four - quark condensate are included for their importance . and the leading order in - medium gluon condensates , @xmath13 , @xmath14 , , @xmath15 , @xmath16 and @xmath17 are derived from the chiral perturbation theory ( chpt ) . following the ref . @xcite , to deal with the determined scalar - scalar four - quark condensate @xmath18 , we introduce a arbitrary parameter @xmath19 to describe its density dependence . the paper is organized as follows . in the subsequent section , the sum rules and the condensates are given . the calculations and analysis are presented in sec . [ result ] . section [ summary ] is a summary . the finite - density qcdsr approach has been well devolved in the series lectures @xcite . as done in @xcite , we can easily extent the @xmath0 sum rules in nuclear matter @xcite to describe the @xmath20 interactions in the pure @xmath0 matter by changing the quark and gluon condensates in nuclear matter to those in pure @xmath0 matter ( the sum rules are listed in appendix a ) . using the obtained sum rules , the baryon scalar self - energy @xmath21 and the vector self - energy @xmath22 and the effective mass @xmath23 can be related to the in - medium quark and gluon condensates at finite - density . then , the @xmath1 nuclear potential @xmath2 can be valued by the formula @xmath24 . the essential quark and gluon condensates are calculated in the subsequent section . to obtain the predictions for the @xmath1 interactions in pure @xmath0 matter from the sum rules described above , we need to know the condensates in pure @xmath0 matter . the first order of the condensates in the nuclear matter can be written as @xmath25 where the ellipsis denote the corrections of higher order density , and @xmath26 is the spin averaged @xmath0 matrix element . as we know , in the qcd hamiltonian density @xmath27 , chiral symmetry is explicitly broken by the current quark mass terms . neglecting the isospin breaking effects , one has the hamiltonian @xcite : @xmath28 where @xmath12 and @xmath29 are the strange and light @xmath30 current quark masses , respectively ; @xmath31 and @xmath32 stand for the @xmath30 quark and strange quark fields , respectively . taking the hamiltonian @xmath33 as a function of @xmath29 , in the hellmann - feyman theorem , one obtains @xmath34 in the above equation , we consider the cases of @xmath35 and @xmath36 , where @xmath36 denotes the ground state of @xmath0 matter with @xmath0 density @xmath37 and @xmath35 denotes the vacuum state . taking the difference of these two cases , and taking into account the uniformity of the system yields @xmath38 where @xmath39 is the energy density of the @xmath0 matter , which is given by @xmath40 where @xmath41 is of the higher order term . recently , the in - medium condensates are studied in refs . @xcite , from their analysis it is found that the contributions of the higher order term @xmath42 to the in - medium condensates are small at low density @xmath43 . thus , the contributions of the higher order term @xmath41 are neglected in the calculations . in the chiral perturbation theory ( see the appendix b of @xcite ) , the @xmath0 mass is given by @xmath44 where @xmath45 , @xmath46 and @xmath47 are real parameters in the chiral lagrangian , which can be seen in many references for example @xcite . then , from the eq . ( [ a1 ] ) , we obtain @xmath48\rho,\end{aligned}\ ] ] where @xmath49 is the @xmath50 sigma term , which is given as @xmath51 following the steps above and those in ref . @xcite , the other dimension 3 and 4 quark and gluon condensates are easily obtained . the results are @xmath52\rho , \\ & & \langle q^\dagger i d_0 q\rangle_\rho = \frac{m_q}{4}\langle\overline{q}q\rangle_\rho + \frac{3}{8}m_\sigma[a^u_2 ( \mu^2)+a^d_2 ( \mu^2 ) ] \rho , \nonumber\\ & & \\ & & \langle s^\dagger i d_0 s\rangle_\rho = \frac{m_s}{4}\langle\overline{s}s\rangle_\rho + \frac{3}{4}m_\sigma[a^s_2 ( \mu^2 ) ] \rho , \\ & & \langle \frac{\alpha_s}{\pi}g^2 \rangle_\rho=\langle \frac{\alpha_s}{\pi}g^2 \rangle_0 -\frac{8}{9 } \ { m_\sigma - [ \sigma_{\pi n } + s + \mathcal{k}]\}\rho , \nonumber \\ & & \\ & & \bigg\langle \frac{\alpha_s}{\pi } \bigg[(u'\cdot g)^2+(u'\cdot\tilde{g})^2\bigg]\bigg\rangle_\rho=-\frac{3}{2\pi}m_\sigma \mathcal{c}(\mu^2 ) \rho , \nonumber \\\end{aligned}\ ] ] where @xmath53 @xcite is the strangeness content of nucleon with a dimensionless quantity @xmath54 , the moments of parton distribution functions @xmath55 in @xmath0 hyperon matter are @xmath56 @xcite , and the value of the @xmath57 is about @xmath58 @xcite , and the parameters @xmath59 express as @xmath60 . the other parameters , such as the vacuum condensates and the current quark masses , are adopted the same as those in our previous work @xcite . finally , the in - medium four - quark condensate , @xmath61 , should be considered justly , because they are numerically important in the finite density sum rules . as pointed out in refs . @xcite , the in - medium four - quark condensates in the @xmath0 sum rules are their factorized forms , which may not be justified in nuclear matter because the four - quark condensates are sensitive to the nuclear density , one might suspect that this is an artifact of the factorization . thus , as done in refs . @xcite we choose to parameterize the scalar - scalar four quark condensates so that they interpolate between their factorized form in free space and their factorized form in @xmath0 matter . that is , in the calculations we need replace @xmath62 in eqs . ( [ a1 ] , [ a2 ] ) by modified form @xmath63 : @xmath64 where @xmath19 is the real parameter . the predictions in refs . @xcite suggest that the four - quark condensate , @xmath62 , should depend weakly on the nuclear density . that is , the artificial parameter @xmath19 is most possibly in the range of @xmath65 . in the calculations , to quantify the fit of the left- and right- sides of the @xmath0 sum rules , we use the logarithmic measure @xcite @xmath66.\end{aligned}\ ] ] here @xmath67 , @xmath68 and @xmath69 denote the right - hand sides of the eqs.([a1][a3 ] ) , respectively . in principle , this three terms are equal to @xmath70 . the predictions for @xmath71 , @xmath72 , @xmath73 , @xmath22 are obtained by minimizing the measure @xmath74 . in the zero - density density , we can obtain the @xmath0 mass in vacuum applying the same procedure to the sum rules . firstly , we should choose a proper borel mass @xmath75 in the calculation . in principle the predictions should be independent of the borel mass @xmath75 . however , in practice one has to truncate the operator product expansion and use a simple phenomenological ansatz for the spectral density , which cause the sum rules to overlap only in some limited range of @xmath75 . the previous studies for the octet baryons show that the sum rules do not provide a particulary convincing plateau . nevertheless , we can assume that the sum rules actually has a region of overlap , although it is imperfect . in order to compensate for at least some of the limitations of the truncated sum rules , we normalize the finite - density predictions for all self - energies to the zero - density prediction for the mass . in refs . @xcite , the optimization region of @xmath75 is suggested as @xmath76 gev@xmath77 , thus , in this work we choose the proper borel mass @xmath75 around this region . to find an optimization region for @xmath75 ( in this region the predictions should be less sensitive to @xmath75 than those in other regions ) , we plot the @xmath0 masses in vacuum and in nuclear medium as a function of borel mass @xmath75 in its possible range @xmath78 in fig . [ pic - vac ] and fig . [ pic - m20.5 ] , respectively . it is found that by normalizing the finite - density predictions in the calculation , a good plateau appears in the range of @xmath79 . this optimal borel mass predicted by us consists with the previous predictions in refs . @xcite . in our later calculations , we choose the medium value @xmath80 . then , the sensitivity of the predictions to the @xmath19 is illustrated in fig . [ picf0.5 ] , where @xmath49 , @xmath82 and @xmath81 are fixed at 56 mev , 0.5 and 270 mev , respectively , as done in @xcite . the fig . [ picq0.5 ] is about the optimum results as a function of the momentum @xmath81 at the normal nuclear density @xmath83 with three different values of @xmath19 . from the figures , it is seen that the predictions for @xmath84 and @xmath85 are sensitive to @xmath19 ( i.e. the four - quark condensate ) but slightly dependent on @xmath81 , they monotonously increase with the increment of the @xmath19 . the @xmath86 is insensitive to both the @xmath19 and @xmath81 . there are large uncertainties of the @xmath50 sigma term @xmath49 and the strangeness content of the nucleon @xmath82 . the recent determinations suggest large values for @xmath87 mev , and hence a large strangeness content of the nucleon , i.e. , @xmath88 are obtained . the @xmath11 sum rule study suggests large strangeness content @xmath88 , which is also in agreement with our recent predictions @xmath88 and @xmath8956 mev in the study of the @xmath90 interaction with qcdsr . while the usual adopted values of @xmath49 and @xmath82 is @xmath91 mev and @xmath92 . to study the effect of the parameters @xmath82 and @xmath49 , we plot @xmath84 and @xmath93 as functions of @xmath82 and @xmath49 , respectively , in fig . [ pic - y ] and fig . [ pic - sigma ] . from the two figures , it can be seen that @xmath93 is insensitive to both y and @xmath49 . however , the scalar self - energy is sensitive to the strange quark content @xmath82 , and slightly depends on the @xmath49 . @xmath84 increases with the increment of the @xmath19 , however , decreases with the increment of the @xmath82 . finally , to study the in - medium properties of the @xmath0-hyperon , the effective mass @xmath84 , the vector self - energy @xmath94 and the potential @xmath86 as functions of densities @xmath37 are plotted . for the uncertainties of the @xmath49 and @xmath82 , two sets of the @xmath49 and @xmath82 are adopted in this work . one set is the new determinations @xmath95 mev and @xmath88 ; and the other set is the usual values @xmath91 mev and @xmath96 . from the figures [ aa ] and [ bb ] , we see the effective mass @xmath84 decreases , whereas the vector self - energy @xmath94 increases monotonously with the increment of the @xmath0 density . the differences of the effective mass @xmath84 between the parameter @xmath97 and @xmath98 are more and more obvious with the increment of the density @xmath37 , while the vector self - energy @xmath94 is insensitive to the parameter @xmath19 at different densities . from the figure , we also find that the potential @xmath2 has strong parameter dependence in the whole density region @xmath99 . the differences between the two sets @xmath97 and @xmath98 become more and more obvious with the increment of the density . when we set @xmath95 mev , @xmath88 , and @xmath100 , the effective mass and vector self - energy at @xmath101 are @xmath102 in the range @xmath65 ( see fig . [ aa ] ) ; and the potential @xmath2 is strongly attractive , the strength increases monotonously with the increment of the @xmath0 density @xmath37 . at @xmath103 , the potential can reach to @xmath104 which is much stronger than the nuclear potential of nucleon at normal density . similarly , the attractive @xmath1 potential is also predicted in @xcite , which is even stronger than the @xmath4 one . if we set @xmath91 mev , @xmath96 ( see fig . [ bb ] ) , it is seen that the effective mass @xmath84 , the vector self - energy @xmath94 at @xmath101 are @xmath105 and the potential is @xmath106 in this case , the medium value of the potential @xmath107 mev is also strongly attractive . comparing with the predictions of the two parameter sets , we find the vector self - energies of them are almost equal , however , the nuclear potential with @xmath108 mev are much weaker than that with @xmath109 mev . although the sum - rule predictions for the scalar self - energy are quite sensitive to the four - quark condensates in nuclear medium and parameter @xmath82 , according to the analysis of the four - quark condensates in the series papers , we could predict that the @xmath1 potential is most likely strongly attractive . this potential is much stronger than the @xmath90 potential @xcite in the same conditions . thus , when we deal with the strange nuclear matter , if many @xmath0 hyperons appear , the interactions between @xmath0 hyperons should play crucial roles . according to our predictions , the bound state of double-@xmath0 maybe exist . in this paper , the @xmath1 interactions are analyzed carefully with the finite - density qcdsr approach . the sum - rule analysis indicates that the vector self - energy @xmath22 is insensitive to the sum rule parameters . however , the potential @xmath2 and the scalar self - energy @xmath21 have strong parameter dependence , especially , they are very sensitive to the four quark condensates . although the predictions strongly depend on the undetermined parameters @xmath19 and @xmath82 , it can predict that the @xmath1 potential @xmath2 is most likely strongly attractive , which could be @xmath3 mev or even more attractive at normal nuclear density . if this prediction is the case , the interactions between @xmath0 hyperons should play crucial roles in the strange nuclear matter , when there are multi-@xmath0 hyperons . the bound state of double-@xmath0 maybe exist . this is a preliminary attempt to study the @xmath1 interactions in finite @xmath0 density . more studies are needed to describe the details of the potential . the four quark condensate in medium should be studied further also . this work is supported , in part , by the natural science foundation of china ( grant 10575054 , 10775145 ) , china postdoctoral science foundation , and k. c. wong education foundation , hong kong . the sum rules for the @xmath0 hyperon in the nuclear matter had been deduced by xuemin jin and marina nielsen@xcite , which are given by @xmath111\bigg\rangle_\rho l^{-4/9 } } \nonumber \\ & & { } + \frac{m_s}{18 \pi^2 } m^2 ( 5 e_0 - 2 \frac{\textbf{q}^2}{m^2 } ) \langle\overline{s}s\rangle_\rho l^{-4/9 } + \frac{m^2}{32 \pi^2 } \times \langle\frac{\alpha_s}{\pi}g^2\rangle_\rho e_0 l^{-4/9 } \nonumber \\ & & { } - \frac{4 m^2}{9 \pi^2 } ( e_0 - \frac{\textbf{q}^2}{m^2 } ) \times \langle q^\dagger i d_0 q\rangle_\rho l^{-4/9 } - \frac{m^2}{9 \pi^2 } ( e_0 - 4 \frac{\textbf{q}^2}{m^2 } ) \langle s^\dagger i d_0 s\rangle_\rho l^{-4/9 } \nonumber \\ & & { } + \frac{\overline{e}_q}{6 \pi^2 } m^2 e_0 ( \langle q^\dagger q \rangle_\rho + \langle s^\dagger s\rangle_\rho ) l^{-4/9 } + \frac{4}{3}\langle q^\dagger q \rangle_\rho \langle s^\dagger s\rangle_\rho l^{-4/9 } + \frac{2}{3}\langle\overline{q}q\rangle_\rho^2 l^{4/9},\end{aligned}\ ] ] y. fujiwara , y. suzuki and c. nakamoto , prog . part . phys . * 58 * , 439 ( 2007 ) [ arxiv : nucl - th/0607013 ] . y. fujiwara , m. kohno , c. nakamoto and y. suzuki , phys . c * 64 * , 054001 ( 2001 ) [ arxiv : nucl - th/0106052 ] . c. nakamoto , y. suzuki and y. fujiwara , prog . * 97 * , 761 ( 1997 ) . z. y. zhang , y. w. yu , p. n. shen , l. r. dai , a. faessler and u. straub , nucl . a * 625 * ( 1997 ) 59 . p. n. shen , z. y. zhang , y. w. yu , x. q. yuan and s. yang , j. phys . g * 25 * , 1807 ( 1999 ) [ arxiv : nucl - th/9901068 ] . h. polinder , j. haidenbauer and u. g. meissner , phys . b * 653 * , 29 ( 2007 ) [ arxiv:0705.3753 [ nucl - th ] ] . h. polinder , j. haidenbauer and u. g. meissner , nucl . a * 779 * , 244 ( 2006 ) [ arxiv : nucl - th/0605050 ] . h. polinder , arxiv : nucl - th/0612042 . m. j. savage , arxiv : nucl - th/0612063 . k. sasaki , e. oset and m. j. vicente vacas , phys . rev . c * 74 * , 064002 ( 2006 ) [ arxiv : nucl - th/0607068 ] . j. haidenbauer and u. g. meissner , phys . c * 72 * , 044005 ( 2005 ) [ arxiv : nucl - th/0506019 ] . v. g. j. stoks and t. a. rijken , phys . c * 59 * , 3009 ( 1999 ) [ arxiv : nucl - th/9901028 ] . t. d. cohen , r. j. furnstahl and d. k. griegel , phys . * 67 * , 961 ( 1991 ) . x. m. jin and m. nielsen , phys . c * 51 * , 347 ( 1995 ) [ arxiv : hep - ph/9405331 ] . x. m. jin and r. j. furnstahl , phys . rev . c * 49 * , 1190 ( 1994 ) . x. m. jin , m. nielsen , t. d. cohen , r. j. furnstahl and d. k. griegel , phys . rev . c * 49 * , 464 ( 1994 ) . r. j. furnstahl , d. k. griegel and t. d. cohen , phys . c * 46 * , 1507 ( 1992 ) . t. d. cohen , r. j. furnstahl , d. k. griegel and x. m. jin , prog . phys . * 35 * , 221 ( 1995 ) [ arxiv : hep - ph/9503315 ] . t. d. cohen , r. j. furnstahl and d. k. griegel , phys . c * 45 * , 1881 ( 1992 ) . d. b. leinweber , annals phys . * 198 * , 203 ( 1990 ) .

the properties of @xmath0-hyperons in pure @xmath0 matter are studied with the finite - density quantum chromo - dynamics sum rule ( qcdsr ) approach . the @xmath1 nuclear potential @xmath2 is most likely strongly attractive , it could be about @xmath3 mev or even more attractive at normal nuclear density . if this prediction is the case , the interactions between @xmath0-hyperons should play crucial roles in the strange nuclear matter , when there are multi-@xmath0 hyperons . the bound state of double-@xmath0 maybe exist .

introduction the method calculations and analysis summary the @xmath0 sum rules

arxiv

silicon is mostly locked in sis , sio and sic@xmath1 in the circumstellar envelope ( cse ) of the carbon - rich star irc+10216 , as evidenced observationally and predicted by models @xcite . these molecules are efficiently formed in the gas phase , close to the stellar photosphere as a consequence of chemical processes enabled under thermodynamical equilibrium @xcite . in the dust formation region ( @xmath2520 ) , the si - bearing molecules are likely to condense onto the dust grains due to their highly refractory nature . the silicon contained in the dust grains can form molecules through grain - surface reactions . also , the interaction of shocks produced by the pulsation of the star with the dust grains can extract certain amounts of silicon from the grains and incorporate that silicon into the gas - phase to react and form other species ( see e.g. * ? ? ? * ) . beyond this region , the abundances of si - bearing molecules are expected to decrease up to the outermost shells of the envelope , where the interstellar ultraviolet ( uv ) radiation field dissociates all the remaining molecules . previous interferometer observations showed the spatial distribution of these molecules in irc+10216 . the sis @xmath3=5 4 , @xmath3=6 5 , @xmath3=8 7 , @xmath3=9 8 and @xmath3=12 11 brightness distributions display a quasi - circular shape with a diameter of @xmath220 elongated along the nebular axis ( p.a.@xmath220@xmath4 , * ? ? ? * ; * ? ? ? * ) . recent observations with the combined array for research in millimeter - wave astronomy ( carma ) of the sis @xmath3=14 13 v=0 and v=1 lines have been reported by @xcite , where the v=0 line shows a circular and compact brightness distribution of @xmath22 and displays maser emission . the v=1 brightness distribution shows a compact source centered at the star position . sio @xmath3=5 4 v=0 brightness distribution maps carried out with the submillimeter array ( sma ) were reported in @xcite . they show circular symmetry with a diameter of @xmath26 at the systemic velocity of the source , which is -26.5 ( e.g. * ? ? ? the @xmath3=6 5 v=0 brightness distribution reported in @xcite displays a quasi - circular symmetry centered at the position of the star , with a diameter of @xmath23 elongated along the nebular direction ( ne sw ) . sic@xmath0 observations carried out with the plateau de bure interferometer ( pdbi ) and carma , show a brightness distribution composed of : ( @xmath5 ) an elongated compact component located at the innermost regions of the cse @xcite and , ( @xmath6 ) a hollow shell structure located at @xmath215 from the star @xcite . the formation mechanism for this outer component was suggested in @xcite , where the reaction between si and c@xmath0h@xmath0 yielding sic@xmath0 , could be responsible for the sic@xmath0 enhancement in the outer envelope . in this work we present the cycle0 observations carried out with the atacama large millimeter array ( alma ) toward irc+10216 . we detected emission of sis @xmath3=15 14 lines of vibrationally excited states , from v=0 up to v=7 , and tentatively of v=8 , 9 and 10 . @xmath3=15 14 lines of different isotopologues are also detected : @xmath7sis ( v=05 ) , @xmath8sis ( v=04 ) , si@xmath9s ( v=03 ) , si@xmath10s ( v=04 ) , @xmath7si@xmath9s ( v=0 ) and @xmath7si@xmath10s ( v=0 ) . we also detected emission of sio @xmath3=6 5 ( v=02 ) , @xmath7sio @xmath3=6 5 ( v=0 ) and several lines of sic@xmath0 in the ground vibrational state . the observations were carried out with alma[multiblock footnote omitted ] between 2012 april 8 and 23 during cycle0 . irc+10216 was observed in the frequency range 255.3 to 274.8ghz ( band 6 ) covered by four different setups with a bandwidth of @xmath25ghz , a channel spacing of 0.49mhz and an effective resolution of 0.98mhz . detailed information of each setup is summarized in table[tab : obs ] . the observations were performed using sixteen antennas covering baselines up to 402 m that allowed us to obtain an angular resolution of @xmath20.6 . the shortest baselines used were @xmath220 m which allow us to recover structures with a size up to @xmath1112 . two runs of 72 minutes each were performed , of which 26 minutes correspond to correlations on source . further details about calibration and imaging restoration can be found in @xcite . the coverage of the uv plane achieved with the setup 6 provides low contributions of the sidelobes ( @xmath1210% of the primary beam ) to the dirty beam . for the rest of the setups the uv coverage is worse and large contributions of the sidelobes ( up to 2030% of the primary beam ) appear in the dirty beam . the continuum comes from a point - like source located at @xmath13=9@xmath1447@xmath1557446 and @xmath16=13@xmath416@xmath174386(j2000 ) , which is in good agreement with the position of irc+10216 measured with the very large array ( vla ) with 40 mas resolution @xcite . we measured an intensity peak of 650 mjybeam@xmath18 with an uncertainty of @xmath28% . the calibration of the data was performed using casa[multiblock footnote omitted ] and data analysis with gildas[multiblock footnote omitted ] . .observational parameters.[tab : obs ] [ cols="^,^,^,^,^ " , ] in figs.[fig : si - bear ] and [ fig:29sio ] , we show maps of the emission of the lines sic@xmath0 @xmath19=11@xmath20 10@xmath21 , @xmath7sio @xmath3=6 5 and @xmath7sis @xmath3=15 14 in their ground vibrational state at different offset velocities with respect to the systemic velocity of the source . sic@xmath0 ( fig.[fig : si - bear ] ) displays a central component elongated in the ne sw direction with a size of @xmath245 along the nebular axis and @xmath234 in the perpendicular direction . the elongation is also observed in the @xmath7sio emission ( see below ) and in the sio and sic@xmath0 maps by @xcite where the authors invoke a possible bipolar outflow to explain it . at the systemic velocity of the source and at + 6 offset , a ringlike , clumpy and weak component is seen at @xmath21011 from the central star . the angular distance between the position of the star and the ring structure , considering a distance of @xmath2130pc to the star from us @xcite , corresponds to a linear distance of @xmath22@xmath2210@xmath23 cm . this ringlike component is consistent with the peak abundance of sic@xmath0 in the outer envelope of irc+10216 reported in @xcite and the chemical model of @xcite . although , this ringlike structure is probably filtered in our data given the shortest baselines used . between the central and the ringlike structure , emission of sic@xmath1 is either very low or absent @xcite . finally , the redshifted emission ( + 12 ) displays a quasi - circular distribution with a diameter of @xmath256 . these brightness distributions could be interpreted as the sic@xmath1 is formed in regions close to the star , then it condenses onto the dust grains , and eventually it reappears at the outer shells of the cse , perhaps as a hollow shell , as the consequence of the interaction between the uv galactic radiation field and the cse @xcite . for @xmath7sio ( figs.[fig : si - bear ] and [ fig:29sio ] ) , the bulk of the emission arises from a compact central component with a size of 2 . this line also displays an extended and clumpy distribution , elongated in the ne sw and with a size of @xmath267 . at velocities close to the terminal expansion velocity , @xmath214.5@xcite , the brightness distribution is elongated in the ne sw direction with a size of @xmath234 . the blueshifted emission at -13 displays a decrease just in front of the star , which can be interpreted as self - absorption and probably absorption of the continuum emission mostly coming from the star . there is no conclusive explanation for the elongation ; nevertheless , some authors pointed out that it could evidence the presence of a bipolar outflow in the cse @xcite . the possible presence of a binary companion to cwleo could also play a decisive role in this scenario @xcite . the observed brightness distributions of the vibrationally excited sis lines are expected to be compact and centered on the star since the involved levels are excited at the high temperatures prevailing close to the star . the sis @xmath3=15 14 v=0 line , which displays maser emission @xcite , shows a circular brightness distribution with a diameter of @xmath22 at the systemic velocity of the source . for the sis @xmath3=15 14 v@xmath241 lines , the observed distributions are not spatially resolved . fig.[fig : si - bear ] shows the emission of the @xmath7sis @xmath3=15 14 v=0 line , which displays a brightness distribution of a compact source surrounding the central star with a diameter of @xmath22 . a large - scale artificial modulation can be seen below the 25@xmath25 level . since it is related to the visibilities at short baselines , we do not expect it to modify the shape and flux of the compact central component of the brightness distribution . also , the quality of the several sis and @xmath7sis maps is affected owing to the low uv coverage and non - neglible contribution of the sidelobes for the setups 35 ( see table[tab : obs ] ) . the lines that lie in the range covered by the setup 6 , are those of high vibrationally excited states ( e.g. sis @xmath3=15 14 v@xmath2410 , @xmath7sis @xmath3=15 14 v@xmath246 ) which are tentative and spatially unresolved . [ a ] [ b ] [ c ] the cycle0 observations with alma allowed us to detect several @xmath3=15 14 lines of high vibrationally excited states of sis isotopologues , in particular , up to v=7 for the main isotopologue ( see fig.[fig : sis_all ] ) . the sis @xmath3=15 14 v=8 and v=10 lines are probably blended with unidentified lines , so we consider them to be tentative . additionally , the sis @xmath3=15 14 v=9 line is considered tentatively detected because even though its fwhm measured seems to follow the trend shown in fig.[fig : sis_all ] , its measured integrated intensity is underestimated considering the population diagram of fig.[fig : sis_all ] . all these lines display a compact unresolved emission peaking at the central star . the fwhm of the lines measured from the spectra at the stellar position decreases with increasing upper level energy ( see fig.[fig : sis_all ] ) . we verified this behavior for sis , @xmath7sis , @xmath8sis , si@xmath9s and si@xmath10s . in the dust formation region , the gas displays a velocity gradient as a function of the radial distance to the star , i.e. , the closer to the star the lower the expansion velocity ( * ? ? ? * and references therein ) . hence , those lines involving higher vibrational states , which are excited in inner and warmer regions , are narrower . thermal broadening for the sis lines excited in the dust formation region is @xmath21 , and thus this mechanism could only account partially for the fwhm variation of the lines . we analyzed the excitation conditions of sis with the rotational diagram technique @xcite using the spectra at the stellar position ( see fig.[fig : sis_all ] ) . we considered two different linear trends for the observational data : one for the transitions with e@xmath26@xmath112500k ( i.e. v=0 , 1 and 2 ) , and a different trend for the rest ( i.e. v=37 ) . the v@xmath248 lines are excluded from the fit . the values derived from the v=0 to v=2 fit are uncertain owing to the maser nature of the sis @xmath3=15 14 v=0 line and also due to possible optically thick emission . both data series show a linear behavior consistent with a single vibrational ( excitation ) temperature for each of the fits . the vibrational temperature derived from the fit involving the transitions in high - energy vibrational states is higher than the temperature derived for the low vibrational levels . therefore , the emission produced by sis transitions in high - energy vibrational states can only arise from regions close to the photosphere . this result is similar to the one obtained by @xcite for hcn . we used a large velocity gradient ( lvg ) code to model the sis emission @xcite . further details about the spectroscopic data used in the calculations are given in [ sec : vibmom ] . the sis collisional data were taken from @xcite and extrapolated to high rovibrational levels . we adopted a distance to the star of @xmath2130pc , an effective temperature of @xmath22330k and a stellar radius of @xmath24@xmath2210@xmath27 cm as input for our model @xcite . we used two different models : ( @xmath5 ) the model of @xcite , which was used to reproduce the molecular abundances in the inner layers of irc+10216 , that we call the `` 2012 model '' , and , ( @xmath6 ) this work , which is a modification of the model ( @xmath5 ) . we modified the h@xmath0 density , decreasing it by a factor @xmath22 , as described in @xcite ( this was used to reproduce the dust nucleation zone , 110 , of irc+10216 ) . additionally , the sis abundance in the dust nucleation zone needs to be balanced with a similar increase ( factor of @xmath22 ) to avoid an underestimation of the sis emission . from these models ( see fig.[fig : sis_all ] ) we obtain a good agreement with the vibrational temperature derived from the vibrational diagram and moderate discrepancies with the total column density within a factor of @xmath22 . these discrepancies in the column density may be explained by the dilution due to the size of the emitting region compared to the half power beam width of the synthetic beam , which would increase the optical depth of the lines . the size of the emitting region should decrease with the vibrational state owing to the energies needed to excite those lines . with our models , we found moderate to high optical depths @xmath28(v=1)@xmath210 to @xmath28(v=4)@xmath20.8 for the abundance profile ( @xmath5 ) and @xmath28(v=1)@xmath247 to @xmath28(v=8)@xmath21.0 for the abundance profile ( @xmath6 ) . the potential energy function used to describe the internuclear motions of the sis isotopologues is a born oppenheimer ( bo ) potential properly extended to accommodate born oppenheimer breakdown ( bob ) corrections . the effective potential has the form @xcite @xmath29\frac{j(j+1)}{r^{2 } } \label{pefdmm1}\ ] ] where @xmath30 and @xmath31 are the silicon and sulfur atomic masses and @xmath32 is the reduced mass . the bo potential is given by @xmath33^{2 } \label{pefdmm2}\ ] ] where @xmath34 @xmath35 and @xmath36 and the bob potential and centrifugal correction terms are represented by the power expansions @xmath37 @xmath38 @xmath39 the potential parameters were obtained by nonlinear least squares fitting to the observed infrared and microwave line positions of the sis isotopologues up to v=12 . the final data set included a total of 2863 lines , 414 rotational transitions @xcite and 2449 rovibrational transitions @xcite . mass - independent dunham coefficients , u@xmath40 , have been derived by @xcite . the fit to the potential energy function was performed using the levenberg marquardt algorithm @xcite to minimize the @xmath41 function , with the line positions weighted by the square of the experimental uncertainties . the rovibrational energy levels of the isotopologues needed to calculate the line positions were computed by solving the radial schrdinger equation using the variational method of @xcite along with harmonic - type basis functions . the potential parameters obtained in the fit are given in the comments of the table[vdip ] . the final @xmath41 value was 1.767 . the unweighted standard deviations for the rotational and , vibrational - rotational line positions were 0.0229mhz and 0.000657cm@xmath18 , respectively . the dipole moment function used for sis was determined semiempirically by @xcite , and the dipole moment matrix elements were computed up to v=4 . in table[vdip ] we provide them up to v=12 . the agreement between our calculations and those of @xcite for v@xmath434 is excellent . sic was detected in irc+10216 by @xcite with line profiles indicating that the molecule was produced in an external shell , probably as a product of the photodissociation of sic@xmath0 . our observations did not cover any frequency range where sic lines could arise ; however , lines of @xmath7sic and sic v=1 lie in this range , but they were not detected . for a temperature in the photosphere of 2300k a significant number of sic molecules could be in the v=1 state ( e@xmath44@xmath21400k ) and higher vibrational levels . we derived an upper limit to the sic column density of 4.4@xmath2210@xmath45@xmath46 from the @xmath7sic upper limit , where we used an isotopic @xmath47si/@xmath7si ratio of 20 @xcite . hence , sic@xmath0 is the main carrier of sic bonds in the gas phase in the dust formation zone of irc+10216 . alma has proved to be an outstanding tool to study the molecular emission from cses of evolved stars , even at the early stages of its development . in particular , alma allowed us to detect sis rotational lines in high - energy vibrational states that have been analyzed to constrain the physical conditions of the innermost shells of irc+10216 . we found that these lines should be excited in regions close to the photosphere of irc+10216 . it also has served to unveil the different brightness distributions of si - bearing molecules . we expect that future alma science , with its full suite of capabilities ready for the next observation cycle , would give us the chance to map the brightness distributions of these si - bearing molecules in greater detail , allowing us to understand their formation mechanisms . we thank the spanish mineco / micinn for funding support through grants aya2009 - 07304 , aya2012 - 32032 , the astromol consolider project csd2009 - 00038 and the european research council ( erc grant 610256 : nanocosmos ) .

we report the detection of sis rotational lines in high - vibrational states as well as sio and sic@xmath0 lines in their ground vibrational state toward irc+10216 during the atacama large millimeter array cycle0 . the spatial distribution of these molecules shows compact emission for sis and a more extended emission for sio and sic@xmath0 , and also proves the existence of an increase in the sic@xmath0 emission at the outer shells of the circumstellar envelope . we analyze the excitation conditions of the vibrationally excited sis using the population diagram technique , and we use a large velocity gradient model to compare with the observations . we found moderate discrepancies between the observations and the models that could be explained if sis lines detected are optically thick . additionally , the line profiles of the detected rotational lines in the high energy vibrational states show a decreasing linewidth with increasing energy levels . this may be evidence that these lines could be excited only in the inner shells , i.e. , the densest and hottest , of the circumstellar envelope of irc+10216 .

introduction observations results conclusion

arxiv

thermonuclear x - ray bursts are produced when helium on the surface of an accreting neutron star ignites unstably ( see * ? ? ? * for a review ) . the accreted material itself is usually hydrogen . the helium is produced because the accreted hydrogen is compressed and heated by the column of material above it , and begins to burn steadily via the cno cycle . eventually , the critical temperature and density are reached such that he can burn via a triple-@xmath2 process . the burning is unstable , and engulfs the entire surface of the neutron star in less than a second . this results in a @xmath3 erg s@xmath4 flash of x - rays that out - shines the emission from the accretion flow for tens of seconds . these bursts recur on time scales of hours to days , and so multiple bursts have been observed from the neutron stars in @xmath5 low - mass x - ray binaries . the unstable burning is likely to begin in a small region on the neutron star , and so it has long been expected that the resulting hot spot should produce modulations in the burst flux at the spin period of the neutron star . indeed , `` burst oscillations '' have now been observed with _ rxte _ from 13 neutron star lmxbs ( table [ tab : init ] ) . there are several reasons to believe that the burst oscillations occur at the spin frequencies of the neutron stars . first , and most importantly , two of these sources are x - ray pulsars , which exhibit periodic modulations in the persistent emission between bursts at the same frequency as the burst oscillations @xcite . the frequencies of all of the oscillations are characteristic to each source , and are distributed uniformly between 270 and 620 hz . second , once one accounts for a small frequency drift ( described below ) the burst oscillations are nearly coherent @xcite . in one case , the oscillations are observed to be coherent for @xmath6 cycles during a carbon superburst @xcite . third , the maximum frequencies of the oscillations are stable to within a few parts in a thousand in bursts separated by several years @xcite . fourth , the oscillations are strongest in the rises of bursts , when the nuclear burning is likely to be confined to small areas on the surfaces of the neutron stars @xcite . finally , in the tails of the bursts , the amplitudes of the oscillations as a function of energy are consistent with those expected from temperature variations of @xmath7 kev across the surface of the neutron star @xcite . as signals from the surfaces of neutron stars , these oscillations can be used to study the evolution of the spin frequencies of accreting neutron stars , the spacetime around the star , and how thermonuclear burning proceeds on the stellar surface . however , the simplest models of inhomogeneous burning fail to explain two aspects of the oscillations , which are illustrated in figure [ fig : freqev ] . first , the oscillations persist for up to 15 s during a burst , long after the burning should have engulfed the entire surface of the neutron star . second , they drift upward in frequency by up to 5 hz during the course of a burst , which suggests that the brightness pattern moves opposite the sense of the rotation , such that @xmath8 @xcite . several models have been proposed to explain one or both of these aspects of the oscillations . the upward sense of the frequency drift could be explained by the conservation of angular momentum in an expanding burning layer @xcite . under this model , the energy released in the first second of the burst causes the burning layer to expand and slow relative to the rotation of the neutron star . the frequency drift is observed as the burning layer cools and re - couples to the rest of the neutron star , causing the frequency of the oscillations to increase . unfortunately , it appears that too little energy is released during a burst to cause the burning layer to expand to the height required to explain the observed frequency drifts ( compare * ? ? ? * ; * ? ? ? an additional frequency drift could be produced by accounting for the propagation of the cooling front after the fuel has been exhausted @xcite . if the burst ignites near the rotational equator , then as the burst cools the pressure at the poles is likely to be larger than at the equator . the pressure gradient and the coriolis force would then combine to generate a zonal flow opposite the rotation of the neutron star , in the same manner as the trade winds are formed on earth . as the entire surface cools , the velocity of the flow should slow . any brightness patterns .sources of burst oscillations [ cols="<,^,^ " , ] observations with _ rxte _ have clearly established that the oscillations observed during thermonuclear x - ray bursts occur at the spin periods of the underlying neutron stars , and have provided tantalizing clues as to how nuclear burning proceeds on the surfaces of neutron stars . in the short term , it would be helpful to address outstanding theoretical questions , such as the expected frequencies of rossby - alfvn modes , and the amount the oscillation signals are attenuated by scattering as they propagate away from the neutron star . however , a future x - ray timing mission truly is needed to bring these initial observations to their full potential by making several crucial observations . first , oscillations are currently only observed from 12 of @xmath5 bursting lmxbs , and are only observed from about half of the bursts from any given source . the difference between the amplitudes of the detected oscillations and the upper limits to the non - detections is only a factor of two @xcite , so a future mission with a larger effective area could greatly increase the number of neutron stars with burst oscillations , and consequently with known spin periods . a larger sample would be important for understanding the distribution of observed spin periods , and thus for exploring why all neutron stars appear to be rotating significantly below their break - up frequency @xcite . second , if the burst oscillations are indeed produced by modes in the surface layers , a future x - ray timing mission should detect a spectrum of signals with different latitudinal and radial wave numbers . the spacing of these modes would allow us to measure the pressure , density , and composition of the burning layers on a neutron star ( e.g. , * ? ? ? * ) . oscillations . for instance , a dipolar ( @xmath9 ) temperature distribution on a neutron star should produce oscillations that are slightly - non sinusoidal , because the flux then would be distributed as @xmath10 ( figure [ fig : dipole ] ) . these harmonics would be just below the detection threshold of _ rxte _ , but would be easily detectable with a timing mission with larger area . as is discussed by tod strohmayer in these proceedings , the harmonic content of the oscillations is crucial to constraining the compactness of the neutron star , and hence its equation of state ( see also * ? ? ? * ; * ? ? ? ? * ; * ? ? ? * ; * ? ? ? i would like to thank d. chakrabarty , d. fox , d. galloway , j. hartman , f. zel , and d. psaltis for their significant contributions to the work i have participated in on this topic . this review was written with support from a hubble fellowship from the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , under nasa contract nas 5 - 26555 . 0 bhattacharyya , s. , strohmayer , t. e. , miller , m. c. , & markwardt , c. b. 2004 , submitted to _ apj _ , astro - ph/0402534 bildsten , l. 1998 , _ apj _ , 501 , l89 braje , t. m. , romani , r. w. , & rauch , k. p. 2000 , _ apj _ , 531 , 447 chakrabarty , d. , morgan , e. h. , muno , m. p. , galloway , d. k. , wijnands , r. , van der klis , m. , & markwardt , c. b. 2003 , _ nature _ , 424 , 42 cumming , a. & bildsten , l. 2000 , _ apj _ , 544 , 453 cumming , a. , morsink , s. m. , bildsten , l. , friedman , j. l. , & holz , d. e. 2002 , _ apj _ , 564 , 343 ford , e. c. 1999 , _ apj _ , 519 , l73 galloway , d. k. , chakrabarty , d. , muno , m. p. , & savov , p. 2001 , _ apj _ , 549 , l85 giles , a. b. , hill , k. m. , strohmayer , t. e. , & cummings , n. 2002 , _ apj _ , 568 , 279 heyl , j. s. 2003 , to appear in _ apj _ , astro - ph/0108450 kaaret , p. , in t zand , j. m. m. , heise , j. , & tomsick , j. a. 2002 , _ apj _ , 575 , 1018 kaaret , p. , in t zand , j. m. m. , heise , j. , & tomsick , j. a. 2003 , _ apj _ , 589 , 481 lee , u. 2003 , astro - ph/0309746 miller , m. c. 2000 , _ apj _ , 531 , 458 miller , m. c. & lamb , f. k. 1998 , _ apj _ , 499 , l37 muno , m. p. , chakrabarty , d. , galloway , d. k. , & psaltis , d. 2002b , _ apj _ , 580 , 1048 muno , m. p. , zel , f. , & chakrabarty , d. 2002b , _ apj _ , 581 , 550 muno , m. p. , zel , f. , & chakrabarty , d. 2003 , _ apj _ , 595 , 1066 muno , m. p. , galloway , d. k. , & charkrabarty , d. 2004 , submitted to _ apj _ , astro - ph/0310726 nath , n. r. , strohmayer , t. e. , & swank , j. h. 2002 , _ apj _ , 564 , 353 smith , d. a. , morgan , e. h. , & bradt , h. 1997 , _ apj _ , 479 , l137 spitkovsky , a. , levin , y. , & ushomirsky , g. 2002 , _ apj _ , 566 , 1018 strohmayer , t. e. 2001 , adv . space . , 28 , 511 strohmayer , t. e. & bildsten , l. 2003 , to appear in compact stellar x - ray sources , eds . w. h. g. lewin & m. van der klis , cambridge university press , astro - ph/0301544 strohmayer , t. e. , jahoda , k. , giles , a. b. , & lee , u. 1997 , _ apj _ , 486 , 355 strohmayer , t. e. & markwardt , c. b. 1999 , _ apj _ , 516 , l81 strohmayer , t. e. & markwardt , c. b. 2002 , _ apj _ , 577 , 337 strohmayer , t. e. , markwardt , c. b. , swank , j. h. , & in t zand , j. 2003 , _ apj _ , 596 , l67 strohmayer , t. e. , zhang , w. , & swank , j. h. 1997 , _ apj _ , 487 , l77 strohmayer , t. e. , zhang , w. , swank , j. h. & lapidus , i. 1998 , _ apj _ , 503 , l147 strohmayer , t. e. , zhang , w. , swank , j. h. , smale , a. , titarchuk , l. , day , c. , & lee , u. 1996 , _ apj _ , 469 , l9 weinberg , n. , miller , m. c. , & lamb , d. q. 2001 , _ apj _ , 546 , 1098 white , n. e. & zhang , w. 1997 , _ apj _ , 490 , l87 wijnands , r. , strohmayer , t. , & franco , l. m. 2001 , _ apj _ , 549 , l71 , w. , lapidus , i. , swank , j. h. , white , n. e. , & titarchuk , l. 1997 , _ iauc _ , 6541 , w. , jahoda , k. , kelley , r. l. , strohmayer , t. e. , swank , j. h. , & zhang , s. n. 1998 , _ apj _ , 495 , l9

i review the basic phenomenology and theory of the millisecond brightness oscillations observed during thermonuclear x - ray bursts from 13 of @xmath0 accreting neutron stars in low - mass x - ray binaries . compelling observations indicate that the oscillations are produced by surface brightness patterns on the rapidly rotating neutron stars . however , it remains to be understood ( 1 ) why the brightness patterns producing them persist for up to 15 s during an x - ray burst , whereas the burning should cover the entire surface in less than 1 s , and ( 2 ) why the frequencies drift upward by @xmath1 hz during the course of the burst . these peculiarities can probably be explained by taking into account the expansion of the surface layers caused by the burning , zonal flows that form due to pressure gradients between the equator and poles , and rossby - alfvn modes that are excited in the surface ocean . further progress toward understanding how burning progresses on the surface of the neutron star can be made with a next - generation x - ray timing mission , which would provide a larger sample of sources with oscillations , detect sideband signals produced by the spectrum of modes that should be excited in the neutron star ocean , and measure harmonic structure in the profiles of the oscillations . these observations would be crucial for measuring the distribution of the rotation rates of neutron stars , the progression of unstable nuclear burning in the accreted ocean , and the curvature of the space - time around the neutron star . address = hubble fellow , department of physics and astronomy , university of california , los angeles , ca 90095

introduction models of the oscillations the future

arxiv

a few years ago an approach was put forward @xcite in which heavy mass scales in four dimensions could be replaced by lighter mass scales in higher dimensions . such a class of theories is nowadays conventionally considered in the context of the brane paradigm . in one class of models extra dimensions are felt only by gravity ( as well as other fields transforming as singlet under the standard - model gauge group ) ; in the other class they are felt also by gauge fields . in the former case , the standard - model fields are confined to a ( 3 + 1)-dimensional subspace of a higher - dimensional space some dimensions of which are compactified with a relatively large radius . the absence of any observed deviation from ordinary newtonian gravity in cavendish - type laboratory experiments implies that the largest compactification radius is smaller than around 0.2 mm @xcite . the main goal in both classes of models is to provide a unified theory in which the electroweak scale @xmath5 and the high energy scales ( planck @xcite , string @xcite , and gut scales @xcite ) can coexist . the same scenario has also been successfully applied to neutrino as well as to axion phenomenology . namely , a higher - dimensional seesaw mechanism may provide light neutrino masses without heavy mass scales @xcite . similarly , axion invisibility can be achieved in extra dimensions even with a low fundamental peccei - quinn ( pq ) scale @xcite . the cern axion solar telescope ( cast ) is designed to search for solar axions of a broad energy spectrum which peaks at about 4 kev , through their conversion into real photons inside the transverse magnetic field @xcite . this telescope may improve the current laboratory bounds on the axion - photon coupling , @xmath6 for @xmath7 ev and @xmath8 for @xmath9 @xcite , by a factor of ten or even more . it also has the potential to extend for the first time the axion searches beyond the limit @xmath10 arising from astrophysical constraints on anomalous energy loss by stars @xcite . although the cast telescope could in principle be sensitive to axion masses in the range of a few kev , the coherence - loss constraints @xcite reduce the sensitivity down to around 1 ev . the first goal of the present note is to interpret prospects of cast in the light of the theory with large extra spatial dimensions . we focus on the case when the limit on the size of two large extra compact dimensions is set by direct tests of gravity @xcite . our second goal is to explore the potential of cast for testing the presence of large extra dimensions . axions are pseudoscalars arising in models which resolve the strong _ cp _ problem in quantum chromodynamics ( qcd ) by the pq mechanism @xcite . owing to their potential abundance in the early universe , they are also well - motivated candidates for the dark matter of the universe . in both classes of ( conventional ) invisible axion models referred to as ksvz or hadronic axion models @xcite ( where axions do not couple to electrons at tree level ) and dfsz or grand unified theory ( gut ) models @xcite , the axion - photon coupling strength is given by the relation @xmath11 here @xmath12 is a model - dependent numerical parameter for hadronic axions , while for dfsz axions @xmath13 . furthermore , the mass of the ( qcd ) axion @xmath4 is related to the pq symmetry breaking scale @xmath14 by @xmath15 in order to avoid ambiguities owing to the model - dependence of the parameter @xmath12 for hadronic axions , it proved more convenient to make constraints on the axion - photon coupling than on the pq energy scale or on the axion mass . in contrast , cosmological considerations and astrophysical arguments ( i.e. , axion emission due to nucleon - nucleon bremsstrahlung from the supernova sn 1987a ) bound the axion mass into two possible ranges @xcite . the first window is @xmath16 , in which case the axion could constitute the cold dark matter of the universe . the second one , being around ten to twenty electronvolts , appears to be of interest for hot dark matter . however , such astrophysical constraints on @xmath4 , although the most stringent , suffer from statistical weakness ( with only 19 neutrinos being observed ) as well as from all uncertainties related to the axion emission from a hot / dense medium . it is therefore of crucial importance to probe the axion properties in a model - independent way @xcite . currently , laboratory searches for solar axions @xcite are being extended by the cast experiment at cern . this telescope uses a decommissioned lhc prototype magnet with a field of 9 t and a length @xmath17 of 10 m. the magnet contains two straight beam pipes with an effective cross sectional area @xmath18 , and is mounted on a moving platform with low - background x - ray detectors on either end allowing it to track the sun about 3 hours per day . hadronic axions could be produced abundantly in the core of the sun by the primakoff conversion of the blackbody photons in the coulomb fields of nuclei and electrons in the solar plasma . the outgoing axion flux is robust and does not depend on subtle details of the solar model . it is approximately given by @xcite @xmath19 here @xmath20 is the axion flux at the earth , differential with respect to axion energy ( @xmath21 ) , and expressed as a function of axion mass ( @xmath22 ) . the quantities @xmath21 , @xmath23 , and @xmath22 are to be taken in kev . the probability for an axion - to - photon conversion in the presence of a transverse magnetic field ( @xmath24 ) and a refractive medium ( i ) is given by @xcite @xmath25 , \label{eq4}\end{aligned}\ ] ] where @xmath26 is the momentum difference between photons in the medium and axions , and @xmath27 denotes the inverse absorption length for x - rays . the effective mass ( plasma frequency ) for an x - ray in he can be described in terms of the operating pressure @xmath28 ( at 300 k ) as @xmath29 . the coherence condition @xmath30 @xcite requires @xmath31 for a photon energy of 4.2 kev ( the average axion energy ) and a coherence length of 10 m in vacuum . to search for axions more massive , coherence can be restored by filling the magnetic conversion region with buffer gas . integrating over all axion energies , the expected number of photons @xmath32 , being detected during the times of solar alignment with the magnet ( @xmath33 ) , is finally @xmath34 assuming 100% detection efficiency for the conversion x - rays . at a fixed pressure @xmath28 , the response of cast will be a sharply peaked function of the actual axion mass @xmath22 , with the fractional resolution @xmath35 . a general analysis of the experimental prospects @xcite explores the full two - dimensional ( @xmath36 ) space for qcd axions rather than the narrow band defined by conventional axion models ( although it remains the best - motivated region ) , as it is shown in fig . the experiment is being operated in a scanning mode in which the gas pressure is varied in appropriate steps ( 1yr with vacuum , an additional 1yr with a he gas pressure increased from 0 - 1atm in 100 increments , and an additional 1yr with 1 - 10atm in 365 increments ) to cover a range of possible axion masses up to 0.82 ev . large extra dimensions aim to stabilize the mass hierarchy ( i.e. , the hierarchy between the planck scale and the electroweak scale ) by producing the hugeness of the planck mass @xmath37 via the relation @xmath38 where @xmath39 is the full volume of the compactified space , and the fundamental scale is set at @xmath40 . as already stressed , a singlet higher - dimensional axion field is also free to propagate into the bulk and therefore a similar volume - suppressed formula can be used to lower the fundamental peccei - quinn symmetry - breaking scale @xmath41 @xcite @xmath42 where @xmath43 is the string scale , @xmath44 . since the phenomenologically allowed region for @xmath14 ( also generating the coupling between the axion and matter ) is such that @xmath45 , the axion must be restricted to a subspace of the full higher - dimensional bulk ( @xmath46 ) , if @xmath41 is to reside in the tev - range @xcite . still , @xmath47 is possible for @xmath48 @xcite . in ref . @xcite the full generalization of the higher - dimensional pq mechanism was given , including a thorough discussion of how extra space dimensions may contribute to the invisibility of the pq axion . all new phenomena contributing to the invisibility of the axion and found there rely on a nontrivial axion mass matrix . such a matrix is induced by a mixing between the four - dimensional axion and the infinite tower of kk excitations . the most interesting phenomenological consequence implied by such a mixing is a decoupling of the mass eigenstate of the axion from the pq scale ( for @xmath49 ) . since this feature is crucial for our considerations here , we discuss it in more detail below . now , we can focus on the higher - dimensional case by considering first the kk decomposition of the axion field . as a major step , we need to calculate the estimated number of x - rays at the pressure @xmath50 as a function of the kk axion mass . the masses of the kk modes are given by @xmath51 where we assume that all @xmath52 extra dimensions are of the same size @xmath1 . when the mass splitting for the size @xmath1 ( @xmath53 ) is sufficiently small , one is allowed to use integration instead of summation @xcite . we have already mentioned that because of the nontrivial axion mass matrix , neither the four - dimensional axion nor the kk states represent the mass eigenstates . instead , the eigenvalues are given as solutions to the transcendental equation @xcite @xmath54 hence , in order to estimate the number of modes with the kk index between @xmath52 and @xmath55 , one should parameterize the whole set of eigenvalues of ( [ eq9 ] ) . this can be done by solving ( [ eq9 ] ) for two limiting cases , @xmath56 and @xmath57 . we find for the eigenvalues @xmath58 and @xmath59 the results from ref . @xcite , where only the mass of the axion zero - mode was estimated , can now be easily reproduced from our expressions ( [ eq10 ] ) and ( [ eq11 ] ) . in fig . [ fig1 ] we show the mass of the first kk state as a function of @xmath60 . it can be seen how the mass quickly approaches its limiting value ( 3/2)@xmath61 . a similar feature was found in ref . @xcite for the zero - mode . , where @xmath4 is given by eq . ( [ eq2]).,width=302 ] finally , the total number of x - rays due to all modes of the kk tower reads @xmath62 and @xmath63 where @xmath64 is the surface of a unit radius in @xmath65 dimensions and @xmath66 is defined as @xmath67 with @xmath68 and @xmath69 . the function @xmath66 arises from the mixing between the kk axion modes entering the kk decomposition of the higher - dimensional axion field and the corresponding normalized mass eigenstates @xcite . it also implies both production and detection of kk axions to occur on our standard - model brane . the function @xmath66 also incorporates the effect of rapid decoherency @xcite of the only linear combination of kk states of the bulk axion which couples to standard - model fields . this means that the production and subsequent detection of this particular linear combination of kk states are strongly suppressed . as a consequence our results always reflect a volume - suppressed coupling @xmath70 . if it was not for the decoherency , the linear combination would be coupled to photons with an unsuppressed coupling @xmath71 . in order to achieve an upper limit on the coupling of the axion to photons from the prospects of cast in the framework of large extra dimensions , we apply the central limit theorem at 3@xmath72 level @xmath73 where @xmath74 is the background of the x - ray detector ( with numerical values taken from ref . @xcite ) . for the sake of simplicity , it is assumed that all axions have an average energy of 4.2 kev . combining eq . ( [ eq14 ] ) with eqs . ( [ eq3])-([eq5 ] ) and ( [ eq12])-([eq13a ] ) for the case of two extra dimensions ( we take the largest compactification radius of 0.150 mm as set by direct tests of newton s law @xcite ) , we have derived limits on the axion - photon coupling @xmath75 as a function of the fundamental pq mass , as shown in fig . [ fig2 ] . as a function of the fundamental pq mass @xmath4 . the solid line , corresponding to prospects of cast for qcd axions , is obtained using numerical values from ref . the dashed region marks the theoretically favored relation between @xmath75 and @xmath4 in axion models in four dimensions . the dashed and dot - dashed line correspond to prospects of cast for kk axions in the case of two extra dimensions ( @xmath76 mm ) for @xmath77 and for @xmath78 , respectively.,width=302 ] although the multiplicity of kk states to which cast could be sensitive is large ( @xmath79 for @xmath77 and @xmath80 for @xmath78 ) , one can see from fig . [ fig2 ] that the upper limit on @xmath75 is only at most an order of magnitude more stringent than that obtained in conventional theories . this is due to the fact that cast is a tuning experiment , i.e. , the coherence condition at a fixed pressure is fulfilled only within a very narrow window of axion masses around @xmath81 . the corresponding width ranges ( depending on pressure ) from @xmath82 down to @xmath83 . another feature visible in fig . [ fig2 ] is a strong decrease in sensitivity to @xmath75 for @xmath84 if @xmath78 and for somewhat lower values if @xmath77 . this is due to the fact that in the regime @xmath56 , @xmath66 decreases as fast as @xmath85 . it is just the regime in which the obtained limits on @xmath75 can not be coupled with the zero - mode axion mass ( @xmath86 ) via relations ( [ eq1 ] ) and ( [ eq2 ] ) because in higher dimensions the mass of the axion is approximatively given as @xmath87 @xcite . in contrast with the case of ordinary qcd axions , in theories with large extra dimensions zero - mode axions with masses outside the favored band ( as determined by conventional axion models in four dimensions ) arise quite naturally . now we would like to point to a new phenomenon predicted for cast : sensitivity to particular kk axions . we have already noted that physical kk modes are given by eqs . ( [ eq10 ] ) and ( [ eq11 ] ) . it is expected that more than one axion signal may be observed at different pressures of the gas . therefore , the detection of the corresponding x - rays at least at two pressures may be the signal for the presence of large extra dimensions . as the cast experiment is scanning the range of axion masses up to 0.82 ev , this requirement actually defines a sensitivity of the experiment to test the compactification radius . from eqs . ( [ eq10 ] ) and ( [ eq11 ] ) we obtain @xmath88 if @xmath89 and @xmath90 if @xmath91 , with @xmath92 . it should be noted here that with the cast sensitivity to @xmath75 the former result holds only for @xmath93 ; for @xmath56 the sensitivity rapidly decreases due to the suppression from the @xmath66 function . the present modifications of cast may increase its sensitivity to @xmath75 by a factor of 1.5 @xcite , providing the sensitivity as mentioned above is of the order of that derived from the solar age consideration @xcite . note that in a recent review of the particle data group @xcite the bound on @xmath1 for the case of two extra dimensions , coming from astrophysics , was listed to have a value within the range 90 to 660 nm ( for the most stringent constraints , see the recent work @xcite ) . in conclusion , we have explored the potential of the cast experiment for observing kk axions coming from the solar interior . because of the restrictive coherence condition , in theories with two extra dimensions ( with @xmath76 mm ) a sensitivity in axion - photon coupling improves at most one order of magnitude in both data taking phases . in this case , the obtained limit on @xmath75 can not be coupled with the mass of the axion , which is essentially given by the ( common ) radius of the extra dimensions . in addition , we have demonstrated that the cast experiment , being a tuning experiment with respect to axion masses , may not be sensitive only to an integrated effect of kk modes up to the kinematical limit but also to particular kk axions . with a requirement to have at least two signals while changing pressure of the gas , we have found that cast is capable of probing ( two ) large extra dimensions with a compactification radius @xmath1 down to around 250 nm if @xmath2 , and down to around 370 nm if @xmath3 . 100 i. antoniadis , phys . b * 246 * , 377 ( 1990 ) ; n. arkani - hamed , s. dimopoulos , and g. dvali , phys . b * 429 * , 263 ( 1998 ) ; phys . d * 59 * 086004 ( 1999 ) ; i. antoniadis , n. arkani - hamed , s. dimopoulos , and g. dvali , nucl . * b516 * , 70 ( 1998 ) . c. d. hoyle _ et al_. , phys . 86 * , 1418 ( 2001 ) ; e. g. adelberger for the et - wash group , arxiv : hep - ex/0202008 . e. witten , nucl . b * 471 * , 135 ( 1996 ) ; j. d. lykken , phys . d * 54 * , 3693 ( 1996 ) . k. r. dienes , e. dudas , and t. gherghetta , phys . b * 436 * , 55 ( 1998 ) ; nucl . b * 537 * , 47 ( 1999 ) . k. r. dienes , e. dudas , and t. gherghetta , nucl . b * 557 * , 25 ( 1999 ) ; n. arkani - hamed , s. dimopoulos , g. dvali , and j. march - russell , phys . d * 65 * , 024032 ( 2002 ) . k. r. dienes , e. dudas , and t. gherghetta , phys . d * 62 * , 105023 ( 2000 ) . k. zioutas _ et al_. , nucl . instr . and meth . a * 425 * , 480 ( 1999 ) . c. e. aalseth _ et al_. , ( cast collaboration ) , nucl . b ( proc . suppl . ) * 110 * , 85 ( 2002 ) ; i. g. irastorza _ et al_. , ( cast collaboration ) , nucl . b ( proc . suppl . ) * 114 * , 75 ( 2003 ) ; j. i. collar _ et al_. , ( cast collaboration ) , arxiv : hep - ex/0304024 ; c. eleftheriadis _ et al_. , ( cast collaboration ) , arxiv : astro - ph/0305534 . s. moriyama _ et al_. , phys . b * 434 * , 147 ( 1998 ) ; y. inoue _ et al_. , phys . b * 536 * , 18 ( 2002 ) . g. g. raffelt , arxiv : hep - ph/0207144 . k. van bibber , p. m. mcintyre , d. e. morris , and g. g. raffelt , phys . d * 39 * , 2089 ( 1989 ) . d. m. lazarus _ et al_. , phys . lett . * 69 * , 2333 ( 1992 ) . r. d. peccei and h. r. quinn , phys . lett . * 38 * , 1440 ( 1977 ) ; phys . d * 16 * , 1791 ( 1977 ) . j. e. kim , phys . * 43 * , 103 ( 1979 ) ; m. a. shifman , a. i. vainshtein , and v. i. zakharov , nucl . phys . * b166 * , 493 ( 1980 ) . m. dine , w. fischler , and m. srednicki , phys . b * 104 * , 199 ( 1981 ) ; a. r. zhitnitski , yad . fiz . * 31 * , 497 ( 1980 ) [ sov . . phys . * 31 * , 260 ( 1980 ) ] . see , e.g. , g. g. raffelt , _ stars as laboratories for fundamental physics _ ( the university of chicago press , chicago , 1996 ) . _ et al_. , phys . b * 442 * , 38 ( 1998 ) . m. krmar _ et al_. , phys . d * 64 * , 115016 ( 2001 ) . f. t. avignone iii _ et al_. , phys . * 81 * , 5068 ( 1998 ) . r. bernabei _ et al_. , phys . b * 515 * , 6 ( 2001 ) . l. di lella , a. pilaftsis , g. raffelt , and k. zioutas , phys . rev . d * 62 * , 125011 ( 2000 ) . s. chang , s. tazawa , and m. yamaguchi , phys . d * 61 * , 084005 ( 2000 ) . r. horvat , m. krmar , and b. laki , phys . d * 65 * , 087701 ( 2002 ) . g. f. guidice , r. rattazzi , and j. d. wells , nucl . b544 * , 3 ( 1999 ) ; t. han , j. d. lykken , and r. j. zhang , phys . d * 59 * , 105006 ( 1999 ) . by simply scaling the solar age limit of qcd axions ( @xmath94 ) @xcite with @xmath95 due to the multiplicity of kk modes and observing that an allowed maximum mass is @xmath96 before the solar flux gets suppressed by the kinematic threshold , one obtains for two extra dimensions : ( i ) for @xmath97 mm : @xmath98 if @xmath77 and @xmath99 if @xmath78 , and ( ii ) for @xmath100 nm : @xmath101 if @xmath77 and @xmath102 if @xmath78 . k. hagiwara _ et al_. 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we explore the potential of the cern axion solar telescope ( cast ) for testing the presence of large extra dimensions . the cast experiment has originally been proposed to search for solar axions with a sensitivity supposed to provide a limit on the axion - photon coupling @xmath0 or even lower . the expected bound on the coupling constant is by a factor of ten more stringent than the current experimental results . this bound extends for the first time beyond the limit dictated by astrophysical considerations . as a tuning experiment planning to explore the axion mass region up to about 1 ev , cast would also be sensitive to the existence of kaluza - klein massive states . therefore , the detection of x - rays at least at two pressures may be the signature of large extra dimensions . from this requirement we find that cast may test ( two ) large extra dimensions with a ( common ) compactification radius @xmath1 down to around 250 nm if @xmath2 , and down to around 370 nm if @xmath3 , where @xmath4 is the peccei - quinn mass .

[1]introduction [2]qcd axions and cast [3]cast and large extra dimensions [4]discussion

arxiv

entanglement is the distinguishing feature of quantum mechanics and is the physical phenomenon according to which only the properties of the entire system have precise values , while the physical properties of a subsystem can be assigned only in reference to those of the other ones . it is now intensively studied because it corresponds to peculiar nonlocal correlations which allows performing communication and computation tasks with an efficiency which is not achievable classically @xcite . furthermore , for a deeper understanding of the boundary between the classical and quantum world , it is important to investigate up to which macroscopic scale one can observe quantum behavior , and in particular under which conditions entanglement between macroscopic objects , each containing a large number of the constituents , can arise . entanglement between two atomic ensembles has been successfully demonstrated in ref . @xcite , while entanglement between two josephson - junction qubits has been detected in refs . more recently , macroscopic entanglement has been demonstrated in electro - mechanical systems @xcite : continuous variable ( cv ) entanglement , similar to that considered by einstein podolski and rosen ( epr ) @xcite , has been generated and detected between the position and momentum of a vibrational mode of a @xmath0 @xmath1m - diameter al membrane , and the quadratures of a microwave cavity field , following the theory proposal of ref . @xcite . entanglement between two mechanical resonators ( mrs ) has been instead demonstrated only at the microscopic level , in the case of two trapped ions @xcite , and between two single - phonon excitations in nano - diamonds @xcite . the realization of this kind of entanglement at the more macroscopic level of micromechanical resonators would be extremely important both for practical and fundamental reasons . in fact , on the one hand , entangled mrs at distant sites could represent an important building block for the implementation of quantum networks for long - distance routing of quantum information @xcite ; on the other hand , these nonclassical states represent an ideal playground for investigating and comparing decoherence theories and modifications of quantum mechanics at the macroscopic level @xcite . many different schemes have been proposed in the literature for entangling two mrs , especially exploiting optomechanical and electromechanical devices @xcite , in which the two mrs simultaneously interact with one or more electromagnetic cavity fields . @xcite considered the steady state of different systems of driven cavities : ref . @xcite focused on two mirrors of a ring cavity , while ref . @xcite assumed to drive two independent linear cavities with two - mode squeezed light transferring its entanglement to the cavity end - mirrors . @xcite instead considered a double - cavity scheme in which one cavity couples to the relative motion of two mrs , and the second cavity to their center - of - mass ; when the system is appropriately driven by squeezed light , such squeezing is transferred to the two mrs which are then prepared in a stationary epr - like state . actually , steady - state entanglement can be achieved , even if at a smaller value , also without squeezed driving , either between two movable mirrors in a fabry - perot cavity @xcite , between two mechanical modes of a single movable mirror @xcite , or in the case of two semi - transparent membranes interacting with two driven cavity modes @xcite . a different approach for generating entangled mrs exploits conditional measurements on light modes entangled or correlated with mechanical degrees of freedom @xcite . in this case , entanglement is generated at the measurement and it has a finite lifetime which may be severely limited by the interaction of the mrs with their reservoirs . a similar strategy has been provided to enhance the entanglement of two mrs @xcite . more recent proposal applied reservoir engineering ideas @xcite to optomechanical scenarios , by exploiting suitable multi - frequency drivings and optical architectures in order to achieve more robust generation of steady state entanglement between two mrs @xcite , eventually profiting from mechanical nonlinearities and/or parametric driving @xcite . in the present paper we propose a novel optomechanical / electromechanical scheme for the generation of remarkably large cv entanglement between two mrs with different frequencies , which is also extremely robust with respect to thermal noise . the scheme is particularly simple , involving only a single , bichromatically - driven , optical cavity mode , and optimally works in a rotating wave approximation ( rwa ) regime where counter - rotating , non - resonant , terms associated with the bichromatic driving are negligible . the scheme shares some analogies with the reservoir - engineering schemes of refs . @xcite , but it may be used to generate robust entanglement also in a pulsed regime , in the special case of equal effective couplings at the two sidebands , where the system becomes analogous to the srensen - mlmer scheme for entangling trapped ions in a thermal environment @xcite . this latter scheme has been already considered in an optomechanical scenario by kuzyk _ et al . _ @xcite for entangling dynamically two optical modes via their common interaction with a single mr . the paper is organized as follows . in section ii we derive the effective quantum langevin equations ( qle ) describing the dynamics of the system in the rwa . in section iii we solve the dynamics in terms of the mechanical bogoliubov modes of the system @xcite , derive the steady state of the system in the stable case , and provide simple analytical expressions for the achievable mechanical entanglement , showing its remarkable robustness with respect to temperature . in section iv we instead consider the special case of equal couplings , when the system can be mapped to the srensen - mlmer scheme @xcite , in which mechanical entanglement is generated only dynamically and slowly decays to zero at long times . in section v we solve and discuss the exact dynamics of the system in order to establish the conditions under which the rwa does not seriously affect the robust generation of large mechanical entanglement . in section vi we discuss the experimental detection of such entanglement and present some concluding remarks . in the appendices we provide some detail on the dynamical evolution of the system , and present a careful derivation of the linearized qle in the rwa regime . as shown in fig . [ detect ] , we consider an optical cavity mode with resonance frequency @xmath2 and annihilation operator @xmath3 interacting via the usual optomechanical interaction with two different mrs , with frequencies @xmath4 and @xmath5 and annihilation operators @xmath6 and @xmath7 respectively . the cavity mode is bichromatically driven at the two frequencies @xmath8 and @xmath9 , with the reference frequency @xmath10 detuned from the cavity resonance by a quantity @xmath11 . if we describe the cavity field in a reference frame rotating at the frequency @xmath10 , then the system hamiltonian is given by @xmath12\hat{a}^\dagger \hat{a } \nonumber \\ & & + \hbar{\left [ { \left ( e_1 e^{-i\omega_1t}+e_2 e^{i\omega_2 t } \right)}\hat{a}^\dagger + { \rm h.c . } \right]}\ . \label{haml}\end{aligned}\ ] ] this means that the cavity mode is simultaneously driven on the blue sideband associated with the mr with annihilation operator @xmath6 , and on the red sideband associated with the mr with @xmath7 . the nonzero detuning @xmath13 makes the present scheme different from the one studied in the supplementary material of ref . @xcite which restricts to the resonant case @xmath14 . our model is instead related to the scheme proposed by kuzyk _ et al . _ @xcite for entangling dynamically two optical modes via their common interaction with a single mr : here we will dynamically entangle two mrs via their common interaction with an optical mode . the system dynamics can be efficiently studied by linearizing the optomechanical interaction in the limit of large driving field . in this case the average fields for both cavity , @xmath15 , and mechanical degrees of freedom , @xmath16 , are large , and one can simplify the interaction hamiltonian at lowest order in the field fluctuations @xmath17 differently from the typical optomechanical settings in which the steady state average fields are time - independent , here the bichromatic driving induces a time - dependent , periodic steady state average field which , in turn , implies time - dependent effective coupling strengths for the linearized dynamics of the fluctuations . as originally discussed in @xcite , and detailed in appendix [ linearization ] , approximated dynamical equations for the fluctuation operators @xmath18 and @xmath19 can be derived , in the interaction picture with respect to the hamiltonian @xmath20 , by neglecting the non - resonant / time - dependent components of the effective linearized interactions . it is possible to prove that this approach is justified when ( see eq . ( [ cond1 ] ) ) @xmath21 the corresponding qle including thermal noise and dissipation at rates @xmath22 and @xmath23 for the cavity and the mechanical mode @xmath24 respectively , are @xmath25 where @xmath26 are the ( generally complex ) linear optomechanical couplings , and @xmath27 and @xmath28 are standard input noise operators with zero mean , whose only nonzero correlation functions are @xmath29 , @xmath30 and @xmath31 , where @xmath32^{-1}$ ] is the mean thermal phonon number of the @xmath33-th mr , which we assume to stay at the same environmental temperature @xmath34 . moreover , we note that here the new cavity detuning @xmath35 includes the time - independent frequency shift induced by the optomechanical interaction , proportional to the dc component of the average mechanical oscillation amplitude @xmath16 , that we here denote with @xmath36 ( see appendix [ linearization ] ) . specifically @xmath37}\ .\end{aligned}\ ] ] we will see that the dynamics described by these equations allows to generate large and robust entanglement between the two mrs , either in the steady state or , in a particular parameter regime , during the time evolution with a flat - top pulse driving . we first notice that the system is stable when all the eigenvalues associated with the linearized dynamics of eqs . ( [ deltaa2])-([b2st ] ) have negative real parts . the stability condition is quite involved in the general case , but it assumes a particularly simple form in the case of equal mechanical dampings , @xmath38 . in such a case , the system is stable if and only if @xmath39.\ ] ] this stability condition reduces to the one derived in the supplementary material of ref . @xcite in the case @xmath40 . we see that a nonzero detuning generally helps in keeping the system stable . the coherent dynamics corresponding to the eqs . ( [ deltaa2])([b2st ] ) , is described by the effective linearized hamiltonian @xmath41 we can always adjust the phase reference of each mr ( which will be determined by a local oscillator which must be used to measure the mechanical quadratures for verifying entanglement ) so that we can take both @xmath42 and @xmath43 real . ( [ heff ] ) naturally suggests to introduce two effective mechanical modes allowing to simplify the system dynamics . we assume for the moment @xmath44 , which is a sufficient condition for stability ( see eq . ( [ stab ] ) ) , and define @xmath45 where @xmath46 eqs . ( [ bet1])-([bet2 ] ) define a bogoliubov unitary transformation of the mechanical mode operators , which can also be written as @xmath47 with @xmath48 the two - mode squeezing operator . the bogoliubov mode @xmath49 describes the `` mechanical dark mode '' , which does not appear in @xmath50 , i.e. , is decoupled from the cavity mode and therefore is a constant of motion in the absence of damping , while @xmath51 is the `` bright '' mode interacting with the cavity mode . this is equivalent to say that the dark mode @xmath49 is the normal mode of the hamiltonian dynamics with eigenvalue equal to zero . the other two normal modes of the system will be linear combinations of @xmath51 and @xmath52 . the bogoliubov mode description has been already employed in cavity optomechanics , associated to two optical modes in refs . @xcite , and to two mechanical modes in refs . @xcite ( see appendix a for a derivation of the normal modes of the system and a study of its hamiltonian dynamics ) . for a realistic description of the system dynamics we must include cavity decay and mechanical dissipation . it is convenient to rewrite the qle in terms of the bogoliubov modes , which in the case when @xmath53 assume the simple form @xmath54 where @xmath55 , @xmath56 , are two correlated thermal noise operators whose only nonzero correlation functions are @xmath57\delta(t - t ' ) , \\ { \left\langle \hat{\beta}_j^{\rm in}(t){^\dagger}\,\hat{\beta}_j^{\rm in}(t ' ) \right\rangle}&= & \bar{n}^{\rm eff}_j(r)\delta(t - t ' ) , { { \nonumber}}\\ { \left\langle \hat{\beta}_1^{\rm in}(t)\,\hat{\beta}_2^{\rm in}(t ' ) \right\rangle}&=&{\left\langle \hat{\beta}_1^{\rm in}(t){^\dagger}\,\hat{\beta}_2^{\rm in}(t'){^\dagger}\right\rangle}=\bar m(r)\delta(t - t'),{{\nonumber}}\end{aligned}\ ] ] with the effective mean thermal phonon numbers @xmath58 and the inter - mode correlation @xmath59 if @xmath60 a dissipative coupling term between the two bogoliubov modes appears , which however does not have relevant effects because it is proportional to @xmath61 which is typically very small with respect to all other damping rates . the dynamics associated with eqs . ( [ deltaa3])([beta2 ] ) is simple : the bright mechanical mode @xmath51 is cooled by the cavity , while the correlated reservoir create finite correlations between dark and bright modes . in particular , the matrix of correlation for the vector of operators @xmath62 , whose elements are @xmath63 is given , at the steady state , by @xmath64 with the number of excitation of the cooled bright mode and the correlations between the two bogoliubov modes respectively given by @xmath65},\end{aligned}\ ] ] and @xmath66 where @xmath67 and @xmath68 can be seen as an effective collective optomechanical cooperativity . the steady state correlation matrix can be expressed in terms of the original modes @xmath69 and @xmath70 by inverting the bogoliubov transformation introduced in eqs . ( [ bet1 ] ) and ( [ bet2 ] ) . the result is@xmath71 with @xmath72 and where now @xmath73 the entanglement between modes @xmath69 and @xmath70 , measured by means of the logarithmic negativity @xcite , can be easily expressed in terms of these matrix elements as @xcite @xmath74 } , { { \nonumber}}\\ \nu&=&1+\bar n_{b1}+\bar n_{b2}-\sqrt{4{\left|{\bar m_b}\right|}^2+{\left ( \bar n_{b1}-\bar n_{b2 } \right)}^2}\ .\end{aligned}\ ] ] when the collective cooperativity @xmath68 is sufficiently large , i.e. , @xmath75 , then @xmath76 is negligible ( see eq . ( [ mbeta ] ) ) . this is the working regime in which we are particularly interested , because in this case , the second bogoliubov mode can be cooled close to its ground state ( @xmath77 ) , corresponding to an entangled state for the original mechanical modes . in this case the steady state correlation matrix for the bogoliubov modes , in eq . ( [ ccbeta ] ) , reduces to the correlation matrix of a state given by the product of two thermal states with occupancies @xmath78 and @xmath79 respectively . for the two mr of interest , associated with the operator @xmath6 and @xmath7 , such a state is just a two - mode squeezed thermal state @xcite @xmath80 where @xmath48 is given in eq . ( [ twomode1 ] ) , and @xmath81 is the density matrix of the thermal equilibrium state of a resonator with occupancy @xmath82 . such a state is entangled for sufficiently large @xmath83 and not too large mean thermal excitation number . this prediction of large stationary entanglement is confirmed in fig . [ entdyn1 ] , where we plot the time evolution of the entanglement between the two mrs , quantified in terms of the logarithmic negativity @xmath84 , obtained from the solution of eqs . ( [ deltaa2])-([b2st ] ) . figure 2 refers to an experimentally achievable set of parameters , @xmath85 s@xmath86 , @xmath87 s@xmath86 , @xmath88 s@xmath86 , @xmath89 s@xmath86 , and to different values of mean thermal phonon numbers @xmath90 , @xmath91 , and of the ratio @xmath92 . we see that remarkable values of @xmath84 are achieved at low temperatures , and that stationary mechanical entanglement is quite robust with respect to temperature because one has an appreciable value of @xmath93 even for @xmath94 , @xmath95 . the time to reach the steady state is essentially given by the inverse of the cooling rate of the bright bogoliubov mode , which is approximately given by @xmath96 ( see eqs . ( [ deltaa3])([beta2 ] ) ) . starting from an initial uncorrelated state with the optical mode fluctuations @xmath52 in the vacuum state and each mr in its thermal state with mean phonon number : i ) @xmath97 , @xmath98 ( black line ) ; ii ) @xmath99 , @xmath100 , @xmath101 ( blue line ) ; iii ) @xmath102 , @xmath103 , @xmath104 ( green line ) ; @xmath94 , @xmath95 , @xmath105 ( red line ) ; the other parameters are @xmath85 s@xmath86 , @xmath87 s@xmath86 , @xmath88 s@xmath86 , @xmath89 s@xmath86.,width=268 ] eq . ( [ twomode2 ] ) suggests that one could achieve large stationary entanglement between the two mrs by taking a large two - mode squeezing parameter @xmath83 , and a large collective cooperativity @xmath106 in order to significantly cool the bright bogoliubov mode . however the corresponding optimization of the system parameters , and especially of the two couplings @xmath42 and @xmath43 , is far from being trivial . in fact , @xmath83 increases when @xmath107 , which however implies , at a fixed value of @xmath43 , a decreasing value of @xmath108 and therefore of @xmath68 ( see eq . ( [ calg - r ] ) and eq . ( [ cm ] ) ) ; moreover increasing @xmath83 has also the unwanted effect of increasing @xmath76 that is the correlations between the two bogoliubov modes ( see eqs . ( [ barm ] ) and ( [ mbeta ] ) ) . however , a judicious choice of parameters is possible , allowing to get very large stationary mechanical entanglement , even in the presence of non - negligible values of the thermal occupancies @xmath90 and @xmath91 . at a given value of @xmath42 , this is obtained by taking a sufficiently large value of the associated single - mode cooperativity , @xmath109 , and correspondingly optimizing the value of @xmath43 , i.e. , of @xmath83 . in fact , the logarithmic negativity associated with the stationary state of eq . ( [ twomode2 ] ) can be evaluated in terms of the parameter @xmath110}{\left ( \cosh^2 r+\sinh^2 r \right ) } \\&&-\sqrt{\bar n_-(r)^2 + 4{\left [ \bar n_+(r)+1 \right]}^2\sinh^2 r\cosh^2 r}\ , { { \nonumber}}\label{lognegtmsts1}\end{aligned}\ ] ] where @xmath111 , and @xmath79 can be explicitly rewritten in terms of the cooperativity @xmath112 as @xmath113 { { \nonumber}}\\&&\times { \left [ 1-\frac{{\left ( 1-\epsilon \right)}\ , c_1}{\sinh^2 r{\left ( 1+\delta^2 \right)}+c_1 } \right]}\ .\end{aligned}\ ] ] the dependence of @xmath84 versus @xmath83 , for given values of @xmath112 , @xmath90 and @xmath91 , shows a maximum and then decays to zero for large @xmath83 ( see fig . [ logneg1 ] which refers to @xmath114 and @xmath99 , @xmath100 ) . this behavior is described by a very simple approximated expression valid in the limit @xmath115 , with not very large @xmath116 , and when @xmath117 ( corresponding to @xmath118 ) , @xmath119 which exhibits a minimum ( hence corresponding to maximum entanglement ) as a function of @xmath83 at @xmath120 given by @xmath121 for values of @xmath83 much larger or much smaller than this value , the resonators may not be entangled . when @xmath83 is increased to very large values @xmath122 , @xmath123 is reduced and the cooling dynamics becomes slow as compared to the standard mechanical dissipation , which takes place at rate @xmath124 , so that the correlations between the mrs can not be efficiently generated . on the other hand , at small @xmath125 the bogoliubov modes are essentially equal to the original modes , so that the cavity cools only the second resonator , and also in this case mechanical entanglement can not be observed . [ logneg1 ] also shows that the simplified expression of eq . ( [ nuappr ] ) provides a simple but valid approximation for large @xmath112 and a very good estimate of the optimal value of the two - mode squeezing parameter @xmath83 , i.e. , of @xmath92 , given by eq . ( [ rmax ] ) . the corresponding value of the logarithmic negativity is @xmath126\end{aligned}\ ] ] and shows that once that the ratio @xmath92 is optimized , the achievable stationary entanglement between the two mrs increases with increasing @xmath127 . at the steady state versus @xmath83 for @xmath85 s@xmath86 , @xmath87 s@xmath86 , @xmath128 s@xmath86 , @xmath40 , implying a cooperativity @xmath129 , and @xmath99 , @xmath100 . the full red line refers to the steady state solution of the qle in eqs . ( [ deltaa3])([beta2 ] ) , that is given by eq . ( [ enexact ] ) , the blue dashed line is evaluated with the approximated value of @xmath130 reported in eq . ( [ nubeta0 ] ) , and the black dashed line corresponds to the approximation in eq . ( [ nuappr ] ) . , width=268 ] the above analysis of the stationary entanglement of the two mrs extends the results of ref . @xcite in various directions . first of all , our model extends to the case of nonzero detuning @xmath35 a model discussed in the supplementary material of ref . we see that a nonzero detuning has a limited effect of the dynamic of entanglement generation , providing only an effective increase of @xmath131 , which however becomes negligible as soon as @xmath132 ( see eq . ( [ ncool2 ] ) ) . moreover , ref . @xcite provided an explicit expression for @xmath84 only for the case of negligible thermal occupancies and not too large values of @xmath83 , while the present discussion applies for arbitrary values of @xmath83 , @xmath90 and @xmath91 . in the special case of equal couplings @xmath133 , i.e. , @xmath134 , the bogoliubov modes can not be defined anymore and the description of the preceding section can not be applied . the dynamics is nonetheless interesting and still allows for the generation of appreciable entanglement between the two mrs , even though only at finite times and not in the stationary state . we notice that in this special case , our scheme becomes analogous to that of ref . @xcite , that showed that two appropriately driven optical modes can be entangled with a pulsed scheme by their common interaction with a mr . more precisely , the qle of eqs . ( [ deltaa2])-([b2st ] ) are the same as those studied in ref . @xcite but now referred to two mechanical modes coupled to the same optical mode , i.e. , with exchanged roles between optical and mechanical degrees of freedom . the physical mechanism at the basis of the generation of dynamical entanglement can be understood by looking at the hamiltonian evolution of the system at equal couplings . such mechanism essentially coincides with the one proposed for entangling internal states of trapped ions by milburn @xcite and by srensen and mlmer @xcite , and first applied to an optomechanical setup by kuzyk _ et al . _ @xcite . in the present case , the common interaction with the bichromatically driven optical mode dynamically entangles the two mrs , and at special values of the interaction time the optical mode is decoupled from the two mrs and mechanical entanglement can be strong . at equal couplings it is convenient to rewrite the effective hamiltonian after linearization of eq . ( [ heff ] ) in terms of mechanical and optical quadratures , using the expressions @xmath135 , @xmath56 , and @xmath136 . one gets @xmath137 where @xmath138 , @xmath139 are linear combinations of the two position and momentum operators of the two mrs . the heisenberg evolution of these latter mechanical operators can be solved in a straightforward way , by exploiting the fact that @xmath140 and @xmath141 are two commuting conserved observables . one gets ( see also refs . @xcite ) @xmath142 relevant interaction times are those when the mr dynamics decouple from that of the optical cavity , and this occurs at @xmath143 , @xmath144 , where @xmath145 this map describes a stroboscopic evolution in which the two mrs become more and more entangled , because it corresponds to the application of the unitary operator @xmath146 \\ & & = \exp\left[-i\frac{2\pi g^2 m}{\delta^2}\left(\delta\hat{b}_1^{\dagger}\delta\hat{b}_1+\delta\hat{b}_2^{\dagger}\delta\hat{b}_2 + 1+\delta\hat{b}_1^{\dagger}\delta\hat{b}_2^{\dagger}+\delta\hat{b}_1\delta\hat{b}_2\right)\right ] \nonumber .\end{aligned}\ ] ] this ideal behavior is significantly modified by the inclusion of damping and noise , especially the one associated with the cavity mode , which acts on the faster timescale @xmath147 and seriously affects the cavity - mediated interaction between the two mrs , as soon as @xmath22 becomes comparable to @xmath35 . mechanical entanglement is large for large @xmath148 and we expect well distinct peaks for @xmath84 at interaction times @xmath149 , in the ideal parameter regime @xmath150 . in the more realistic regime in which @xmath151 , @xmath35 and @xmath22 are comparable , the peaks will be washed out , but we still expect an appreciable value for the mechanical entanglement for a large interval of interaction times . this is confirmed by the numerical solution of the time evolution associated with the qle shown in fig . [ smdynamics ] , which refers to the parameter set @xmath85 s@xmath86 , @xmath87 s@xmath86 , @xmath152 s@xmath86 , @xmath99 , @xmath100 , and to three different values of the detuning , @xmath153 s@xmath86 ( black dashed line ) , @xmath154 s@xmath86 ( red full line ) , and @xmath155 s@xmath86 ( blue full line ) . we see that an appreciable value of @xmath84 ( even though smaller than the one achievable at the same @xmath90 and @xmath91 after the optimization of @xmath92 of the previous section ) is reached for a large interval of interaction times @xmath156 . therefore even at equal couplings ( and nonzero detuning ) one can entangle the two resonators with a pulsed experiment . mechanical entanglement instead vanishes in the stationary state . in the case of equal couplings for the parameter set @xmath85 s@xmath86 , @xmath87 s@xmath86 , @xmath152 s@xmath86 , @xmath99 , @xmath100 , and for three different values of the detuning : @xmath153 s@xmath86 ( black dashed line ) , @xmath154 s@xmath86 ( red full line ) , and @xmath155 s@xmath86 ( blue full line).,width=268 ] the derivation of the effective linearized dynamics of appendix [ linearization ] suggests that the counter - rotating terms that we have neglected may play an important role when the mechanical frequencies are not too large with respect to the other parameters ( see also the comments in the supplementary material of ref . it is therefore interesting to study their effect by comparing the above predictions , both in the case of @xmath157 and in the case of equal couplings , to the solution of the exact qle obtained without neglecting the various time - dependent terms . in appendix [ linearization ] we describe the derivation of the effective linearized equations that we have studied in the preceding sections and that is based on the elimination of fast rotating terms and on the expansion of the linearized coupling strength at lowest order in @xmath158 . here we analyze the limit of validity of these approximations by solving numerically the system dynamics with the inclusion of the non - resonant terms expanded at different orders in powers of @xmath158 . in fig . [ res1 ] and [ res2 ] the red lines are evaluated without the non - resonant terms ( i.e. , the treatment of the preceding sections ) , while the green and the blue ones take into account the full dynamics . in particular the green lines are computed by expanding the average fields @xmath15 and @xmath16 ( that have been introduced in eq . ( [ displacement ] ) and discussed in appendix [ linearization ] ) , at the lowest relevant order in powers of @xmath159 , while for the blue ones they have been expanded at sixth order in powers of @xmath159 . moreover , the green line results are found considering only the steady state solution for @xmath15 and @xmath16 , while the blue lines are computed taking into account their full dynamics ( that includes also the transient regime before the steady state is reached ) with initial condition @xmath160 . in fig . [ res1 ] we compare the time evolution of the entanglement evaluated with and without the time - dependent terms when @xmath44 . the parameters used in these plots are consistent with those used in fig . [ entdyn1 ] . specifically the three red curves in figs . [ res1 ] ( a ) , ( b ) and ( c ) , that are barely visible because almost entirely covered by the green curves , are equal to the three lowest curves in fig . [ entdyn1 ] . we observe that the green and the red lines are always very close , meaning that the linearized rwa treatment is a very good approximation of the full dynamics when @xmath15 and @xmath16 can be expanded at lowest order in @xmath159 . nevertheless , we note that if the mechanical frequencies are not large enough and higher order terms are taken into account together with the full dynamics of @xmath15 and @xmath16 , then the results can be significantly different as described by the blue curves . specifically , the solid - blue lines are evaluated for sufficiently large values of the mechanical frequencies so that the condition in eq . ( [ cond01 ] ) is well fulfilled , and the effective linearized rwa dynamics recovers with significant accuracy the one determined with the inclusion of the non - resonant terms . the dashed - blue lines are instead evaluated for smaller frequencies . in this case it is evident that the non - resonant terms have a significant role in the system dynamics and that the lowest order expansion of the coefficients @xmath15 and @xmath16 does not provide an accurate description . we note that according to eq . , in order to eliminate the fast rotating terms , the ratios @xmath161 have to be much larger than one . although the dashed - blue curves are evaluated for a ratio @xmath162 of roughly 25 , which can be considered significantly large , we have found , indeed , that it is not enough for a faithful approximation of the system dynamics with the model discussed in the preceding sections . the conclusive analysis of these cases would , possibly , require a non - perturbative approach that is beyond the scope of the present work . a final remark is in order . we have verified that the discrepancy between the dashed - blue lines and the red ones is due to the combined effect of the higher order terms and of the transient initial dynamics of @xmath15 and @xmath16 . specifically , when we consider either the lowest order terms and the transient dynamics , or the higher order terms and only the steady state of @xmath15 and @xmath16 , the corresponding results for the entanglement dynamics are very similar to the red lines . in fig . [ res2 ] we study the case of equal couplings @xmath163 . in this case solid and dashed lines differ in the values of the cavity detuning @xmath35 . in general larger @xmath35 ( dashed lines ) corresponds to smaller entanglement , and the results evaluated by including the counter - rotating terms tends to exhibit larger entanglement than the corresponding ones obtained without the non - resonant terms . the solid curves are found with smaller @xmath35 . in this case red , green and blue lines are very close when the mechanical dissipation is sufficiently large as in fig . [ res2 ] ( a ) . larger discrepancies are found when the mechanical dissipation is reduced as in fig . [ res2 ] ( b ) and ( c ) , especially at relatively large time . we observe in fact that , while the red curves for the entanglement decay to zero at large time , the corresponding green and blue lines seem to approach a finite sizable value . as shown by the insets , when this different behaviour is observed , the average photon number in the cavity tends to diverge . this is a signature of the fact that the full dynamics including counter - rotating terms is actually unstable , even though the rwa dynamics without these terms is stable ( see eq . ( [ stab ] ) ) . we have confirmed the unstable nature of the time - dependent dynamics by calculating the floquet exponents of the dynamical equations of the system . in fact , when @xmath15 and @xmath16 are considered in their steady state , one has a system of linear differential equations with periodic , time - dependent coefficients ( see appendix [ linearization ] ) , and the floquet theory can be applied in this case @xcite ; we have verified that for the parameters of fig . [ res2 ] there is always at least one positive floquet exponent , meaning that the system is unstable . this implies that , in general , the corresponding results are well - grounded only for relatively short time until the populations are not exceedingly large . on the other hand , our results show that in a pulsed experiment with the parameters of fig . [ res2 ] , these instabilities do not constitute a serious hindrance to the creation of significant entanglement at finite times . therefore , when the mechanical frequencies are sufficiently large ( @xmath164 ) ( and , limited only to the case of equal couplings , when also mechanical damping is not too small ) , the effective linearized rwa dynamics obtained by neglecting the counter - rotating terms approximates with very good accuracy the full system dynamics . we finally discuss how to detect the generated mechanical entanglement between the two mrs at different frequencies . the present entanglement describes epr - like correlations between the quadratures of the two mrs and therefore we need to perform homodyne - like detection of these quadratures . in the linearized regime we are considering the state of the two mrs is a gaussian cv state , which is fully characterized by the matrix of all second - order correlations between the mechanical quadratures . therefore from the measurement of these correlations one can extract the logarithmic negativity @xmath165 . one does not typically have direct access to the mechanical quadratures , but one can exploit the currently available possibility to perform low - noise and highly efficient homodyne detection of optical and microwave fields , and implement an efficient transfer of the mechanical phase - space quadratures onto the optical / microwave field . as suggested in ref . @xcite and then implemented in the electromechanical entanglement experiment of ref . @xcite , the motional quadratures of a mr can be read by homodyning the output of an additional `` probe '' cavity mode . in particular , if the readout cavity mode is driven by a much weaker laser so that its back - action on the mechanical mode can be neglected , and resonant with the first red sideband of the mode , i.e. , with a detuning @xmath166 , @xmath56 , the probe mode adiabatically follows the mr dynamics , and the output of the readout cavity @xmath167 is given by ( see fig . [ detect ] ) @xcite @xmath168 with @xmath169 the very small optomechanical coupling with the probe mode . therefore using a probe mode for each mr , changing the phases of the corresponding local oscillator , and measuring the correlations between the probe mode outputs , one can then detect all the entries of the correlation matrix and from them numerically extract the logarithmic negativity @xmath165 . we have studied in detail a general scheme for the generation of large and robust cv entanglement between two mrs with different frequencies through their coupling with a single , bichromatically driven cavity mode . the scheme extends and generalizes in various directions similar schemes exploiting driven cavity modes @xcite for entangling two mrs or two cavity modes . the scheme is able to generate a remarkably large entanglement between two macroscopic oscillators in the stationary state , i.e. , with virtually infinite lifetime , and it is quite robust because one can achieve appreciably large cv entanglement even with thermal occupancies of the order of @xmath170 . the scheme is particularly efficient in the limit where counter - rotating terms due to the bichromatic driving of the cavity mode are negligible , and we have verified with a careful numerical analysis that this is well justified when the two mechanical frequencies are sufficiently large @xmath164 . this work has been supported by the european commission ( itn - marie curie project cqom and fet - open project iquoems ) , by miur ( prin 2011 ) . it is straightforward to see that the diagonal form of the interaction hamiltonian of eq . ( [ heff ] ) is @xmath171 where @xmath172 define the other two normal modes together with the dark mode @xmath49 , introduced in eq . ( [ bet1 ] ) , with @xmath173 defined by the condition @xmath174 , while the eigenvalues are given by @xmath175 , @xmath176 , @xmath177 , with @xmath178 . the normal modes allows to understand the dynamics in the absence of optical and mechanical damping processes . in fact , from eq . ( [ heffdia ] ) one can easily derive the heisenberg evolution of the mechanical bosonic operators . by inverting eqs . ( [ bet1])-([bet2 ] ) one has , @xmath179 , @xmath180 and using @xmath181 , @xmath56 , and @xmath182 , one gets @xmath183\left[\cos\frac{\tilde{\delta}t}{2}-i\cos 2\theta \sin\frac{\tilde{\delta}t}{2}\right]\hat{\beta}_2^\dagger ( 0 ) \nonumber \\ & & -i\sinh r \sin 2\theta \exp\left[i\frac{\delta t}{2}\right ] \sin\frac{\tilde{\delta}t}{2 } \delta\hat{a}(0)\nonumber \\ & & \delta\hat{b}_2(t ) = -\sinh r \hat{\beta}_1^{\dagger}(0 ) \label{b2 } \\ & & + \cosh r \exp\left[-i\frac{\delta t}{2}\right]\left[\cos\frac{\tilde{\delta}t}{2}+i\cos 2\theta \sin\frac{\tilde{\delta}t}{2}\right]\hat{\beta}_2(0 ) \nonumber \\ & & + i\cosh r \sin 2\theta \exp\left[-i\frac{\delta t}{2}\right ] \sin\frac{\tilde{\delta}t}{2 } \delta\hat{a}(0 ) . \nonumber\end{aligned}\ ] ] we now look for special time instants at which the two mechanical modes can be strongly entangled . a necessary condition for such dynamical entanglement is that at these times , the cavity mode must be decoupled from the mechanical modes and eqs . ( [ b1])-([b2 ] ) show that it occurs when @xmath184 , i.e. , @xmath185 , @xmath186 . at these time instants one has @xmath187\delta\hat{b}_1(0 ) \label{b1tp } \\ & & + \sinh r \cosh r \left(1-e^{i\phi_p}\right)\delta\hat{b}_2^\dagger ( 0 ) , \nonumber \\ & & \delta\hat{b}_2^{\dagger}(t_p ) = \left[e^{i\phi_p}\cosh^2 r -\sinh^2 r\right]\delta\hat{b}_2^\dagger ( 0 ) \label{b2tp } \\ & & - \sinh r \cosh r \left(1-e^{i\phi_p}\right ) \delta\hat{b}_1(0),\nonumber\end{aligned}\ ] ] where @xmath188 . in particular , if @xmath189 one gets @xmath190 i.e. , the state of the two mrs at time @xmath191 is the result of the application of the two - mode squeezing operator with squeezing parameter @xmath192 , @xmath193 ( see eq . ( [ twomode1 ] ) ) to their initial state . in the usual case of an initial thermal state for the two mrs with mean thermal phonon numbers @xmath194 , the state at time @xmath191 is therefore a two - mode squeezed thermal state @xcite ( see eq . ( [ twomode2 ] ) ) , with logarithmic negativity @xcite @xmath195 , \nonumber\end{aligned}\ ] ] where @xmath196 . for the relevant case of not too small values of the squeezing parameter @xmath83 , @xmath84 can be well approximated with its value at equal mean thermal phonon number @xmath197 , @xmath198,\ ] ] showing that at this interaction time , the entanglement between the mr can be very large , even if starting from a relatively hot state , by properly tuning the ratio @xmath199 , i.e. , the intensity of the two tones . this large mechanical entanglement is achieved when the condition @xmath189 is also satisfied for a given integer @xmath200 . this is obtained for any odd @xmath200 when @xmath201 , or by properly adjusting the value of @xmath108 for a given @xmath202 , i.e. , if @xmath203 this dynamical scheme for the generation of continuous variable mechanical entanglement is similar to the bogoliubov scheme proposed in ref . @xcite for entangling two optical cavity modes . it is extremely hard however to use it for entangling two mechanical modes as in the present case , because the cavity decay rate is comparable to @xmath108 and @xmath35 in typical situations , thereby strongly affecting the ideal hamiltonian dynamics described here . the system dynamics is described by the following qle @xmath204}\hat{a}-i\left[e_1 e^{-i\omega_+ t}+e_2 e^{i\omega_+ t}\right ] + \sqrt{2\kappa } \hat{a}^{\rm in}{{\nonumber}}\\ & & -i \left[g_1\left(\hat{b}_1 + \hat b_1^\dagger \right)+g_2\left(\hat{b}_2+\hat b_2^\dagger \right)\right]\hat{a } , \nonumber \\ \dot{\hat{b}}_j & = & -{\left ( \frac{\gamma_j}{2}+i\omega_{j } \right ) } \hat{b}_j - i g_j \hat{a}^{\dagger } \hat{a } + \sqrt{\gamma_j } \hat{b}_j^{\rm in } \ \;\;\;j=1,2,{{\nonumber}}\\\end{aligned}\ ] ] where , here , differently from the description used in sec . [ system ] , we are representing the cavity field in a reference frame rotating at the frequency @xmath205 , and we have introduced the frequencies @xmath206 the other parameters and operators are defined in the main text . if we perform a time dependent displacement , for both cavity and mechanical degrees of freedom , of the form @xmath207 the qle reduce to the form @xmath208}+2i g_2{\rm re}{\left [ \beta_2(t ) \right ] } \right\ } } \delta \hat{a } { { \nonumber}}\\ & & -ia(t ) + \sqrt{2\kappa } \hat{a}^{\rm in}{{\nonumber}}\\ & & -i \alpha(t){\left [ g_1{\left ( \delta\hat b_1+\delta\hat b_1{^\dagger}\right)}+g_2{\left ( \delta\hat b_2+\delta\hat b_2{^\dagger}\right ) } \right ] } { { \nonumber}}\\ & & -i \left[g_1\left(\delta \hat{b}_1 + \delta \hat b_1^\dagger \right)+g_2\left(\delta \hat{b}_2+\delta \hat b_2^\dagger \right)\right]\delta \hat{a } , \\ \dot{\delta \hat{b}}_j & = & -{\left ( \frac{\gamma_j}{2}+i\omega_{j } \right ) } \delta \hat{b}_j - ib_j(t)+\sqrt{\gamma_j } \hat{b}_j^{\rm in } { { \nonumber}}\\ & & -ig_j{\left [ \alpha(t)\delta\hat a{^\dagger}+\alpha(t)^*\delta\hat a \right]}-i g_j \delta \hat{a}^{\dagger } \delta \hat{a } , \label{qle2}\end{aligned}\ ] ] where the new driving terms , @xmath209 and @xmath210 read @xmath211 -\frac{\partial \alpha(t)}{\partial t } -{\left [ \kappa + i{\left ( \delta_0+\omega_- \right ) } \right]}\alpha(t ) { { \nonumber}}\\ & & -2i{\left ( g_1{\rm re}{\left [ \beta_1(t ) \right]}+g_2{\rm re}{\left [ \beta_2(t ) \right ] } \right ) } \alpha(t ) , { { \nonumber}}\\ b_j(t)&= & -\frac{\partial \beta_j(t)}{\partial t}-{\left ( \frac{\gamma_j}{2}+i\omega_{j } \right)}\beta_j(t)-ig_j{\left|{\alpha(t)}\right|}^2\ .\end{aligned}\ ] ] when @xmath212 and @xmath213 are sufficiently small and we chose @xmath15 and @xmath16 such that @xmath214 and @xmath215 , then the non - linear terms , i.e. the last terms in the two equations ( [ qle1 ] ) and ( [ qle2 ] ) , can be neglected . the equations @xmath214 and @xmath215 define a set of non - linear differential equations with periodic driving for the parameters @xmath15 and @xmath16 . the solution can be evaluated perturbatively in the small parameters @xmath212 and @xmath213 @xcite . here we assume @xmath216 and we observe that the solutions for @xmath15 and @xmath16 , with initial condition @xmath160 , contain , respectively , only even and odd powers of @xmath159 , @xmath217 the equations for each component of these expansions can be written in the form @xmath218 where @xmath219 and the driving terms are defined recursively as @xmath220}+{\rm re}{\left [ \beta_2{^{(p - q-1)}}(t ) \right ] } \right ) } , { { \nonumber}}\\ \xi_\beta{^{(p)}}(t ) & = & -i\sum_{q=0}^{p-1 } \alpha{^{(q)}}(t)\ { \alpha{^{(p - q-1)}}(t)}^*\ , \end{aligned}\ ] ] with the initial condition @xmath221 in particular they can always be rewritten as sums of exponential functions of the form @xmath222 with @xmath223 , @xmath224 , @xmath225 and @xmath226 time - independent complex coefficients , whose specific form can be computed iteratively . moreover , the expression for @xmath227 and @xmath228 are found integrating eq . ( [ dotalphabeta ] ) and are given by @xmath229 we note that all the coefficients @xmath225 and @xmath226 have non - positive real parts , @xmath230 } , { \rm re}{\left [ \zeta_\beta{^{(p , n ) } } \right]}\leq 0 $ ] , thus the large - time solutions @xmath231 and @xmath232 are found from eq . ( [ solalphabeta ] ) by keeping only the terms for which @xmath225 and @xmath226 are purely imaginary , that can be shown to be equal to @xmath233 with @xmath234 odd and even integer respectively . in particular @xmath235 and @xmath236 are periodic functions ( with period @xmath237 ) which contains frequency components that are , respectively , odd and even multiples of @xmath238 , @xmath239 where @xmath240 and @xmath241 are defined in eq . ( [ zw ] ) , and @xmath242 and @xmath243 are the coefficients that correspond to those particular parameters @xmath225 and @xmath226 that are imaginary . the qle , in the interaction picture with respect to the hamiltonian @xmath244 , reduce to @xmath245}+ g_2{\rm re}{\left [ \beta_2(t ) \right ] } \right ) } \delta \hat{a } { { \nonumber}}\\ & & -i\alpha(t){{\rm e}}^{i\omega_- t } { \left [ g_1{\left ( \delta\hat b_1{{\rm e}}^{-i\omega_1t}+\delta\hat b_1{^\dagger}{{\rm e}}^{i\omega_1 t } \right ) } \right.}{{\nonumber}}\\ & & \rpq { + g_2{\left ( \delta\hat b_2{{\rm e}}^{-i\omega_2t}+\delta\hat b_2{^\dagger}{{\rm e}}^{i\omega_2 t } \right ) } } , { { \nonumber}}\\ \dot{\delta \hat{b}}_j&=&-{\frac{\gamma_j}{2 } } \delta \hat{b}_j+\sqrt{\gamma_j } \hat{b}_j^{\rm in } -ig_j{{\rm e}}^{i\omega_jt}{\left [ { { \rm e}}^{i\omega_- t}\alpha(t ) \delta\hat a{^\dagger}+h.c . \right]}\ . { { \nonumber}}\label{qle2_b0}\end{aligned}\ ] ] before proceeding , we note that we can include the dc component of @xmath16 into the cavity detuning , hence we introduce @xmath246}$ ] , according to the notation introduced in eq . ( [ bardelta ] ) , @xmath247 moreover we can isolate the resonant terms of the qle , namely the terms with time - independent coefficients , by considering the lowest order frequency components of @xmath15 , i.e. , @xmath248 corresponding to the frequencies @xmath249 , and defining @xmath250 thereby we find @xmath251 where @xmath252 and @xmath253 account for the terms with time - dependent coefficients and are given by @xmath254}+ g_2{\rm re}{\left [ \bar\beta_2(t ) \right ] } \right ) } \delta \hat{a } { { \nonumber}}\\ & & -i\alpha(t){{\rm e}}^{i\omega_- t } { \left ( g_1\delta\hat b_1{{\rm e}}^{-i\omega_1 t } + g_2\delta\hat b_2{^\dagger}{{\rm e}}^{i\omega_2 t } \right ) } { { \nonumber}}\\ & & -i\bar\alpha_-(t){{\rm e}}^{i\omega_- t } g_1\delta\hat b_1{^\dagger}{{\rm e}}^{i\omega_1 t } -i\bar\alpha_+(t){{\rm e}}^{i\omega_- t}g_2 \delta\hat b_2{{\rm e}}^{-i\omega_2 t } , { { \nonumber}}\\ f_{b_1}(t)&=&-ig_1{{\rm e}}^{i\omega_1t}{\left [ { { \rm e}}^{i\omega_- t}\bar\alpha_-(t ) \delta\hat a{^\dagger}+{{\rm e}}^{-i\omega_- t}\alpha^*(t ) \delta\hat a \right ] } , { { \nonumber}}\\ f_{b_2}(t)&=&-ig_2{{\rm e}}^{i\omega_2t}{\left [ { { \rm e}}^{i\omega_- t}\alpha(t ) \delta\hat a{^\dagger}+{{{\rm e}}^{-i\omega_- t}\bar\alpha_+(t)}^ * \delta\hat a \right]}\ .\end{aligned}\ ] ] in particular we can introduce the linearized coupling strength @xmath255 and @xmath256 , with @xmath257 defined in eq . ( [ alpha_pm ] ) . the expressions introduced in eq . ( [ g1g2 ] ) correspond to the expansion of these parameters at zeroth order in @xmath159 ( see also eq . ( [ firstorder ] ) ) . we are interested in the regime in which the terms in eqs . ( [ ff ] ) with time - dependent coefficients are negligible . they can be neglected when @xmath258 in particular this condition is true when it is valid for the lowest order term in the expansion in power of @xmath159 . in details , the non resonant terms can be neglected when @xmath259 when this condition is fulfilled the parameters @xmath15 and @xmath16 can be safely expanded at the lowest order in @xmath159 . specifically they can be approximated as @xmath260 } , { { \nonumber}}\\ \bar\beta_{j , st}(t)&\simeq&g_j\,\beta_{j , st}{^{(1)}}(t)-\beta_j^{\rm dc } { { \nonumber}}\\&=&-i g_j{\left [ \frac { \alpha_-{\alpha_+^*}}{w_j-2i\omega_+}{{\rm e}}^{-2i\,\omega_+\,t } \frac { \alpha_+{\alpha_-^*}}{w_j+2i\omega_+}{{\rm e}}^{2i\,\omega_+\,t } \right]}\ . { { \nonumber}}\end{aligned}\ ] ] moreover the parameters @xmath261 defined in eq . ( [ baralpha ] ) are zero . using these expressions the qle in eq . ( [ qle1_b1 ] ) can be rewritten as @xmath262 with @xmath42 and @xmath43 defined in eq . ( [ g1g2 ] ) and @xmath263}+ g_2 ^ 2{\rm re}{\left [ \bar\beta_{2,st}{^{(1)}}(t ) \right ] } \right ) } \delta \hat{a } { { \nonumber}}\\ & & -i{{\rm e}}^{i\,\omega_-\,t}\alpha_{st}{^{(0)}}(t ) { \left ( g_1\delta\hat b_1{{\rm e}}^{-i\omega_1 t } + g_2\delta\hat b_2{^\dagger}{{\rm e}}^{i\omega_2 t } \right ) } , { { \nonumber}}\\ f_{b_1}(t)&\simeq&-ig_1{{\rm e}}^{i(\omega_1+\omega_-)t}{\alpha_{st}{^{(0)}}}^*(t)\delta\hat a , { { \nonumber}}\\ f_{b_2}(t)&\simeq&-ig_2{{\rm e}}^{i(\omega_2+\omega_-)t } \alpha_{st}{^{(0)}}(t ) \delta\hat a{^\dagger}\ .\end{aligned}\ ] ] when the time - dependent coefficients are neglected these equations reduce to eqs.([deltaa2])([b2st ] ) . o. romero - isart , a. c. pflanzer , f. blaser , r. kaltenbaek , n. kiesel , m. aspelmeyer , and j. i. cirac , phys . * 107 * , 020405 ( 2011 ) ; o. romero - isart , a. c. pflanzer , m. l. juan , r. quidant , n. kiesel , m. aspelmeyer , and j. i. cirac , phys . a * 83 * , 013803 ( 2011 ) . g. teschl , _ ordinary differential equations and dynamical systems _ , graduate studies in mathematics vol . 140 , american mathematical society , providence , rhode island ( 2012 ) . + http://www.mat.univie.ac.at/ gerald / ftp / book - ode/

we investigate a general scheme for generating , either dynamically or in the steady state , continuous variable entanglement between two mechanical resonators with different frequencies . we employ an optomechanical system in which a single optical cavity mode driven by a suitably chosen two - tone field is coupled to the two resonators . significantly large mechanical entanglement can be achieved , which is extremely robust with respect to temperature .

introduction system hamiltonian and derivation of the effective langevin equations dark and bright bogoliubov modes dynamical evolution in the case of equal couplings effect of the counter-rotating terms. study of the exact dynamics strategies for the experimental detection of mechanical entanglement acknowledgments normal modes and hamiltonian dynamics linearization of the optomechanical dynamics with two-frequency drives

arxiv

the detection of optical broad - emission - lines @xcite and high brightness - temperature ( @xmath5 k ) radio cores @xcite in the nuclei of a well - defined sample of nearby bright galaxies ( ho , filippenko , & sargent 1997a ) indicates that at least 20% of all nearby bright galaxies have an accreting massive black hole . these nearby nuclei have been christened low - luminosity active galactic nuclei or llagns . their low nuclear luminosities require either very low accretion rates ( @xmath610@xmath7l@xmath8 ; e.g. * ? ? ? * ) or radiative efficiencies ( the ratio of radiated energy to accreted mass ) much lower than the typical value of @xmath610% ( e.g. chapter 7.8 of * ? ? ? * ) assumed for powerful agns . this has led to renewed interest in spherical accretion models which produce low radiated luminosities , e.g. advection - dominated accretion flows ( adafs ; * ? ? ? * ) , which may have associated outflows @xcite , and convection - dominated accretion flows ( cdafs ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the sub - parcsec radio emission in llagns may originate in an adaf or cdaf inflow , and the predictions of these models are discussed in the following section . the sub - parsec radio emission may alternatively originate in synchrotron radiation from discrete plasma components or from the base of a continuous jet ejected from the central engine . in the former case , the presence of different self - absorption frequencies for individual components results in a flat spectral - shape ( @xmath9 ; @xmath10 ) for the overall system ( e.g. * ? ? ? in the latter case , i.e. for relativistic electrons at the base of a continuous , freely expanding jet @xcite , the variation of the electron density ( n@xmath11 ) , electron temperature ( @xmath12 ) , and magnetic field ( @xmath13 ) , where @xmath14 is the distance along the jet axis , results in a flat overall radio spectrum . slightly inverted spectra ( up to @xmath15 ) may result from the bulk acceleration of the jet plasma @xcite , and even higher ( temporary ) values of @xmath16 may be measured during radio outbursts ( e.g. * ? ? ? * ) . on larger scales the synchrotron emission from the ejecta or jet becomes optically - thin ( @xmath17 ) . thus the spectral index of the overall synchrotron emission from a ` jet ' is expected to be between 0.2 and @xmath180.7 , depending on the relative contributions of the base and extended components . in their simplest form adaf and cdaf self - similar spherical models invoke standard @xmath19 viscosity , a two - temperature thermal plasma , and a fixed ratio ( @xmath20 ) of magnetic to gas pressure . let us first use the analytic scaling - law approximations of ( * ? ? ? * hereafter m97 ) to estimate the expected radio spectral index in a self - similar flow . if we follow the derivation in sec . 4.1 of m97 using magnetic field @xmath21 , electron temperature @xmath22 , and synchrotron emission factor @xmath23 then the radio emission is self - absorbed , with spectral index @xmath24 at frequencies below a critical frequency @xmath25 . here @xmath26 is the radius in units of the schwarzschild radius and @xmath27 is 5/4 in an adaf @xcite and 3/4 in a cdaf @xcite . when @xmath28 , the dominance of synchrotron cooling forces @xmath29 to 0 over the radio emitting region ( @xmath3010@xmath31r ) in an adaf or cdaf ( see * ? ? ? * ) . however , when @xmath32 the electrons are adiabatically compressed so that @xmath33 in an adaf @xcite and @xmath34 ( i.e. virial ) in a cdaf @xcite . the value of @xmath14 is more difficult to estimate . m97 find @xmath351/15 in adafs with @xmath28 ( their appendix b ) . solving equation b1 of m97 with typical values of @xmath36 k and @xmath37 shows that for an adaf @xmath38 when @xmath39 and @xmath40 when @xmath41 . here @xmath42 is the black hole mass in solar masses and @xmath43 is the accretion rate in units of the eddington rate ( assuming 10% radiative efficiency ) . if we simplistically modify the form of @xmath44 and @xmath45 in an adaf ( equations 5 of m97 ) only in the power law dependence of @xmath26 to convert them to the equivalent expresssions for a cdaf ( see narayan et al . 2000 for some justification of this ) , then we can solve the equivalent of m97 s equation b1 to find that for a cdaf @xmath46 when @xmath39 and @xmath47 when @xmath48 . thus , for an adaf one expects 0.2@xmath49 and for a cdaf one may expect @xmath50 when @xmath28 and @xmath51 when @xmath32 . given the many approximations in this analytical method the above results should be considered only roughly indicative . adafs with outflows have density profile @xmath52 @xcite with @xmath53 varying from 0 ( no outflow ) to 1 ( strong outflow ) . the radio emission from the accretion flow corresponds to the adaf case for @xmath54 and to the cdaf case for @xmath55 ; radio emission from the outflow has not been modeled and is not considered in this paper . more accurate numerical modelling of the radio emission from an adaf gives @xmath56 when @xmath28 ( m97 ) and @xmath57 when @xmath32 with @xmath16 decreasing but still positive as @xmath53 increases from 0 to 0.6 @xcite . to our knowledge , there has been no published work dealing explicitly with numerical modelling of the radio emission from a cdaf . in summary , the radio emission below @xmath25 from the accretion inflow of an adaf or cdaf is expected to have a moderately to highly inverted spectrum except for a ) cdafs in systems with @xmath58 and b ) adafs with strong ( @xmath4 ) outflows . the inflow is optically thin at frequencies greater than @xmath59 so the radio emission falls off exponentially above @xmath25 ( m97 ) . for adafs without outflows @xmath60 and @xmath61 are relatively independent of @xmath42 for @xmath62@xmath63m@xmath64 ( fig . 2 of m97 ) . thus , for typical values of @xmath19 ( 0.3 ) , @xmath20 ( 0.5 ) , and @xmath43 ( @xmath65@xmath66 ) , one has @xmath67 , i.e. the turnover frequency is greater than 15 ghz for all black holes considered here . in adafs with outflows and in cdafs the flatter density profile results in a decrease in the values of @xmath60 and @xmath45 near @xmath68 , and consequently a decrease in @xmath25 ( and l@xmath69 ) ; the introduction of a moderately strong ( @xmath70 ) outflow to an adaf can lower the value of @xmath25 by almost two orders of magnitude @xcite . sixteen of the 96 nearest ( d @xmath0 19 mpc ) llagns from the palomar sample @xcite have a highly compact ( @xmath0 2 mas ) , high brightness - temperature ( @xmath71 k ) radio core ( see * ? ? ? * ; * ? ? ? these sixteen llagns , listed in table 1 , were observed with the very large array ( vla ) at 5 ghz ( 6 cm ) , 8.4 ghz ( 3.6 cm ) , and 15 ghz ( 2 cm ) . the observations were made on september 5 and september 10 , 1999 , while the vla was in `` a''-configuration ( see thompson et al . each llagn observation was sandwiched between two observations of a nearby phase - calibrator with typical cycle times , in minutes , of 1 - 7 - 1 , 1 - 6 - 1 , and 2 - 7 - 2 , at 5 ghz , 8.4 ghz , and 15 ghz , respectively . the observations at the three frequencies are simultaneous to @xmath030 min for each llagn . data were calibrated and mapped using the aips software , following the standard procedures outlined in the aips cookbook . observations of 3c 147 and 3c 286 were used to set the 5 ghz flux - density scale , and observations of 3c 286 were used to set the 8.4 ghz and 15 ghz flux - density scales . the 15 ghz observation of 3c 286 was made at a single ( 1.4 ) airmass , and all 15 ghz observations were made at airmasses of 1.06 to 1.4 , so the 15 ghz flux calibration error from elevation effects is expected to be less than 0.2% @xcite . for this reason we did not make elevation - dependent gain corrections to the 15 ghz data . the vla documentation suggests that the flux calibration at 5 ghz and 8.4 ghz should be accurate to 1%2% , and that at 15 ghz should be accurate to 3%5% ; we conservatively use the higher numbers as the respective @xmath72 errors . for sources with flux greater than 3 mjy , we were able to iteratively self - calibrate ( both phase - only and amplitude - and - phase ) and image the data so as to increase the signal - to - noise ratio in the final map . the root mean square noise in the final uniformly weighted maps was typically 100 @xmath73jy , 60 @xmath73jy , and 170 @xmath73jy at 5 ghz , 8.4 ghz , and 15 ghz , respectively . the resolution at these three wavelengths was typically 05 , 027 and 015 , respectively . we also made 15 ghz and 8.4 ghz maps with the same resolution ( 05 ) as the 5 ghz maps , by appropriately tapering the @xmath74 data . all sources except ngc 4168 at 15 ghz were clearly detected in initial ( non - self - calibrated ) maps . the 15 ghz observation of ngc 4168 was made during very bad weather , and we were able to make a noisy map only after self - calibration with a point - source model . the newly measured flux - densities are listed in table 1 . the 15 ghz data are noisy because of bad weather and high humidity . therefore the three nuclei for which we could not self - calibrate the 15 ghz data have true 15 ghz fluxes somewhere between the measured values and 3 mjy . for all but three of the objects , the radio emission at all three frequencies is compact ; a gaussian fit to the source does not give a deconvolved size more than half a beam - size . the three sources with detected extended structure are all previously known to have such structure : ngc 4278 @xcite , ngc 4472 @xcite , and ngc 4486 ( m 87 ; e.g. * ? ? ? the unresolved emission dominates the extended emission in our maps of these three sources except in ngc 4472 , which has a very weak core . most nuclei have roughly similar fluxes in the full resolution 15 ghz maps and the 05 resolution tapered maps ( table 1 ) ; the same is true at 8.4 ghz . the peak flux - density in the 05 resolution , 5 ghz vla maps is @xmath60.82.1 times the total ( but not necessarily core ) flux in the central @xmath020 mas of the 5 ghz ( non - simultaneous ) vlba maps for all sources which were observed in our june 1997 and april 1999 vlba runs ( column 14 of table 1 ) . the variation of the core spectral index ( from the peak fluxes in matched resolution maps ) with black hole mass is shown in fig . [ figmdo_alpha]a . we have distinguished between black hole masses derived directly from stellar- , gas- , and maser - dynamics @xcite from those inferred from central velocity dispersions ( using the relationship derived by * ? ? ? * ) and galaxy bulge masses ( using the relationship derived by * ? ? ? only ngc 3031 ( m 81 ) and ngc 4772 consistently show the highly inverted radio spectrum expected in adaf models with or without weak to moderately - strong outflows . the core radio emission from the latter galaxy is probably dominated by extended emission ( see above ) , and as discussed below the radio emission from the former is probably from a jet . apart from ngc 4472 , the elliptical galaxies have a similar spectral shape above and below 8.4 ghz . most of the non - ellipticals have a spectrum which falls more rapidly at frequencies above 8.4 ghz than below ( fig . [ figmdo_alpha]b ) , with ngc 3718 and ngc 4258 being the exceptions . when high ( @xmath02 ) resolution 2 - 10 kev x - ray luminosities are available ( e.g. * ? ? ? * ) , the ratio of @xmath75l@xmath76 between x - ray and radio is @xmath77 , suggestive of strong outflows in an adaf scenario ( di matteo , carilli , & fabian 2001 ) . interestingly , this ratio is @xmath78100 for the seyfert 1s and @xmath30100 for the other nuclei . very low accretion rates ( perhaps due to convection or strong outflows ) may cause @xmath25 to fall close to 515 ghz for the objects in the sample . such a scenario is supported by the evidence for turnover frequencies in the 1030 ghz range for a few ellipticals @xcite . if this is the case , then eqn . [ eqnnu_p ] implies lower values of @xmath25 for more massive black holes . that is , within our sample we would expect nuclei with less massive black holes to have more inverted spectra than nuclei with more massive black holes . however , even though we sample more than two orders of magnitude in @xmath42 , fig . 1 does not support such a trend . therefore , unless non - ellipticals have different micro - physical parameters , or higher accretion rates , or a different accretion mechanism , as compared to ellipticals , it is unlikely that a turnover frequency in the 515 ghz range is the cause of the observed flat radio spectrum in most of the sample . if @xmath79 ghz for most nuclei in the sample , then any inverted spectrum radio component must be dominated by other sources at 5 ghz and 8.4 ghz ( and perhaps even at 15 ghz ) . one potential source is non - thermal electrons within the adaf @xcite . significant emission from star - formation related processes can be ruled out as the radio core has a high brightness - temperature at 5 ghz , and at this wavelength most of the flux within the central 05 is also detected on mas - scales ( table 1 ) . on the other hand , the observed distribution of spectral indices is consistent with the 515 ghz radio emission originating in synchrotron - emitting jets . whether or not an accretion flow contributes to the nuclear radio emission , the detections of what appear to be collimated pc - scale jets in the five llagns of table 1 with the highest core flux - ngc 3031 ( m 81 ; * ? ? ? * ) , ngc 4278 @xcite , ngc 4486 ( m 87 ; * ? ? ? * ) , ngc 4374 ( m 84 ; * ? ? ? * ; * ? ? ? * ) , and ngc 4552 ( m 89 ; * ? ? ? * ) - does indicate that synchrotron emission from jets is a significant contributor to the sub - arcsecond radio emission . in fact , in all three sample nuclei which have been comprehensively studied at high resolution in the radio , the radio flux from the jet dominates that from the unresolved `` core . '' in ngc 4486 the jet component within 30 mas ( 2.5 pc ) of the nucleus contributes three times the radio flux of the unresolved ( 1 mas x 0.2 mas ) `` core '' @xcite . given that this `` core '' continues to be further resolved at higher resolutions ( junor , biretta , & livio 1999 ) , and that the jet is a strong radio emitter on scales of 30 mas to 1 , the jet is certainly the dominant sub - arcsecond radio emitter . in ngc 3031 , which has a spectral shape consistent with an adaf model , sub - mas multi - epoch observations reveal that the sub - parsec jet contributes at least three times the radio flux of the unresolved `` core '' @xcite . deep radio observations of ngc 4258 not only reveal a sub - parsec jet , but also indicate an absence of continuum emission from the putative location ( as traced by the water - vapour maser disk ) of the nucleus @xcite . in the context of any low - luminosity spherical accretion model , the presence of outflows and the smaller radio luminosities attributable to the inflow both point to accretion rates at least an order of magnitude lower than than earlier predicted using adaf models ( e.g. 10@xmath8010@xmath81 ; * ; * ? ? ? a jet can also cause considerable disruption of the high - frequency radio emitting region ( @xmath82 ) of the inflow . @xcite find a wide ( @xmath83 ) initial opening angle for the ( potentially relativistic ) radio jet in ngc 4486 , with collimation only occurring at @xmath84 . for a moderately - strong outflow ( e.g. @xmath70 ) about 25% of the material accreted at 100@xmath85 is lost to the outflow by @xmath86 . thus it may not be accurate to model the radio emitting region as a spherical self - similar flow in which the only effect of the outflow is a modification of the accretion rate and central density .

we present simultaneous , sub - arcsecond ( @xmath0 50 pc ) resolution 5 ghz , 8.4 ghz , and 15 ghz vla observations of a well - defined sample of sixteen low - luminosity active galactic nuclei ( llagns ) . the radio emission in most of these nuclei does not show the rising spectrum ( 0.2@xmath1 , @xmath2 ) expected from thermal electrons in an advection dominated accretion flow ( adaf ) with or without weak to moderately - strong outflows . rather , the flat radio spectra are indicative of either synchrotron self - absorbed emission from jets , convection - dominated accretion flows ( cdafs ) with l@xmath3 , or adafs with strong ( @xmath4 ) outflows . the jet interpretation is favored by three factors : a ) the detection of pc - scale radio extensions , morphologically reminiscent of jets , in the five nuclei with the highest peak radio flux - density ; b ) the domination of parsec - scale jet radio emission over unresolved ` core ' emission in the three best - studied nuclei ; and c ) the lack of any clear correlation between radio spectral shape and black hole mass as would be expected from the dependence of the radio turnover frequency on black hole mass in adaf and cdaf models . a jet domination of nuclear radio emission implies significantly lower accretion rates in adaf - type models than earlier estimated from core radio luminosities .

introduction radio spectral predictions of adafs and cdafs observations and data reduction results discussion

arxiv

the investigation of observables having singular or highly oscillating behaviors at microscopic scales , which become smooth after appropriate coarse graining is a typical occurrence in many research areas of modern physics . a detailed theoretical investigation and description is therefore highly desirable and important . quasiclassical green s functions have been introduced in the transport theory of spatially nonuniform superconductors and superconducting hybrid structures as an envelop approximation for exact green s functions quickly oscillating at fermi wavelength scales , see ref . @xcite for a review . one result of this coarse - gaining of the order parameter is the landau - khalatnikov theory of second order phase transitions @xcite . envelop approximations of highly oscillating solutions of hydrodynamic equations gave an opportunity to achieve analytical and numerical advances in hydrodynamics @xcite . in the study of granular superconducting materials , josephson junction arrays ( jja ) and strongly disordered superconducting films near the superconductor insulator transition , the current - voltage characteristics and the dynamic conductivity show singular behaviors described by weighted superpositions of delta - functions @xcite . the imaginary part of the two - particle green s function in ultrasmall metallic granules , where the electron spectrum is essentially discrete , have the form of a series of delta - functions . the transport characteristics of ultrasmall tunnel junctions exchanging energy with a quantum bath with discrete spectrum have a similar singular behavior @xcite . an important and in general unsolved problem is the construction of the envelop approximation for an observable ( which is typically measured ) in case the theoretically derived expression is a countable superposition of delta - functions . remarkably , this problem has a general analytical solution going beyond standard approximations that allows transforming the discrete series into integrals . we found the envelop approximation for singular observables analytically , using the jacoby theta - function formalisms @xcite . our calculations can be physically interpreted using the language of the landau - hopf turbulence @xcite , at least qualitatively . in this paper we derive the envelop approximation for the real - valued function @xmath1 with real argument @xmath2 : @xmath3 here , @xmath4 , is a @xmath5-dimensional vector of quantum numbers , @xmath6 is a scalar real function of @xmath4 and the summation is performed over all configurations of @xmath4 with weights ( probabilities ) @xmath7 . in physical applications @xmath2 is usually a variable with dimension of energy , @xmath6 is closely related to ( the difference of ) the energy levels of a quantum system , while @xmath8 is the gibbs distribution probability . in the most general situation when all components of @xmath4 belong to a countable set , standard approximations like the euler - maclaurin asymptotic @xcite do not work anymore . in that case one can not naively integrate out the @xmath0-functions in . our main result deals with this case . we find the conditions when the density of the @xmath0-functions starts to increase similar to the development of landau - hopf chaos and find the envelope approximation for @xmath1 in that case . we discuss the use of our results for solutions of specific physical problems mentioned above . the case with linear @xmath6 and @xmath8 with bilinear exponent is one of the most relevant for applications . we focus on the situation when @xmath9 here @xmath10 is a vector of integer numbers , @xmath11 is a ( fixed ) vector and @xmath12 is a symmetric positively defined @xmath13 matrix . we will also show that our investigation is not crucially dependent on the specific form of @xmath8 in . the case where @xmath8 decays quickly with @xmath14 ( all eigenvalues @xmath15 of the matrix @xmath12 are much larger than unity ) is the `` quantum '' limit . then the set of numbers @xmath4 is essentially discrete . moreover @xmath8 effectively restricts @xmath4 to a subset with small @xmath14 . in this case the graph of @xmath1 looks like a sparse sequence of isolated @xmath0-function peaks . rather straight - forward is the opposite `` classical '' case where @xmath8 depends only slightly on @xmath4 ( all eigenvalues @xmath15 of the matrix @xmath12 are much smaller than unity and @xmath5 is sufficiently large ) . then the euler - maclaurin approximation is applicable and one can replace the discrete sum over @xmath4 in eq . by an integration over @xmath16 . the result is @xmath17 , where @xmath18 is constant . the most interesting case is the `` mesoscopic '' case when the eigenvalues @xmath15 of the matrix @xmath12 are slightly smaller or of the order of unity and @xmath5 is not too large . this case is in between the quantum and the classical case . then the restriction for the choice of @xmath4 is rather weak and there are a many @xmath4-vectors solving the inequality @xmath19 where @xmath20 is the width of the @xmath0-function . [ @xmath0-functions in physical applications have always some small width , @xmath21 , due to , e.g. , interaction with a heat bath ( dissipation ) . ] if the components of the vector @xmath22 are integer numbers ( commensurate ) then only if @xmath2 is close to an integer [ the fractional part of @xmath2 is smaller than @xmath21 ] there is vector @xmath4 the solution of inequality . however , in case the components of vector @xmath22 are not commensurate as it is most natural in physical systems the formal solution of eq . for arbitrary @xmath2 is only applicable in the classical regime where no ( integer ) restriction of @xmath14 exists . in the mesoscopic regime , @xmath24 , the set of @xmath2 for which eq . has a solution is restricted . but the effective measure of this set is much larger then in the case when @xmath22 has commensurable components . this implies that the effective density of @xmath0-functions in eq . strongly increases in a given @xmath2 interval , but is still discrete . we will further refer to this observation as the @xmath0-function `` condensation '' . as the result the singular part of @xmath1 becomes relatively small and @xmath1 has a smooth envelop approximation , @xmath25 . we emphasize that the functional behavior of @xmath1 is different from the classical limit due to the nontrivial factor @xmath26 . we find that @xmath27 is the one - dimensional jacoby theta - function , @xmath28 , where the real parameter @xmath29 depends on @xmath12 and @xmath11 and qualitatively shows to what degree the components of @xmath22 are incommensurate . we explicitly calculate @xmath30 , which behaves in the classical limit as @xmath31 resulting in @xmath32 . in order to investigate @xmath33 in the regime where the @xmath0-functions condensate , it is quite ineffective to use the @xmath0-function representation of @xmath33 , eq . , directly . instead , we rewrite @xmath33 in terms of matrix @xmath34-function , @xmath35 [ generalizing the approach suggested in ref . @xcite ] , where the real symmetric matrix @xmath36 has one zero eigenvalue , while the other eigenvalues are positive . this zero mode is the manifestation of @xmath0-functions in eq . . we investigate the class of the @xmath34-functions with the ( nearly)degenerate @xmath36 ; it is an interesting problem itself . using the matrix @xmath34-function representation of @xmath33 we can analytically and numerically investigate @xmath1 in the regime where the @xmath0-function condensation takes place . in a limiting case we reproduce the results of ref . we relate the nature of the strong increase of the @xmath0-function density to the chaotic behavior of quasi - periodic functions . our results can help understanding the transport theory of josephson junction arrays and the superconductor - insulator transition . finally , we discuss how stable our results related to the @xmath0-function condensation in eq . are with respect to the choice of the shape of the weight functions @xmath8 other than given in eq . . the structure of our paper is the following : in sec . [ sec : theta ] we show how problem can be reformulated in terms of matrix theta - functions ; in sec . [ sec2 ] we investigate the properties of the matrix theta - function in the mesoscopic regime and in particular formulate the conditions for the delta - function condensation ; in appendix [ sec3 ] we give a numerical receipt for an efficient calculation of the matrix theta - function ; finally in the discussion section [ disc ] , we show that the problem we address in this paper has a number of important physical applications . we rewrite the sum in eqs . - using the poisson s formula for summation @xmath37 where @xmath38 is a continuous integrable function . function @xmath33 in eq . depends on the vector @xmath39 . therefore , we have to introduce also @xmath40 and @xmath41 accordingly . approximating the @xmath42-function by a smeared gaussian function @xmath43 with @xmath44 we get after the integration over @xmath41 : @xmath45+\left[\frac{\omega}{\sigma^2}\right]^2e_ie_j\right)\frac 12\langle x_i x_j\rangle\right\}.\end{gathered}\ ] ] here @xmath46_{ij}.\end{aligned}\ ] ] we simplify eq . by expanding it over @xmath21 . then we find , @xmath47 where the explicit analytical expressions for @xmath48 , @xmath49 $ ] and @xmath50 can be found in appendix , eqs . - . introducing the riemann theta - function @xcite , @xmath51 we rewrite @xmath33 as follows : @xmath52 the proof that @xmath53 is given in appendix , eq . . therefore , the @xmath34-function matrix parameter is degenerate resulting in the appearance of the @xmath0-functions in eq . . it is worth noting that @xmath54 is generally not an eigenvector of @xmath36 ( such that @xmath55 ) . an important role play the eigenvalues @xmath56 of matrix @xmath36 . we will distinguish the cases when @xmath56 are smaller or larger than unity below . without loss of the generality we can assume that the eigenvalue corresponding to the eigenvector @xmath22 has the index one : @xmath57 . then for the other eigenvalues follows : @xmath58 . we kept @xmath59 above which resulted in @xmath57 . in typical physical realizations , the @xmath0-function always have a finite width @xmath21 as was mentioned in the introduction . if we preserve the sub - leading terms in @xmath60 in @xmath33 then @xmath61 . perturbations of the other parameters in eq . produce sub - leading corrections to the shape of @xmath1 . .,title="fig : " ] + there are special cases when @xmath62 and the parameters @xmath18 , @xmath63 , and @xmath36 have very simple forms . one case corresponds to a diagonal matrix @xmath12 as follows from the proof given in appendix [ apb ] . the case , when @xmath12 has the additional structure : @xmath64 , where @xmath65 is a parameter and @xmath66 , is important for transport physics in jjas and dirty superconducting films near the superconductor - insulator transition @xcite . we will refer to this case as to the `` minimal model '' . it should be noted that @xmath22 in this case is not an eigenvector of @xmath12 in general . this case is of theoretical interest since it is possible to find @xmath67 exactly and @xmath68 analytically in all orders of @xmath69 : @xmath70_{ij}\beta=-\frac{1}{\beta\gamma+\sum_{k}e_k}+\frac{\delta_{ij}}{e_i } , \\\label{eq : detg } \det g=\beta^n\frac{(\beta\gamma+\sum_i e_i)\prod_k e_k}{\beta\gamma}.\end{gathered}\ ] ] then we get provided @xmath71 : @xmath72 similarly we find using eq . that @xmath73 expressions eqs . - generalize similar results in ref . @xcite obtained for the limit where @xmath74 , @xmath75 , @xmath76 , and @xmath22 is close to the vector @xmath77 . we took @xmath59 in eq . such that @xmath57 . if we take into account @xmath20 then we should correct all matrix elements , @xmath78 , by an additional term , @xmath79 $ ] corresponding to @xmath80 . below we study how the properties of the function @xmath34 in eq . depend on the parameters @xmath22 , @xmath12 , and especially @xmath5 . we start our analysis with the simplest case , @xmath81 . then we get from eq . that @xmath34 reduces to the usual jacobi theta - function : @xmath82 for @xmath83 , @xmath84 and for @xmath85 , @xmath86 , see fig . [ fig_theta3d ] for an illustration . the case @xmath81 is marginal to some extent because then @xmath87 and so @xmath27 is always a set of @xmath0-functions ; formally there is no classical limit for @xmath81 . in the following we focus on the case @xmath88 only . for @xmath89 . without loss of the generality we assume that @xmath22 ( direction singled out by the green line ) is close to @xmath90 . the dotted lines are parallel to @xmath91 . the red points belong to the subset of @xmath92 that gives the leading contribution to @xmath34 in the mesoscopic regime , eq . .,title="fig : " ] + the classical limit as follows from eq . can be found by setting @xmath93 in eqs . - . then @xmath94 and as follows from eq . @xmath95 we recall that in the mesoscopic regime @xmath96 ( @xmath97 ) . in the mesoscopic ( quantum ) regime @xmath1 strongly differs from eq . because the prefactor , @xmath98 , in eq . behaves in a non - trivial way . the definition of the matrix @xmath34-function , eq . , includes the sum over @xmath92 . the components of @xmath92 are integer numbers . so @xmath92 can be treated as edge vectors of the nodes of an effective cubic crystal in a @xmath5-dimensional space , see fig . [ condition ] . it follows from the definition of the @xmath34-function , eq . , that the leading contribution to @xmath34 gives @xmath92 nearly collinear to @xmath22 . we focus first on the case when @xmath22 is close to the crystallographic axis @xmath91 of the effective crystal . then the subset of @xmath99 , @xmath100 , where @xmath101 , gives the leading contribution to @xmath34 : @xmath102 and @xmath103 where @xmath104 . the vectors @xmath22 and @xmath91 should be close such that the following condition is fulfilled : @xmath105 which ensures a complete delta - function overlap ( we recall that the @xmath0-function width is @xmath106 , while @xmath107 ) . here the integer @xmath108 $ ] , where @xmath109 is the integer part . this condition indicates how far from the origin the distance between the points on the lines o@xmath22 and o@xmath91 becomes of the order of the effective crystal period , see fig . [ condition ] . for @xmath110 [ black color ] . the minimal model has been used . taking the vector from the set of the gaussian random numbers with zero average and the dispersion @xmath111 we generated @xmath112 , where @xmath113 . the red line corresponds to @xmath28 the envelop function . the inset shows that the @xmath0-function width @xmath21 is the minimal characteristic frequency of the matrix @xmath34-function.,title="fig : " ] + expression is in fact the envelop approximation for the matrix theta - function . the @xmath34-function quickly oscillates as the function of @xmath2 on the scale of the order of @xmath21 on top of the envelop function , see fig . [ fig_theta ] . the amplitude of the oscillations does not exceed @xmath114 . it is worth noting that condition is sufficient but not necessary . even beyond the limitations set by condition the envelop function eq . usually still approximates @xmath34 quite well . this case is illustrated in fig . [ fig_theta ] . numerical calculations @xcite show that the envelope of the @xmath34-function follows well eq . while @xmath115 , where @xmath116 is a constant of order unity . this exponential factor is closely related to the density of the @xmath0-functions in eq . , which have in the mesoscopic regime weights of the same order . if @xmath117 then @xmath118 reduces to a superposition of @xmath0-functions , see eq . and fig . [ fig_theta3d ] . these small values of @xmath30 appear when @xmath119 , see eq . . in case the components of @xmath22 are integer numbers ( they are commensurable ) then the @xmath0-function singularities of @xmath1 remain in the mesoscopic regime as well . however , if @xmath22 and @xmath91 are even slightly different , @xmath30 can easily become of order unity and @xmath27 becomes a smooth function of @xmath2 . returning to eq . we will have a strong increase of the @xmath0-function density and their final overlap , when the components of @xmath22 become not commensurable . to conclude this section , we emphasize that eq . is one of our main results . it shows that in the mesoscopic regime @xmath1 is nearly continuous ( plus a relatively small and quickly oscillating background ) and the form of the function @xmath1 is nontrivial . we find the envelop approximation of @xmath1 analytically : @xmath120 the accuracy of the prefactor in eq . is illustrated in fig . [ fig_theta ] . the density graph [ fig3d ] shows the evolution of the matrix @xmath34-function in the minimal model when the parameter @xmath65 switches the model from the quantum to the mesoscopic regime . the numerical calculations are briefly described in appendix [ sec3 ] . for @xmath110 . the axes @xmath121 corresponds to @xmath2 while @xmath122 axes corresponds to @xmath123 . when @xmath124 the theta function behaves as the discrete set of @xmath0-peaks . with decreasing @xmath65 , more peaks become visible , for @xmath125 the peaks start merging and for @xmath126 the matrix theta function becomes nearly smooth.,title="fig : " ] + the time representation helps to understand the properties of @xmath1 from another point of view . for a special case it was shown in @xcite that the landau - hopf turbulence scenario can account for the merging of the @xmath0-functions in eq . in the mesoscopic regime . here we apply the ideas developed in ref . @xcite for the general case . to this end we express @xmath1 , eq . , in time representation , @xmath127 , and find : @xmath128 function @xmath129 belongs to the set of _ quasi - periodic _ functions @xcite . if we take @xmath130 as coordinates in an @xmath5-dimensional space then the trajectory @xmath131 is modeled by a curve on a torus @xmath132 that wraps around without ever exactly coming back on itself if the @xmath133 are _ incommensurate _ , see fig . [ fig3 ] for @xmath89 . the path covers the torus surface densely everywhere . if we return to eq . then this property would mean that the @xmath0-function positions are densely distributed . in the quantum regime only @xmath4 with @xmath134 contribute to @xmath129 such that not more than one frequency appears in the exponent in eq . . the classical limit formally corresponds to a torus with an infinite number of dimensions . our consideration for the condensation of the @xmath0-functions in eq . can be extended to the more general case of arbitrary weights @xmath135 which decay quickly with @xmath4 @xcite . in this case , the time representation of @xmath33 would consist of quasi - periodic functions as well and the topological argument of the path covering a torus surface densely would be applicable again . in this more general case @xmath1 can not be reduced to the matrix theta - function . however , the poisson transformation of @xmath33 with an appropriate @xmath21-expansion still allows to reduce @xmath33 to an `` easy to handle '' form for analytical and numerical investigation in the regime when the @xmath0-functions start to merge . it is worth noting that expressions similar to eq . appear in many applications . for example , it describes the velocity field @xmath136 of a liquid when turbulence develops according to the landau - hopf scenario @xcite , @xmath137 . here the developed turbulence corresponds to the case of large @xmath5 @xcite . quasi - periodic functions describe quasi - periodic motion of mechanical systems @xcite and often appear in the theory of differential equations , see e.g. , ref . @xcite . + in the study of granular superconducting materials and josephson junction arrays ( jja ) transport properties are of central importance . the most interesting case corresponds to the quantum limit when the josephson coupling @xmath138 between the granules is much smaller than the characteristic coulomb energy @xmath139 of the cooper pair located on a granule . the transport problem was solved within linear response approaches , see e.g. , @xcite . it was shown that the conductivity @xmath140 of the granular superconductor is a singular function in the leading order over @xmath138 . it behaves as the countable superposition of @xmath0-functions with the gibbs weights . for example , the conductivity in a jja according to ref . @xcite has the following form [ which can also be derived from eq . in appropriate limiting cases ] : @xmath141 where @xmath2 is the frequency , @xmath132 is the temperature , the indices @xmath142 , @xmath143 label the islands of the jja , @xmath144 is the inverse capacitance matrix of the jja and @xmath145 . here @xmath146 is the charge of the cooper pair and @xmath147 is the electron charge [ we will use the system of units where @xmath148 . the integer numbers @xmath149 show the effective number of cooper pairs sitting on the island @xmath142 . we assume here and below the einstein sum convention . the exponential weights in eq . come from the gibbs distribution . the delta - functions ensure energy conservation during the processes of cooper pair tunneling from one granule of the jja to its neighbor . they can not be easily integrated out in eq . since changes of the coulomb energy are essentially discrete because of charge quantization within granules . the size of the jja arrays can be quite small , especially in the case of one - dimensional ( 1d ) arrays . therefore , the number of junctions @xmath5 in the jja is typically finite , @xmath150 . on the other hand the interaction matrix @xmath151 can be rather quickly decaying with @xmath152 . generally , we can not use standard statistical physics approaches based on the thermodynamic limit @xmath153 trying to smear out the @xmath0-function singularities in the observables , as it was done in eq . . an important question is to understand the nature of the @xmath0-function singularities in the observables and finding systematic ways of their regularization . one way to overcome the difficulties related to the @xmath0-functions in transport observables of the jja was proposed in ref . that calculation was based on the assumption of an energy band for cooper - pair tunneling . however , this band can form in jja with nearly identical granules but should be suppressed by the disorder in typical disordered jjas @xcite . in refs . @xcite an attempt was made to find the transport characteristics in the disordered jjas . the model of refs . @xcite leads to the conductivity behaving according to eq . with @xmath154 and @xmath22 close to the vector @xmath77 ( a limiting case of our minimal model ) . it was shown in ref . @xcite that although @xmath22 has non - commensurate components , the delta - functions merge and the conductivity may become a smooth function of its parameters . in this paper we do not restrict the choice of @xmath12 and @xmath22 like in ref . @xcite . the properties of the matrix-@xmath34-function strongly depend on the number @xmath155 of the nonzero components of @xmath54 . this number can be treated as an effective dimension of the @xmath34-function . generally @xmath156 . without loss of the generality we can assume that the first @xmath155 components of @xmath54 are nonzero while the others take zero values . we can apply a reduction procedure if @xmath157 : to take the sum over @xmath158 with @xmath159 in eq . . we finally obtain a function very similar to the matrix - theta function but now with a summation over @xmath155-dimensional integer vectors . for example , @xmath160 for the problem considered in ref . @xcite and @xmath161 for ref . @xcite . the problem of the evolution of a quantum system interacting with a ( quantum ) environment of ( soft ) modes is being studied for more than 60 years , but more important to be solve than ever , see e.g. , refs . usually it is implied that the environment has a continues spectrum of modes . that condition is important since it typically avoids the the appearance of the @xmath0-functions like in eq . in observables of the quantum system . a discrete environment , on the other hand , can not simply absorb arbitrary amounts of energy , but rather only discrete energy portions in quantums on the order of its level spacing . in other words , the set of quantum modes can serve as a `` bath '' only when its spectrum is continuous @xcite , which we will clarify below . . the wavy lines show schematically the exchange of the energy between the junction and the environment @xcite . b ) the environment is not necessary described by the equilibrium density matrix @xcite . here we sketched the environment interacting with a thermal bath and the quantum system . , title="fig : " ] + an important problem where a quantum system interacts with an environment , is the problem of quantum transport through an ultrasmall tunnel junction , see fig . [ figeh ] , and ref . @xcite for a review . if the contacts of the junction are superconducting , the supercurrent is given by @xmath162 where @xmath163 is the probability to exchange energy @xmath2 with the environment and @xmath138 is the josephson energy of the junction . next , we concentrate on the shape of @xmath164 . according to ref . @xcite @xmath165\exp[-ie^*\hat\phi(0)]\rangle e^{i\omega t},\ ] ] where @xmath166 is the `` charge transfer '' operator , @xmath167 for cooper pairs and @xmath168 denotes averaging over the density matrix @xmath169 of the environment . [ we recall that we have chosen units where @xmath170 . since the environment is isolated , it can be described by the hamiltonian @xmath171 with a set of energies @xmath172 and eigenfunctions @xmath173 . in general we have @xmath174 . writing the phase operator in heisenberg representation explicitly , we get @xmath175=\hat u^{-1}(t ) \exp[ie^*\hat\phi(0)]u(t)$ ] , where @xmath176 . this way we find : @xmath177 thus @xmath163 is represented as the superposition of @xmath0-functions . only if the spectrum of the environment is continuous we can introduce a continuous density of states and integrate out the @xmath0-functions in eq . . as long as the spectrum of the environment is discrete , we can assume without loss of generality that the greek indices labeling the levels are a set of integer numbers . if the environment is represented by oscillator modes ( or quantum rotators ) , @xmath178 is linear in the indices labeling the environment states like in eq . . if , in addition , the environment is in local equilibrium , such that @xmath179 , where @xmath180 is the environment temperature , the structure of eq . is the same as the structure of eq . . to complete our investigation we focus on the quasi - particle current @xmath181 through the tunnel junction shown in fig . [ figeh ] . according to refs . @xcite we have @xmath182 where @xmath183 ( @xmath184 ) is the tunneling rate from the left ( right ) to the right ( left ) , and , for a single junction , @xmath185 where @xmath186 are the electronic distribution functions within the left ( right ) electrodes , and @xmath187 is the bare tunnel resistance , representing the interaction of electrons with the bath . here @xmath163 is the probability for electron quasi - particle to lose the energy @xmath2 to the environment ; it is given by eq . with @xmath188 . the backward scattering rate is given by @xmath189 . [ if the contacts are superconducting , we can account for that by introducing the quasi - particle densities of the states in the contacts @xcite in eq . . would then give the quasi - particle current . ] if the environment is absent and the relaxation is provided by a phonon bath , @xmath190 and eq . reproduces the conventional ohm law . it follows from eqs . - that the integration over the energy removes the @xmath0-functions of the environment and the quasiparticle current is not as singular as the supercurrent when the environment has a discrete spectrum of modes . to summarize , the problem we solve in this paper is closely related to the problem of a discrete environment exchanging energy with a quantum conductor . it should be emphasized that the environment should not be necessary located somewhere outside the quantum conductor . on the contrary , it can be part of the conductor itself . such a situation is realized in jjas , see e.g. , ref . @xcite , or in very dirty conductors where the transition between discrete and continuous `` built - in '' environments is closely related to many - body localization , refs . we have shown that observables having a form as described by eq . can be rewritten in terms of the matrix theta - function . we demonstrated that the @xmath0-functions typically condensate in the mesoscopic regime such that the observable @xmath33 we focus on becomes a nearly continuous function . we found the envelop function for @xmath33 analytically and therefore , our results can help to understand and describe transport properties in a number of strongly correlated systems . the authors thank a. petkovic and v. vinokur for active discussions in the initial stage of the work ; we also thank m. fistul for helpful comments and t. baturina for interest to our work . this work was supported in part by the russian foundation for basic research ( grant no . 11 - 02 - 00341 ) and by the u.s . department of energy office of science under the contract no . de - ac02 - 06ch11357 . the matrix @xmath12 can be diagonalized by the orthogonal transformation @xmath191 , such that @xmath192 , where @xmath193 is the diagonal matrix of the eigenvalues . then expanding the determinant of @xmath194 over @xmath69 we get the following relation : @xmath195 where @xmath196 . similarly we can prove , using an induction procedure , that @xmath197 and @xmath198= \\ \frac{\det e}{g}=\left(\sum_{i } \frac{(\tilde e_{i})^2}{\lambda_{i}}\right)^{-1}. \ ] ] it should be noted that @xmath199 . @xmath200 we will see that the vector @xmath54 is the characteristic direction of the @xmath27-function . it is worth noting that generally @xmath22 and @xmath54 are not parallel in euclidean space . the vector @xmath54 can be found explicitly : @xmath201 it follows from eq . that @xmath202 finally , we find the inverse of the matrix @xmath194 at @xmath44 . this is somewhat tricky since @xmath194 becomes singular while , following eq . , @xmath67 becomes degenerate . nevertheless there is a finite nontrivial limit for @xmath67 at @xmath44 : @xmath203 the diagonal elements have the structure : @xmath204 where @xmath205 . we assume that @xmath22 is an eigenvector of matrix @xmath12 corresponding without loss of generality to eigenvalue @xmath206 . then @xmath207 , and therefore @xmath208 where @xmath209 . most important in this case is @xmath210 this identity can be proven in a diagonal representation of @xmath12 . without loss of the generality @xmath211 corresponds to the eigenvalue @xmath206 of @xmath12 . then we can take again @xmath212 and @xmath213 so , @xmath214 . an important task is to verify the accuracy of eq . and the envelope approximation of the matrix @xmath34-function numerically . the required calculations of the matrix @xmath34-function for @xmath88 requires care since one should sum up many quickly oscillating functions . we have found an effective numerical algorithm explicitly relying on the existence of the soft mode @xmath22 of the matrix @xmath36 . the matrix @xmath36 can be expressed explicitly through its eigenvalues @xmath215 and the corresponding eigenvectors @xmath22 and @xmath216 , @xmath217 [ @xmath218 @xmath219 the set of @xmath92 , see eq . , form a @xmath5-dimensional cubic crystal . we construct a cylinder in this space with generating lines @xmath22 and an elliptical support with main directions @xmath216 . doing numerical calculations we take into account only points of the crystal , @xmath92 , in eq . that belong to the cylinder volume , see fig . [ fig_tube ] . the proportions of the cylinder depend on the calculation accuracy . the ratio of the cylinder height to its characteristic diameter is of the order of @xmath220 . the case @xmath97 is the most interesting since then the @xmath0-functions in eq . are expected to condensate . then only @xmath92 , the nearest neighbors to the line directed along @xmath22 , should be taken into account , see green diamonds in fig . [ fig_tube ] and red dots in fig . [ condition ] . this property strongly reduces the numerical efforts compared to a direct calculation of @xmath221 using eq . . , in eq . , that belong to the `` cylinder '' volume and satisfy the condition @xmath222 . here the solid line is directed along @xmath223 . the green diamonds distinguish the points of the effective crystal that give the leading contribution to @xmath34 for @xmath224 ; the black points should be added if @xmath225 while the small blue points become important for @xmath226 . , title="fig : " ] + a typical graph of @xmath98 for @xmath110 is shown in fig . [ fig_theta ] [ black line ] , the red line shows the envelop function represented according to eq . by the one dimensional ( jacoby ) @xmath34-function . the inset shows that the @xmath0-function width @xmath21 is the minimal characteristic `` frequency '' of the matrix @xmath34-function . it follows also that the matrix @xmath34-function can be approximated by the one - dimensional ( jacoby ) @xmath34-function with specially chosen parameters . for producing the graph [ fig_theta ] we have used @xmath36 from eq . . this minimal model we have chosen for two reasons : 1 ) @xmath36 is parameterized by a minimal set of parameters , 2 ) this model has some relation to physical applications , see refs . @xcite,@xcite . the vector @xmath22 we expressed as @xmath227 , where we chose @xmath228 according to ref . the components of @xmath229 are generated by a gaussian random number generator with zero average and the variance @xmath111 . for simplicity we used the restriction , @xmath230 . then we get @xmath231 $ ] that agrees with ref . @xcite . if @xmath119 , the components of @xmath22 are commensurate and @xmath232 behaves just as the superposition of @xmath0 functions shifted by a constant period of the order unity along the @xmath233-axis as follows from eq . . but if @xmath234 then the components of @xmath22 are not commensurate , there is no periodicity in the @xmath0-function set , as it is illustrated in fig . [ fig_theta ] for @xmath235 . we implied nearly everywhere above that @xmath236 . if this is not the case and @xmath124 , @xmath237 in eq . quickly decay with growing @xmath14 and we can disregard all @xmath4 with @xmath238 within acceptable error bars . it is clear that in that case @xmath1 [ as well as @xmath27 ] behaves always as a set of @xmath0-functions with some finite distance from each other . disorder in @xmath22 only slightly shifts the positions of the @xmath0-functions on the @xmath2-axis and there is no @xmath0-function condensation . what happens when we go from @xmath236 to @xmath124 is illustrated in fig . [ fig3d ] . when @xmath124 the theta function behaves as a discrete set of the @xmath0-peaks . with decreasing @xmath65 more peaks develop , for @xmath125 the peaks start merging and for @xmath126 the matrix theta - function becomes nearly smooth . l. d. landau and i. m. khalatnikov , collected papers of l.d . landau , d. ter haar , ed . ( new york : gordon and breach , 1965 ) ; l. d. landau and i. m. khalatnikov , dokladii academii nauk cccp 96 , 469 ( 1954 ) ; v.l . ginzburg , and l. d. landau , zh . fiz . * 20 * , 1064 ( 1950 ) . the function @xmath240 is the quasi - periodic function with periods @xmath241 if @xmath242 where @xmath243 is periodic function of @xmath244 with the periods , @xmath241 , respectively . all the periods @xmath241 are strictly positive , and their reciprocals @xmath245 are rationally linearly independent . the quasiperiodic function can be represented by the series : @xmath246 . d.v.averin and k.likharev , in _ quantum efects in small disordered systems _ , ed . by b.l.altshuler , p.a.lee , and r.a.webb ( elsevier , amsterdam , 1991 ) ; g.schn and a.d.zaikin , phys . rep . * 198 * , 237 ( 1990 ) .

we investigate dynamic properties of inhomogeneous nano - materials , which appear in analytical descriptions typically as series of @xmath0-functions with corresponding gibbs weights . we focus on observables relevant for transport theories of josephson junction arrays and granular systems near the superconductor insulator transition . furthermore , our description applies to the theory of tunnel junctions exchanging energy with a `` bath '' , the latter having a discrete spectrum . using the matrix theta - function formalism we find an analytical expression for the transport characteristics capturing the complete temperature driven transition from the quantum to the classical regime .

introduction[sec:intro] generalized matrix theta-function matrix @xmath34-function in the mesoscopic regime discussion conclusions acknowledgments parameters of the @xmath34-function. analytical expressions. diagonal @xmath12-matrix[apb] @xmath1, numerical investigation

arxiv

mixed conductors- which transport charge by the motion of ionic species- are essential ingredients in next - generation energy sources such as fuel cells and for separating hydrogen from mixed gas streams in fossil fuel power plants.@xcite such applications have demanding requirements for mixed conductors that can withstand high temperatures and reactive atmospheres . proton conducting ceramic membranes will also play a vital role in challenging industrial processes such as gas - to - liquid conversion , selective dehydrogenation , water - gas - shift reaction and ammonia synthesis thanks to their integration in catalytic membranes reactors.@xcite the crystal structure of mixed conductors plays a crucial role . a combination of a crystalline sublattice , to provide rigidity , and a defective , short - range correlated sublattice of charge carriers would be ideal.@xcite oxygen vacancies in particular are needed for proton conduction , and are usually created by doping . however , this synthetic strategy faces a delicate balancing act@xcite since the creation of defects by doping causes structural inhomogeneities , such as cation size and charge mismatches or grain boundaries , which may trap charge carriers.@xcite high - quality grain - boundary free thin films are thus needed for optimum performance.@xcite proton conduction is also favored by high symmetry lattices . however , in the commonly studied abo@xmath3 perovskite materials , the tolerance factor required for cubic structures presupposes large a - site cations such as barium , which result in chemical instabilities in the acidic atmospheres found in typical applications . this is especially true for high - temperature applications , such as the removal of hydrogen from syngas mixtures , which are needed to support the transition to a hydrogen - based economy and to improve the efficiency of current sustainable energy production.@xcite rare earth tungstates with the general formula ln@xmath2wo@xmath5 ( ln = rare earth ) are promising materials as they show several ordered crystal structures as a function of rare earth size , and an excellent combination of proton and electron conductivity.@xcite the ceramic la@xmath6wo@xmath1 ( 0.3 < _ x _ < 0.7 ) provides the highest proton conductivity and fulfills the requirements for an efficient mixed conductor . the physical properties shown by la@xmath0wo@xmath1 membranes are ideal for applications.@xcite despite many attempts , the polymorphism and extremely subtle structural distortions shown by this family of materials have prevented solution of the crystal structure . various reports of disordered pyrochlore or ordered defect fluorite - like structures have previously been published.@xcite most recently , magraso et al.@xcite have published three different structures for la@xmath0wo@xmath1 using neutron and x - ray powder diffraction . these authors eventually converged on a cubic fluorite - type structure with a 2__a _ _ x 2__a _ _ x 2__a _ _ superlattice caused by cation ordering , and noted that single - phase specimens were only observed for la / w ratios between 5.3 and 5.7.@xcite dft calculations and recent high temperature neutron diffraction experiments have updated the former model.@xcite however , the exact relationship between the average and local crystal structure , microstructure and proton conductivity remains unresolved . most fundamentally , it is also presently unclear why the cubic phases of _ _ re__@xmath7wo@xmath1 stoichiometry are so disordered , given the presence of completely ordered phases very close by in the phase diagram . for example , orthorhombic phases of stoichiometry _ _ re__@xmath8w@xmath9o@xmath10 and rhombohedral _ _ re__@xmath2wo@xmath5 phases form for smaller rare earths . both of these structure types have ordered arrangements of anion defects and hence are very poor mixed conductors.@xcite here , we synthesize pure samples with a composition of la@xmath0wo@xmath1 that show exceptional device performance and stability.@xcite complementary x - ray and neutron diffraction techniques are used to solve the average crystal structure , which goes beyond recently proposed models . we establish the structure of la@xmath0wo@xmath1 , e.g. lattice symmetry , degree of cation ordering and the location of oxygen vacancies . real - space pair distribution function analysis of high - energy synchrotron x - ray diffraction data is applied to confirm the model of the average crystal structure and to develop a model of the local crystal structure since total scattering analysis provides a tool to model bragg and diffuse scattering simultaneously.@xcite x - ray absorption spectroscopy ( exafs ) at la k - edge and w l@xmath11-edge was performed to confirm the local crystal structure . _ sample synthesis _ : the powder material was prepared using the corresponding anhydrous oxides as precursors . the reagents were milled in ethanol for 24 hours , dried and calcined at 800@xmath12c and 1000@xmath12c for 15 hours respectively . subsequently the mixture was milled again , pressed and then sintered at t = 1400@xmath12c . a stoichiometry of la@xmath0wo@xmath1 was selected to achieve phase - pure materials since nominal la@xmath2wo@xmath5 has been shown to segregate la@xmath9o@xmath3 . + _ compositional analysis _ : chemical composition was studied by neutron activation analysis ( naa ) at the berii reactor in berlin and by inductively coupled plasma optical emission spectrometry ( icp - oes ) . + _ phase and structural analysis _ : phase analysis was performed applying x - ray diffraction using a bruker d8 diffractometer in bragg - brentano geometry and cu - k@xmath13 radiation . the crystal structure was studied in detail by different diffraction methods . high - resolution synchrotron x - ray diffraction ( hrsxrd ) was performed at esrf beamline id31 ( @xmath14 = 0.3962 , t = 10 k and 295 k ) . neutron powder diffraction ( nd ) data were collected on the d1a and d2b diffractometers at the institut laue - langevin , grenoble , at wavelengths of @xmath14 = 1.909 ( t = 573 k ) and @xmath14 = 1.595 ( t = 300 k ) , respectively . prior to the diffraction experiments the sample was dried for 96 hours at 1173 k under argon flow and loaded in a controlled helium atmosphere using a glove box . the combination of x - ray and neutron diffraction data proved to be crucial as the former is sensitive to the heavy cations , while the neutron scattering lengths for w ( 4.86 fm ) , la ( 8.24 fm ) and o ( 5.80 fm ) are comparable . rietveld refinement of crystal structure models against the experimental data was performed using gsas with the expgui graphical user interface.@xcite high energy powder x - ray diffraction data were collected using id15b also at the esrf . a wavelength of 0.1422 was used and the scattered x - rays were detected by a mar345 image plate . the pair distribution function was calculated using in - house software ( ipdf ) developed by one of the authors ( sajk ) , which runs as a graphical user interface within igor pro . briefly , data were corrected for background , compton scattering , and the atomic form factor . the compton shift , detector efficiency and incoherent fluorescence were also taken into account , before fourier transformation according to : @xmath15sin(qr)dq\ ] ] here q.[s(q)-1 ] represents the properly corrected and normalized intermediate structure factor and the _ r_-grid used in real space had a spacing of 0.01 . models were fitted to the pdf data using the pdfgui package.@xcite model pdfs were calculated using : @xmath16 - 4\pi r \rho_{0}\ ] ] the indices _ i _ and _ j _ run over all atoms in the sample . the scattering powers of the different atoms are _ b@xmath17 _ and _ b@xmath18 _ , and _ r _ represents the radial distance in real space . pdfgui uses the so - called small box approximation , which implies that the first summation above only runs over the atoms within one unit cell as defined by the average crystallographic structure . this approximation makes data modeling tractable out to relatively large distances in real space . + _ exafs experiments _ : extended x - ray absorption fine structure spectroscopy ( exafs ) was performed on beamline kmc-2 at bessy ii , berlin , on the w l@xmath3 absorption edge ( e = 10207 ev ) at room temperature and on beamline _ x _ at hasylab , hamburg , on the la k absorption edge ( e = 38925 ev ) at 10 k. the measurements were performed in transmission mode and the data was processed and analyzed using the ifeffit code and the corresponding user interfaces athena and artemis.@xcite after background subtraction , the normalized and k@xmath19 weighted spectra were fourier transformed over a photoelectron wave number range of k = 3.4 - 12.6 @xmath20 for the w l@xmath3-edge and k = 3.66 - 16.46 @xmath20 for the la k - edge . the theoretical exafs signal was calculated by performing ab - initio calculations using the code feff8.2.@xcite the model was least squares fitted to the data in q - space with the following fitting parameters : a single amplitude reduction factor _ s@xmath21@xmath22 _ and an overall energy parameter _ @xmath23e@xmath21 _ for each dataset , fractional changes in bond length @xmath13 and a mean squared displacement parameter @xmath24@xmath22 for each coordination shell . _ initial steps _ : + the results of our neutron activation analysis showed that our sample had a composition of la@xmath25wo@xmath26 . analysis of the icp - oes measurements gave a similar result . our preliminary laboratory x - ray powder diffraction data showed that a doubled fluorite cell was found , with systematic absences consistent with face centred space groups . wo@xmath1 , highlighting the region of the ( 10,6,2 ) reflection . the peak asymmetry is well fitted by an anisotropic broadening model as discussed in the text . ] . ( b ) fourier difference maps calculated from neutron powder diffraction data collected at 573 k displaying the oxygen site splitting around w. the map size is 4 x 4 @xmath22 and the centre is at _ x _ = _ y _ = 0 and _ z _ = 0.125 . contour lines are drawn at 0.05 , 0.10 , 0.15 , 0.20 , 0.25 , and 0.30 fm@xmath27 . ( b ) resulting model for the oxygen site splitting around w ( gray ) . the oxygen split site ( red ) is shown with its anisotropic atomic displacement parameters ( adps ) at 573 k. ( d ) [ 111 ] oxygen vacancy pairs around w and the resulting oxygen displacements . ] due to the limited q - range data and medium resolution of the laboratory x - ray diffraction data , the structure solution was performed using the high - resolution synchrotron powder diffraction data ( hrsxrd ) . trial le bail intensity extractions using the doubled fluorite unit cell resulted in poor fit - statistics ( @xmath28r@xmath29=12.14 % and @xmath30=6.50 ) and close examination of the diffraction profile showed that this was caused by anisotropic peak broadening , which affected all reflections equally ( fig . 1 ) . as various models have been proposed in the literature@xcite , we performed an exhaustive search of distorted cells using le bail intensity extractions . this was compared with a minimal anisotropic broadening model which consisted of two cubic unit cells with an approximately 0.02% difference in lattice parameter ( _ _ a__@xmath31=11.1689(1 ) , _ _ a__@xmath9=11.1673(1 ) at 10 k ) . none of the distorted cells tested , including tetragonal ( _ i4/mmm _ , no . 139 ) , orthorhombic ( _ immm _ , no . 71 ) or monoclinic ( _ i112 _ , no . 5 ) , gave better fit - statistics than the phase separation model . this result is significant because the lower symmetry cells have more variables and so a higher degree of freedom in the intensity extractions . note that id31 is perhaps the highest resolution powder diffractometer currently available , and that this tiny effect is invisible with other instrumentation . we note that the clear peak splittings observed in the id31 data of magraso _ et al _ are almost certainly the result of measuring a phase separated sample , rather than a tetragonal distortion as reported in ref . + _ cation disorder _ : + during the structure solution and refinement , we initially used two _ fm@xmath32 m _ cubic phases differing only in lattice parameter . internal parameters were constrained to be the same for both phases . electron density maps identified three heavy atom positions , corresponding to the wyckoff sites 4__a _ _ ( 0,0,0 ) , 4__b _ _ ( @xmath33,@xmath33,@xmath33 ) and 24__d _ _ ( 0,@xmath34,@xmath34 ) . these sites were attributed to the cations and , in the first cycles of rietveld refinement , assumed to be statistically occupied with a la / w ratio of 5.4 as determined by naa . we initially assumed fluorite - type oxygen sites ( 32__f _ _ ( _ x_,_x_,_x _ ) position , x@xmath35 @xmath36 0.125 and x@xmath37 @xmath36 0.375 ) . after these initial steps , we then exploited the differing contrasts of the x - ray and neutron data sets to order the cations such that the 4__a _ _ site with the highest electron density was occupied by w , with the 4__b _ _ site occupied by la and the 24__d _ _ site occupied with 98% la and 2% w. for the 24__d _ _ site , the refined atomic displacement parameters ( adps ) were large and anisotropic . our electron density maps show that this site is actually split ( fig . 2a ) onto the 48__h _ _ ( 0,_y_,_y _ ) site with 50% occupancy . this disorder gave a much - improved refinement and realistic adps . these @xmath40.22 displacements are static and localized as they were identical in the 10 k and 295 k data - sets . + _ anion disorder _ : + finally , we were able to develop a disordered model for the oxygen sites coordinating the w 4__a _ _ site . difference fourier maps were calculated from the neutron powder diffraction data ( fig . these showed that oxygen occupies the 96__k _ _ site , with a total of 24 possible nearest neighbours arranged around the corners of a cube ( fig . 2c ) . the refined occupancy was found to be 25% , which is equivalent to an average coordination of 6.0(1 ) , as expected for octahedral w@xmath38 . the oxygen displacement parameters were found to be anisotropic , and elongated in the direction transverse to the w - o bonds . wo@xmath1 is shown . red spheres represent oxygen atoms , gray cubes w polyhedra with their partially occupied oxygen sites , green cubes the la(1 ) sites and yellow spheres the half occupied la(2 ) split sites . about one out of 48 la(2 ) positions is occupied by w , yielding the correct composition of la@xmath0wo@xmath1 . ] the local coordination can be explained by consideration of the defect chemistry . assuming two vacancies per site , and that it is energetically favourable to locate them as far apart as possible , gives the picture shown in fig . the vacancy pair is located on one [ 111 ] axis of the cube . the nearest - neighbour anions can then relax toward the defect in the [ 100 ] direction by 0.5 , forming a nearly regular octahedra . in combination with the cation disorder described above , this model then gave an excellent fit to both the synchrotron x - ray and neutron powder diffraction profiles ( figures 3a and 3b ) . the refined atomic coordinates , displacement parameters and residuals from the fit to the former are given in table i. the final model for the average crystal structure of la@xmath0wo@xmath1 is shown in fig . the structure is derived from a fluorite structure with doubled lattice parameter due to cation ordering . the tungsten positions form an _ fcc _ lattice and nearest w neighbors are linked by the split la(2)/w(2 ) site , which has an average of 7 fold oxygen coordination . in the structural model developed here , one w is substituted onto the la(2)/w(2 ) site 48__h _ _ to stabilize the la@xmath0wo@xmath1 phase . this is in complete contrast to ln@xmath2wo@xmath5 materials containing rare earth atoms with small ionic radii , which show complete ordering of cations and oxygen vacancies.@xcite .[tab : table2]refined atomic coordinates for la@xmath0wo@xmath1 at 295 k from synchrotron x - ray powder diffraction in space group _ fm@xmath32m_. the site occupancies are w(1 ) = 1 , la(1 ) = 1 , la(2)/w(2 ) = 0.48/0.02 , o(1 ) = 0.25 , o(2 ) = 0.9825 . the refined average lattice parameter is @xmath39 = 11.1844(1 ) and the residuals are @xmath28r@xmath29 = 7.36% , @xmath30 = 2.34 and r@xmath40 = 3.40% . [ cols="^,^,^,^,^",options="header " , ] the above experiments have identified a complex hierarchy of structure , which spans multiple length scales in la@xmath0wo@xmath1 . in what follows , we briefly review our average structure in comparison to earlier work , and discuss the local structure in more detail . + the lattice symmetry of la@xmath0wo@xmath1 has previously been called into question based on very high resolution synchrotron x - ray powder diffraction experiments performed at the esrf.@xcite however , our results show that la@xmath0wo@xmath1 is actually cubic , but with slightly asymmetric peak broadening that implies a tiny ( @xmath40.02 % ) lattice parameter distribution . we emphasise again that this effect is masked entirely by the resolution of standard x - ray and neutron diffraction instrumentation . having accounted for this effect , we were able to propose a new average structure model in space group _ fm@xmath41 m _ , which has two improvements on previously published work . firstly , we identified positional disorder on parts of the metal sublattice , which has an appealing explanation in terms of the defect chemistry in la@xmath0wo@xmath1 . we find ( fig . 2a ) that the la(2 ) site is split into two 50:50 occupied sites with displacement along the vectors joining neighbouring tungsten sites . when the possible location of oxygen vacancies in the lattice ( fig . 2 and fig . 6 ) are accounted for , this finds a natural explanation , as the la sites relax away from the defects to maintain seven - fold coordination . as we comment on more fully below , this is naively expected to propagate order through the lattice at temperatures below the onset of oxygen ion conduction . the second improvement of our model on previous work is the quantification of anti - site disorder . the possibility of a small amount ( ca . 5% ) of tungsten subsitution on the la(2 ) sites was suggested from dft calculations . however , no direct evidence of this has yet been presented . here we used a combination of x - ray and neutron diffraction , and exploited the differing contrasts of these techniques to get a consistent estimate of the anti - site disorder . when we refined the occupancy of the la(2 ) site against the x - ray data , it rose to above unity , consistent with a small amount of heavier w. indeed the refined x - ray scattering power was equivalent to 53.32 e@xmath42 , where f@xmath31(la@xmath43 ) = 52.79e@xmath42 and f@xmath31(w@xmath38 ) = 68.24e@xmath42 ) . for the refinement against the neutron diffraction data , we found an average scattering length of 8.12 fm , reduced from the value expected for la ( _ _ b__@xmath44=8.24 fm ) , again consistent with substitution by w ( _ _ b__@xmath44 = 4.86 fm ) . these measurements thus enabled us to calculate an anti - site disorder of 4.4(2)% . in addition , out exafs measurements on the w l@xmath3-edge , which are much more sensitive to this small structural perturbation , gave an additional confirmation . the unique element specificity of exafs means that extra sites are immediately apparent . here we found an order of magnitude decrease in the fitting residuals of our structure refinement when we included the trace amount of disorder inferred from the diffraction measurements . + while the details described above complete the description of the average structure of la@xmath0wo@xmath1 , our results go much further on local length scales . we were able to obtain qualitative insight into the local structure by using simple chemical concepts . for example , from an electrostatic point of view , the approximately octahedral coordination of tungsten shown in fig . 2d is by far the most likely as it minimises the repulsion between ligands . similar effects have been inferred for other simple functional materials , such as scandium and yttrium stabilised zirconias.@xcite this is shown by diffuse scattering observed in neutron powder diffraction experiments , which are sensitive to oxygen order . here , a simple model of two linked octahedra accounts for the peak positions of this scattering ( fig . this result suggests the intriguing possibility that order might propagate over a limited range , below the coherence length required to produce sharp bragg peaks . assuming that all octahedra have fixed orientations produces our _ pa@xmath41 _ structure ( fig . once again , this insight from diffraction is strongly confirmed by our exafs data sets , which were also satisfactorily fitted by this model . + we completed our picture of the local structure using real - space refinements against our x - ray pair distribution function data . this provides a definitive model for the _ pa@xmath41 _ structure which will be of significant use for dft investigations of the mechanism for mixed conduction . furthermore , since we were able to fourier transform our data out to at least 60 , we could estimate the size of the coherent _ pa@xmath41 _ domains . this analysis yielded a local coherence length of ca . 3.5 nm . we anticipate that future pdf experiments performed at high temperatures , will enable determination of the link between this length scale and mixed conduction performance in la@xmath0wo@xmath1 . + finally , the observation of local order of a limited length scale raises interesting questions . for example , why is this high entropy ground state favoured in la@xmath0wo@xmath1 ? furthermore , what factors prevent long - range order from establishing itself in this material ? we conclude our discussion by briefly speculating about connections with a seemingly unrelated field , that of so - called frustrated materials.@xcite these are materials , in which pairwise interactions ( for example an antiferromagnetic exchange interaction , j ) compete due to the lattice connectivity . long - range order is not established even at temperatures where _ _ k__@xmath45 @xmath46 j. the canonical example is water ice , where local rules enforcing the formation of bent water molecules create a massively degenerate ground state manifold . here the picture is similar . in our local structure , the tungsten cations are placed on a non - bipartite _ fcc _ lattice . when pairwise interactions are present ( such as our local rule linking the orientation of wo@xmath2 octahedra ) this lattice is frustrated . we speculate that this also results in a hugely degenerate number of structures for la@xmath0wo@xmath1 , which results in the glassy ground state we observe . indeed , our average structure model is nothing more than the result of spatially averaging the orientation of all of the wo@xmath2 octahedra in the sample . we note in passing , that much as the proton displacements in water ice map to an ising spin model ( spin ice),@xcite the la displacements in la@xmath0wo@xmath1 map to a well - studied ising model on the octahedral lattice.@xcite this comparison may be relevant for future investigations of defect dynamics in la@xmath0wo@xmath1 . for example , the propagation of defects in frustrated models is often quite different to that in standard materials . for example , protons diffuse through ice,@xcite rather than the hopping motion found in ceramic proton conductors.@xcite we believe future investigations of the local interactions in la@xmath0wo@xmath1 and other mixed conductors are thus well - merited and speculate that so - called function through frustration may be an appealing route to designing new mixed conductors in the future . in summary , we have reported a detailed study of both the average and the local structure of la@xmath0wo@xmath1 . we show that former has both positional and substitutional disorder of metal sites , as well as partially occupied anions sites . however , our diffuse scattering experiments show that on a local scale , rigid wo@xmath2 octahedra are found and we propose a simple defect rule which links the orientation of neighbouring sites . our local structure is confirmed by exafs experiments and will be of significant use in further modelling the mixed conduction of this class - leading material . we thank the helmholtz association for funding through the helmholtz alliance mem - brain ( initiative and networking fund ) and the spanish government ( grant ene2011 - 24761 ) . we thank d. alber and g. bukalis for naa measurements and e. suard and g. nenert for assistance with neutron diffraction data collection . the european synchrotron radiation facility , bessy - ii , hasylab and the institut laue langevin are thanked for the provision of beam time . we thank d.a . tennant , t. fennell , d.j.p . morris and s.j.l . billinge for useful discussions .

we report a comprehensive investigation of the average and local structure of la@xmath0wo@xmath1 , which has excellent mixed proton , electron and oxide ion conduction suitable for device applications . synchrotron x - ray and neutron powder diffraction show that a cubic fluorite supercell describes the average structure , with highly disordered lanthanum and oxide positions . on average the tungsten sites are six - fold coordinated , and we detect a trace ( 4.4(2 ) % ) of anti - site disorder . in addition to sharp bragg reflections , strong diffuse neutron scattering is observed , which hints at short - range order . we consider plausible _ local _ configurations , and show that the defect chemistry implies a simple chemical exchange interaction that favours ordered wo@xmath2 octahedra . our local model is confirmed by synchrotron x - ray pair distribution function analysis and exafs experiments performed at the la k and w l@xmath3-edges . we show that ordered domains of around @xmath43.5 nm are found , implying that mixed conduction in la@xmath0wo@xmath1 is associated with a defective glassy - like anion sublattice . the origins of this ground state are proposed to lie in the non - bipartite nature of the _ fcc _ lattice and the pairwise interactions which link the orientation of neighbouring octahedral wo@xmath2 sites . this function through frustration could provide a means of designing new mixed conductors .

[sec:level1]introduction [sec:level1]experimental [sec:level1]results [sec:level1]discussion [sec:level1]conclusions [sec:level1]acknowledgements

arxiv

nonequilibrium phenomena are observed throughout nature and society , and can often be thought of as some kind of driven diffusive system @xcite . examples include chemical kinetics , the motion of interacting molecular motors along a substrate , population dynamics in an ecology , the exchange of wealth or commodities in an economy , traffic flow and many more . the ongoing interest in driven systems has been sustained by the wide variety of behaviour the model systems exhibit , which incorporate not only elements familiar from equilibrium statistical mechanics , such as phase transitions , cooperative behaviour , and so on , but also many dramatic new features . these include phase transitions induced by external driving and even in one dimension ; long - range ( i.e. power - law ) correlations , not only at criticality , but generically in the steady state ; and very rich dynamics . model driven diffusive systems are typically defined on a lattice on which particles hop from site to site , where the precise definition of the stochastic particle dynamics is motivated on physical grounds . nonequilibrium steady states are constructed by driving a current of particles through the system . despite being simply stated , they still exhibit the wide range of phenomena . indeed , a virtue of their simplicity is that in some cases the models can be solved exactly . however , most cases elude exact solution ; moreover , even in cases where the steady state can be written down , the task of computing macroscopic observables may be too difficult . for these reasons , it is necessary to develop approximate techniques . a technique that has been hugely successful in equilibrium statistical mechanics is the renormalisation group procedure . here , we show how one can adapt these equilibrium methods to nonequilibrium systems , where we devise a way to accommodate the feature that distinguishes the nonequilibrium steady state from its equilibrium counterpart the existence of currents in the renormalisation group prescription . in particular , our approach includes the effects of fluctuations , which , in certain equilibrium and nonequilibrium systems , destroy the transitions predicted by mean field theory . thus our approach builds on previous prescriptions @xcite , and it complements real - space techniques devised for quantum - spin chains , and adapted to stochastic models @xcite . we test our approach using the asymmetric simple exclusion process with open boundaries , which we introduce and review in section [ asep ] . in section [ mp ] , we outline the matrix product formulation of the master equation . this formulation provides a convenient notation for the purposes of the steady state renormalisation which we describe in section [ ssrg ] . we recover the exact critical point and find qualitative agreement for the phase diagram . we extend the renormalisation to the dynamics in section [ drg ] and again we obtain qualitative agreement with the exact predictions for the relaxation dynamics . the asymmetric simple exclusion process with open boundaries is one of the most widely studied driven diffusive systems . exact results show that it undergoes both first- and second order phase transitions in the steady state and exhibits different relaxation dynamics in each of its phases . for these reasons , it represents an ideal test bed with which to assess the efficacy of approximate methods . the model is defined on a chain of @xmath0 sites , labelled @xmath1 . the occupation variable @xmath2 if site @xmath3 is vacant and @xmath4 if it is occupied ; multiple occupancy is forbidden . the stochastic dynamics are illustrated in figure [ asep dynamics ] : in continuous time , a particle at site @xmath5 hops to its nearest neighbour site to the right with rate @xmath6 , provided it is empty . if site 1 is vacant , it becomes occupied with rate @xmath7 , and if there is a particle at site @xmath0 , it leaves the system with rate @xmath8 . the phase diagram for this model is known exactly @xcite . in the limit of large @xmath0 , depending on the values of the ratios @xmath9 and @xmath10 , the system may reside in one of three possible phases ( see figure [ phase diagram ] ) : for @xmath11 and @xmath12 , the system is in a low density phase with particle density @xmath13 ; for @xmath14 and @xmath15 , the system is in a high density phase , @xmath16 ; for @xmath17 and @xmath18 the current , @xmath19,through the system assumes its maximum value @xmath20 and the density @xmath21 . the transition between the high and low density phases , indicated by the dotted line in the phase diagram , is first order along the transition line a low density region coexists with a high density region and the system density @xmath21 . the transition between these low current phases and the maximum current phase , indicated by the solid line in the phase diagram , is second order . the relaxation dynamics have also been obtained recently using the bethe ansatz @xcite : in the maximum current phase the relaxation time scales as @xmath22 where the dynamic exponent @xmath23 ; along the coexistence line @xmath24 ; in the two low current phases away from the coexistence line , the relaxation times are finite . the exact solution for the steady state of the asep was obtained using a matrix product technique @xcite . in this approach , the aim is to represent the probability that one observes the system in a configuration @xmath25 at time @xmath26 , @xmath27 , as @xmath28 where @xmath29 and @xmath30 are ( in general time - dependent ) matrices , and @xmath31 and @xmath32 are ( time - independent ) vectors . here , @xmath33 is a normalisation given by @xmath34 where we have introduced the ( time - independent ) matrix @xmath35 . thus , a string of matrices is used to represent a configuration of the particles in the system , where the matrix @xmath29 is used to represent an occupied site and @xmath30 a vacant site . in the limit @xmath36 , it was shown in @xcite that ( [ probabilities ] ) and ( [ normalisation ] ) do indeed give the steady state probabilities provided the matrices @xmath29 and @xmath30 and the vectors @xmath31 and @xmath32 satisfy the algebra @xmath37 moreover , matrix representations were found which satisfy ( [ bulk algebra])-([rh boundary condition ] ) . we remark however that in order to compute the probabilities to observe a particular configuration , it is not necessary to find such matrix representations ; it is instead sufficient to use ( [ bulk algebra ] ) to order all the matrices @xmath30 to the left of the matrices @xmath29 , and then use the boundary conditions ( [ lh boundary condition ] ) and ( [ rh boundary condition ] ) to obtain probabilities in terms of @xmath7 , @xmath8 and @xmath6 . in @xcite , it was shown how the steady state algebra can be generalised to a full dynamic algebra such that ( [ probabilities ] ) and ( [ normalisation ] ) represent the probabilities at all times @xmath26 . in this case , the matrices and vectors must satisfy @xmath38\;,\label{eqn of motion}\\ \lambda c^{-1}d = dc^{-1}\lambda\;,\\ \langle w|(\alpha / p - c^{-1}\lambda ) = 0\;,\\ ( \beta / p - \lambda c^{-1}|v\rangle = 0\label{dynamic rh bc}\;,\end{aligned}\ ] ] where @xmath39 is a current operator ( capturing the fact that in order for a particle to hop there must be a particle to the left of a vacancy ) . the key feature which distinguishes nonequilibrium steady states from equilibrium ones is the presence of currents . moreover , in driven diffusive systems with open boundaries , the boundary dynamics determine the particle current and density throughout the bulk of the system . this is in marked contrast to equilibrium systems where boundary effects are typically localised to the vicinity of the boundary . in this section , we present a renormalisation group method which accounts for these currents by employing a blocking procedure designed to accommodate the influence of the boundaries throughout the bulk of the system . a blocking transformation is implemented by grouping sites of the lattice into blocks where each block contains @xmath40 sites , as illustrated for the case @xmath41 in figure [ blocking ] . sites . ] these blocks are then identified with the sites of a renormalised lattice with a number of sites @xmath42 . this must be supplemented by a prescription which identifies particle configurations within a block with a single coarse - grained block variable . we achieve this by using a majority rule identification , which for @xmath41 is @xmath43 the issue for any such renormalisation group treatment then is what quantities are held fixed under the blocking and which are rescaled . we keep the current across each bond @xmath19 and the bulk density @xmath44 fixed , and rescale the parameters @xmath9 and @xmath10 . to achieve this , we consider pairs of adjacent blocks separately , and replace the effect of particle exchange with the rest of the system with particle injection and ejection at the boundaries of the pair of blocks , with rates @xmath9 and @xmath10 , as illustrated in figure [ asep blocking ] . . ] thus the current and density in the rescaled system are written @xmath45 under the majority rule matching between configurations , ( [ maj d ] ) , ( [ maj e ] ) , the block current @xmath19 and the block density @xmath44 assume the form @xmath46 note that we impose that neither the vectors and @xmath31 and @xmath32 nor the matrix algebra ( [ bulk algebra ] ) to ( [ rh boundary condition ] ) are changed under the rescaling . this reflects the fact that both the scaled and the unscaled configurations are taken to evolve under the same stochastic dynamics : any extension of the parameter space is prohibited under the rescaling . finally , the rescaling of the parameters @xmath9 and @xmath10 is achieved by setting @xmath47 and @xmath48 . up to this point , the use of the matrix product formulation has been nothing more than a choice of notation . the task now is to evaluate the correlation functions ( [ j ] ) to ( [ rho ] ) and this is relatively straightforward if one makes use of the algebra ( [ bulk algebra ] ) to ( [ rh boundary condition ] ) , though we remark that even here the algebra only simplifies the calculations : one could evaluate the relevant correlations directly from the master equation , for example . for the case @xmath50 one obtains ( still for @xmath41 ) @xmath51 which is also the result obtained in the exact solution for @xmath52 , and @xmath53 from matching @xmath47 , where @xmath54 . stable fixed points of this equation , @xmath55 , are found at @xmath56 and @xmath57 , and these are separated by an unstable fixed point at @xmath58 . at the unstable fixed point , @xmath59 . thus the @xmath56 fixed point represents the low - current phase , and the transition between this phase and the maximum current phase is given by the unstable fixed point , for which we recover the exact value @xmath58 . moreover the current and density at this point are also given by their exactly known values throughout the maximum current phase : @xmath59 and @xmath60 . for @xmath61 the scaling equations are straightforward to obtain , but rather lengthy so we do not include them here . their fixed point structure is given in table [ fixed points ] . .fixed points of the steady state blocking transformation for @xmath41 . the values given as decimals are evaluated numerically all the rest are determined analytically . the fixed points depicted in the flow diagram shown in figure [ flow diagram ] are labelled a to g. [ cols="^,^,^,^,^,^,^,^,^,^,^,^",options="header " , ] we also show the exact values which have only recently been obtained @xcite . our results appear to capture the crossover from @xmath23 dynamics in the maximum current phase ( represented by point a ) to @xmath24 dynamics along the coexistence line ( represented by point d ) . one can also obtain scaling equations for the case @xmath62 in order to probe the dynamics at points b and c in the flow diagram , points which represent the high and low density phases , respectively . one finds that at both b and c , @xmath63 for all @xmath64 and @xmath65 . this implies there is no system size dependence in the leading contribution to the gap for large @xmath0 , which is consistent with the exact prediction of finite relaxation times in the low current phases away from the coexistence line @xcite . we have developed a renormalisation group procedure which we have applied to the boundary driven asymmetric simple exclusion process . in the steady state transformation , exact results for the critical point are recovered as well as a flow diagram that shares the qualitative structure of the exact phase diagram . the dynamic scaling also captures the crossovers in the exactly computed relaxation dynamics in each phase of the asep . these scaling methods seem to be widely applicable : they can be generalised to higher dimensions , multiple particle species and multiple site occupancy . since only small clusters give , through the scaling , even the highly correlated properties , the scaling does not depend on the availability of exact results , and may be particularly useful in cases where the exact steady state can be written down but any calculation of macroscopic properties remains difficult . they are also direct and straightforward to interpret , without requiring any translation from quantum spins or abstract fields as in field - theoretic methods . t. h. would like to thank the supa and the epsrc for support under programme grant gr / s10377/01 .

we introduce a real - space renormalisation group procedure for driven diffusive systems which predicts both steady state and dynamic properties . we apply the method to the boundary driven asymmetric simple exclusion process and recover exact results for the steady state phase diagram , as well as the crossovers in the relaxation dynamics for each phase .

introduction the asymmetric simple exclusion process (asep) matrix product formulation of the asep steady state renormalisation of the asep conclusion

arxiv

the standard model of particle physics is believed to be an effective theory valid up to energies close to 1 tev . however , no new physics beyond the standard model ( sm ) has been observed yet , and it is critical to explore a wide range of possible signatures . a promising avenue lies in final states that involve the heaviest of the particles presumed to be elementary , the top quark . this document summarizes the status of these searches using the atlas @xcite and cms @xcite detectors at the cern large hadron collider ( lhc ) . the results are based on proton - proton collision data corresponding to integrated luminosities between 1 and 5 collected at a center - of - mass energy of 7 tev in 2011 . the fact that the top quark is the heaviest elementary particle might be a hint that it plays a special role in the theory of electroweak symmetry breaking . the so - called hierarchy problem refers to the fact that the large quantum contributions to the square of the higgs - boson mass should make the higgs mass many orders of magnitude larger than the electroweak scale . either there is an incredible , unnatural fine - tuning cancellation or nature chose another mechanism to protect the higgs mass to keep it at the observed low value @xcite . many models introduce top partners to cancel these quantum corrections . finally , there are tantalizing hints of new physics in the forward - backward asymmetry in events at the tevatron @xcite . benchmark signal models are used to define signature - based searches in final states involving one or two leptons ( electrons and muons ) , jets , of which typically one or more are required to be identified as @xmath0-jets , and missing transverse momentum ( ) . the benchmarks include models of fourth - generation and vector - like quarks @xcite , top partners in little higgs models @xcite , as well as non - sm production of four top quarks @xcite , same - sign top - quark pair ( @xmath1 ) production @xcite , top+jet resonances in + jet events @xcite , @xmath2 resonances @xcite , and flavor changing neutral currents ( fcnc ) in single top - quark production @xcite . searches for resonances , third generation supersymmetry and new physics in top - quark decays and properties are covered elsewhere in these proceedings @xcite . the experimental approaches are similar between atlas and cms . searches are based on the l+jets or dilepton channels . jets are reconstructed from three - dimensional calorimeter energy clusters using the anti-@xmath3 jet clustering algorithm @xcite with a radius parameter of @xmath4 for atlas and @xmath5 for cms . leptons are required to pass quality criteria and to be isolated @xcite . the transverse momenta of jets and leptons are typically required to be larger than 20 or 25 gev or more . multivariate tagging algorithms @xcite are used to identify @xmath0-jets . typically , the statistical tool bumphunter @xcite is used to check for deviations , an excess or deficit , from the background hypothesis . in the absence of a signal , cross - section and mass limits are derived for benchmark models using cls @xcite or occasionally bayesian methods @xcite . all limits quoted in this document are obtained at the 95% confidence level . unless mentioned otherwise the new particles are assumed to decay with 100% branching fraction to the corresponding final state under study . the largest background typically originates from sm production and is estimated from monte carlo ( mc ) simulation using mc@nlo @xcite or powheg @xcite at atlas , and madgraph @xcite at cms . data - driven multi - jet background estimates are based on the matrix method and on binned likelihood fits to the distribution . data - driven @xmath6+jets estimates use the inherent @xmath6 charge asymmetry in @xmath7 collisions . alpgen @xcite ( atlas ) and madgraph ( cms ) are used to model @xmath6+jets in mc simulation . the composition of the flavor of the quarks produced in association with the @xmath6 boson is measured in the low jet multiplicity bins and extrapolated using mc simulations . small backgrounds , like the production of single top quarks , boson pairs , + @xmath6 , + @xmath8 , and @xmath8+jets are typically estimated from mc simulations . the leading systematic uncertainties typically originate from the uncertainty in the jet energy scale , the @xmath0-tagging efficiency , and the mc modeling . in certain cases the impact of systematic uncertainties on the sensitivity of the search is reduced by using gaussian constraints or other marginalization techniques , see e.g. ref . . an atlas analysis @xcite searches for the pair production of heavy non - sm quarks @xmath9 with decays according to @xmath10 with @xmath11 , @xmath12 , @xmath0 for up - type @xmath9 or @xmath13 , @xmath14 for down - type @xmath9 . the search is performed with 1.04 of integrated luminosity . dilepton final states are selected , requiring large and at least two jets . no @xmath0-tagging is applied . mass reconstruction of heavy quark candidates is performed by assuming that the @xmath6-boson decay products are nearly collinear . the resulting mass reconstruction is shown in fig . [ atlas_qqdilep ] . no deviation from the sm expectation is observed . heavy non - sm quark masses below 350 gev are excluded . comparison between the data and the simulated background for @xmath15 @xcite . the expected distribution for a signal with @xmath16 gev is also shown . the signal region is defined by @xmath17 gev . ] comparison between the data and the simulated background for @xmath15 @xcite . the expected distribution for a signal with @xmath16 gev is also shown . the signal region is defined by @xmath17 gev . ] a similar cms search @xcite for pair production of heavy top - like quarks @xmath18 has been performed in the decay mode @xmath19 . the search uses 5.0 of integrated luminosity . again dilepton final states are selected , requiring large and at least two jets , but this time exactly two of the jets have to be identified as @xmath0-jets . the minimum value of the four possible masses of the system defined by one of the two leptons and one of the two @xmath0-jets ( @xmath15 ) is found to be a good variable for distinguishing @xmath20 from @xmath21 events , as can be seen in fig . [ cms_mlbmin ] . the observed number of events agrees with the expectation from sm processes . heavy @xmath18 quarks with a mass less than 557 gev are excluded . searches for heavy top - like quarks @xmath18 given the hypothesized decay mode @xmath22 are also conducted in final states with a single charged lepton , and , depending on the analysis , at least three or four jets , of which at least one must be identified as a @xmath0-jet @xcite . cms @xcite uses the reconstructed @xmath18 mass @xmath23 , obtained from a kinematic fit of the reconstructed four - momenta to the decay hypothesis @xmath24 , as well as , defined as the scalar sum of the transverse momenta of the objects associated to the @xmath18 and @xmath25 decay products . the two - dimensional distributions of versus @xmath23 are fitted with analytic functions for the signal s and the background b. all two - dimensional bins are then sorted in increasing order of the expected s / b ratio , using the functions . the resulting distribution of this so - called s / b rank is shown in fig . [ cms_sobrank ] and is used to exclude @xmath18 masses below 570 gev with 5.0 of integrated luminosity . observed ( red filled area ) and expected ( red dashed line ) exclusion limits in the plane of br(@xmath26 ) vs br(@xmath27 ) for different values of the vector - like @xmath18 quark mass @xcite . the grey solid area corresponds to the unphysical region where the sum of the branching ratios exceeds unity . ] observed ( red filled area ) and expected ( red dashed line ) exclusion limits in the plane of br(@xmath26 ) vs br(@xmath27 ) for different values of the vector - like @xmath18 quark mass @xcite . the grey solid area corresponds to the unphysical region where the sum of the branching ratios exceeds unity . ] atlas @xcite uses the reconstructed mass @xmath23 of the candidate @xmath18 as the discriminant . a tight selection is applied targeting @xmath18 masses above 400 gev . in this mass range the decay products have large momenta . the reconstruction of hadronically - decaying @xmath6 bosons @xmath28 takes advantage of this . they are either defined as a single jet with @xmath29 gev and jet mass in the range of 60 - 110 gev or as a dijet system with @xmath30 gev , angular separation @xmath31 , and mass within the range of 60 - 110 gev . additional kinematic selection criteria include ( defined as the scalar sum of the transverse momenta of the lepton , , and the jets from the hypothesized @xmath18 decays ) to be larger than 750 gev , @xmath32 , @xmath33-jet@xmath34 , and @xmath35-jet@xmath34 . with 4.7 of integrated luminosity a @xmath18 quark with mass lower than 656 gev is excluded . in addition , in light of the recent discovery of a new boson of mass 126 gev at the lhc , upper limits are derived , as shown in fig . [ atlas_vlqplane ] , for vector - like quarks of various masses in the two - dimensional plane of br(@xmath26 ) versus br(@xmath27 ) , where @xmath36 is the sm higgs boson ( br(@xmath37 ) @xmath38 br(@xmath26 ) @xmath39 br(@xmath27 ) . ) a search for pair - produced , heavy , vector - like charge-@xmath40 quarks is performed by cms @xcite , assuming br(@xmath37)@xmath41 . events are selected by requiring two charged leptons from the @xmath8-boson decay , as well as an additional isolated charged lepton . using 1.14 of integrated luminosity @xmath18 quarks with a mass less than 475 gev are excluded . both atlas and cms conduct searches for heavy pair - produced bottom - like quarks . these @xmath42 quarks are assumed to decay exclusively to @xmath43 . the @xmath44 final state has the distinctive signature of three or more leptons or two leptons of same charge which is exploited . cms uses 4.9 of integrated luminosity to select trilepton and same - sign - dilepton events with @xcite . at least one jet must be identified as a @xmath0-jet . events are rejected where any two leptons of the same flavor have an invariant mass consistent with the @xmath8-boson mass . furthermore , , defined as the scalar sum of the transverse momenta of the leptons , , and the jets , is used to reject background , as shown in fig . [ cms_trilep ] . @xmath42 quarks with mass below 611 gev are excluded . expected and observed lower limits on the @xmath45 signal as a function of the @xmath45 mass and the coupling constant @xmath46 @xcite . the shaded area is excluded . ] expected and observed lower limits on the @xmath45 signal as a function of the @xmath45 mass and the coupling constant @xmath46 @xcite . the shaded area is excluded . ] a similar analysis by atlas focuses on same - sign - dilepton events using 4.7 of integrated luminosity @xcite . the most important background is the contribution arising from fake leptons . another significant background is from various sm processes where two real leptons are produced , but in which one of the leptons has a mis - identified charge . additional signal hypotheses are considered for the interpretation of the results . both single and pair production of new heavy quarks @xmath45 , with charge @xmath47 , are considered with @xmath48 . for the single production the assumed coupling constant @xmath46 of the @xmath49 vertex is varied as shown in fig . [ atlas_t5/3 ] . assuming only @xmath50 or @xmath51 production quark masses below 670 gev are excluded . in addition , first limits are set on non - sm production of four top quarks , yielding @xmath52 fb . in an earlier version of the atlas same - sign - dilepton search @xcite using 1.04 of integrated luminosity the results are also interpreted for same - sign top - quark pair production . the results leave little room to explain the measurement of the forward - backward asymmetry in top - quark pair production at the tevatron by a flavor - changing @xmath53 boson . cms excludes @xmath45 masses below 645 gev by analyzing very similar same - sign - dilepton final states , assuming @xmath51 production and using 5.0 of integrated luminosity @xcite . also the 1-lepton channel is used by atlas to set limits on @xmath50 production with @xmath54 @xcite . similar to ref . @xcite hadronically decaying @xmath6 bosons are reconstructed . the limits are not competitive since only 1.04 of integrated luminosity was used for these results . a combined search in the 1-lepton , same - sign - dilepton and trilepton final states is presented by cms by assuming both single and pair production of both @xmath18 and @xmath42 @xcite . by analyzing and the invariant @xmath55 mass in bins of number of reconstructed hadronically decaying @xmath6 bosons and @xmath0-jets , limits are presented as a function of the mass difference between @xmath18 and @xmath42 and as a function of the matrix element @xmath56 . both atlas @xcite and cms @xcite set limits on @xmath50 production with at least one @xmath42 decaying to a @xmath8 boson and a bottom quark . using 2.0 of integrated luminosity atlas excludes vector - like singlet @xmath42 quarks mixing solely with the third sm generation with masses below 358 gev . assuming br(@xmath57)@xmath41 cms excludes @xmath42 masses below 550 gev using 4.9 . searches for signatures of pair production of supersymmetric top partners also have sensitivity to spin-@xmath58 top partners in little higgs models as shown by atlas analyses presented in refs . @xcite . searches for new heavy resonances , a color singlet @xmath59 or a color triplet @xmath60 , produced in association with a top quark are motivated by top - flavor violating processes designed to explain the @xmath21 forward - backward asymmetry observed at the tevatron . two - dimensional limits are set on the mass and the coupling of @xmath59 and @xmath60 by atlas @xcite by analyzing the @xmath61+jet and the @xmath62+jet invariant mass , respectively , in @xmath21+jet candidate events . similar limits are set by cms @xcite . the limits leave little room for top - flavor violating processes to explain the @xmath21 forward - backward asymmetry observed at the tevatron . both atlas @xcite and cms @xcite search for resonances in the @xmath63 ( and c.c . ) spectrum and set lower limits on the mass of right handed @xmath2 of 1.85 tev . atlas uses the invariant @xmath63 mass as the discriminant , while cms makes use of boosted decision trees . atlas uses a neural network analysis to search for fcnc single top - quark production @xcite . two - dimensional limits are set in the plane of br(@xmath64 ) and br(@xmath65 ) . top quarks play an important role in atlas and cms searches for physics beyond the sm . no hints of new phenomena could be established yet . results from lhc @xmath7 collisions at a center - of - mass energy of 8 tev and eventually close to 14 tev are anticipated with great suspense . 9 atlas collaboration , phys . b * 716 * , 1 ( 2012 ) [ arxiv:1207.7214 [ hep - ex ] ] . cms collaboration , phys . b * 716 * , 30 ( 2012 ) [ arxiv:1207.7235 [ hep - ex ] ] . t. aaltonen _ et al . _ [ cdf collaboration ] , phys . d * 83 * , 112003 ( 2011 ) [ arxiv:1101.0034 [ hep - ex ] ] . v. m. abazov _ et al . _ [ d0 collaboration ] , phys . d * 84 * , 112005 ( 2011 ) [ arxiv:1107.4995 [ hep - ex ] ] . atlas collaboration , phys . d * 86 * , 012007 ( 2012 ) [ arxiv:1202.3389 [ hep - ex ] ] . cms collaboration , phys . b * 716 * , 103 ( 2012 ) [ arxiv:1203.5410 [ hep - ex ] ] . cms collaboration , phys . b * 718 * , 307 ( 2012 ) [ arxiv:1209.0471 [ hep - ex ] ] . atlas collaboration , phys . lett . * 108 * , 261802 ( 2012 ) [ arxiv:1202.3076 [ hep - ex ] ] . atlas collaboration , arxiv:1210.5468 [ hep - ex ] ( 2012 ) . cms collaboration , phys . lett . * 107 * , 271802 ( 2011 ) [ arxiv:1109.4985 [ hep - ex ] ] . cms collaboration , jhep * 1205 * , 123 ( 2012 ) [ arxiv:1204.1088 [ hep - ex ] ] . atlas collaboration , atlas - conf-2012 - 130 , https://cds.cern.ch/record/1478217 ( 2012 ) . atlas collaboration , phys . * 109 * , 032001 ( 2012 ) [ arxiv:1202.6540 [ hep - ex ] ] . cms collaboration , arxiv:1209.1062 [ hep - ex ] ( 2012 ) . atlas collaboration , phys . lett . * 109 * , 071801 ( 2012 ) [ arxiv:1204.1265 [ hep - ex ] ] . cms collaboration , cms - pas - exo-11 - 066 , https://cds.cern.ch/record/1460386 ( 2011 ) . atlas collaboration , phys . * 108 * , 041805 ( 2012 ) [ arxiv:1109.4725 [ hep - ex ] ] . atlas collaboration , jhep * 1211 * , 094 ( 2012 ) [ arxiv:1209.4186 [ hep - ex ] ] . atlas collaboration , arxiv:1209.6593 [ hep - ex ] ( 2012 ) . cms collaboration , phys . b * 717 * , 351 ( 2012 ) [ arxiv:1206.3921 [ hep - ex ] ] . atlas collaboration , phys . * 109 * , 081801 ( 2012 ) [ arxiv:1205.1016 [ hep - ex ] ] . cms collaboration , arxiv:1208.0956 [ hep - ex ] ( 2012 ) . atlas collaboration , phys . b * 712 * , 351 ( 2012 ) [ arxiv:1203.0529 [ hep - ex ] ] . see contributions in these proceedings by j .- f . arguin , s. fleischmann and d. pagano . m. cacciari , g. p. salam and g. soyez , jhep * 0804 * , 063 ( 2008 ) [ arxiv:0802.1189 [ hep - ph ] ] . atlas collaboration , eur . j. c * 72 * , 1909 ( 2012 ) [ arxiv:1110.3174 [ hep - ex ] ] . atlas collaboration , atlas - conf-2011 - 063 , https://cds.cern.ch/record/1345743 ( 2011 ) . i. bertram _ et al . _ [ d0 collaboration ] fermilab - tm-2104 ( 2000 ) . s. frixione and b. r. webber , jhep * 0206 * , 029 ( 2002 ) [ hep - ph/0204244 ] . s. frixione , p. nason and b. r. webber , jhep * 0308 * , 007 ( 2003 ) [ hep - ph/0305252 ] . s. frixione , p. nason and c. oleari , jhep * 0711 * , 070 ( 2007 ) [ arxiv:0709.2092 [ hep - ph ] ] . f. maltoni and t. stelzer , jhep * 0302 * , 027 ( 2003 ) [ hep - ph/0208156 ] . j. alwall , p. demin , s. de visscher , r. frederix , m. herquet , f. maltoni , t. plehn and d. l. rainwater _ et al . _ , jhep * 0709 * , 028 ( 2007 ) [ arxiv:0706.2334 [ hep - ph ] ] . m. l. mangano , m. moretti , f. piccinini , r. pittau and a. d. polosa , jhep * 0307 * , 001 ( 2003 ) [ hep - ph/0206293 ] .

searches are presented for physics beyond the standard model involving top - quark and related signatures . the results are based on proton - proton collision data corresponding to integrated luminosities between 1 and 5 collected at a center - of - mass energy of 7 tev with the atlas and cms detectors at the large hadron collider in 2011 . the data are found to be consistent with the standard model . the non - observation of a signal is converted to limits at the 95% confidence level on the production cross section times branching ratio and on the masses of the hypothesized new particles for appropriate benchmark models .

introduction experimental techniques results conclusions references

arxiv

in hierarchical cosmologies , structure forms on broad mass scales from the growth of density perturbations in the early universe . the rate at which structure grows is governed by the cosmological parameters , particularly the density parameter @xmath7 ( white & rees 1978 ; kaiser 1986 ) . dark matter halos on cluster mass scales grow slowly at redshifts below @xmath8 in low density , hierarchical model universes and flat , lambda - dominated universes , while rapid and continuous growth is expected between @xmath9 and the present in a flat , @xmath10 universe with zero cosmological constant ( richstone , loeb , & turner 1992 , luppino & gioia 1995 , eke , cole , & frenk 1996 , bahcall , fan , & cen 1997 , mathiesen & evrard 1998 , viana & liddle 1998 ) . to the extent that x - ray emission traces deep cluster potential wells , the abundance of x - ray clusters at redshifts @xmath11 should probe @xmath7 directly and with high sensitivity . accordingly , several x - ray - selected samples of distant clusters have been drawn from the @xmath3 archive to investigate this issue ( e.g. collins et al . 1997 , jones et al . 1998 , rosati et al . 1998 , vikhlinin et al . 1998a , henry et al . 2001 , gioia et al . these surveys have yielded no evidence for evolution of cluster abundances for clusters with x - ray luminosities below @xmath12 and redshifts below @xmath13 . however , there is a growing body of evidence for a mild but significant decrease in the abundance of the most x - ray - luminous distant clusters ( henry et al . 1992 , vikhlinin et al . 1998b ) . the degree to which clusters have matured into galaxy - rich , virialized structures as a function of look - back time ( redshift ) likewise depends on the cosmological parameters , and is therefore an important aspect of cluster evolution ( kaiser 1986 ; frenk et al . 1996 ) . however , it is difficult to select samples of clusters to study the systematic properties of cluster galaxy populations the degree to which clusters have evolved into rich concentrations of galaxies , for example while avoiding fatal selection biases . because the optical detectability of distant clusters is itself coupled to their degree of galaxy concentration , optically - selected samples are not ideally suited to such studies . on the other hand , x - ray - selected clusters are largely unbiased with regard to the galaxy population , and are therefore suited to studies of cluster galaxy evolution . furthermore , the existence of reasonably well - defined relationships between x - ray luminosity , temperature , and mass ( evrard , metzler , & navarro 1996 , mushotzky & scharf 1997 , markevitch 1998 ) make it possible to use x - ray flux - limited samples of clusters in the nearby and distant universe to determine the degree to which the galaxy populations have settled into their cluster potential wells as function of time and cluster mass . one measure of this process is cluster richness ( yee and lopez - cruz 1999 ) . in this paper we examine the richness distribution ( i.e. the net number of galaxies encircled by the abell radius within two magnitudes of the third brightest galaxy ) for 14 of the 28 most distant clusters found in our @xmath3 160 square degree cluster survey ( vikhlinin et al . we compare our cluster richnesses to those of nearby clusters with similar x - ray selection criteria , and to those for distant , optically - selected clusters ( e.g. bower et al . 1994 , holden et al . 1997 , couch et al 1991 , postman et al . 1996 ) , and we discuss our results in the context of cluster evolution . the clusters were discovered using the x - ray selection function described in vikhlinin et al . ( 1998a ) . of the two to three dozen cluster candidates beyond @xmath14 discovered in our survey , the 14 for which we have the best optical data are listed in table 1 . the spectroscopic redshifts measured for the brightest one to three galaxies nearest the x - ray centroid are taken from mullis et al . ( 2001 , in preparation ) . we obtained r - band harris filter images of candidate clusters with the 1.2 m telescope of the f.l . whipple observatory , the 1 m las campanis telescope , the danish 1.5 m telescope , and the 3.6 m telescope at la silla during many observing runs from june 1995 to february 1999 . the optical images were originally obtained to determine whether the extended x - ray sources correspond to optical galaxy concentrations , and to measure photometric redshifts for the clusters . several ccd cameras were used over the course of the survey . exposure times ranged from 10 min to over an hour , and seeing was typically 1.3 arcsec . the fields of view were 10.6 arcmin , 13.3 arcmin , 23 arcmin , and 5.5 arcmin for the 1.2 m , 1.5 m , 1 m , and 3.6 m telescopes respectively . for reference , 2.3 arcmin corresponds to a linear size of 1 mpc at @xmath15 for @xmath16 and @xmath17 , which is assumed throughout this paper . deep images of cl 0529 - 5848 and cl 1311 - 0551 were taken with the efosc camera on the 3.6 m telescope at la silla . these images provided a deep measurement of the galaxy luminosity function within a 1 mpc radius for the two clusters . we used the average of these luminosity functions to calculate magnitude depth incompleteness corrections when estimating richnesses for the remaining clusters . all ccd image frames were de - biased , overclock - corrected , flat - fielded , and combined using iraf routines to form the science images for each cluster . the galaxies were selected for the analyses described below as follows . using a wavelet detection routine , we determined the locations of all sources on the science ccd frames . we measured the full width at half maximum ( fwhm ) in right ascension and declination of each object , and rejected all point - like objects and bright foreground stars and galaxies . we measured instrumental magnitudes for each galaxy using 3 arcsec and 6 arcsec diameter apertures , which correspond to 22 and 44 kpc linear diameters at @xmath15 . we then ranked the galaxies by magnitude relative to the brightest galaxy nearest the x - ray centroid . the brightest cluster galaxy and one or two companions were used to determine the spectroscopic redshift of the cluster ( mullis et al . 2001 , in preparation ) . galaxies within the abell radius brighter than @xmath18 , where @xmath19 is the third brightest galaxy magnitude , were selected from the catalog . we counted galaxies in the magnitude - selected subsample of cluster and field galaxies within centered apertures of @xmath20 and @xmath21 mpc radius . we determined the background by averaging the counts in 2030 apertures placed at random locations away from the cluster . the net number of cluster galaxies was estimated by subtracting the mean background counts , adjusted appropriately for area , from the counts in the cluster aperture . we restricted our search for the third brightest cluster galaxy to a radius of 0.5 mpc to minimize foreground galaxy contamination that can introduce errors into the richness measurements ( postman et al . 1996 ) . using the integrated luminosity function discussed in 6.1 , we find that a half magnitude error in the counting depth can cause a @xmath22 error in the richness estimate . errors of this magnitude can be introduced when a bright foreground galaxy is misidentified as the third brightest cluster galaxy ( e.g. postman et al . 1996 ) . this would cause @xmath19 to be underestimated , which in turn would cause us to mine to erroneously shallow depths . this chain of errors would then result in erroneously small cluster richness estimates . applying the abell radius criterion to distant cluster fields would produce this error most of the time ( e.g. postman et al . therefore , we mitigated this effect by selecting the third brightest galaxy within a half mpc radius . we determined whether we were successful at mitigating the third brightest galaxy problem by comparing our distribution of @xmath23 to the distribution in the aco catalog . we found good agreement between the two samples . the mode for our 14 cluster sample is @xmath24 magnitudes , which compares well to the aco value of @xmath25 magnitudes . we therefore do not appear to be significantly biased in this respect . nevertheless , using another method , we were able to circumvent this process entirely by comparing galaxy counts in nearby and distant clusters within restricted apertures and depths in 4 and 5 . the analysis was done using our own routines written in idl and fortran . we compared our galaxy selection algorithm to the `` sextractor '' routine and found similar results . we show an optical image , with x - ray contours superposed , of a typical richness class 0 cluster discovered in our survey in figure 1 . the brightest cluster galaxy nearest the x - ray centroid , indicated by an arrow , has a spectroscopic redshift of @xmath26 . because we were required to optically vet over 200 cluster candidates , we were initially unable to obtain deep images for all clusters . we therefore obtained optical images to a sufficient depth to determine whether the x - ray cluster candidates are indeed associated with galaxy concentrations , and to measure brightest cluster galaxy magnitudes for photometric redshift estimation ( vikhlinin et al . 1998 a , b ) . in most instances , our data do not completely sample cluster galaxies two magnitudes fainter than the third brightest galaxy . therefore , we were required to correct our galaxy counts to a consistent depth , as discussed in 6.1 . however , first we will demonstrate that galaxy concentrations are indeed present at the locations of the extended x - ray sources using the contrast parameter introduced by couch et al . ( 1991 ) . the contrast parameter @xmath27 ( couch et al . 1991 ) , is a measure of the observed over - density of bright cluster galaxies against the background . here , @xmath28 is the net number of galaxies within an aperture centered on the cluster , after subtracting the average background surface density measured in 2030 randomly placed apertures surrounding the cluster . the parameter @xmath29 is the standard deviation about the mean background value for the surrounding apertures . the contrast parameter provides a crude measure of cluster richness that is relatively insensitive to the imaging depth , but depends strongly on redshift , central galaxy concentration , and detection aperture size . we demonstrate the effects of increasing aperture size and increasing imaging depth on @xmath30 in figures 2 & 3 . in figure 2 we examine the effect of aperture size on the contrast parameter by plotting histograms of @xmath30 against aperture size for our sample . @xmath30 reaches a maximum with the 0.250.5 mpc apertures and declines with increasing aperture size . within these apertures , the modal contrast is @xmath31 . the contrast for an isothermal distribution of galaxies should decline with aperture radius roughly as @xmath32 ( shown as the broken line in figure 2 ) , while @xmath33 for a @xmath34 profile . the contrast parameter appears to decline with increasing aperture size in a manner that is roughly consistent with an isothermal profile . in figure 3 we plot the contrast parameter within a 1 mpc radius as a function of imaging depth . the imaging depth , @xmath35 , is the limiting magnitude depth relative to the brightest cluster galaxy s magnitude . the data are for two richness class 12 clusters in our sample with deep imaging obtained with the eso 3.6 m telescope . no strong dependence is found between contrast significance and imaging depth for either cluster . figures 2 & 3 demonstrate that the optical contrast of clusters against the background is relatively constant with imaging depth , but declines rapidly with aperture size . both properties are due to the rapidly increasing background galaxy counts . we find that a 1 mpc aperture strikes a good compromise between the competing demands of good cluster counting statistics and minimizing the background noise . in general , the background counts are factors of 23 larger than the cluster counts within a 1 mpc aperture . figures 2 & 3 also show that relatively shallow imaging is effective at detecting galaxy concentrations at redshift @xmath9 . however , as is seen in table 1 and discussed below , the contrast parameter is not a reliable measure of cluster richness . for instance , couch et al . found that , within a 1.5 mpc diameter aperture using a passband similar to ours , @xmath36 corresponds approximately to a richness class 1 cluster , and @xmath37 corresponds to a richness class 2 to 3 cluster at redshifts @xmath38 . we do not find such a large increase in contrast with richness for our clusters ( cf . table 1 ) . on average , our clusters show a lower contrast than couch s , and fall in the regime where their catalog becomes seriously incomplete . in fact , a cluster such as cl 2146 + 0422 , shown in figure 1 , would probably have been omitted from the couch et al . ( 1991 ) catalog . here we compare the galaxy counts from our sample to those in nearby clusters by comparing the net number of galaxies within a 1 mpc radius as a function of imaging depth . this approach has two significant advantages over direct richness estimates . first , galaxy counts within a 1 mpc radius are more accurate than counts within a 3 mpc radius , because the background corrections are smaller . second , we avoid entirely the uncertain depth and aperture corrections required to estimate abell richnesses . the primary disadvantage is that galaxy counts as a function of cluster radius and magnitude depth are generally unavailable for clusters . we remedied this situation by measuring galaxy counts toward 14 nearby abell clusters from digital sky survey ( dss ) images . the clusters , with redshifts between @xmath39 , were selected from the rosat brightest cluster sample ( bcs , ebeling et al . their x - ray luminosities , @xmath40 , roughly match the range for our distant sample , after correcting for the bandpass offset . the galaxies on the dss images were selected and counted automatically , after correcting for photographic nonlinearity , in a nearly identical manner to our distant cluster method . for example , the galaxy selection algorithm and background field corrections were applied in a similar fashion . the expression used to compute the galaxy and background fluxes that relates intensity , @xmath41 , to photographic density , @xmath42 , is : @xmath43 . the photographic nonlinearity was removed through the coefficients @xmath44 , @xmath45 , and @xmath46 , which were derived by comparing the digitized photographic magnitudes of stars and galaxies from the dss to calibrated ccd images of the same fields . we avoided the potential errors associated with selecting the third brightest galaxy ( cf . 3 ) by measuring depth with respect to the magnitude of the bcg . the largest uncertainty associated with this approach would be differential luminosity evolution of the bcg with respect to the remaining cluster galaxies . however , we do nt believe this to be a serious problem . the bcgs in our sample are very good standard candles to @xmath38 ( vikhlinin et al . the variation about their mean @xmath47-band luminosity is @xmath48 magnitudes . while a variation at this level would affect individual clusters , it would not significantly affect the comparison , as both samples were treated in a similar fashion . nevertheless , this approach provides a good check on our application of the abell richness criteria . the details of the automated dss galaxy counts for the nearby clusters will be presented in a future paper ( whitman , mcnamara , & vikhlinin , in prep ) . it is worth noting in advance , however , that our automated counts agree with the aco catalog to within the counting statistics for all clusters within this luminosity and richness range . this check provides a measure of reassurance that our comparison in figure 4 for the 1 mpc apertures is reliable . the galaxy count comparison between the bcs and 160 square degree survey clusters as a function of imaging depth is shown in figure 4 . the average number of galaxies as a function of depth within a 1 mpc radius for the 14 nearby bcs abell clusters is shown as connected , filled rectangles . the one @xmath49 deviation about the mean is enclosed by the shaded region . the galaxy counts from the deep imaging of cl 1311 - 0551 and cl 0529 - 5848 are shown as connected points . the remaining 160 square degree clusters are shown as solid points . the error bars are one sigma poisson estimates using the net cluster and background counts in each aperture , calculated as @xmath50 . figure 4 shows that nearly all of the 160 square degree clusters lie within @xmath51 of the bcs clusters . although a few outliers are good candidate galaxy - poor clusters , the overall agreement is quite good . we can conclude that the 160 degree survey clusters are similarly rich to the nearby bcs clusters with the same range of x - ray luminosity . in the next section , we estimate the richnesses of the distant clusters according to the abell criteria by applying depth and aperture corrections to the galaxy counts . our images often do not reach a sufficient depth to sample completely to @xmath52 . therefore , we measure an incompleteness parameter for each cluster by taking the difference between the location of the peak of the instrumental magnitude histogram for all galaxies in each field and @xmath52 . the incompleteness parameter , @xmath53 , is listed in table 1 . we corrected the galaxy counts assuming a cumulative luminosity function of the form @xmath54 , where @xmath55 is the slope of the cumulative luminosity function ( sarazin 1986 ) . the function was normalized by the detected number of cluster galaxies , @xmath56 , as @xmath57 the cluster galaxy luminosity function is poorly known for distant clusters . we therefore estimated @xmath55 for the sample using deep @xmath47-band images of cl 1311 - 0551 and cl 0529 - 5848 , which we obtained with the european southern observatory s 3.6 m telescope . the cumulative distributions of galaxies , after background subtraction , are shown for each cluster in figure 5 . the curves shown are normalized to the magnitude of the brightest cluster galaxy in each cluster . our data sample completely three magnitudes below the brightest galaxy for cl1311 - 0551 and four magnitudes below the brightest galaxy for cl1311 - 0551 . as the modal difference between the brightest and third brightest galaxies for our 14 cluster sample is @xmath24 magnitudes ( the aco value is @xmath25 magnitudes ) , these two clusters , if representative of our sample as a whole , reach a depth well below @xmath18 . the slope of the linear fit to the data is @xmath58 . we assume from now on that the luminosity functions of the remaining clusters have this slope , and we use @xmath55 to correct their galaxy counts for imaging depth incompleteness as described above . after correcting our counts for imaging depth incompleteness , we computed cluster richnesses , @xmath59 , by extrapolating counts in each aperture to the 3 mpc abell radius . the extrapolations assume both @xmath60 and @xmath61 surface density profiles , normalized by the corrected counts within each aperture . extrapolation , rather than direct measurement , is required because the galaxy background increases more rapidly with increasing aperture than the cluster counts . when the size of the poisson variations in the background counts approaches the cluster signal , the cluster counts become unreliable . although the backgrounds in the 0.25 and 0.5 mpc apertures are only @xmath62 of the cluster counts , only a few cluster galaxies are generally present there . therefore , the poisson errors in the cluster counts are relatively large . on the other hand , as the aperture grows to 1 mpc , the background becomes factors of @xmath63 larger than the cluster counts . nonetheless , the one to three dozen net cluster galaxies present can be counted with greater precision . whether or not a universal galaxy surface density profile for distant clusters exists is unknown . the density profile of galaxies for a sample of 14 intermediate redshift ( @xmath13 ) , x - ray - selected clusters was found by carlberg et al . ( 1997 ) to follow a navarro , frenk , & white ( 1997 ) profile . this profile implies a steep @xmath64 decline in galaxy surface density at very large radii , but a somewhat shallower decline over much of the 3 mpc region of interest . on the other hand , the @xmath65 clusters selected optically in the palomar distant cluster survey appear to have considerably shallower , @xmath61 profiles within the radial range considered in this sample . we therefore analyzed our sample by applying both @xmath61 and isothermal , @xmath60 surface density profiles to our aperture corrections . in the upper panel of figure 6 , we present richnesses for 13 clusters with @xmath66 . in the lower panel , we plot richnesses for a subset of 9 clusters with smaller depth corrections , @xmath67 . the solid distribution of clusters has been corrected assuming the @xmath60 profile , and the dashed distribution has been corrected for the @xmath61 profile . the shallower , @xmath61 profile correction gives somewhat higher richnesses compared to the @xmath60 profile , as would be expected . however , the distributions do not differ greatly . therefore , for simplicity , we adopt the @xmath60 profile when deriving richnesses and richness classes ( e.g. , figure 7 and table 1 ) , although our conclusions do not depend significantly on this assumption . the cluster richnesses and richness classes were determined by applying the depth and aperture corrections to the @xmath68 , and 1 mpc aperture counts prior to averaging them . cluster richnesses are binned into richness classes ( @xmath69 ) mapped following the aco catalog : @xmath70 @xmath71 , respectively . the errors in the richnesses are dominated by systematics , so we adopted the extreme values from the three apertures as an error estimate ( shown in figure 7 and discussed below ) . figure 6 shows that our clusters are primarily richness class 02 . while three clusters may lie in richness classes @xmath72 , their richness estimates depend on large and uncertain depth corrections ( see table 1 ) . their richness estimates are therefore suspect . we compare our distant cluster richnesses to the nearby @xmath73 , x - ray - flux - limited sample of edge et al . ( 1990 ) , shown as the filled histogram in figure 6 . the richness classes and counts for most clusters in the edge et al . sample were taken from the aco catalog . others were taken from bahcall ( 1980 ) and owen et al . only clusters with x - ray luminosities exceeding our lower limit of @xmath74 are included . edge s x - ray luminosities were adjusted downward by typically a factor of two to match the @xmath3 band . richnesses for five edge et al . clusters ( 3c129 , 0745191 , a3158 , ophiucus , triangulum australis ) are unavailable in the literature and were thus excluded . overall , the richness distributions for the 160 degree survey clusters are similar to the nearby cluster distribution , if not somewhat poorer ( the modal richness of the edge et al . clusters is @xmath75 , while for the 160 degree survey clusters it is @xmath76 ) . this difference turns , however , on only a few clusters . our sample is simply too small to reliably discriminate between these distributions . we conclude , as we did with figure 4 , that to within the sampling precision , the nearby and distant clusters have similar richness distributions . in figure 7 , we plot the 14 clusters in our sample on the @xmath6richness plane . for comparison , we plotted the mean @xmath6richness relation for @xmath77 clusters from the @xmath3 all sky survey ( briel & henry 1993 ) . our clusters are plotted as solid points , while the briel - henry relation is the solid line . in addition , we plotted optically - selected , @xmath78 clusters from the palomar distant cluster survey ( pdcs ; postman et al . 1996 , holden et al . 1997 ) as open points . the optical cluster x - ray data from holden et al . 1997 and the briel & henry relation have been adjusted slightly to register the x - ray passbands . the horizontal broken line at @xmath79 shows the approximate luminosity of a @xmath80 cluster corresponding to our x - ray flux detection threshold of @xmath81 . figure 7 shows the larger fraction of our clusters lying above the briel & henry mean relation in such a way as to appear poorer than average , as was found in figure 6 . nevertheless , our clusters lie well within the large scatter of the briel & henry relation . given the combination of the small sample size , the large uncertainties in the richness corrections , and the ample spread in the briel henry relation , we can only conclude that our clusters may be a bit poorer than average , but they are within the expected range for nearby clusters with similar x - ray luminosities . on the other hand , the optically - selected , pdcs clusters have a similar richness distribution to our x - ray clusters , yet few , if any , are detected in x - rays . only 3 of 16 similarly distant pdcs clusters were detected in x - rays , and even these may be background sources ( holden et al . a similar trend was noted by bower et al . ( 1994 ) using clusters selected from the couch et al . ( 1991 ) catalog . the distributions of our clusters and the pdcs clusters shown in figure 7 are clearly segregated by x - ray luminosity . therefore , based on the analysis of these small samples , our survey and the pdcs are apparently not detecting similar mass concentrations . the reason why this would be so is not entirely clear . however , a serious and well - known problem associated with optically - detected clusters would be the tendency to misidentify superposed but physically - distinct groups of galaxies along a sight line as a physical cluster ( see postman et al . 1996 , oke , postman , & lubin 1998 ) . furthermore , optical catalogs may be prone to contain real , galaxy - rich mass concentrations which have not yet virialized ( peebles 1993 ) , yet they can be confused with rich , virialized clusters . on the other hand , optical selection techniques would be sensitive to optically rich but x - ray - faint galaxy clusters that would otherwise be missed by x - ray surveys . this last hypothesis , intriguing as it may be , is difficult to understand , as significant x - ray emission is expected from clusters in most cosmological scenarios ( bower et al 1997 ) . in a recent study of the couch et al . ( 1991 ) catalog of clusters , bower et al . ( 1997 ) found concordant galaxy velocities for most clusters . based on an analysis of the radial velocity histograms , they argued that the couch objects were real mass concentrations . however , their velocity dispersions were nearly a factor of two larger than nearby clusters with comparable x - ray luminosities . conversely , their x - ray luminosities were an order of magnitude smaller than nearby clusters with similar velocity dispersions . bower et al . ( 1997 ) concluded that the couch cluster velocity dispersions do not reflect the clusters virial temperature . rather , the velocity dispersions are either inflated by infalling galaxies on nearly radial orbits , or the clusters are embedded in large - scale filaments oriented along the line of sight . since the infall scenario seems to imply an unrealistically rapid rate of infall , the latter scenario seems more likely to be true . the essential point is that optical clusters may poorly represent the general population of distant clusters ( cf . donahue et al . 2001 ) . the comparison between our clusters and the pdcs clusters , shown in figure 7 , supports this interpretation . most importantly , the new results in this paper demonstrate that x - ray emission is a reliable tracer of cluster - scale mass concentrations in the distant universe . we have estimated richnesses for some of the most distant clusters from the 160 square degree cluster survey ( vikhlinin et al . 1998a , b ) using three methods . we found that our survey is efficient at detecting rich galaxy contrations at redshifts beyond @xmath15 . these clusters have x - ray luminosities between @xmath1 , and abell richness classes typically between @xmath4 . the richness distribution of the distant , 160 square degree clusters is similar to the distribution of nearby clusters with similar x - ray luminosities . clusters have evidently evolved by less than @xmath82 richness classes between redshifts of @xmath83 and today . we find marginal evidence that our distant clusters are somewhat poorer than the average nearby cluster , although deep , multicolor imaging would be required to confirm this . we compared our sample richnesses to those for a sample of equally distant , optically - selected clusters from the palomar distant cluster survey that have been observed by @xmath3 ( holden et al . although the pdcs clusters are comparably rich , they have significantly lower x - ray luminosities compared to our clusters . this segregation in @xmath84richness plane is not understood , but may reflect the tendency for optical surveys to detect superpositions of galaxies , and unvirialized mass concentrations associated with large - scale filaments . thanks larry david , craig sarazin , marijn franx , and tony tyson for helpful discussions , and acknowledges support from stsci grant go07533.0196a and nas839073 . we thank the referee for suggestions that substantially improved this paper

we have measured the surface density of galaxies toward 14 x - ray - selected cluster candidates at redshifts @xmath0 , and we show that they are associated with rich galaxy concentrations . these clusters , having x - ray luminosities between @xmath1 , are among the most distant and luminous in our @xmath2 square degree @xmath3 pspc cluster survey . we find that the clusters range between abell richness classes @xmath4 , and have a most probable richness class of one . we compare the richness distribution of our distant clusters to those for three samples of nearby clusters with similar x - ray luminosities . we find that the nearby and distant samples have similar richness distributions , which shows that clusters have apparently not evolved substantially in richness since redshift @xmath5 . there is , however , a marginal tendency for the distant clusters to be slightly poorer than nearby clusters , although deeper , multicolor data for a large sample would be required to confirm this trend . we compare the distribution of distant x - ray clusters in the @xmath6richness plane to the distribution of optically - selected clusters from the palomar distant cluster survey . the optically - selected clusters appear overly rich for their x - ray luminosities when compared to x - ray - selected clusters . apparently , x - ray and optical surveys do not necessarily sample identical mass concentrations at large redshifts . this may indicate the existence of a population of optically rich clusters with anomalously low x - ray emission . more likely , however , it reflects the tendency for optical surveys to select unvirialized mass concentrations , as might be expected when peering along large - scale filaments . # 1 accepted for publication in the astrophysical journal +

introduction observations galaxy selection and photometry the cluster contrast parameter comparison of galaxy counts between our distant x-ray clusters and nearby abell clusters estimating cluster richnesses conclusions

arxiv

heterogeneous mixtures of granular materials tend to segregate by size or shape under conditions where one might naively expect mixing . in 1939 , oyama noticed that a binary mixture of large- and small - grained sand segregated into axial bands when tumbled in a horizontal `` drum mixer''.@xcite after a hiatus of several decades , this counterintuitive phenomenon has captured the attention of many researchers,@xcite and it has become something of a standard problem in the study of granular materials . the experiment is deceptively simple . in a typical arrangement , equal amounts of two grain sizes , often of contrasting colors , are uniformly mixed and used to partly fill a horizontal transparent tube . the tube is then rotated about its long axis at a rotation rate which is sufficient to cause the grains to stream smoothly . axial bands of segregation first appear after a few tens of rotations and are faint at first , but intensify and saturate . after a few hundred to a few thousand rotations , an apparently stable array of axial bands of roughly equal spacing is formed . bands may merge or split on long time scales , but the general tendency is towards merging@xcite . with very extended rotation time , some experiments show that the bands exhibit a remarkably long period of metastability , but eventually all merge and finally achieve complete segregation , where the two species occupy opposite ends of the tube.@xcite the occurrence of axial segregation seems to require that the dynamic angle of repose ( roughly , the angle of the flowing surface ) be different for the two species @xcite . this has been taken to be evidence that the geometry of surface flows is responsible for the segregation . this observation has inspired theoretical models@xcite that treat the onset of segregation as a kind of reverse - diffusion driven instability in which initial concentration fluctuations are amplified . many researchers have studied the number of bands that appear . it is claimed that more bands appear when more small particles are used @xcite , and that more bands appear when the rotation rate is slower@xcite . recently , mri studies have shown that there exist subsurface bands that may not be visible from the surface@xcite . this suggests that visual band counting may be missing some bands , and casts doubt on a purely surface - driven segregation mechanism . in some systems , the segregation also appears to be affected by coherent surface avalanche waves@xcite or fingering instabilities@xcite . these would provide non - diffusive mechanisms for axial segregation . in this paper , we report our observations on the dynamics of segregation in a mixture of sand and salt in a long pyrex tube . we have focussed on the very early stages of the evolution , in an effort to shed light on the nature of the instability that leads to segregation . the primary improvements we made over previous experiments were in control of the initial conditions and improved resolution in the image processing and fourier analysis . by employing a very long tube , a larger number of bands can be studied , yielding fourier peaks that are well separated from dc and have good resolution . we digitized images of the flowing surface and used averaging and image - division techniques to observe faint bands . we studied the initial growth of the bands by observing the exponential growth of fourier modes . at the threshold of saturation , we examined the spectrum of preferred wavelengths of the band pattern . we measured the rate at which the bands grew and identified the dominant wavelength , as a function of the rotation rate of the tube . by composing images of the spacetime evolution of the surface contrast , we were able to discern transient traveling waves@xcite and other structures . in this paper , we characterize the traveling wave speed as a function of wavelength , rotation period , filled fraction of the tube , and the relative proportions of the two granular components . these traveling waves are surprizing because they are not expected from any simple diffusion model of segregation . this paper is organized as follows . in section [ expt ] , we describe the apparatus and experimental procedures . in section [ results ] , we describe the detailed observations . section [ discussion ] and [ summary ] contain some discussion and a brief summary of our conclusions . in this section we describe the apparatus , the experimental procedures and the data acquisition and analysis . the experiment was optimized for the quantitative observation of the early stages of segregation . many parameters effect the details of the phenomena , and we were only able to systematically explore a few of them . the drum mixer consisted of a @xmath01 m long tube partially filled with the granular mixture which was rotated about its long axis at a constant angular frequency @xmath1 using a computer - controlled stepper motor . the tube was made as long and narrow as practical so that the number of bands was large . this maximized the resolution of fourier analysis , described below . three different tubes were used . each had inner diameter 27 mm . most of the observations were made using one of two pyrex tubes which had aspect ratios ( length / diameter ) of [email protected] and [email protected] . the third tube was made of plastic , and was only used to investigate charging effects . friction with the endcaps tends to produce segregation ; we used teflon - covered end plugs in an attempt to minimize this effect . the speed of the traveling waves we observed with some mixtures was sensitive to the precise leveling of the tube . waves traveled downhill more quickly than uphill . we leveled the tube to better than @xmath3 by matching the speeds of the left- and right - going waves . the components of the granular material were selected to maximize the color contrast between the two species . by doing this , we were able to observe very faint bands . this capability was especially important for studying the traveling waves . the characteristics of the grains were by far the most difficult parameters to specify and control . the smaller black component was `` hobby '' sand , while the larger white component was common table salt.@xcite under magnification , we found that most of the salt grains had a cubic crystalline shape and only a very small fraction were irregularly shaped . the black sand grains , on the other hand , were all somewhat rounded and were very irregularly shaped . we also performed some tests with light colored ottawa test sand in place of the salt . this sand was very rounded in shape . two different types of mixtures were used for the experiments , which differed in the internal size distributions of each component . these distributions are shown in figure [ sandsalt - distribution ] , and were determined using standard sieves . in all cases , the sand was smaller than the salt . mixture b had a larger variety of particle sizes , and had larger salt grains on average than mixture a. these sizes were selected to be roughly similar to those used by das gupta _ et al_. @xcite we define the size composition fraction @xmath4 to be the volume of the salt component divided by the sum of the volumes of the sand and salt components , where all volumes were measured _ before _ mixing . because of packing effects , the volumes of mixtures were typically about 10% smaller than the sum of the volumes of the unmixed components . thus , @xmath4 is not exactly equivalent to the volume concentration of the salt in any mixture , but is experimentally easier to specify and vary systematically . in the runs described below , mixture a was used at a fixed @xmath5 , while type b s mixture ratio was varied between 0.33 and 0.79 . filling the long tube with the mixture to the desired amount was a nontrivial task . any sort of pouring , flowing or shaking operation causes segregation . we also wanted to avoid nonuniformities in the initial filled volume fraction along the length of the tube . the tube was filled by first gently placing the sand / salt mixture in a u - shaped aluminum channel as long as the tube , inserting it lengthwise into the tube , and then rotating it to dump its contents in place . in most runs , two such operations were used , corresponding to a filled volume fraction @xmath6 of 0.28 @xmath2 0.02 . this quantity is necessarily somewhat ill - defined , as packing effects , dilation under shear and segregation all change the volume of the granular mixture during the course of the experiment . in some runs , we varied the filled volume fraction by filling the tube with different numbers and sizes of u - shaped channels . in these cases , we simply used the total mass of material in the tube as a measure of the filled volume fraction . mixture a with @xmath5 and @xmath7 was used with randomly premixed initial conditions for studies of the dynamics of segregation into normal bands . the initial conditions were prepared by manually mixing the grains in small quantities and loading the tube as described above . in spite of our best efforts to mix the ingredients , there were typically still some concentration fluctuations that were large enough to be visible to the eye . mixture b with a range of @xmath4 and @xmath6 was used for studies of the dynamics of the transient traveling waves . for this purpose , presegregated initial conditions were prepared for a series of runs . these were obtained by placing thin partitions in the u - shaped channel and then alternately filling each segment with one component of the mixture . when the channel was full , the partitions were removed and the tube was loaded as before . by varying the position of the partitions , we could prepare any desired wavelength of initial segregation , while holding the size composition fraction @xmath4 nearly constant . to determine the exact value of @xmath4 , the mixture was sieved after each presegregated run . we used 100% presegregated bands as initial conditions for our study , but any other degree of presegregation could also be made by filling the segments of the channel with suitably premixed material . given the highly insulating nature of all the components , electrostatic effects must be carefully considered . charge transferred between the grains or onto the tube can modify the segregation patterns by introducing long - range forces.@xcite rubbing the exterior of the rotating tube or repeated rapid pouring of the mixture when it was outside the tube caused a few of the grains to stick to glass surfaces . we monitored the humidity of the enclosure surrounding the experiment with a hygrometer , and it was found to vary from 18 - 34% . no sticking due to static charging or humidity - dependent effects were observed in normal runs , which lasted @xmath0 1 hr . in very long runs , which we do not discuss here , static charge did eventually build up enough that sticking became apparent . to investigate the effect of charge on the tube , we made a few runs using a plastic tube in place of the pyrex one . with the plastic tube , charging effects were much greater , but all of the same phenomena , including the traveling waves , were observed . this indicates that although charging effects may be present at some level in the pyrex tube , they do not significantly effect our results . the grinding effect of repeated tumbling caused slow but systematic changes in the quantitative behavior of the mixtures , particularly of traveling waves . the salt became progressively greyer in color as the polishing of the grain s surfaces made them more transparent . there was also a tendency for the corners and edges of the salt grains to become damaged , making the grains slightly rounded . there was no obvious change in the sand component with grinding . in no case was grinding sufficient to cause any measurable change in the size distributions , as determined by sieving . to avoid systematic effects , we used mixtures that were `` run - in '' briefly to avoid initial transients and then discarded them after @xmath8 hours of tumbling . during rotation , the surface brightness of the flowing material was visualized using a monochrome ccd camera that could image the entire length of the tube using a wide angle lens ( focal length 3.7 mm ) . the tube was uniformly lit by a single long fluorescent tube , which was located just above it . the video signal was acquired by a computer via an 8-bit frame grabber . the camera was positioned so that it was pointed perpendicular to the flowing surface . only the portion of the video frame covering the flowing surface was digitized . after acquisition , the resulting long rectangular image was corrected in software for the geometrical distortion introduced by the wide angle lens so that length measurements could be accurately made . in this step , we also interpolated the 400 pixel length of the array to 512 pixels , to facilitate later fourier analysis . spatial intensity variations introduced by nonuniform lighting were removed by dividing by a reference image . each rectangular image was then spatially averaged across the height of the streaming surface to produce a one - dimensional greyscale array . in this step , we also interpolated the 400 pixel length of the array to 512 pixels , to facilitate later fourier analysis . arrays taken at regular time intervals were stacked to form spacetime images which recorded an entire run . calibration of the greyscale with respect to the local volume concentration of black sand indicates that , except for low black concentrations ( @xmath9 20% ) , the greyscale is , to a good approximation , linearly related to the local salt concentration . the spacetime images were analyzed with one- and two - dimensional fourier transforms . a one - dimensional fourier transform , applied separately to each row ( i.e. , in the space direction only ) , was used to study the time evolution of the fourier modes over the course of a run . two - dimensional fourier transforms of the whole spacetime image were used to analyze traveling wave states to obtain wave speeds and for simple image processing to extract left- and right - going waves . we observed radial segregation , end segregation , axial segregation and transient traveling waves , under various experimental conditions . the latter two were studied in detail and are described in the following sections . except for traveling waves@xcite , all of these effects have previously been observed in other experiments . @xcite for our system , radial segregation occurs rapidly , on a time scale of @xmath0 10 rotations . we can not observe this directly , but it is evident from a change in the overall surface color as a portion of the smaller black component migrates to the axis of rotation . as in many previous studies , we also observed the formation of end segregated bands that were also composed of the smaller black component . these formed in spite of our teflon - covered end plugs . however , if a white salt band was started against an end cap in a presegregated run , then the band stayed white . the end segregated bands formed more quickly than other bands , but did not subsequently change over the duration of the run . in order to study the growth and saturation of normal bands , we used mixture a at a fixed composition @xmath5 and filled volume fraction @xmath7 . the angular rotation frequency @xmath1 was varied between 1.57 and 8.38 rad / s . this interval covers essentially the entire range over which the mixture streams smoothly down the flowing surface . at lower @xmath1 , intermittent avalanching is observed , while for larger @xmath1 , the mixture spends part of each rotation airborne . a run typically lasted for 600 seconds , with 5 seconds elapsing between image acquisitions . we used only premixed initial conditions . a spacetime image of a sample run is shown in figure [ sample - dynamics ] . several images are averaged together to form each line of the picture . the first faint axial segregation bands can be seen after just a few tens of rotations . these bands usually intensified until the black bands saturated at 100% concentration . since earlier radial segregation carries smaller black sand to the axis of rotation , it seems likely that the saturated black bands seen on the surface consist only of that species all the way through . on the other hand , the apparently saturated white bands probably still contain a black core that is not visible from the surface . in the following , we will focus on the dynamics of the growth of the black bands . at the highest rotation rates , @xmath10rad / s , the black bands apparently saturate but do not reach 100% concentration . the bands stayed faint . apparently the randomizing diffusive effects of high rotation rates limit the final concentration of the bands . on the other extreme , at very low rotation rates , @xmath11 rad / s , the time to saturation sometimes exceeded our observation time . bands do not all saturate at the same time . if an area has a high local black concentration at the start of a run , saturation is reached earlier there . saturated black bands are metastable over several thousand rotations . they were never observed to split . only a few merging events were observed between saturated bands , and they occurred very slowly . saturated black bands can sometimes be seen to absorb unsaturated ones . the traveling waves discussed in the next section also tend to be absorbed by saturated bands . before saturation , bands exhibit more interesting dynamics . bands can appear and disappear , and they can merge and split . examples are shown in figure [ sample - dynamics ] . fourier spectra with multiple modes with no one mode dominant are common . the growth of the total power in the fourier spectrum is well - described as being initially exponential , as illustrated in figure [ log - totalpow - with - spacetime ] . we omit the power in the first 5 fourier bins to eliminate the dc component . the total power ceased to grow exponentially at the same time as saturated bands appeared . figure [ growthrate - vs - freq ] shows the exponential growth rate of the total power _ vs. _ @xmath1 . we express the growth rate in units of the rotation period in order to eliminate the obvious tendency for growth to be faster for larger @xmath1 . interestingly , the growth per tube rotation appears to reach a maximum at @xmath12 rad / s . to look for characteristic spacings of the saturated bands , the fourier spectra were examined at the onset of saturation , for various @xmath1 . the onset of saturation was determined by the loss of exponentiality of the total power . examining the fourier spectra before saturation is not particularly informative as the relative strengths of fourier modes change rapidly due to band dynamics . the spectra at saturation are noisy ; to get some insight , we simply averaged together spectra for all @xmath1 in the range 1.57 - 8.38 rad / s . each spectrum was normalized by dividing by the total power outside dc before being included in the average . figure [ average - fft ] shows the result . there is a clear peak near a band spacing of 50 mm . there is a rather broad range of growing modes about this peak , however , with the hint of other peaks . for each @xmath1 , we also examined the spectra to ascertain which fourier mode , if any , dominated . as there were sometimes multiple peaks of almost equal size , centre - of - mass wavelengths were also obtained to more accurately take the spread into account . the centre - of - mass was obtained over a range of wavenumbers from just above dc to twice the wavenumber of the highest peak . the result is shown in figure [ selected - wavelength ] . it shows that , to a first approximation , the selected wavelength of the axial bands is independent of @xmath1 . under certain conditions , mixture b exhibited traveling waves during the transient that precedes the formation of the usual stationary axial bands . here we review this puzzling phenomenon , which was first described in ref . @xcite , and make some further observations . traveling wave transients occur only in a restricted region of the parameter space in which axial segregation is found . outside this region , one finds generally merging transients like those shown in fig . [ sample - dynamics ] . our objective was to determine , as far as possible , which parameters are important for the existence of wave - like transients and which are not . surprisingly , the size composition fraction @xmath4 turns out to be an important parameter . we varied @xmath4 over the range 0.33 - 0.79 in mixture b , while @xmath1 and @xmath6 were held constant . the waves were not observed unless @xmath13 , and , for a given wavelength , their speed increased with @xmath4.@xcite large @xmath4 corresponds to mixtures rich in the larger salt component . the traveling waves obey a well - defined dispersion relation . on the other hand , we show below that the traveling waves are rather insensitive to variations in @xmath1 or @xmath6 . with premixed initial conditions and @xmath14 , patches of spontaneous left- and right- traveling waves are observed , which eventually give way to the usual stationary bands . the waves have a preferred wavelength of 45 @xmath2 5 mm . the waves move at nearly constant velocities in either direction and usually pass through each other with little or no interaction , until band formation , which has been occurring in parallel , grows strong enough to interfere with and destroy them . the traveling waves are dissipated in the vicinity of black bands which are saturated at 100% sand . all these features are illustrated in fig . 1(a ) of ref . @xcite . in order to study homogeneous regions of the traveling waves , we employed presegregated initial conditions in which a chosen wavelength was launched all along the length of the tube . the resulting uniform traveling waves pass through each other to form a standing wave which persists for several oscillations before it is disrupted by stationary band formation . figure [ seeded ] shows a spacetime image of a portion of such a standing wave . in order to study how the traveling velocity varies with wavelength , numerous runs were performed with @xmath15 , @xmath7 , @xmath16 rad / s and different presegregated initial wavelengths . the velocities of the resulting waves were measured by locating the times when the standing waves pass through zero . to avoid early transients and later nonlinear effects , we used only the positions of the first few zero - crossing times.@xcite we also measured the speeds of some spontaneous waves occurring with premixed initial conditions at these values of @xmath1 , @xmath4 and @xmath6 to confirm that they were similar to the presegregated waves . the speeds of spontaneous waves were estimated from the slopes of their worldlines in spacetime images , and by extracting the positions of peaks in 2d fourier transforms of spacetime images . figure [ dispersion - relation ] shows the dispersion relation obtained from velocity and wavelength measurements . an interesting feature of the dispersion relation is that the waves do not travel when presegregated below a certain cutoff wavenumber . we refer to these as `` frozen '' bands . the following observations indicate a possible explanation for this behavior . when the rotation is started , there is a brief relaxation phase during which the initially sharp presegregated dark bands become diffuse and narrower as the material near the black / white interfaces mixes . it seems that traveling waves occur only when this mixing is sufficient to completely submerge the initial presegregated black bands . for wavelengths which do not travel , it was found that the central portion of the black bands remained 100% sand . salt can not move through such pure regions of sand , and thus broad sand bands can not travel . the appearance , immobility and stability of these frozen black bands suggests that they are similar to the 100% segregated bands that eventually grow up spontaneously and halt the traveling wave transient regime . the transition region between frozen bands and traveling waves is difficult to pin down experimentally . in this vicinity we find that some bands remain frozen , while others travel a short distance , but merge when they encounter another band ( figure [ merging ] ) . these mergings appear to be similar to the late - stage absorption of traveling waves by 100% segregated bands , seen at shorter wavelengths . using the greyscale as a measure of the local concentration , we can examine the amplitude linearity of the traveling waves . their amplitude is quickly damped away with rotation , as shown in fig . [ vamplitude](a ) . despite this large change in amplitude , their velocity is constant , as shown in fig . [ vamplitude](b ) . thus , the waves are remarkably linear . this is also indicated by the fact that waves traveling in opposite directions pass through each other with little or no distortion , i.e. , they obey superposition . the merging behavior seen in fig . [ merging ] can be interpreted as due to a loss of linearity near the transition from traveling waves to frozen bands . the existence and velocity of the traveling waves are robust over a wide range of filled volume fractions @xmath6 , as shown in fig . [ speed - vs - filling ] . the wave speed is essentially constant , even when @xmath6 is changed by a factor of two . the accessible range of @xmath6 is limited by the tendency of the material to slip along the walls of the tube for small @xmath6 , and the need to leave enough free volume to fill the tube using the u - shaped channel at large @xmath6 . we have also measured the traveling wave speeds as a function of the angular frequency of rotation @xmath1 . we found that bands in a presegregated run freeze for @xmath17 rad / s , but travel for smaller @xmath1 . [ speed - vs - period ] shows the wave speed vs. the period of rotation @xmath18 . when velocity is expressed in terms of distance traveled per rotation period , we find that wave speeds do not have a strong dependence on @xmath19 . to check for momentum effects , we stopped the rotation of the tube during the propagation of a standing wave , and then restarted it when all grain motion had ceased . this did not significantly alter the behavior of the waves , which continued to travel once the rotation restarted . we examined how the traveling wave phenomenon depends on the details of the two size distributions and on grain shape . in general , almost any change in either of these quantities causes the traveling waves to change or disappear altogether . substitution of the cubical salt component with the same size distribution of rounded ottawa sand resulted in segregation but had transients with no traveling waves . similarly , even the rather minor rounding effect that prolonged grinding has on the salt grains eventually results in the disappearance of the traveling waves . variations of the internal size distribution of either of the components of mixture b has a similar effect . these observations suggest that traveling waves occupy only a small portion of the parameter space over which segregation itself exists . some other interesting dynamical objects can be observed in mixture b. fig . [ fountain ] shows a `` fountain '' band which sent out traveling waves in both directions . the fountain appears to constantly send out traveling waves that disappear as they near the 100% segregated bands on either side . if the 100% segregated bands indeed absorb traveling waves , then it is possible that , eventually , the fountain would become depleted and cease . however , that did not happen during the @xmath20 rotations that we followed the evolution of the fountain . the fountain occurred spontaneously in a presegregated run ; it is not known how to reproducibly make one . using presegregated initial conditions , it is straightforward to launch pulses . [ pulse ] shows the evolution of one such pulse , which was prepared by laying down three bands , two white and one black , of 100% presegregated material in an otherwise premixed background with @xmath15 . when the rotation was begun , each of the two prepared white bands pulled in black sand on their outer boundaries , and proceeded to travel away from each other . these two waves appear to be accompanied by fainter , co - traveling waves on their sides . it would be interesting to systematically study various kinds of pulses and their interactions . in the absence of a general theory of granular matter , any explanation of axial segregation must be developed heuristically . since the grains are more - or - less locked in solid rotation for most of the rotation period , models have naturally focussed on identifying sorting processes occurring in the dilated , flowing surface layer . in the simplest picture , the surface is characterized by a single number , the dynamic angle of repose @xmath21 . it is readily observed in most segregating mixtures that axial segregation results in a modulation of the surface slope along the axis of the tube.@xcite thus , one assumes that @xmath21 depends on both @xmath1 and the local concentration @xmath22 , and seeks a dynamical equation for @xmath22.@xcite in one dimension , the conservation of sand implies the continuity equation @xmath23 where @xmath24 is the axial coordinate of the tube and @xmath25 is the axial concentration current . the current @xmath26 is assumed to have the form @xmath27 where @xmath28 is the usual fick diffusion coefficient and @xmath29 depends on the difference in the dynamic angles of repose of the two components . if @xmath30 , eqn . [ continuity ] is a diffusion equation with a negative diffusion coefficient , so that initial concentration fluctuations grow . several observations support this general approach . das gupta _ _ @xcite and hill _ @xcite have observed that segregation stops , and mixing occurs , when @xmath1 is such that the difference in @xmath21 between the two species is zero . segregation is also observed in two - dimensional poured piles@xcite , where it has been explained by various models based on differing angles of repose.@xcite an elaborate theory of axial segregation based on angle of repose effects has been proposed by zik _ _ @xcite in which an expression for @xmath31 was derived from geometrical considerations and continuity . although this model has its successes , such as obtaining an s - shaped cross - sectional profile of the flowing sand that is qualitatively similar to experimental observation , it can do no more than indicate that an instability occurs . without nonlinear terms , it can not make any prediction about the saturation or preferred wavelength of the band pattern . however , if we assume that saturation effects with a short - wavelength cut - off eventually limit the growth of the fluctuations , it follows that axial segregation proceeds in a manner analogous to spinoidal decomposition . the initial stages exhibit an exponential growth phase , followed by a slow merging of nearly 100% segregated bands . this is qualitatively what we observe . our data suggest that there exists a fastest growing fluctuation , and it has a wavelength of about 50 mm , or about twice the tube diameter . this wavelength is essentially independent of @xmath1 , while its growth rate is a maximum for @xmath32rad / s . the irreversible evolution of the bands favours merging and will result in complete segregation after a sufficiently long time . such behavior has been observed in some mixtures : after two weeks of rotation , chicharro _ et al . _ @xcite reached a fully segregated configuration in which the two species each occupied one end of the tube . they also observed that the almost regular initial spacing of the bands was surprisingly metastable , however . other experiments@xcite have clearly observed late - stage dynamics in which segregation is not complete . in the negative - diffusion model , segregation is accomplished by diffusive differential surface transport down axial gradients of the angle of repose . it is clear from various experiments that this picture is highly simplistic , however . it ignores the radially segregated core of the smaller component which lies below the surface . recent mri experiments@xcite have imaged the core , and shown that subsurface segregated bands exist , which presumably can not be due to purely surface sorting . transport to and from the core can be driven by the same mechanism which drives the fast radial segregation . this mechanism has been elucidated in a series of experiments@xcite and simulations@xcite on the simpler case of two - dimensional drums partly filled with disks . while it is not at all obvious how material in the core might move axially , it is clear that the core represents a reservoir of material which is coupled to the surface via a channel which is not taken into account in the negative - diffusion model . the traveling waves we observed in mixture b when @xmath4 is sufficiently large are also irreconcilable with the negative - diffusion model . it is not possible to support bidirectional , linear traveling waves in a one dimensional pde which is first order in time . no such equation can describe counter - propagating waves which together form a standing wave , as we observe . this is most easily seen by considering the moment when the standing wave passes through zero as an initial condition for the rest of its evolution . obviously , the subsequent motion must depend on a momentum - like quantity which is specified at that instant as well as on the zero displacement initial condition . since equations which are first order in time lack any analog of momentum initial conditions , they can not be sufficient . such waves are not , however , inconsistent with first order dynamics in more dimensions , or , for example , with coupled first order equations . such equations might arise in models in which the core is taken into account , or which involved a more realistic two - dimensional surface . collective motions of the grains provide another mechanism which might affect transport in a non - diffusive way and cause momentum - like terms in the concentration dynamics . in experiments with homogeneous sand@xcite , oscillatory cellular instabilities of the streaming surface and undulations modes of the lower contact have been observed . these occurred on a timescale comparable to one rotation period and are associated with waves of avalanching propagating axially along the flowing surface . the time - scale of our traveling concentration waves we observed is at least 100@xmath33 longer . in other experiments@xcite , long sequences of avalanche waves are clearly correlated to segregation . the avalanche waves can have a strong effect on slow axial segregation phenomena in mixtures where one component exhibits avalanche waves and the other does not . this is true despite the very large separation of timescales involved . so far as we can determine , none of these avalanche waves are present in our experiment , although they are conceivably present at some amplitude which is too small to detect visually . stopping and starting the tube rotation did not prevent the continued propagation of the traveling waves , but this only interrupts the instantaneous momentum of the grains and does not necessarily mean that avalanche waves are not involved . very thin layers of sand , corresponding to filled volume fractions @xmath34 have been discovered@xcite to have a rich dynamical behavior which is reminiscent of that of fluids in a similar configuration@xcite . various collective modes are observed , including traveling waves . thus , it seems possible that non - diffusive transport due to subtle collective motions of the grain may couple to the concentration field and give rise to some of the traveling wave effects we observe . this possibility deserves further investigation . we studied the initial growth and saturation of patterns of axial segregation of a binary granular mixture in a rotating tube . by fourier techniques , we followed the mode structure of the pattern as it emerged from random , premixed initial conditions . we measured the growth rates of various modes and determined that the system experiences initially exponential growth across a broad spectrum of modes peaked at a wavelength of 50 mm . this preferred wavelength was largely independent of rotation rate . we have also found that the overall growth rate of structures varies with rotation rate , with a maximum at @xmath354 rad / s . in addition to normal axial segregation , we have further examined the phenomenon of transient traveling segregation waves that occurred in certain mixtures . we have measured the velocity of the traveling waves and its dependence on the size composition fraction , the wavelength , the rotational frequency and the filled volume fraction . we have found transitions from traveling to stationary bands as a function of some of these parameters . between the traveling regime and the stationary regime , we find merging behavior . we established the amplitude linearity of the waves . we have also observed traveling pulses and sources . we would like to thank elaine lau , eamon mckernan and holly cummins for development work , and zahir daya , wayne tokaruk and troy shinbrot for useful discussions . this work was supported by the natural sciences and engineering research council of canada .

we have studied the early time evolution of granular segregation patterns in a horizontal rotating cylinder partially filled with a sand / salt mixture . the growth of concentration fluctuations starting from premixed initial conditions was analyzed using fourier techniques . in one mixture , we observed generally merging dynamics in the segregated bands up to the onset of saturation . at the threshold of saturation , we found a spectrum of wavelengths with a broad peak . the peak position was nearly independent of the tube rotation rate . the overall growth rate of fourier modes had a maximum at a particular value of the angular rotation frequency . in a slightly different mixture , we observed transient traveling waves when the larger grains were in the majority . we measured the wave speed as a function of several parameters using presegregated initial conditions to launch waves of various wavelengths .

introduction experiment results discussion summary and conclusion

arxiv

the transient source s belongs to the numerous subclass of the accreting x - ray pulsars with be optical companions ( bexrp ) . it was discovered by _ sas-3 _ observatory during the galactic plane survey in 1975 @xcite . later on the transient nature of the source was established as well as strong pulsations with the period of 9.3 s and amplitude of @xmath8 were found @xcite . using the same data an orbital period of the binary system was measured @xmath9 d and a suggestion that s is likely a be / x - ray binary system has been made @xcite . it is important to note that no optical counterpart was directly determined for s so far ( however see below ) . second time after the discovery s came into the view of x - ray instruments during the outburst in 20072008 @xcite . an intensive monitoring of this outburst with the _ rxte _ observatory allowed to improve orbital parameters of the system and to trace the spectral evolution in the energy range 2.530 kev @xcite . particularly it was shown that the source spectra in all available intensity states can be well fitted with the combination of a black - body component ( with temperature varying between 2.5 and 4 kev ) and a broken power - law component . the iron emission line at @xmath10 kev and a strong photoelectric absorption corresponding to hydrogen column density of @xmath11 @xmath12 were also required by the fit . the third episode when s has undergone an outburst was observed in the beginning of 2015 . the increase of the flux seen in the _ maxi_/gsc data was reported by @xcite . an estimated starting date of the activity was around 2015 january 28 ( mjd 57050 ) . in this paper we describe results of the comprehensive spectral and temporal analysis of the high quality data collected by the _ nustar _ observatory during the declining phase of this outburst ( mjd 57115.48 ) . main goal was to explore the source properties at high energies ( above 30 kev ) for the first time . the _ nuclear spectroscopic telescope array ( nustar ) _ @xcite , launched on 13 june 2012 , is the first orbital x - ray focusing telescope operating at energies above 10 kev . the observatory consists of two co - aligned identical x - ray telescope systems operating in a wide energy range from 3 to 79 kev with angular resolution of 18(fwhm ) and half - power diameter ( hpd ) of 58 . spectral energy resolution of 400 ev ( fwhm ) at 10 kev is provided by independent solid state cdznte pixel detector units for each telescope , usually referred as focal plane module a and b ( fpma and fpmb ) . ) . times of the _ nustar _ and _ chandra _ observations are marked . _ bottom : _ evolution of the pulsar spin period over the outburst as seen by the _ fermi _ gamma - ray burst monitor ( gbm ) . asterisk shows corrected for the orbital motion period measured by _ nustar _ ( see text for the details ) . ] _ nustar _ performed a too observation of s in the declining phase of the outburst ( mjd 57115.4834 , see fig . [ fig : lc ] ) with a total exposure time of 27 ks ( obs . i d 90101002002 ) . the source was relatively bright demonstrating net count rate of about 20 cts s@xmath5 on both fpma and fpmb . we did not notice any special issues related to the high count rate of the source in the following analysis . the source covered large area of the fov ( fig . [ fig : ima ] ) because the _ nustar _ psf has wide wings @xcite . the data were reduced using the nustardas pipeline version v1.4.1 ( 28 may 2014 ) and caldb version 20150612 . in order to perform a barycentric corrections it is necessary to know the source position with a good accuracy . observations with the _ chandra _ observatory , performed on 2015 may 21 ( mjd 57163.68 ; obsid . 17662 ) , allowed us to measure it as r.a.= 15@xmath1357@xmath14483 , dec.= @xmath754@xmath1524531 ( j2000 ) with an 1 uncertainty ( 90% ) . note , that these _ chandra _ observations as well as observations with the _ swift_/xrt telescope were used to determine the optical counterpart in the system and to measure the absorption value @xcite . the latter is important for the analysis of the _ nustar _ data as it works above 3 kev and not very sensitive to the small or moderate absorption values . we would like to remind here that a quite high photoabsorption ( @xmath16 @xmath12 ) was required by @xcite to describe the source spectra , obtained with the _ rxte _ observatory ( it operates also above 3 kev ) . such high values are quite atypical for bexrps , whose spectra do not demonstrate usually a strong absorption . our analysis of _ chandra _ and _ swift_/xrt data in soft x - ray band shows lower photoabsorption value in the source spectrum around @xmath17 @xmath12 @xcite , which was fixed in the following spectral analysis of the _ nustar _ data . note , that it is only slightly higher , than the estimations of the galactic interstellar absorption in this direction @xmath18 @xmath12 . we extracted spectra using nuproducts script provided by the nustardas pipeline . the source spectrum was extracted within 120 aperture around the source position as shown in fig . [ fig : ima ] , which constitutes 92% of psf enclosed energy ( see , e.g. , * ? ? ? the _ nustar _ background varies across the fov due to the different stray light ( also called `` aperture '' ) background components : individual bright x - ray sources , isotropic extragalactic cosmic x - ray background ( cxb ) and galactic x - ray ridge emission ( grxe ; * ? ? ? the field around s does not contain stray light from nearby galactic sources , but it has cxb and grxe components in the background , because s is located in the galactic plane ( @[email protected] , @[email protected] ) . we used suite of idl routines nuskybgd @xcite to model all the known background components ( instrumental , cxb and grxe ) in the source - free region outside the green dashed circle ( r=330 ) shown in fig . [ fig : ima ] . this model was utilized to estimate background spectrum at the position of the source . kev energy band . the images have been smoothed by a gaussian kernel with 9 width and color - coded in logarithmic scale for convenience . the color bar on the right indicates the units of the images expressed in @xmath21 cts pix@xmath5 s@xmath5 . the solid white circle ( 120 radius ) denotes region for the source extraction . the nuskybgd background model has been calibrated in the area outside the green dashed circle ( 330 radius ) . ] _ fermi _ gamma ray burst monitor ( gbm ) is an all sky monitor whose primary objective is to extend the energy range over which gamma - ray bursts are observed in the large area telescope ( lat ) on _ fermi _ @xcite . gbm consists of 12 nai detectors with a diameter of 12.7 cm and a thickness of 1.27 cm and two bgo detectors with a diameter and thickness of 12.7 cm . the nai detectors have an energy range from 8 kev to 1 mev while the bgo s extend the energy range to 40 mev . the _ swift _ burst alert telescope ( bat ) is a hard x - ray monitor that has a field - of - view of 1.4 steradians and its array of cdznte detectors are sensitive in 15150 kev range @xcite . it is a coded aperture instrument with a detector area of 5200 @xmath22 . we use the bat transient monitor results ( 1550 kev ) , provided by the bat team in order to model the torque imparted to the neutron star by the accreted material . gbm channel 1 ( 1225 kev ) ctime data from 5702057145 mjd is binned to 250 ms and fit to a semi - empirical background model . the background model is subtracted and pulsed flux and frequencies for 2s 1553 - 542 are extracted from the data by modeling its fourier components @xcite . as can be seen from fig . [ fig : lc](a ) total duration of the analysed outburst is around 3.5 months covering more than 3 complete binary orbital cycles . such a duration is typical for type ii outbursts from bexrps . these events are caused by the non - stationary increase of the amount of matter in the be circumstellar disc . peak luminosities can be much higher than @xmath23 . we have determined an orbital model for 2s 1553@xmath7542 using the _ fermi_/gbm and the _ swift_/bat data . typically , after correcting the pulse arrival times for earth s motion , one can use the doppler boosted frequencies of the pulses to fit a model for the binary system . material accreted onto a neutron star surface or collected in an accretion disk threaded by the neutron star s magnetic field transfers angular momentum to the neutron star . disentangling this intrinsic spin - up from the orbital signature is challenging . a solution to this problem is to model the intrinsic spin - up using a proxy for the system s x - ray luminosity . the luminosity is a function of mass accretion which is related to the torque imposed on the neutron star . @xcite showed that , at high luminosity , the intrinsic spin - up , @xmath24 , is proportional to @xmath25 when accretion is mediated through a disk . the proportionality constant is a function of mass , radius , moment of inertia , distance and magnetic field of the neutron star along with a few parameters describing accretion and emission efficiency . when connecting multiple outburst with the same torque model , it is necessary to include a spin - down term to account for angular momentum losses during quiescence . a search for frequency and frequency rate is performed using pulse profiles from gbm folded over a two day interval . each frequency epoch is chosen as the mean exposure - weighted observation time . the epochs are barycentered using the jpl planetary ephemeris de200 . bat survey data ( 1550 kev ) for 2s 1553@xmath7542 are used as a proxy for the source luminosity and to model the intrinsic spin - up rate . in order to eliminate under - constrained and over - constrained rates , only bat rates with errors greater than @xmath26 and less than 0.05 are used . this spin - up model along with the line of sight delay associated with the binary orbit from @xcite is used to model the barycentered arrival times . minimization of the @xmath27 fit of the barycentric frequencies and the bat rates is performed using the levenberg - marquart method and is of the form : @xmath28 where @xmath29 is the orbital redshift factor at time @xmath30 which is a function of the orbital elements , @xmath31 is the measured barycentric frequency at time @xmath32 which is the frequency epoch of the search interval @xmath33 . each epoch is chosen as the mean exposure - weighted observation time . @xmath34 is the average value of @xmath35 between @xmath30 and @xmath36 and @xmath37 is the bat rate . the model parameter @xmath38 is the orbitally corrected frequency at time @xmath30 and @xmath39 is @xmath40 . the updated orbital model is used in a new search for frequencies and frequency rates recursively until the orbital solution becomes stationary . the final fit resulted in a @xmath41 with 38 d.o.f . variability in the pulse profile within the two day integration interval contributed to errors in the measured frequency . in addition , changes in the emission beam within the integration interval or systematically throughout the outburst is expected to produce systematic errors in the bat rates used to model the spin - up . in order to adjust the errors on the orbital parameters to account for these issues , the errors are increased by 1.35 which results in a orbital fit with a reduced @xmath42 . .orbital ephemeris for 2s 1553@xmath7542 . [ cols="<,^,^ " , ] ( bbodyrad + cutoffpl+ gauss ) @xmath43 cyclabs . ( b ) residuals from the phabs @xmath43 cutoffpl continuum model . ( c ) residuals from phabs @xmath43 ( bbodyrad + cutoffpl+ gauss ) @xmath43 cyclabs model . black crosses represent averaged background level . ] as it was mentioned above , spectra of x - ray pulsars can vary significantly over the pulse . the observed variations of spectral parameters can give information about changes of the physical conditions or parameters of the emission regions near the neutron star . in order to study their evolution over the pulse period in the case of s we performed a pulse - phase resolved spectroscopy . the spin period was divided into 10 equal phase bins . the pulse profile has a more or less smooth shape ( fig.[fig : pprof ] ) , therefore such a division allows us both to obtain a good statistic for each spectrum and to trace well evolution of spectral parameters . to describe phase - resolved spectra we used the same spectral model as for the analysis of the averaged spectrum ( with cyclabs prescription for the cyclotron line ; model i ) it is necessary to note that the width of the cyclotron line can not be firmly determined in some phase bins due to insufficient photon statistics . therefore , its value was fixed at 8 kev the value which was measured in phase bins near the pulse maximum with the better statistics . the results of the pulse phase - resolved spectroscopy are shown in fig . [ fig : resspec ] . the average pulse profile of s across the entire _ nustar _ energy range is presented in each panel to visualize better the variations of spectral parameters with the phase . the photon index is varying quite significantly ( from 2 to 0 ) whereas the cutoff energy is less variable staying in the range between 4 and 6 kev . we can mention also a possible tentative correlation between these parameters . the temperature of the black body component is around @xmath44 kev and is quite stable over the pulse with some variations from @xmath45 to @xmath46 kev in first three bins . variations of the cyclotron line energy and the line depth are the most interesting and important for us , as both of them are changing significantly over the pulse . in particular , the line centroid energy has minimum ( @xmath47 kev ) near the pulse profile maximum , whereas the line depth has maximum near this phase . in general , the line energy @xmath48 is varying between @xmath49 and @xmath50 kev ( using cyclabs model for the cyclotron line ) , that is the main reason for the significant broadening of the line measured in the averaged spectrum ( @xmath51 kev ) . note that the described behaviour of the cyclotron line parameters over the pulse is very similar to the observed one in well studied bexrp v0332 + 53 when this object had nearly the same luminosity @xmath52 @xcite . from another side this behaviour is opposite to what observed in sub - critical pulsars due to different beaming properties . to demonstrate that the observed spectral variations are real ones we show the ratio of two spectra obtained in the first ( minimal line depth ) and forth ( maximal line depth ) phase bins in fig . [ fig : sprat ] . the upper panel demonstrates corresponding models for these spectra . the ratio of the observed spectra is shown in the bottom panel with filled circles . the solid line in this panel represents the ratio of the corresponding models . a strong absorption like feature , caused by difference in line depths for these phase bins is clearly seen in this figure between 20 and 30 kev . the contribution of different continuum components ( black body and power - law with cutoff ) to the total source luminosity demonstrates a variability of their ratio over the pulse . particularly , the black body flux remains virtually constant at the level of @xmath53 , whereas power - law flux is dominating at all phases and is determining the overall pulse profile shape . the contribution of the black body emission to the total flux is varying between @xmath54 and @xmath55 per cent . using estimation of the distance to the source @xmath56 kpc ( see section 4 ) the measured black body flux corresponds to the emitting area with the radius of @xmath57 km . this value is comparable with the neutron star size , that means that the black body emission can emerge from the neutron star atmosphere heated up by the intercepted emission from the accretion column @xcite . so big radius of the illuminated ns surface area can explain virtual constancy of the black body emission component over the pulse . this hypothesis can be verified in future by a set of observations of s at different luminosities and , hence , with different illuminated areas on the neutron star . in this work we presented detailed spectral and temporal analysis of the emission from the poorly studied bexrp s using the _ nustar _ data collected during the presumably type ii outburst in 2015 . the energy spectrum of the source can not be fitted satisfactory with any simple continuum model but requires inclusion of the absorption feature centered at @xmath1 kev . this absorption component has a clear physical meaning and represents the cyclotron absorption line known to be main evidence for the strong magnetic field on the neutron star surface . the absorption line in the s spectrum can be fitted both by the gaussian ( gabs model ) or lorenzian ( cyclabs model ) profiles with approximately the same quality . the line energy ( @xmath58 kev for gabs and @xmath59 kev for cyclabs ) corresponds to the magnetic field strength at the neutron star surface @xmath2 g after the correction for the gravitational redshift . the existence of the cyclotron feature is confirmed not only by the spectral fitting , but also by temporal properties of the source . particularly , the pulsed fraction dependence on the energy has a broad feature ( local decrease ) around 23 kev ( see fig . [ fig : ppfr ] ) that coincides with the position of crsf in the source spectrum . such non - monotonic dependencies of the pulsed fraction on energy were observed in several x - ray pulsars with cyclotron line in their spectra and even were proposed as a tool to search for crsfs . further evidence for the cyclotron absorption feature comes from the behaviour of the pulse profile with energy . a phase lag around the spectral feature can be clearly seen in figs [ fig : pprof ] and [ fig:2dpprof ] expressed in a wave - like structure near 2030 kev . such behaviour has been shown to be typical for a few others transient bexrps @xcite and can be explained by a natural assumption of an energy - dependent beaming of the radiation from the emitting region . the knowledge of the magnetic field strength gives us an opportunity to estimate the absolute value of the mass accretion rate needed to provide the spin - up rate measured by _ fermi_/gbm at the moment of our _ nustar _ observation @xmath60 s s@xmath5 . for that we used the accretion torque theories developed by different authors @xcite . these models explain the observed spin - up / down rate as a function of the neutron star parameters and depend on physics of `` accretion disc magnetosphere '' interaction . for our calculations we used the ns magnetic dipole moment @xmath61 g @xmath62 , derived from the measured value of the magnetic field strength , @xmath63 and 10 km as the neutron star mass and radius , respectively , keeping the mass accretion rate as a free parameter . type ii outbursts in bexrps are usually accompanied by the formation of the temporary accretion disk around the neutron star revealing itself in a strong spin - up rate and properties of the noise power spectrum . therefore , as a first approximation we used eq . ( 15 ) from @xcite to estimate the bolometric luminosity @xmath64 needed to support the measured spin - up rate . given the flux from s during the _ nustar _ observation @xmath65 the distance to the system can be estimated as @xmath66 kpc . the torque model by @xcite is not the only one developed for the disc accretion . in all these models the total torque can be expressed in the form @xmath67 . the only parameter of the dimensionless angular momentum @xmath68 is so called fastness parameter @xmath69 , where @xmath70 and @xmath71 are magnetospheric and corotation radii , correspondingly . we used different prescriptions for the dimensionless angular momentum @xmath72 which describes physical properties of the accretion disc interaction with the magnetosphere . particularly , utilizing approaches from @xcite and @xcite we got distances @xmath73 kpc and @xmath66 kpc , correspondingly . such not large dispersion of the distances derived from different models is due to the source being far from the spin equilibrium , where the difference in the above - mentioned models is maximal ( see , e.g. , fig . 2 from * ? ? ? radius of the magnetosphere in these models is assumed to be a fraction of the alfv@xmath74n radius @xmath75 . in the calculations above we assumed the parameter @xmath76 , however its exact value is not known and supposed to be between 0.5 and 0.7 ( see , e.g. , * and references therein ) . to estimate a possible influence of this parameter for our results we recalculated distances with @xmath77 this affected the derived values very insignificantly shifting the distance estimations based on models by @xcite and @xcite to @xmath7824 and @xmath7821 kpc , respectively . these estimations make s one of the most distant high - mass x - ray binary in the galaxy @xcite , putting it to the opposite side of the milky way . it is important to note , that the distance estimations based on the temporal properties of x - ray pulsars have quite large systematic uncertainties ( of the order of 1520 per cent ) due to model dependency , an unknown efficiency of the accretion and effects of a possible emission beaming , therefore should be considered with the caution . nevertheless , a large distance to the source ( @xmath7915 kpc ) is estimated from the optical data as well @xcite . the dispersion of the estimated distance values can be considered as a systematic uncertainty of this method . it is interesting to note , that if our estimations of distance ( @xmath80 kpc ) are correct then s exceeds the so - called critical luminosity above which the accretion column begins to grow above the neutron star surface @xcite . the value of the critical luminosity as well as conditions for growing of the accretion column are still under debates and depend on the physical models ( see , e.g. , * ? ? ? * and references therein ) . a set of observations in different intensity states during the outburst is needed to verify this hypothesis by , e.g. , observing of the anti - correlation between the cyclotron energy and source luminosity , as seen in at least one bright transient x - ray pulsar v0332 + 53 @xcite . presence of such an anti - correlation in another similar source , 4u0115 + 63 , is still under debates . the recent outburst from bexrp s was only the third transient event when the source came into the view of x - ray instruments . thanks to the _ nustar _ wide energy coverage and high sensitivity we were able to discover a cyclotron absorption line with centroid energy @xmath81 kev , corresponding to the neutron star magnetic field strength @xmath2 g typical for the known x - ray pulsars @xcite . the presence of the cyclotron line in the source spectrum is also supported by the behaviour of the pulse profile and pulsed fraction with the energy . the pulse - phase resolved spectroscopy revealed significant variations of the cyclotron line parameters over the pulse . particularly , the line centroid energy is anti - correlating with the intensity , whereas the line depth shows a correlation . thanks to the _ fermi_/gbm data we were able to substantially improve the orbital parameters of the system . the intrinsic spin period value and its evolution observed by _ fermi_/gbm and _ nustar _ during the current outburst are similar to those measured by the _ rxte _ observatory during the previous outburst @xcite . virtual constancy of the period since the source discovery in 1975 and a significant spin - up observed during both outbursts ( up to @xmath82 s s@xmath5 ) implies action of deceleration torques between outbursts . taking into account the last measured value of the pulse period during the previous outburst @xcite and its first measured value during the current outburst ( fig.[fig : lc ] ) we can estimate roughly a spin - down between 2007 and 2015 outbursts as @xmath83 s s@xmath5 . this value is more than an order of magnitude lower than the spin - up during the outbursts and comparable with the spin - down observed between outbursts in other bexrbs ( see , e.g. , * ? ? ? the knowledge of the magnetic field and spin - up rate allowed us to estimate the distance to the system @xmath6 kpc using the standard accretion torque models . so large distance agrees well with the fact that the optical counterpart was not directly detected so far . this research has made use of data obtained with _ nustar _ , a project led by caltech , funded by nasa and managed by nasa / jpl , and has utilized the nustardas software package , jointly developed by the asdc ( italy ) and caltech ( usa ) . this research has made also by using _ data provided by the chandra x - ray center . the publication makes use of software provided by the chandra x - ray center ( cxc ) in the application package ciao . st , al and sm acknowledge support from russian science foundation ( grant 14 - 12 - 01287 ) . jp thanks the academy of finland for financial support ( grant 268740 ) . partial support comes from the eu cost action mp1304 `` newcompstar '' .

we report results of a spectral and timing analysis of the poorly studied transient x - ray pulsar s using data collected with the _ nustar _ and _ chandra _ observatories and the _ fermi_/gbm instrument during an outburst in 2015 . properties of the source at high energies ( @xmath0 kev ) are studied for the first time and the sky position had been essentially improved . the source broadband spectrum has a quite complicated shape and can be reasonably described by a composite model with two continuum components a black body emission with the temperature about 1 kev at low energies and a power law with an exponential cutoff at high energies . additionally an absorption feature at @xmath1 kev is discovered both in phase - averaged and phase - resolved spectra and interpreted as the cyclotron resonance scattering feature corresponding to the magnetic field strength of the neutron star @xmath2 g. based on the _ fermi_/gbm data the orbital parameters of the system were substantially improved , that allowed us to determine the spin period of the neutron star @xmath3 s and a local spin - up @xmath4 s s@xmath5 due to the mass accretion during the _ nustar _ observations . assuming accretion from the disk and using standard torque models we have estimated the distance to the system @xmath6 kpc . [ firstpage ] accretion , accretion discs magnetic fields stars : individual : 2s 1553@xmath7542 x - rays : binaries .

introduction observations and data reduction results discussion conclusion acknowledgments

arxiv

a challenge placed before @xmath1cdm hydrodynamic cosmological simulations is to form realistic galaxies while matching such quantities as the stellar - mass to halo - mass relation and average star formation history as a function of halo mass and redshift ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in the simulations , the interplay between stellar feedback processes , originating in the interstellar medium ( ism ) , and filamentary and galaxy accretion , originating in the intergalactic medium ( igm ) , give rise to extended metal - enriched gaseous structures surrounding galaxies , i.e. , the circumgalactic medium ( cgm ) . the exact role of the cgm in governing the observed properties of galaxies is not yet well established ; however , it is quite possible that the cgm is a highly regulating component of galaxies and , if better understood , could provide powerful insights into global galaxy relations ( cf . * ; * ? ? ? * ) thus , it is an additional challenge for successful simulations to also statistically match the observed distributions of gas density , temperature , kinematics , and chemical and ionization conditions of the cgm . in general , there are two main approaches to simulating galaxies and the cgm . the first uses smoothed particles hydrodynamics ( sph ) , which discretizes gas mass into particles , and the second uses adaptive mesh refinement ( amr ) , which spatially discretizes the gas using grid cells . sph simulations generally trade off high spatial and mass resolution in favor of modeling the hydrodynamics in thousands of galaxies in a simulation box . a great strength of sph simulations is that the statistical characteristics of the cgm can be studied over a wide range of halo mass and cosmological environment . the draw back is that the detailed physics of stellar formation and feedback are generally in the form of scaling relations , such as constant velocity winds , or winds with launch velocities proportional to the stellar velocity dispersion , which scales with the gravitational potential ( e.g. * ? ? ? * ; * ? ? ? these relations do not directly address the underlying physics of feedback . a strength of amr simulations is that the star formation and feedback models directly target the underlying physical processes of star formation and feedback . though the processes are still unresolved , the physics is highly detailed and the simulations can be employed to study star formation at the scale of molecular cloud physics and stellar feedback at the scale of radiation pressure physics and photo - heating physics @xcite . with such detail , the draw back is that only a few galaxies can be simulated at a time . however , with amr , greater insight into the complex interplay between star formation and feedback that regulates the cgm , and therefore galaxy formation and evolution , can be gleaned . observationally , the primary method for studying the gas properties of the cgm is the technique of quasar absorption lines . the most commonly studied cgm absorption lines are @xmath2 ( ) and @xmath3 ( ) ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the lithium isosequence zero - volt resonant doublets and ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and the sodium isosequence zero - volt resonant doublet ( e.g. , * ? ? ? * ; * ? ? ? * and references therein ) . virtually all of the physical conditions we have learned about the cgm are derived from absorption line analysis . as such , studying the cgm properties of simulated galaxies and quantitatively comparing these properties to those derived from observations is clearly best accomplished using absorption line measurement and analysis methods ( also see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . absorption line analysis of simulations will not only directly increase our knowledge of the cgm and its role in galaxy evolution , but will provide a profound insight into how effective and accurate commonly applied observational analysis methods recover the `` true '' gas properties ( densities , temperatures , chemical and ionization conditions , and kinematics ) . to accomplish these objectives , we must quantitatively compare the inferred gas properties derived from observational techniques applied to synthetic absorption lines in simulations directly to the physical properties of the simulation gas from which the absorption arises . for example , ionization models ( e.g. , cloudy , * ? ? ? * ; * ? ? ? * ) are used to determine the gas density , @xmath4 , and metallicity , as well as the ionization parameter , @xmath5 , where @xmath6 is the number density of hydrogen ionizing photons . many assumptions are invoked in applying these models , such as single - phase absorbing clouds ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , or simple two - phase absorbing clouds ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? how appropriate are these assumptions ? quantifying the relationships between absorption profiles , the inferred properties of the absorbing gas from the absorption line data , and the true properties of the gas giving rise to the absorption in the simulations can provide insight to this question . a second example is the inferred kinematics , which is either assumed to be provided by the profile of the column density with velocity via the apparent optical depth ( aod ) column density method ( e.g. , * ? ? ? * ) , or reflected in voigt profile ( vp ) decomposition of the absorption profile ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . however , inferring the properties of the gas using either the aod profile method and vp decomposition implicitly relies on the assumption that gas at a given velocity arises from a single unique spatial location along the line of sight . furthermore , vp decomposition models the data as isothermal clouds , each having a different peculiar velocity . using simulations , the relative spatial locations of the absorbing gas along the los can be examined to quantify the degree to which absorption with similar ( or aligned ) kinematics arises in the same spatial gas structures . this latter information has important implications for the assumptions applied to the ionization modeling and kinematic analysis of observed data . to be effective , the application of absorption line analysis to simulations should emulate observational work as closely as possible so that the selection methods , spectral resolution , sensitivity limitations , and analysis techniques of various absorption - line surveys can be accurately duplicated . in general , surveys target a limited number of absorption line transitions , each which probe a relatively narrow range of gas phase , i.e. , they arise in gas with favorable density , temperature , ionization conditions , and metallicities . this will also be true for absorption lines in synthetic spectra obtained by passing sightlines through the cgm in hydrodynamic simulations . for example , along a single sightline , some gas structures may contribute to and/or absorption , whereas other unique gas structures may contribute to absorption , but not to or absorption . but , importantly , all our knowledge of the cgm from absorption line analysis is filtered through the instruments that modify the data . which instrument and telescope facility is used to observe a given absorption line depends upon the rest wavelength of the transition and the redshift of the absorbing structure , and this strongly governs the design , sensitivity , and data quality for observational surveys . since which absorption lines get observed depend upon which facilities capture the wavelength range of the redshifted transition , this governs which gas phases can be probed and to what level of sensitivity they can be probed . furthermore , the range of physical gas properties that contribute to detectable absorption will differ as a function of the detection thresholds of the spectra , which depends on the signal - to - noise ratio , resolution , and pixelization of the data . shallower detection thresholds result in probing gas with higher column densities , which presumably means that higher density , higher metallicity gas , or favorable ionization conditions are preferentially being studied . this holds true for both real - world observations and synthetic spectra from simulations , provided that the synthetic spectra are carefully designed to emulate real - world spectra . with the aforementioned considerations in mind , in this paper we describe the methods we developed to generate `` realistic '' synthetic spectra through simulated galaxies in amr cosmological simulations and to analyze the spectra , all in a manner that emulates real - world observations . in section [ sec : sims ] we describe the simulated galaxy we employ for this study . in section [ sec : ionmodel ] , we briefly review the ionization model @xcite used to obtain the gas ionization conditions . we detail our methods for absorption line analysis in section [ sec : simabslines ] and discuss selected preliminary findings in section [ sec : discuss ] . in section [ sec : conclude ] , we summarize our work . we employ the @xmath7-body plus gasdynamics amr code hydroart @xcite . the simulations were run using the `` zoom - in '' technique @xcite , which allows us to resolve the formation of a single galaxy consistently in its full cosmological context . the high - resolution region around the galaxy is typically @xmath82 mpc across and the hydrodynamics is resolved with @xmath9 grid cells , with a minimum cell size of roughly @xmath10 pc at @xmath11 ( the proper size of the cells decrease with increasing redshift ) . physical processes implemented in the code include star formation , stellar feedback , type ii and ia metal enrichment , thermal and radiation pressure , and metallicity - dependent cooling and heating . gas is self - shielded , advects metals , is heated by a homogeneous ultraviolet background , and can cool to 300 k due to metal and molecular line cooling . gas flows , shock fronts , and metal disbursement follow self - consistently from this physics . for star formation , we use observations of molecular clouds @xcite to guide our model . the star particles form in the dense , cold molecular phase ( @xmath12 , @xmath13 k ) . the star formation rate is proportional to the gas density divided by the free fall time of the molecular cloud . we use an observationally motivated low ( 2% percent ) efficiency per free fall time for converting gas into stars . compared to previous methods @xcite , this treatment results in a greater number of individual star particles , but with smaller masses , between @xmath14 and @xmath15 m@xmath16 . runaway young , hot stars are included according to @xcite by providing one third of the newly - formed star particles with a random velocity kick . for the feedback model , mechanical energy from stellar winds and sn type ii is assumed to thermalize and is injected into the gas as thermal energy around young stars following the rates predicted by starburst99 @xcite for the chabrier imf @xcite . we also incorporated photoionization heating , radiation pressure , and shocked stellar winds from massive stars ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * for details ) . these feedback processes disrupt the cold molecular gas and regulate the formation of stars @xcite . matching observations of regions @xcite , we treat photoheating by adding a non - thermal pressure ( @xmath17 k @xmath18 ) to the gas surrounding young stars . this pressure decreases rapidly in order to reproduce the declining density within a growing region . for radiation pressure , which we treat similarly to @xcite , @xcite , and @xcite , we include momentum from the radiation field from young massive , which is coupled to the gas and dust through scattering and absorption . absorption of uv photons scales as @xmath19 . scattering due to trapped ir photon scales as @xmath20 . these values are adopted from the suggested values of @xcite based upon @xmath21 scaling with column density @xcite . this star formation and feedback model was shown to reproduce many properties of low - mass galaxies at @xmath22 without fine tuning ( see * ? ? ? * ) ; the stellar to halo mass ratio , cold gas fraction , baryon content , star formation history , rotation curves and morphologies of the simulated galaxies agree remarkably well with observations . the heating and cooling balance of the gas is determined using heating and cooling functions obtained from cloudy @xcite and incorporates equilibrium photoionization and collisional ionization . these include metal and molecular line cooling , and a uniform @xcite ionizing background with self - shielding of high column density gas , and stellar radiation for gas in the vicinity of stars . for additional details see @xcite . at each grid cell , the hydroart code follows the evolution of the density , temperature , velocity , and metal mass fraction . the metals produced in type ii and ia supernovae are followed separately and are self - consistently advected with the gas flow . to compute the relative abundances of the ions in the gas , we employ an equilibrium ionization model as a post - processing step . we briefly describe the ionization model in section [ sec : ionmodel ] ; full details are given in @xcite . for the analysis in this paper , we adopt the simulation of the low - mass ( dwarf ) galaxy designated dwall_8 presented in @xcite . for this simulation , the host dark matter halo evolved into an isolated dwarf galaxy with a virial mass @xmath23 m@xmath16 , a virial radius @xmath24 kpc , a stellar mass @xmath25 m@xmath16 , and a maximum circular velocity @xmath26 at @xmath11 . as described above , the dwall_8 model incorporates feedback in which the optical depth of the gas and dust to ir photons is small and the pressure of the gas due to photo - heating in regions is @xmath27 k @xmath18 . these assumptions are within the range favored by observations of star forming regions . the dark matter particle resolution is @xmath28 m@xmath16 . we study the galaxy when it is at redshift @xmath0 . at this redshift , the galaxy has virial mass @xmath29 m@xmath16 , virial radius @xmath30 kpc , and stellar mass @xmath25 m@xmath16 . in the galaxy interstellar medium , the minimum gas cell proper size is @xmath31 pc . in the circumgalactic medium , the proper cell sizes range from @xmath31 pc to @xmath32 pc . within two virial radii , there are @xmath33 gas cells . in figure [ fig : simshots ] , we present thin slices through the gas distribution centered on the galaxy at @xmath0 , showing ( a ) gas density , ( b ) temperature , ( c ) metallicity , and ( d ) the @xmath34-component of velocity . the scale is indicated in physical kpc at the top of each panel and the dashed circles show the virial radius . we developed an ionization model specifically designed for post - processing application with the hydroart cosmological simulations . the details of the code , including comparisons with the industry standard ionization code cloudy @xcite are presented in @xcite . the code has also been successfully applied for observational work ( e.g. , * ? ? ? * ; * ? ? ? , we briefly summarize the ionization modeling of the gas in the simulations . for each individual gas cell in the simulation box , the ionization model calculates the equilibrium ionization state of the gas . the ionization model treats photoionization , auger ionization , direct collisional ionization , excitation auto - ionization processes , charge exchange ionization , radiative recombination , dielectronic recombination , and charge exchange recombination . if desired , the effects of each of these processes can be isolated by turning the process `` off '' or `` on '' . the output of the ionization code is a simulation box containing the cell equilibrium electron densities , and the number densities of all ions . we thus have the ability to study the spatial distribution of the ions . if desired , the photoionization rates and recombination and collisional ionization rate coefficients in a given cell can be recorded for specified ions . three cell properties constrain the gas physics ( 1 ) the total hydrogen density of the cell , @xmath4 , ( 2 ) the equilibrium temperature of the cell , @xmath35 , and ( 3 ) the mass fractions of the atomic species in the cell , which is given as the type ii and ia supernovae yields from the stellar feedback and metal transport . metals up to and including zinc are incorporated into the ionization model . the fourth quantity that governs the gas physics is the spectral energy distribution of ionizing photons . the ionization model accounts for the ultraviolet background ( uvb ) , and/or radiation from the stellar particles ( populations ) in the simulated galaxy . a limitation of the current version of the ionization model is that only optically thin gas can be treated because we presently do not treat radiative transfer through the grid cells . as discussed in @xcite , we employ the definition for `` optically thin '' to mean that the optical depth is less than unity at the hydrogen and helium ionization edges , which dominate modification of the ionizing sed . we showed that the cloud models are constrained to have upper limits on the column densities of @xmath36 @xmath37 for neutral hydrogen , @xmath38 @xmath37 for neutral helium , and @xmath39 @xmath37 for singly ionized helium . via the relationship @xmath40 , where @xmath41 is the column density of atomic species x in ionization stage @xmath42 , @xmath43 is the ionization fraction of ion @xmath44 , and @xmath45 is the number density of species x , the upper limits on column density translate into upper limits on the cell size , @xmath46 , for validity of the optically thin constraint . in @xcite , we showed that the cell size upper limits can be written , @xmath47 l_{\rm max}({\hei } ) & \simeq & 9 \cdot ( { 0.01}/{f\subhei } ) ( { 0.01}/{n\subh } ) & { \rm kpc } \\[3pt ] l_{\rm max}({\heii } ) & \simeq & 20 \cdot ( { 0.01}/{f\subheii } ) ( { 0.01}/{n\subh } ) & \ , , \end{array } \label{eq : maxcell}\ ] ] assuming a relative abundance of helium to hydrogen of 10% . the proper minimum cell size for the simulations at @xmath0 ( the redshift of the galaxy we study in this work ) is @xmath48 kpc . from eq . [ eq : maxcell ] , we see that only in cases where the product of the ionization fraction and the hydrogen number density exceed @xmath49 does the maximum cell size decrease from the fiducial values of 0.5 , 9 , and 20 kpc for the respective ionization edges . in @xcite , we further showed that the maximum allowed cell size is below the 30 pc resolution minimum for @xmath50 when @xmath51 , and for @xmath52 when @xmath53 . for @xmath54 , the maximum allowed cell size is never less than 30 pc . as such , our ionization model is currently not entirely valid for `` cold '' cells ( @xmath55 ) with densities @xmath56 nor for `` warm / hot '' cells ( @xmath57 ) with densities @xmath58 . the former cells are found to reside almost exclusively in the ism of the simulated dwarf galaxies , and the latter cells are virtually non - existent because the warm / hot gas is associated with densities in the range @xmath59 . direct comparisons between our ionization model and cloudy 13.03 @xcite showed that the two codes are in good agreement @xcite . the ionization fractions of neutral hydrogen are virtually identical over the density and temperature ranges @xmath60 and @xmath61 . the helium ionization fractions are also in full agreement except for a factor of 2 - 3 overestimate for neutral helium for @xmath62 . comparison of the ionization corrections , @xmath63 , were in agreement within @xmath64 for the commonly observed ions mg@xmath65 , c@xmath66 , and o@xmath67 over the majority of the @xmath68@xmath35 parameter space . most importantly , as we discuss @xcite , the region of agreement is always within @xmath69 where the ionization fractions of the target ions are the largest , meaning that dominant ionization stages where absorption will be most affected are in agreement with cloudy . given that @xmath70 , we argued that since typical uncertainties in observed column density measurements range between @xmath64 to @xmath71 in the logarithm , the difference in the ionization corrections between the two ionization models would be consistent with typical measurement errors in @xmath72 obtained from absorption line analysis . the redshift of the simulation box is denoted @xmath73 . the position of the center of the galaxy in the box , @xmath74 , is obtained by locating the center of mass of the stellar particles surrounding the minimum of the gravitational potential . the peculiar velocity of the galaxy in the box is the velocity of the center of mass of the stellar particles , and is denoted @xmath75 . first , the ionization model is run on the simulation box , from which the number densities of all ions are determined for each gas cell . to generate `` observed '' quasar spectra ( see section [ sec : mkspectra ] ) , a line of sight ( los ) is passed through the simulation box from the vantage point of an `` observer '' viewing the galaxy on the plane of the sky . each los is defined by ( 1 ) an impact parameter , @xmath76 , ( 2 ) a position angle on the plane of the sky , @xmath77 , which ranges from @xmath78 to @xmath79 , and ( 3 ) the inclination , @xmath80 , of the simulated galaxy with respect to the los direction . the orientation of the galaxy in the simulation box is defined by the angular momentum vector of the star particles . once @xmath76 , @xmath77 , and @xmath80 are specified , we determine the direction cosines @xmath81 of the los with respect to the box coordinate system . for an individual simulated galaxy , we can create and study an arbitrary number of randomly oriented or parallel los . this formalism allows the opportunity study the relationship between galaxy orientation and absorption line properties ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the position along the los for cell @xmath80 is @xmath82 ^{1/2 } , \ ] ] where @xmath83 , measured in kiloparsecs , is the position of cell @xmath80 intercepted by the los . the plane of the sky is defined as the plane perpendicular to the los intersecting @xmath84 . since the los unit vector is @xmath85 , the cell los velocity with respect to the simulation box is @xmath86 , where @xmath87 is the cell velocity vector , and the observed redshift of the cell is @xmath88 . the los systemic velocity of the galaxy is @xmath89 , and the `` observed '' redshift of the galaxy is @xmath90 . the column density of ion @xmath91 for each cell along the los is @xmath92 , the product of the number density of the ion in the cell and the pathlength of the los through the cell , @xmath93 , which is the true length of the los vector through the cell computed as the pathlength from the entry to the exit points on the cell walls . for this investigation , we ran 1000 los through the simulated galaxy from the perspective of a face - on orientation . the los range from @xmath94 kpc , corresponding to @xmath95 . in figure [ fig : los ] ( lower set of panels ) , we illustrate selected cell physical quantities as a function of the los position , @xmath96 , over the range @xmath97 kpc ( corresponding to @xmath98 ) for los 0092 through the simulated galaxy shown in figure [ fig : simshots ] . this los is at @xmath99 kpc , which corresponds to @xmath100 . we selected this particular los for illustration purposes because it gives rise to both low ionization and high ionization absorption , and therefore provides insights to both gas phases . overall , los 0092 is typical of the many los that probe the simulated galaxy in the range @xmath101 . from top to bottom in figure [ fig : los ] , we present the gas cell hydrogen number density , @xmath102 , temperature , @xmath103 , gas - phase metallicity , where @xmath104 is the mass fraction of all metals and @xmath105 is the mass fraction of hydrogen . ] , @xmath106 , line of sight velocity , @xmath107 , and ion column density , @xmath108 . the curves are presented as histograms that show the gas cell pathlengths along the los . recall that @xmath109 is the plane of the sky ( defined by the galaxy center ) . multiple panels of each property are repeated from left to right in figure [ fig : los ] in order to illustrate which cells contribute to detected absorption for the , , , and transitions ( we discuss how these cells are determined in section [ sec : isolatingcells ] ) . the synthetic absorption lines are shown in the top set of panels ( generation of the synthetic spectra is described in section [ sec : mkspectra ] ) . these absorbing cells are marked with @xmath110 points overplotted on the histograms ( see section [ sec : kin - space ] for further discussion ) . for this los , the cgm of the simulated dwarf has density range @xmath111 , temperature range @xmath112 , and metallicity range @xmath113 . positive temperature spikes of @xmath114 dex occur in regions of @xmath1151 dex reductions in density . from the behavior of the los velocity , there is a clear outflow for @xmath116 kpc , with a velocity inversion in the range @xmath117 kpc , which is characterized by a drop in density and increase in temperature . note the @xmath115 dex drop in metallicity at @xmath118 kpc that proceeds the velocity inversion . visual inspection of density , temperature , metallicity , and velocity slices of the simulated galaxy shown in figure [ fig : simshots ] clearly show that the cgm of this galaxy is as highly variable , dynamic , and complex as this single los example indicates . the computation of the cell column densities ( bottom panels of figure [ fig : los ] ) rely on the ionization modeling . based upon our criteria for optically thin gas as determined by the optical depth at the hydrogen and helium ionization edges , we find that even in the cells where the h@xmath119 and mg@xmath120 column densities are highly peaked , only one gas cell along the los barely violates the optically thin criteria . note that the c@xmath66 and o@xmath67 column densities have a relatively flat distribution along the los out to @xmath121 kpc . the simulated absorption spectra are generated with our code specsynth with the assumption that each cell , @xmath80 , contributes to the optical depth as if it were an isothermal `` cloud '' . for a given transition from ion @xmath91 , the optical depth as a function of observed wavelength is computed from @xmath122 where the physical constants have their usual meaning , @xmath123 is the rest - frame wavelength of the transition , @xmath124 is the transition oscillator strength , and @xmath125 is the column density of the ion . the term @xmath126 is the doppler width , where @xmath127 is the mass of the ionic species . the voigt function , @xmath128 , is computed as the real part of the complex probability function using the code cpf12 @xcite with unitless parameters @xmath129 where the factor @xmath130 ensures that the wavelength difference , @xmath131 , is co moving corrected , and where @xmath132 is the redshifted central wavelength of the absorbing cell . the resulting normalized counts in the spectrum prior to being recorded by an instrument are obtained by @xmath133 \ , , \ ] ] where @xmath134 is the number of cells along the los . the spectrum is then `` passed through '' an instrument . the choice of instrument would be dictated by which ions and redshifts are being studied so that the spectra can be directly comparable to observational data . for example , lyman series lines and the doublet at low redshifts , i.e. , @xmath135 , are observed in cos g130 m spectra , whereas the doublet at @xmath136 is studied in keck / lris , keck / hires and/or vlt / uves spectra . having chosen an instrument , we first convolve @xmath137 with the instrument spread function ( isf ) , @xmath138 , yielding the instrument convolved normalized spectrum , @xmath139 the convolved normalized spectrum is then sampled with the pixelization @xmath140 , of the chosen instrument . finally , gaussian deviate noise is added on a pixel by pixel basis assuming a fixed signal - to - noise ratio per pixel , @xmath141 , and adopting the instrumental read noise , @xmath142 . the read noise is applied in units of electrons , and not in digital number ( which is smaller by the readout amplifier gain factor ) because properly modeling the poisson statistics requires electron counts . the adopted @xmath141 ideally should be selected to reflect the average @xmath141 of the observed spectra comprising surveys targeting the transition being studied in the simulations . in this way , the detection sensitivities of observational surveys are emulated for a direct comparison between spectra from simulations and real world spectra . the normalized uncertainty spectrum due to poisson statistics is given by @xmath143 ^{1/2 } } { { \cal i}_c } \ , , \ ] ] where @xmath144\ ] ] approximates the continuum counts for the desired @xmath141 @xcite . to account for the additional uncertainty due to placement of the continuum fit ( as required for observational spectra ) , we adopt an approximation that the uncertainty due to continuum placement is proportional to the poissonian uncertainty in the continuum , i.e. , @xmath145 , where @xmath146 ( see * ? ? ? the final normalized uncertainty spectrum is then @xmath147 \!\!\!\ ! & = & \!\!\!\ ! \displaystyle \frac{\displaystyle \sqrt { { \cal i}_c { \cal i}(\lambda ) + { \rm rn}^2 + h^2\left [ { \cal i}_c + { \rm rn}^2 \right ] } } { \displaystyle { \cal i}_c } \ , , \end{array}\ ] ] to determine the proportionality constant , @xmath148 , we undertook a blind experiment . we generated 100 synthetic spectra from several common instruments with a range of @xmath141 ( from 5 to 50 ) and added continuum shapes ranging from from 2nd order to 7th order legendre polynomials . each spectrum had 2048 pixels . we then interactively continuum fit the spectra and computed the mean standard deviation ( over all pixels ) between the noiseless input continua and the blindly fitted smooth continua . this yielded @xmath149 , which , in continuum regions , corresponds to an 8% increase for the final uncertainty relative to the poisson - only uncertainty . the final pixelated normalized spectrum with noise is computed from @xmath150 where @xmath151 is a random unit gaussian deviate generated on a per pixel basis . for each los ( and there can be an arbitrary number per simulated galaxy ) , an individual synthetic spectrum and uncertainty spectrum is generated for each ion / transition that we aim to study . the observed wavelength range of a given synthetic spectrum is set such that the spectrum covers @xmath152 with respect to the system velocity of the simulated galaxy . however , this range can be adjusted and redefined with ease . in figure [ fig : profiles ] , we present example synthetic spectra for four los ( 0027 , 0092 , 0150 , and 0152 , from left to right in order of increasing impact parameter ) through the the simulated galaxy shown in figure [ fig : simshots ] . the impact parameter range from @xmath153 to 30 kpc ( @xmath154 to 0.5 ) . from top to bottom , we show the the commonly studied @xmath155 , @xmath156 , @xmath157 , @xmath158 , @xmath159 , @xmath160 transitions , and the transition . for this example , we adopt @xmath161 per pixel and show only the `` blue '' member of the doublets . since the simulated galaxy is at @xmath0 , the lines are redshifted into the optical at 4306 so we adopt the hires instrument ( @xmath162 ) for this doublet . we model the hires isf as a unit - area gaussian with @xmath163 , corresponding to 3 pixels per resolution element ( @xmath164 ) . the , and lines are observed in the fuv g160 m grating of cos ( @xmath165 ) , the and lines are observed in the nuv g225 m grating of cos ( @xmath166 ) , and the line is observed with the nuv g185 m grating of cos ( @xmath167 ) . for the cos isfs we employ the on - line lifetime position 2 tabulated theoretical line spread functions @xcite . we determine the isf at a given observed wavelength using cubic spline interpolation . los 0092 ( second from left ) is the los illustrated in figure [ fig : los ] . note that as impact parameter is increased the low ionization absorption diminishes and the and absorption develops somewhat greater kinematic complexity . we further discuss the nature of the absorption for los 0092 in section [ sec : discuss ] . in order to emulate observational analysis common to quasar absorption line studies , we adopt objective methods used ( or that should be used ) for observed spectra . we use a fully automated version of our graphical interactive code sysanal , which we have applied for several other works ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? after all measured quantities are calculated , the final step of the process is the generation of a single table for each ion that contains the absorption properties for all los . these tables can then be analyzed to study the cgm properties of the simulated galaxies . to locate statistically significant absorption features in the synthetic spectra , we employ the methods described in @xcite and @xcite , which are derived from @xcite . first , the spectrum is converted to an equivalent width spectrum , @xmath168 , and the uncertainty spectrum is converted to an equivalent width uncertainty spectrum , @xmath169 . the equivalent width spectrum provides the observed ( not rest - frame ) equivalent width per resolution element for an unresolved absorption feature as a function of observed wavelength and the equivalent width uncertainty spectrum provides the @xmath170 observed equivalent width detection threshold as a function of observed wavelength . the computation of @xmath168 and @xmath169 requires convolving the normalized and pixelized isf with the flux decrements in each pixel of the spectrum ( see section 3.1.1 and equations 1 and 2 of * ? ? ? an absorption feature is objectively defined in spectral regions where @xmath171 , where we adopt @xmath172 for singlet absorption lines . for doublets , we adopt @xmath172 for the blue ( higher oscillator strength ) member of a doublet and @xmath173 for the red member of a doublet . for a doublet to be `` detected '' , both members must satisfy the above detection criteria . however , to facilitate direct comparison with observational data from various surveys that may adopt different criteria , the detection criteria can easily be tailored to those of any survey . the lower and upper wavelength limits over which the absorption feature is defined are taken to be where @xmath174 is first recovered while scanning the spectrum blueward and then redward of the wavelength first satisfying the detection criterion . using this method , several `` sub - features '' , absorption separated by continuum , can be uniquely defined and analyzed ( see figure 1 of * ? ? ? * ) . in figure [ fig : sysanal ] , we illustrate the detection method for objectively defining absorption features . in the upper panels , we present the rest - frame @xmath156 synthetic absorption lines for the cos g185m/1921 grating for stripe b. we vary the @xmath141 values ( left ) @xmath161 , ( center ) @xmath175 , ( right ) @xmath176 per pixel . the blue spectrum provides the @xmath170 uncertainty in the normalized flux . in the lower panels of figure [ fig : sysanal ] , we present the rest - frame equivalent width spectrum , @xmath168 . we also present the @xmath177 and @xmath178 rest - frame equivalent width detection thresholds for unresolved absorption lines as the bracketing blue spectra . note that the @xmath179 detection threshold changes from @xmath180 for @xmath161 to @xmath181 for @xmath176 . for @xmath161 and 15 , several pixels across the profiles are clearly significant at greater than the @xmath179 level . however , for @xmath176 , the absorption feature is on the verge of detection ; had the noise characteristics been different , this feature may not have been formally detected . the green vertical lines mark the wavelength range over which the absorption profiles are defined for the purpose of quantifying the absorption . sysanal works simultaneously on all ion / transitions for a given los . once all synthetic spectra for a los are objectively searched for absorption features , and the spectral ranges of the detected absorption features are determined , the code then computes the observed and rest - frame equivalent widths , doublet ratios ( when applicable ) , the flux decrement weighted velocity centers , velocity widths , and velocity asymmetries . formal uncertainties in these quantities are also computed . in the case of nondetections , the @xmath182 upper limits are computed for the observed and rest - frame equivalent widths . in the case where multiple sub - features are detected , they are individually measured in addition to the measurement of the `` total system '' quantities ( all sub - features treated as a single absorption feature ) . the mathematical computation of these quantities , originally based on the work of @xcite , is given in @xcite and in the appendix of @xcite . additional details are provided in sections 3.13.4 of @xcite . sysanal also creates an apparent optical depth ( aod ) column density spectrum and uncertainty aod spectrum for each ion / transition @xcite . from these spectra , we determine the `` best '' aod column density spectrum for a given ion . for ions with multiple transitions , such as and the common metal - line doublets , we employ the following procedure . for a given velocity pixel , we take the optimal weighted mean aod column density of the transitions if ( 1 ) more than one transition for a given ion has measured values ( not lower or upper limits ) of the aod column density , and ( 2 ) unresolved saturation ( see * ? ? ? * ) is not present . the uncertainties are propagated in the standard fashion . if all but one transition yields a limit , the transitions with measured values are adopted . in the case of unresolved saturation , the adopted aod column density is taken to be that with the highest oscillator strength . in the case of upper limits in all transitions , the highest oscillator strength transition is adopted , and in the case of lower limits in all transitions , the lowest oscillator strength transition is adopted . this process yields the aod column density profile for a given ion , @xmath183 [ atoms @xmath37 ( ) @xmath184 , from which we compute the integrated aod column densities , @xmath185 for each ion , @xmath186 where the integration is over the spectral region over which the absorption profile is detected ( see the vertical green lines in figure [ fig : sysanal ] ) . if saturation persists over a minimum of three adjacent pixels , i.e. , @xmath187 is a lower limit over an extended velocity range approaching a resolution element , we quote a @xmath182 lower limit for @xmath188 . we discuss comparison of the aod spectra to the simulation gas properties in section [ sec : kin - space ] . the final spectral analysis step is vp decomposition , which yields the column densities , @xmath7 , doppler @xmath189 parameters , and observed redshifts , @xmath190 , of multiple vp components . to obtain an initial model for each ion , we run autovp on the spectra . redshifted transitions from different ions can fall in wavelength ranges appropriate for different spectrograph / grating combinations ( i.e. , cos , hires , lris , etc . ) , so that different transitions from a single los are likely to be measured with different resolutions . we properly account for the isf appropriate for the instrument with which a given synthetic absorption line was created . due to the different kinematics of the lower and higher ionization gas , we have found that fully automating the vp decomposition of the synthetic spectra to be challenging and we are still developing our approach . currently , we run autovp on the lower ionization profiles and select the profile with the largest number of vp components as the initial kinematic template for the low ionization transitions . we repeat the process for the higher ionization transitions . to obtain the final model , we then run our minimization code minfit @xcite . minfit refines the model using the maximum likelihood modification of the levenberg - marquardt algorithm dnls1 @xcite , which minimizes the sum of the squares of @xmath191 nonlinear functions in @xmath7 variables . we adopt the model with the fewest vp components that are statistically significant at the 97% confidence level by applying an @xmath192-test on the @xmath193 distribution . components with the largest fraction errors , @xmath194 are tested for significance in descending order of @xmath195 . if a component is not significant , it is removed from the model , and the process is repeated until all components are significant . full details of the most up - to - date version of minfit and the fitting process are described in @xcite . the difference for our application is that we adopt a unique vp model for the lower ionization gas and for the higher ionization gas . though not perfect as a description of the complex multiphase gas structures that give rise to the absorption lines , it does provide us a formalism for segregating the absorption into two gas phases . further development is under way . in figure [ fig : vpfit ] , we show an example of vp decomposition for los 0092 . this is the los illustrated in figure [ fig : los ] . for this los , the lower ionization species are taken to be , , , and ( shown as the blue fits ) , whereas the higher ionization species are taken to be , , and ( shown as the red fits ) . the individual vp components are shown as magenta ( except for , where , for clarity , one component is magenta and the second is green ) . the los velocities of the low ionization vp components are tied together and all lower ionization transitions are fitted simultaneously . the same holds for the higher ionization transitions . we assume purely thermal line broadening for each vp component . in this mode , the component temperatures are the actual fitting parameters , from which the @xmath189 parameters are computed from @xmath196 for ion x@xmath197 , where @xmath45 is the mass of ion x. for reference , we list the vp component parameters in table [ tab : vpfit ] . presented are ( 1 ) the ion and transition fitted , ( 2 ) the los velocity , @xmath198 , ( 3 ) the column density , @xmath199 , ( 4 ) the doppler @xmath189 parameter , and ( 5 ) the temperature . upper limits are quoted at the @xmath182 level . we discuss comparison of the vp fit parameters to the simulation gas properties along this los in section [ sec : kin - space ] . lrrrr + & 3.20 & @xmath200 & @xmath201 & @xmath202 + & 20.01 & @xmath203 & @xmath204 & @xmath205 + & 3.20 & @xmath206 & @xmath207 & @xmath208 + @xmath209 & 20.01 & @xmath210 & @xmath211 & @xmath205 + & 3.20 & @xmath212 & @xmath213 & @xmath202 + @xmath156 & 20.01 & @xmath214 & @xmath215 & @xmath205 + & 3.20 & @xmath216 & @xmath217 & @xmath202 + @xmath157 & 20.01 & @xmath218 & @xmath219 & @xmath205 + + & @xmath220 & @xmath221 & @xmath222 & @xmath223 + @xmath158 & 10.69 & @xmath224 & @xmath225 & @xmath226 + & 27.80 & @xmath227 & @xmath228 & @xmath229 + & 38.05 & @xmath230 & @xmath231 & @xmath232 + & @xmath220 & @xmath233 & @xmath222 & @xmath223 + @xmath234 & 10.69 & @xmath235 & @xmath225 & @xmath226 + & 27.80 & @xmath236 & @xmath228 & @xmath229 + & 38.05 & @xmath237 & @xmath231 & @xmath232 + & @xmath220 & @xmath238 & @xmath239 & @xmath239 + @xmath240 & 10.69 & @xmath241 & @xmath242 & @xmath226 + & 27.80 & @xmath243 & @xmath239 & @xmath239 + & 38.05 & @xmath244 & @xmath245 & @xmath232 one of our aims is to develop methods to directly compare the `` true '' properties of the simulated cgm to those inferred from observations . comparisons of the simulated cgm and the observed cgm must account for the gas being probed by the absorption lines . isolating the gas that is responsible for detected absorption allows for _ direct _ comparison between the measured `` observed '' quantities from synthetic absorption line analysis and the physical properties of the gas . as such , it is centrally important to identify which grid cells along a given los are `` detected '' in simulated absorption lines . in order to isolated the cells that contribute to the absorption profile from a given ion , we adopt a differencing technique . for a given ion , the cells with los velocities aligned within the objectively defined velocity range of the absorption profiles are sorted into descending column density order . an identification number for each cell from the simulation box is included in the list . for a given ion , we then iteratively regenerate synthetic spectra for the transition with the largest oscillator strength by progressively omitting one gas cell at a time until the equivalent width of the profile stops changing by a defined percent difference . the spectra for this exercise are appropriately convolved with the instrumental spread function and pixelated according to the selected spectrograph and grating settings ; however , the test spectra are noiseless . after testing various percent differences , we adopted 5% difference . to elaborate , we start with the highest column density cell in the velocity range of the absorption profile , remove the cell , and recompute the profile . if the equivalent width is reduced by more than a 5% difference , @xmath246 , we consider this cell to contribute significantly to the absorption . we then advance to the next highest column density cell , regenerate the profile , and determine if the equivalent width has decreased by more than 5% difference . we repeat the process on successively smaller column density cells until the change in the equivalent width is less than a 5% difference . the result is that we identify all cells that contribute more than a 5% difference to the equivalent width of the profile . cells that reside in the velocity range of the absorption deemed to not contribute to absorption typically have column densities a factor of 50 to 100 below the cell contributing the highest column density . referring back to figure [ fig : los ] , we overplotted colored data points showing the properties of the absorption selected gas as a function of los position , @xmath247 . as can be seen , the and absorption lines arise in many of the same cells within close proximity of @xmath248 . the absorption also arises in these cells ; but the vast majority of the absorbing cells are distributed over @xmath249 kpc , a region over which the gas phases change substantially along the los with a 2 dex variation in @xmath4 and 0.4 dex variation in @xmath35 . note the cells at @xmath250 kpc , where @xmath251 dips below @xmath252 , do not contribute to absorption ( nor absorption ) . many of the cells that give rise to absorption also give rise to absorption , however , the physical extent of absorbing cells along the los is greater than that of the absorbing cells . following the methods described in sections [ sec : mklos ] and [ sec : mkspectra ] , we ran 1000 random los through the simulated galaxy ( face - on orientation ) and generated synthetic absorption line spectra of and , @xmath253 , @xmath156 , @xmath254 and @xmath255 , , , and transitions . the synthetic spectra have @xmath161 and the characteristics of the hires instrument and the appropriate cos nuv or fuv high resolution gratings for a @xmath256 absorption line system . the @xmath179 rest - frame detection threshold of these spectra is @xmath257 ( see figure [ fig : sysanal ] ) . the impact parameters of the los cover the range @xmath95 , corresponding to random sky coverage within a projected separation from the galaxy of @xmath258 kpc . we then analyzed the absorption line spectra , measuring their equivalent widths , velocity widths , and column densities , etc . , as described in section [ sec : sysanal ] . finally , we determined which cells contributed to absorption following the methods described in section [ sec : isolatingcells ] . in figure [ fig : phases ] , we plot the phase diagrams ( hydrogen density , @xmath4 , versus temperature , @xmath35 ) showing the relative number of cells giving rise to detected absorption in the synthetic spectra for the four ions , h@xmath119 ( ) , mg@xmath120 ( ) , c@xmath66 ( ) , and o@xmath67 ( ) . for these phase diagrams , we include only cells that partake in absorption lines having equivalent widths greater than @xmath259 . this threshold is typical of the sensitivity threshold of the cos - halos and cos - dwarfs surveys @xcite , though some of the sightlines in the surveys have deeper detection thresholds . for , , and , the absorbing cells with phases in the range @xmath260 and @xmath261 are located in the galaxy ism ( roughly within @xmath262 ) ; the remaining cells are located in the cgm . there is no absorption from the ism , only from the cgm . we note that the phase properties of many of the absorbing cells in the ism do not satisfy our optically thin criteria and should be viewed with caution , whereas all cells in the cgm do meet the criteria . for this simulated dwarf galaxy , we see that the majority of the cgm , as selected by absorption with greater than 0.1 , is characterized by temperatures in the range @xmath263 ; the absorbing gas is primarily what we might call the `` cool / warm '' cgm . interestingly , the gas that gives rise to absorption also resides in this temperature range . it would seem that the cgm of this dwarf galaxy is primarily photoionized gas ( in section [ sec : ion - eq ] , we quantify this and also discuss the appropriateness of the assumption of equilibrium ionization modeling ) . a minority of the absorbing cells have @xmath62 , and it is possible that these cells are dominated by collisional ionization processes . the densities of the absorbing gas reside in the range @xmath264 and the temperatures are confined to a narrow range at @xmath265 . note that a substantial fraction of the absorbing cells also reside in this temperature range . however , the majority of the and absorbing gas primarily has lower densities in the range @xmath266 . the relative population of absorbing cells in phase space is such that and absorption is selecting out significantly more cells than and absorption . to the degree to which the cgm of this simulated dwarf galaxy reflects that of real - world dwarf galaxies , this would immediately suggest the observed covering fraction for and absorption in dwarfs should be substantially higher than the covering fraction for and absorption . indeed , the covering fraction of absorption for @xmath267 galaxies in the cos - dwarfs sample is greater than 40% for @xmath268 for a detection threshold of @xmath269 @xcite . we defer study of how the covering fractions and the equivalent width and column density impact parameter distributions respond to different feedback recipes for future work ( vander vliet 2014a , in preparation ) . the absorbing gas phase distributions for this simulated dwarf galaxy show some similarities and some differences with those obtained by @xcite , who present , , and absorption phase diagrams for a @xmath270 simulated galaxy with a halo mass of @xmath271 m@xmath16 . their simulations are performed with sph , whereas we have used amr simulations . for and , our absorbing gas phase distributions , obtained for 1000 los with @xmath272 kpc , are qualitatively consistent with the phase distributions for the los with @xmath273 kpc and @xmath274 mpc from @xcite . this is likely because the overdensity of the cgm within @xmath273 kpc in a dwarf galaxy halo is similar to that of a @xmath271 m@xmath16 halo at @xmath275 kpc . for , the temperature distribution for our dwarf is similar to that found by @xcite , however , the distribution of @xmath4 in the cgm of our dwarf galaxy peaks at @xmath276 @xmath37 , whereas , the peak found by @xcite is @xmath277 @xmath37 for @xmath275 kpc , and is @xmath278 @xmath37 for @xmath279 kpc . we can not compare because @xcite did not present a @xmath4@xmath35 phase diagram for this ion . here , we present insights into the application of observational analysis techniques by examining the kinematic and spatial relationship between the absorption profiles and the cgm gas giving rise to absorption in simulations . for our discussion , we again focus on the , , , and absorption for los 0092 . of foremost interest ( also see figure [ fig : los ] ) is that , whereas the absorption - selected cells are coincident with the absorbing cells , these cells represent 15% of the selected cells and none of the cells selected by absorption . this non - coincidence of and absorption - selected gas would not be perceptible in real - world observations . since some of the absorption has los velocities coincident with the absorption , it is likely that when employing common observational analysis methods some fraction of the column density would be attributed to the absorbing gas phase ( what fraction depends upon a somewhat subjective educated estimate or `` artfully '' modeling of the absorption , such as vp decomposition ) . therefore , it is almost assured that the metallicity of the absorbing phase would then be underestimated in observational studies . this would suggest that there may be systematic bias in the metallicity determinations derived from observed systems that exhibit low - ionization absorption conditions , since the column density might arise only within the low - ionization phase . note that , to a large degree , the same argument holds for the absorption , for which the majority of the absorbing cells are not selected by absorption . in their sph simulations , @xcite also find that and absorption does not trace the same baryons , even though they are both present in the cgm . they also conclude that , if true , ths could result in underestimatations of the true metallicity , since @xmath280 . it is possible that the lack of coincident and absorbing gas could be a resolution effect in amr simulations ( and sph simulations ) . in low density regions , where the grid - cell sizes are larger , thermal and dynamic instabilities leading to condensations of cooler higher density gas can not be resolved . thus , multiphase structures that might form embedded in the hotter lower density regions of the simulated cgm would be suppressed . such structures may have been detected observationally @xcite . and absorption arise exclusively in cells with lengths of @xmath281 kpc , where as and arise in cells with lengths ranging from 311 kpc and 645 kpc , respectively . even in these extended cells , the ionization fraction of is so small that the path length can not compensate to produce detectable absorption . we note , that even if condensations were resolved and successfully modeled in the simulations , they would still comprise a separate phase from the absorbing gas and the and absorbing cells would not be coincident . thus , we caution that absorption may rarely be associated with the same gas structures giving rise to absorption . in figure [ fig : aod - times](a ) we re - plot the absorption profiles . in figures [ fig : aod - times](c ) , [ fig : aod - times](d ) , and [ fig : aod - times](e ) and we plot the absorbing cell hydrogen number densities , @xmath4 , temperatures , @xmath35 , and metallicities @xmath282 in solar units as a function of los velocity . what is immediately notable is that the gas phases sampled by the low - ionization ions cover a narrow range clustered around @xmath283 and @xmath284 . on the other hand , there is a fairly steep metallicity gradient with los velocity , @xmath285 dex ( ) @xmath286 . note that the velocity range over which the absorption is strongest ( where the highest column density cells are selected ) coincides with the lowest metallicity absorption selected cells . the relatively strong absorption is due to the slightly higher @xmath4 and lower @xmath35 values , suggesting that the ionization balance dominates the profile shape , not the metallicity . stronger absorption _ within _ a profile does not necessarily suggest higher metallicity . in contrast , the gas phases of the higher - ionization ions represent a much broader range of phases , @xmath287 and @xmath288 for absorption selected gas and @xmath289 and @xmath290 for absorption selected gas . likewise , the metallicity ranges for these ions is quite large , @xmath291 . unlike the low - ionization ions , there is no trend in these highly variable quantities with los velocity . in figure [ fig : aod - times](f ) , we plot the los spatial locations , @xmath247 , of the absorption - selected cells as a function of los velocity . both the and absorption arise in what might be considered a `` cloud '' . by this term , we mean that the absorption arises in spatially contiguous cells over a few kiloparsec los pathlength and that there is little variation in the number densities and temperatures of this gas over this short pathlength [ see figures [ fig : aod - times](c ) and [ fig : aod - times](d ) ] . the point here is that modeling the and absorption with vp decomposition , though not precisely appropriate , would not be entirely without justice . as shown in figure [ fig : vpfit ] , the vp profile model comprises two components , which reflect the velocity structure of the , , and absorption and seems appropriate given the two clusters of low - ionization absorbing cells in los velocity . furthermore , the very narrow range of cell temperatures is consistent with the isothermal assumption inherent in vp decomposition . when we examine the physical properties of the cells selected by and absorption , we see an entirely different physical situation . the densities and temperatures in these cells cover a 2.5 dex and 0.5 dex range , respectively . the spatial locations are spread out on the scale of 100 kpc . clearly , the gas selected by higher - ionization absorption profiles for this simulation do not arise in a cloud - like structure . we revisit this below . we now compare the aod column density and vp fitting parameters to the column densities of the absorption - selected cells . the @xmath292 profiles are shown in figure [ fig : aod - times](b ) . the dotted histograms are the asymmetric @xmath293 uncertainties in each velocity pixel . the horizontal lines outside the velocity range of the detected absorption provides the upper limit on the aod column density in the continuum of the spectra . due to the black - bottomed saturation of the absorption , the @xmath294 profile provides a lower limit on the column density in the velocity range @xmath295 . the individual points ( black @xmath110 symbols ) provide the column densities of the absorption - selected cells for each transition as a function of los velocity . also shown are the vp component column densities ( colored open circles ; see figure [ fig : vpfit ] for the vp model profiles and table [ tab : vpfit ] for the vp component fitting parameters ) . we remind the reader that the vp models were segregated by ionization level , with and ( along with and ) being simultaneously fit for the lower ionization phase , and and ( along with ) being simultaneously fit for the higher ionization phase . the low - ionization model is a two - component fit and the high - ionization model is a four component fit . consider the absorption . the column densities of a majority of the cells _ exceed _ the @xmath296 profiles . this is a fairly well understood expectation that is due to the intrinsic absorption profiles from the gas being unresolved by the spectrograph ( e.g. , * ? ? ? * ) for which correction methods have been developed @xcite . for @xmath297 , the intrinsic absorption profile width is @xmath298 ( fwhm ) for , whereas the fwhm of the instrumental function for the simulated spectra is @xmath299 . note that discrepancy between @xmath187 and the cell column densities is also seen for the coolest gas selected by and absorption , since these cells give rise to the narrowest intrinsic line shapes . the vp component column densities for the absorption are fairly consistent with the cell column densities , though the @xmath300 component is @xmath301 dex below the dominant absorbing cells . the column densities of the vp components are also @xmath302 dex higher than the peak column densities of the absorbing cells . if we examine the inferred temperatures of the gas from the vp components [ figure [ fig : aod - times](d ) ] , we see that the component temperatures are also overestimated . since the lower ionization profiles are decomposed into only two components , the doppler @xmath189 parameters are likely to be systematically large ; that is , the kinematic morphology of the absorption is due to a steep velocity gradient across the los ( see the @xmath198 panels of figure [ fig : los ] ) , which has been modeled as two thermally broadened `` clouds '' . for the and absorption , the vp component column densities and temperatures are significantly overestimated , roughly by @xmath303 dex and @xmath302 dex , respectively . in fact , the inferred temperatures from the vp components are @xmath304 , and are approaching temperatures commonly assumed to be indicative of collisional ionization for these two ions ( in the following subsection , we show that the and absorbing gas is dominated by photoionization equilibrium ) . the very broad component suggests @xmath305 , but the vp component broadening is clearly modeling the bulk velocity flow . these considerations serve as a warning that vp decomposition can be highly misleading about the ionization condition of the gas , and may simply be a flawed approach to modeling and absorbing gas ( as we discuss further below ) . these discrepancies with the vp decomposition would undoubtedly systematically skew estimated of the gas density and metallicity inferred from ionization modeling of the vp components . however , when we examine the _ total _ column densities , we find good agreement between the aod columns , @xmath306 , the vp column densities , and the absorbing cell column densities . in table [ tab : cols ] , we list the total column densities for , , , and . the vp total column density is the sum of the components , the aod total column density is given by eq . [ eq : naod ] , and the total for the `` cells '' is the sum of the absorption selected cell column densities . that the totals are in good agreement is promising , for the consistency between observational methods for obtaining the total column densities and the actual total column density of the gas selected by absorption would imply that mass estimates of the cgm using integrated aod profiles and summed vp column densities accurately recover the gas mass for both the low and high ionization cgm ( for example , as done by * ? ? ? * ; * ? ? ? lrrr & @xmath307 & @xmath308 & @xmath309 + & @xmath310 & @xmath311 & @xmath312 + & @xmath313 & @xmath314 & @xmath315 + & @xmath316 & @xmath317 & @xmath318 interpretation of both aod column density profiles and vp decomposition is predicated on the assumption that the absorption at a given los velocity arises from the same physical gas complex , or cloud . whereas the @xmath296 profile reasonably recovers the distribution of absorbing gas column densities with los velocity , the same can not be said for the gas giving rise to and absorption . for these ions , we see that multiple absorbing gas structures arise with virtually identical velocities and that these structures can have up to a @xmath8 dex spread in column densities . that is , the flux decrement at any given velocity can result from the sum of the column densities from multiple cells that happen to have the same los velocity . when we further consider the spatial locations of absorbing cells with the same los velocity , we find that the assumption underlying standard observational analysis of the absorption profiles is not validated by the kinematic - spatial distribution of the absorbing gas for all ions . as mentioned above while discussing figure [ fig : aod - times](f ) , the absorption and the absorption is distributed over a range of velocities in gas that forms a single contiguous structure at @xmath319 ( i.e. , a single `` cloud '' ) . however , at most los velocities , the and absorbing gas originates from multiple groupings of cells with physical separations ranging from a few to 150 kpc . for example , the absorption in the range of @xmath320 arises in six physically distinct los locations spread over 100 kpc with typical los separations of @xmath321 kpc , whereas the cells in each of these spatial groupings are contiguous over one to a few kpc . not only does the absorption arise in gas distributed over @xmath14 kpc , the gas is characterized by a complex velocity field that reverses direction along the los several times that results in los velocity caustics where the column densities of physically unconnected gas structures are summed . the absorption exhibits similar behavior . we note that in a double line - of - sight experiment , @xcite concluded that the absorption arises in large extended structures that are consistent with our simulation results . the fact that and arise in these extended structures with complex spatial and velocity fields is no doubt the reason that the vp component doppler @xmath189 parameters yield inferred temperatures that are @xmath8 dex too high [ see figure [ fig : aod - times](d ) ] . in summary , current observational analysis techniques may be valid for low - ionization gas . as mentioned above , the and absorption arises in a single contiguous gas structure . the density and temperature ranges of this structure are @xmath322 and @xmath323 , respectively . based upon standard observational techniques and cloudy modeling applied to cos spectra , ( * ? ? ? * @xmath324 galaxies ) and ( * ? ? ? * @xmath325 galaxies ) constrained the properties of the cool / warm photoionized cgm to reside near these values and to be generally consistent with the phase diagrams of and absorbing gas shown in figure [ fig : phases ] . however , becasue of the complex spatial - kinematic distrbution of the higher - ionization gas , current observational techniques of and absorption likely do _ not _ correctly model the underlying physics of the absorbing gas . the implication of the kinematic - spatial distribution of the absorbing gas is that observational analysis techniques ( i.e. , aod column density and/or vp decomposition coupled with ionization modeling ) that are founded on the absorbing gas at a given los velocity arising in a physically contiguous gas structure may be quite invalid ( especially for high - ionization species ) . it remains to be determined to what degree of accuracy the inferred densities , temperatures , and metallicites from observational modeling techniques are recovered when applied to the simulated spectra and compared to the gas cell properties . in the near future , we aim to undertake such a study ( vander vliet 2014b , in preparation ) . we now turn to the question as to whether cgm absorbing gas can be modeled with equilibrium ionization models . we also investigate methods for determining if the absorbing gas is photoionized or collisionally ionized . since most all ionization modeling of observational data is based upon the assumption of ionization equilibrium , it is important to understand to what degree this assumption holds . our ionization modeling , and therefore all subsequent absorption line analysis of the simulations , is also is based upon this assumption . investigations into non - equilibrium ionization had been investigated by @xcite and by @xcite . non - equilibrium collisional ionization results in a reduction of the column density relative to the equilibrium solution , though @xcite find that non - equilibrium effects are not as pronounced when photoionization is taken into account . whether the gas is predominantly photoionized or collisionally ionized has implications for correctly estimating the density , temperature , physical size , cooling time , and estimated mass of the absorbing gas ( in otherwords , understanding the physical nature of the cgm , and subsequently its role in galaxy evolution ) . in observational work one can not directly deduce whether the gas is dominated by photoionization or collisional ionization . often , inferences of collisional iopnization are based upon the presence of large thermal velocity widths in the absorption profiles , or the column density ratios of different ions in different ionization stages ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? however , from the ionization modeling one can directly examine the degree to which the ionization balance in the absorbing gas is dominated by photoionization or collisional ionization . the method we employ can also be applied to observational work . the condition of ionization equilibrium is that the gas must be thermally stable over the time required for the photoionization , recombination , and collisional ionization rates to achieve a steady - state balance . if the time required to achieve ionization equilibrium is denoted @xmath326 , the condition is often expressed , @xmath327 where @xmath328 is the cooling timescale . here , @xmath329 is the energy density of the gas [ erg @xmath18 ] , @xmath331 is the electron density , @xmath332 is the total number density of all ions , @xmath333 is the heating rate per unit volume , and @xmath334 is the cooling rate per unit volume [ erg s@xmath286 @xmath18 ] . both @xmath333 and @xmath334 are functions of the spectral energy distribution of ionizing photons , the gas density , @xmath4 temperature , @xmath35 , and the relative abundances of the metals . the ionization timescales are different for each ion in the gas . as such , it is possible that some ions are in ionization equilibrium , while other ions are not . for our work , which focuses on the absorption properties of ions that are commonly observed in absorption lines studies , we aim to determine if the assumption of ionization equilibrium applies for these ions in particular . here , we examine three timescales , photoionization , @xmath335 , recombination , @xmath336 , and collisional ionization , @xmath337 , for the four ions h@xmath119 , mg@xmath120 , c@xmath338 and o@xmath67 in the absorption selected gas shown in figures [ fig : los ] and [ fig : aod - times ] for los 0092 . the ionization timescales for ion x@xmath197 are obtained by rearranging the rate equations ( see * ? ? ? * for full details ) into the form @xmath339 , where @xmath340 is the rate [ s@xmath184 at which @xmath341 is either increasing or decreasing . the solution is of the form @xmath342 . thus , the ionization timescales are the @xmath343-folding time for a change in the numder density . the rate @xmath340 is a function of the ionizing photon field , @xmath344 , ion densities , electron density , and gas temperature , which are all assumed to change by no more than a negligible amount so that @xmath340 can be approximated as a constant ( thus , greatly simplifying the integrals ; as such , the timescales are approximations ) . integrating over the time interval @xmath345 with the condition @xmath346 yields @xmath347 . the absolute value reflects the fact the change in @xmath341 may be decreasing or increasing over the time interval , but we wish to know the timescale only , regardless of the sign of @xmath348 . accounting only for photoionization rates ( omitting auger ionization ) , we obtain the photionization timescale , @xmath349 where @xmath350 is the photoionization rate [ s@xmath184 for ion x@xmath197 , and the density units are @xmath18 . in the case of the neutral ion ( @xmath351 ) , the rates indexed by @xmath352 do not enter into the derivation and are omitted from eq . [ eq : tau - photo ] . accounting for both photo and dielectronic recombination , we obtain the recombination timescale , @xmath353 \right| ^{-1 } \quad { \rm sec } \ , , \label{eq : tau - recomb}\ ] ] where @xmath354 and @xmath355 are the photo and dielectronic recombination rate coefficients [ @xmath356 s@xmath184 for ion x@xmath197 . accounting for both direct and excitation - auto collisional ionization processes , we obtain the collisional ionization timescale using eqs . [ eq : tau - recomb ] and [ eq : tau - coll ] . ] , @xmath357 \right| ^{-1 } \quad { \rm sec } \ , , \label{eq : tau - coll}\ ] ] where @xmath358 and @xmath359 are the direct and excitation - auto ionization rate coefficients [ @xmath356 s@xmath184 for ion x@xmath197 . for the derivation of eq . [ eq : tau - coll ] , we have omitted charge exchange reactions . for each cell in the simulation , we compute the energy density , @xmath329 , and the photoionization , recombination , and collisional ionization timescales directly from the photoionization rates , equilibrium electron density , and the recombination and collisional ionization rate coefficients . the computation of the rates , rate coefficients , and the equilibrium ionization condition for each gas cell is described in @xcite . to compute the cooling timescale , we adopt the cooling functions of @xcite . these cooling functions were generated with cloudy 07.02 @xcite for model clouds as a function of hydrogen number density , temperature , metallicity , and redshift . the ionizing radiation is the @xcite ultraviolet background ( uvb ) . the clouds are assumed to be dust free and optically thin . as with our ionization model , the gas is assumed to be in ionization equilibrium . we note here that it is important the cooling function accounts for photoionization processes . at a given gas density and temperature , photoionization will yield a gas that is more highly ionized than collisionally ionized gas . this results in cooling rates that are diminished as compared to the rates for collisional ionized gas ( * ? ? ? * ; * ? ? ? * and references therein ) . thus , the standard collisional ionization equilibrium cooling functions ( e.g. , * ? ? ? * ) can substantially understimate the cooling timescales in photoionized gas . we also note that the @xcite cooling functions are highly consistent with the cooling functions used by hydroart @xcite . this is important because the cooling functions dictate the equilibirum temperature of the gas , and the gas temperature determines the collision based ionization and recombination rates in equilibrium ionization models . the cooling functions are determined by interpolating across the high - resolution tables of @xcite , which return the normalized net cooling function , @xmath360 . we determined @xmath361 for each gas cell in the simulation box . the cell data required for the interpolation are the hydrogen number density , @xmath4 , temperature , @xmath35 , redshift , @xmath73 ( needed for the uvb ) , and the abundances of he , c , n , o , ne , mg , si , s , ca , and fe , relative to hydrogen . the latter are detemined from the type ii and ia supernovae mass fractions in the gas cell . note that the _ individual _ contributions of the aforementioned ions , based on their number densities relative to hydrogen , are accounted for in the cooling functions . we then apply eq . [ eq : coolingtime ] to compute @xmath362 for each cell in the simulation box . in figure [ fig : aod - times](g ) , we plot the timescales for photoionization ( green open squares ) , recombination ( red filled triangles ) , and collisional ionization ( open blue circles ) for the four ions h@xmath119 , mg@xmath120 , c@xmath66 , and o@xmath67 in the absorption selected cells . we also plot the cooling timescales ( black @xmath110 symbols ) . the h@xmath119 ion is clearly in photoionization equlibrium . the photoionization and recombination timescales are equal and both of these are roughly three orders of magnitude shorter than the collisional ionization timescales . furthermore , the cooling timescales are also roughly three orders of magnitude longer than the photoionization timescales , which satisfies the criterion given by eq . [ eq : eqcool ] for ionization equilibrium . the collisional ionization processes of hydrogen do not achieve equilibrium , since the gas cools on similar timescales , but photoionization completely dominates the ionization of hydrogen . the mg@xmath120 ion , on the otherhand , is in near balance between photoionization , collisional ionization , and recombination . however , the assumption of ionization equilibrium is well satisfied for the majority of gas in that the cooling timescales exceed the ionization timescales . in the cases of c@xmath66 and o@xmath67 , the ions are in equilibrium between the recombination and photoionization timescales . for both ions , the collisional ionization timescales exceed @xmath363 gyr ; the gas giving rise to and absorption is clearly dominated by photoionization . furthermore , the assumption of ionization equilibrium is well supported by the long cooling timescales . at los velocities of @xmath364 , the gas may be only marginally in ionization equilibrium for these ions . in summary , the assumption of ionization equlibrium for the commonly observed ions h@xmath119 , mg@xmath120 , c@xmath66 , and o@xmath67 is sound for modeling the ionization conditions of the absorbing gas ( at least for the cgm of this simulated galaxy ) . this lends a great deal of confidence that observational data can be accurately analyzed using ionization equilibrium for these ions . futhermore , we have shown that it is straight forward to determine whether a given ion is photoionized or collisionally ionized in the absorbing gas of the simulated cgm . in principle , the method can be applied to observational data if the photoionization , collisional ionization , and recombination rate coefficients from the ionization modeling are retreivable . however , we caution that using column densities of individual vp components may not yield an ionization model that reflects the absorbing gas conditions ( see section [ sec : kin - space ] ) . we have presented a case as to why a comprehensive understanding of the cgm via absorption line studies requires objective absorption line analysis of high resolution hydrodynamic cosmological simulations that mirrors observational absorption line methods . most importantly , we argued and demonstrated that the synthetic absorption line spectra of the simulated cgm should emulate the observed spectra in all of its characteristics , as should the analysis of the absorption line measurements obtain directly from the spectra . primarily , this assures that the gas selected by absorption in the simulated cgm is directly comparable to the gas selected by absorption in the observed cgm . the point is that , when comparing the simulated cgm gas properties ( densities , temperatures , kinematics , ionization and chemical conditions , and spatial distributions ) to the properties of the cgm gas inferred from observational absorption line studies , only the gas that is selected by the simulated absorption lines should be compared . only in this fashion are direct insights into observational analysis and the physical nature of the cgm gleaned . we generated absorption profiles for the and transitions , and for the spectra of and , @xmath365 , @xmath156 , @xmath366 and @xmath157 , , , and metal - line transitions for 1000 los through the cgm of a simulated dwarf galaxy from the work of @xcite . though in this paper we focused on a single los for illustrating the relationships between commonly observed absorption lines and the underlying properties of the absorbing gas , the results discussed for this los are generally true from los to los . highlights of our findings for the cgm of this simulated galaxy are : 0.1 in ( 1 ) the simulations indicate that low ionization gas , as probed in absorption with ions having ionization potentials near the hydrogen ionization edge , likely arise from what might be called `` clouds '' . along the los , these structures are characterized by a narrow range of densities and temperatures , suggesting they can be modeled as single phase structures . in addition , the absorption lines arise in gas that is localized along the los ( the grid cells in the simulations are spatially contiguous ) . as such , commonly applied observational analysis methods that incorporate aod column densities , vp decomposition , and single phase ionization modeling are likely to be sound . 0.1 in ( 2 ) the simulations indicate that higher ionization gas likely arises in extended structures that sample gas distributed up to 50100 kpc along the los . furthermore , the absorption at a given los velocity can arise in several smaller scale regions of the gas separated by tens of kiloparsecs that circumstantially align in los velocity . the gas phases ( densities and temperatures ) that give rise to absorption at a given los velocity can vary up to an order of magnitude or more . as such , aod profiles and vp decomposition do not properly reflect the high - ionization absorbing gas properties . ionization modeling of higher ionization gas as `` cloud''-like structures should be viewed with healthy skepticism . however , aod profiles and vp decomposition can accurately reflect the total gas mass and average gas properties , because the _ total _ column densities from these methods are highly consistent with the total gas column along a los . 0.1 in ( 3 ) higher ionization gas , such as absorption selected gas , can have los absorption velocities that overlap with the los absorption velocity of absorption selected gas , but the simulations indicate that in not all cases is the absorption associated with the higher ionization absorbing gas . in addition to the caveats for point ( 2 ) above , this presents further challenges for ionization modeling and metallicity estimates of the high ionization cgm . 0.1 in ( 4 ) estimates of the ionization timescales and the cooling timescales of the gas that gives rise to detected absorption indicate that the gas can be successfully modeled as being in ionization equilibrium . broad absorption in high - ionization ions does not necessarily indicate high temperature gas that is often taken to imply collisional ionization dominates . in fact , we found that vp decomposition of and absorption yielded temperatures consistent with collisional ionization when in fact the absorbing gas is roughly 1 dex cooler and the broadening is due to the complexity of the spatial and velocity fields over the extended absorbing structure . indeed , analysis of the ionization modeling showed that the and absorbing gas is photoionized . 0.1 in to the extent that the cgm of this simulated galaxy reflects the real world , the points we have discussed here provide our first qualitative insights into the effectiveness or possible misapplication of standard absorption line analysis methods applied to studies of the cgm . however , we have studied the simulated cgm of a single isolated low - mass ( dwarf ) galaxy . the star formation history is highly stochastic and `` bursty '' @xcite , and there are no satellite galaxies and filamentary accretion is negligible . the effects of both these characteristics are certainly manifest in the global cgm properties of this simulated galaxy . it remains to be seen if the cgm of higher mass galaxies in more complex cosmological environments presents a substantially different or broader set of insights . in the future , we aim to undertake a comprehensive quantitative analysis of the simulated cgm , comparing the inferences derived from various observational works to the inferences derived from the application of observational analysis methods to simulated absorption data . examples of observational data that can be compared are the covering fractions , the column density and equivalent width impact parameter distributions , and the kinematics . in particular , these quantities may promise to provide insight into how the cgm properties reflect different stellar feedback recipes . using the absorption selected gas properties , we aim to also quantify the degree to which the observational analysis methods accurately recover the `` true '' properties of absorbing gas . we aim to determined to what degree of accuracy the inferred densities , temperatures , and metallicities from observational modeling techniques are recovered when applied to the simulated spectra and compared to the gas properties . this work will require full application of the aod column density , vp decomposition , and ionization modeling techniques . it is also an immediate goal to include and test shielding and basic radiative transfer effects in our ionization model @xcite so that we can relax the optically thin constraint . we thank the referee for helpful comments that improved this manuscript . cwc , st - g , and ak were partially supported through grants hst - ar-12646 and hst - go-13398 provided by nasa via the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , under nasa contract nas 5 - 26555 . ggk was supported by an australian research council future fellowship ft140100933 . jrv acknowledges support through a nasa new mexico space grant consortium ( nmsgc ) graduate research fellowships .

we study the circumgalactic medium ( cgm ) of a @xmath0 simulated dwarf galaxy using hydroart simulations . we present our analysis methods , which emulate observations , including objective absorption line detection , apparent optical depth ( aod ) measurements , voigt profile ( vp ) decomposition , and ionization modeling . by comparing the inferred cgm gas properties from the absorption lines directly to the gas selected by low ionization and , and by higher ionization and absorption , we examine how well observational analysis methods recover the `` true '' properties of cgm gas . in this dwarf galaxy , low ionization gas arises in sub - kiloparsec `` cloud '' structures , but high ionization gas arises in multiple extended structures spread over 100 kpc ; due to complex velocity fields , highly separated structures give rise to absorption at similar velocities . we show that aod and vp analysis fails to accurately characterize the spatial , kinematic , and thermal conditions of high ionization gas . we find that absorption selected gas and absorption gas arise in totally distinct physical gas structures , calling into question current observational techniques employed to infer metallicities and the total mass of `` warm - hot '' cgm gas . we present a method to determine whether and absorbing gas is photo or collisionally ionized and whether the assumption of ionization equilibrium is sound . as we discuss , these and additional findings have strong implications for how accurately currently employed observational absorption line methods recover the true gas properties , and ultimately , our ability to understand the cgm and its role in galaxy evolution .

introduction the simulations the ionization model absorption line analysis of the cgm discussion conclusions

arxiv