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gamma - ray bursts ( grbs ) are brief , intense flashes of @xmath0-rays originating at cosmological distances , and they are the most luminous objects in the universe . they also have broadband afterglows long - lasting after the @xmath0-ray radiation has ceased . it has been established that the bursts and afterglows are emitted from outflows moving towards us at highly relativistic speeds @xcite , and at least some grbs are associated with the collapse of massive stars ( e.g. , * ? ? ? * ; * ? ? ? observations suggest that the burst is produced by internal dissipation within the relativistic jet that is launched from the center of the explosion , and the afterglow is the synchrotron emission of electrons accelerated in a collisionless shock driven by the interaction of the jet with the surrounding medium ( for recent reviews , * ? ? ? * ; * ? ? ? * ; * ? ? ? in spite of extensive observational and theoretical efforts , several key questions concerning the nature of the central engines of the relativistic jets and the jets themselves remain poorly understood . in fact , some of these questions are very difficult or even impossible to answer with the spectral and lightcurve information currently collected . on the other hand , polarization information , if retrieved , would lead to unambiguous answers to these questions . in particular , polarimetric observations of grbs can address the following : _ magnetic composition of grb jets _ it is highly speculated that strong magnetic fields are generated at the grb central engine , and may play an essential role in the launch of the relativistic jets . however , it is unclear whether the burst emission region is penetrated by a globally structured , dynamically important magnetic field , and whether the burst is due to shock dissipation or magnetic reconnection ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? . _ emission mechanisms of the bursts _ the leading model for the emission mechanism of the prompt burst emission is synchrotron emission from relativistic electrons in a globally ordered magnetic field carried from the central engine , or random magnetic fields generated in - situ in the shock dissipation region @xcite . other suggestions include compton drag of ambient soft photons @xcite , synchrotron self - compton emission @xcite , and the combination of a thermal component from the photosphere and a non - thermal component ( e.g. , synchrotron ) @xcite . _ geometric structure of grb jets _ although it is generally believed that grb outflows are collimated , the distribution of the jet opening angles , the observer s viewing direction , and whether there are small - scale structures within the global jet are not well understood @xcite . to date , robust positive detections of grb polarization have been made only in the optical band in the afterglow phase . varying linear polarizations have been observed in several optical afterglows several hours after the burst trigger , with a level of @xmath6 , which is consistent with the synchrotron emission mechanism of grb afterglow ( for reviews , see * ? ? ? * ; * ? ? ? an upper limit @xmath7 has been obtained for the early @xmath8 optical afterglow of grb 060418 @xcite . also for radio afterglows , we have several upper limits for the polarization degree @xcite ( for some implications , see * ? ? ? as for the prompt burst emission , strong linear polarization of the @xmath0-ray emission at a level of @xmath9 was claimed for grb 021206 based on an analysis of _ rhessi _ data @xcite , although this claim remains controversial because of large systematic uncertainties @xcite . several other reports of high levels of polarization in the prompt burst emission are also statistically inconclusive @xcite . recently , more sensitive observational techniques for x - ray and @xmath0-ray polarimetry have been developed , and there are several polarimeter mission concepts . these include polarimeters for energetic transients ( _ poet _ , * ? ? ? * ; * ? ? ? * ) , polarimeter of gamma - ray observer ( _ pogo _ , * ? ? ? * ) , _ polar _ @xcite , advanced compton telescope ( _ act _ , * ? ? ? * ) , gravity and extreme magnetism ( _ gems _ , * ? ? ? * ) _ xpol _ @xcite , gamma - ray burst investigation via polarimetry and spectroscopy ( _ grips _ , * ? ? ? * ) , and so on . several of these missions , if launched , would provide definitive detections of the burst polarizations and enable us to discuss the statistical properties of the polarization degrees and polarization spectra . although there are several polarimetry mission concepts described in the literature , _ poet _ is the only one to date that incorporates a broadband capability for measuring the prompt emission from grbs , and for this reason it provides a good case study for our simulations . _ poet _ will make measurements with two different polarimeters , both with wide fields of view . the gamma - ray polarimeter experiment ( grape ; 60 - 500 kev ) and the low energy polarimeter ( lep ; 2 - 15 kev ) provide a broad energy range for the observations . suborbital versions of both _ poet _ instruments are currently being prepared for flight within the next few years . grape will fly on a sub - orbital balloon in 2011 , and the gamma - ray burst polarimeter ( grbp , a smaller version of lep ) will fly on a sounding rocket . theoretically , it has been shown that similarly high levels of linear polarization can be obtained in several grb prompt emission models ; the synchrotron model with a globally ordered magnetic field , the synchrotron model with a small - scale random magnetic field @xcite , and the compton drag model @xcite . thus the detections of grb prompt emission polarization would support these three models . in this paper , we show that these models can be distinguished by their statistical properties of observed polarizations . we performed detailed calculations of the distribution of polarization degrees by including realistic spectra of grb prompt emission and assuming realistic distributions of the physical parameters of grb jets , and show that _ poet _ , or other polarimeters with similar capabilities , can constrain the grb emission models . we use the limits of _ poet _ for grb detection and polarization measurements as realistic and fiducial limits . this paper is organized as follows . we first introduce the _ poet _ mission concept in [ sec : poet ] . in [ sec : theory ] , we summarize the properties of the observed linear polarization from uniform jets within the three emission models . based on these models , we perform monte carlo simulations of observed linear polarizations and show how the statistical properties of observed polarization may constrain grb emission mechanisms in [ sec : simulation ] . a summary and discussion are given in [ sec : summary ] . _ poet _ ( polarimeters for energetic transients ) is a small explorer ( smex ) mission concept , that will provide highly sensitive polarimetric observations of grbs and can also make polarimetry measurements of solar flares , pulsars , soft gamma - ray repeaters , and slow transients . the payload consists of two wide field of view ( fov ) instruments : a low energy polarimeter ( lep ) capable of polarization measurements in the 2 - 15 kev energy range and a high energy polarimeter ( gamma - ray polarimeter experiment ; grape ) that will measure polarization in the 60 - 500 kev energy range . _ poet _ can measure grb spectra from 2 kev up to 1 mev . the _ poet _ spacecraft provides a zenith - pointed platform for maximizing the exposure to deep space and spacecraft rotation provides a means of effectively dealing with systematics in the polarization response . _ poet _ provides sufficient sensitivity and sky coverage to detect up to 200 grbs in a two - year mission . lep and grape determine polarization by measuring the number of events versus the event azimuth angle ( eaa ) as projected onto the sky . this is referred to as a modulation profile and represents a measure of the polarization magnitude and direction of polarization for the incident beam . depending on the type of polarimeter , the eaa is either the direction of the ejected photoelectron ( lep ) or the direction of the scattered photon ( grape ) . the response of a polarimeter to @xmath10 polarized photons can be quantified in terms of the modulation factor , @xmath11 , which is given by : @xmath12 where @xmath13 and @xmath14 are the maximum and minimum of the modulation profile , respectively . the polarization fraction ( @xmath2 ) of the incident flux is obtained by dividing the measured modulation by that expected for @xmath10 polarized flux . the polarization angle ( @xmath15 ) corresponds either to the maximum of the modulation profile ( lep ) or the minimum of the modulation profile ( grape ) . to extract these parameters from the data , the modulation histograms are fit to the functional form : @xmath16 the sensitivity of a polarimeter is defined in terms of the minimum detectable polarization ( mdp ) , which refers to the minimum level of polarization that is detectable with a given observation ( or , equivalently , the apparent polarization arising from statistical fluctuations in unpolarized data ) . the precise value of the mdp will depend on the source parameters ( fluence , spectrum , etc . ) and the polarimeter characteristics . at the @xmath17 confidence level , the mdp can be expressed as , @xmath18 where @xmath19 is the observed source strength ( @xmath20 ) , @xmath21 is the total observed background rate ( @xmath20 ) , and @xmath22 is the observing time ( s ) . the ultimate sensitivity , however , may not be limited by statistics but by systematic errors created by false modulations that arise from azimuthal asymmetries in the instrument . lcc polarimetry & 60500 kev & 215 kev + detectors & bgo / plastic scintillator ( 62 ) & @xmath23 gas ( 8) + spectroscopy & 15 kev 1 mev & 2 15 kev + detectors & nai(ti ) scintillator ( 2 ) & as above + field - of - view & @xmath24 & @xmath25 + at energies from @xmath26 kev up to several mev , photon interactions are dominated by compton scattering . the operational concept for grape is based on the fact that , in compton scattering , photons are preferentially scattered at a right angle to the incident electric field vector ( the polarization vector ) @xcite . if the incident beam of photons is polarized , the azimuthal distribution of scattered photons will be asymmetric . the direction of the polarization vector is defined by the minimum of the scatter angle distribution . the grape performance characteristics are shown in table [ tabparam ] . the design of the grape instrument is very modular , with 62 independent polarimeter modules and 2 spectroscopy modules . each polarimeter module incorporates an array of optically independent 5x5x50 @xmath27 non - hygroscopic scintillator elements aligned with and optically coupled to the 8x8 scintillation light sensors of a 64-channel mapmt . two types of scintillators are employed . low - z plastic scintillator is used as an effective medium for compton scattering . high - z inorganic scintillator ( bismuth germanate , bgo ) is used as a calorimeter , for absorbing the full energy of the scattered photon . the arrangement of scintillator elements within a module has 28 bgo calorimeter elements surrounding 32 plastic scintillator scattering elements . valid polarimeter events are those in which a photon compton scatters in one of the plastic elements and is subsequently absorbed in one of the bgo elements . these events can be identified as a coincident detection between one plastic scintillator element and one bgo calorimeter element . the azimuthal scatter angle is determined for each valid event by the relative locations of hit scintillator elements . it is not necessary to know where within each element the interaction takes place ( e.g. , the depth of interaction ) . it is sufficient to know only the lateral location of each element to generate a histogram of photon scatter angles . at energies below @xmath28 kev , the most sensitive technique for broadband polarimetry is the photoelectric effect . the lep measures the polarization of incident photons with the innovative operation of a time projection chamber ( tpc ) @xcite . the lep polarimeter enclosure consists of four dual - readout detector modules each with an isolated gas volume contained by a be x - ray window . each detector module contains two 6 x 12 x 24 @xmath29 ( lxwxh ) tpcs that share a single x - ray transparent drift electrode . each tpc is comprised of a micropattern proportional counter , consisting of a shared drift electrode and a high - field gas electron multiplier ( gem ) positioned 1 mm from a strip readout plane . when an x - ray is absorbed in the gas between the drift electrode and the gem , a photoelectron is ejected in a preferential direction with a @xmath30 distribution , where @xmath31 is the azimuthal angle measured from the x - ray polarization vector . as the photoelectron travels through the gas it creates a path of ionization that drifts in a moderate , uniform field to the gem where an avalanche occurs . the charge finally drifts to the strip detector where it is read out . to estimate realistic mdp values for grbs detected by grape and lep , we perform an analytical calculation for lep and a monte carlo simulation for grape using the current instrument configuration ( table [ tabparam ] ) . the input spectrum in the calculation and the simulation is a typical grb spectrum which can be described as a smoothly broken power - law spectra characterized by photon energy at the @xmath32 spectral peak , @xmath33 , and lower and higher indices of the @xmath34 spectrum , @xmath35 and @xmath36 , respectively @xcite . ( we treat the spectral indices of the specific energy flux @xmath34 , while @xcite define @xmath37 and @xmath38 as the indices of the photon number flux , i.e. , @xmath39 and @xmath40 , since we will calculate the net polarizations by using specific energy fluxes ( equation ( [ eq : band_spec ] ) ) . ) the various @xmath33 and time - averaged flux in 2 - 400 kev , @xmath41 , are investigated with fixed @xmath42 , and a burst duration of @xmath43 s. we also assume the incident angle of bursts to be 30 degree off - axis . we interpret simulated events with @xmath44 as @xmath2-measurable events. figure 1 shows the contour of the mdp values in the @xmath45 plane for grape and lep . as can be seen in the figure , with the combination of lep and grape , it is possible to measure the polarization of grbs with @xmath46 ranging from a few kev to mev with reasonable sensitivity . we calculate the linear polarization for instantaneous emission from a thin spherical shell moving radially outward with a bulk lorentz factor @xmath47 and an opening angle @xmath48 . the comoving - frame emissivity has the functional form of @xmath49 , where @xmath50 is the normalization which may depend on direction in the comoving frame and other physical quantities of the shell and @xmath51 represents the spectral shape . a prime represents the physical quantities in the comoving frame . the delta functions describe the instantaneous emission at @xmath52 and @xmath53 . the normalization , @xmath50 , has units of @xmath54 . using the spherical coordinate system @xmath55 in the lab frame , where @xmath56 is the line of sight , we obtain the spectral fluence @xcite : @xmath57 where @xmath58 and @xmath59 are the redshift and the luminosity distance of the source , respectively , and @xmath60 . the integration is performed within the jet cone , so that it depends on the viewing angle @xmath61 , i.e. , the angle between the jet axis and the line of sight . the corresponding stokes parameters of the local emission ( i.e. , the emission from a given point on the shell ) are given by @xmath62 and @xmath63 , where @xmath64 and @xmath65 are the polarization degree and position angle of the local emission measured in the comoving frame , respectively . the stokes parameters of the emission from the whole shell can be obtained by integrating those of the local emission similarly to the intensity @xmath66 : @xmath67 the polarization degree is lorentz invariant , i.e. , @xmath68 . the position angle @xmath69 is calculated by taking account of the lorentz transformation of the electromagnetic waves , and it is measured from a fixed direction , which we choose to be the direction from the line of sight to the jet axis . then by calculating @xmath70 , we obtain the time - averaged linear polarization in the given wavebands @xmath71 $ ] : @xmath72 we consider synchrotron and compton drag ( cd ) mechanisms for the grb prompt emission . in the synchrotron case , the magnetic field consists of a globally ordered field , @xmath73 , and small - scale random field , @xmath74 , i.e. , @xmath75 . the field @xmath73 may originate from the central engine , while @xmath74 may be produced in the emission region itself . here we consider two extreme cases ; synchrotron model with an ordered field ( so ) , in which @xmath76 , and a synchrotron model with a random field ( sr ) , in which @xmath77 . for the so model , in particular , we assume a toroidal magnetic field . in the following sub - sections , we describe @xmath50 , @xmath51 , @xmath78 , and @xmath69 as functions of @xmath79 for each model , and calculate the linear polarization for given parameters @xmath0 , @xmath48 , @xmath61 , and @xmath58 . the prompt emission of grbs could be explained by synchrotron emission from accelerated electrons that have a non - thermal energy spectra by some dissipation process within the jet , e.g , internal shocks . synchrotron emission from the relativistically moving shell within a globally ordered magnetic field results in a net observed linear polarization , reflecting the direction of the field @xcite . let us assume that the jet is permeated by a toroidal field . this is a likely configuration if a magnetic field is advected by the jet with a constant speed from the central engine ( e.g. , * ? ? ? * ; * ? ? ? a general formula for calculating the observed linear polarization for synchrotron emission from a uniform jet , in which the electrons have a single power - law energy spectrum and an isotropic pitch angle distribution and the magnetic field is ordered globally , is derived by @xcite and @xcite . here we adopt their formulation and extend it for the electrons having a broken power - law energy spectrum in order to reproduce the typical observed spectra of grbs @xcite . we adopt the following form for the radiation spectrum : @xmath80 where @xmath81 and @xmath82 where @xmath83 , @xmath35 , and @xmath36 are the break frequency and low - energy and high - energy spectral indices of the comoving spectrum , respectively . ) is thought to be produced by the broken power - law energy spectrum of electrons : @xmath84 for @xmath85 and @xmath86 for @xmath87 , where @xmath88 and @xmath89 . this formulation also includes the case of @xmath90 , in which @xmath91 , @xmath92 , and @xmath93 for @xmath94 @xcite . ] if we assume that the energy spectrum of the electrons and the strength of the magnetic field are uniform in the emitting shell , then we may write @xmath95 , where @xmath96 is the angle between the direction of the emitted radiation and the local direction of the magnetic field @xcite . the local polarization degree is given by : @xmath97 for a globally ordered magnetic field , the faraday depolarization effect may be strong within the emitting region ( e.g. , * ? ? * ; * ? ? ? * ; * ? ? ? * ) , but we neglect it here for simplicity . by using a new variable @xmath98 , we obtain ( see appendix [ subsec : app_so ] ) : @xmath99^{1/2 } , \label{eq : tsinthetab}\ ] ] @xmath100 where @xmath101 . then the formulation of the net polarization degree in the observed frequency region @xmath71 $ ] becomes : @xmath102^{-1 } , \end{array } \label{eq : atorob}\ ] ] where @xmath103 , and @xmath104 @xmath105 & { \rm otherwise}. \end{array } \right . \label{eq : delta_phi}\ ] ] the polarization degree , @xmath2 , in the waveband [ @xmath106 can be calculated if the geometrical parameters , @xmath107 , the spectral parameters , @xmath108 , and the redshift , @xmath58 , are given . figure [ fig : atorob ] shows the polarization degree in the @xmath109 kev band as a function of @xmath110 for several values of @xmath111 . the other parameters are @xmath112 kev , @xmath113 , @xmath114 , and @xmath115 . the polarization degree is negligible for @xmath116 , because in this case the local polarization vectors are axisymmetric around the line of sight , i.e. , @xmath117 ( see appendix [ subsec : app_so ] ) , and the local polarizations are canceled out . for @xmath118 , a high level of polarization is obtained for @xmath119 ( i.e. , @xmath120 ) . in this case , only a fraction of the emitting shell ( i.e. , @xmath121 ) is bright because of the relativistic beaming effect , and the direction of the magnetic field is quite ordered in the bright region . the contribution of the emission from high latitude , @xmath122 , is negligible , especially for @xmath123 , so that the net polarization degree is determined only by the emission from the bright region with @xmath121 and then it is nearly constant . the results of our calculations for the case of @xmath124 and @xmath123 are consistent with the results of @xcite and @xcite . for @xmath125 , a high level of polarization is obtained for @xmath126 ( i.e. , @xmath127 ) . in this case , the bright region on the emitting shell is small , also . the polarization is higher for softer spectra ( i.e. , larger @xmath35 and @xmath36 ) . for example , for @xmath128 , @xmath112 kev , and @xmath115 , the polarization degree at the plateau for @xmath129 is @xmath130 for @xmath131 and @xmath132 , while it is @xmath133 for @xmath134 and @xmath135 . this is caused mainly by the dependence of the synchrotron polarization on the spectral indices ( equation [ eq : synch_pi ] ) . the maximum polarization degree obtained in the so model is @xmath136 for @xmath137 , @xmath138 , and @xmath139 if the magnetic field is produced at the shock itself within the jet , the directions of the field would be random on a scale as small as the plasma skin depth @xcite . it is quite plausible that the directions of the field are not completely random , but have symmetry around the direction normal to the shock . the less isotropic the magnetic field directions behind the shock , the higher the local polarization . we consider the extreme case in which the field is random strictly within the plane of the shock . in this model , the directions of the local polarization vectors on the shell are axisymmetric around the line of sight ( see below ) , so that no net polarization remains if the visible region , @xmath121 , is wholly within the jet cone . however , if the observer views the jet from an off - axis angle and the symmetry is broken a high level of polarization remains @xcite . similarly to the so model , we adopt the broken power - law form of the spectrum : @xmath140 where @xmath141 and @xmath142 is given by equation ( [ eq : band_spec ] ) . we assume that the energy distribution of the electrons and the strength of the magnetic field are uniform in the emitting shell . the local stokes parameters are given by averaging them with respect to the magnetic field directions within the shock plane ( see appendix [ subsec : app_sr ] ) . thus we may write @xmath143 , where @xmath144 represents the average . the local polarization degree is given by @xmath145 , where : @xmath146^{(\alpha+1)/2},\ ] ] @xmath147^{(\alpha-1)/2 } \right . \\ \times \left . \left[\sin^2\eta ' - \left(\frac{1-y}{1+y}\right)^2 \cos^2\eta ' \right ] \right\}. \end{array}\ ] ] the local polarization position angle measured in the lab frame is given by @xmath117 , therefore we obtain the formulation for the net polarization in the observed frequency region @xmath71 $ ] : @xmath148^{-1 } , \end{array } \label{eq : arndb}\ ] ] where @xmath149 , @xmath150 , @xmath151 , and @xmath152 and @xmath153 are given by equations ( [ eq : synch_pi ] ) and ( [ eq : delta_phi ] ) , respectively . figure [ fig : arndb ] shows the polarization degree in the @xmath109 kev band as a function of @xmath110 for several values of @xmath111 . the other parameters are @xmath154 kev , @xmath155 , and @xmath115 . the results of our calculations for the case of @xmath124 are consistent with those of @xcite and @xcite . a high level of polarization is obtained for @xmath126 ( i.e. , @xmath127 ) for each value of @xmath111 . since the local polarization vectors are axisymmetric around the line of sight , the local polarizations are canceled out if the line of sight is within the jet cone . if the jet is observed from an off - axis angle , the net polarization remains . the local polarization degree is highest for emission where @xmath156 , so that the net polarization has a maximum value . the maximum @xmath2 is higher for smaller @xmath111 , because the contribution of the emission from high latitude points ( @xmath122 ) , with a low level of local polarization , is smaller . similarly to the so model , the polarization is higher for softer spectra , mainly because of the dependence of the local polarization degree on frequency ( equation [ eq : synch_pi ] ) . for example , for @xmath157 , @xmath158 kev , and @xmath115 , the maximum polarization is @xmath159 for @xmath160 and @xmath161 , while it is @xmath162 for @xmath134 and @xmath135 . for @xmath137 , @xmath138 , and @xmath163 the maximum polarization degree in the sr model is @xmath136 . the prompt emission from grbs could be produced by bulk inverse comptonization of soft photons from the relativistic jet @xcite . the local polarization position angles are symmetric around the line of sight , similarly to the sr model . therefore this model also requires an off - axis observation of the jet to achieve a high level of polarization . however , the cd model is different from the sr model in the fact that the cd model can in principle achieve @xmath164 under the most optimistic geometric configurations , whereas the maximum @xmath2 is @xmath165 in the sr model . we assume that the seed radiation is unpolarized and has a non - thermal , isotropic spectrum , and the scattered radiation has the broken power - law spectrum @xmath166 , where @xmath81 and @xmath142 is given by equation ( [ eq : band_spec ] ) . if the intensity of the seed radiation and the electron number density of the shell are assumed to be uniform then we may write @xmath167 , and @xmath168 @xcite . the polarization vectors in the comoving frame are perpendicular to both incident and scattering directions of photons , so that we obtain @xmath169 in the lab frame . therefore we achieve the formulation for the net linear polarization in the observed frequency region @xmath71 $ ] : @xmath170^{-1 } , \end{array } \label{eq : acomp}\ ] ] where @xmath149 , @xmath150 , @xmath151 , and @xmath153 is given by equation ( [ eq : delta_phi ] ) . figure [ fig : acomp ] shows the polarization degree in the @xmath109 kev band as a function of @xmath110 for several values of @xmath111 . the other parameters are @xmath154 kev , @xmath171 and @xmath115 . the results of our calculations for the case of @xmath124 are consistent with those of @xcite . the results are similar to those of the sr model , but the polarization degree is higher than in the sr model . the polarization is higher for softer spectra , although the local polarization degree is not dependent on the frequency in this model . for instance , for @xmath172 , @xmath173 kev , and @xmath115 , the maximum polarization is @xmath174 for @xmath131 and @xmath161 , while it is @xmath175 for @xmath134 and @xmath135 , but the variation is smaller than for the synchrotron models ( see [ subsec : so ] and [ subsec : sr ] ) . this variation is caused by the kinematic effect . the local polarization degree is a maximum for @xmath156 ( i.e. , @xmath176 ) . thus the net polarization is higher when the contribution of the emission from higher latitude with @xmath122 is smaller . the high latitude emission is dimmer as the radiation spectrum is softer . therefore the net polarization is higher when the spectrum is softer . this effect also arises in the so and sr models , although in those models the intrinsic dependence of polarization on the spectrum ( equation [ eq : synch_pi ] ) is rather strong ( see [ subsec : so ] and [ subsec : sr ] ) . for @xmath137 , @xmath138 , and @xmath163 the maximum polarization degree for the cd model is @xmath177 . in this section we show the results of our monte carlo simulation of the grb prompt emission polarization . first , in [ subsec : flu ] , we give the values of the model parameters so that the observed fluences and peak energies of simulated bursts are consistent with the data obtained with the _ hete-2 _ satellite . in [ subsec : sum ] , we examine the properties of the polarization distribution of bursts detectable by the _ poet _ satellite , regardless of instrument mdp . next , in [ subsec : cum ] , we show the distribution of polarizations that can be measured by _ poet _ , and discuss how we may constrain the emission models . we performed monte carlo simulations to obtain the distribution of the observed spectral energies and fluences in the three emission models . such simulations have been developed to discuss the empirical correlation between spectral peak energies in the cosmological rest frame and isotropic @xmath0-ray energies among grbs and x - ray flashes in several models of geometrical structure of grb jets @xcite . we generated 10,000 grb jets with lorentz factor , @xmath0 , and opening angle , @xmath48 , and a random viewing angle for each jet according to the probability distribution of @xmath178 with @xmath179 rad . rad in our simulation are not detected by _ hete-2 _ or _ poet _ with the parameters we adopt in this paper . we can therefore discuss the distribution of several quantities of the detectable bursts and the event rate ratio of bursts for which polarizations can be measured to the detectable bursts without considering the bursts with @xmath180 rad . ] for each burst generated we calculate the @xmath181 spectrum to obtain the spectral peak energy , @xmath33 , and the fluence , @xmath182 , in the @xmath183 kev range by using equation ( [ eq : flu ] ) . since @xmath33 s and @xmath182 s calculated for each @xmath149 in the three models are different only by factors less than 2 , @xmath33 s and @xmath182 s of the simulated bursts may be calculated using just one model , for which we chose the cd model . the distributions of @xmath0 and @xmath48 for grb jets are highly uncertain . we make a simple assumption for the distribution and in [ subsec : cum ] we perform some simulations for different assumptions . we fix @xmath184 . we assume the distribution of @xmath48 as @xmath185 where @xmath186 and @xmath187 . the value of @xmath188 is inferred from the observations of the steepening breaks ( i.e. , jet breaks ) of some optical afterglows @xcite and from analysis of _ batse _ data using some empirical relations @xcite . there are several suggestions of events with very small @xmath48 ( e.g. , * ? ? ? * ; * ? ? ? * ) , although the value of @xmath189 is highly uncertain . the spectral parameters @xmath190 and @xmath36 are assumed as follows the first two parameters are given so that the rest - frame spectral peak energies and isotropic @xmath0-ray energies calculated for a jet viewed with @xmath191 are consistent with those of typical grbs . such an on - axis emission has approximately @xmath192 and @xmath193 . the parameters @xmath194 and @xmath195 are given through the empirical relations @xmath196 erg and @xmath197 kev ( e.g. , * ? ? ? * ; * ? ? ? we assume that the coefficients @xmath198 and @xmath199 obey the log - normal distribution @xcite with averages of @xmath200 and logarithmic variances of @xmath201 and @xmath202 , respectively . the last two parameters are fixed by @xmath203 and @xmath114 , which are typical values for grb prompt emission @xcite . the distribution of the source redshift , @xmath58 , is assumed to be in proportional to the cosmic star formation rate . we adopt the model sf2 in @xcite , i.e. , the comoving grb rate density is assumed to be proportional to @xmath204 we take the standard cosmological parameters of @xmath205 and @xmath206 . figure [ fig : ep_flu ] shows the results of @xmath207 and time - averaged flux , @xmath41 . the time - averaged flux is calculated by @xmath208 , where @xmath209 is the duration of a burst . we fix @xmath43 s , which is a typical value for long grbs ( e.g. , * ? ? ? we show only the simulated bursts that have fluxes above the detectable limit of the _ hete-2 _ satellite . they are consistent with the data obtained by _ hete-2 _ the scatter of the simulated bursts is due both to the scatter of the assumed jet parameters and the viewing angle effect @xcite . ) for @xmath210 kev in the uniform jet model , but the @xmath211 diagram we derive is still consistent with the observed dataset . ] we calculated the linear polarization , @xmath2 , by using equations ( [ eq : atorob ] ) , ( [ eq : arndb ] ) , and ( [ eq : acomp ] ) to obtain the polarization distribution of the simulated bursts that can be detected by grape and lep . the detection limits of grape and lep are set to be the mdp contours of 1.0 . ( see figure [ fig : mdp ] ) . are similar but not identical to the mdp contours of 1.0 . thus our setting for the detection limits is just for simplicity . ] figures [ fig : ep_pi_1_g ] and [ fig : ep_pi_1_l ] show the @xmath212 diagrams of all the simulated bursts that can be detected by grape and lep , respectively , in the so ( _ red open circles _ ) , sr ( _ green filled circles _ ) , and cd ( _ blue plus signs _ ) models . in the so model , most of the detectable bursts have @xmath213 in the grape band ( 60 - 500 kev ) , while they have @xmath214 in the lep band ( 2 - 15 kev ) . in the sr and cd models , most of the detectable bursts have @xmath215 in both grape and lep bands . the events with @xmath216 are distributed uniformly with @xmath217 and @xmath218 for the sr and cd models , respectively . this result can be roughly explained by the polarizations calculated as functions of @xmath111 and @xmath149 for @xmath112 kev and @xmath115 ( see figures [ fig : atorob ] , [ fig : arndb ] , [ fig : acomp ] ) and the distribution of @xmath48 and @xmath110 for the detectable bursts in this simulation , shown in figure [ fig : thj_q ] . the detectable events are dominated by the events with @xmath129 , since events with @xmath129 are much brighter than those with @xmath219 because of the relativistic beaming effect . for events with @xmath219 , narrower jets are easier to detect since they have intrinsically higher emissivities by our assumption . most of the detectable events have @xmath129 and @xmath220 ( i.e. , @xmath221 ) . for these events the so model gives @xmath213 in most cases , while the sr and cd models give @xmath215 , for the grape band as shown in figures [ fig : atorob ] , [ fig : arndb ] , and [ fig : acomp ] . the remaining detectable events mainly have @xmath219 and @xmath222 ( i.e. , @xmath223 ) . these events have @xmath224 in the so model , @xmath225 in the sr model , and @xmath218 in the cd model , for the grape band as shown in figures [ fig : atorob ] , [ fig : arndb ] , and [ fig : acomp ] . the results for the lep band can be explained similarly . in all the three models , the results show @xmath226 for almost all the detectable bursts with @xmath227 . this is due to the fact that typically the contribution of the high - energy photons with spectral index @xmath36 is larger in the grape band than in the lep band . the emission with softer spectrum has higher polarization because of the intrinsic property of the synchrotron polarization ( equation [ eq : synch_pi ] ) for the so and sr models and the kinematic effect for the cd model ( see [ subsec : compton ] ) , respectively . in the so model , the polarization of grbs with @xmath129 is higher for lower @xmath33 for the grape band . this is because the contribution from high - energy photons , with energy spectral index @xmath36 , is larger . in the sr and cd models , the higher @xmath2 grbs can be obtained for smaller @xmath48 . the maximum @xmath2 is obtained for @xmath228 . we obtain the distribution of polarization that can be measured , by using the mdp values we derived for @xmath203 , @xmath114 , and @xmath43 s ( see [ sec : poet ] ) . we interpret the simulated events with @xmath44 as ` @xmath2-measurable events ' . figure [ fig : cum_1 ] shows the cumulative distribution of @xmath2 that can be measured by grape and lep in the so , sr , and cd models . we have set the number of detectable events @xmath229 . since the polarization in the lep band is lower than in the grape band for almost all the cases as discussed in [ subsec : sum ] , the number of events for which polarization can be measured by lep is smaller than for grape . in the so model , the number of @xmath2-measurable bursts is @xmath230 , and the cumulative distribution of measurable @xmath2 varies rapidly at @xmath231 for the grape band . in the sr model , @xmath232 , and the maximum polarization is @xmath233 . in the cd model , @xmath234 , and @xmath235 . to investigate general properties of the cumulative distribution that do not depend on the model parameters , we performed simulations for other values of @xmath236 @xmath237 and @xmath238 , the lorentz factor of the jets and the power - law indices of the distribution of the opening angles of the jets , respectively . we refer to the parameters adopted for the above simulation as ` typical ' parameters . we now consider a range of parameters : @xmath239 , @xmath240 and @xmath241 , which are quite reasonable for grbs ( e.g. , * ? ? ? * ; * ? ? ? . within these parameter ranges we obtain the lower ( upper ) limit of @xmath1 for the so model ( the sr / cd models ) . figure [ fig : cum_g ] shows the results for @xmath242 and the same ` typical ' values for the other parameters . the number @xmath243 is larger in the so model and smaller in the sr and cd models than the case of @xmath184 . as @xmath0 is larger , the beaming effect is stronger and the ratio of the bursts with @xmath129 for detectable bursts is larger . thus the number of bursts with a high degree of polarization is larger in the so model and smaller in the sr and cd models . figure [ fig : cum_q1 ] shows the results for @xmath244 and the same ` typical ' values for the other parameters . since the ratio of the number of the bursts with smaller @xmath111 to that of detectable bursts is smaller , @xmath243 is slightly smaller than that for the ` typical ' parameters in the sr and cd models . figure [ fig : cum_q2 ] shows the results for @xmath245 and the ` typical ' values for the other parameters . in this case @xmath243 is slightly larger than that for the ` typical ' parameters in the sr and cd models . the number @xmath243 in the so model is similar for figure [ fig : cum_1 ] , [ fig : cum_q1 ] , and figure [ fig : cum_q2 ] in the grape band . to summarize , for the parameters @xmath246 , @xmath247 , @xmath241 , @xmath203 and @xmath114 , we can say that @xmath3 for grape and the cumulative distribution of measurable @xmath2 varies rapidly from @xmath231 in the so model . for the sr model , @xmath248 for grape , with a maximum polarization @xmath233 . for the cd model , @xmath4 for grape , and @xmath235 . since the dependence of the polarization degree on the spectral indices is relatively large in the so and sr models , we should take account of the distribution of @xmath35 and @xmath36 . the observed spectral parameters @xmath35 and @xmath36 are distributed roughly as @xmath249 and @xmath250 @xcite . within these ranges of @xmath35 and @xmath36 , the polarization degree for @xmath251 , @xmath129 , and @xmath252 kev is @xmath253 in the so model . thus the measurable polarizations are clustered at @xmath253 . the maximum polarization obtained in the so model for @xmath254 and @xmath255 is @xmath136 ( see [ subsec : so ] ) . in this case @xmath1 will be larger than 30% . in the cd model , the result will not be significantly different from the case of fixed @xmath35 and @xmath36 . in the sr model , the polarization degree does not exceed those calculated in the cd model , and thus @xmath4 . the maximum polarization obtained in the sr model for @xmath254 and @xmath255 is @xmath136 ( see [ subsec : sr ] ) . in conclusion , we can constrain the emission mechanism of grbs by using the cumulative distribution obtained by grape . if @xmath3 , the sr and cd models may be ruled out , and in this case if the measured polarizations are clustered at @xmath253 , the so model will be favored . if @xmath4 , the so model may be ruled out , but we can not distinguish between the sr and cd models with different distributions of @xmath111 , @xmath35 , and @xmath36 . if several bursts with @xmath5 are detected , however , the cd model which includes adequate number of small @xmath111 bursts will be favored . recently there has been an increasing interest in the measurement of x - ray and @xmath0-ray polarization , and the observational techniques can now achieve significant sensitivity in the relevant energy bands . several polarimetry mission concepts , such as _ poet _ , are being planned . the _ poet _ concept has two polarimeters , grape ( 60 - 500 kev ) and lep ( 2 - 15 kev ) both of which have wide fields of view . if launched , missions of this type would provide the first definitive detection of the polarization of grb prompt emission . this would enable the discussion of the statistical properties of the polarization degree and polarization spectra , which will give us diagnostic information on the emission mechanism of grbs and the nature of the grb jets that can not be obtained from current spectra and lightcurve observations . we have performed monte carlo simulations of the linear polarization from grb jets for three major emission models : synchrotron model with globally ordered magnetic field ( so model ) , synchrotron model with small - scale random magnetic field ( sr model ) , and compton drag model ( cd model ) . we assumed that the physical quantities for the emission of the jets are uniform on the emitting surface and that the jets have sharp edges . our jet angle distribution allows the detections of grbs with very small opening angles ( i.e. , smaller than 1 degree ) as suggested by several _ bursts @xcite . we have shown that the _ poet _ mission or other polarimeters with similar capabilities , i.e. , broadband spectral capabilities for the determination of @xmath33 and sensitive broadband polarimetric capabilities to minimize mdp , can constrain the emission models of grbs . furthermore , these simulations indicate that an increase in the lep effective area would be beneficial to compensate for the lower expected polarization at lower energies . as shown in figures [ fig : atorob ] , [ fig : arndb ] , and [ fig : acomp ] , the sr and cd models require off - axis observations of the jets to achieve a high level of polarization , while the so model does not . in this sense the sr and cd models are categorized as _ geometric _ models , and the so model as an _ intrinsic _ model @xcite . the distribution of observed polarizations obtained by our simulations show that the geometric sr / cd models will be ruled out if the number ratio of the @xmath2-measurable bursts to detected bursts is larger than @xmath256 , and in this case the so model will be favored if the measurable polarizations are clustered at @xmath253 . if the number ratio is smaller than @xmath257 , the so model may be ruled out , but we can not distinguish between the sr and cd models with different distributions of @xmath150 , @xmath35 , and @xmath36 , where @xmath0 and @xmath48 are the bulk lorentz factor and the opening angle of the grb jet , respectively , and @xmath35 and @xmath36 are lower and higher indices of the energy spectrum . however , if several bursts with @xmath5 are detected , the cd model which includes an adequate number of small @xmath111 bursts will be favored . if the cumulative distribution of the measurable polarizations favors the so model , the globally ordered magnetic field would be advected from the central engine . if we understand the strength of the magnetic field in the emitting region from the luminosity and the spectrum of the emission , we can constrain the strength of the field at the central engine . if the geometric sr / cd models are favored from the observations , it will be established , independently of the afterglow observations , that grb outflows are not spherical but highly collimated . if the cd model is favored by the observations , we may constrain the distribution of the parameter @xmath258 of grb jets . the cd model needs a dense optical / uv photon field interacting within the relativistic jets @xcite . we have made some simplifications in our simulations , and there are some caveats . we have assumed that the jets are uniform on the emitting surfaces and have sharp edges . to compare the simulations and the observations further , more sophisticated modeling is required ( e.g. , * ? ? ? * ; * ? ? ? we have interpreted bursts as a simple combination of pulses , without taking account of the temporal variation of the lorentz factor @xmath0 of the jet . if this is accounted for , each pulse may have different @xmath150 but the same @xmath149 . we should then average the polarization with respect to fluence of each pulse having different @xmath111 @xcite . however , in the so model , the cumulative distribution of measurable @xmath2 will not be changed significantly as long as @xmath259 , because @xmath2 is clustered into a small range for @xmath129 and @xmath251 . to average the polarization in the case of @xmath219 , the relation between the luminosity and the lorentz factor for each pulse is required to predict the polarization distribution . for the sr model we have assumed that the directions of the magnetic field are confined within the shock plane . they may be more isotropic in reality , in which case the polarization degree in the sr model will be reduced . in the synchrotron model with a combination of the globally ordered magnetic field and the locally random field , @xmath260 , the linear polarization can be calculated by @xmath261 , where @xmath262 and @xmath263 are the stokes parameters from the ordered and random fields , respectively . @xmath264 and @xmath265 are described by equations ( [ eq : atorob ] ) and ( [ eq : arndb ] ) , and @xmath266 . this model will reduce the number ratio of @xmath2-measurable bursts to detected bursts to less than 30% and the clustering of measurable polarizations will be at @xmath267 . this work is supported in part by the grant - 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ronning , n. m. , & mszros , p. 2004 , apj , 601 , l119 zhang , b. 2007 , chin . j. astron . astrophys . , we consider the synchrotron radiation from the shell moving radially outward with a bulk lorentz factor @xmath47 , and the magnetic field in the shell is globally ordered within the plane parallel to the shock plane . if the matter of the shell expands with a constant speed , the strength of magnetic field with radial direction scales as @xmath268 while that with transverse direction scales as @xmath269 . thus the field advected with the shell is likely to have the direction parallel to the shock plane . we set the line of sight ( i.e. , the direction from the central engine to the earth ) in the lab frame to be @xmath58 axis , and the direction of the magnetic field on a given point of the shell , projected onto the plane perpendicular to @xmath58 axis , to be @xmath270 axis . the given point can be described by spherical coordinates @xmath271 . then the components of the velocity vector of the given point and the unit wave vector can be described by the right - handed coordinate system @xmath272 as @xmath273 and @xmath274 , respectively . the unit wave vector in the comoving frame is @xmath275.\ ] ] since the direction of the magnetic field in the comoving frame is perpendicular to the velocity vector of the fluid , @xmath276 . then we may calculate @xmath277 , and we obtain @xmath278 in the limit @xmath47 . the direction of the polarization vector of the synchrotron radiation is calculated by @xmath279 . then we obtain the direction of the polarization vector in the lab frame by @xmath280 the results are @xmath281 and @xmath282 the angle @xmath283 is the polarization position angle measured from the axis @xmath270 ( i.e. , the direction of the local magnetic field ) . the above equation can be rewritten as @xmath284 $ ] . this result is consistent with that of @xcite . based on the above results , we consider the case that the magnetic field is axisymmetric around the jet and has a toroidal configuration . we set the direction from the line of sight to the jet axis to be @xmath285 axis . below we will rewrite the above results by using the azimuthal angle @xmath286 measured from @xmath285 axis . in the coordinate system of @xmath287 , the jet axis and the coordinates of a given point on the shell are described as @xmath288 , and @xmath289 , respectively . the magnetic field at the given point is given by @xmath290 . let the unit vectors of the directions of @xmath291 and @xmath292 projected onto @xmath293 plane be @xmath294 and @xmath295 , and @xmath296 . then we obtain @xmath297 where @xmath298 . equation ( [ eq : tsinthetab ] ) is given by inserting equation ( [ eq : cos2phi ] ) into equation ( [ eq : sinthetab ] ) . if we measure the position angle from the @xmath285 axis , we obtain equation ( [ eq : tchi ] ) , i.e. , @xmath299 . these results are consistent with those of @xcite . here we consider that the directions of the magnetic fields are confined within the plane parallel to the shock plane and that they are completely random . this field configuration is possible if the field is generated by the shock . in the comoving frame of the shell , we set the direction of @xmath300 to be axis @xmath301 , and set a right - handed coordinate system @xmath302 . let the polar and azimuthal angles of @xmath292 be @xmath96 and @xmath303 , respectively . in this coordinate system , the stokes parameters of synchrotron emissivity are given by @xmath304 next we set another right - handed coordinate system @xmath287 of which @xmath58 axis is along the velocity vector of the fluid and @xmath305 plane includes @xmath306 . then the angle between @xmath306 and @xmath58 axis is @xmath307 . here the magnetic field @xmath308 is confined within @xmath293 plane . let the azimuthal angle of @xmath308 be @xmath309 , and we obtain the relations between the components of @xmath308 in the systems @xmath302 and @xmath287 . @xmath310 then we obtain @xmath311^{1/2 } , \\ \cos(2\phi'_b ) = \frac{1}{\sin^2\theta'_b } \left [ \sin^2\eta ' - \left(\frac{1-\gamma^2\theta^2}{1+\gamma^2\theta^2 } \right)^2 \cos^2\eta ' \right ] . \end{array}\ ] ] to obtain the polarization degree of synchrotron radiation from the random field , we average the stokes parameters with respect to @xmath309 . this leads to @xmath312 . then we can calculate the polarization degree by @xmath313 , and the polarization vector is along axis 1 , i.e. , the direction perpendicular to @xmath306 and within the plane including @xmath306 and @xmath314 .

the emission mechanism and the origin and structure of magnetic fields in gamma - ray burst ( grb ) jets are among the most important open questions concerning the nature of the central engine of grbs . in spite of extensive observational efforts , these questions remain to be answered and are difficult or even impossible to infer with the spectral and lightcurve information currently collected . polarization measurements will lead to unambiguous answers to several of these questions . recent developments in x - ray and @xmath0-ray polarimetry techniques have demonstrated a significant increase in sensitivity enabling several new mission concepts , e.g. _ poet _ ( polarimeters for energetic transients ) , providing wide field of view and broadband polarimetry measurements . if launched , missions of this kind would finally provide definitive measurements of grb polarizations . we perform monte carlo simulations to derive the distribution of grb polarizations in three emission models ; the synchrotron model with a globally ordered magnetic field ( so model ) , the synchrotron model with a small - scale random magnetic field ( sr model ) , and the compton drag model ( cd model ) . the results show that _ poet _ , or other polarimeters with similar capabilities , can constrain the grb emission models by using the statistical properties of grb polarizations . in particular , the ratio of the number of grbs for which the polarization degrees can be measured to the number of grbs that are detected ( @xmath1 ) and the distributions of the polarization degrees ( @xmath2 ) can be used as the criteria . if @xmath3 and @xmath2 is clustered between 0.2 and 0.7 , the so model will be favored . if instead @xmath4 , then the sr or cd model will be favored . if several events with @xmath5 are observed , then the cd model will be favored .

let @xmath0 be a finite simplicial graph and let @xmath1 be the induced flag complex , i.e. , the maximal simplicial complex whose @xmath2-skeleton is @xmath0 . the associated _ right angled artin group _ @xmath3 is the group presented by @xmath4 because one can import topological properties of the associated flag complex @xmath1 into the group @xmath3 , these groups have provided important examples of exotic behavior . ( see for example @xcite , @xcite and @xcite . ) here we refine the understanding of the end topology of right angle artin groups by giving an explicit formula for the cohomology of @xmath3 with group ring coefficients in terms of the cohomology of @xmath1 and links of simplices in @xmath1 . if @xmath5 is a simplicial complex let @xmath6 denote the set of closed simplices including the empty simplex in @xmath5 . the dimension of a simplex is denoted @xmath7 ; the link is denoted @xmath8 ; the star of @xmath9 is @xmath10 . by definition @xmath11 and @xmath12 . let @xmath0 be a finite simplicial graph , let @xmath1 be the associated flag complex and @xmath3 the associated right angled artin group . as long as @xmath1 is not a single simplex , @xmath13\ .\ ] ] if @xmath1 is a single simplex then @xmath3 is free abelian and @xmath14 is simply @xmath15 in top dimension . let @xmath1 be @xmath16 . then the reduced cohomology of @xmath17 is concentrated in dimension @xmath18 where it is @xmath19 . the link of any other simplex @xmath9 is a @xmath20-sphere hence its reduced cohomology is concentrated in dimension @xmath21 , where it is @xmath15 . thus @xmath14 is trivial except in dimension @xmath22 where it is the sum of a countably generated free abelian group and a countable sum of @xmath19 s . there are at least two approaches to establishing the main theorem . one can modify the techniques of @xcite that were developed for computing the cohomology of coxeter groups with group ring coefficients as well as the cohomology with compact supports of any locally finite building to compute this cohomology for right angled artin groups . in fact , the formula given in the main theorem is quite similar to the formulas for cohomology with compact supports of locally finite buildings ( theorem 5.8 in @xcite ) . we take a more efficient route , and use the fact that right angled artin groups are commensurable with certain right angled coxeter groups @xcite , and appeal to the formula for the cohomology of a right angled coxeter group with group ring coefficients ( @xcite or @xcite ) . in the last section we explain how the formula of the main theorem extends results of @xcite on the end topology of right angled artin groups . one of the classical approaches to the study of asymptotic properties of a group @xmath23 is via its cohomology with @xmath24-coefficients . for example , from proposition 7.5 and exercise 4 of @xcite , if @xmath23 is a discrete group and @xmath25 is a contractible @xmath23-complex with finite cell stabilizers and finite quotient , then @xmath26 where @xmath27 is the cohomology of @xmath25 with compact supports . in particular , one can take as @xmath25 either of the classifying spaces @xmath28 or @xmath29 provided they have finite quotients @xmath30 or @xmath31 ( cf . cohomology with group ring coefficients determines the cohomological dimension of @xmath23 ( * ? ? ? * viii.6.7 ) : if @xmath23 is of type fp then @xmath32 it is also closely related to connectivity at infinity and duality properties as is described at the end of the next section . right angled artin groups admit cat(0 ) @xmath33s formed as the union of tori . if @xmath0 is a finite simplicial graph , let @xmath34 be the complex formed by joining tori in the manner described by the flag complex @xmath1 . that is , for each simplex @xmath35 , let @xmath36 be the torus formed by identifying parallel faces of a unit @xmath37-cube . ( the torus @xmath38 is a single vertex . ) the complex @xmath34 is then the union of these tori , subject to @xmath39 when @xmath40 in @xmath1 . for a proof that these @xmath34 s are cat(0 ) classifying spaces , see @xcite . we denote the universal cover of @xmath34 by @xmath41 . the complex @xmath41 is also the davis complex for an appropriate right angled coxeter group . given a finite simplicial graph @xmath0 the _ right angled coxeter group _ @xmath42 is the quotient of @xmath3 formed by declaring that each generator is an involution @xmath43 for a finite simplicial graph @xmath0 let @xmath44 be the graph whose vertices are given by @xmath45 where @xmath46 for @xmath47 or @xmath48 . [ thm : commensurable ] the artin group @xmath3 and the coxeter group @xmath49 are commensurable and in fact the complexes @xmath50 and the davis complex for @xmath49 are identical . ( because @xmath41 is the davis complex for @xmath49 we do not actually define the davis complex for a coxeter group ; see @xcite for a definition . ) one can now derive a formula for @xmath14 from known results in the literature . namely , because 1 . cohomology with group ring coefficients can be expressed in terms of cohomology with compact supports of an @xmath29 , and 2 . @xmath41 is both an @xmath51 and an @xmath52 , and 3 . the cohomology of a coxeter group with group ring coefficients has been computed , and can be expressed in terms of the cohomology of subcomplexes of links of vertices in the davis complex ( @xcite or @xcite ) , we have the following formula for the cohomology of @xmath3 with @xmath53 coefficients . [ cor : firstformula ] each @xmath54 has an associated simplex @xmath55 such that @xmath56 each simplex @xmath57 occurs countably many times in this sum , while @xmath58 occurs exactly once . although the formula above is correct , it obfuscates the connection between @xmath14 and the cohomology of the flag complex @xmath1 . as a first step toward expressing the right hand side in terms of the flag complex @xmath1 , we give an alternate description of the flag complex @xmath59 . for each @xmath60 let @xmath61 be the full subcomplex of @xmath1 induced by the vertices @xmath62 . thus @xmath61 is a deformation retract of @xmath1 with the vertex @xmath63 removed . let @xmath64 be the coxeter system where @xmath65 is abelian and the generating set has been identified with the vertices of the graph @xmath0 . hence @xmath65 is simply @xmath66 let the @xmath61 be a set of mirrors related to this coxeter system and form the associated @xmath65-complex in the following manner . for each @xmath67 let @xmath68 be the subgroup of @xmath65 generated by the set of @xmath60 such that @xmath69 belongs to @xmath61 . in other words , @xmath68 is generated by those @xmath63 such that @xmath69 is not in the open neighborhood of @xmath63 in @xmath1 . define @xmath70 where @xmath71 if and only if @xmath72 . the complex @xmath59 shows up in the formula of corollary [ cor : firstformula ] because it is isomorphic to the link of any vertex in @xmath41 . one can find the following result in @xcite . the complex @xmath59 is isomorphic to @xmath73 , and is isomorphic to the link of the vertex in @xmath34 . if @xmath74 one can form a subcomplex @xmath75 by defining @xmath76 to be the subgroup of @xmath65 generated by @xmath77 , and forming @xmath78 where as before @xmath71 if and only if @xmath72 . in particular , if @xmath58 ( the empty simplex ) then @xmath79 . , the associated complex @xmath73 , and subcomplex @xmath80 [ fig : link],width=240 ] [ exmp : link ] let @xmath81 be the simplicial arc indicated in figure [ fig : link ] . the group @xmath65 is then generated by four elements associated with the vertices . switching to greek letters we denote these generators as @xmath82 and @xmath83 , where the mirror associated to @xmath84 is the subgraph induced by @xmath85 , and similarly for the other three generators . the complex @xmath73 is then as is indicated in figure [ fig : link ] . the generator @xmath84 acts on @xmath73 by exchanging the vertices labeled @xmath86 and @xmath87 , and leaves all other vertices fixed . similarly @xmath88 exchanges @xmath89 and @xmath90 , fixing all other vertices , and so on . finally , if @xmath91 then @xmath92 is the bottom complex in figure [ fig : link ] . for any @xmath93 let @xmath94 so that @xmath95 is a deformation retract of @xmath1 with the barycenter of @xmath9 removed . [ lem : homologyformula ] the cohomology groups of @xmath96 are given by @xmath97 where in a small abuse of notation we let @xmath98 denote all closed simplices of @xmath1 except those with non - empty intersection with @xmath9 . in @xcite mike davis gives a formula for the homology of a complex on which a coxeter group acts . one can switch this to a formula for cohomology using universal coefficients , or via a minor rewriting of davis s original argument . in our case the formula is rather simple . since @xmath76 is abelian , each @xmath99 is determined by the set of generators @xmath100 that are necessary to express @xmath101 . temporarily following davis s notation , define @xmath102 ( if @xmath103 then @xmath104 and so @xmath105 is empty as well . ) davis s formula then gives @xmath106 this can be simplified . if @xmath100 is not the vertex set of a simplex in @xmath1 , then @xmath107 ; if @xmath108 for some @xmath74 , then @xmath109 . thus the formula above can be rewritten as @xmath110 from corollary [ cor : firstformula ] we know that @xmath111\ ] ] where we know there are infinitely many copies of @xmath112 since by its construction there are no non - trivial finite conjugacy classes in @xmath49 . to arrive at our main theorem we need a formula for @xmath113 where @xmath9 is any simplex in @xmath114 . thus our key lemma is : the complex @xmath1 embeds in @xmath59 in a number of ways . standard embedding _ @xmath118 have image the subcomplex induced by @xmath119 . define @xmath120 to be the subcomplex induced by @xmath121 . if @xmath9 is a simplex in @xmath114 then @xmath9 is defined by a set of vertices in @xmath0 along with choices of @xmath122 . if @xmath123 then the automorphism @xmath124 takes @xmath9 to the simplex @xmath125 ( here we have used the same convention on naming generators of @xmath65 as in example [ exmp : link ] . ) thus in discussing the topology of @xmath116 for @xmath115 , we may without loss of generality assume @xmath126 . but the space formed by removing the closed simplex @xmath127 from @xmath59 deformation retracts onto the subcomplex formed by making all possible reflections of @xmath1 that do not involve the generators of @xmath65 that correspond to vertices of @xmath9 . in other words , @xmath116 deformation retracts onto @xmath92 , which implies our first claim . from lemma [ lem : homologyformula ] we know @xmath128 , thus it suffices to establish @xmath129 first , if @xmath130 , @xmath131 and @xmath132 , so we get @xmath133 . if @xmath134 then by excision , @xmath135)$ ] where @xmath136 is the closed star of @xmath137 and @xmath138 $ ] denotes the @xmath139 suspension . because the star @xmath136 is contractible , the long exact sequence in cohomology shows @xmath140 ) = \overline{h}^{*-1}(s^{|\tau|}\left[{\mbox{lk}}(\tau)\right])\ .\ ] ] but the cohomology of a suspension is just a shifted copy of the cohomology of the original complex @xmath141 ) = \overline{h}^ { * - |\tau|-1}\left({\mbox{lk}}(\tau)\right)\ ] ] and the result follows . in example [ exmp : link ] we considered @xmath142 a simplicial arc , and two associated complexes , @xmath73 and @xmath80 . ( the first claim of lemma [ lem : puncturedcoho ] states that @xmath80 is homotopy equivalent to @xmath73 with the closed edge @xmath143 removed . ) the formula of lemma [ lem : puncturedcoho ] says , for example , that @xmath144 this then becomes @xmath145 using the convention that @xmath146 . let @xmath0 be a finite simplicial graph , let @xmath1 be the associated flag complex and @xmath3 the associated right angled artin group . as long as @xmath1 is not a single simplex , @xmath13\ .\ ] ] if @xmath1 is a single simplex then @xmath3 is free abelian and @xmath14 is simply @xmath15 in top dimension . from corollary [ cor : firstformula ] we have @xmath149 by lemma [ lem : puncturedcoho ] this gives @xmath150\ .\ ] ] if @xmath1 is not a single simplex , then each @xmath151 will show up in the product inside the square brackets for infinitely many @xmath54 , and the formula in the theorem follows . on the other hand , if @xmath1 is a single simplex @xmath9 , then @xmath9 only occurs in the summand corresponding to @xmath152 . all other simplices occur infinitely often , but if @xmath153 then @xmath154 is contractible , and @xmath155 is zero . thus @xmath156 , consistent with the fact that @xmath157 . as was alluded to in the previous section , cohomology with group ring coefficients is closely related to asymptotic properties . a group @xmath23 that admits a finite @xmath158 is _ @xmath159-acyclic at infinity _ if roughly speaking , complements of compact sets in the universal cover have trivial homology through dimension @xmath159 ( see @xcite for a precise definition . ) it was from this perspective that brady and meier determined when a right angled artin group was @xmath159-acyclic at infinity . their approach was via a combinatorial morse theory argument using the @xmath34 complexes . however , there is an algebraic characterization that says a group @xmath23 is @xmath159-acyclic at infinity if and only if @xmath160 for @xmath161 and @xmath162 is torsion - free ( see @xcite ) . the group @xmath23 is an _ @xmath159-dimensional duality group _ if there is a dualizing module @xmath163 such that @xmath164 for all @xmath165 and all @xmath23-modules @xmath166 . this too can be recast in terms of cohomology with group ring coefficients : @xmath23 is an @xmath159-dimensional duality group if its cohomology with group ring coefficients is torsion - free and concentrated in dimension @xmath159 @xcite . thus our main theorem implies three results of @xcite . it is important to remember that @xmath167 , and the formal dimension of @xmath168 is @xmath48 . since @xmath8 is @xmath170-acyclic it follows by universal coefficients that its cohomology is trivial up to dimension @xmath171 and that @xmath172 is torsion - free . thus the formula of the main theorem implies that @xmath173 is zero for @xmath161 and @xmath174 is torsion - free . a simplicial complex @xmath5 is _ cohen - macaulay _ if for any simplex @xmath175 , the cohomology of @xmath8 is concentrated in top dimension ( and is torsion free ) . it follows from the formula of the main theorem that @xmath14 is torsion free and concentrated in top dimension if and only if @xmath1 is cohen - macaulay . recall that an @xmath159-dimensional duality group is called a _ duality group if and only if @xmath176 @xcite . after the statement of theorem c in @xcite it was remarked that a theorem of strebel combined with theorem c implies that a right angled artin group @xmath3 is a poincar duality group if and only if @xmath3 is free abelian . this characterization follows directly from the formula in our main theorem .

we give an explicit formula for the cohomology of a right angled artin group with group ring coefficients in terms of the cohomology of its defining flag complex .

the fundamental group of the complement of plane curves is a very important topological invariant , which can be also computed for line arrangements . we list here some applications of this invariant . chisini @xcite , kulikov @xcite and kulikov - teicher @xcite have used the fundamental group of complements of branch curves of generic projections in order to distinguish between connected components of the moduli space of smooth projective surfaces , see also @xcite . moreover , the zariski - lefschetz hyperplane section theorem ( see @xcite ) states that @xmath3 where @xmath4 is an hypersurface and @xmath5 is a generic 2-plane . since @xmath6 is a plane curve , the fundamental groups of complements of curves can be used also for computing the fundamental groups of complements of hypersurfaces in @xmath7 . a different need for fundamental groups computations arises in the search for more examples of zariski pairs @xcite . a pair of plane curves is called _ a zariski pair _ if they have the same combinatorics ( to be exact : there is a degree - preserving bijection between the set of irreducible components of the two curves @xmath8 , and there exist regular neighbourhoods of the curves @xmath9 such that the pairs @xmath10 are homeomorphic and the homeomorphism respects the bijection above @xcite ) , but their complements in @xmath11 are not homeomorphic . for a survey , see @xcite . it is also interesting to explore new finite non - abelian groups which serve as fundamental groups of complements of plane curves in general , see for example @xcite . an arrangement of lines in @xmath12 is a union of copies of @xmath13 in @xmath12 . such an arrangement is called _ real _ if the defining equations of the lines can be written with real coefficients , and _ complex _ otherwise . note that the intersection of the affine part of a real arrangement with the natural copy of @xmath14 in @xmath12 is an arrangement of lines in the real plane . for real and complex line arrangements @xmath0 , fan @xcite defined a graph @xmath15 which is associated to its multiple points ( i.e. points where more than two lines are intersected ) : given a line arrangement @xmath0 , the graph @xmath15 of multiple points lies on @xmath0 . it consists of the multiple points of @xmath0 , with the segments between the multiple points on lines which have at least two multiple points . note that if the arrangement consists of three multiple points on the same line , then @xmath15 has three vertices on the same line ( see figure [ graph_gl](a ) ) . if two such lines happen to intersect in a simple point ( i.e. a point where exactly two lines are intersected ) , it is ignored ( and the lines are not considered to meet in the graph theoretic sense ) . see another example in figure [ graph_gl](b ) ( note that this definition gives a graph different from the graph defined in @xcite ) . fan @xcite proved some results concerning the projective fundamental group : [ fan ] let @xmath0 be a complex arrangement of @xmath16 lines and@xmath17 be the set of all multiple points of @xmath0 . suppose that @xmath18 , where @xmath19 is the first betti number of the graph @xmath15 ( hence @xmath18 means that the graph @xmath15 has no cycles ) . then : @xmath20 where @xmath21 is the multiplicity of the intersection point @xmath22 and @xmath23 . in @xcite , similar results were achieved for the affine and projective fundamental groups by different methods . fan @xcite has conjectured that the inverse implication is also correct , i.e. if the fundamental group @xmath24 can be written as a direct sum of free groups and infinite cyclic groups , then the graph @xmath15 has no cycles . in an unpublished note , fan @xcite shows that if the fundamental group of the affine complement is a free group , then the arrangement consists of parallel lines . recently , eliyahu , liberman , schaps and teicher @xcite proved fan s conjecture completely . these results motivate the following definition : let @xmath25 be a fundamental group of the affine or projective complements of some line arrangement with @xmath16 lines . we say that @xmath25 has _ a conjugation - free geometric presentation _ if @xmath25 has a presentation with the following properties : * in the affine case , the generators @xmath26 are the meridians of lines at some far side of the arrangement , and therefore the number of generators is equal to @xmath16 . * in the projective case , the generators are the meridians of lines at some far side of the arrangement except for one , and therefore the number of generators is equal to @xmath27 . * in both cases , the relations are of the following type : @xmath28 where @xmath29 is an increasing subsequence of indices , where @xmath30 in the affine case and @xmath31 in the projective case . note that for @xmath32 we get the usual commutator . note that in usual geometric presentations of the fundamental group , most of the relations have conjugations ( see section [ mt ] ) . based on the last definition , fan s result yields that if the graph associated to the arrangement is acyclic , then the corresponding fundamental group has a conjugation - free geometric presentation . the following natural problem arises : which line arrangements have a fundamental group which has a conjugation - free geometric presentation ? the aim of this paper is to attack this problem . the importance of this family of arrangements is that the fundamental group can be read directly from the arrangement or equivalently from its incidence lattice ( where the _ incidence lattice _ of an arrangement is the partially - ordered set of non - empty intersections of the lines , ordered by inclusion , see @xcite ) without any computation . hence , for this family of arrangements , the incidence lattice determines the fundamental group of the complement . we start with the easy fact that there exist arrangements whose fundamental groups have no conjugation - free geometric presentation : the fundamental group of the affine ceva arrangement ( also known as the _ braid arrangement _ , appears in figure [ ceva ] ) has no conjugation - free geometric presentation . this fact was checked computationally by a package called _ testisom _ @xcite , which looks for isomorphisms ( or proves a non - isomorphism ) between two given finitely - presented group . note that the ceva arrangement is the minimal arrangement ( with respect to the number of lines ) with this property . our main result is : the fundamental group of following family of real arrangements have a conjugation - free geometric presentation : an arrangement @xmath0 , where @xmath15 is a union of disjoint cycles of any length , has no line with more than two multiple points , and the multiplicities of the multiple points are arbitrary . we also give the exact group structure ( by means of a semi - direct product ) of the fundamental group for an arrangement of @xmath1 lines whose graph is a cycle of length @xmath2 ( i.e. a triangle ) , where all the multiple points are of multiplicity @xmath2 : let @xmath0 be the real arrangement of @xmath1 lines , whose graph consists of a cycle of length @xmath2 , where all the multiple points are of multiplicity @xmath2 . moreover , it has no line with more than two multiple points . then : @xmath33 where @xmath34 is the free product . as mentioned above , for the family of arrangements with a conjugation - free geometric presentation of the fundamental group , the incidence lattice of the arrangement determines its fundamental group . there are some well - known families of arrangements whose lattice determines the fundamental group of its complement - the families of _ nice _ arrangements ( jiang - yau @xcite ) and _ simple _ arrangements ( wang - yau @xcite ) . it is interesting to study the relation between these families and the family of arrangements whose fundamental groups have conjugation - free geometric presentations , since for the latter family , the lattice determines the fundamental group of the complement too . we have the following remark : the fundamental group of the arrangement @xmath35 ( appears in figure [ a_n ] ) has a conjugation - free geometric presentation ( this fact was checked computationally ) , but this arrangement is neither nice nor simple . it will be interesting to find out whether our family of arrangements is broader than the family of simple arrangements , or whether there exists a simple arrangement whose fundamental group has no conjugation - free geometric presentation . [ rem_dehornoy ] it is worth to mention that conjugation - free geometric presentations are complemented positive presentations ( defined by dehornoy @xcite , see also @xcite ) . some initial computations show that in general conjugation - free geometric presentations are not complete ( since the cube condition is not satisfied for some triples of generators ) . nevertheless , we do think that there exist conjugation - free geometric presentations which are complete and hence have all the good properties induced by the completeness ( see the survey @xcite ) . we will discuss this subject in a different paper . the paper is organized as follows . in section [ mt ] , we give a quick survey of the techniques we are using throughout the paper . in section [ length_three ] , we show that the fundamental group of a real arrangement whose graph has a unique cycle of length 3 has a conjugation - free geometric presentation . in this section , we also deal with the exact structure of the fundamental group of a real arrangement whose graph consists of a cycle of length @xmath2 , where all the multiple points have multiplicity @xmath2 . section [ length_n ] deals with the corresponding result for a real arrangement whose graph has a unique cycle of length @xmath16 . we also generalize this result for the case of arrangements whose graphs are a union of disjoint cycles . in this section , we present the computation of the fundamental group of the complement of real line arrangements . this is based on the moishezon - teicher method @xcite and the van kampen theorem @xcite . some more presentations and algorithms can be found in @xcite . if the reader is familiar with this algorithm , he can skip this section . to an arrangement of @xmath36 lines in @xmath14 one can associate a _ wiring diagram _ @xcite , which holds the combinatorial data of the arrangement and the position of the intersection points . a wiring diagram is a collection of @xmath36 wires ( where a _ wire _ in @xmath37 is a union of segments and rays , homeomorphic to @xmath38 ) . the induced wiring diagram is constructed by choosing a new line ( called the _ guiding line _ ) , which avoids all the intersection points of the arrangement , such that the projections of intersection points do not overlap . then , the @xmath36 wires are generated as follows . start at the @xmath39 end of the line with @xmath36 parallel rays , and for every projection of an intersection point , make the corresponding switch in the rays , as in figure [ latowd ] . to a wiring diagram , one can associate a list of _ lefschetz pairs_. any pair of this list corresponds to one of the intersection points , and holds the smallest and the largest indices of the wires intersected at this point , numerated locally near the intersection point ( see @xcite and @xcite ) . for example , in the wiring diagram of figure [ wdtolp ] , the list of lefschetz pairs is ( we pass on the intersection points from right to left ) : @xmath40}},{{\left[{2},{4}\right]}},{{\left[{1},{2}\right]}},{{\left[{4},{5}\right]}},{{\left[{2},{3}\right]}},{{\left[{3},{4}\right]}},{{\left[{4},{5}\right]}},{{\left[{2},{3}\right ] } } ) .\ ] ] let @xmath41 be a closed disk in @xmath14 , @xmath42 a set of @xmath36 points , and @xmath43 . let @xmath44 be the group of all diffeomorphisms @xmath45 such that @xmath46 is the identity and @xmath47 . the action of such @xmath48 on the disk applies to paths in @xmath41 , which induces an automorphism on @xmath49 . the _ braid group _ , @xmath50 $ ] , is the group @xmath44 modulo the subgroup of diffeomorphisms inducing the trivial automorphism on @xmath49 . an element of @xmath51 $ ] is called a _ braid_. for simplicity , we will assume that @xmath52 , and that @xmath53 . choose a point @xmath54 ( for convenience we choose it to be below the real line ) . the group @xmath55 is freely generated by @xmath56 , where @xmath57 is a loop starting and ending at @xmath58 , enveloping the @xmath59th point in @xmath60 . the set @xmath61 is called a _ geometric base _ or _ g - base _ of @xmath55 ( see figure [ fig_1 ] ) . let @xmath62}},\dots,{{\left[{a_p},{b_p}\right]}})$ ] be a list of lefschetz pairs associated to a real line arrangement @xmath63 with @xmath36 lines . the of the complement of the arrangement is a quotient group of @xmath55 . there are @xmath64 sets of relations , one for every intersection point . in each point , we will compute an object called a _ skeleton _ , from which the relation is computed . in order to compute the skeleton @xmath65 associated to the @xmath59th intersection point , we start with an _ initial skeleton _ corresponding to the @xmath59th lefshetz pair @xmath66}}$ ] which is presented in figure [ fig3 ] , in which the points correspond to the lines of the arrangement and we connect by segments adjacent points which correspond to a local numeration of lines passing through the intersection point . 0.7 cm to the initial skeleton , we apply the lefschetz pairs @xmath67 } } , \cdots,{{\left[{a_1},{b_1}\right]}}$ ] . a lefschetz pair @xmath68}}$ ] acts by rotating the region from @xmath69 to @xmath70 by @xmath71 counterclockwise without affecting any other points . for example , consider the list @xmath72}},{{\left[{2},{4}\right]}},{{\left[{4},{5}\right]}},{{\left[{1},{3}\right]}},{{\left[{3},{4}\right]}})$ ] . let us compute the skeleton associated to the 5th point . the initial skeleton for @xmath73}}$ ] is given in figure [ fig4](a ) . by applying @xmath74}}$ ] and then @xmath75}}$ ] , we get the skeleton in figure [ fig4](b ) . then , applying @xmath76}}$ ] yields the skeleton in figure [ fig4](c ) , and finally by acting with @xmath77}}$ ] we get the final skeleton in figure [ fig4](d ) . 4.5 cm from the final skeletons we compute the relations , as follows . we first explain the case when @xmath78}}$ ] corresponds to a simple point , @xmath79 . then the skeleton is a path connecting two points . let @xmath41 be a disk circumscribing the skeleton , and let @xmath60 be the set of points . choose an arbitrary point on the path and pull it down , splitting the path into two parts , which are connected in one end to @xmath80 and in the other to the two end points of the path in @xmath60 . the loops associated to these two paths are elements in the group @xmath81 , and we call them @xmath82 and @xmath83 . the corresponding elements commute in the of the arrangement s complement . figure [ av_bv ] illustrates this procedure . now we show how to write @xmath82 and @xmath83 as words in the generators @xmath84 of @xmath85 . we start with the generator corresponding to the end point of @xmath82 ( or @xmath83 ) , and conjugate it as we move along @xmath82 ( or @xmath83 ) from its end point on @xmath60 to @xmath58 as follows : for every point @xmath86 which we pass from above , we conjugate by @xmath87 when moving from left to right , and by @xmath88 when moving from right to left . for example , in figure [ av_bv ] , @xmath89 and so the induced relation is : @xmath90 one can check that the relation is independent of the point in which the path is split . for a multiple intersection point of multiplicity @xmath91 , we compute the elements in the group @xmath81 in a similar way , but the induced relations are of the following type : @xmath92 we choose an arbitrary point on the path and pull it down to @xmath58 . for each of the @xmath91 end points of the skeleton , we generate the loop associated to the path from @xmath58 to that point , and translate this path to a word in @xmath93 by the procedure described above . in the example given in figure [ av_bv_mul ] , we have : @xmath94 , @xmath95 and @xmath96 , so the relations are @xmath97 in this section , we prove the following proposition : [ triangle - prop ] the fundamental group of a real affine arrangement without parallel lines , whose graph which has a unique cycle of length @xmath2 and has no line with more than two multiple points , has a conjugation - free geometric presentation . in the first subsection we present the proof of proposition [ triangle - prop ] . the second subsection will be devoted to studying the group structure of the fundamental group of the simplest arrangement of this family . for simplicity , we will assume that all the multiple points have the same multiplicity @xmath98 , but the same argument will work even if the multiplicities are not equal . by rotations and translations , one can assume that we have a drawing of an arrangement which has a unique cycle of multiple points of length 3 and has no line with more than two multiple points , as in figure [ arrange_mult_3 ] . we can assume it due to the following reasons : first , one can rotate a line that participates in only one multiple point as long as it does not unite with a different line ( by results 4.8 and 4.13 of @xcite ) . second , moving a line that participates in only one multiple point over a different line ( see figure [ triangle - line ] ) is permitted in the case of a triangle due to a result of fan @xcite that the family of configurations with @xmath1 lines and three triple points is connected by a finite sequence of smooth equisingular deformations . each of the blocks 1,4,5 contains simple intersection points of two pencils . in block 1 , one can assume that all the intersections of any horizontal line are adjacent , without intervening points from the third pencil . in blocks 4 and 5 , one can assume that all the intersections of any vertical line are adjacent ( in block 4 , the vertical lines are those with positive slopes ) . all the intersection points of block 5 are to the left of all the intersection points of block 4 . hence , we get the list of lefschetz pairs as in table [ tab1 ] ( we put a double line to separate between the pairs related to different blocks ) . @xmath99 & & 2n(n-1)+3 & [ n , n+1 ] \\ 2 & [ n-1,n ] & & 2n(n-1)+4 & [ n-1,n ] \\ \vdots & \vdots & & \vdots & \vdots \\ n & [ 1,2 ] & & n(2n-1)+2 & [ 1,2 ] \\ \hline n+1 & [ n+1,n+2 ] & & n(2n-1)+3 & [ n+1,n+2 ] \\ n+2 & [ n , n+1 ] & & n(2n-1)+3 & [ n , n+1 ] \\ \vdots & \vdots & & \vdots & \vdots \\ 2n & [ 2,3 ] & & n(2n)+2 & [ 2,3 ] \\ \hline \vdots & \vdots & & \vdots & \vdots\\ \hline ( n-2)n+1 & [ 2n-2,2n-1 ] & & ( 3n-1)(n-1)+3 & [ 2n-2,2n-1 ] \\ ( n-2)n+2 & [ 2n-3,2n-2 ] & & ( 3n-1)(n-1)+4 & [ 2n-3,2n-2 ] \\ \vdots & \vdots & & \vdots & \vdots\\ ( n-1)n & [ n-1,n ] & & 3n(n-1)+2 & [ n-1,n ] \\ \hline\hline ( n-1)n+1 & [ n,2n ] & & 3n(n-1)+3 & [ n,2n ] \\ \hline\hline ( n-1)n+2 & [ 2n,3n ] & & & \\ \hline\hline ( n-1)n+3 & [ 2n-1,2n ] & & & \\ ( n-1)n+4 & [ 2n-2,2n-1 ] & & & \\ \vdots & \vdots & & & \\ ( n-1)(n+1)+2 & [ n+1,n+2 ] & & & \\ \hline ( n-1)(n+1)+3 & [ 2n,2n+1 ] & & & \\ ( n-1)(n+1)+4 & [ 2n-1,2n ] & & & \\ \vdots & \vdots & & & \\ ( n-1)(n+2)+2 & [ n+2,n+3 ] & & & \\ \hline \vdots & \vdots & & & \\ \hline ( 2n-1)(n-1)+3 & [ 3n-2,3n-1 ] & & & \\ ( 2n-1)(n-1)+4 & [ 3n-3,3n-2 ] & & & \\ \vdots & \vdots & & & \\ 2n(n-1)+2 & [ 2n,2n+1 ] & & & \\ \hline \end{array}\ ] ] by the moishezon - teicher algorithm ( see section [ mt ] ) , we get the following skeletons : * for point @xmath91 , where @xmath100 , the corresponding final skeleton appears in figure [ braid1 - 6](a ) , where @xmath101 and @xmath102 . * for point @xmath103 , the corresponding final skeleton appears in figure [ braid1 - 6](b ) . * for point @xmath104 , the corresponding final skeleton appears in figure [ braid1 - 6](c ) . * for point @xmath91 , where @xmath105 , the corresponding final skeleton appears in figure [ braid1 - 6](d ) , where @xmath106 and @xmath107 . * for point @xmath91 , where @xmath108 , the corresponding final skeleton appears in figure [ braid1 - 6](e ) , where @xmath109 and @xmath110 . * for point @xmath111 , the corresponding final skeleton appears in figure [ braid1 - 6](f ) . before we proceed to the presentation of the fundamental group , we introduce one notation : instead of writing the relations ( where @xmath22 are words in a group ) : @xmath112 we will sometimes write : @xmath113 $ ] . by the van kampen theorem ( see section [ mt ] ) , we get the following presentation of the fundamental group of the line arrangement s complement : generators : @xmath114 + relations : + 1 . @xmath115=e$ ] , where @xmath101 and@xmath102 . 2 . @xmath116 $ ] . 3 . @xmath117 $ ] . @xmath118=e$ ] where @xmath106 and @xmath107 . 5 . @xmath119=e$ ] where @xmath109 and @xmath110 . @xmath120 $ ] . now , we show that all the conjugations in relations ( 1),(2),(3 ) and ( 4 ) can be simplified . we start with relations ( 1 ) , and then relations ( 2 ) . we continue to relations ( 4 ) and we finish with relations ( 3 ) . we start with the first set of relations : for @xmath121 , we get that for all @xmath101 we have : @xmath122=e$ ] . now , we proceed to @xmath123 . for @xmath124 , we get : @xmath125=e$ ] . by the relation @xmath126=e$ ] , it is simplified to @xmath127=e$ ] . in this way , we get that for @xmath123 , we have : @xmath128=e$ ] for @xmath101 . by increasing @xmath129 one by one , we get that all the conjugations disappear and we get @xmath130=e$ ] , where @xmath101 and @xmath102 , as needed . relations ( 2 ) can be written as : @xmath131 @xmath132 @xmath133 by the simplified version of relations ( 1 ) , we can omit all the generators @xmath134 . hence we get : @xmath135 as needed . we proceed to relations ( 4 ) . we start with @xmath136 . taking @xmath124 , we get : @xmath137=e.\ ] ] by relations ( 6 ) , we have : @xmath138 , and hence we get : @xmath139=e.\ ] ] for @xmath140 , we get : @xmath141=e.\ ] ] by relations ( 6 ) again and the simplified version of relations ( 1 ) , we get : @xmath142=e.\ ] ] using the simplified relation @xmath143=e$ ] , we get @xmath144=e$ ] . in the same way , we get that for @xmath136 and @xmath145 , we have : @xmath146=e$ ] . we continue to @xmath147 . taking @xmath124 , we have : @xmath148=e.\ ] ] by the simplified version of relations ( 1 ) , we can omit all the generators @xmath149 . hence we get : @xmath150=e.\ ] ] by relations ( 5 ) , we can omit @xmath151 too , and therefore : @xmath152=e$ ] . for @xmath140 , we have : @xmath153=e.\ ] ] by the simplified version of relations ( 1 ) , we can omit all the generators @xmath149 . hence we get : @xmath154=e.\ ] ] by relations ( 5 ) , we can omit @xmath151 too , and therefore : @xmath155=e.\ ] ] by @xmath152=e$ ] , we get @xmath156=e$ ] . in the same way , we get that for @xmath147 and @xmath145 , we get : @xmath157=e$ ] . in the same way , by increasing @xmath129 one by one , we will get that for all @xmath158 and @xmath145 , we get : @xmath130=e$ ] as needed . relations ( 3 ) can be written : @xmath159 @xmath160 @xmath161 by relations ( 1 ) , we can omit the generators @xmath134 , so we get : @xmath162 @xmath163 @xmath164 by relations ( 2 ) , we can omit also the generators @xmath165 in order to get : @xmath166 hence , we get the following simplified presentation : generators : @xmath114 + relations : 1 . @xmath130=e$ ] , where @xmath101 and @xmath102 . 2 . @xmath167 $ ] . 3 . @xmath168 $ ] . @xmath130=e$ ] where @xmath106 and @xmath107 . @xmath119=e$ ] where @xmath109 and @xmath110 . 6 . @xmath120 $ ] . therefore , we have a conjugation - free geometric presentation , and hence we are done . cohen and suciu @xcite give the following presentation of @xmath169 , which is known @xcite to be the fundamental group of the complement of the affine ceva arrangement ( see figure [ ceva ] ) : @xmath170 the actions of the automorphisms @xmath171 and @xmath172 are defined as follows : @xmath173 + @xmath174 @xmath175 + @xmath176 @xmath177 + @xmath178 notice that if we rotate clockwise the lowest line in the affine ceva arrangement ( figure [ ceva ] ) , we get an arrangement @xmath0 whose graph consists of a unique cycle of length @xmath2 , where all the multiple points are of multiplicity @xmath2 . by a simple check , the effect of this rotation is the addition of the commutator relation @xmath179=e$ ] to the presentation of the group . hence , we get that the actions of the automorphisms @xmath171 and @xmath172 are changed as follows : @xmath173 + @xmath174 @xmath180 @xmath177 + @xmath178 this is the presentation of the group : @xmath181 , where @xmath34 is the free product . to summarize , we get the following result : let @xmath0 be the arrangement of @xmath1 lines without parallel lines whose graph is a unique cycle of length @xmath2 , where all the multiple points are of multiplicity @xmath2 . then : @xmath33 it is interesting to check how this proposition can be generalize to arrangements whose graphs are cycles of length @xmath182 . in this section , we show that the fundamental group of a real affine arrangement whose graph is a unique cycle of any length and has no line with more than two multiple points , has a conjugation - free geometric presentation . at the end of this section , we generalize this result to arrangements whose graphs are unions of disjoint cycles . we start by investigating the case of a cycle of length @xmath183 and then we generalize it to any length . in figure [ cycle_multiple5 ] , we present a real arrangement whose graph is a cycle of @xmath183 multiple points ( note that any real arrangement whose graph is a unique cycle of @xmath183 multiple points and has no line with more than two multiple points , can be transferred to this drawing by rotations , translations and equisingular deformations ) . based on figure [ cycle_multiple5 ] , we get the list of lefschetz pairs presented in table [ tab2 ] . @xmath184 & 2 & & 13 & [ 6,7 ] & 2 & & 25 & [ 8,9 ] & 2 \\ 2 & [ 5,6 ] & 2 & & 14 & [ 7,8 ] & 2 & & 26 & [ 6,7 ] & 2 \\ 3 & [ 7,8 ] & 2 & & 15 & [ 3,4 ] & 2 & & 27 & [ 5,6 ] & 2 \\ 4 & [ 6,7 ] & 2 & & 16 & [ 4,5 ] & 2 & & 28 & [ 7,8 ] & 2 \\ 5 & [ 4,5 ] & 2 & & 17 & [ 5,6 ] & 2 & & 29 & [ 6,7 ] & 2 \\ 6 & [ 3,4 ] & 2 & & 18 & [ 6,7 ] & 2 & & 30 & [ 4,6 ] & 3 \\ 7 & [ 5,6 ] & 2 & & 19 & [ 4,6 ] & 3 & & 31 & [ 3,4 ] & 2 \\ 8 & [ 4,5 ] & 2 & & 20 & [ 3,4 ] & 2 & & 32 & [ 4,5 ] & 2 \\ 9 & [ 2,3 ] & 2 & & 21 & [ 4,5 ] & 2 & & 33 & [ 2,3 ] & 2 \\ 10 & [ 1,2 ] & 2 & & 22 & [ 8,9 ] & 2 & & 34 & [ 1,2 ] & 2 \\ 11 & [ 2,4 ] & 3 & & 23 & [ 7,8 ] & 2 & & 35 & [ 2,4 ] & 3 \\ 12 & [ 4,6 ] & 3 & & 24 & [ 9,10 ] & 2 & & & & \\ \hline\hline \end{array}\ ] ] by the moishezon - teicher algorithm ( see section [ mt ] ) , one can compute the skeletons of the braid monodromy . after the computation , one should notice that actually we can group the intersection points into blocks according to their braid monodromies ( see figure [ cycle_multiple5_block ] ) , since the structure of the skeletons is similar . following this observation , we can deal with each block separately . so , we get the following sets of skeletons : * quadruples of type q1 : see figure [ quadruple_case1](a ) for @xmath185 . * quadruples of type q2 : see figure [ quadruple_case2](a ) for @xmath186 , @xmath187 , @xmath188 . * a triple of type t1 : see figure [ triple_case1](a ) . * triples of type t2 : see figure [ triple_case2 ] for @xmath189 . * a triple of type t3 : see figure [ triple_case3](a ) . * a triple of type t4 : see figure [ triple_case4](a ) . now we pass to the general case . one can draw an arrangement of @xmath190 lines whose graph is a unique cycle of length @xmath16 and has no line with more than two multiple points in a similar way to the way we have drawn the arrangement of @xmath191 lines whose graph is a cycle of length @xmath183 . hence , one can compute the braid monodromy of the general arrangement in blocks similar to what we have done in the case of @xmath192 : * quadruples of type q1 : for @xmath193 , see figure [ quadruple_case1](b ) . * quadruples of type q2 : for @xmath194 , @xmath187 , @xmath195 , see figure [ quadruple_case2](b ) . * a triple of type t1 : see figure [ triple_case1](b ) . * triples of type t2 : for @xmath196 , see figure [ triple_case2 ] . * a triple of type t3 : see figure [ triple_case3](b ) . * a triple of type t4 : see figure [ triple_case4](b ) . by the van - kampen theorem ( see section [ mt ] ) , we get the following presentation of the fundamental group of the complement of the arrangement : generators : @xmath197 + relations : * from quadruples of type q1 : 1 . @xmath198=e$ ] where @xmath193 2 . @xmath199=e$ ] where @xmath193 3 . @xmath200=e$ ] where @xmath193 4 . @xmath201=e$ ] where @xmath193 * from quadruples of type q2 : for @xmath194 , @xmath187 , @xmath195 : 1 . @xmath202=e$ ] 2 . @xmath203=e$ ] 3 . @xmath204=e$ ] 4 . @xmath205=e$ ] * from the triple of type t1 : 1 . @xmath206=e$ ] 2 . @xmath207=e$ ] 3 . @xmath208 * from triples of type t2 : 1 . @xmath209 where @xmath196 2 . @xmath210=e$ ] where @xmath196 3 . @xmath211=e$ ] where @xmath196 * from the triple of type t3 : 1 . @xmath212 2 . @xmath213=e$ ] 3 . @xmath214=e$ ] * from the triple of type t4 : 1 . @xmath215=e$ ] 2 . @xmath216=e$ ] 3 . @xmath217 we now show that all the conjugations can be simplified , and hence we have a conjugation - free geometric presentation for the fundamental group . we have conjugations in the relations coming from triples of points and quadruples of points . we start with the relations which correspond to triples of points . the conjugation in relation ( 3 ) of the triple of type t1 can be simplified using relations ( 1 ) and ( 2 ) of the triple of type t1 . the conjugation in relation ( 1 ) of triples of type t2 can be simplified using relations ( 2 ) and ( 3 ) of the corresponding triples of type t2 . the conjugation in relation ( 3 ) of the triple of type t3 can be simplified using relation ( 3 ) of the triple of type t4 . the conjugation in relation ( 2 ) of the triple of type t3 can be simplified using relations ( 1 ) and ( 2 ) of the triple of type t4 . the conjugation in relation ( 1 ) of the triple of type t3 can be simplified using relations ( 2)(3 ) of the triple of type t3 and relations ( 1)(3 ) of the triple of type t4 . we continue to the relations induced by to the quadruples of type q1 . by the first two relations of the quadruples of type q1 , one can easily simplify the conjugations which appear in the last two relations of the quadruples of type q1 . so we get that the relations correspond to the quadruples of type q1 can be written without conjugations . now , we pass to the relations correspond to the quadruples of type q2 . we start with @xmath218 : we have the following relations : 1 . @xmath219=e$ ] 2 . @xmath220=e$ ] 3 . @xmath221=e$ ] 4 . @xmath222=e$ ] by relations ( 2 ) and ( 3 ) of triple t2 ( for @xmath223 ) , we have the relations @xmath224=e$ ] and @xmath225=e$ ] . hence , the conjugations in relation ( d ) are canceled and we get @xmath226=e$ ] . by the same relations and the simplified version of relation ( d ) , we get the following relation from relation ( a ) : @xmath227=e$ ] . substituting @xmath223 in relation ( 1 ) of triple t2 yields @xmath228 by this relation , relation ( c ) becomes @xmath229=e$ ] , and relation ( b ) becomes @xmath230=e$ ] . the same argument holds for any @xmath231 , where @xmath232 and @xmath233 . hence , one can simplify the conjugations in these cases . now , we pass to the case where @xmath234 and @xmath233 . let @xmath235 . we have the following relations : 1 . @xmath236=e$ ] 2 . @xmath237=e$ ] 3 . @xmath238=e$ ] 4 . @xmath239=e$ ] by relations ( 2 ) and ( 3 ) of triple t2 ( for @xmath223 ) , we have the relations @xmath224=e$ ] and @xmath225=e$ ] . hence , relations ( a ) and ( d ) become : 1 . @xmath240=e$ ] 2 . @xmath241=e$ ] by relations ( a ) and ( d ) above , we get @xmath242=e$ ] , and therefore we also get @xmath243=e$ ] . substituting @xmath223 in relation ( 1 ) of triple t2 yields @xmath228 by this relation , relations ( b ) and ( c ) become : 1 . @xmath244=e$ ] 2 . @xmath245=e$ ] by relations ( b ) and ( c ) above , we get @xmath246=e$ ] and hence @xmath247=e$ ] . it is easy to show by a simple induction that we can simplify the conjugations for any @xmath231 , where @xmath248 and @xmath233 . the remaining case is @xmath249 . we start with @xmath250 . we have the following relations : + \(a ) @xmath251=e$ ] + ( b ) @xmath252=e$ ] + ( c ) @xmath253=e$ ] + ( d ) @xmath254=e$ ] we will show that these conjugations can be simplified . by relation ( 3 ) of triple t4 and relation ( 3 ) of triple t3 , relations ( c ) and ( d ) can be written as : + ( c ) @xmath255=e$ ] + ( d ) @xmath256=e$ ] by relation ( 3 ) of triple t2 for @xmath257 , we have @xmath258=e$ ] , and hence relation ( c ) becomes @xmath259=e$ ] . by relation ( 1 ) of triple t2 for @xmath257 , we have : @xmath260 and then relation ( d ) becomes : @xmath261=e$ ] . now , we simplify relation ( a ) . again , by relations ( 2 ) and ( 3 ) of triple t2 for @xmath257 , we have @xmath258=e$ ] and @xmath262=e$ ] , and hence : @xmath263=e\ ] ] by relation ( 3 ) of triple t4 , we have : @xmath264=e.\ ] ] by relations ( 1 ) and ( 2 ) of triple t4 , we have : @xmath265=e.\ ] ] by relations ( 1 ) of triple t3 , we finally have : @xmath266=e$ ] . now , we simplify relation ( b ) . by relation ( 3 ) of triple t4 , we have : @xmath267=e.\ ] ] by relations ( 1 ) and ( 2 ) of triple t4 , we get : @xmath268=e.\ ] ] by relations ( 2 ) and ( 3 ) of triple t3 , we get : @xmath269=e.\ ] ] finally , by relation ( 1 ) of triple t2 for @xmath257 , we have @xmath260 so we get : @xmath270=e$ ] . by similar tricks , one can simplify the conjugations for all the cases where @xmath249 and @xmath271 . hence , we have a presentation based on the topological generators without conjugations in the relations , and hence we are done . the above proof is based on the fact that the multiplicity of each multiple point is @xmath2 . we now explain why it can be generalized to any multiplicity . in case of higher multiplicities , the quadruples from the previous case will be transformed to a block of @xmath272 simple points . it can be easily checked that all the conjugations can be simplified in this case . moreover , the triples from the previous case will be transformed into blocks similar to the blocks we had in the case of a cycle of length @xmath2 ( see proposition [ triangle - prop ] ) , and in this case too , it can be easily checked that all the conjugations can be simplified , and hence we have shown that arrangements whose graph is a unique cycle and have no line with more than two multiple points , have a conjugation - free geometric presentation . using the following decomposition theorem of oka and sakamoto @xcite , we can generalize the result from the case of one cycle to the case of a union of disjoint cycles : * ( oka - sakamoto ) * let @xmath273 and @xmath274 be algebraic plane curves in @xmath275 . assume that the intersection @xmath276 consists of distinct @xmath277 points , where @xmath278 are the respective degrees of @xmath273 and @xmath274 . then : @xmath279 hence , we have the following result : if the graph of the arrangement @xmath0 is a union of disjoint cycles of any length and the arrangement has no line with more than two multiple points , then its fundamental group has a conjugation - free geometric presentation . we would like to thank patrick dehornoy , uzi vishne and eran liberman for fruitful discussions . we owe special thanks to an anonymous referee for many useful corrections and advices and for pointing out the connection between our presentations and dehornoy s positive presentations ( remark [ rem_dehornoy ] ) . m. eliyahu , e. liberman , m. schaps and m. teicher , _ characterization of line arrangements for which the fundamental group of the complement is a direct product _ , alg . topo . , to appear . [ available online : http://arxiv.org/abs/0810.5533 ] . d. garber , _ on the connection between affine and projective fundamental groups of line arrangements and curves _ , singularits franco - japonaises ( j .- p . brasselet and t. suwa , eds . ) , sminaires & congrs * 10 * , 6170 ( 2005 ) . d. garber and m. teicher , _ the fundamental group s structure of the complement of some configurations of real line arrangements _ , complex analysis and algebraic geometry , edited by t. peternell and f .- o . schreyer , de gruyter , 173 - 223 ( 2000 ) . kulikov and m. teicher , _ braid monodromy factorizations and diffeomorphism types _ , izv . nauk ser . mat . * 64*(2 ) , 89120 ( 2000 ) [ russian ] ; english translation : izv . 64*(2 ) , 311341 ( 2000 ) . b. moishezon and m. teicher , _ braid group technique in complex geometry ii : from arrangements of lines and conics to cuspidal curves _ , in algebraic geometry , lect . notes in math . * 1479 * , 131180 ( 1990 ) .

we introduce the notion of a _ conjugation - free geometric presentation _ for a fundamental group of a line arrangement s complement , and we show that the fundamental groups of the following family of arrangements have a conjugation - free geometric presentation : a real arrangement @xmath0 , whose graph of multiple points is a union of disjoint cycles , has no line with more than two multiple points , and where the multiplicities of the multiple points are arbitrary . we also compute the exact group structure ( by means of a semi - direct product of groups ) of the arrangement of @xmath1 lines whose graph consists of a cycle of length @xmath2 , and all the multiple points have multiplicity @xmath2 .

the m-@xmath8 relation ( e.g. , * ? ? ? * ) has suggested that the evolution of galaxies and super - massive nuclear black holes ( smbhs ) are linked . both the stellar population and the smbh of a galaxy are thought to grow and evolve by merging of smaller gas - rich galaxies and their nuclear smbhs @xcite . during this process , the smbh may be buried " by thick molecular gas , which feeds the smbh at high rates , causing the birth of an obscured compton thick ( ct , * ? ? ? * ; * ? ? ? * ) active galactic nucleus ( agn ) . ct agns are characterized in the x rays by a hard high energy continuum , a reflection " flat continuum in the @xmath9 kev range , and a very high equivalent width ( ew ) @xmath10 kev 6.4 fe - k@xmath11 line . examples of this merger - driven evolution are given by the pairs of nuclei discovered in the 6.4 kev fe - k line with _ chandra _ in the merger infrared ( ir ) luminous galaxy ngc 6240 @xcite and in the ct agn ngc 3393 @xcite . at a distance of 76 mpc @xcite , arp 220 ( ic 4553/4 ) is both a merger , and the nearest ultra - luminous ir galaxy . near ir high - resolution ( 0.1 ) nicmos - hst imaging identifies the two nuclear regions of the merging galaxies , which are coincident with the two components of a double radio source @xcite . at a separation of 0.98 `` ( 361 pc at a distance of 76 mpc and @xmath12 @xcite . ] ) , these nuclei are closer together than the nuclei of ngc 6240 ( @xmath13 pc separation , @xcite ) , and therefore should be subject to even stronger gravitational interaction and possible accretion . indeed , the presence of an agn with a contribution to the bolometric luminosity between @xmath14 and @xmath15 , and a best estimate of 18% , is suggested by the spitzer mid - ir spectrum of the central 8 '' region of arp 220 ( @xcite ; more recent _ herschel _ results @xcite and modeling of the nuclear spectra @xcite agree with this conclusion . the presence of a maser and a rotating massive molecular disk suggests a massive nuclear black hole in the west nucleus of arp 220 . a high resolution study with a @xmath16 ks chandra acis observation ( obsid 869 ; * ? ? ? * ) failed to secure the firm identification of nuclear agn emission , reporting the presence of three hard x - ray sources in the region , of which two ( x-1 and x-4 , see their figure 2 ) are near , but not coincident with , the nuclear radio sources . the x - ray spectrum extracted from the central @xmath17 region showed complexity , with a possible hard component and fe - k line . subsequent _ xmm - newton _ observation detected fe - k line emission centered at 6.7 kev with ew @xmath18 kev @xcite suggesting highly photoionized , low - density gas illuminated by a hidden ct agn . a re- analysis of the _ chandra _ data @xcite only manages to set an upper limit on the fe - k@xmath11 ew assuming a 6.4 kev line energy . the nature of the x - ray emission of arp 220 is therefore still elusive . in this paper we re - examine the question of the x - ray agn emission of arp 220 , by means of sub - pixel imaging of chandra acis data in narrow spectral ranges . this technique has been used successfully to study crowded emission regions of nearby seyferts ( e.g. in ngc 4151 @xcite ; mrk 573 , @xcite ) ; in the nearby ct agn ngc 3393 , it has led to the discovery of two ct nuclei , with 150 pc separation @xcite . our new look at the nuclear region of arp 220 , has resulted in the discovery of two sources in the 6 - 7 kev fe - k band , strongly suggestive of ct nuclei . these sources are spatially coincident with the near - ir and radio positions . below we discuss our technique and results . arp 220 was observed by _ chandra _ on 2000 june 24 for 57 ks ( obs . i d 869 , pi : clements ) . level 2 event data were retrieved from the _ chandra _ data archive and reduced with the ciao ( @xcite ) 4.4 software and the _ chandra _ calibration data base ( caldb ) 4.5.3 , adopting standard procedures . after excluding time intervals of background flares exceeding @xmath19 with the lc_sigma_clip task , we obtained a low - background exposure time of @xmath20 ks . the nucleus has no significant pile up , as measured by the ciao pileup_map tool . imaging analysis was performed without pixel randomization to take advantage of the telescope dithering in event positioning and with the sub - pixel event repositioning ( ser ) procedure @xcite . we used a pixel size 1/4 of @xmath21 , the native _ chandra_/acis detector pixel . using the same orion acis - s data as in the calibration of @xcite , we find a significant @xmath22 ( improvement in psf fwhm as defined in @xcite ) from sub - pixel repositioning for an on - axis source at 6 - 7 kev ( @xmath23 at @xmath17 kev because of the narrower psf ) . most of the imaging improvement is from sub - pixel binning , which uses the sampling of the psf by the well characterized spacecraft dither motion . because of the similarly ` peaked ' inner psf this is similarly effective at 2 and 6 kev . the resulting full band ( 0.5 - 10 kev ) acis image is presented in the left panel of figure [ fullband ] . this figure shows complexity in the central region of arp 220 , but there is no x - ray feature that can be univocally associated with the radio / ir nuclei . instead , narrow band imaging ( 6 - 7 kev containing the fe - k lines ) reveals these hidden nuclei ( figure [ fullband ] , right panel ) . the only sources of emission in this spectral band are localized in two regions separated by @xmath24 ( corresponding to @xmath0 pc at a distance of 76 mpc ) and co - located with the nir @xmath25 @xcite and radio nuclei @xcite . following @xcite , we have shifted the _ vla _ 6 cm sources in the ne direction by @xmath26 in order to match the position of the western lobe with the western nucleus . we note that the eastern radio lobe results somewhat dislocated from eastern narrow - band nucleus . however , due to the low counts in this region , the location of this emission is consistent within uncertainties with the radio lobe . under this assumptions , radio , nir and fe - k nuclei are consistent within astrometric uncertainties . deeper _ chandra _ x - ray observations are needed for convincingly evaluate if the position of the e nucleus is in better agreement with the ir or radio position . we note that the extension of the narrow - band emission in the east direction is not due to psf asymmetries , as indicated by the make_psf_asymmetry_region tool that shows psf artifacts in the north - west direction . we extracted counts from the 0.5 circles centered at ra=15:34:57.252 dec=+23:30:11.64 ( w ) and ra=15:34:57.326 dec=+23:30:11.84 ( e ) ( figure [ hardbands ] , upper panel ) ; _ chandra _ absolute astrometric uncertainty is 0".6 . we find in the 6 - 7 kev band 12 counts associated with the western nucleus ( w ) , and 3 counts associated with the eastern nucleus ( e ) . note that the background emission in this band is 0.01 counts in the same area of the extraction regions , so even a 3 counts detection is a highly significant source ( @xmath27 of chance detection , corresponding to a @xmath28 gaussian significance ) . given the energy dependent grade branching ratio for bi acis ccd @xcite , and since sub - pixel repositioning is more uncertain for 1 pixel ( grade=0 ) events than for 2 pixel ( grade=2,3,4 ) and 4 pixel ( grade=6 ) events , we checked the event grade distribution in w and e regions to ensure reliability of events position with ser . the majority of w region events are 2 and 4 pixel , while no 1 pixel event is found in e region . therefore sub - pixel analysis improves the positioning of these events . the flux in this band , however , is highly uncertain due to the low counts . in this case , approximate narrow - band model - independent fluxes can be estimated with the ciao aprates tool , to yield @xmath29 ( e ) and @xmath30 ( w ) . these correspond to 6 - 7 kev observed luminosities @xmath31 ( e ) and @xmath32 ( w ) . the locations of these unique emission regions strongly argue for an identification of these sources with the nuclei of the merging galaxies . although the w source is positionally coincident with the x-4 hard x - ray source reported by @xcite ( see figure [ fullband ] ) , the sub - pixel narrow band imaging suggests a different picture . figure [ hardbands ] ( middle panel ) shows the 3 - 6 kev band image , where the most prominent source is found to the west of both w and of the reported x-4 position @xcite . figure [ hardbands ] ( lower panel ) shows the emission in the 7 - 10 kev band , suggesting hard continuum emission from the w nucleus ( 3 cts ) with @xmath33 of chance detection . the ciao aprates tool gives in this region @xmath34 cts , while we derive a firm 2 counts upper limit in the e nucleus region . these images suggest the presence of highly obscured ct agn in the nuclei . the narrow - band images ( figure [ hardbands ] ) also show that the continuum emission from the nuclei is contaminated by unrelated sources even at _ chandra _ resolution . nevertheless , we attempted a spectral characterization of the emission , extracting 3 - 8 kev spectra from the two circular regions ( 0.5 " radius ) indicated in figure [ hardbands ] with the ciao specextract task , applying point - source aperture correction . we then fitted simultaneously the two spectra employing the cash statistic . we used a model typical of ct agn emission @xcite , comprising an absorption component fixed to the galactic value @xmath35 , a pure neutral reflection component ( pexrav ) with a spectral index fixed to 1.8 and a gaussian fe - k line . we used both xspec ( ver . 12.7.1 ) and sherpa with identical results . the extracted spectra and the best - fit parameters are presented in figure [ spectra ] and table [ table2]-@xmath8 confidence level for one parameter of interest . ] , respectively . given the contamination of the continuum emission by non - nuclear sources , our estimate of the nuclear continuum luminosity is an upper limit , and the fe- k ews must be considered as lower limits . the spectral analysis detects fe - k line features in both regions , with comparable ew ( 1.1 kev and 0.9 kev in w and e region , respectively ) . as expected from the imaging , the fe - k line is more luminous in the w nucleus ( @xmath36 ) with respect to the e ( @xmath37 ) ; we note that these values are consistent within errors with the narrow - band ( 6 - 7 kev ) luminosities obtained with aprates tool . as already discussed , the _ observed _ 2 - 10 kev luminosities , @xmath38 ( w ) and @xmath39 ( e ) , should be considered as upper limits because of contamination . our detection of large fe - k ews from both nuclei may seem at odds with the results of @xcite , who did not detect fe - k emission using the same chandra acis data . to investigate this discrepancy , we repeated the analysis of the entire central emission of arp 220 , following the procedure of these authors . using a circular 4.5 " radius count extraction region centered at ra=15:34:57.194 dec=+23:30:12.40 , and their fitting model , we confirm their results ( table [ table ] , left column of results ) . however , the authors fixed the fe - k line rest - frame energy fixed at 6.4 kev . if , instead , we allow the gaussian line rest - frame energy to vary , we detect a line at @xmath40 kev with and equivalent width of @xmath41 kev , compatible with the _ xmm - newton _ detection at @xmath42 kev @xcite ; all the other model parameters remain unchanged ( see the right column in table [ table ] ) . as demonstrated by figures [ fullband ] and [ hardbands ] there is no 6 - 7 kev emissions in regions outside the nuclei , even when the overall x - ray emission is more prominent . we have also extracted the spectrum of the softer luminous regions west of the w nucleus shown in figure [ hardbands ] ( middle - right panel ) centered at ra=15:34:57.207 dec=+23:30:11.84 , and find no evidence of line emission . we analyzed the four archival _ xmm - newton _ observations of arp 220 ( the two discussed in @xcite , and two new ones performed in 2005 ) in order to test the possible contribution of a neutral iron emission line to the 6 - 7 kev emission we see in figure [ hardbands ] . the data were reduced following a standard procedure , analogous to the one described by @xcite . the results are also in agreement : in a continuum plus single line model we obtain a best fit peak energy @xmath43 . however , if we fit the data with two lines with fixed peak energies @xmath44 and @xmath45 , we obtain the results shown in figure [ contours ] : a neutral component accounting for up to 40% of the observed line flux can not be ruled out at a 90% confidence level . if we then consider that 40% of the fe - k line flux we estimate from _ chandra _ spectra is due to fe - k@xmath11 neutral 6.4 kev emission line , the 2 - 10 kev _ emitted _ luminosities inferred from the fe - k luminosities are @xmath46 ( w ) and @xmath47 ( e ) . we note that these corrections are calibrated on standard " obscured seyfert galaxies , with an x - ray reflection efficiency of a few percent @xcite . hard x - ray observation of ulirgs have demonstrated that on average this efficiency is much lower for these sources @xcite . consequently , the intrinsic x - ray luminosity of the two agn detected here could be significantly higher . considering the values for a standard reflection efficiency , the inferred x - ray luminosity is at least a factor of 3 higher than that expected from a pure starburst with the bolometric luminosity of arp 220 . the chandra acis sub - pixel narrow - band imaging of the central region of the ulirg merger arp 220 provides compelling evidence of two ct agns , in both nuclei of the merging galaxies . within the central 5x5 , there are _ only _ two sources detected in the 6 - 7 kev band , containing the fe - k lines . although the e nucleus is detected with only 3 counts , the very low field background ( @xmath48 ) makes this a @xmath49 detection ( section [ data ] ) . the centroids of these sources , @xmath1 apart , are consistent , within _ chandra _ astrometric uncertainty of 0.6 , with the position of the two nir nuclear clusters identified by @xcite , and each coincident with a 6 cm vla radio source @xcite ( figure [ fullband ] ) . we note that the w source is also consistent with the variable radio sources reported by . we stress that _ no emission _ in the 6 - 7 kev band is detected from other parts of the central region of arp 220 , even where the diffuse emission from the starburst is most intense . while the detections are highly significant for both nuclei , the fluxes are more uncertain , given the small number of detected photons ( see section [ data ] ) . the spectral analysis of both the central 4.5 region , and of the individual e and w nuclei , results in the detection of emission lines . the rest energy of the line is larger than the 6.4 kev of the k@xmath11 line , and suggests a contribution from 6.7 kev shock - ionized fe xxv line as concluded by @xcite , which could be associated with the starburst - induced shock ( see e.g. the strong extended fe xxv emission in ngc 6240 @xcite ) . however , if we _ assume _ that both 6.4 kev and 6.7 kev lines are present in the spectrum , we obtain an acceptable fit to the _ xmm - newton _ data which allow for 40% 6.4 kev contribution . the statistics , however , do not allow us to disentangle these two contributions to the observed emission in _ chandra _ data . however , as discussed in section [ data ] , the line emission is only connected with the nuclear region , not with the more extended starburst . although the uncertainties are large , we find large @xmath50 kev equivalent widths for the fe - k lines ; simulated data show that shock - ionized gas can yield comparable ews for these lines , but would also yield continuum contribution much higher than we observe . besides a definite detection of hard photons ( @xmath51 kev ) from the w nucleus , which is also consistent with it being a ct agn , the continuum emission from the nuclear regions is not easy to establish because of contamination from the surrounding emission in the 3 - 6 kev band . in particular , the w nucleus is significantly contaminated by the softer emission of a source or extended emission area nearby , which could be connected with the starburst phenomenon . this complex circumnuclear emission impedes the measurement of the nuclear continuum by means of spectral fitting . any determination of the nuclear continuum with lesser spatial resolution data will give an even higher overestimation of its strength . considering the above caveats , we can only set an upper limit of @xmath52 to the ratio between the observed and the inferred intrinsic luminosities for the two nuclear sources ( see section [ data ] ) . this limit is consistent with the ratio between the observed luminosities of less obscured and ct agns . the spectra of the two nuclear sources presented in figure [ spectra ] are in fact typical of ct agns , with observed fe - k ew @xmath10 kev . the possibility of arp 220 harboring one heavily obscured agn has been discussed by @xcite with _ _ beppo__sax and _ xmm - newton _ data ; these authors conclude that , due to lack of hard emission above 10 kev , the ct agn should be enshrouded in absorbing clouds with column density exceeding @xmath53 and covering factor close to unity . even ignoring the contamination of the continuum by extra - nuclear sources , the low counts in the two regions do not allow us to estimate absorption through spectral fitting . we can however compare the x - ray spectral energy distribution ( sed ) of the w source , for which we detect the hard nuclear continuum , with simulated spectra of obscured agns from . for this purpose we use only the spectrum at energies @xmath54 kev , where the continuum is low and less contaminated by the nearby source ( see figure [ spectra ] ) . the sed of the w nucleus is consistent with emission absorbed by @xmath55 . for comparison we extracted the spectrum from the bright region visible in figure [ hardbands ] ( middle - right panel ) west of w region ; besides not showing any appreciable fe - k feature , it is also compatible with absorbing column densities @xmath56 . the two nuclei have radio fluxes of 112 ( w ) and 88.2 ( e ) mjy @xcite , and x - ray to optical ratios @xmath57 . from the inferred 2 - 10 kev emitted luminosities of the two nuclear sources we evaluate bolometric luminosities assuming typical agn seds @xcite , and the x - ray reflection efficiency of seyfert galaxies , as discussed in section [ data ] . the estimated agn bolometric luminosities , which should be regarded as lower limits , are @xmath58 ( w ) and @xmath59 ( e ) . these represent only a few percent of arp 220 bolometric luminosity , confirming that overall the emission of arp 220 is dominated by the starburst component . this result is in broad agreement with the estimates from the mid - ir spectroscopy @xcite . lower limits on associated bh masses can be evaluated assuming eddington limited accretion ( with a standard 10% accretion rate to luminosity conversion efficiency ) , yielding @xmath60 ( w ) and @xmath61 ( e ) . our results add to the evidence of ct agns arising as a result of the later stages of the merging evolution in galaxies , which may trigger accretion onto the supermassive black holes ( e.g. , * ? ? ? after ngc 6240 @xcite and ngc 3393 @xcite , arp 220 provides the third clear case of the occurrence of this phenomenon in the near universe . as ngc 6240 @xcite , arp 220 is a highly disturbed system of galaxies engaged in a major merging interaction . the physical projected separation of the ct nuclei is @xmath62 pc in ngc 6240 and @xmath63 pc in arp 220 , suggesting that the latter may be in a more advanced stage of merging . the third case , the apparently regular early- type spiral ngc 3393 , with two ct nuclei separated by 150 pc @xcite , suggested instead a more evolved merger , or perhaps a minor merger of unequal size galaxies . interestingly , following the discovery of the double nuclear x - ray source , evidence of a merger past has surfaced in the optical spectra of this galaxy @xcite . we have found compelling evidence for two ct agns , associated with the nuclei of the merging galaxies in the nearest ulirg arp 220 , making this galaxy the third case of detection of close pair ct agns ( a few 100 pc apart ) , after ngc 6240 and ngc 3393 . these nuclei are the sole regions of significant 6 - 7 kev emission in the central 2 kpc of arp 220 . the data are consistent with fe - k line emission although at an energy possibly higher than 6.4 kev of the fe - k@xmath11 line , suggesting substantial contribution from fe xxv emission lines ( see also * ? ? ? our analysis of the entire _ xmm - newton _ dataset confirms the presence of fe xxv emission line , but allows 40% of the narrow - band emitted flux form the neutral 6.4 kev line . albeit uncertain - the spectral analysis of these regions suggests large @xmath1 kev ews . the w nucleus was also detected at hard ( @xmath51 kev ) energies implying absorption @xmath64 ; at such energies no continuum emission was detected from the e nucleus . our results are consistent with previous multi - wavelength indications of nuclear activity in arp 220 ( see section [ intro ] ) , and strengthen the evolutionary association of merging and nuclear activity in galaxies ( e.g. , * ? ? ? * ; * ? ? ? based on the fe - k detections , we infer lower limits on the bolometric luminosity of the agns of @xmath65 for the w agn , and @xmath66 for the e agn . these are a few percent of the total ulirg bolometric luminosity , confirming that overall the emission of this source is dominated by the starburst component , as estimated from the mid - ir spectroscopy @xcite . these results have only been possible because of the unmatched _ chandra _ spatial resolution , and the use of sub - pixel imaging in narrow spectral bands , which expose the telltale fe - k emission of ct agns , and give us a clear picture of the nuclear surroundings . although the fe - k band detections of the two nuclei are highly significant , the paucity of photons results in large uncertainties for all derived quantities . a significantly longer _ chandra_/acis exposure will be needed to firmly measure the emission parameters of the two nuclei . aalto s. , wilner d. , spaans m. , wiedner m. c. , sakamoto k. , black j. h. , caldas m. , 2009 , a&a , 493 , 481 baan w. a. , haschick a. d. , 1995 , apj , 454 , 745 batejat f. , conway j. e. , rushton a. , parra r. , diamond p. j. , lonsdale c. j. , lonsdale c. j. , 2012 , a&a , 542 , l24 clements d. l. , mcdowell j. c. , shaked s. , baker a. c. , borne k. , colina l. , lamb s. a. , mundell c. , 2002 , apj , 581 , 974 contini m. , 2012 , mnras , 247 di matteo t. , springel v. , hernquist l. , 2005 , natur , 433 , 604 downes d. , eckart a. , 2007 , a&a , 468 , l57 elvis m. , et al . , 1994 , apjs , 95 , 1 elvis m. , risaliti g. , zamorani g. , 2002 , apj , 565 , l75 fabbiano g. , wang j. , elvis m. , risaliti g. , 2011 , natur , 477 , 431 fruscione , a. , mcdowell , j. c. , allen , g. e. , et al . 2006 , , 6270 gilli r. , comastri a. , hasinger g. , 2007 , a&a , 463 , 79 harris , d. e. , mossman , a. e. , & walker , r. c. 2004 , , 615 , 161 hopkins a. m. , mcclure - griffiths n. m. , gaensler b. m. , 2008 , apj , 682 , l13 iwasawa k. , matt g. , guainazzi m. , fabian a. c. , 2001 , mnras , 326 , 894 iwasawa k. , sanders d. b. , evans a. s. , trentham n. , miniutti g. , spoon h. w. w. , 2005 , mnras , 357 , 565 kim d .- c . , sanders d. b. , 1998 , apjs , 119 , 41 komatsu , e. , smith , k. m. , dunkley , j. , et al . 2011 , , 192 , 18 komossa s. , burwitz v. , hasinger g. , predehl p. , kaastra j. s. , ikebe y. , 2003 , apj , 582 , l15 lamassa s. m. , heckman t. m. , ptak a. , martins l. , wild v. , sonnentrucker p. , hornschemeier a. , 2011 , apj , 729 , 52 levenson n. a. , heckman t. m. , krolik j. h. , weaver k. a. , ycki p. t. , 2006 , apj , 648 , 111 li , j. , kastner , j. h. , prigozhin , g. y. , & schulz , n. s. 2003 , , 590 , 586 magorrian j. , et al . , 1998 , aj , 115 , 2285 maiolino r. , salvati m. , bassani l. , dadina m. , della ceca r. , matt g. , risaliti g. , zamorani g. , 1998 , a&a , 338 , 781 matt g. , et al . , 1997 , a&a , 325 , l13 matt g. , fabian a. c. , guainazzi m. , iwasawa k. , bassani l. , malaguti g. , 2000 , mnras , 318 , 173 nardini e. , risaliti g. , watabe y. , salvati m. , sani e. , 2010 , mnras , 405 , 2505 nardini e. , risaliti g. , 2011 , mnras , 415 , 619 paggi a. , wang j. , fabbiano g. , elvis m. , karovska m. , 2012 , apj , 756 , 39 palmeri p. , mendoza c. , kallman t. r. , bautista m. a. , melndez m. , 2003 , a&a , 410 , 359 perlman , e. s. , padgett , c. a. , georganopoulos , m. , et al . 2010 , , 708 , 171 ranalli p. , comastri a. , setti g. , 2003 , a&a , 399 , 39 rangwala , n. , maloney , p. r. , glenn , j. , et al . 2011 , , 743 , 94 risaliti g. , maiolino r. , salvati m. , 1999 , apj , 522 , 157 sanders d. b. , mirabel i. f. , 1996 , ara&a , 34 , 749 scoville n. z. , et al . , 1998 , apj , 492 , l107 siemiginowska , a. , stawarz , . , cheung , c. c. , et al . 2007 , , 657 , 145 soifer b. t. , sanders d. b. , madore b. f. , neugebauer g. , danielson g. e. , elias j. h. , lonsdale c. j. , rice w. l. , 1987 , apj , 320 , 238 teng s. h. , veilleux s. , baker a. j. , 2012 , aas , 220 , # 409.03 van wassenhove s. , volonteri m. , mayer l. , dotti m. , bellovary j. , callegari s. , 2012 , apj , 748 , l7 veilleux s. , et al . , 2009 , apjs , 182 , 628 wang j. , et al . , 2011a , apj , 729 , 75 wang j. , fabbiano g. , elvis m. , risaliti g. , mundell c. g. , karovska m. , zezas a. , 2011b , apj , 736 , 62 wang j. , et al . , 2011c , apj , 742 , 23 wang j. , et al . , 2013 , in preparation counts . the large circles are our count extraction areas . right frame : same as left frame but with a 2x2 pixel fwhm gaussian filter smoothing applied . ( middle panel ) same as upper panel but in the 3 - 6 kev band . ( lower panel ) same as upper panel but in the 7 - 10 kev band . ]

narrow - band spectral imaging with sub - pixel resolution of the _ chandra_-acis archival observation of the ulirg merger arp 220 strongly suggests two compton thick nuclei , spatially coincident with the infrared and radio emitting nuclear clusters , and separated by 1 ( @xmath0 pc at a distance of 76 mpc ) . these previously undetected highly obscured agns - west ( w ) and east ( e ) - are imaged , and separated from neighboring sources , in the 6 - 7 kev band , where the fe - k lines dominate the emission . the western nucleus is also detected at energies above 7 kev . we estimate fe - k equivalent width @xmath1 kev or possibly greater for both sources , and observed 2 - 10 kev luminosities @xmath2 ( w ) and @xmath3 ( e ) . from the observed fe - k lines luminosities , and assuming on the basis of the _ xmm - newton _ spectrum that 40% of this may be from the 6.4 kev component , we evaluate 2 - 10 kev intrinsic luminosities @xmath4 ( w ) and @xmath5 ( e ) . the inferred x - ray luminosity is at least a factor of 3 higher than that expected from a pure starburst with the bolometric luminosity of arp 220 . for a typical agn sed the bolometric luminosities are @xmath6 ( w ) and @xmath7 ( e ) .

a priori , the description of any dynamic process in atomic physics should be gauge invariant . yet , there is a caveat . namely , one usually calculates atomic transitions using wave functions obtained from the solution of the unperturbed schrdinger equation , and ignores both the gauge transformation of the wave function as well as the fact that the physical interpretation of the wave function changes under a gauge transformation from the velocity to the length gauge . lamb @xcite has shown that if one insists on using ordinary schrdinger wave functions in off - resonant one - photon transition matrix elements , then the length gauge form has to be used for the laser field . here , we refer to off - resonant transitions as those where the frequency of the incident radiation is not exactly equal to the resonance frequency of the atom ; generalized to two - photon transitions , this implies that the sum frequency of the two photons does not exact match the energy ( frequency ) difference of the ground and excited level . for off - resonant two - photon transitions , one might think that the transition matrix element could be equal in the length and velocity gauges if all possible intermediate , virtual states are included in the calculation . here , inspired by refs . @xcite , we aim to reinvestigate the status of the gauge invariance of two - photon transition matrix elements ( `` length '' versus `` velocity '' gauges ) . note that invariance under the change from the `` length '' to the `` velocity '' gauges implies that the gauge transformation of the wave function is ignored ( we also refer to the invariance of matrix elements under the neglect of the wave function transformation and under the neglect of any necessary reinterpretation of physical operators as the `` extended gauge invariance '' ) . in refs . @xcite , the two - photon transition matrix element has been examined without any additional condition enforced upon the laser frequency ; the laser might be off - resonant or on resonance . as will be explained below , the arguments given in refs . @xcite appear to be applicable only at exact resonance . the role of the intermediate , virtual quantum states in the gauge invariance will be analyzed here in terms of general identities , applicable to various physical processes . second , from a conceptual point of view , the role of the gauge transformation of the wave function and the concomitant change in its interpretation appears to profit from further explanatory remarks beyond the problem at hand . the subject matter of this article is rather basic quantum mechanics , supplemented with explicit analytic results for the length and velocity forms of the @xmath1@xmath2 two - photon transition matrix element in hydrogen . we start by investigating gauge transformations and physical interpretations of operators in sec . [ sec2 ] , before reexamining the `` gauge invariance '' of the ac stark shift under a change of the interaction hamiltonian from the length to the velocity form ( sec . [ sec3 ] ) . here , `` gauge invariance '' has to taken with a grain of salt ; the invariance of the theoretical expressions for the ac stark shift holds even if the mandatory gauge transformation of the wave function is neglected , a fact on which we shall comment in sec . [ sec4 ] . in sec . [ sec3 ] , we shall examine , based on explicit analytic and numerical calculations , the behavior of a typical two - photon transition matrix element ( namely , of the @xmath1@xmath2 two - photon transition in hydrogen ) off resonance . we aim to show that the inclusion of the intermediate states does not solve the problem of gauge invariance , but the gauge dependence is due to a change in the physical interpretation of the wave function under the presence of a nonvanishing vector potential . the physically correct result for the transition rate off resonance is obtained in the length gauge . in order to illustrate that gauge transformations can change the physical interpretation of operators , let us start from a trivial example . we consider a wave function @xmath3 where @xmath4 is the normalization volume ; it describes a particle at rest . a unitary `` gauge '' transformation of the form @xmath5 is applied . the momentum operator in the free hamiltonian @xmath6 , with @xmath7 , transforms as @xmath8 the gauge - transformed hamiltonian thus reads as @xmath9 , and the interpretation of the momentum operator @xmath10 has changed : namely , the kinetic momentum operator no longer is @xmath11 , but @xmath12 . indeed , @xmath13 is the conjugate variable of the position operator @xmath14 ( see also the appendix of ref . @xcite ) . a similar situation is encountered in electrodynamics @xcite . the time - dependent wave function receives a unitary gauge transform , @xmath15 the momentum operator transforms as @xmath16 where @xmath17 is the electron charge . the following hamiltonian describes the atom - electromagnetic field dynamical system consisting of an electron coupled to a vector potential @xmath18 , in the binding ( coulomb ) potential @xmath19 , @xmath20 the product @xmath21 of electron charge and binding scalar potential is often denoted as @xmath4 , because it acts as a potential term in the hamiltonian . typicall , the vector potential @xmath18 describes a laser . the unitary gauge transformation , applied to @xmath22 , leads to the transformed hamiltonian @xmath23 , @xmath24 where @xmath25 is the gauge - transformed vector potential . conversely , for @xmath26 , and @xmath27 , one asserts that the interpretation of the momentum operator @xmath28 changes ; it no longer describes the kinetic momentum . the place of the latter is taken by the conjugate variable of position , namely , @xmath29 . however , under the gauge transformation , the physical interpretation of the hamiltonian also changes . a priori , the hamiltonian @xmath22 is equal to the the time derivative operator @xmath30 . after the gauge transformation , it is equal to a unitarily transformed time derivative operator , @xmath31 the new time derivative operator is thus obtained by setting equal the unitarily transformed hamiltonian and the unitarily transformed time derivative operator , and reads as @xmath32 where @xmath33 is the gauge - transformed scalar potential . from the above derivation , which in principle recalls well - known facts , it is immediately obvious that the gauge - transformed hamiltonian @xmath34 can not be obtained from @xmath22 by a unitary transformation . conversely , the unitarily transformed @xmath23 can not be interpreted any more as the time derivative operator when acting on the gauge - transformed wave function . in order to fix ideas , it is useful to examine the velocity - gauge one - photon transition matrix element @xmath35 , @xmath36 \to & \ ; \left < \phi_f \left| -e \ , \frac{\vec a \cdot \vec p}{m } \right| \phi_i \right > = \left < \phi_f \left| -e \ , \frac{{{\mathrm i}}}{\hbar } \ , \vec a \cdot [ h , \vec r ] \right| \phi_i \right > \nonumber\\[0.133ex ] = & \ ; -e \ , \frac{{{\mathrm i}}}{\hbar } ( e_f - e_i ) \ , \left < \phi_f \left| \vec a \cdot \vec r \right| \phi_i \right>\end{aligned}\ ] ] of an initial atomic state @xmath37 and a final state @xmath38 . [ we have made the dipole approximation @xmath39 and assumed that the seagull term @xmath40 does not contribute because of the angular symmetry of the initial and final states involved in the dipole transition . ] in view of the identity @xmath41 , the length - gauge one - photon matrix elements @xmath42 is related to its velocity - gauge counterpart @xmath35 as follows , @xmath43 \to & \ ; - { { \mathrm i}}\omega \ , \frac{e}{m } \ , \left < \phi_f \left| \vec a \cdot \vec r \right| \phi_i \right > = \frac{\hbar \omega}{e_f - e_i } \ , { \mathcal{m}}_v \,.\end{aligned}\ ] ] the velocity - gauge expression @xmath35 differs from its length - gauge counterpart @xmath42 by an additional factor @xmath44 . hence , in a remark on p. 268 of ref . @xcite , lamb observed that the physical interpretation of the wave function is preserved only in the length gauge , and `` no additional factor @xmath44 actually occurs '' . indeed , the physical interpretation of the momentum operator in the schrdinger coulomb hamiltonian is preserved only in the length gauge after a laser field is switched on . in other words , the problems off resonance with the velocity gauge result from the fact that one uses two wave functions , which are eigenstates of a hamiltonian that involves the momentum operator @xmath11 , and formulates the interaction hamiltonian @xmath45 with an expression that also involves the momentum operator @xmath11 , but in a situation where @xmath11 loses the original physical interpretation that it had in the unperturbed schrdinger coulomb hamiltonian . alternatively , one can also argue that the electric field used in the length - gauge interaction is gauge invariant , while the vector potential in the velocity - gauge term is not @xcite . within the dipole approximation , the atomic hamiltonian reads as follows [ we denote the laser field by @xmath46 and the binding coulomb potential by @xmath4 ] @xmath48 under a gauge transformation @xmath49 , this hamiltonian is transformed to @xmath50 ^ 2}{2 m } + v \,.\ ] ] as we have seen , the two hamiltonians @xmath51 and @xmath52 are not related by a unitary transformation , and furthermore , matrix elements of the interaction hamiltonians @xmath53 and @xmath54 differ off resonance . despite this fact , a number of processes such as the ac stark shift are `` gauge - invariant '' ( in an extended sense ) under a replacement of the interaction @xmath55 by @xmath56 , even without any gauge transformation of the wave function . before we discuss the reasons why `` extended gauge invariance '' holds or fails for a given physical problem , we shall reexamine the extended gauge invariance of the ac stark shift , and the failure of the extended gauge invariance in the case of the two - photon transition matrix element . the derivation of the ac stark shift is easiest in a second - quantized formalism , where the @xmath57-polarized laser field in the dipole approximation is modeled by a field operator , resulting in a length - gauge interaction [ hli ] @xmath58 h_\ell = & \ ; -e \ , z \ , e_l \,.\end{aligned}\ ] ] here , the @xmath59 and @xmath60 are the annihilation and creation operators for laser photons . the velocity - gauge interaction is given as @xmath61 h^{(1)}_v = & \ ; -\frac{e \ , a_l \ , p_z}{m } \ , , \qquad h^{(2)}_v = \frac{e^2 \ , a_l^2}{m } \,.\end{aligned}\ ] ] the unperturbed hamiltonian is the sum of an atomic hamiltonian @xmath62 and the hamiltonian @xmath63 which describes the electromagnetic field ( laser mode @xmath64 and modes @xmath65 other than the laser field ) , @xmath66 h_a = & \ ; \sum_m e_m \ , | \phi_m \rangle \ , \langle \phi_m | \ , , \\[0.133ex ] h_{em } = & \ ; \sum_{\vec k \ , \lambda \neq l } \hbar \omega_{\vec k \ , \lambda } a_{\vec k \ , \lambda}^+ \ ; a_{\vec k \ , \lambda } + \hbar \omega \ ; a^+_l \ ; a_l \,.\end{aligned}\ ] ] the unperturbed state @xmath67 with the atom in state @xmath68 and @xmath69 laser photons fulfills the relationships @xmath70 h_0 \ , | \phi_0 \rangle = & \ ; e_0 \ , | \phi_0 \rangle \ , , \qquad e_0 = e + n_l \hbar \omega \,.\end{aligned}\ ] ] the reduced green function for the combined system of atom and radiation field is given by @xmath71 where @xmath72 contains both atomic as well as laser - field terms and the prime on the green function denotes the omission of the reference state @xmath73 of the combined atom@xmath74field system from the sum over intermediate state . in the length gauge , the second - order ac stark shift can be expressed as @xmath75 the laser - field intensity is @xmath76 and the polarizability is given as @xmath77 ( we here assume a radially symmetric reference state @xmath78 . ) in the language of second quantization @xcite , the two terms with an opposite sign of @xmath79 in the denominator are generated by paired photon annihilation and photon creation operators from eq . . the extended gauge invariance of the ac stark shift in atomic hydrogen relies on the fact that @xmath80 given in eq . can alternatively be expressed as @xmath81 after treating the photons , the extended gauge invariance is easily shown to be equivalent to the identity @xmath82 = \omega^2 \ , \sum_\pm \left < \phi \left| \vec r \ , \frac{m}{h_a - e \pm \omega } \ , \vec r \right| \phi \right > \,,\end{gathered}\ ] ] where @xmath83 is the normalization integral ; the corresponding term originates from the seagull hamiltonian @xmath84 . in order to show this identity , we generalize the problem somewhat and write the following two matrix elements , @xmath85 q(\omega ) = & \ ; \left < \phi_f \left| \vec r \ , \frac{1}{h_a - e_n - \omega } \ , \vec r \right| \phi_i \right > \,,\end{aligned}\ ] ] for two ( not necessarily equal ) atomic states @xmath86 and @xmath87 . repeated application of the commutator relations @xmath88 \ , , \qquad \qquad h_a = \frac{\vec{p}^{\,2}}{2 m } - \frac{e^2}{4 \pi \epsilon_0 r}\,,\ ] ] results in the equality @xmath89 = ( e_f - e_i - \hbar \omega ) ( -\hbar \omega ) \left < \phi_f \left| \vec r \ , \frac{1}{h_a - e_i - \omega } \ , \vec r \right| \phi_i \right > \nonumber\\[0.1133ex ] + ( \hbar \omega - e_f ) \ , \left < \phi_f \left| \vec r^{\,2 } \right| \phi_i \right > + \left < \phi_f \left| \vec r \ , h_a \ , \vec r \right| \phi_i \right > \,.\end{gathered}\ ] ] one rewrites this expression using the operator identity @xmath90 + \vec r^{\,2 } \ , h_a + h_a \ , \vec r^{\,2 } \right ) \,,\ ] ] applies the hamiltonian @xmath62 on either side to an eigenstate , and concludes that the following `` master identity '' holds , @xmath91 & \ ; + \left(\hbar\omega - \tfrac12 \ , ( e_f - e_i ) \right ) \ , \left < \phi_f \left| \vec r^{\,2 } \right| \phi_i \right > \nonumber\\[0.1133ex ] & \ ; + \frac{3 \hbar^2}{2 m } \left < \phi_f | \phi_i \right > \,.\end{aligned}\ ] ] for the ac stark shift , one sets @xmath92 , @xmath93 , replaces @xmath94 , and adds two terms with @xmath95 . one can thus easily show that eq . follows from eq . , demonstrating the `` extended gauge invariance '' of the ac stark shift . recently , an analogous derivation has been shown to lead to the `` extended gauge invariance '' of one - loop correction to the imaginary part of the polarizability @xcite . here , the situation is different from the ac stark shift ; the initial state consists of a combined atom@xmath74field state where the atom is in the ground state and @xmath96 photons are in the laser mode , @xmath97 the final state has the atom in state @xmath98 and two photons less in the laser mode , @xmath99 the matrix element for the transition is @xmath100 \mathop{=}^{\mbox { ? } } & \ ; \left < \phi_2 \left| h^{(1)}_v \ , g'(\omega ) \ , h^{(1)}_v \right| \phi_1 \right > + \left < \phi_2 \left| h^{(2)}_v \right| \phi_1 \right > \,.\end{aligned}\ ] ] here , in contrast to the ac stark shift , only one term contributes in the electric field , namely , the one with the annihilation operators . furthermore , because @xmath101 ( the two states are manifestly different ) , the seagull term makes no contribution . after treating the photon degrees of freedom , the equality of the length and velocity gauge expressions for the two - photon matrix element is easily shown to be equivalent to the relation @xmath102 \mathop{=}^{\mbox { ? } } \pm \omega^2 \ , \left < \phi_f \left| \vec r \frac{1}{h_a - e - \omega } \ , \vec r \right| \phi_i \right > \,.\end{gathered}\ ] ] we have allowed for a sign ambiguity on the right - hand side ; both signs would lead to the same rabi frequency , which is proportional to the absolute modulus of the transition matrix element . in order to investigate whether the identity holds , we specialize our general `` master identity '' given in eq . to the case @xmath103 , @xmath104 & \ ; = ( e_f - e_i - \hbar \omega ) ( -\hbar \omega ) \left < \phi_f \left| \vec r \ , \frac{1}{h_a - e_n - \omega } \ , \vec r \right| \phi_i \right > \nonumber\\[0.1133ex ] & \ ; \qquad + \left(\hbar \omega - \tfrac12 \ , ( e_f - e_i ) \right ) \ , \left < \phi_f \left| \vec r^{\,2 } \right| \phi_i \right > \,.\end{aligned}\ ] ] at exact resonance , i.e. , for @xmath105 one has indeed @xmath106 = -\omega_r^2 \left < \phi_f \left| \vec r \ , \frac{1}{h_a - e_n - \omega_r } \ , \vec r \right| \phi_i \right > \,,\end{gathered}\ ] ] which is exactly of the required form given in eq . , for the case @xmath107 . again , the minus sign does not influence the calculation of the rabi frequency and is physically irrelevant . however , for @xmath108 , i.e. , off resonance , the two - photon transition rate as calculated in the length gauge differs from the corresponding result in the velocity gauge . the identity does not hold for @xmath108 . as already discussed in sec . [ sec2 ] , the `` gauge - noninvariance '' of _ one_-photon transitions is well known [ see the discussion surrounding eqs . ] . for two - photon transitions , the role of the inclusion of the entire spectrum of virtual atomic states has been somewhat unclear ( see refs . the initial conjecture regarding the equality of the length and velocity - gauge expressions can be traced to the paper by geltman @xcite , which treats a manifestly resonant process , namely , the two - photon absorption and ionization of a ground - state hydrogen atom . geltman writes an expression which corresponds to the seagull term in eq . ( 4 ) of his paper , adds it to the length - gauge expression which he gives in his eq . ( 1 ) , and asserts that the result is equal to the velocity - gauge result given in eqs . ( 3 ) and ( 5 ) of ref . @xcite . from the presentation , it is clear that geltman s argument applies to a resonant process , namely , the ionization rate of an initially ground - state atom by the absorption of two photons of frequency @xmath109 , into a continuum state of energy @xmath110 ( in the notation of ref . @xcite ) . comparison of the functions @xmath111 ( dashed line ) and @xmath112 ( solid line ) in the range @xmath113 . the overlap occurs at @xmath114 , which is the two - photon resonance condition . ] in view of advances in handling the schrdinger coulomb propagator @xcite , and the calculation of energy - dependent matrix elements of the nonrelativistic propagator , which are necessary for analytic lamb shift calculations @xcite , it is feasible to write analytic expressions for the two - photon transition matrix element in the velocity and length gauges . we may define the dimensionless matrix element @xmath115 = & \ ; \frac{(\alpha m c)^4}{3 m \hbar^2 } \ , \left < 2s \left| \vec r \ , \frac{1}{h_a - e_n - \omega } \ , \ , \vec r \right| 1s \right > \ , , \nonumber\end{aligned}\ ] ] for which we may write the expression @xmath116 \times \left ( 419 t^7 + 134 t^6 - 15 t^5 + 30 t^4 \right . + 60 t^3 - 120 t^2 - 32 t + 64 \right ) \\[2ex ] - \frac{4096 \ , \sqrt{2 } \;\ ; { } _ 2 f_1\left(1 , -t , 1-t , \frac{(1-t ) \ , ( 2-t)}{(1+t ) \ , ( 2+t ) } \right ) } { 3 \ , ( t^2 - 2)^3 \ , ( t^2 - 1)^2 } \,\end{gathered}\ ] ] where @xmath117 ( all formulas pertain to atomic hydrogen , where we set the nuclear charge number equal to @xmath118 ) . the dimensionless matrix element in the velocity gauge is @xmath119 = & \ ; \frac{1}{3 m } \ , \left < 2s \left| \vec p \ , \frac{1}{h_a - e_n - \omega } \ , \vec p \right| 1s \right > \ , , \nonumber\end{aligned}\ ] ] for which the expression reads @xmath120 - \frac{256 \ , \sqrt{2 } \;\ ; { } _ 2 f_1\left(1 , -t , 1-t , \frac{(1-t ) \ , ( 2-t)}{(1+t ) \ , ( 2+t ) } \right)}{3 \ , ( t-2)^2 \ , ( t^2 - 1 ) \ , ( t+2)^2 } \ , .\end{gathered}\ ] ] in fig . [ fig1 ] , we compare the expressions @xmath121 f_2 = & \ ; \frac{(e_{2s } - e_{1s } -\hbar\omega ) \ , ( -\hbar\omega)}{(\alpha^2 \ , m \ , c^2)^2 } \ , { \mathcal{q}}_{2s;1s}(\omega)\end{aligned}\ ] ] in the range @xmath122 , as a function of @xmath123 . the difference of the results given in eqs . and , @xmath124 = & \ ; \frac{m}{\hbar^2 } \left ( \hbar\omega - \tfrac12 ( e_{2s}-e_{1s } ) \right ) \left < 2s \left| \vec r^{\,2 } \right| 1s \right > \nonumber\\[0.1133ex ] = & \ ; -\frac{512\ , \sqrt{2}}{729 } \left(x - \frac{3}{16}\right ) \,,\end{aligned}\ ] ] is plotted in fig . [ fig2 ] . at exact resonance , one has @xmath125 as well as @xmath126 = -7.853\,655\,422 \,.\end{gathered}\ ] ] here , @xmath127 is the lerch @xmath128 transcendent . gauge difference @xmath129 . the zero occurs at @xmath114 . ] with lamb @xcite and kobe @xcite , we note that the electric field is a gauge - independent quantity , use the length - gauge expression and supply the prefactors in si units in order to write the following expression for the rabi frequency @xcite , @xmath130 \qquad \beta_{2s;1s}(\omega ) = & \ ; - \frac{e^2 \hbar}{\alpha^4 m^3 c^5 ( 4 \pi \epsilon_0 ) } \ , { \mathcal{q}}_{2s;1s}(\omega ) \,.\end{aligned}\ ] ] an expansion of the rabi frequency about resonance leads to the result @xmath131 = \left ( 3.68111 \times 10^{-5 } + 2.32293 \times 10^{-4 } ( x - x_r ) \right ) \ , \frac{{\rm hz } \ , { \rm m}^2}{{\rm w}^2 } \,,\end{gathered}\ ] ] where @xmath132 and @xmath133 . a remark on two - color absorption is in order . if an atom is simultaneously subjected to two laser fields of different frequencies @xmath134 and @xmath135 , which fulfill the resonance condition @xmath136 , then gauge invariance is restored . on the basis of eq . , this is verified as follows , @xmath137 \ , \left < \phi_f \left| \vec r^{\,2 } \right| \phi_i \right > \\ + \left[\hbar\omega_2 - \tfrac12 ( e_f - e_i ) \right ] \ , \left < \phi_f \left| \vec r^{\,2 } \right| \phi_i \right > \\ = - ( \hbar\omega_1 ) ( \hbar\omega_2 ) \ , [ q(\omega_1 ) + q(\omega_2 ) ] \,.\end{gathered}\ ] ] the relation @xmath138 $ ] is equivalent to the gauge invariance of the resonant two - color , two - photon transition . in tables i and ii of ref . @xcite , the authors present resonant two - color , two - photon matrix elements which in our notation would read @xmath139 \,,\ ] ] where , again , @xmath140 . for example , the gauge - invariant resonant two - color result at frequency @xmath141 reads as @xmath142 \\ = -62.659\,473\,633 \,,\end{gathered}\ ] ] verifying the fifth entry in the last row of tables i and ii of ref . diagrammatically , the two terms in eq . correspond to photon absorption processes with two different possible time orderings of the absorptions of photons with frequencies @xmath134 and @xmath143 . in the current paper , we ( re-)examine the transformation from the length to the velocity gauge in sec . [ sec2 ] , and recall that the length - gauge and velocity - gauge hamiltonians are not related by a unitary transformation . furthermore , we show that the physical interpretation of a quantum mechanical operator depends on the gauge , vindicating arguments given by lamb @xcite and kobe @xcite regarding the applicability of the length gauge off resonance . in sec . [ sec3 ] , we consider the ac stark shift as a paradigmatic example of a physical process invariant under an `` extended '' gauge transformation . specifically , in atomic hydrogen , we rederive the known result @xcite that the ac stark shift formulated in the length gauge is equal to the velocity - gauge expression , even if the gauge transformation of the wave function is ignored . the derivation is based on the `` master identity '' given in eq . . [ sec4 ] , we investigate the two - photon transition matrix element , where the `` extended gauge invariance '' does not hold off resonance . the derivation again profits from the general identity , which can be applied to both of the problems studied in secs . [ sec3 ] and [ sec4 ] ; its validity is verified on the basis of analytic and numerical calculations [ see eqs . , as well as figs . [ fig1 ] and [ fig2 ] ] . in retrospect , it would have seemed somewhat surprising if extended gauge invariance had been applicable to two - photon transitions ( under the inclusion of all possible virtual , intermediate states ) but failed for one - photon transition [ see eqs . we conclude that for two - photon transitions , the length gauge needs to be used off resonance , just as for one - photon absorption . yet , for two - color , two - photon absorption with the sum of the two photon frequencies adding up to the exact resonance frequency , extended gauge invariance again holds ( see sec . [ sec4 ] and ref . @xcite ) . a few explanatory remarks are in order . we have seen that extended gauge invariance is restored at exact resonance , for both one- as well as ( one - color and two - color ) two - photon transitions . mathematically , extended gauge invariance is restored at resonance in view of commutation relations , notably , @xmath144 $ ] , where @xmath62 is the atomic schrdinger coulomb hamiltonian . physically , extended gauge invariance holds because processes at exact resonance , or , processes which involve energy shifts , can be formulated using a form of the interaction where the fields and potentials are adiabatically switched off in the infinite future and in the infinite past , using a damping term of the form @xmath145 . the gauge transformation of the wave functions ( in and out states ) then proceeds in the distant past and future , where the fields are switched off and the gauge transformation is just the identity . for one- and two - photon transitions , the necessity to introduce the damping terms is inherent to the formulation of fermi s golden rule , which describes transition rates at exact resonance , where the initial and final states fulfill an energy conservation condition [ see refs . @xcite . the ac stark shift can be formulated using the gell mann low theorem [ see eqs . ( 19 ) and ( 21 ) of ref . @xcite ] , in which case one uses a time evolution operator that evolves the wave function from the infinite past to the present , with the interactions being switched off for @xmath146 . within the gell mann low formalism , the gauge transformation of the wave function in the infinite past amounts to the identity transformation , because the interactions are adiabatically switched off in this limit . the extended gauge invariance of those physical processes whose description allows such as adiabatic damping , thus finds a natural explanation . for one- and two - photon transitions off resonance , however , the quantum dynamics are instantaneous , and the physical interpretation of the operators must be carefully restored . in this case , only the length gauge provides a consistent physical description ( see the discussion in sec . [ sec2 ] ) . one might thus ask if the velocity gauge has any advantages in the physical description of laser - related processes . the answer can be given as follows . there are @xmath147-matrix elements in the so - called strong - field approximation whose evaluation becomes easier in the velocity gauge . in this case , the in- and out - states are asymptotic states [ the @xmath147 matrix is a time evolution operator from the infinite past to the infinite future ] . indeed , as stressed by reiss in eqs . ( 29 ) and ( 31 ) of ref . @xcite , the volkov state in a strong laser field is much easier to formulate in the velocity gauge , and consequently , @xmath147-matrix calculations should preferentially be done in this gauge [ see also refs . @xcite ] . in the formulation of the @xmath147 matrix , one canonically uses infinitesimal damping parameters [ see ref . @xcite ] , and thus , extended gauge invariance is restored . we conclude that the choice of gauge in these cases should be made according to practical considerations , and in strong laser fields , the velocity gauge provides for the most simple computational framework . this research has been supported by the national science foundation ( grant phy1403973 ) . 28ifxundefined [ 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 * * , ( ) in @noop _ _ , ( , , ) pp . @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( )

we reexamine the invariance of two - photon transition matrix elements and corresponding two - photon rabi frequencies under the `` gauge '' transformation from the length to the velocity gauge . it is shown that gauge invariance , in the most general sense , only holds at exact resonance , for both one - color as well as two - color absorption . the arguments leading to this conclusion are supported by analytic calculations which express the matrix elements in terms of hypergeometric functions , and ramified by a `` master identity '' which is fulfilled by off - diagonal matrix elements of the schrdinger propagator under a the transformation from the velocity to the length gauge . the study of the gauge dependence of atomic processes highlights subtle connections between the concept of asymptotic states , the gauge transformation of the wave function , and infinitesimal damping parameters for perturbations and interaction hamiltonians that switch off the terms in the infinite past and future [ of the form @xmath0 . we include a pertinent discussion .

in recent time people are too much interested to find some flavor symmetry in order to generate mass and mixing pattern of fermions . continuous symmetry like @xmath9 @xcite , @xmath10 @xcite symmetry and most popular discrete symmetry , @xmath11 exchange symmetry ( @xmath12@xcite have got some success to describe mass and mixing pattern in leptonic sector . to avoid mass degeneracy of @xmath13 and @xmath14 under @xmath15 symmetry , e. ma and g. rajasekaran in @xcite have introduced first time the @xmath1 symmetry . after this paper , a lot of work have done with this symmetry @xcite-@xcite . after introduction of tri - bi maximal mixing pattern ( @xmath16 , @xmath17 , @xmath18)@xcite , people have tried to fit this mixing pattern through the @xmath1 symmetry . in an well motivated extension of the standard model through the inclusion of @xmath1 discrete symmetry tri - bi maximal mixing pattern comes out in a natural way in the work of altarelli and feruglio @xcite . more precisely , the leptonic mixing arises solely from the neutrino sector since the charged lepton mass matrix is diagonal . the model @xcite also admits hierarchical masses of the three charged leptons whereas the neutrino masses are quasi - degenerate or hierarchical . although the model gives rise to @xmath19 ( @xmath20 ) which is consistent with the chooz - palo verde experimental upper bound ( @xmath21 at 3@xmath22 ) , however , the non - zero and complex value of @xmath0 leads to the possibility to explore _ violation in the leptonic sector which is the main goal of many future short and long baseline experiments . within the framework of @xmath23 model , non - zero @xmath0 is generated either through the radiative correction @xcite or due to the introduction of higher dimensional mass terms @xcite . generation of non zero complex @xmath0 and possibility of non - zero cp violation has been extensively studied in @xcite for the proposed model of altarelli - feruglio @xcite with explicit soft breaking of @xmath1 symmetry @xcite . in the model @xcite the authors showed that the tri - bi maximal mixing pattern is also generated naturally in the framework of see - saw mechanism with @xmath23 symmetry . exact tri - bi maximal pattern forbids at low energy cp violation in leptonic sector . the textures of mass matrices in @xcite could not generate lepton asymmetry also . in the present work , we investigate the generation of non - zero @xmath0 through see saw mechanism by considering a small perturbation in @xmath2 , the dirac neutrino mass matrix , keeping the same texture of the right - handed majorana neutrino mass matrix as proposed in ref.@xcite . at first , we have studied in detail perturbation of @xmath2 by adding a small parameter at different entries of @xmath2 and see the variations of three mixing angles in terms of other model parameters considering all of them real . we extend our analysis to the complex case for a suitable texture . we study detailed phenomenology of neutrino mass and mixing including cp violation at low energy , neutrinoless double beta decay and leptogenesis . our approach to get nonzero @xmath0 is minimal as we break @xmath1 symmetry explicitly by single parameter in single element of @xmath2 . generation of cp violation at low energy as well as high energy is also minimal as we consider only one parameter complex . we consider the model proposed in @xcite , which gives rise to diagonal @xmath2 and @xmath24 ( the charged lepton mass matrix ) along with a competent texture of @xmath25 and after see - saw mechanism and diagonalisation gives rise to tri - bimaximal mixing pattern . the model consists of several scalar fields to generate required vacuum alignment to obtain tri - bimaximal mixing . in table i. , we have listed the scalar fields and their vev s and representation content under all those symmetries . .list of fermion and scalar fields used in this model , @xmath26 . [ cols="^,^,^,^",options="header " , ] the model is fabricated in such a way that after spontaneous breaking of @xmath1 symmetry , the @xmath15 symmetry remains on the neutrino sector and the charged lepton sector is invariant under @xmath27 symmetry . consider the lagrangian of the model @xcite , @xmath28 after spontaneous symmetry breaking , the charged lepton mass matrix comes out diagonal with @xmath29 , @xmath30 , and @xmath31 . the neutrino sector gives rise to the following dirac and majorana matrices @xmath32 where @xmath33 , @xmath34 . the structure of light neutrino mass matrix can be obtained from see - saw formula : @xmath35 where , @xmath36 this is clear from eq.[ssf ] that @xmath37 is the diagonalising matrix for light neutrino mass matrix @xmath38 . the form of @xmath37 is in eq.[tbmix ] which is nothing but the so called tribimaximal mixing matrix . from eq.[ssf ] we have the eigenvalues of @xmath38 : @xmath39 from eq.[tbmix ] we have the mixing angles @xmath16 , @xmath40 and @xmath18 and from eq.[a4ev ] we get the solar and atmospheric mass squared differences as @xmath41 where @xmath42 , @xmath43 and all parameters are real . from the experiments we know @xmath3 is positive and dictates either @xmath44 or @xmath45 . if @xmath44 , then it has to be small in order to generate small value of @xmath3 provided @xmath46 is not too small as @xmath3 . but small positive @xmath47 corresponds to same order of magnitude of @xmath3 and @xmath4 which is not acceptable according to the experimental results . now @xmath44 only acceptable for @xmath48 and hierarchy of @xmath3 and @xmath4 obtained with the singular nature of @xmath4 as in eq.[a4msd ] near @xmath49 . this corresponds to normal hierarchical mass spectrum . again for @xmath50 , @xmath45 is the physical region . this region of @xmath47 makes @xmath51 which is so called inverted ordering of neutrino mass pattern . again @xmath52 should take small value in order to generate small value of @xmath3 . for one complex parameter @xmath53 , we can write the mass differences in the following form @xmath54 in the complex case , positivity of @xmath3 can be obtained either with @xmath44 and @xmath55 or with @xmath56 and @xmath57 . for the first case with @xmath48 and with @xmath58 one can have normal hierarchical mass spectrum . for the second case hierarchy will be inverted and @xmath59 have to be small . in both case @xmath47 should take the value such that the @xmath60 range also satisfy . the mixing pattern is tri - bi maximal eq.[tbmix ] and it is independent to the fact whether the parameters are real or complex . in this mixing pattern @xmath61 and non - zero complex @xmath0 is a basic requirement to see the non - zero dirac cp violation . now we concentrate on the issue of leptogenesis in this model . the decay of right handed heavy majorana neutrinos to lepton(charged or neutral ) and scalar(charged or neutral ) generate non - zero lepton asymmetry if i ) c and cp are violated , ii)lepton number is violated and iii ) decay of right handed neutrinos are out of equilibrium . we are in the energy scale where @xmath1 symmetry is broken but the sm gauge group remains unbroken . so , the higgs scalars both charged and neutral are physical . the cp asymmetry of decay is characterized by a parameter @xmath62 which is defined as @xmath63 spontaneous @xmath1 symmetry breaking generates right handed neutrino mass and the mass matrix @xmath25 obtained is shown in eq . we need to diagonalize @xmath25 in order to go into the physical basis ( mass basis ) of right handed neutrino . this form of @xmath25 gives the diagonalising matrix in the tri - bi maximal form @xmath37 in eq.[tbmix ] : @xmath64 however , the eigenvalues are not real . we need to multiply one diagonal phase matrix @xmath65 with @xmath37 . hence , diagonalising matrix @xmath66 relates the flavor basis to eigen basis of right handed neutrino : @xmath67 in this basis the couplings of @xmath68 with leptons and scalars are modified and it will be : @xmath69 at the tree level there there are no asymmetry in the decay of right handed neutrinos . due to the interference between tree level and one loop level diagrams , the asymmetry is generated . there are vertex diagram and self energy diagram to contribute to the asymmetry @xcite . the vertex contribution is : @xmath70 \label{vertex}\end{aligned}\ ] ] and the self energy part is : @xmath71 where @xmath72 and @xmath73 the key matrix , whose elements are necessary to calculate leptogenesis , is @xmath74 . in this model @xmath2 is diagonal and proportional to identity . hence , @xmath74 matrix is real diagonal and it is also proportional to identity matrix and it is independent of the form of @xmath75 . the terms for decay asymmetry generated by `` i '' th generation of right handed neutrino @xmath76 for both vertex and self energy contributions are proportional to @xmath77 ( where @xmath78 ) as in eq.[vertex ] and eq . all off - diagonal elements of @xmath74 are zero . so , decay of all three generation of right handed majorana neutrinos could not generate lepton asymmetry . so , in this model of @xmath1 symmetry tri - bi maximal mixing pattern is not compatible with the low energy dirac cp violation as well as high energy cp violation . in order to obtain non - zero @xmath79 , low energy dirac cp violation and leptogenesis we need to break the @xmath1 symmetry through not only spontaneously but also explicitly introducing some soft @xmath1 symmetry breaking ( soft in the sense that the breaking parameter is small to consider @xmath1 as an approximate symmetry ) terms in the lagrangian . we consider minimal breaking of @xmath1 symmetry through a single parameter in a single element of @xmath2 keeping @xmath25 unaltered as @xmath80 we introduce the breaking by small dimensionless parameter @xmath81 to the @xmath82 element of dirac type yukawa term for neutrino . after spontaneous @xmath83 symmetry breaking it modifies only one element @xmath82 of @xmath2 of neutrino . there are nine possibilities to incorporate the breaking parameter @xmath81 in @xmath2 . we know that after spontaneous @xmath1 symmetry breaking , a residual @xmath15 symmetry appears in neutrino sector . there is a special feature of @xmath15 symmetry which ensures one @xmath84 and one maximal @xmath85 mixing angles . there is one task to check whether our newly introduced explicit breaking term can break @xmath15 symmetry or not . this is important because we need non - zero @xmath79 . we have seen that in one case out of the nine possibilities , residual @xmath15 symmetry remains invariant . this is @xmath86 case . in other cases @xmath15 symmetry is broken and one expect non - zero @xmath79 from those cases . primarily , we consider that all parameters are real . we want to study the mixing pattern and want to see its deviation from tri - bi maximal pattern considering experimental value of mass squared differences of neutrinos . we explore all nine cases including @xmath86 case . although @xmath86 case could not generate non - zero @xmath79 , however , we want to see whether this breaking can reduce the tri - bi maximal value of @xmath87 ( @xmath88 ) to its best fit value ( @xmath89 ) or not along with the special feature @xmath90 and @xmath91 . here , we explicitly demonstrate the procedure for a single case and for the other cases expressions for eigenvalues and mixing angles are given in apendix . \(i ) breaking at 22 element : in this case , the structure of @xmath2 is given by @xmath92 and after implementation of see - saw mechanism keeping the same texture of @xmath25 , three light neutrino mass eigenvalues come out as @xmath93 and the three mixing angles come out as @xmath94\nonumber\\ \sin\theta_{13 } & = & { \frac{\epsilon}{3 } } \left({\frac{d}{{\sqrt 2}a } } -{\frac{{\sqrt 2}d}{4a - 2d}}\right ) \label{angel22 } \end{aligned}\ ] ] assuming a relationship between the parameters @xmath95 and @xmath96 as @xmath97 we rewrite in a convenient way the above three mixing angles as @xmath98 \qquad\sin\theta_{13 } = \frac{\e k(1-k)}{3{\sqrt 2}(2-k ) } \label{mangel22 } \end{aligned}\ ] ] and the mass - squared differences are @xmath99\nonumber\\ \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-2\e(2k^2 - 4k-1)\right ] \label{masd22}\end{aligned}\ ] ] where @xmath100 . defining the ratio @xmath101 in terms of mass - squared differences we get @xmath102 } { \left[3k(2-k)-2\e(2k^2 - 4k-1)\right ] } \label{r22 } \end{aligned}\ ] ] which in turn determines the parameter @xmath81 as @xmath103 } { \left[(k-1)^2(2k^2 + 4k+1)+ r(k+1)^2(2k^2 - 4k-1)\right ] } \label{ep22 } \end{aligned}\ ] ] similarly , we have evaluated all other possible cases which we have listed in the appendix . plot of @xmath87 with respect to @xmath47 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] plot of @xmath106 with respect to @xmath47 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] plot of @xmath107 with respect to @xmath47 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] now the @xmath108 parameter is determined in terms of @xmath101 and @xmath47 and we substitute it to the expressions for mixing angles . thus it is possible to explore all three mixing angles @xmath87 , @xmath106 and @xmath107 in terms of @xmath101 and @xmath47 . particularly , the deviation from tri - bimaximal mixing depends only on @xmath101 and @xmath47 . for the best fit values of the solar and atmospheric mass squared differences ( @xmath109 ) , we have shown in fig . [ th12 ] , fig . [ th23 ] and fig . [ th13 ] the variations of @xmath87 , @xmath106 , @xmath107 verses @xmath47 , respectively . we have studied all nine possible cases and shown in the plots . first of all , non - zero value of @xmath107 is obtained if we allow @xmath1 symmetry breaking terms explicitly in any one of the 12 , 13 , 21 , 31 , 22 , 33 element of the dirac neutrino mass matrix and those are the cases of our interest . in the present analysis , we have shown that non - zero @xmath107 is generated in a softly broken @xmath1 symmetric model which leads to deviation from the tri - bimaximal mixing . in @xmath1 symmetric model @xmath107 is zero because of residual @xmath15 symmetry in neutrino sector after spontaneous breaking of @xmath1 symmetry . apart from the explicit breaking of @xmath1 symmetry at 11 element , @xmath15 is broken for all 12 , 13 , 21 , 31 , 22 , 33 , 23 , 32 cases . furthermore , perturbation around 23 , 32 elements also lead to zero value of @xmath107 at the leading order although @xmath15 symmetry is broken , non - zero value is generated if we consider higher order terms of @xmath110 which are too tiny and hence , discarded from our analysis . we include 11 case for completeness which preserves @xmath15 symmetry and hence generates @xmath111 and @xmath91 . it only shifts @xmath87 from the tri bimaximal value , but it can not be able to go towards the best fit value of solar angle , @xmath112 . if @xmath113 , then we get @xmath114 , and thereby , the value of @xmath107 is very small also @xmath87 will hit the exact tri - bimaximal value in some cases . the effect of variation on the mixing angles around @xmath115 are asymmetric . for some cases ( for example 23 , 32 ) @xmath106 changes very fast in the @xmath116 region . so , we explore the mixing angles with the range @xmath117 . we choose the most feasible cases in which perturbation is applied around 12 , 13 elements , because in those cases , variation of @xmath47 encompasses the best - fit values of @xmath87 and @xmath106 . although , in the 21 , 31 cases , the value of @xmath106 touches the best - fit value @xmath85 , however , @xmath87 far apart from the best - fit value . in order to achieve large @xmath107 , we have to choose the 21 , 31 cases , but we have to allow the variation of @xmath87 around as large as @xmath118 . in case of 22 , 33 , the structure of @xmath2 is still diagonal and also we can get larger @xmath107(upto @xmath119 ) and also @xmath87 is within @xmath120 , however , @xmath106 will reach @xmath121 value . in summary , we have shown that non - zero @xmath107 is generated in a @xmath1 symmetric model which leads to deviation from the tri - bimaximal mixing through see - saw mechanism due to the incorporation of an explicit @xmath1 symmetry breaking term in @xmath2 . the breaking is incorporated through a single parameter @xmath108 and we have investigated the effect of such breaking term in all nine elements of @xmath2 . some of them generates still zero value of @xmath107 and rest of the others generated non - zero @xmath107 . we expressed all three mixing angles in terms of one model parameter and showed the variation of all three mixing angles with the model parameter @xmath47 . we find breaking through 12 and 13 elements of @xmath2 are most feasible in view of recent neutrino experimental results . in this section , we consider one of the parameter is complex and out of all nine cases as mentioned earlier , we investigate one suitable case arises due to 13 element perturbation . this is one of the suitable positions of breaking justified from real analysis . again this extension is minimal to generate non - zero cp violation because we consider only one parameter complex . we take @xmath95 as complex : @xmath53 . hence , the form of @xmath2 and @xmath25 under explicit @xmath1 symmetry breaking with complex extension are : @xmath122 using the see - saw mechanism we get the light neutrino mass matrix as @xmath123 we need to diagonalize the @xmath38 to obtain the masses and mixing angles . the eigenvalues are same as we have in the real case and only difference is that the @xmath95 is complex now . we explicitly write down the complex phase in the mass matrix . the obtained eigenvalues are : @xmath124 where we keep terms upto first order in @xmath81 . now with @xmath42 , @xmath125 and keeping term upto first order in @xmath81 we get the three neutrino mass squared as @xmath126 using those expressions we get the mass squared differences and their ratio which are , @xmath127 and @xmath128 . \label{rc13 } \end{aligned}\ ] ] the mixing angles are obtained from diagonalisation of @xmath38 . we solve the equations of the form @xmath129 . these @xmath130 will give the columns of the diagonalising unitary matrix @xmath131 . throughout our calculation we assume that breaking parameter @xmath81 is small . we have the nonzero @xmath132 which is proportional to @xmath81 . so , the values of @xmath133 and @xmath134 will give the solar and atmospheric mixing angles , respectively . the expressions for the mixing angles come out as @xmath135 plot of @xmath87 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in 13 element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values.,height=302 ] @xmath137 @xmath138^{1/2 } \label{t13c13}. \end{aligned}\ ] ] from the expression of mixing angles it is clear that the deviations from tri - bi maximal are first order in @xmath81 . the independent parameters in this model are @xmath96 , @xmath95 , @xmath139 , @xmath81 and @xmath136 . alternatively the independent parameters are @xmath140 , @xmath141 , @xmath47 , @xmath81 and @xmath136 ( where @xmath142,@xmath125 , @xmath143 ) . in the above analysis of light neutrino mass and mixing , scale @xmath140 did not appear explicitly . we have four well measured observable which are @xmath3 , @xmath4 , @xmath87 and @xmath106 , and , thus , in principle it is possible to determine four parameters @xmath141 , @xmath47 , @xmath81 and @xmath136 and we are able to predict the other less known observable such as angle @xmath107 , cp violating parameter @xmath144 etc . it is difficult to get inverse relations of those observable . from the expression of @xmath101 in eq . [ rc13 ] we easily obtain the expression for @xmath81 as plot of @xmath106 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in 13 element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] plot of @xmath107 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in `` 13 '' element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] @xmath145}{4\left[r(1+k^2 - 2k\cos\phi)(1+k^2 + 2k\cos\phi)+(2+k^2 + 2k\cos\phi)(1+k^2 - 2k\cos\phi)\right]}.\nonumber\\ \label{epc13 } \end{aligned}\ ] ] now using the relation of @xmath4 with the parameters eq . [ msdc13 ] we get the expression for @xmath46 : @xmath146 where @xmath81 is in the form of eq . [ epc13 ] . thus , @xmath81 and @xmath46 depend on the parameters @xmath47 , @xmath136 and experimentally known @xmath101 . extraction of @xmath47 and @xmath136 from other two known mixing angles is little bit difficult . rather we have plotted @xmath87 and @xmath106 with respect to @xmath47 and @xmath136 and obtain the restriction on the parameter space of @xmath47 and @xmath136 . from the expression of @xmath147 in eq . [ t12c13 ] we are seeing that there is a factor @xmath59 in the denominator . for @xmath116 there will be a @xmath136 for which the quantity @xmath59 becomes zero . hence , we should keep @xmath45 . again the factor @xmath59 should be small to ensure that @xmath81 is also small . it justifies our whole analysis because we consider first order perturbation as we considered symmetry @xmath1 remains approximate . we consider the range @xmath148 and @xmath149 . from fig . [ th12c13 ] we see that @xmath87 changes from the tri - bi maximal value @xmath150 to @xmath151 . near @xmath152 it crosses the best fit value @xmath112 . we have plotted @xmath106 in fig . [ th23c13 ] . the variation of @xmath106 is from @xmath153 to @xmath154 for the same range of @xmath47 and @xmath136 . the best fit value @xmath155 is remain within range of variation and it is in the low @xmath47 low @xmath136 region . the plot of @xmath107 is in fig . [ th13c13 ] . value of @xmath107 remains within @xmath156 for the same range of @xmath47 and @xmath136 and the model predicts @xmath107 is very small but non - zero . question may arise whether such small value of @xmath107 can generate observable cp violation or not . keeping all those constraints in view next we explore the parameter space of cp violation parameter @xmath157 . the parameter @xmath157 defined as @xcite @xmath158 } { \delta m^2_{21}\delta m^2_{31}\delta m^2_{32 } } \label{jcp1}\end{aligned}\ ] ] where @xmath159 , @xmath160 is dirac phase . this @xmath161 is associated with cp violation in neutrino oscillation and is directly related to dirac phase of mixing matrix . plot of @xmath157 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in 13 element of @xmath2 in the unit of @xmath162 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] plot of @xmath163 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in `` 13 '' element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] the from eq . ( [ mnuc ] ) we can express @xmath38 matrix in terms of @xmath81 and @xmath141 and @xmath47 and @xmath136 . with the expressions for @xmath81 and @xmath141 we can easily obtain the @xmath38 and hence @xmath164 completely in terms of @xmath47 and @xmath136 . hence , similar to the mixing angles @xmath144 will be also only function of @xmath47 and @xmath136 . we have plotted @xmath144 in fig . [ fjcp1 ] where the values are normalized by a factor @xmath162 . for the same range of @xmath47 and @xmath136 the model predicts @xmath144 up to the order of @xmath165 which is appreciable to observe through the forthcoming experiments . inverting the expression of @xmath144 , the phase @xmath160 is extracted in terms of @xmath47 and @xmath136 and it is plotted in fig . [ dcp13 ] . we see that the value of @xmath160 is large upto @xmath166 and , thereby , compensates small @xmath107 effect in @xmath144 and makes it observable size . one important discussion we have to make about the range of @xmath136 and @xmath47 . one can ask why we are keeping ourselves small range of those parameters where larger @xmath136 can enhance the @xmath107 and @xmath144 . we have studied that the larger value of @xmath136 and also @xmath47 in negative become responsible for breaking the analytic bound @xmath167 . so , we keep ourselves in shrinked parameter space which keep @xmath87 and @xmath106 in exceptionally good values according to the experiment and also can able to generate observable cp violation instead of small @xmath107 . another thing we want to point out that negative value of @xmath136 equally acceptable as far as it is small . it could not change mixing angles because their expressions depend on @xmath136 through @xmath168 and @xmath169 . only @xmath144 and @xmath160 will change in sign which are unsettled according to the experiments . at the end we want to check whether the range of @xmath47 and @xmath136 can satisfy the double beta decay bound @xmath170 ev . in our model expression for this quantity is : @xmath171 and it will be also only function of @xmath47 and @xmath136 . we plot this in fig . [ mee13 ] and it remains well below the experimental upper bound . again we want to discuss about the mass pattern . throughout our whole analysis in real as well as complex case we keep @xmath3 and @xmath4 to their best fit value and take the negative sign of @xmath4 . it corresponds to so called inverted ordering of neutrino mass . it is the feature near @xmath115 . it is necessary to keep @xmath172 . why we so fond of this region of @xmath47 instead of region @xmath173 which can give the normal hierarchical mass spectrum . the reason is that the inverted ordering corresponds to the light neutrino mass scale @xmath174 where @xmath175 for normally ordered mass spectrum . so , from the point of view of observable cp violation , it is inevitable to choose larger value of @xmath141 because @xmath176 . so , inverted hierarchical mass spectrum compatible with the observable cp violation . now we extend our study of this model to leptogenesis . we want to see whether we can have appreciable leptogenesis compatible with baryon asymmetry for the same parameter space @xmath47 and @xmath136 after successful low energy data analysis for the feasible value of scale @xmath140 . plot of @xmath177 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in `` 13 '' element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] after successful predictions of low energy neutrino data we want to see whether this model can generate non - zero lepton - asymmetry with proper size and sign to describe baryon asymmetry . we keep the same right handed neutrino mass matrix @xmath25 as before . the change only appear in dirac type yukawa coupling and hence in @xmath2 also . the diagonalisation of @xmath25 gives @xmath178 and hence the masses of the right handed neutrino are @xmath179 and the phases are @xmath180 the explicit form of diagonalising matrix is @xmath181 where the expressions for the phases are given in eq . ( [ majp ] ) . the @xmath2 matrix is no longer diagonal after explicit breaking of @xmath1 symmetry . in the mass basis of right handed neutrino the modified dirac mass term is @xmath182 . hence the relevant matrix for describing leptogenesis is : @xmath183 to calculate lepton asymmetry as in eq . ( [ vertex ] ) and eq . ( [ self ] ) we need to calculate following quantities from matrix @xmath74 : @xmath184 calculating @xmath72 from eq . [ mrev ] , @xmath185 from eq . ( [ hc13 ] ) and taking @xmath77 from eq . ( [ imh ] ) we calculate the self energy part of lepton asymmetry from eq . ( [ self ] ) and vertex part of lepton asymmetry from eq . ( [ vertex ] ) . adding both we obtain the following decay asymmetry of right handed neutrinos for all three generations @xmath186.\nonumber\\ \label{las1 } \end{aligned}\ ] ] @xmath187 \label{las2 } \end{aligned}\ ] ] @xmath188\nonumber\\ \label{las3 } \end{aligned}\ ] ] cp asymmetry parameters @xmath62 are related to the leptonic asymmetry parameters through @xmath189 as @xcite @xmath190 where @xmath191 is the lepton number density , @xmath192 is the anti - lepton number density , @xmath193 is the entropy density , @xmath194 is the dilution factor for the cp asymmetry @xmath62 and @xmath195 is the effective number of degrees of freedom @xcite at temperature @xmath196 . value of @xmath195 in the sm with three right handed majorana neutrinos and one extra higgs doublet is @xmath197 . the baryon asymmetry @xmath198 produced through the sphaleron transmutation of @xmath189 , while the quantum number @xmath199 remains conserved , is given by @xcite @xmath200 where @xmath201 is the number of fermion families and @xmath202 is the number of higgs doublets . the quantity @xmath203 in eq . ( [ barasym ] ) for sm with two higgs doublet . now we introduce the relation between @xmath198 and @xmath204 , where @xmath204 is the baryon number density over photon number density @xmath205 . the relation is @xcite @xmath206 where the zero indicates present time . now using the relations in eqs.([leptasym],[barasym ] , [ yetar ] ) , @xmath203 and @xmath207 we have @xmath208 this dilution factor @xmath194 approximately given by @xcite @xmath209 where @xmath210 is the decay width of @xmath211 and @xmath212 is hubble constant at @xmath196 . their expressions are @xmath213 where @xmath214 , @xmath215gev and @xmath216gev . thus we have @xmath217 for our model @xmath218 , @xmath219 and @xmath220 are @xmath221 plot of baryon asymmetry @xmath222 in unit of @xmath223 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in `` 13 '' element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values and have plotted for mass scale of right handed neutrino @xmath224 gev , height=302 ] plot of baryon asymmetry @xmath222 in unit of @xmath223 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in `` 13 '' element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values and have plotted for mass scale of right handed neutrino @xmath225 gev , height=302 ] plot of baryon asymmetry @xmath222 in unit of @xmath223 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in `` 13 '' element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values and have plotted for mass scale of right handed neutrino @xmath226 gev , height=302 ] we combine all three plots of baryon asymmetry for three right handed neutrino mass scale along with the wmap value of baryon asymmetry @xmath7 which is the plane surface.,height=302 ] where @xmath227 , @xmath228 . using eq . ( [ k1k2k3 ] ) into eq . ( [ kppa ] ) and from the expression of @xmath62 we can say that apart from logarithmic factor , @xmath229 and @xmath230 . so , baryon asymmetry will be independent of @xmath141 and @xmath231 . they only appear through logarithmic factor in @xmath194 . we consider @xmath232 . substituting @xmath141 from eq . ( [ m013c ] ) , @xmath81 from eq . ( [ epc13 ] ) and considering @xmath140 some specific value into the expressions of @xmath62 , @xmath194 we can have baryon asymmetry as function of @xmath47 and @xmath136 only . in fig . [ basymp1 ] , fig . [ basymp2 ] and fig . [ basymp3 ] we have plotted @xmath204 as function of @xmath47 and @xmath136 in the unit of @xmath223 for three @xmath140 values , @xmath224 gev , @xmath225 gev and @xmath226 gev rspectively . we have seen that the experimental value of @xmath233 is obtainable in our model within the same range of @xmath234 and @xmath136 as in the low energy case . to see more explicitly the baryon asymmetry plots we combine all three plots along with the observed baryon asymmetry value @xmath7 which corresponds the plane surface in fig . [ basym0 ] . the observed wmap value of the baryon asymmetry curve intersect the lower curve ( for @xmath224 gev ) near the boundary of @xmath234 and @xmath136 variation . so , lower value of @xmath140 could not generate observable baryon asymmetry . in the intersection region @xmath235 and @xmath236 . if we allow that much of variation of @xmath87 and @xmath106 , we can have large low energy cp violation as well as baryon asymmetry with proper size and sign with @xmath224 gev which is near the upper bound of right handed neutrino mass scale for generation of lepton as well as baryon asymmetry . if we more relax , we can easily see that the intersection of experimental and theoretical curve for @xmath225 gev and @xmath226 gev is in the lower value of @xmath47 and @xmath136 where well known neutrino mixing angles are more closer to their best fit value . let us give a close look to the plot of @xmath204 in fig . [ basymp1 ] , fig . [ basymp2 ] and fig . [ basymp3 ] near very low @xmath136 and @xmath237 region . this is the region where @xmath238 become very large . from the expression of @xmath239 and @xmath240 it is clear that @xmath239 and @xmath240 both are singular at @xmath241 . this corresponds to equality of the masses @xmath242 or @xmath243 . this singularity can be avoided considering finite decay width of right handed neutrinos . we can able to maximize @xmath239 and @xmath240 and hence @xmath204 using resonant condition @xmath244 . but , as we have already obtained the observed baryon asymmetry without resonance , it is not necessary to think about so finely tuned condition . again in the region where the resonant condition is applicable , the @xmath144 is miserably small to observe through any experiments . contour plot of baryon asymmetry @xmath222 , @xmath87 and , @xmath106 in @xmath136-@xmath47 plane for @xmath104 and @xmath105 to their best fit values and for mass scale of right handed neutrino @xmath224 gev.,height=226 ] contour plot of baryon asymmetry @xmath222 , @xmath87 and , @xmath106 in @xmath136-@xmath47 plane for @xmath104 and @xmath105 to their best fit values and for mass scale of right handed neutrino @xmath225 gev.,height=226 ] contour plot of baryon asymmetry @xmath222 , @xmath87 and , @xmath106 in @xmath136-@xmath47 plane for @xmath104 and @xmath105 to their best fit values and for mass scale of right handed neutrino @xmath226 gev.,height=226 ] we end our analysis with the help of three contour plots of baryon asymmetry @xmath245 for three scales @xmath224 gev , @xmath225 gev and , @xmath226 gev in @xmath136-@xmath47 plane . we insert the contours of @xmath106 and @xmath87 and manage to find the intersection of three contours for some reasonable value of @xmath106 and @xmath87 . case(i ) : @xmath224 gev , from fig . [ fig : contplot1 ] , we are seeing that three contours @xmath246 , @xmath247 and , @xmath245 are intersecting at a point ( @xmath248 , @xmath249 ) in @xmath136-@xmath47 plane . so , the mixing angles are within nearly @xmath156 variation about the best fit values . obtained @xmath47 , @xmath136 value gives @xmath250 , @xmath251 , @xmath252 , @xmath253 ev , @xmath254 ev , @xmath255 and @xmath256 . case ( ii ) : @xmath225 gev , now from fig . [ fig : contplot2 ] , we have the intersection of the contour @xmath245 with the contours @xmath257 , @xmath258 at ( @xmath259 , @xmath260 ) in @xmath136-@xmath47 plane . so , @xmath87 and @xmath106 are more closer to their best fit values ( nearly @xmath261 deviation from their best fit value ) . at this point we have @xmath262 , @xmath263 , @xmath264 , @xmath265 ev , @xmath266 ev , @xmath267 and @xmath268 . case ( iii ) : @xmath226 gev , from the fig . [ fig : contplot3 ] , the contours @xmath245 , @xmath269 and , @xmath270 have crossed in @xmath136-@xmath47 plane at ( @xmath271 , @xmath272 ) . so , higher scale of right handed neutrino mass helps to have very good value of @xmath87 and @xmath106 . at the intersection point , we get @xmath273 , @xmath274 , @xmath275 , @xmath276 ev , @xmath277 ev , @xmath278 and @xmath279 . the small value of @xmath81 compatible with all experimental results . so , we can demand that @xmath1 is an approximate symmetry . in this model everything is determinable in terms of parameter @xmath47 and @xmath136 . well measured quantities fix the value of those parameters . so , value of the rest of the physical quantities ( some of them not so well measured in experiment like @xmath107 , @xmath280 and some of them yet to measure in experiment like @xmath144 , @xmath160 and the majorana phases also ) are obtainable in this model . question may arise whether we can have any relations among the phases in this model or not . first of all , there are two kinds phases , low energy and high energy phases . high energy phases are responsible to generate the lepton asymmetry . the low energy phases are responsible for determining low energy leptonic cp violation . low energy phases are in two type , one is the lepton no preserving cp violating phase @xmath160 and another two are the lepton no breaking cp violating phases . in general , all phases are independent , meaning that there are no correlation among the phases between high and low energy sector , and also phases inside a particular sector are not correlated . for three generations of neutrinos , there are three key phases which are responsible for leptogenesis . those are phases in @xmath281 , @xmath282 , and @xmath283 . from matrix @xmath74 given in eq . ( [ hc13 ] ) , we have @xmath284 it leads to the relation , @xmath285 for a given @xmath47 and @xmath136 the @xmath286 and @xmath287 are known from eq . ( [ majp ] ) . hence @xmath288 , @xmath289 and @xmath290 are individually determinable phases . but , in this model values of those high energy phases follow the relation given in eq . ( [ hprl ] ) . now , let us give a fresh look to the leptonic mixing matrix . in eq . ( [ mnuc ] ) we have given the tri - bimaximal rotated form of the neutrino mass matrix . keeping terms upto first order in @xmath81 we obtain the diagonalising matrix in the following form , @xmath291 where the @xmath292 , @xmath293 , @xmath294 and the associated phases are completely known function of @xmath47 and @xmath136 . an additional phase matrix @xmath295 is needed to make masses of light neutrino real , from eq . ( [ msevc ] ) we have the phase matrix @xmath296 to obtain the ckm form of mixing matrix we need to rotate @xmath297 by two diagonal phase matrix , let @xmath298 and @xmath299 . so , we have @xmath300 now the with @xmath107 small we can write @xmath301 and more six relations . but these three are sufficient for our discussions . the phases associated to @xmath297 elements , like @xmath302 , @xmath303 , and @xmath304 associated to @xmath305 , @xmath306 and @xmath307 respectively , are completely determinable in terms of @xmath47 , @xmath136 using functional form of @xmath81 , @xmath292 , @xmath293 , @xmath294 and their associates phases . now from the eq . ( [ ckm ] ) we obtain the following phase relations @xmath308 now the form of total mixing matrix is , @xmath309 this phase part in the parenthesis can be absorbed to charged lepton fields and the remaining part gives the leptonic mixing matrix of the form @xmath310 , where the @xmath311 and @xmath312 are the two majorana phases of leptonic mixing matrix . from eq . ( [ ffmv ] ) and using relations in eq . ( [ prl1 ] ) and eq . ( [ phh ] ) , we have the majorana phases , @xmath313 where @xmath314 and @xmath315 are known function of @xmath47 and @xmath136 . in eq . ( [ majp2 ] ) we have the relation of low energy and high energy phases . so , in our model we have correlation among the cp violating phases . we have shown that non - zero @xmath0 is generated in a softly broken @xmath1 symmetric model through see - saw mechanism incorporating single parameter perturbation in @xmath2 in single element . first , we have studied all possible nine cases to explore the mixing angles considering all model parameters real . the extent of @xmath107 investigated , keeping the experimental values of present solar and atmospheric mixing angles . among all nine possible texture of @xmath2 some of them generates non - zero @xmath107 . out of those non - zero @xmath107 generating textures of @xmath2 we find that breaking at 12 and 13 elements are encompassing the best values of @xmath87 and @xmath106 . however , the reach of @xmath107 in those cases are around @xmath261 . considering one of the parameter complex we extend our analysis with one of the most suitable texture of @xmath2 with breaking at 13 element . we have calculated mixing angles and neutrino mass squared differences in terms four model parameters ( @xmath141 , @xmath81 , @xmath47 , @xmath136 ) . we restrict model parameters utilising the well measured quantities @xmath3 , @xmath4 , @xmath87 and @xmath106 and we have obtained @xmath107 ( upto @xmath156 ) and large @xmath144 ( @xmath316)and @xmath317 well below the present experimental upper bound . in addition to that a large @xmath160 is also obtained . further study on leptogenesis is also done and the present wmap value of baryon asymmetry is obtained for a right handed neutrino mass scale @xmath8 gev . in our model , we have seen that the phases responsible for the leptogenesis are correlated . we also find out the relations among low energy cp violating phases and the lepton asymmetry phases . small @xmath1 symmetry breaking parameter @xmath81 , is sufficient to describe the all low energy neutrino data and high energy cp violation ( leptogenesis ) . so , @xmath1 symmetry is an approximate symmetry . here we consider breaking of @xmath1 symmetry in all other entries of @xmath2 . case ( i ) is already discussed in the text . 0.1 in ( ii)breaking at 11 element : in this case @xmath2 is given by @xmath318 the mass eigenvalues are @xmath319 and the three mixing angles come out as @xmath320 the solar and atmospheric mass differences and their ratio are @xmath321\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-4\e(k-1)^2\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+4\e(k^2 + 2k-1)\right ] } { \left[3k(2-k)-4\e(k-1)^2\right ] } \label{msd11 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath322 } { \left[r(k+1)^2+k^2 + 2k-1\right ] } \label{ep11 } \end{aligned}\ ] ] iii ) breaking at 33 element : in this case , the structure of @xmath2 is given by @xmath323 mass eigenvalues are @xmath324 and the angels are @xmath325 \quad \sin\theta_{13 } = -\frac{\e k(1-k)}{3{\sqrt 2}(2-k ) } \label{angel33 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath326\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-2\e(2k^2 - 4k-1)\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+2\e(2k^2 + 4k+1)\right ] } { \left[3k(2-k)-2\e(2k^2 - 4k-1)\right ] } \label{msd33 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath327 } { \left[(k-1)^2(2k^2 + 4k+1)+ r(k+1)^2(2k^2 - 4k-1)\right ] } \label{ep33 } \end{aligned}\ ] ] iv ) breaking at 12 element : in this case , the structure of @xmath2 is given by @xmath328 mass eigenvalues are @xmath329 and the three mixing angles come out as @xmath330 \quad \sin\theta_{13 } = -\frac{\e}{3\sqrt 2}\frac{k^2-k-3}{(k-2)}\nonumber\\ \label{angel12 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath331\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-4\e(1-k)^2)\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+4\e(k^2 + 2k+2)\right ] } { \left[3k(2-k)-4\e(1-k)^2\right ] } \label{msd12 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath332 } { \left[(k^2 + 2k+2)+ r(k+1)^2\right ] } \label{ep12 } \end{aligned}\ ] ] v ) breaking at 13 element : in this case , the structure of @xmath2 is given by the mass eigenvalues are @xmath329 and the three mixing angles come out as @xmath334\nonumber\\ \sin\theta_{13}&= & -\frac{\e}{3\sqrt 2}\frac{k^2-k-3}{(k-2 ) } \label{angel13 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath331\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-4\e(1-k)^2)\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+4\e(k^2 + 2k+2)\right ] } { \left[3k(2-k)-4\e(1-k)^2\right ] } \label{msd13 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath332 } { \left[(k^2 + 2k+2)+ r(k+1)^2\right ] } \label{ep13 } \end{aligned}\ ] ] vi ) breaking at 23 element : in this case , the structure of @xmath2 is given by @xmath335 the mass eigenvalues are @xmath336 and the three mixing angles come out as @xmath337 \quad\quad \sin\theta_{13 } = 0 \label{angel23 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath326\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-2\e(2k^2 - 4k+5)\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+2\e(2k^2 + 4k+1)\right ] } { \left[3k(2-k)-2\e(2k^2 - 4k+5)\right ] } \label{msd23 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath327 } { \left[r(1+k)^2(2k^2 - 4k+5)+ ( k-1)^2(2k^2 + 4k+1)\right ] } \label{ep23 } \end{aligned}\ ] ] vii ) breaking at 21 element : in this case , the structure of @xmath2 is given by @xmath338 the mass eigenvalues are @xmath329 and the three mixing angles come out as @xmath339\nonumber\\ \sin\theta_{13 } & = & \frac{\epsilon}{3\sqrt 2}\frac{(3+k^2 - 4k ) } { ( 2-k ) } \label{angel21 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath331\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-4\e(1-k)^2\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+4\e(k^2 + 2k+2)\right ] } { \left[3k(2-k)-4\e(1-k)^2\right ] } \label{msd21 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath340 } { \left[r(1+k)^2+k^2 + 2k+2 \right ] } \label{ep21 } \end{aligned}\ ] ] viii ) breaking at 31 element : in this case , the structure of @xmath2 is given by @xmath341 the mass eigenvalues are @xmath329 and the three mixing angles come out as @xmath342\nonumber\\ \sin\theta_{13 } & = & -\frac{\e}{3\sqrt{2 } } \frac{3+k^2 - 4k}{2-k } \label{angel31 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath331\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-4\e(1-k)^2\right]\nonumber\\ & & r = \frac{\ds}{\da}= \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+4\e(k^2 + 2k+2)\right ] } { \left[3k(2-k)-4\e(1-k)^2\right ] } \label{msd31 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath332 } { \left[r(1+k)^2+k^2 + 2k+2 \right ] } \label{ep31 } \end{aligned}\ ] ] ix ) breaking at 32 element : in this case , the structure of @xmath2 is given by @xmath343 the mass eigenvalues are @xmath336 and the three mixing angles come out as @xmath344 \quad \sin\theta_{13 } = 0 \label{angel32 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath326\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-2\e(2k^2 - 4k+5)\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+2\e(2k^2 + 4k+1)\right ] } { \left[3k(2-k)-2\e(2k^2 - 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we have shown that non - zero @xmath0 is generated in a see - saw type softly broken @xmath1 symmetric model through a single parameter perturbation in @xmath2 in a single element . we have explored all possible 9 cases to study the neutrino mixing angles considering the best fitted values of @xmath3 and @xmath4 with all parameters real . we have extended our analysis for the complex case and demonstrated large low energy cp violation ( @xmath5 ) and @xmath6 in addition to mixing and mass pattern . we have also investigated leptogenesis and for a reasonable choice of model parameters compatible with low energy data , wmap value of baryon asymmetry @xmath7 is obtained for right handed neutrino mass scale @xmath8 gev . we have obtained a relation among the phases responsible for leptogenesis and have shown that those phases also have correlations with low energy cp violating phases . pacs number(s ) : 14.60.pq , 11.30.hv , 98.80.cq

organic molecular crystals , namely crystals composed of organic molecules held together by weak van der waals forces , are emerging as excellent candidates for fabricating nanoscale devices . these have potential application in electronics and optoelectronics in particular in areas such as solar energy harvesting , surface photochemistry , organic electronics and spintronics @xcite . a feature common to such class of devices is that they are composed from both an organic and inorganic component , where the first forms the active part of the device and the second provides the necessary electrical contact to the external circuitry . clearly the electronic structure of the interface between these two parts plays a crucial role in determining the final device performance and needs to be understood carefully . in particular it is important to determine how charge transfers between the organic and the inorganic component and the energies at which the transfer takes place . this is a challenging task , especially in the single - molecule limit . upon adsorption on a substrate , the electron addition and removal energies of a molecule change value from that of their gas phase counterparts . this is expected since , when the molecule is physisorbed on a polarisable substrate , the removal ( addition ) of an electron from ( to ) the molecule gives rise to a polarisation of the substrate . the image charge accumulated on the substrate in the vicinity of the molecule alters the addition or removal energy of charge carriers from the molecule . a common way to calculate the addition and removal energies is to use a quasiparticle ( qp ) description . within the qp picture , one ignores the effects of relaxation of molecular orbitals due to addition or removal of electrons and consequently takes the relative alignment of the metal fermi level , @xmath1 , with either the lowest unoccupied molecular orbital ( lumo ) and highest occupied molecular orbital ( homo ) of a molecule as removal energy . this effectively corresponds to associate the electron affinity and the ionization potential respectively to the lumo and homo of the molecule . the adequacy of the qp description then depends on the level of theory used to calculate the energy levels of the homo and lumo . if the theory of choice is density functional theory ( dft ) @xcite , then a number of observations should be made . firstly , it is important to note that except for the energy of the homo , which can be rigorously interpreted as the negative of the ionization potential @xcite , in general the kohn - sham orbitals can not be associated to qp energies . this is , however , commonly done in practice and often the kohn - sham qp levels provide a good approximation to the true removal energies , in particular in the case of metals . for molecules unfortunately the situation is less encouraging with the local and semi - local approximations of the exchange and correlation functional , namely the local density approximation ( lda ) and the generalized gradient approximation ( gga ) , performing rather poorly even for the homo level . such situation is partially corrected by hybrid functionals @xcite or by functional explicitly including self - interaction corrections @xcite , and extremely encouraging results have been recently demonstrated for range separated functionals @xcite . the calculation of the energy levels alignment of a molecule in the proximity of a metal , however , presents additional problems . in fact , the formation of the image charge , although it is essentially a classical electrostatic phenomenon , has a completely non - local nature . this means that unless a given functional is explicitly non - local it will in general fail in capturing such effect . the most evident feature of such failure is that the position of the homo and lumo changes very little when a molecule approaches a metallic surface @xcite . such failure is typical of the lda and gga , and both hybrid and self - interaction corrected functionals do not improve much the situation . a possible solution to the problem is that of using an explicit many - body approach to calculate the qp spectrum . this is for instance the case of the gw approximation @xcite , which indeed is capable of capturing the energy levels renormalization due to the image charge effect @xcite . the gw scheme , however , is highly computationally demanding and can be applied only to rather small systems . this is not the case for molecules on surfaces , where the typical simulation cells have to include several atomic layers of the metal and they should be laterally large enough to contain the image charge in full . this , in addition to the gw necessity to compute a significant fraction of the empty states manifold , make the calculations demanding and it is often not simple even to establish whether convergence has been achieved . in this paper we approach the problem of evaluating the charge transfer energies of an organic molecule physisorbed on an inorganic substrate with the help of a much more resource - efficient alternative , namely constrained density functional theory ( cdft ) @xcite . in cdft , one transfers one electron from the molecule to the substrate ( and vice versa ) and calculates the difference in energy with respect to the locally charge neutral configuration ( no excess of charge either on the molecule or the substrate ) @xcite . as such cdft avoids the calculation of a qp spectrum , which is instead replaced by a series of total energy calculations for different charge distributions this approach is free of any interpretative issues and benefits from the fact that even at the lda level the total energy is usually an accurate quantity . finally , it is important to remark that , for any given functional , cdft is computationally no more demanding than a standard dft calculation , so that both the lda and the gga allow one to treat large systems and to monitor systematically the approach to convergence . here we use the cdft approach to study the adsorption of molecules on a 2-dimensional ( 2d ) metal in various configurations . it must be noted that in contrast to a regular 3d metal , in a 2d one the image charge induced on the substrate is constrained within a one - atom thick sheet . this means that electron screening is expected to be less efficient than in a standard 3d metal and the features of the image charge formation in general more complex . in particular we consider here the case of graphene , whose technological relevance is largely established @xcite . most importantly for our work , recently graphene has been used as template layer for the growth of organic crystals @xcite . it is then quite important to understand how such template layer affects the level alignment of the molecules with the metal . as a model system we consider a simple benzene molecule adsorbed on a sheet of graphene . this has been studied in the past @xcite , so that a good description of the equilibrium distance and the corresponding binding energy of the molecule in various configurations with respect to the graphene sheet are available . furthermore , a @xmath2 study for some configurations exists @xcite , so that our calculated qp gap can be benchmarked . our calculations show that the addition and removal energies decrease in absolute value as the molecule is brought closer to the graphene sheet . such decrease can be described with a classical electrostatic model taking into account the true graphene dielectric constant . as it will be discussed , a careful choice of the substrate unit cell is necessary to ensure the inclusion of the image charge , whose extension strongly depends on the molecule - substrate distance . we also reveal that the presence of defects in the graphene sheet , such as a stone - wales one , does not significantly alter the charge transfer energies . in realistic situations , _ e.g. _ at the interface between a molecular crystal and an electrode , a molecule is surrounded by many others , which might alter the level alignment . we thus show calculations , where neighboring molecules are included above , below and in the same plane of the one under investigation . interestingly , our results suggest that the charge transfer states are weakly affected by the presence of other molecules . in order to find the ground state energy of a system , kohn - sham dft minimises a universal energy functional @xmath3=\sum_{\sigma}^{\alpha,\beta}\sum_{i}^{n_\sigma}\langle \phi_{i \sigma}|-\frac{1}{2}\nabla^{2}|\phi_{i \sigma}\rangle+\int d\mathbf{r}v_n(\mathbf{r})\rho(\mathbf{r})+j[\rho]\\ + e_\mathrm{xc}[\rho^{\alpha},\rho^{\beta}]\ : , \end{split}\ ] ] where @xmath4 , @xmath5 and @xmath6 denote respectively the hartee , exchange - correlation ( xc ) and external potential energies . the kohn - sham orbitals , @xmath7 , for an electron with spin @xmath8 define the non - interacting kinetic energy @xmath9 , while @xmath10 is the total number of electrons with spin @xmath8 . the electron density , is then given by @xmath11 . in contrast to regular dft , in cdft one wants to find the ground state energy of the system subject to an additional constraint of the form @xmath12 where @xmath13 is a weighting function that describes the spatial extension of the constraining region and @xmath14 is the number of electrons that one wants to confine in that region . in our case @xmath15 is set to 1 inside a specified region and zero elsewhere . in order to minimise @xmath16 $ ] subject to the constraint , we introduce a lagrange multiplier @xmath17 and define the constrained functional @xcite @xmath18=e[\rho]+v_\mathrm{c}\left(\sum_{\sigma}\int w_\mathrm{c}^{\sigma}(\mathbf{r})\rho^{\sigma}(\mathbf{r})d\mathbf{r}-n_\mathrm{c}\right)\ ] ] now the task is that of finding the stationary point of @xmath19 $ ] under the normalization condition for the kohn - sham orbitals . this leads to a new set of kohn - sham equations @xmath20\phi_{i\sigma}\\ = \epsilon_{i\sigma}\phi_{i\sigma}\ : , \end{split}\ ] ] where @xmath21 is the exchange and correlation potential . equation ( [ equ4 ] ) does not compute @xmath17 , which remains a parameter . however , for each value of @xmath17 it produces a unique set of orbitals corresponding to the minimum - energy density . in this sense we can treat @xmath19 $ ] as a functional of @xmath17 only . it can be proved that @xmath19 $ ] has only one stationary point with respect to @xmath17 , where it is maximized @xcite . most importantly the stationary point satisfies the constraint . one can then design the following procedure to find the stationary point of @xmath19 $ ] : ( i ) start with an initial guess for @xmath22 and @xmath17 and solve eq . ( [ equ4 ] ) ; ( ii ) update @xmath17 until the constraint eq . ( [ equ2 ] ) is satisfied ; ( iii ) start over with the new @xmath17 and a new set of @xmath23s . here we use cdft to calculate the charge transfer energy between a benzene molecule and a graphene sheet . for any given molecule - to - substrate distance , @xmath24 , we need to perform three different calculations : 1 . a regular dft calculation in order to determine the ground state total energy @xmath25 and the amount of charge on each subsystem ( _ i.e. _ on the molecule and on the graphene sheet ) 2 . a cdft calculation with the constraint that the graphene sheet contains one extra electron and the molecule contains one hole . this gives the energy @xmath26 . 3 . a cdft calculation with the constraint that the graphene sheet contains one extra hole and the molecule one extra electron . this gives the energy @xmath27 . the charge transfer energy for removing an electron from the molecule and placing it on the graphene sheet is then @xmath28 . similarly , that for the transfer of an electron from the graphene sheet to the molecule is @xmath29 . since in each run the cell remains charge neutral , there is no need here to apply any additional corrections . however , we have to keep in mind that this method is best used when the two subsystems are well separated so that the amount of charge localized on each subsystem is a well defined quantity . in our calculations we use the cdft implementation @xcite for the popular dft package siesta@xcite , which adopts a basis set formed by a linear combination of atomic orbitals ( lcao ) . the constrain is introduced in the form of a projection over a specified set of basis orbitals and in particular use the lowdin projection scheme . throughout this work we adopt double - zeta polarized basis set with an energy cutoff of 0.02 ry . the calculations are done with norm - conserving pseudopotential and the lda is the exchange - correlation functional of choice . a mesh cutoff of 300 ry has been used for the real - space grid . we impose periodic boundary conditions with different cell - sizes and the @xmath30-space grid is varied in accordance with the size of the unit cell . for instance , an in plane 5@xmath315 @xmath30-grid has been used for a 13@xmath3113 graphene supercell . we begin this section with a discussion on the equilibrium distance for a benzene molecule adsorbed on graphene . this is obtained by simply minimizing the total energy difference @xmath32 , where @xmath33 is the total energy for the cell containing benzene on graphene , while @xmath34 ( @xmath35 ) is the total energy of the same cell when only the benzene ( graphene ) is present . this minimization is performed for two different orientations of the benzene molecule with respect to the graphene sheet : the _ hollow _ ( h ) configuration , in which all the carbon atoms of the benzene ring are placed exactly above the carbon atoms of graphene , and the _ stack _ ( s ) configuration , in which alternate carbon atoms of the benzene molecule are placed directly above carbon atoms of the graphene sheet [ see fig . [ fig : figure1](a , b ) ] . for the h configuration we find an equilibrium distance of 3.4 , while for the s one this becomes 3.25 . these results are in fair agreement with another lda theoretical study @xcite ( predicting 3.4 and 3.17 respectively for for the h and s orientations ) . note that a more precise evaluation of such distances requires the use of van der waals corrected functionals . this exercise , however , is outside the scope of our work and here we just wish to establish that the equilibrium distance is large enough for our constrain to remain well defined . it can also be noted that the equilibrium distance of 3.6 obtained with a vdw - df study @xcite is not very different from our lda result . , for different unit cell sizes of graphene sheet . the results are presented for two different molecule - to - graphene distances : 3.4 and 6.8 .,scaledwidth=45.0% ] we then study the dependence of the charge transfer energies on the size of the graphene unit cell used . this is achieved by looking at the charge transfer gap , @xmath36 , as a function of the unit cell size at various molecule - to - graphene distances ( see fig . [ fig : figure2 ] ) . when the molecule is very close to the graphene sheet , after transferring an electron , the image charge is strongly attracted by the oppositely charged molecule and thereby remains highly localized . however , as the molecule moves away from the substrate , the attraction reduces since the coulomb potential decays with distance , resulting in a delocalization of the image charge . this will eventually spread uniformly all over the graphene sheet in the limit of an infinite distance . if the unit cell is too small , the image charge will be artificially over - confined , resulting in an overestimation of @xmath37 and @xmath38 and , as a consequence , of the charge transfer energies . this effect can be clearly seen in fig . [ fig : figure2 ] , where we display the variation of the charge transfer energies as a function of the cell size . clearly , for the shorter distance ( 3.4 corresponding to the average equilibrium distance ) , the energy gap converges for supercells of about 10@xmath3110 ( 10@xmath3110 graphene primitive cells ) . at the larger distance of 6.8 the same convergence is achieved for a 13@xmath3113 supercell . next we compute the charge transfer energies as a function of the distance between the sheet and the molecule . in order to compare our results with the gap expected in the limit of an infinite distance , we need to evaluate first the ionization potential , @xmath39 , and the electron affinity , @xmath40 , of the isolated benzene molecule . this is also obtained in terms of total energy differences between the neutral and the positively and negatively charged molecule , namely with the @xmath41scf method . this returns a quasiparticle energy gap , @xmath42 , of 11.02 ev , in good agreement ( within 4.5% ) with the experimental value @xcite . likewise we also determine the fermi level ( @xmath43 ) of graphene , which is found to be 4.45 ev . in fig . [ fig : figure3](a ) we show the change in the charge transfer energy gap with the distance of the benzene from the graphene sheet for the h configuration . as expected , when the molecule is close to the surface , there is a considerably large attraction between the image charge and the opposite charge excess on the molecule , resulting in an additional stabilization of the system and a reduction in magnitude of @xmath44 and @xmath37 . hence , in such case the charge transfer energies have a reduced magnitude and the charge transfer gap is smaller than that in the gas phase . then , as the molecule moves away from the graphene sheet , the charge transfer energies increase and so does the charge transfer energy gap until it eventually reaches the value corresponding to the homo - lumo gap of the isolated molecule in the limit of an infinite distance . in figs . [ fig : figure3](b ) , ( c ) , ( d ) and ( e ) we show the excess charge - density , @xmath45 , in different parts of the system after transferring one electron for two different molecule - to - graphene distances . the excess charge - density @xmath45 is defined as @xmath46 , where @xmath47 and @xmath48 are respectively the charge densities of the system before and after the charge transfer . thus the portion of @xmath45 localized on the graphene sheet effectively corresponds to the image charge profile . clearly , due to the stronger coulomb attraction , the image charge is more localized for @xmath49 than for @xmath50 . at equilibrium for the s configuration , @xmath51 , the charge transfer energy gap is calculated to be 8.91 ev , which is in good agreement ( within 4% ) with the gap obtained by @xmath52 @xcite . in table [ table : configuration ] , for the purpose of comparison , we have listed the charge transfer energies and charge transfer gaps for two different heights , 3.4 and 6.8 , and in different configurations . the most notable feature is that for the case of a pristine graphene substrate the specific absorption site plays little role in determining the charge transfer levels alignment . in general actual graphene samples always display lattice imperfections @xcite . in order to determine the effect of such structural defects on the ct energies , we consider a reference system where a stone - wales ( sw ) defect ( in which a single c - c bond is rotated by 90@xmath53 ) is present in the graphene sheet . we have then calculated @xmath54 for two different positions of the molecule with respect to the defect on the sheet , namely the @xmath55 position , in which the molecule is placed right above the defect and the @xmath56 position , in which it is placed above the sheet far from the defect ( see fig . [ fig : figure1 ] ) . our findings are listed in tab . [ table : configuration ] , where we report the charge transfer energies for both the configurations , assuming the molecule is kept at the same distance from the graphene sheet . from the table it is evident that the structural change in graphene due to presence of such defect does not alter the charge transfer energies of the molecule . this is because the image charge distribution on graphene is little affected by presence of the sw defect . in addition , the density of states ( dos ) of graphene remains almost completely unchanged near its fermi energy after introducing such defect as can be seen in fig . [ figure4 ] , which shows that the partial density of states ( pdos ) of the atoms forming the sw defect has no significant presence near the fermi level . thus , after the charge transfer , the electron added to ( or removed from ) the graphene sheet has the same energy that it would have in the absence of the defect , i.e. it is subtracted ( added ) from a region of the dos where there is no contribution from the sw defect . in this context , it is noteworthy that a @xmath52 study @xcite has concluded that altering the structure of pristine graphene by introducing dopant ( which raises the fermi level of graphene by 1 ev ) also has minor effect on the qp gap of benzene , reducing it by less than 3% . .@xmath57 , @xmath58 and @xmath59 for various configurations of a benzene molecule on pristine and defective graphene . h and s denote adsorption of benzene on graphene in the _ hollow _ and _ stack _ configuration , respectively . @xmath55 and @xmath56 correspond to adsorption on graphene with sw defect , with the former corresponding to adsorption exactly on top of the defect and the latter corresponding to adsorption away from the site of the defect . the configurations @xmath60 and @xmath61 both correspond to adsorption of two benzene molecules in _ hollow _ configuration- one at height 3.4 and another at a height 6.8 . while in @xmath60 , the ct is calculated for the lower molecule , in @xmath61 , the ct is calculated for the upper one . finally @xmath62 represents the case in which we have a layer of non - overlapping benzenes adsorbed on graphene and one is interested in calculating the ct energy for one of them , which is placed in the _ hollow _ configuration . [ cols="^,^,^,^,^",options="header " , ] in real interfaces between organic molecules and a substrate , molecules usually are not found isolated but in proximity to others . it is then interesting to investigate the effects that the presence of other benzene molecules produce of the charge transfer energies of a given one . to this end we select three representative configurations . in the first one , @xmath60 , the graphene sheet is decorated with two benzene molecules , one at 3.4 while the other is placed above the first at 6.8 from the graphene plane . we then calculate the charge transfer energies of the middle benzene ( the one at 3.4 from the sheet ) . the excess charge on different parts of the system ( image charge ) , after transferring one electron to the sheet , is displayed in fig . [ fig : he](a ) and fig . [ fig : he](b ) . the second configuration , @xmath61 , is identical to the first one but now we calculate the charge transfer energies of the molecule , which is farther away from the graphene sheet , namely at a distance of 6.8 . for this configuration , the excess charge after a similar charge transfer is shown in fig . [ fig : he](c ) and fig . [ fig : he](d ) . in the third configuration , @xmath62 , we arrange multiple benzene molecules in the same plane . the molecules are in close proximity with each other although their atomic orbitals do not overlap . charge transfer energies are then calculated with respect to one benzene molecule keeping the others neutral and an isovalue plot for similar charge transfer is shown in fig . [ fig : he](e ) and fig . [ fig : he](f ) . the charge transfer energies calculated for these three configurations are shown in tab . [ table : configuration ] . if one compares configurations where the molecule is kept at the same distance from the graphene plane , such as the case of h(@xmath49 ) , @xmath60 and @xmath62 or of h(@xmath50 ) and @xmath61 , it appears clear that the presence of other molecules has some effect on the charge transfer energies . in particular we observe than when other molecules are present both @xmath57 and @xmath58 get more shallow , i.e. their absolute values is reduced . interestingly the relative reduction of @xmath57 and @xmath58 depends on the details of the positions of the other molecules ( e.g. it is different for @xmath60 and @xmath62 ) but the resulting renormalization of the homo - lumo gap is essentially identical [ about 25 mev when going from h(@xmath49 ) to either @xmath60 or @xmath62 ] . this behaviour can be explained in terms of a simple classical effect . consider the case of @xmath60 for example . when one transfers an electron from the middle benzene to the graphene sheet the second benzene molecule , placed above the first , remains neutral but develops an induced charge dipole . the moment of such dipole points away from the charged benzene and lowers the associated electrostatic potential . importantly , also the potential of graphene will be lowered . however , since the potential generated by an electrical dipole is inversely proportional to the square of distance , the effect remains more pronounced at the site of the middle benzene than at that of the graphene sheet . a similar effect can be observed for an electron transfer from the graphene sheet to the middle benzene and for the @xmath62 configuration . in the case of @xmath61 , the system comprising the topmost benzene ( from which we transfer charge ) and the graphene plane can be thought of as a parallel - plate capacitor . the work , @xmath64 , done to transfer a charge @xmath65 from one plate to the other is @xmath66 where @xmath67 is the capacitance , which in turn is proportional to the dielectric constant of the medium enclosed between the plates . hence , at variance with the case of h(d=6.8 ) , the space in between the molecule and the graphene sheet is occupied by a molecule with finite dielectric constant and not by vacuum . this results in a reduction of @xmath64 , so that the charge transfer energies for @xmath61 are smaller than those for h(d=6.8 ) . finally , we show that our calculated energy levels alignment can be obtained from a classical electrostatic model . if one approximates the transferred electron as a point charge and the substrate where the image charge forms as an infinite sheet of relative permittivity @xmath68 then , for a completely planar distribution of the bound surface charge , the work done by the induced charge to take an electron from the position of the molecule ( at a distance @xmath24 ) to infinity is @xmath69 hence , this electrostatic approximation predicts that the presence of the substrate lowers the @xmath70 of the molecule by @xmath71 with respect to the corresponding gas - phase value . however , the actual image charge is not strictly confined to a 2d plane but instead spills out over the graphene surface . we can account for such non - planar image charge distribution by introducing a small modification to the above expression @xcite and write the lumo at a height @xmath24 as @xmath72 where @xmath73 is the distance between the centre of mass of the image charge and the substrate plane and @xmath74 is the gas - phase lumo ( the electron affinity ) . a similar argument for the homo level shows an elevation of same magnitude due to the presence of the substrate . in fig . [ fig : classical_plot ] we plot the charge transfer energies and show that they compare quite well with the curves predicted by the classical model by using an effective dielectric constant of 2.4 for graphene @xcite . when drawing the classical curves we have used an approximate value , @xmath75 , which provides an excellent estimate for smaller distances , @xmath24 . it is worth noting that for larger distances , though the actual value of @xmath73 should be much less , the overall effect of @xmath73 is very small and almost negligible . in the same graph , we have also plotted the classical curves corresponding to benzene on a perfectly metallic ( @xmath76 ) surface . this shows that the level renormalization of benzene for physisorption on graphene is significantly different from that on a perfect metal , owing to the different screening properties of graphene . ( circles ) and @xmath77 ( squares ) calculated for different molecule - to - substrate distances . the cdft results are seen to agree well with the classically calculated curve given in red . the horizontal lines mark the same quantities for isolated an molecule ( gas - phase quantities ) . the continuous black line shows the position of the classically calculated level curve for adsorption on a perfect metal @xmath76.,scaledwidth=44.0% ] we have used cdft as implemented in the siesta code to calculate the energy levels alignment of a benzene molecule adsorbed on a graphene sheet . in general the charge transfer energies depend on the distance between the molecule and the graphene sheet , and this is a consequence of the image charge formation . such an effect can not be described by standard kohn - sham dft , but it is well captured by cdft , which translates a quasi - particle problem into an energy differences one . with cdft we have simulated the energy level renormalization as a function of the molecule - to - graphene distance . these agree well with experimental data for an infinite separation , where the charge transfer energies coincide with the ionization potential and the electron affinity . furthermore , an excellent agreement is also obtained with @xmath0 calculations at typical bonding distances . since cdft is computationally inexpensive we have been able to study the effects arising from bonding the molecule to a graphene structural defect and from the presence of other benzene molecules . we have found that a stone - wales defect does not affect the energy level alignment since its electronic density of state has little amplitude at the graphene fermi level . in contrast the charge transfer energies change when more then a molecule is present . all our results can be easily rationalized by a simple classical electrostatic model describing the interaction of a point - like charge and a uniform planar charge distribution . this , at variance to the case of a perfect metal , takes into account the finite dielectric constant of graphene . this work is supported by the european research council , quest project . computational resources have been provided by the supercomputer facilities at the trinity center for high performance computing ( tchpc ) and at the irish center for high end computing ( ichec ) . additionally , the authors would like to thank dr . ivan rungger and dr . a. m. souza for helpful discussions . 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 @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1002/cphc.200700177 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrev.136.b864 [ * * , ( ) ] link:\doibase 10.1103/physrevb.18.7165 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.88.165112 [ * * , ( ) ] @noop * * , ( ) @noop ( ) \doibase http://dx.doi.org/10.1016/j.apsusc.2010.07.069 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.96.146107 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.88.235437 [ * * , ( ) ] @noop * * , ( ) \doibase http://dx.doi.org/10.1016/0022-1902(81)80486-1 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( )

constrained density functional theory ( cdft ) is used to evaluate the energy level alignment of a benzene molecule as it approaches a graphene sheet . within cdft the problem is conveniently mapped onto evaluating total energy differences between different charge - separated states , and it does not consist in determining a quasi - particle spectrum . we demonstrate that the simple local density approximation provides a good description of the level aligmnent along the entire binding curve , with excellent agreement to experiments at an infinite separation and to @xmath0 calculations close to the bonding distance . the method also allows us to explore the effects due to the presence of graphene structural defects and of multiple molecules . in general all our results can be reproduced by a classical image charge model taking into account the finite dielectric constant of graphene .

since its discovery in 2004 graphene @xcite draws much attention because of unique features of this two - dimensional system . graphene is composed of a sp@xmath2-bonded carbon atoms forming honeycomb structure . it became famous for its very interesting electronic structure with characteristic , linear energy dispersion near k point of brillouin zone and many other features @xcite . shortly after , experimental techniques allowed fabrication of other new two - dimensional materials , like bn and mos@xmath3 honeycomb structures @xcite . the discovery of such stable two - dimensional materials triggered search for similar structures made from different compounds . up to now many of these hypothetical structures constructed from silanene ( 2d si ) and germanene ( 2d ge ) @xcite , iii - v compounds @xcite , sic @xcite or zno @xcite have been studied theoretically . also , calculations show @xcite , that graphene - like type of structure is not the only one possible for two - dimensional material . this new class of boron sheets , composed of triangular and hexagonal motifs can be stabilized by interplay of three- and two - center bonding scheme @xcite . another example of triangular sheet could be found in already known material , which is li@xmath4n in its @xmath5 phase . li@xmath6n is a bulk material known to be a fast ion conductor @xcite . li@xmath6n is also known as a candidate for hydrogen storage material due its high theoretical h@xmath0 capacity @xcite . bulk li@xmath4n crystallizes in hexagonal structure which is characterized by @xmath7 symmetry group , each nitrogen atom is surrounded by eight lithium atoms . it has layered structure , one layer is li@xmath0n and the other is of li atoms only . previous theoretical studies confirm ionic nature of bonding in this compound @xcite . since n - containing layer is rather weakly bound with two li - only layers , it would be interesting to study electronic properties of such two - dimensional structure ( 2dli@xmath0n ) - fig [ fig0]a . since this structure would have n atoms with dangling bonds , it would give opportunity to study influence of different atoms addition on them . for example addition of hydrogen atoms in case of graphene resulted in new material which is graphane @xcite . graphene and other nano - scale materials are recognized as future building blocks of new electronics technologies @xcite , including spintronics @xcite . in the case of low ( one- and two- ) dimensional structures problem arises because of famous mermin - wagner theorem @xcite , which prevents ferro- or antiferromagnetic order to occur in finite temperatures , which is essential for practical application . this started the theoretical and experimental search for magnetism in graphene and other two - dimensional structures . one of the most promising directions is emergence of magnetism in such structures as an effect of presence of local defects @xcite . according to works of palacios et al . @xcite and , independently , of yazyev @xcite single - atom defects can induce ferromagnetism in graphene based materials . in both cases , the magnetic order arises as an effect of presence of single - atom defects in combination with a sublattice discriminating mechanism . in the case of @xmath8role of such defect could play non - hydrogenated n atom in hydrogenated structure . it would be then instructive to check influence of hydrogenation level on magnetic moment of the structure . in this paper electronic and magnetic structure of pure and hydrogenated 2dli@xmath0n have been analyzed by means of @xmath9-@xmath10 calculations . to investigate electronic and magnetic properties of two - dimensional li@xmath4n structures a series of @xmath9-@xmath10 calculations have been conducted with use of dft vasp code @xcite with paw potentials @xcite . for both spin - unpolarized and spin - polarized cases exchange - correlation potential has been approximated by generalized gradient approximation ( gga ) using pw91 functional @xcite . kinetic energy cutoff of 500 ev for plane - wave basis set has been used . in all cases for self - consistent structure optimizations , the brillouin zone ( bz ) was sampled by @xmath11 special k points . all structures have been optimized for both , spin - unpolarized and spin - polarized cases unless feynman - hellman forces acting on each atom become smaller than 10@xmath12 ev/@xmath13 . a vacuum spacing of 12 was applied to hinder the interactions between @xmath8monolayers in adjacent cells . kiedy supercell i jak liczone magn . ) bandstructure and density of states ( dos ) calculations have been confirmed by use of wien2k code @xcite which implements the full - potential linearized augmented plane wave ( flapw ) method @xcite . in this case for exchange and correlation generalized gradient approximation was used in the perdew - burke - ernzerhoff ( pbe ) parameterization @xcite . to study electronic properties of @xmath8 , at first comparison has been made with bulk material . for both cases lattice constants have been determined by total energy calculations and are found to be equal to 3.65 for bulk ( experimental value 3.63 ) and 3.57 for @xmath8 . in agreement with @xcite bulk li@xmath4n is a semiconductor with non - direct bandgap equal to 1.15 ev between a ( valence band ) and @xmath14 ( conduction band ) points . in contradiction to this , @xmath8 has metallic nature . two - dimensional structure is rather weakly bound - binding energy ( defined as @xmath15 where @xmath16 is the energy of isolated atom(s ) and @xmath17 is the total energy of two - dimensional structure ) is equal to 10.36 ev , while binding energy of bulk structure is equal to 14.25 ev . also , two dimensional sheet would have n atoms with dangling bonds , such structure would be then rather unstable with respect to foreign atoms addition . graphane case suggests that it would be instructive to examine influenece of hydrogenation on electronic structure in such cases as well as addition of lithium atoms . the nature of li - n bond is ionic , as it can be seen from fig . [ fig1 ] showing charge density projected on [ 110 ] plane . since every bond has both ionic and covalent character the level of ionicity can be estimated using difference between electronegativities of bonded atoms @xcite . in the case of li - n bond this difference equal to 2 suggests , that the bond is about 65@xmath18 ionic and 35@xmath18covalent . this fact together with rather large lattice constant suggest that the structure of two - dimensional @xmath8 can be low - buckled ( lb ) rather than plane ( pl ) , according to puckering mechanism described in @xcite . to check this the series of calculations has been done , each with different distance in z direction between li atoms and the plane on which n atoms lie ( @xmath19 ) . the structure with minimal energy has been then optimized . calculations show , that the buckled structure with @xmath19 = x lies 0.54 ev lower that the plane , which means that indeed the puckering mechanism stabilizes the structure . both , plane and low - buckled structures can be seen on fig [ fig0 ] . four structures have been then studied in two conformations , plane and low - buckled two ( pl and lb ) with single h atom attached on top of each n atom ( 2dli@xmath0n+h ) , two with two h atoms attached on both sides of n ( 2dli@xmath0n+2h ) , two with single li atom attached on top of each n atom ( 2dli@xmath0n+li ) , and two with two li atoms attached on both sides of n ( 2dli@xmath0n+2li ) . [ cols="^,^ " , ] @xmath20-@xmath10 calculations have been conducted for hypothetical two - dimensional material @xmath8 to investigate electronic and magnetic properties . calculations show , that structure is much more stable when dangling bonds of nitrogen atoms are functionalized with hydrogen atoms . this hydrogenation has very strong influence on on bandstructure , changing it from wide - gap semiconductor to metal . magnetic properties are also interesting . in analogy to graphene and other two - dimensional materials it is possible to generate non - zero magnetic moment by introduction of distorsion . in the case of @xmath8 the distorsion would be a two - hydrogen or hydrohen - lithium vacancy around the same nitrogen atom . this generates magnetic moment of 1 @xmath1 . since bulk li@xmath4n material has ususally 1 - 2% li vacancies in li@xmath0n layers @xcite such two - dimensional hydrogenated sheet would be almost naturally magnetic . these results may give a hint for experimentalists seeking for two - dimensional ( magnetic ) materials , which would be interesitng addition to growing family of two - dimensional materials . 99 k. s. novoselov et al . , science 306 , 666 ( 2004 ) . k. s. novoselov et al . , proc . usa , 102 , 10451 ( 2005 ) . a. h. castro neto et al . , rev . 81 , 109 ( 2009 ) . s. lebegue et al . , phys . rev . b 79 , 115409 ( 2009 ) . s. cahangirov et al . , phys . 102 , 236804 ( 2009 ) . h. sahin et al . b 80 , 155453 ( 2009 ) . e. bakaroglu et al . , phys . rev . b 81 , 075433 ( 2010 ) . m. topsakal et al . , phys rev . b 80 , 235119 ( 2009 ) . j. kunstmann et al . , phys rev . b 74 , 035413 ( 2006 ) . h. tang et al . , phys . 99 , 115501 ( 2007 ) . g. kerker , phys rev . b , 23 , 12 ( 1981 ) . t. t. fister et al . , phys . , 129 , 044702 ( 2008 ) . h. m. jin et al . , app . lett . , 90 , 084101 ( 2007 ) . s. wu et al . , j. phys . chem , 114 , 16706 ( 2010 ) . j. sofo et al . , phys rev . b 75 , 153401 ( 2007 ) . p. avouris et al . , nat . nanotech . 2 , 605 ( 2007 ) . i. zutic et al . , rev . 76 , 323 ( 2004 ) . n. d. mermin , h. wagner phys . 17 , 1133 ( 1966 ) . p. esquinazi , phys . 91 , 227201 ( 2003 ) . j. j. palacios et al . , phys . rev . b 77 , 195428 ( 2008 ) . o. yazyev , phys . rev . lett . 101 , 037203 ( 2008 ) . o. yazyev et al . , phys . b 75 , 125408 ( 2007 ) . e. h. lieb , phys . 62 , 1201 ( 1989 ) ; 62 , 1927(e ) ( 1989 ) . c. ataca et al . , phys . rev . b 82 , 165402 ( 2010 ) . g. kresse , j. hafner , phys . b 47 , 558 ( 1993 ) . g. kresse , j. furthmuller , phys . rev . b 54 , 11169 ( 1996 ) . e. blochl , phys . b 50 , 17953 ( 1994 ) . j. p. perdew , et al . , phys . rev . b 46 , 6671 ( 1992 ) . p. blaha , k. schwarz , g. k. h. madsen , d. kvasnicka , and j.luitz , in wien2k , an augmented plane wave plus local orbitals program for calculating crystal properties , edited by k. schwarz , techn . universitt wien , austria , 2001 . e. wimmer et al . , b 24 , 864 ( 1981 ) . j. p. perdew , k. burke , and m. ernzerhof , phys . 77 , 3865 ( 1996 ) . ref . needed l. pauling `` the nature of covalent bond '' , ?

using first - principles plane - wave calculations study of electronic and magnetic properties of hypothetical two - dimensional structure of li@xmath0n compound have been conducted . calculations show , that electronic properties of this this structure can be inflenced by hydrogenation , which may change the system from wide - gap semiconductor to metal . also , non - zero magnetic moment , equal to 1 @xmath1 can be generated by intruduction of h vacanies in hydrogenated structure .

any cooperation among agents ( players ) being able to make strategic decisions becomes a _ coalition formation game _ when the players may for various personal reasons wish to belong to a relative _ small coalition _ rather than the `` grand coalition '' . partitioning players represents the crucial question in the game context and a stable partition of the players is referred to as an equilibrium . in @xcite , the authors propose an abstract approach to coalition formation that focuses on simple merge and split stability rules transforming partitions of a group of players . the results are parametrized by a preference relationship between partitions from the point of view of each player . on the other hand , a coalition formation game is called to be _ hedonic _ if * _ the gain of any player depends solely on the members of the coalition to which the player belongs _ , and * _ the coalitions form as a result of the preferences of the players over their possible coalitions set_. accordingly , the stability concepts aiming _ hedonic conditions _ can be summarized as following @xcite : a partition could be _ individually stable , nash stable , core stable , strict core stable , pareto optimal , strong nash stable , strict strong nash stable_. in the sequel , we concentrate on the nash stability . the definition of the nash stability is quite simple : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ a partition of players is nash stable whenever there is no player deviating from his / her coalition to another coalition in the partition_. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we refer to @xcite for further discussions concerning the stability concepts in the context of hedonic coalition formation games . in @xcite , the problem of generating nash stable solutions in coalitional games is considered . in particular , the authors proposed an algorithm for constructing the set of all nash stable coalition structures from players preferences in a given additively separable hedonic game . in @xcite , a bargaining procedure of coalition formation in the class of hedonic games , where players preferences depend solely on the coalition they belong to is studied . the authors provide an example of nonexistence of a pure strategy stationary perfect equilibrium , and a necessary and sufficient condition for existence . they show that when the game is totally stable ( the game and all its restrictions have a nonempty core ) , there always exists a no - delay equilibrium generating core outcomes . other equilibria exhibiting delay or resulting in unstable outcomes can also exist . if the core of the hedonic game and its restrictions always consist of a single point , it is shown that the bargaining game admits a unique stationary perfect equilibrium , resulting in the immediate formation of the core coalition structure . in @xcite , drze and greenberg introduced the hedonic aspect in players preferences in a context concerning local public goods . moreover , purely hedonic games and stability of hedonic coalition partitions were studied by bogomolnaia and jackson in @xcite . in this paper , it is proved that if players preferences are additively separable and symmetric , then a nash stable coalition partition exists . for further discussion on additively separable and symmetric preferences , we refer the reader to @xcite . our work aims at considering stable strategies for hedonic coalition formation games . we first develop a simple decentralized algorithm finding the nash stability in a game if at least one exists . the algorithm is based on _ the best reply strategy _ in which each player decides serially his / her coalition . thus , the problem is considered as a non - cooperative game . we consider a _ random round - robin _ fashion where each player determines its strategy in its turn according to a _ scheduler _ which is randomly generated for each round . under this condition , we prove that the algorithm converges to an equilibrium if it exists . then the fundamental question that comes is to determine which utility allocation methods may ensure a nash - stable partition . we address this issue in the sequel . we first propose the definition of _ the nash - stable core _ which is the set of all possible utility allocation methods resulting in nash - stable partitions . we show that efficient utility allocations where the utility of a group is completely shared between his / her members , may have no nash - stable partitions with some exceptions . rather , we proved that relaxing the efficiency condition may ensure the non - emptyness of the core . indeed we prove that if the sum of players gains within a coalition is allowed to be less than the utility of this coalitionn then a nash - stable partition always exist . a coalition formation game is given by a pair @xmath0 , where @xmath1 is the set of @xmath2 _ players _ and @xmath3 denotes the _ preference profile _ , specifying for each player @xmath4 his preference relation @xmath5 , i.e. a reflexive , complete and transitive binary relation . a coalition structure or a _ coalition partition _ is a set @xmath6 which partitions the players set @xmath7 , i.e. , @xmath8 , @xmath9 are disjoint coalitions such that @xmath10 . given @xmath11 and @xmath12 , let @xmath13 denote the set @xmath14 such that @xmath15 . in its partition form , a coalition formation game is defined on the set @xmath7 by associating a utility value @xmath16 to each subset of any partition @xmath11 of @xmath7 . in its characteristic form , the utility value of a set is independent of the other coalitions , and therefore , @xmath17 . the games of this form are more restricted but present interesting properties to achieve equilibrium . practically speaking , this assumption means that the gain of a group is independent of the other players outside the group . hedonic coalition formation games fall into this category with an additional assumption : [ def : hedonic ] a coalition formation game is called to be _ hedonic _ if * _ the gain of any player depends solely on the members of the coalition to which the player belongs _ , and * _ the coalitions form as a result of the preferences of the players over their possible coalitions set_. the preference relation of a player can be defined over a _ preference function_. let us denote by @xmath18 the preference function of player @xmath12 . thus , player @xmath12 prefers the coalition @xmath19 to @xmath20 iff , @xmath21 we consider the case where the preference relation is chosen to be the utility allocated to the player in a coalition , then @xmath22 where @xmath23 refers to the utility received by player @xmath12 in coalition @xmath19 . in the case of transferable utility games ( tu games ) we are considering in this paper , the utility of a group can be tranfered among users in any way . thus , an utility allocation is set relatively efficient if for each coalition @xmath19 , the sum of individual utilities is equal to the coalition utility : @xmath24 . now , if the preferences of a player are _ additively separable _ , the preference can be even stated with a function characterizing how a player prefers another player in each coalition . this means that the player s preference for a coalition is based on individual preferences . this can be formalized as follows : the preferences of a player are said to be additively separable if there exists a function @xmath25 s.t . @xmath26 @xmath27 where , according to @xcite , @xmath28 is normalized and set to @xmath29 . a profile of additively separable preferences , represented by @xmath30 , satisfies _ symmetry _ if @xmath31 . the question we address in this paper concerns the stability of this kind of games . the stability concept for a coalition formation game may receive different definitions . in the litterature , a game is either said _ individually stable , nash stable , core stable , strict core stable , pareto optimal , strong nash stable , strict strong nash stable_. we refer to @xcite for a thorough definition of these different stability concepts . in this paper , we concentrate only on the nash stability because we are interested in those games where the players do nt cooperate to take their decisions which means that only individual moves are allowed . the definition of the nash stability for an _ hedonic coalition formation game _ is simply : [ def : nashstability ] a partition of players is nash stable whenever no player is incentive to unilaterally change his or her coalition to another coalition in the partition . which can be mathematically formulated as : a partition @xmath11 is said to be nash - stable if no player can benefit from moving from his coalition @xmath13 to another existing coalition @xmath32 , i.e. : @xmath33 nash - stable partitions are immune to individual movements even when a player who wants to change does not need permission to join or leave an existing coalition @xcite . in the literature ( @xcite ) , the stability concepts being immune to individual deviation are _ nash stability , individual stability , contractual individual stability_. nash stability is the strongest within above . we concentrate our analysis on the partitions that are nash - stable . the notion of _ core stability _ has been used already in some models where immunity to coalition deviation is required @xcite . but the nash stable core has not been defined yet at the best of our knowledge . this is what we derive in the next section . under this definition we propose in this paper to evaluate the existence of nash stability and to propose an approach that ensures the convergence to a nash equilibrium of an approximated convex game . let us consider a hedonic tu game noted @xmath34 ( since @xmath35 is transferable to the players , we consider hedonic coalition formation games based on transferable utility ) . for the sake of simplicity , the preference function of player @xmath12 is assumed to be the gain obtained in the corresponding coalition , i.e. , @xmath36 as well as let @xmath37 denote the _ allocation method _ which directly sets up a corresponding preference profile . the corresponding space is equal to the number of set , i.e. @xmath38 where @xmath39 . we now define the operator @xmath40 , where @xmath41 is the set of all possible partitions . clearly , for any preference function , the operator @xmath42 finds the set of nash - stable partition @xmath11 , i.e. @xmath43 . if a nash - stable partition can not be found , the operator maps to empty set . moreover , the inverse of the operator is denoted as @xmath44 which finds the set of all possible preference functions that give the nash - stable partition @xmath11 . thus , the nash - stable core includes all those efficient allocation methods that build the following set : @xmath45 to know if the core is non empty , we need to state the set of constraints the partition function as to fulfill . under the assumption of _ efficiency _ , we have a first set of constraints @xmath46 then a given partition @xmath11 is nash - stable with respect to a given partition function if the following constraints hold : @xmath47 where @xmath48 is the unique set in @xmath11 containing @xmath12 . then , the nash - stable core is non - empty , iif : @xmath49 the nash - stable core can be further defined as : @xmath50 which let us to conclude : the nash - stable core can be non - empty . algorithmically , the nash - stable core is non - empty if the following linear program is feasible : @xmath51 however , it is not possible to state about the non - emptiness of the nash - stable core in the general case . further , searching in an exhaustive manner over the whole partitions is np - complet as the number of parititions grows according to the bell number . typicall , with only @xmath52 players , the number of partition is as large as @xmath53 . we now analyse some specific cases in the following . in the case the grand coalition is targeted , the stability conditions are the following . let @xmath54 , then the following constaints hold : @xmath55 resulting in the following : @xmath56 those cooperative tu games that satisfy this condition are said to be _ we now propose to formulate a special game where the utility is shared among players with an equal relative gain . let us denote the gain of player @xmath12 in coalition @xmath19 as @xmath57 in which @xmath58 is called _ the relative gain_. note that for an isolated player @xmath12 , one have @xmath59 . the preference relation can be determined w.r.t . the relative gain : @xmath60 the total allocated utilities in coalition @xmath19 is @xmath61 . therefore , @xmath62 , where @xmath63 is the _ marginal utility _ due to coalition @xmath19 . the symmetric relative gain sharing approach relies on equally dividing the marginal utility in a coalition , i.e. @xmath64 this choice means that each player in coalition @xmath19 has the same gain ; thus the effect of coalition @xmath19 is identical to the players within it . [ cor : equivalentevaluation ] * equivalent evaluation * : assume that @xmath65 . due to eq . ( [ eq : equallydivmarutility ] ) , the following must hold @xmath66 it means that all players in @xmath67 prefer coalition @xmath19 to @xmath20 whenever the relative gain in @xmath19 is higher than @xmath20 . for this particular case we obtained the following theorems : [ lma : nashcoretwoplayersincaseofsymmetricrelativegain ] there is always a nash - stable partition when @xmath68 in case of symmetric relative gain . see appendix [ lma : nashcorethreeplayersincaseofsymmetricrelativegain ] there is always a nash - stable partition when @xmath69 in case of symmetric relative gain . see appendix thus , we can conclude that symmetric relative gain always results in a nash - stable partition when @xmath70 . however , this is not the case when @xmath71 . we can find many counter examples that justify it such as the following one : [ cexample : nashcoremorethanthreeplayersincaseofsymmetricrelativegain ] let the marginal utility for all possible @xmath72 be as following : @xmath73 let us now generate the preference profile according to these marginal utility values . notice that we could eliminate those marginal utilities which are negative since a player will prefer to be alone instead of a negative relative gain . further , ranking the positive relative gains in a descending order results in the following sequence : @xmath74.\ ] ] according to the ranking sequence , we are able to generate the preference list of each player : @xmath75 note that this preference profile does not admit any nash - stable partition . thus , we conclude that symmetric relative gain allocation does nt provide always a nash - stable partition when @xmath76 . we now turn out to the case of separable and symmetric utility case . consider eq . ( [ eq : additivelyseparable ] ) meaning that player @xmath12 gains @xmath77 from player @xmath78 in any coalition . in case of symmetry , @xmath79 such that @xmath80 . further , we denote as @xmath81 the utility that player @xmath12 gains in coalition @xmath19 . then , the sum of allocated utilities in coalition @xmath19 is given by @xmath82 let us point out that @xmath83 ( for example , @xmath84 , @xmath85 $ ] ) . therefore , the following determines the existence of additively separable and symmetric preferences when the utility function @xmath35 is allocated to the players : @xmath86 where @xmath87 is the _ marginal utility _ due to coalition @xmath19 . albeit we will come back later on this point , it is true noting these constraints are strongly restrictive and are rarely oberved in real problems . let it be illustrated for example @xmath88 the constraints imposed are : @xmath89,\notag\\ v(1,3 ) & = \frac{1}{2}[u(1,3)-u(1)-u(3)],\notag\\ v(2,3 ) & = \frac{1}{2}[u(2,3)-u(2)-u(3)],\notag\\ v(1,2)+v(1,3)+v(2,3 ) & = \frac{1}{2}[u(1,2,3)-u(1)-u(2)-u(3)].\end{aligned}\ ] ] we have 3 variables and 4 constraints , thus only special problem may fit with all constraints . a more general approach allows to state the following theorem : the nash - stable core of the additively separable and symmetric utility hedonic game is non - empty if there exist balanced weights of the dual problem ( balancedness conditions ) : @xmath90 \quad \forall s\in 2^n.\end{aligned}\ ] ] according to bondareva - shapley theorem @xcite when the gains of players are allocated according to the additively separable and symmetric way we can note : * @xmath91 the all possible bipartite coalitions such that @xmath92 . note that @xmath93 . * @xmath94 the index set of all possible bipartite coalitions . so , @xmath95 is the @xmath96th bipartite coalition . * @xmath97 which is the vector demonstration of all @xmath98 . * @xmath99 . * @xmath100 such that @xmath101 where @xmath102 . * @xmath103 is a matrix such that @xmath104 . by using these definitions , the nash - stable core is non - empty whenever the following linear program is feasible @xmath105.\end{aligned}\ ] ] the linear program that is dual to @xmath106 is given by @xmath107,\end{aligned}\ ] ] where @xmath108 denote the vector of dual variables . let @xmath109 denote the @xmath96th column of @xmath110 . then @xmath111 implies @xmath112 this result means that the feasible solutions of @xmath113 exactly correspond to the vectors containing balancing weights for balanced families . more precisely , when @xmath114 is feasible in @xmath113 , @xmath115 is a balanced family with _ balancing weights _ @xmath116 . according to the _ weak duality theorem _ , the objective function value of the primal @xmath117 at any feasible solution is always greater than or equal to the objective function value of the dual @xmath113 at any feasible solution , i.e. @xmath118 which implies @xmath119 and @xmath120 combining these results , we have the following balancedness conditions of @xmath35 : @xmath121 \quad \forall s\in 2^n.\end{aligned}\ ] ] however , these conditions are very restrictive . for a given set of players @xmath122 , the number of variables is strictly equal to @xmath123{c } 2\\ n \end{array } \right)=n\cdot(n-1)/2 $ ] while the number of constraints is equal to the number of sets , i.e. @xmath124 . for @xmath125 , we have @xmath126 variables and as much as @xmath127 constraints . considering the former result we propose to transform an initial hedonic game under an additively separable and symmetric utility case , by relaxing the efficiency constraint . clearly , we propose to relaw the constraint of having the sum of allocated utilities in a coalition to be strictly equal to the utility of the coalition , i.e. @xmath128 . we assume that the system can not provide any coalition with additional utility and therefore the unique way is to taxe a group to ensure the convergence , which leads to having : @xmath129 now the following theorem may be stated : the nash - stable core is always non - empty in case of relaxed efficiency . a feasible solution of the following linear program guarantees the non - emptiness of the nash - stable core : @xmath130 which is equivalent to @xmath131.\end{aligned}\ ] ] note that @xmath132 is always feasible since * there are no any inconsistent constraints , i.e. there are no at least two rows in @xmath110 that are equivalent , * the polytope is bounded in the direction of the gradient of the objective function @xmath133 . in the case where the system is able to provide some redistribution of utilities to a coalition , we can also allow the sum of individual utilities in a coalition to be higher than the coalition s utility . in this case , the system may be intersted however to find an additively separable and symmetric utility while minimizing the total deviation from the utilities . we can then propose to select the symmetric preferable preferences @xmath98 according to the following optimization problem : @xmath134 this formulation leads to an analytical solution : @xmath135 . however the system may be interested in adding hard constraints on some specific sets , typically the grand coalition , to avoid the risk of having to pay some additional costs . this can be done by adding additional constraints to the problem above , but if the constraints are expressed as linear equalities or inequalities the problem remains convex . in this section , we now develop a decentralized algorithm to reach a nash - stable partition whenever one exists in a hedonic coalition formation game . we in fact model the problem of finding a nash stable partition in a hedonic coalition formation by formulating it as a non - cooperative game and we state the following : a hedonic coalition formation game is equivalent to a non - cooperative game . let us denote as @xmath136 the set of strategies . we assume that the number of strategies is equal to the number of players . this is indeed sufficient to represent all possible choices . indeed , the players that select the same strategy are interpreted as a coalition . based on this equivalence , it is possible to reuse some classical results from game theory . we consider a _ random round - robin _ algorithm where each player determines its strategy in its turn according to a _ scheduler _ which is randomly generated for each round . a scheduler in round @xmath137 is denoted as @xmath138 where @xmath139 is the turn of player @xmath12 . it should be noted that a scheduler is a random permutation of the set of players @xmath7 and therefore , our problem turns out to a _ weakly acyclic games_. a non - cooperative game is classified as weakly acyclic if every strategy - tuple is connected to some pure - strategy nash equilibrium by a best - reply path . weakly acyclic games have the property that if the order of deviators is decided more - or - less randomly , and if players do not deviate simultaneously , then a best - reply path reaches an equilibrium @xcite if there exists at least one . the type of scheduler chosen in this work is by default considered as _ memoryless _ since the identity of the deviator in each round does not depend on the previous rounds . however , it may be more efficient to design a scheduler according to the past observations . thus , we come up with so called _ algorithmic mechanism design _ that could enable to converge an equilibrium in less number of rounds . this kind of optimization is kept out of the scope of this paper . a _ strategy tuple _ in step @xmath140 is denoted as @xmath141 , where @xmath142 is the strategy of player @xmath12 in step @xmath140 . the relation between a round and a step can be given by @xmath143 . in each step , only one dimension is changed in @xmath144 . we further denote as @xmath145 the partition in step @xmath140 . define as @xmath146 the set of players that share the same strategy with player @xmath12 . thus , note that @xmath147 for each step . the preference function of player @xmath12 is denoted as @xmath148 and verifies the following equivalence : @xmath149 where player @xmath12 is the one that takes its turn in step @xmath140 . any sequence of strategy - tuple in which each strategy - tuple differs from the preceding one in only one coordinate is called a _ path _ , and a unique deviator in each step strictly increases the utility he receives is an _ improvement path_. obviously , any _ maximal improvement path _ which is an improvement path that can not be extended is terminated by stability . an algorithm for hedonic coalition formation can be given as following : set stability flag to zero generate a scheduler according to the scheduler , each player chooses the best - reply strategy check stability set stability flag to one the proposed algorithm [ alg : nashstabilityestablisher ] ( nash stability establisher ) always converges to a stable partition whenever there exists one . the proof exploits the propery mentioned above , relative to weakly acyclic games : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ every weakly acyclic game admits always a nash equilibrium @xcite ; since _ nash stability establisher _ is exactly a weakly acyclic game , it always converges to a partition which is nash - stable_. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ let us denote as @xmath150 the initial partition where each player is alone . it corresponds to the case where each player chooses different strategy ; thus each player is alone in its strategy : @xmath151 . the transformation of strategy tuple and partition in each step can be denoted as following : @xmath152 where @xmath153 represents the the step in which the stable partition occurs . in fact , the stable partition in @xmath154 is exactly the nash equilibrium in a weakly acyclic game . we propose now to evaluate the use of the former algorithm on hedonic games transformed to additively separable and symmetric . such approach needs two steps . during the first step , the system computes the relative symmetric gains @xmath98 according to one of the suboptimal approximations proposed above . then , during the second step , the players do their moves according to the algorithm and using the modified utilities until an equilibrium is reached . the social optimum is the maximum total global utility , i.e. what is the partitioning of the players such that the total global utility is maximized . it can be formulated as a set partitioning optimization problem which can be given by @xmath155 by which we find a partition @xmath156 maximizing the global utility . note that the total social utility in case of a nash - stable partition will always be lower or equal to the one obtained by social optimum , i.e. @xmath157 . we could say that an approach is socially optimal if it reaches the social optimum . the distance between the social optimum and the equilibrium solution achieved is the price of anarchy . we study here the utility allocation based on relaxed efficiency according to the marginal utilities given in counter example [ cexample : nashcoremorethanthreeplayersincaseofsymmetricrelativegain ] . we also suppose that @xmath158 , @xmath159 , @xmath160 , @xmath161 . we calculate the social optimum which is equivalent to the set partitioning problem s optimization version . a utility allocation method can be found by solving the following linear program : @xmath162 which produces the following values : @xmath163 thus , the preference profile is obtained @xmath164 which admits the nash - stable partition @xmath165 . the total social utility can be calculated as @xmath166 . the social optimum in the considered example is found to be @xmath167 which is a result of partition @xmath168 . we suggested a decentralized algorithm for finding the nash stability in a game whenever there exists always at least one . the problem of finding the nash stability is considered as a non - cooperative game . we consider a _ random round - robin _ fashion where each player determines its strategy in its turn according to a _ scheduler _ which is randomly generated for each round . under this condition , we proved that the algorithm converges to an equilibrium which is the indicator of the nash stability . moreover , we answer the following question : is there any utility allocation method which could result in a nash - stable partition ? we proposed the definition of the nash - stable core . we analyzed the cases in which the nash - stable core is non - empty , and prove that in case of the relaxed efficiency condition there exists always a nash - stable partition . according to corollary [ cor : equivalentevaluation ] , @xmath173 . thus , combining all constraint sets of all possible partitions , we have the following result constraint set : @xmath174 $ ] . it means that for any value of @xmath175 , symmetric relative gain always results in a nash - stable partition for two players case . note that there are @xmath176 possible partitions in case of @xmath69 . thus , according to equally divided marginal utility , the following variables occur : @xmath177 , @xmath178 , @xmath179 , @xmath180 . enumerating all possible partitions results in the following conditions : note that the constraint set @xmath191 covers all values in @xmath192 in case of @xmath193 further , it also covers all values when @xmath194 . we are able to draw it since there are three dimensions : @xmath195{nashcorethreeplayersincaseofsymmetricrelativegain.pdf}\ ] ] 1 k. r. apt and a. witzel . `` a generic approach to coalition formation , '' _ international game theory review _ , vol.11 , no.3 , pp . 347367 , 2009 . h. aziz and f. brandl , `` existence of stability in hedonic coalition formation games , '' _ in proceedings of the 11th international conference on autonomous agents and multiagent systems ( aamas 2012 ) _ , jun . 2012 . h. keinnen , `` an algorithm for generating nash stable coalition structures in hedonic games , '' _ in proceedings of the 6th international conference on foundations of information and knowledge systems ( foiks10 ) _ , 2010 . j. drze and j. greenberg , `` hedonic coalitions : optimality and stability , '' _ econometrica _ , vol . 9871003 , jan . a. bogomonlaia and m. jackson , `` the stability of hedonic coalition structures , '' _ games and economic behavior _ , vol . 38 , pp . 201230 , jan . n. burani , and w. s. zwicker , `` coalition formation games with separable preferences , '' _ mathematical social sciences , elsevier _ , vol . 1 , pp . 2752 , feb . 2003 . young , `` the evolution of conventions , '' _ econometrica _ , vol . 61 , pp . 5784 , 1993 . i. milchtaich , `` congestion games with player - specic payoff functions , '' _ games and economic behavior _ vol . 13 , pp . 111124 , 1996 . j. hajdukov , `` coalition formation games : a survey , '' _ international game theory review _ , vol . 4 , pp . 613641 , 2006 . f. bloch and e. diamantoudi , `` noncooperative formation of coalitions in hedonic games , '' _ int j game theory _ 40 , pp . 263280 , 2011 . o. n. bondareva,``some applications of linear programming methods to the theory of cooperative games , '' _ problemy kybernetiki _ , vol . 10 , 119139 , 1963 . l. s. shapley , `` on balanced sets and cores , '' _ naval research logistics quarterly _ , vol . 14 , pp . 453460 , 1967 .

this paper studies _ the nash stability _ in hedonic coalition formation games . we address the following issue : for a general problem formulation , is there any utility allocation method ensuring a nash - stable partition ? we propose the definition of _ the nash - stable core _ and we analyze the conditions for having a non - empty nash - stable core . more precisely , we prove that using relaxed efficiency in utility sharing allows to ensure a non empty nash - stable core . then , a decentralized algorithm called _ nash stability establisher _ is proposed for finding the nash stability in a game whenever at least one exists . the problem of finding the nash stability is formulated as a non - cooperative game . in the proposed approach , during each round , each player determines its strategy in its turn according to a _ random round - robin scheduler_. we prove that the algorithm converges to an equilibrium if it exists , which is the indicator of the nash stability .

the first step in describing the interaction between many particles is to determine their pair potential or the forces among a single pair . if the governing physical equations are linear ( like for gravity or electrostatics ) , this approach yields a quantitatively reliable description of the physical system considered , based on the linear superposition principle . however , if nonlinearities are present , linear superposition of pair potentials is no longer accurate and nonadditivity gives rise to many - body effects . these latter effects can lead , e.g. , to a strengthening or weakening of the total force acting on a particle surrounded by more than a single other one , a change of sign of that force , or the appearance of stable or unstable configurations . many - body effects appear in rather diverse systems such as nuclear matter , superconductivity @xcite , colloidal suspensions @xcite , quantum - electrodynamic casimir forces @xcite , polymers @xcite , nematic colloids @xcite , and noble gases with van der waals forces acting among them @xcite . each of these systems is characterized by a wide range of time and length scales . integrating out the degrees of freedom associated with small scales ( such as the solvent of colloidal solutes or polymers ) for fixed configurations of the large particles , generates effective interactions among the latter , which are inherently not pairwise additive . this is the price to be paid for achieving a reduced description of a multicomponent system . driven by these effective interactions the large particles of the system may exhibit collective behavior of their own ( like aggregation or phase separation , see refs . @xcite and @xcite and references therein ) , which can be described much easier if it is governed by pair potentials . in order to be able to judge whether this ansatz is adequate one has to check the relative magnitude of genuine many - body forces . in this paper we assess the quantitative influence of such many - body effects on critical casimir forces ( ccfs ) @xcite . these long - ranged forces arise as a consequence of the confinement of the order parameter fluctuations in a critical fluid @xcite . they have been analyzed paradigmatically by studying the effective interaction between a _ single _ colloidal particle and a homogeneous @xcite or inhomogeneous @xcite container wall as well as between _ two isolated _ colloidal particles @xcite upon approaching the critical point of the solvent . here we add one sphere to the sphere - wall configuration , which is the simplest possibility to study many - body forces . ( the wall mimics a third , very large sphere . ) in order to be able to identify the latter ones one has to resort to a theoretical scheme which allows one to compute the forces between individual pairs and the three - body forces on the same footing . since these forces are characterized by universal scaling functions , which depend on the various geometrical features of the configuration and on the thermodynamic state , we tackle this task by resorting to field - theoretic mean field theory ( mft ) , which captures the universal scaling functions as the leading contribution to their systematic expansion in terms of @xmath1 spatial dimensions . experience with corresponding previous studies for simple geometries tells that this approach does yield the relevant qualitative features of the actual universal scaling functions in @xmath2 ; if suitably enhanced by renormalization group arguments these results reach a semi - quantitative status . we point out that even within this approximation the numerical implementation of this corresponding scheme poses a severe technical challenge . thus at present this approach appears to be the only feasible one to explore the role of many - body critical casimir forces within the full range of their scaling variables . accordingly we consider the standard landau - ginzburg - wilson hamiltonian for critical phenomena of the ising bulk universality class , which is given by @xmath3 = \int_v{\rm d}^d\mathbf{r } \left\ { \frac{1}{2}\left ( \nabla\phi \right)^2 + \frac{\tau}{2}\phi^2 + \frac{u}{4!}\phi^4 \right\ } ~,\ ] ] with suitable boundary conditions ( bcs ) . in the case of a binary liquid mixture near its consolute ( demixing ) point , the order parameter @xmath4 is proportional to the deviation of the local concentration of one of the two species from the critical concentration . @xmath5 is the volume accessible to the fluid , @xmath6 is proportional to the reduced temperature @xmath7 , and the coupling constant @xmath8 stabilizes the statistical weight @xmath9 in the two - phase region , i.e. , for @xmath10 . close to the bulk critical point @xmath11 the bulk correlation length @xmath12 diverges as @xmath13 , where @xmath14 in @xmath2 and @xmath15 in @xmath16 , i.e. , within mft @xcite . the two non - universal amplitudes @xmath17 are of molecular size ; they form the universal ratio @xmath18 for @xmath2 and @xmath19 for @xmath16 @xcite . the bcs reflect the generic adsorption preference of the confining surfaces for one of the two components of the mixture . for the critical adsorption fixed point @xcite , the bc at each of the confining surfaces is either @xmath20 or @xmath21 , to which we refer as @xmath22 or @xmath23 , respectively . within mft the equilibrium order parameter distribution minimizes the hamiltonian in eq . for the aforementioned bcs , i.e. , @xmath24/\delta\phi = 0 $ ] . far from any boundary the order parameter approaches its constant bulk value @xmath25 for @xmath10 or @xmath26 for @xmath27 . @xmath28 is a non - universal bulk amplitude and @xmath29 ( for @xmath16 ) is a standard critical exponent . in the following we consider the reduced order parameter @xmath30 . the remainder of this paper is organized as follows . in sec . [ section_the_system ] we define the system under consideration and the scaling functions for the ccfs as well as the normalization scheme . in sec . [ section_results ] we present the numerical results obtained for the universal scaling functions of the ccfs , from which we extract and analyze the many - body effects . in sec . [ section_conclusion ] we summarize our results and draw some conclusions . we study the normal and the lateral ccfs acting on two colloidal particles immersed in a near - critical binary liquid mixture and close to a homogeneous , planar substrate . we focus on the critical concentration which implies the absence of a bulk field conjugate to the order parameter [ see eq . ] . the surfaces of the colloids and of the substrate are considered to exhibit a strong adsorption preference for one of the two components of the confined liquid leading to @xmath31 bcs . the forces are calculated using the full three - dimensional numerical analysis of the appropriate mft as given by eq . . specifically , we consider two three - dimensional spheres of radii @xmath32 and @xmath33 with bcs @xmath34 and ( @xmath35 ) , respectively , facing a homogeneous substrate with bc @xmath36 at sphere - surface - to - substrate distances @xmath37 and @xmath38 , respectively ( see fig . [ system_sketch ] ) . the coordinate system @xmath39 is chosen such that the centers of the spheres are located at @xmath40 and @xmath41 so that the distance between the centers , projected onto the @xmath42-axis , is given by @xmath43 . the bcs of the whole system are represented by the set @xmath44 , where @xmath45 , @xmath35 , and @xmath46 can be either @xmath47 or @xmath48 . it is important to point out that we discuss colloidal particles with the shape of a hypercylinder @xmath49 where @xmath50 are the semiaxes ( or radii ) of the hypercylinder and @xmath51 , @xmath52 , @xmath53 . if @xmath54 and @xmath55 , the hypercylinder reduces to a hypersphere . the generalization of @xmath56 to values larger than 3 is introduced for technical reasons because @xmath57 is the upper critical dimension for the relevance of the fluctuations of the order parameter . these fluctuations lead to a behavior different from that obtained from the present mft which ( apart from logarithmic corrections @xcite ) is valid in @xmath16 . we consider two hypercylinders in @xmath16 with @xmath58 and @xmath59 . the two colloids are taken to be parallel along the fourth dimension with infinitely long hyperaxes in this direction . considering hypercylinders , which are translationally invariant along the @xmath60axis , allows us to minimize @xmath61 $ ] numerically using a three dimensional finite element method in order to obtain the spatially inhomogeneous order parameter profile @xmath62 for the geometries under consideration ( see fig . [ system_sketch ] ) . and @xmath33 immersed in a near - critical binary liquid mixture ( not shown ) and close to a homogeneous , planar substrate at @xmath63 . the two colloidal particles with bcs @xmath34 and @xmath64 are located at sphere - surface - to - substrate distances @xmath37 and @xmath38 , respectively . the substrate exhibits bc @xmath36 . the lateral distance between the centers of the spheres along the @xmath42-direction is given by @xmath43 , while the centers of both spheres lie in the plane @xmath65 . in the case of four spatial dimensions the figure shows a cut of the system , which is invariant along the fourth direction , i.e. , the spheres correspond to parallel hypercylinders with one translationally invariant direction , which is @xmath66 . ] in the case of an upper critical demixing point of the binary liquid mixture at the critical concentration , @xmath67 corresponds to the disordered ( i.e. , mixed ) phase of the fluid , whereas @xmath68 corresponds to the ordered ( i.e. , phase separated ) phase . the meaning of the sign is reversed for a lower critical point . in the following we assume an upper critical point . the normal ccf @xmath69 acting on sphere @xmath70 in the presence of sphere @xmath71 ( @xmath72 and @xmath73 ) along the @xmath74-direction takes the scaling form @xmath75 where @xmath76 , @xmath77 , @xmath78 , @xmath79 , and @xmath80 ( i.e. , @xmath81 for @xmath27 and @xmath82 for @xmath10 ) . equation describes the singular contribution to the normal force emerging upon approaching @xmath11 . @xmath83 is the force per length of the hypercylinder due to its extension in the translationally invariant direction . in the spirit of a systematic expansion in terms of @xmath84 around the upper critical dimension we study the scaling functions @xmath85 within mft as given by eq . for hypercylinders in @xmath16 , which captures the correct scaling functions in @xmath16 up to logarithmic corrections occurring in @xmath86 @xcite , which we do not take into account here . since mft renders the leading contribution to an expansion around @xmath16 , geometrical configurations with small @xmath87 , @xmath88 , or @xmath89 are not expected to be described reliably by the present approach due to the dimensional crossover in narrow slit - like regions , which is not captured by the @xmath84 expansion . the colloidal particles will also experience a lateral ccf @xmath90 , for which it is convenient to use the scaling form @xmath91 where @xmath92 ( i.e. , @xmath93 for @xmath27 and @xmath94 for @xmath10 ) . note that the choice of @xmath95 as the scaling variable does not depend on the type of particle the force acts on . equation also describes the singular contribution to the lateral force near @xmath11 . the total ccf acting on particle @xmath70 is @xmath96 where @xmath97 and @xmath98 are the unit vectors pointing in @xmath42- and @xmath74-direction , respectively . due to symmetry all other components of the ccf are zero . as a reference configuration we consider a single spherical colloid of radius @xmath99 with bc @xmath100 at a surface - to - surface distance @xmath101 from a planar substrate with bc @xmath36 . this colloid experiences ( only ) a normal ccf @xmath102 in the following we normalize the scaling functions @xmath103 and @xmath104 by the amplitude @xmath105 of the ccf acting at @xmath11 on a single colloid for @xmath106 bcs at a surface - to - surface distance @xmath107 . accordingly , in the following we consider the normalized scaling functions @xmath108 with @xmath109 and @xmath110 . experimentally it can be rather difficult to obtain @xmath105 . a standard alternative way to normalize is to take the more easily accessible amplitude @xmath111 for the ccf at @xmath11 between two parallel plates with @xmath106 bcs , which is given within mft by ( see ref . @xcite and references therein ) @xmath112 ^ 4}{u } \simeq -283.61 u^{-1 } ~,\ ] ] where @xmath113 is the elliptic integral of the first kind . within mft the amplitude @xmath105 can be expressed in terms of @xmath111 : @xmath114 equation allows for a practical implementation of the aforementioned normalization , which eliminates the coupling constant @xmath115 , which is unknown within mft . we calculate the normal and lateral forces directly from the numerically determined order parameter profiles @xmath62 by using the stress tensor which , within the ginzburg - landau approach , is given by @xcite @xmath116 ~,\ ] ] with @xmath117 . the first index of the stress tensor denotes the direction of a force , the second index denotes the direction of the normal vector of the surface upon which the force acts . therefore one has @xmath118 where @xmath119 is a hypersurface enclosing particle @xmath70 , @xmath120 is the @xmath121-th component ( to be summed over ) of its unit outward normal , and @xmath122 is the length of the @xmath123-dimensional hyperaxis of @xmath124 . in particular we focus on the normal and lateral ccfs acting on colloid ( 2 ) for the configuration shown in fig . [ system_sketch ] , for @xmath125 , and with the binary liquid mixture at its critical concentration . in the following analysis we consider colloid ( 1 ) to be fixed in space at a sphere - surface - to - substrate distance @xmath126 , equally sized colloids ( i.e. , @xmath127 ) , and fixed @xmath128 bc for the substrate . we proceed by varying either the vertical ( @xmath74-direction ) or the horizontal ( @xmath42-direction ) position of colloid ( 2 ) by varying either @xmath38 or @xmath129 , respectively . we also consider different sets of bcs for the colloids . in the following results the numerical error is typically less than @xmath130 , unless explicitly stated otherwise . in fig . [ d2_z ] we show the behavior of the normalized [ eq . ] scaling function @xmath131 of the normal ccf acting on colloid ( 2 ) with @xmath132 bcs close to a homogeneous substrate with @xmath128 bc and in the presence of colloid ( 1 ) with @xmath133 bc . the scaling functions are shown as functions of the scaling variable ratio @xmath134 , i.e. , for @xmath27 . the various lines correspond to distinct fixed values of @xmath135 as the sphere - surface - to - substrate distance in units of the sphere radius . thus fig . [ d2_z ] shows the temperature dependence of the normal ccf on colloid 2 for three different values of @xmath38 and for colloid ( 1 ) fixed in space . from fig . [ d2_z ] ( a ) one can see that , for colloid ( 1 ) with @xmath136 bc and for any given value of @xmath137 , the scaling function of the normal ccf acting on colloid ( 2 ) with @xmath138 bc changes sign upon varying @xmath135 . due to the change of sign of @xmath139 , for any value of @xmath137 there is a certain value @xmath140 at which the normal ccf acting on colloid ( 2 ) vanishes . for sphere - surface - to - substrate distances sufficiently large such that @xmath141 , colloid ( 2 ) is pushed away from the substrate due to the dominating repulsion between the two colloids in spite of the attraction by the substrate , whereas for @xmath142 it is pulled to the substrate due to the dominating attraction between it and the substrate this implies that , in the absence of additional forces , levitation of colloid ( 2 ) ( i.e. , zero total normal force ) at height @xmath143 is not stable against perturbations of the sphere - surface - to - substrate distance . on the other hand , upon varying temperature , any distance @xmath38 can become a stable levitation position for colloid ( 2 ) with @xmath144 bc in the presence of colloid ( 1 ) with @xmath133 bc [ see fig . [ d2_z ] ( b ) , according to which each scaling function corresponding to a certain value of @xmath38 exhibits a zero so that , at this zero , increasing ( decreasing ) @xmath38 at fixed temperature leads to an attraction ( repulsion ) to ( from ) the substrate ] . in this case the attraction between the two colloids is dominating for large sphere - surface - to - substrate distances @xmath135 , while the repulsion between colloid ( 2 ) and the substrate dominates for small values of @xmath135 . of the normal ccf acting on colloid ( 2 ) with bcs @xmath145 in ( a ) and @xmath144 in ( b ) . the scaling functions are shown for @xmath27 as functions of the scaling variable ratio @xmath146 for three fixed values of the scaling variable @xmath77 : @xmath147 ( black lines ) , 1.5 ( red lines ) , and 2 ( green lines ) , while @xmath148 for all curves in ( a ) and ( b ) so that @xmath149 . for @xmath33 fixed the three curves correspond to three different vertical positions of colloid ( 2 ) with colloid ( 1 ) fixed in space . as expected , the forces become overall weaker upon increasing @xmath135 . @xmath150 @xmath151 implies that the colloid is attracted to ( repelled from ) the substrate along the @xmath74-direction . ] figure [ d2_x ] shows the behavior of the normalized scaling function @xmath152 of the lateral ccf acting on colloid ( 2 ) in the presence of colloid ( 1 ) having the same bc , i.e. , @xmath153 . the scaling functions are shown as functions of the scaling variable ratio @xmath154 . from figs . [ d2_x ] ( a ) and ( b ) one can infer that @xmath155 . therefore colloid ( 2 ) is always attracted towards colloid ( 1 ) which has the same bc . hence the substrate does not change the sign of the lateral ccf as compared with the attractive lateral ccf in the absence of the confining substrate . however , the shapes of the scaling functions for @xmath156 bcs [ fig . [ d2_x ] ( a ) ] differ from the ones for @xmath157 bcs [ fig . [ d2_x ] ( b ) ] ; without the substrate , they are identical . in the former case and in contrast to the latter one , the scaling functions exhibit minima above @xmath11 , which is reminiscent of the shape of the corresponding scaling functions in the absence of the substrate . of the lateral ccf acting on colloid ( 2 ) facing a homogeneous substrate with @xmath128 bc and in the presence of colloid ( 1 ) with @xmath158 bcs . the scaling functions are shown for @xmath27 as functions of the scaling variable ratio @xmath154 for three fixed values of the scaling variable @xmath77 : @xmath147 ( black curves ) , 1.5 ( red curves ) , and 2 ( green curves ) , while @xmath148 for all curves in ( a ) and ( b ) so that @xmath149 . for @xmath33 fixed the three curves correspond to three different vertical positions of colloid ( 2 ) with colloid ( 1 ) fixed in space . as expected , the forces become overall weaker upon increasing @xmath135 . @xmath159 implies that colloid ( 2 ) is attracted towards colloid ( 1 ) . two different sets of @xmath160 bcs are considered : @xmath156 in ( a ) and @xmath157 in ( b ) . ] in fig . [ l_x ] we show the results obtained for the normalized scaling functions @xmath161 of the lateral ccf acting on colloid ( 2 ) . in fig . [ l_x](a ) the scaling function is shown as function of the scaling variable ratio @xmath154 ; the black , red , and green curves correspond to @xmath162 , @xmath163 , and @xmath164 , respectively . in the absence of the substrate , the ccf between two colloids with opposite bcs is repulsive . however , as shown in fig . [ l_x](a ) , in the presence of the substrate with @xmath128 bc , there is a change of sign in the scaling function of the lateral ccf . this implies that the lateral ccf acting between the two colloids changes from being attractive to being repulsive ( or reverse ) upon decreasing ( increasing ) the reduced temperature . thus temperature allows one to control both the strength and the sign of the lateral ccf in the case of two colloids with opposite bcs being near a wall . the at first sight unexpected lateral attraction between two colloids with opposite bcs in the presence of the substrate ( i.e. , @xmath165 ) can be understood as follows . in the absence of the two colloids , the order parameter profile @xmath166 is constant along any path within a plane @xmath167 because in this case @xmath168 . since the substrate area is much larger than the surface areas of the colloids , one can regard the immersion of these colloidal spheres as a perturbation of this profile . in figs . [ l_x](a ) and ( b ) , the region within which the scaling function is negative ( corresponding to an attractive force ) indicates that under these circumstances [ i.e. , when the colloids are sufficiently away from each other ; see fig . [ l_x](b ) ] the perturbation generated by the presence of the spheres decreases upon decreasing the lateral distance between them . this causes the colloids to move towards each other in order to weaken the perturbation by reducing its spatial extension ; this amounts to an attraction , i.e. , @xmath165 . on the other hand , when they are sufficiently close to each other the pairwise interaction between the two colloids dominates and the total lateral ccf is positive ( i.e. , repulsive ) . in fig . [ l_x](c ) we show how the equilibrium lateral distance @xmath169 measured in units of @xmath170 varies as function of temperature , i.e. , @xmath170 for fixed @xmath99 . of the lateral ccf acting on colloid ( 2 ) . both colloids are taken to have the same size ( @xmath171 ) and the same sphere - surface - to - substrate distance ( @xmath172 ) . the scaling function is shown for @xmath27 as function of the scaling variable ratio @xmath154 . @xmath165 @xmath151 implies that colloid ( 2 ) is attracted ( repelled ) by colloid ( 1 ) . black , red , and green curves correspond to lateral distances @xmath173 , 1.5 , and 2 , respectively , between the surfaces of the colloids [ see fig . [ system_sketch ] ] . the zero @xmath174 of @xmath175 implies that for any lateral distance @xmath129 there is a reduced temperature such that for @xmath176 this distance represents an equilibrium lateral distance between the two colloids , provided they are located at equal sphere - surface - to - substrate distances . ( b ) @xmath175 as function of the reduced surface - to - surface distance @xmath79 between the two colloids for @xmath177 ( blue line ) , 3.873 ( yellow line ) , 4.472 ( magenta line ) , and 5 ( purple line ) . each curve in ( b ) corresponds to the vertical dashed line with same color in panel ( a ) . the change of sign of the scaling function ( from positive to negative ) as the distance between the colloids is increased indicates that the lateral ccf changes from repulsive to attractive , which means that there is a lateral position @xmath169 corresponding to a stable equilibrium point . in ( c ) we show how this equilibrium position , measured in units of @xmath178 , varies as function of temperature ( i.e. , @xmath178 ) for fixed @xmath33 . note that @xmath169 is not proportional to @xmath178 , since @xmath179 is not a constant . as a guide to the eye the four data points are connected by straight lines.,title="fig : " ] of the lateral ccf acting on colloid ( 2 ) . both colloids are taken to have the same size ( @xmath171 ) and the same sphere - surface - to - substrate distance ( @xmath172 ) . the scaling function is shown for @xmath27 as function of the scaling variable ratio @xmath154 . @xmath165 @xmath151 implies that colloid ( 2 ) is attracted ( repelled ) by colloid ( 1 ) . black , red , and green curves correspond to lateral distances @xmath173 , 1.5 , and 2 , respectively , between the surfaces of the colloids [ see fig . [ system_sketch ] ] . the zero @xmath174 of @xmath175 implies that for any lateral distance @xmath129 there is a reduced temperature such that for @xmath176 this distance represents an equilibrium lateral distance between the two colloids , provided they are located at equal sphere - surface - to - substrate distances . ( b ) @xmath175 as function of the reduced surface - to - surface distance @xmath79 between the two colloids for @xmath177 ( blue line ) , 3.873 ( yellow line ) , 4.472 ( magenta line ) , and 5 ( purple line ) . each curve in ( b ) corresponds to the vertical dashed line with same color in panel ( a ) . the change of sign of the scaling function ( from positive to negative ) as the distance between the colloids is increased indicates that the lateral ccf changes from repulsive to attractive , which means that there is a lateral position @xmath169 corresponding to a stable equilibrium point . in ( c ) we show how this equilibrium position , measured in units of @xmath178 , varies as function of temperature ( i.e. , @xmath178 ) for fixed @xmath33 . note that @xmath169 is not proportional to @xmath178 , since @xmath179 is not a constant . as a guide to the eye the four data points are connected by straight lines.,title="fig : " ] of the lateral ccf acting on colloid ( 2 ) . both colloids are taken to have the same size ( @xmath171 ) and the same sphere - surface - to - substrate distance ( @xmath172 ) . the scaling function is shown for @xmath27 as function of the scaling variable ratio @xmath154 . @xmath165 @xmath151 implies that colloid ( 2 ) is attracted ( repelled ) by colloid ( 1 ) . black , red , and green curves correspond to lateral distances @xmath173 , 1.5 , and 2 , respectively , between the surfaces of the colloids [ see fig . [ system_sketch ] ] . the zero @xmath174 of @xmath175 implies that for any lateral distance @xmath129 there is a reduced temperature such that for @xmath176 this distance represents an equilibrium lateral distance between the two colloids , provided they are located at equal sphere - surface - to - substrate distances . ( b ) @xmath175 as function of the reduced surface - to - surface distance @xmath79 between the two colloids for @xmath177 ( blue line ) , 3.873 ( yellow line ) , 4.472 ( magenta line ) , and 5 ( purple line ) . each curve in ( b ) corresponds to the vertical dashed line with same color in panel ( a ) . the change of sign of the scaling function ( from positive to negative ) as the distance between the colloids is increased indicates that the lateral ccf changes from repulsive to attractive , which means that there is a lateral position @xmath169 corresponding to a stable equilibrium point . in ( c ) we show how this equilibrium position , measured in units of @xmath178 , varies as function of temperature ( i.e. , @xmath178 ) for fixed @xmath33 . note that @xmath169 is not proportional to @xmath178 , since @xmath179 is not a constant . as a guide to the eye the four data points are connected by straight lines.,title="fig : " ] in order to determine the preferred arrangement of the colloids , we have also analyzed the direction of the total ccf @xmath180 acting on colloid ( 2 ) [ see eq . ] for several spatial configurations and bcs . for @xmath156 bcs we have found that the colloids tend to aggregate laterally in such a way that several particles with @xmath22 bc , facing a substrate with the same bc , can be expected to form a monolayer on the substrate . on the other hand , for the case of @xmath157 bcs , we have found that the colloids can be expected to aggregate on top of each other so that a collection of colloids with such bcs is expected to form three - dimensional sessile clusters . these tendencies become more pronounced upon approaching the critical point ( see fig . [ direction_resulting_force_d2 ] ) . similar results have been found by soyka et al . @xcite in experiments using chemically patterned substrates . for them the authors have found indeed that colloids with @xmath23 bc distributed over those parts of the substrate with the same bc [ which is equivalent to @xmath156 bcs ] aggregate and form a single layer . moreover , they have found that colloids distributed over parts of the substrate with opposite bc [ corresponding to @xmath157 bcs ] form three - dimensional clusters . and @xmath157 bcs , with @xmath181 . the black rectangles represent the substrate and blue circles represent colloid ( 1 ) while black , red , and green circles represent colloid ( 2 ) with @xmath182 , and 2 , respectively . the centers of all colloids lie in the plane @xmath65 . ] we have determined the many - body force acting on particle @xmath183 by subtracting from the total force @xmath184 [ see eq . ] the sum of the pairwise forces acting on it , i.e. , the colloid - colloid ( cc ) and the colloid - substrate ( cs ) forces . accordingly the many - body ccf @xmath185 acting on colloid ( @xmath70 ) is given by ( see fig . [ system_sketch ] ) @xmath186 where @xmath187 with @xmath188 is the pairwise colloid - colloid force ( acting on colloid 2 @xmath23 or 1 @xmath22 with 2 having the larger @xmath42-coordinate ) expressed in terms of the absolute value @xmath189 of the force between two colloids at surface - to - surface distance @xmath190 in free space . @xmath191 is the ccf between the substrate and a single colloid @xmath183 . we have studied both the normal @xmath192 and the lateral @xmath193 many - body ccfs which are characterized by corresponding scaling functions [ compare eqs . and ] : @xmath194 and @xmath195 in fig . [ mb_l_z ] we show the normalized [ see eqs . and ] scaling functions @xmath196 of the many - body normal ccf acting on colloid ( 2 ) . this figure reveals similar results for @xmath156 [ fig . [ mb_l_z ] ( a ) ] and @xmath197 [ fig . [ mb_l_z ] ( b ) ] bcs . in these cases , each mb scaling function exhibits both a maximum and at least one minimum , the former one appearing for smaller values of the scaling variable @xmath198 ( i.e. , at temperatures closer to @xmath11 ) . for a certain range of temperatures close to @xmath11 , as the distance @xmath129 between the colloids increases , the many - body normal ccf changes from attractive to repulsive . this shows that for each temperature within this range there is a lateral distance @xmath199 for which the many - body contribution to the normal force acting on colloid ( 2 ) is zero . this means that under such conditions the sum of pairwise forces provides a quantitatively reliable description of the total force acting on colloid ( 2 ) . for temperatures sufficiently far from @xmath11 , the many - body normal ccf is always attractive with a monotonic dependence on @xmath198 . here , as in figs . [ d2_z ] - [ l_x ] , the ccfs decay exponentially for @xmath200 . as expected , the many - body effects are more pronounced if the colloids are closer to each other and/or closer to the substrate . indeed for situations in which the colloids are close to each other [ see , e.g. , the black curves in figs . [ mb_l_z ] ( a ) and ( b ) ] we have found that when the normal many - body ccf reaches its maximal strength , corresponding to the minimum of the scaling function @xmath201 at @xmath202 , the relative contribution of the many - body ccf reaches @xmath0 of the strength of the total normal ccf . for larger distances between the colloids [ see , e.g. , the green curves in figs . [ mb_l_z ] ( a ) and ( b ) ] this relative contribution is smaller ( around @xmath203 for @xmath204 and @xmath205 for @xmath206 ) . of the many - body normal ccf acting on colloid ( 2 ) for @xmath207 and @xmath208 . the scaling functions are shown as functions of the scaling variable ratio @xmath209 for the sets of bcs @xmath156 in ( a ) and @xmath197 in ( b ) . the black , red , and green lines correspond to @xmath210 , and @xmath211 , respectively . figures [ d2_z](a ) and [ mb_l_z](b ) allow a direct comparison between the full ccf and the corresponding many - body contribution ( note the different scales of the ordinates . ) ] we are not aware of results for the quantum - electrodynamic casimir interactions which are obtained along the same lines as our ccf analysis above . nonetheless , in order to assess the significance of our results we compare them with the results in ref . @xcite , which is the closest comparable study which we have found in the literature . therein the authors study theoretically two dielectric spheres immersed in ethanol while facing a plate . depending on the kind of fluid and on the materials of the spheres and of the plate as well as on the distances involved , also the quantum - electrodynamic casimir force can be either attractive or repulsive . it is well known @xcite that for two parallel plates with permittivities @xmath212 and @xmath213 separated by a fluid with permittivity @xmath214 and without further boundaries , the quantum - electrodynamic casimir force is repulsive if @xmath215 within a suitable frequency range . in ref . @xcite it is stated that this also holds for two spheres immersed in a fluid . the authors of ref . @xcite analyze the effect of nonadditivity for the above system by studying the influence of an additional , adjacent substrate on the equilibrium separation @xmath56 between two nanometer size dielectric spheres . to this end , they consider two spheres of different materials with the same radii @xmath207 and the same surface - to - plate distances @xmath216 and analyze how the lateral equilibrium distance @xmath217 between the spheres depends on @xmath101 . by comparing the equilibrium distance @xmath217 with that in the absence of the substrate , @xmath218 , they find that @xmath217 increases or decreases ( depending on the kind of materials of the spheres ) by as much as @xmath219 as the distance from the plate varies between @xmath220 and @xmath221 . they also find that `` the sphere - plate interaction changes the sphere - sphere interaction with the same sign as @xmath101 becomes smaller '' , which means that if the sphere - plate force is repulsive ( attractive ) , the sphere - sphere one will become more repulsive ( attractive ) upon decreasing the distance from the plate @xmath101 . by construction these changes are genuine many - body contributions . in the case of two chemically different spheres , the sign of the many - body force contribution ( i.e. , whether it is attractive or repulsive ) agrees with the sign of the stronger one of the two individual sphere - plate interactions . in fig . [ comparing ] we show schematically the system considered in ref . @xcite [ ( a ) and ( b ) ] and the system considered here [ ( c ) , ( d ) , and ( e ) ] . for the quantum - electrodynamic casimir effect , the dielectric spheres are represented by circles of equal radii , with the green one corresponding to a polystyrene sphere and the red one to a silicon sphere . the semi - infinite plates are represented by gray and yellow rectangles for teflon and gold , respectively . the whole configuration is immersed in ethanol which , for simplicity , is not shown in the figure . the dashed arrows indicate the direction of the strongest of the two pairwise sphere - substrate forces , while the solid arrows indicate the direction of the lateral many - body force . the directions of the arrows in figs . [ comparing ] ( a ) and ( b ) are chosen as to illustrate the findings in ref . @xcite , according to which the sign of the _ lateral _ many - body force is the same as the one of the strongest _ normal _ pairwise sphere - plate force : attractive in ( a ) and repulsive in ( b ) . also in the case of the critical casimir forces , depicted in figs . [ comparing ] ( c ) , ( d ) , and ( e ) , we represent the colloids by circles and the laterally homogeneous semi - infinite substrate by rectangles . the orange filling represents the @xmath22 bc while the blue filling represents the @xmath23 bc . again , the dashed arrows indicate the direction of the stronger one of the two pairwise ( normal ) colloid - substrate forces , while the solid arrows indicate the direction of the lateral many - body contribution to the ccf . as one can infer from fig . [ mb_x ] , the _ lateral _ many - body ccf acting on colloid ( 2 ) is always attractive for the given geometrical configuration , regardless of the bcs . and alluding to the system studied in ref . @xcite , the circles represent the projections of dielectric spheres with equal radii , the green one corresponding to polystyrene and the red one to silicon ; the rectangles represent semi - infinite plates with their surfaces perpendicular to the @xmath222 plane , the gray and the yellow one being teflon and gold , respectively . the system is immersed in ethanol , which is not indicated in the figure . in ( a ) and ( b ) each dashed arrow indicates the direction of the stronger one of the two corresponding pairwise forces between the dielectric spheres and the plate , which turns out to determine the direction of the many - body lateral force acting on the spheres : if the stronger one of the two pairwise forces is attractive [ repulsive ] , the lateral many - body force will also be attractive [ repulsive ] ( see ref . also in the case of the critical casimir interaction [ ( c ) , ( d ) , and ( e ) ] , the circles and rectangles represent projections of spherical colloids and of homogeneous substrates , respectively : orange and blue indicate @xmath22 and @xmath23 bcs , respectively . in ( c ) and ( d ) the pairwise normal forces between each of the two spheres and the substrate are equal : attractive in ( c ) and repulsive in ( d ) . in ( e ) the two pairwise normal forces have opposite directions with the repulsive one being the stronger one @xcite . the corresponding dashed arrows have the same meaning as in ( a ) and ( b ) . according to fig . [ mb_x ] the many - body lateral ccfs are attractive for all three cases ( c ) , ( d ) , and ( e ) . the comparison shows that the systems in ( a ) and ( c ) behave similarly . however , the behavior of system ( b ) has no counterpart for ccfs [ see ( d ) and ( e ) ] . in this figure all surface - to - surface distances equal the sphere radius , which in our notation corresponds to @xmath223 . ] we can also compare our results with those for two atoms close to the surface of a planar solid body . mclachlan @xcite has tackled this problem by treating the solid as a uniform dielectric . by using the image method he derived an expression for the many - body corrections to the pairwise interaction energies , i.e. , the atom - atom ( london ) and the atom - surface energies , in order to obtain the total interaction energy between the two atoms close to the surface . qualitatively , he found that the leading contribution of the many - body correction leads to a repulsion if the atoms are side by side , i.e. , at equal surface - to - substrate distances . rauber et al . @xcite used mclachlan s approach to study the electrodynamic screening of the van der waals interaction between adsorbed atoms and molecules and a substrate . the latter plays a role which is `` analogous to that of the third body in the three - body interaction between two particles embedded in a three - dimensional medium '' . the van der waals interaction between the two atoms at equal distances from the substrate is altered by the presence of the solid substrate and this perturbation is given by @xcite @xmath224 where @xmath225 is the distance between the atoms , @xmath121 is the height above the image plane , which is the same for both atoms , and @xmath226 . the coefficients are given by @xmath227 and @xmath228 with @xmath229 / \left [ \epsilon(i\zeta ) + 1 \right ] ~,\ ] ] where @xmath230 is an imaginary frequency , @xmath231 is the polarizability of the atoms ( with the dimension of a volume ) , and @xmath232 is the dielectric function of the solid ( i.e. , the substrate ) . the lateral force due to the perturbation potential given by eq . , which is the analogue of the many - body contribution to the lateral ccf , follows from differentiating @xmath233 with respect to @xmath225 : @xmath234 ~.\end{gathered}\ ] ] in figure [ mclachan_fig ] we plot the lateral force given by eq . as function of the distance @xmath225 between the two atoms for several ( equal ) distances @xmath121 above the substrate . we use the values provided in ref . @xcite for the coefficients @xmath235 and @xmath236 : @xmath237 ev@xmath238()@xmath239 and @xmath240 ev@xmath238()@xmath239 , which correspond to ne , and @xmath241 ev@xmath238()@xmath239 and @xmath242 ev@xmath238()@xmath239 , which correspond to ar . as one can infer from fig . [ mclachan_fig ] , the many - body contribution to the lateral van der waals force is always repulsive and , as the two atoms approach the substrate , its strength increases . on the other hand , in the case of the many - body contribution to the lateral ccf , we have found that it is attractive for all bcs considered , if the surface - to - surface distances between the spheres and the sphere - surface - to - substrate distances are equal to each other and to the radius of the spheres ( see fig . [ mb_x ] ) . further , we can _ quantitatively _ compare our results with those from refs . @xcite and @xcite . to this end , we assign values to the geometrical parameters characterizing the configuration of the two atoms close to the substrate and compare the results of refs . @xcite and @xcite with those for similar configurations in our model . for example , estimating the many - body contribution to the lateral van der waals force for a configuration of two atoms close to a substrate corresponding to the configuration associated with the black curve the black curve corresponds to a function of temperature but there is no temperature dependence of the van der waals force between the atoms . therefore the comparison has to be carried out by choosing a certain value of @xmath243 . in fig . [ mb_x ] ( i.e. , @xmath244 in the case of the ccf and @xmath245 in the case of the two atoms ) , one obtains from eq . a value for the relative contribution of the many - body force to the lateral force which corresponds to ca . this is comparable with the relative contribution of the many - body force to the lateral ccf sufficiently close to @xmath11 , although in the case of the van der waals force it is repulsive ( fig . [ mclachan_fig ] ) while in the case of the ccf ( fig . [ mb_x ] ) it is attractive . considering the decay of the many - body contribution to the normal ccf as function of the surface - to - surface distance @xmath129 between the spheres ( for @xmath216 ) , we can compare the corresponding decay of the normal many - body force @xmath246 given by the potential @xmath233 in eq . . for small separations @xmath225 between the atoms the many - body contribution to the normal van der waals force increases as @xmath247 , while for large separations it decays as @xmath248 . by analyzing the data shown in fig . [ mb_l_z ] one finds that for @xmath249 the scaling function of the many - body contribution to the normal ccf decays slower than @xmath250 . this means that in this temperature regime the many - body contribution to the normal ccf is much more long ranged than the corresponding contribution to the normal van der waals force in the case of two atoms close to a surface . for fixed @xmath225 the many - body contribution to the lateral van der waals force decays as @xmath251 upon increasing the distance of both atoms from the substrate whereas the normal force on a single atom decays as @xmath252 . as a final remark we point out that we have not found a completely stable configuration for the two colloids ( fig . [ system_sketch ] ) : whenever there is a stable position in the horizontal ( vertical ) direction , the force is nonzero in the vertical ( horizontal ) direction . for example , consider a vertical path with @xmath253 in figs . [ d2_z](b ) and [ d2_x](b ) . from the first one can see that along this path the normal ccf changes from being repulsive to being attractive as the sphere - surface - to - substrate distance for colloid ( 2 ) is increased , implying that there is a vertical position of colloid ( 2 ) in which the normal ccf is zero . however , according to fig . [ d2_x](b ) the lateral ccf is always attractive regardless of the vertical position of colloid ( 2 ) . this means that there is a configuration which is stable only in the normal direction . accordingly , a dumbbell configuration with a rigid thin fiber between the two colloids can levitate over the substrate . whether this configuration is stable with respect to a vertical tilt remains as an open question . of the lateral many - body ccf acting on colloid ( 2 ) for @xmath207 and @xmath254 . the scaling function is shown as function of the scaling variable ratio @xmath255 . the black , red , and green lines correspond to the bcs @xmath157 , @xmath197 , and @xmath156 , respectively . for all three bcs the many - body contribution is not monotonic as function of temperature . quantitatively the green , red , and black curves here should be compared with the black curves in figs . [ d2_x](a ) , [ d2_x](b ) , and [ l_x](a ) , respectively . however , for the data shown in _ this _ figure the error bars ( not shown ) due to limits of the numerical accuracy are between 10@xmath256 and 15@xmath256 . this is the main reason why we refrain from showing what would be an instructive plot such as @xmath257 as a function of @xmath258 for various values of @xmath259 and @xmath170 ( as we did in fig . [ l_x ] ) , which would allow for a direct comparison with the case of atoms . ] , derived from the expression for the excess potential given in refs . @xcite and @xcite [ eq . ] . the forces are plotted as functions of the lateral separation @xmath225 between the two atoms for several equal vertical distances @xmath121 of the atoms from the substrate . the curves correspond to two sets of values for the coefficients @xmath235 and @xmath236 in eq . @xcite , corresponding to ne ( solid lines ) and ar ( dashed lines ) . ] we now turn our attention to the case in which the colloids are vertically aligned with respect to a planar , homogeneous substrate , i.e. , when their centers have the same coordinates in both the @xmath42 and the @xmath260 direction ( see fig . [ vertical_sketch ] ) . we focus on the normal ccf acting on colloid ( 1 ) when the system is immersed in a near - critical binary liquid mixture at its critical concentration . as before we consider @xmath31 bcs corresponding to a strong adsorption preference for one of the two components of the confined liquid . in particular , we consider two three - dimensional spheres of radii @xmath32 and @xmath33 with bcs @xmath34 and ( @xmath35 ) , respectively , facing a laterally homogeneous substrate with bc @xmath36 . colloid ( 1 ) is positioned at a sphere - surface - to - substrate distance @xmath37 and colloid ( 2 ) is at a surface - to - surface distance @xmath261 from colloid ( 1 ) ( see fig . [ vertical_sketch ] ) . the coordinate system @xmath39 is chosen such that the centers of the spheres are located at @xmath262 and @xmath263 so that the distance between the centers , along the @xmath74-axis , is given by @xmath264 . as before , the bcs of the system as a whole are represented by the set @xmath44 , where @xmath45 , @xmath35 , and @xmath46 can be either @xmath47 or @xmath48 . and @xmath33 immersed in a near - critical binary liquid mixture ( not shown ) and close to a homogeneous , planar substrate at @xmath63 . the colloidal particle ( 1 ) with bc @xmath34 is located vertically at the sphere - surface - to - substrate distance @xmath37 , whereas the colloidal particle ( 2 ) with bc @xmath64 is located vertically at the surface - to - surface distance @xmath261 between the spheres . the substrate exhibits bc @xmath36 . the vertical distance between the centers of the spheres along the @xmath74-direction is given by @xmath264 , while the centers of both spheres lie on the vertical axis @xmath265 . in the case of four spatial dimensions the figure shows a three - dimensional cut of the system , which is invariant along the fourth direction , i.e. , the spheres correspond to parallel hypercylinders with one translationally invariant direction , which is @xmath66 [ see eq . ] . ] the normal ccf @xmath266 acting on colloid ( 1 ) along the @xmath74-direction takes the scaling form @xmath267 where @xmath268 ( i.e. , @xmath269 for @xmath27 and @xmath270 for @xmath10 ) , @xmath76 , @xmath271 , and @xmath272 ; @xmath273 . equation describes the singular contribution to the normal force emerging upon approaching @xmath11 . @xmath274 is the force on a hypercylinder divided by its extension in the translationally invariant direction [ see eq . ] . we use the same reference system as the one described by eq . in order to normalize the scaling function defined in eq . according to @xmath275 we calculate the many - body contribution to the normal ccf acting on particle ( 1 ) @xmath276 by subtracting from the total force the sum of the pairwise forces acting on it , i.e. , the colloid - colloid and the colloid - substrate forces [ see eq . ] . this many - body force takes the scaling form @xmath277 in fig . [ mb_vertical_z ] we show the normalized [ see eqs . , , and ] scaling functions @xmath278 of the many - body contribution to the normal ccf acting on colloid ( 1 ) as functions of the scaling variable ratio @xmath279 for two spherical colloids of the same size ( @xmath207 ) . keeping the surface - to - surface distance between the spheres fixed at @xmath280 , we vary the sphere - surface - to - substrate distance @xmath37 for several bcs : @xmath156 in ( a ) , @xmath281 in ( b ) , @xmath197 in ( c ) , and @xmath157 in ( d ) . from figs . [ mb_vertical_z](a ) and ( d ) one can infer that if the colloids have symmetric bcs , the scaling function of the many - body normal ccf acting on colloid ( 1 ) is negative ( i.e. , it is directed towards the substrate ) for any value of @xmath198 . on the other hand , for non - symmetric bcs between the colloids [ figs . [ mb_vertical_z](b ) and ( c ) ] , the many - body contribution to the normal ccf acting on colloid ( 1 ) is positive for any value of @xmath198 . the apparent change of sign in figs . [ mb_vertical_z](b ) and ( c ) is likely to be an artifact occurring within the error bars due to numerical imprecision . the relative contribution of the many - body ccf to the total force is between 10@xmath256 and 15@xmath256 . we point out that this configuration with the the two colloids vertically aligned with respect to the substrate allows for a wide range of interesting aspects which will be further explored in future works . we have investigated critical casimir forces ( ccfs ) for a system composed of two equally sized spherical colloids ( @xmath207 ) immersed in a near - critical binary liquid mixture and close to a laterally homogeneous substrate with @xmath128 boundary condition ( bc ) ( see fig . [ system_sketch ] ) . by denoting the set of bcs of the system as @xmath44 , where @xmath282 corresponds to the bc at colloid @xmath283 and @xmath46 to the bc at the substrate , we have first focused on the total normal and lateral forces acting on one of the colloids [ labeled as `` colloid ( 2 ) '' ] for several geometrical configurations of the system and various combinations of bcs at the colloids . both the normal and the lateral forces are characterized by universal scaling functions [ eqs . and , respectively ] , which have been studied in the one - phase region of the solvent as functions of @xmath284 and @xmath285 . @xmath129 is the surface - to - surface distance between the two colloids , and @xmath178 is the bulk correlation length of the binary mixture in the mixed phase . we have used mean - field theory together with a finite element method in order to calculate the order parameter profiles , from which the stress tensor renders the normalized scaling functions associated with the ccfs . for the scaling function of the total normal ccf acting on colloid ( 2 ) with @xmath145 bc , in the presence of colloid ( 1 ) with @xmath286 , we have found ( fig . [ d2_z](a ) ) that the scaling function changes sign for a fixed value of @xmath284 as the distance @xmath38 between colloid ( 2 ) and the substrate increases , signaling the occurrence of an unstable mechanical equilibrium configuration of a vanishing normal force . for the total normal ccf acting on colloid ( 2 ) with @xmath144 bc and in the presence of colloid ( 1 ) with @xmath133 , we have found ( fig . [ d2_z](b ) ) that the force changes sign upon changing the temperature . for this combination of bcs , the equilibrium configuration of colloid ( 2 ) is stable in the normal direction . without a substrate , at the critical composition of the solvent the ccf between two @xmath22 spheres is identical to the one between two @xmath23 ones . this degeneracy is lifted by the presence of the substrate as one can infer from the comparison of the scaling functions for the lateral ccfs for @xmath156 and @xmath157 bcs [ see figs . [ d2_x](a ) and ( b ) , respectively ] . in the first case , the shape of the scaling function resembles that of the two colloids far away from the substrate , with a minimum at @xmath287 . in the second case this minimum at @xmath288 disappears . these substrate - induced changes are more pronounced if the two spheres are close to the substrate ( fig . [ d2_x ] ) . without a substrate the ccf between spheres of opposite bcs is purely repulsive . in the presence of a substrate the corresponding lateral ccf for @xmath197 bcs turns attractive for large lateral distances @xmath129 ( fig . [ l_x ] ) , which is a pure many - body effect . we have also studied the direction of the total ccf acting on colloid ( 2 ) for various spatial configurations and bcs in order to assess the preferred arrangement of the colloids . for @xmath156 bcs we have found that they tend to aggregate laterally . in this case a collection of colloids with @xmath22 bcs , facing a substrate with the same bc , are expected to form a monolayer on the substrate . for the situation of @xmath157 bcs , we have found that the colloids are expected to aggregate on top of each other . this indicates that a set of several colloids with such bcs are expected to form three - dimensional clusters . these tendencies are enhanced upon approaching the critical point ( fig . [ direction_resulting_force_d2 ] ) . by calculating the pairwise colloid - colloid and colloid - substrate forces and subtracting them from the total force , we have determined the pure many - body contribution to the force acting on colloid ( 2 ) . for the scaling functions associated with the normal many - body ccfs we have found the interesting feature of a change of sign at fixed temperature upon varying the lateral position of colloid ( 2 ) ( fig . [ mb_l_z ] ) . this implies that , for a given temperature , there is a lateral position where the normal many - body ccf is zero , in which case the sum of pairwise forces provides a quantitatively reliable description of the interactions of the system . as expected we have found that the contribution of the many - body ccfs to the total force is large if the colloid - colloid and colloid - substrate distances are small , as well as if the binary liquid mixture is close to its critical point . we have compared our results with corresponding ones for quantum - electrodynamic casimir interactions . to this end we have referred to the results in ref . @xcite for two dielectric spheres immersed in ethanol and facing a plate . these authors analyze the influence of the distance @xmath289 from the plate on the equilibrium separation @xmath217 between the spheres , which are subject to quantum - electrodynamic casimir forces . they find that the lateral many - body force is attractive ( repulsive ) if the stronger one of the two normal sphere - plate forces is attractive ( repulsive ) [ figs . [ comparing](a ) and ( b ) ] . on the other hand , in the case of ccfs we have found that for a configuration in which the surface - to - surface distance between the colloids is equal to the sphere - surface - to - substrate ones and equal to the radius of the spheres , the many - body contribution to the lateral ccf is always attractive , regardless of the bcs ( fig . [ mb_x ] ) . we have also compared our results with the corresponding ones for the case of two atoms close to the planar surface of a solid body . in this respect we have referred to the mclachlan model @xcite for the many - body contribution to the van der waals potential [ see eq . and fig . [ mclachan_fig ] ] and the results from ref . @xcite . from this comparison we have found that if the two atoms are fixed at the same distance from the surface of the solid body , the normal many - body contribution to the total van der waals force decays with the atom - atom distance @xmath225 as @xmath248 for large atom - atom distances . this decay is much faster than the decay we estimate for the many - body contribution to the normal ccf , which within a suitable range appears to be slower than @xmath250 . furthermore , we have found that the many - body contribution to the lateral van der waals force is repulsive while the corresponding many - body ccf is attractive regardless of the set of bcs . finally we have considered the configuration in which the two colloids are vertically aligned with respect to the substrate ( fig . [ vertical_sketch ] ) . we have calculated the many - body contribution to the normal ccf acting on colloid ( 1 ) for two spherical colloids of the same size ( @xmath207 ) keeping the sphere - surface - to - surface distance @xmath280 fixed ( fig . [ mb_vertical_z ] ) . we have varied the sphere - surface - to - substrate distance @xmath37 for several bcs and have found that if the colloids have the same bcs , the many - body contribution to the normal ccf is directed towards the substrate [ figs . [ mb_vertical_z](a ) and ( d ) ] , whereas for colloids with opposite bcs , the many - body contribution to the normal ccf is directed away from the substrate [ figs . [ mb_vertical_z](b ) and ( c ) ] . we have found that the contribution of the many - body ccf to the total force is between 10@xmath256 and 15@xmath256 . t.g.m . would like to thank s. kondrat for valuable support with the computational tools used to perform the numerical calculations . s.d . thanks m. cole for providing ref . @xcite .

within mean - field theory we calculate the scaling functions associated with critical casimir forces for a system consisting of two spherical colloids immersed in a binary liquid mixture near its consolute point and facing a planar , homogeneous substrate . for several geometrical arrangements and boundary conditions we analyze the normal and the lateral critical casimir forces acting on one of the two colloids . we find interesting features such as a change of sign of these forces upon varying either the position of one of the colloids or the temperature . by subtracting the pairwise forces from the total force we are able to determine the many - body forces acting on one of the colloids . we have found that the many - body contribution to the total critical casimir force is more pronounced for small colloid - colloid and colloid - substrate distances , as well as for temperatures close to criticality , where the many - body contribution to the total force can reach up to @xmath0 .

the search for the higgs boson has been the cornerstone of the physics program at modern high energy colliders . the higgs boson of the standard model has well defined production and decay modes that allow for mass dependent searches in a number of channels . one of the key discovery modes at hadron colliders is higgs boson production by gluon - gluon fusion with decay through two leptonically decaying @xmath9-bosons , @xmath10 , giving opposite sign di - leptons plus missing energy . the dominant background in this channel comes from electroweak pair production of @xmath9-bosons , @xmath11 . this background is substantially larger than the higgs boson signal . however , the two processes have somewhat different kinematic properties that may be exploited using either cut based or multi - variate techniques . based on the expected kinematic properties of the signal and dominant di - boson background obtained from simulations , searches in this channel have been carried out at both the tevatron @xcite and large hadron collider ( lhc ) @xcite . in addition to the background from @xmath9-boson pair production , there are a number of other important processes that contribute background to the opposite sign di - lepton plus missing energy channel . while smaller than the dominant background , some can be comparable to the higgs boson signal . among these are a class of backgrounds arising from direct electroweak production of a @xmath9-boson in association with some other object that is mis - reconstructed as a fake lepton . this includes a @xmath9-boson produced along with jets , where a jet fakes a lepton , @xmath12 . another in this class is production of a @xmath9-boson and photon , with the on - shell photon undergoing an asymmetric external conversion to an electron positron pair in the electromagnetic field of an atomic nucleus within the detector material , @xmath13 , where the parentheses indicate the trailing electron or positron . if the conversion is sufficiently asymmetric in momentum , the trailing member of the pair is not reconstructed as an independent object and does not ruin the isolation criteria of the leading one , and the converted photon fakes an electron or positron . these backgrounds are treated in ongoing higgs boson searches [ 15 ] . here we consider a closely related process within this class of backgrounds coming from direct production of a @xmath9-boson and virtual off - shell photon that undergoes an internal asymmetric conversion in vacuum to a lepton anti - lepton pair , @xmath14 , where @xmath15 . initial and final state virtual photon radiation contributions to this process are shown in fig . [ fig : w_conv_fig ] , with additional contributions coming from @xmath9-boson virtual photon radiation near the production or decay vertex . in a manner similar to the external conversions discussed above , if the momentum sharing of the conversion pair is sufficiently asymmetric , the trailing member is not reconstructed as an independent object and does not ruin the isolation criteria of the leading one , and the internal conversion fakes a lepton or anti - lepton . this process may be referred to as loss of a muon or electron following an asymmetric internal conversion ( lame faic ) . -boson at a hadron collider in association with an initial or final state virtual off - shell photon radiation that internally converts in vacuum to a lepton anti - lepton pair . parentheses indicate asymmetric internal conversion in which the trailing converted lepton is not reconstructed as an independent isolated object . diagrams with an off - shell photon radiated from the intermediate @xmath9-boson near the production or decay vertex are not shown . ] it is instructive to compare and contrast lepton anti - lepton pairs arising from external and internal conversion . in both cases in order for the conversion to give rise to a single fake object that is reconstructed as a lepton , the conversion must be sufficiently asymmetric as described above . this effective restriction to the asymmetric region of phase space implies that only a fraction of the conversions yield fake lepton objects . simultaneous reconstruction of a conversion pair with both the lepton and anti - lepton identified could recover most of the remaining symmetric conversion region of the phase space , and possibly give a handle on these backgrounds . another similarity is that charge conjugation symmetry of electrodynamics ensures that conversion photons yield fake leptons of both charges in roughly equal proportion . this equality may provide a simple but powerful tool for characterizing the kinematic properties and distributions of these backgrounds . it is already used to constrain the total magnitude of backgrounds within this class that arise from a @xmath9-boson in association with a mis - reconstructed fake lepton of uncorrelated charge @xcite . external and internal conversions differ in important regards . the probability for an on - shell photon to convert in material to a lepton anti - lepton pair depends strongly on the lepton mass . near the forward direction in the high energy asymmetric limit , the ratio of external conversion probability for a muon anti - muon pair to that for an electron positron pair scales like @xmath16 . so for all practical purposes external conversions give rise only to electron positron pairs . this is in contrast to internal conversions for which there is only a moderate logarithmic enhancement of electron positron over muon anti - muon pairs , as described in the next section . another key difference is that since external conversion takes place in material , the reconstructed lepton track in this case may emerge part - way through the tracking detector . this feature of missing hits on the inner part of a reconstructed track may be utilized as a criterion for identifying external conversions . it is however not useful for identifying leptons from internal conversion since these originate from the collision vertex . in the next section , we present the theory of asymmetric internal conversions . then we study the potential impact of this background on the higgs search with a simulation . our simulation of the background at the generator level is done carefully , but the detector simulation that follows is not particularly sophisticated and is only meant to motivate detailed studies by the higgs search teams . we then conclude with a brief discussion of a possible approach for dealing with the asymmetric internal conversion backgrounds . the probability for a photon to split to a lepton anti - lepton pair by internal conversion may be calculated in the off - shell photon phase space using the optical theorem . the one - loop contribution of a lepton of mass @xmath17 to the discontinuity across the branch cut in the electromagnetic current two - point correlation function gives the conversion probability distribution m _ d p ( ^ * ) d m _ = 2 3 ( 1 - 4 m^2_m^2 _ ) ^1/2 ( 1 + 2 m^2_m^2 _ ) where @xmath18 is the fine structure constant , and @xmath19 is the lepton anti - lepton or equivalently off - shell photon invariant mass . the internal conversion probability has an infrared soft singularity in the lepton anti - lepton invariant mass phase space that is cutoff only by the lepton mass , @xmath20 . the probability per logarithmic lepton anti - lepton invariant mass in the vicinity of the singularity is roughly constant . the total conversion probability integrated between the di - lepton threshold and an ultraviolet matching scale @xmath21 is ( ^ * ) = _ 2 m_^ dm _ d p ( ^ * ) d m _ = 2 3 [ splitprob ] in leading logarithmic approximation , the infrared singular region of the lepton anti - lepton invariant mass phase space gives the dominant contribution to the process of internal conversion . the contributions from non - singular regions of phase space with @xmath22 are formally @xmath23 corrections to the logarithm in the brackets in ( [ splitprob ] ) . in the background processes of interest here , the total probability for a high energy photon to undergo internal conversion to a lepton anti - lepton pair is @xmath24(1% ) . for example , with @xmath25 gev the splitting probability to an electron positron pair is @xmath26 , to a muon anti - muon pair is @xmath27 , and to a tau anti - tau pair is @xmath28 . an important kinematic property of internal conversion is the degree of asymmetry between the lepton and anti - lepton . in order to characterize this asymmetry it is useful to define the momentum fraction carried by the negatively charged lepton in the direction of motion of the off - shell photon , @xmath29 . in the high energy co - linear limit in which the lepton anti - lepton pair emerges in the direction of the off - shell photon , the momentum fraction is related to the lepton decay angle @xmath30 in the off - shell photon frame as measured with respect to its direction of motion by @xmath31 , where @xmath32 is the lepton velocity in this frame . the momentum fraction in the co - linear limit lies in the range @xmath33 . the probability distribution with respect to this momentum fraction depends on the polarization of the off - shell photon . in high energy scattering processes the probability for emission of an off - shell photon with longitudinal polarization is suppressed with respect to transverse polarization by @xmath34 , where @xmath35 is an ultraviolet mass scale associated with the hard scattering process . so for the backgrounds of interest here , longitudinal polarization may be neglected . the normalized probability distribution with respect to the momentum fraction in the co - linear limit for transverse polarization may be calculated in the two - body lepton anti - lepton phase space using the optical theorem as described above , f_t(z , ) = 2 - ^2 + ( 1 - 2z)^2 2 ( 1 - ^2 /3 ) charge conjugation symmetry ensures that this distribution is invariant under @xmath36 . right at threshold , @xmath37 , the conversion probability is symmetric with @xmath38 . however , well above threshold , @xmath39 , the transverse conversion normalized probability distribution becomes f_t(z , ) = 3 4 + o ( m _ / m_)^2 with @xmath40 in the range @xmath41 . in this limit the distribution is maximized for maximally asymmetric momentum fraction , @xmath42 , and is minimized for symmetric momentum sharing , @xmath43 . the total normalized probability for internal conversion with momentum fraction @xmath44 in the co - linear limit and well above threshold is ( ) 2 _ 0^ dz f_t(z,1 ) = 3 - 3 ^2 + 2 ^3 this conversion fraction of course vanishes as @xmath45 , but is not insignificant for moderately small values of asymmetry . for example , the fraction of internal conversions with asymmetry parameter @xmath46 is @xmath47 . so a sizeable fraction of internal conversions can be fairly asymmetric . another important kinematic property of internal conversion is the opening angle between the lepton and anti - lepton . near the high energy co - linear limit this angle may be written in terms of the lepton momentum fraction and lepton anti - lepton invariant mass _ = m _ 2 |_| z(1-z ) + o ( m_^2 / ( |_| m_))^3 where @xmath48 is the laboratory frame total momentum . the opening angle is small for lepton anti - lepton invariant masses in nearly the entire range @xmath49 for conversions not too far from the symmetric limit of @xmath50 . the angle is small for all values of momentum fraction , including near maximal asymmetry @xmath51 , for invariant masses in the range @xmath52 . before proceeding to an evaluation of the higgs lame faic background , we extend eqn . ( [ splitprob ] ) . the differential cross section with respect to any kinematic quantities @xmath53 formed from the four - vectors of the initial and final state particles including the lepton anti - lepton pair is given by _ 2 m_^ dm _ d d m _ dx ( init + ) = p ( ^ * ) ( init + ) |_p _ = p _ + o ( /m)^2 where @xmath54 is the off - shell photon to lepton anti - lepton pair conversion probability presented in ( [ splitprob ] ) , and @xmath35 is an ultraviolet mass scale associated with the hard scattering process . in order to maximally capture the phase space of the lame faic asymmetric conversions in a simulation , free parameters in the simulation package must be chosen with due care . in addition , the spatial proximity of the conversion leptons in the tracker can have large impact on the extent of lame faic background . the acceptance thus critically depends on the detector properties , reconstruction algorithms and kinematic selection criteria . therefore , the conclusions of the study that we describe below are not rigorously quantiative . in order to simulate 7 tev lame faic s , we use madgraph ( v5 ) @xcite to separately generate @xmath55 , @xmath56 and @xmath57 samples . the z pole is removed in order not to doublecount the @xmath58 background . a rapidity cut of @xmath59 was used for all leptons . in each case , the @xmath60 of the hardest and the second hardest @xmath61 is required to be above 5 gev . to capture asymmetric conversions maximally , @xmath60 of the third lepton is allowed to be as low as 0.1 gev and the dilepton invariant mass as low as 2@xmath19 . it was necessary to alter the madgraph source code in order to implement these differential thresholds for the daughter leptons . the generation cross section for the three processes as reported by madgraph are 3878 , 1076 and 228 fb for @xmath62 and @xmath63 , respectively . for comparison purposes , we also generated the leptonic decay modes for @xmath64 signal ( @xmath65 ) and @xmath66 pythia @xcite samples . the cross section for the higgs sample was scaled up to the nnlo value @xcite . these madgraph and pythia samples then underwent a generic lhc detector simulation with the pgs @xcite software package . we altered the pgs source code as follows to implement physics object isolation in a manner similar to the way it is done at lhc . an isolation variable is calculated for each photon candidate centered on an ecal cell @xmath67 as a fractional sum of the transverse energy deposited in ecal cells surrounding @xmath67 within a @xmath68 radius of @xmath69 _ iso , n _ e_t , k |_r < 0.4 e_t,,n identified photons are required to satisfy @xmath70 . this isolation requirement is a good representation of the full photon identification requirements used by the lhc experiments . the sum @xmath60 of all tracks with @xmath60 greater than 0.5 gev within an annulus of inner and outer radius of 0.03 and 0.4 respectively in the @xmath68 plane must be less than 0.15 of the @xmath60 of the candidate muon . t_iso , _ p_t , i p_t , @xmath71 e_iso , e _ e_t , k e_t , e @xmath72 . p_t , iso , e _ p_t , i @xmath73 gev _ e @xmath74 the total transverse calorimeter energy in a ( 3 x 3 ) grid and @xmath75 of @xmath76 , the default value in the pgs parameter set for cms . ] around the candidate electron ( excluding the candidate cell ) is defined as etiso . pgs then imposes the requirement that etiso / @xmath77(candidate ) be less than 10% . the total @xmath78 of tracks with @xmath79 0.5 gev within a @xmath80 0.40 cone is defined as ptiso . in this case , this excludes the leading electron track . it must satisfy : ptiso @xmath81 5 gev . finally the ratio of the calorimeter cell energy to the pt of the candidate track , e / p , should be within 50% to 150% . armed with the lame faic event generation followed by a rudimentary detector simulation , now are in a position to evaluate the higgs background . we apply a set of selection criteria ( `` ww analysis '' ) that mimic typical ww selection criteria . these get the data sample ready for final kinematic selection and discrimination between the signal and the background ( `` higgs analysis '' ) . table 1 lists these selection criteria . . selection criteria and requirements for a representative opposite sign di - lepton plus missing energy @xmath82 analysis , along with additional requirements for a representative 130 gev higgs analysis . jets are defined with a cone algorithm with @xmath83 . the projected missing transverse energy is defined to be the missing transverse energy , @xmath84 if @xmath85 , and to be @xmath86 if @xmath87 , where @xmath88 . [ cols="^,^,^,^ " , ] [ optable2 ] table 2 also shows dilepton yields after the higgs selection . as expected from the accompanying kinematic distributions , the ww background is reduced drastically . the lame faic s , however , still pose an @xmath24(20% ) background to the 130 gev higgs signal . we have shown that the internal conversion background is potentially sizeable and is sufficiently similar in kinematics to the higgs signal that it could throw a wrench in the delicate workings of sophisticated multivariate analysis techniques employed in the higgs searches . similarities between internal conversions and higgs in both yield and kinematic properties warrants experimental studies of this background . furthermore , even sophisticated simulation may prove inadequate in quantifying detector acceptance of the surviving lepton since the impact of the lost low - p@xmath89 lepton on the tracking and isolation of the surviving one is difficult to gauge . in that case , and for the purposes of making searches robust , it behooves the higgs hunters to employ data - based techniques for reducing and then incorporating this background into multivariate schemes . since the lame faic background should yield equal number of same - sign ( ss ) and opposite - sign ( os ) dileptons , the ss dilepton data sample can be used to constrain the lame faic background in the os ( higgs ) analysis . a detailed understanding of the ttbar , electroweak and @xmath90 external conversion backgrounds of the ss sample would constrain the os lame faic s . a monte carlo simulation of the os lame faic s that has been validated with the ss data sample could be integrated in the higgs multivariate analysis techniques . to conclude , the opposite - sign dileptons resulting from asymmetric @xmath91 internal conversion form a potential background for the higgs searches . this background has not been addressed in the recent results released by the tevatron or lhc higgs search teams . while our limited study lacks the quantitative rigor in simulating the intricacies of detector acceptacnce in the presence of a soft conversion lepton , it is possible that the more detailed simulation tools currently used by the higgs search teams are also inadequate in addressing it , thus necessitating development of data - based techniques . we would like to thank johan alwall , emmanuel contreras - campana , yuri gershtein , amit lath , yue zhao and other colleagues at rutgers for their insights and constructive comments . the research of ck , mp and st was supported in part by doe grant de - fg02 - 96er40959 and nsf grant phy-0969020 . the research of rg and ss was supported in part by nsf grant phy-0969282 . t. aaltonen _ et al . _ [ the cdf collaboration ] , `` inclusive search for standard model higgs boson production in the @xmath92 decay channel using the cdf ii detector , '' phys . rev . lett . * 104 * , 061803 ( 2010 ) [ arxiv:1001.4468 [ hep - ex ] ] . v. m. abazov _ et al . _ [ the d0 collaboration ] , `` search for higgs boson production in dilepton and missing energy final states with 5.4 fb@xmath93 of @xmath94 collisions at @xmath95 tev , '' phys . lett . * 104 * , 061804 ( 2010 ) [ arxiv:1001.4481 [ hep - ex ] ] . t. aaltonen _ et al . _ [ cdf and d0 collaborations ] , `` combination of tevatron searches for the standard model higgs boson in the @xmath82 decay mode , '' phys . lett . * 104 * , 061802 ( 2010 ) [ arxiv:1001.4162 [ hep - ex ] ] . s. chatrchyan _ et al . _ [ cms collaboration ] , `` measurement of @xmath82 production and search for the higgs boson in pp collisions at @xmath96 tev , '' phys . b * 699 * , 25 ( 2011 ) [ arxiv:1102.5429 [ hep - ex ] ] . g. aad _ et al . _ [ atlas collaboration ] , `` limits on the production of the standard model higgs boson in @xmath97 collisions at @xmath98 tev with the atlas detector , '' arxiv:1106.2748 [ hep - ex ] . j. alwall , m. herquet , f. maltoni , o. mattelaer , t. stelzer , `` madgraph 5 : going beyond , '' jhep * 1106 * , 128 ( 2011 ) . [ arxiv:1106.0522 [ hep - ph ] ] .

a class of potential backgrounds for higgs boson searches in the @xmath0 channel at both the tevatron and large hadron collider is presented . backgrounds from @xmath1 production with _ external _ conversion of the on - shell photon in detector material to an asymmetric electron positron pair , @xmath2 , with loss of the trailing electron or positron has been treated adequately in higgs searches . here we consider analogous backgrounds from @xmath3 production with _ internal _ conversion of the off - shell photon in vacuum to an asymmetric lepton anti - lepton pair @xmath4 . while the former process yields almost entirely electrons or positrons , the latter can give electron , positron , muon , and anti - muon backgrounds in roughly equal amounts . we estimate that asymmetric internal conversion backgrounds are comparable to the higgs boson signal in the standard signal region of phase space . these processes also represent potential backgrounds for new physics searches in same - sign di - lepton channels . some data driven methods to characterize asymmetric internal conversion backgrounds are suggested . ru - nhetc-2011 - 16 + uttg-20 - 11 + tcc-022 - 11 + 1.0 in * backgrounds to higgs boson searches from * + * @xmath5 asymmetric internal conversion * 0.65 in richard c. gray@xmath6 can kilic@xmath7 michael park@xmath6 0.1 in sunil somalwar@xmath6 and scott thomas@xmath6 0.25 in @xmath6_department of physics + rutgers university + piscataway , nj 08854 _ 0.15 in @xmath8 _ theory group , department of physics and texas cosmology center + the university of texas at austin + austin , tx 78712 _ 0.75 in

it is well known that the lift to m - theory of a system of parallel d6-branes @xcite corresponds to a purely geometric background , the taub - nut metric . when the position of n of these d6-branes coincide , one gets an @xmath8 singularity at a point in the multi taub - nut space . in this paper , we would like to make a step forward in the relation between the physics of d6-branes at strong coupling and purely gravitational backgrounds in eleven dimensional supergravity by studying the lift of a system of coincident d6-@xmath9 branes to m - theory . we shall primarily be concerned with the geometry describing these configurations , its evolution as branes and antibranes annihilate each other and some similarities between the qualitative patterns that we find in this evolution and some recent results on the evolution due to the condensation of localised closed string tachyons in non - supersymmetric orbifold singularities @xcite . in particular , we shall study the lift to m - theory of the generically non - bps configurations found in @xcite preserving @xmath10 . the latter depend on three parameters . the subset of configurations in which we will be interested in corresponds to setting one of them to zero . these particular geometrical configurations look like @xmath11 , for some curved four dimensional manifold . it turns out that @xmath12 has a _ bolt _ type singularity , that is , a locus of conical singularities , whose conical defects depend on the mass and the charge of the configuration . the brane - antibrane annihilation expected in the open string description gives rise to a reduction in the size of the bolt and a desingularization of the conical singularities , by which they become `` less conical '' . in the sector of non - vanishing charge , the bolt becomes a nut , whereas in the vanishing charge sector , the bolt disappears . locally , when the size of the bolt is big , the system looks like @xmath13 . the size of the bolt is proportional to the product of the number of branes , the number of antibranes and @xmath14 . thus , big bolt limit means that @xmath15 is big , i.e. the number of branes and antibranes should be large in order to keep a small string coupling . thus , by reducing along a trivial circle , the original d6-@xmath9 system is related to a @xmath1 orbifold in the forementioned limit . whenever @xmath16 , there are closed string tachyons in the twisted sectors . recent studies @xcite suggest that this system evolves to flat space making the cone `` less conical '' by a sequence of transitions @xmath17 our qualitative comparison in the large bolt limit suggests a relation between brane - antibrane annihilation and twisted tachyon evolution . and in particular , each transition @xmath18 , which reduces the order of the orbifold by two , is related to the annihilation of a d6-@xmath9 pair . in the second part of the paper , and motivated by the previous relation , we start from a non - supersymmetric orbifold acting on @xmath19 in type iia , lift the configuration to m - theory using a trivial transverse circle and reduce it along a non - trivial circle in @xmath19 . one expects such system to be the local description for an unstable system of branes . in particular , we consider @xmath20 , where each abelian group preserves different supersymmetry , so that the full orbifold is non - supersymmetric . the interpretation of the reduced system is in terms of fractional d6-branes living on a @xmath1 singularity . here the closed string tachyonic instabilities can not be mapped to open string tachyons as in the previous case . the organisation of the paper is as follows . in section 2 , we revisit the construction of supergravity solutions given in @xcite , paying attention to the particular case of d6-@xmath9-branes . these solutions depend on three parameters . we discuss the scaling limits leading to bps configurations , generalising the discussion in @xcite . we consider the lift of such configurations to m - theory and argue why it is interesting for us to set one of the parameters to zero . in this way , we get a two parameter family of solutions , where the parameters can be mapped to the ramond - ramond ( rr ) charge and mass of the system . in section 3 , we analyse this solution in detail , both in the charged and uncharged sectors . in section 4 , we discuss the evolution of the system and we compare the open and closed string descriptions . section 5 is devoted to the study of the inverse problem : going from a non - supersymmetric orbifold to a local description of a system of d6-branes . in particular , we consider a @xmath21 non - supersymmetric orbifold . in @xcite , the most general solution to the supergravity equations of motion with @xmath22 symmetry and carrying the appropriate ramond - ramond ( rr ) charge was integrated . it was subsequently interpreted in @xcite as a system of coincident dp-@xmath23 branes . in this work , we shall concentrate on the d6-@xmath9 system . in the einstein frame , the configuration is described by @xmath24 where @xmath25 is the ten dimensional metric , @xmath26 is the dilaton and @xmath27 is the rr seven form potential . the set of scalar functions characterising the above configuration is given by @xmath28 \\ b(r ) & = \log[f_-(r ) f_+(r ) ] -7a(r ) \\ \phi(r ) & = c_1\,h(r ) + 12\,a(r ) \\ e^{\lambda(r ) } & = - \sqrt{c_2 ^ 2 -1 } \frac{\sinh(k h(r))}{\cosh(k h(r ) ) - c_2 \sinh(k h(r ) ) } \end{aligned } \label{setfunctions}\ ] ] where @xmath29 \\ k & = \sqrt{4-\frac{7}{16}c_1 ^ 2 } ~. \end{aligned}\ ] ] thus , it depends on two dimensionless parameters @xmath30 defined in the ranges @xmath31 , @xmath32 , and a third one @xmath33 , with dimensions of length satisfying @xmath34 . the charge ( q ) and mass ( m ) of this solution were computed in @xcite and we shall follow their conventions . they are expressed in terms of @xmath35 as follows @xmath36 \label{mass}\end{aligned}\ ] ] where @xmath37 , @xmath38 being the spacelike volume spanned by the branes and @xmath39 stands for the ten dimensional newton s constant . written in string units , @xmath40 , where @xmath41 is the string coupling constant and @xmath42 is the string length @xmath43 . notice that in general the configuration is non - bps @xmath44 , as expected , and it is useful to introduce the difference between these observables @xmath45 \ , . \label{nonbps}\ ] ] the first natural question to address is how to recover the well - known bps configurations corresponding to n d6-branes ( or @xmath466-branes ) from the general solution . at this point , we would like to point out that there are more possibilities than the one discussed in @xcite . indeed , the idea there was to take a certain scaling limit in the set of parameters @xmath35 , or equivalently in @xmath47 , such that the charge q remains finite while @xmath48 . as discussed in @xcite , one possibility is to consider @xmath49 which can also be formulated in terms of @xmath50 , by @xmath51 . the previous scaling limit is certainly not the only possibility , and as it will turn out important for us later on , we discuss a second possibility . consider the following double scaling limit @xmath52 \label{bpslimit}\ ] ] it is clear that the charge remains finite in the limit and that @xmath53 vanishes , as required . as a further check , it is straightforward to analyse in the above limit to get back the bps metric @xcite from . by rescaling the einstein metric to the string frame and using the standard kaluza - klein ansatz , one derives a family of purely geometrical configurations in eleven dimensions described by the metric @xmath54 \left(dr^2 + r^2d\omega_2 ^ 2\right ) \\ + \left(\frac{f_-(r)}{f_+(r)}\right)^{7c_1/12}\left [ \cosh(k h(r ) ) - c_2 \sinh(k h(r))\right]^{-1}\left(dz + c_1\right)^2 \label{msolution}\end{gathered}\ ] ] where @xmath55 stands for the spacelike coordinate along the m - theory circle with length at infinity @xmath56 and @xmath57 is the magnetic dual one form to the previous rr 7-form @xmath58 $ ] . notice that whenever @xmath59 , the eleven dimensional geometry is not that of seven dimensional minkowski spacetime times some curved manifold , but contains a warped factor . in the limit @xmath60 keeping @xmath61 fixed , the geometry asymptotes to the maximally supersymmetric minkowski spacetime . one non - trivial check @xcite for the above family of solutions concerns the zero charge sector @xmath62 . indeed , it has been known for a while the embedding in eleven dimensions @xcite of the kaluza - klein dipole solution @xcite describing a monopole - antimonopole pair separated by some distance . studying such a solution in the limit of vanishing dipole size , one gets the configuration @xmath63 where the scalar function @xmath64 is defined by @xmath65 , m being some constant parameter . it is clear that the matching between and requires setting @xmath66 to ensure the vanishing of the warped factor and charge , respectively . the same reasoning applies for a system of more than two monopoles . if we want the solution to remain as a seven dimensional minkowski spacetime times some four dimensional manifold where the monopoles are living , one needs @xmath67 . in this subspace , becomes @xmath68 notice that and are equivalent , as expected , under the coordinate transformation : @xmath69 where @xmath70 stands for the radial coordinate in , provided the two constant parameters are identified as @xmath71 notice that the right hand side of the above coordinate transformation is invariant under the transformation @xmath72 . we shall see later that this symmetry is not restricted to the vanishing charge sector @xmath73 , but generalizes to @xmath74 . in the following , we shall concentrate on the @xmath75 $ ] subspace of solutions @xcite @xmath76(dr^2 + r^2 d\omega^2_2 ) + \frac{(f_+ f_-)^2}{\left[\frac{1 - c}{2 } f^4_- + \frac{1 + c}{2 } f^4_+\right ] } ( dz + c_{(1)})^2 \nonumber \\ \label{msolution1}\end{aligned}\ ] ] which includes in the sector of zero charge @xmath77 $ ] . we would like to emphasise that such a subspace of configurations includes both the bps ones , through the scaling limit , and the zero distance monopole - antimonopole pair solution . since it contains a seven dimensional minkowski spacetime , it allows us to concentrate on the physics of the four dimensional curved manifold , which is rather natural if one is interested in relating the physics of d6-@xmath9 at strong coupling with tachyon condensation in orbifold models in @xmath19 , whose local description close to the fixed point ( singularity ) consists of such a seven dimensional minkowski spacetime times some four dimensional manifold . the two parameters @xmath78 appearing in can be mapped to the charge and mass of the system , which satisfy the quadratic relation : @xmath79 showing that the mass is bigger or equal to the charge . these parameters can be expressed in a much more physical way in terms of the number of branes ( n ) and anti - branes @xmath80 as @xmath81 or equivalently , by @xmath82 as we can see from these formulae the radius of the bolt and the value of @xmath83 are discrete , as only an integer number of branes is allowed . notice that the measure for the non - bps character of the configuration is proportional to the ratio @xmath84 where @xmath85 is the radius of the m - theory circle and @xmath86 is the eleven dimensional planck length . a natural way of measuring the non - bps character of the configuration in terms of d6-branes data is by the quotient @xmath87 if there are only branes or antibranes , the quotient equals @xmath88 , which can only be achieved if @xmath89 . notice that to keep the charge fixed in that limit , one must take at the same time @xmath90 , which matches our discussion on bps limits , in particular the scaling limit . as we shall discuss more extensively in the next section , there is a _ bolt _ type singularity at @xmath33 , both in the charged and non - charged sectors , for non - zero values of @xmath33 . when approaching the supersymmetric configuration , the fate of the _ bolt _ singularity depends on the sector in which we are : * if @xmath74 , it gives rise to the usual _ nut _ singularity at @xmath91 where the monopoles ( or antimonopoles ) are sitting this is the source for the naked singularity of the d6-branes ( or @xmath9-branes ) at the origin @xcite . * if @xmath92 , it gives rise to flat space . let us analyse the geometry of solution . first of all , it is exactly the taub - bolt singularity without imposing the absence of conical singularities @xcite . that can be seen explicitly by the change of radial coordinate @xcite : @xmath93 and identifying the parameters in both solutions as @xmath94 and @xmath95 . if we keep the charge fixed and take @xmath90 , or equivalently , we take the double scaling limit , we end up with the taub - nut metric : @xmath96 where @xmath97 , as expected for the bps configuration ( m = q ) . in the limit close to the origin , the metric reproduces the singularity of a @xmath98 orbifold ( @xmath99 singularity ) , where n is the number of branes defined previously , i.e. in d - brane units @xmath100 . indeed , close to the singularity located at @xmath91 , one can make the coordinate transformation @xmath101 which allows us to write the metric as @xmath102~.\ ] ] taking into account that @xmath55 has a period of @xmath103 one gets that the circle parametrised by @xmath55 has a conical behaviour like a @xmath98 orbifold . the solution is defined for @xmath104 , the interior of the sphere @xmath105 not belonging to the solution . however , it is interesting to point out the existence of an isometry , the in@xmath106out symmetry , that relates @xmath107 with @xmath108 , @xmath109 the geometry far away from @xmath110 has the same assymptotic behaviour as in the supersymmetric configuration . thus , any source of instability reflected in the geometry has to be in the region @xmath110 , at which we shall now look in detail . let us start our analysis in the charged sector @xmath111 . whenever the configuration is non - bps , the metric has a _ bolt _ singularity at @xmath105 . the bolt is a sphere of radius proportional to @xmath33 with conical singularities on it . to study these singularities , we can examine the metric close to the bolt , by introducing the distance to the bolt as a coordinate @xmath112 and concentrating on the region @xmath113 . after a trivial rescaling of the new radial coordinate , the four dimensional metric looks like @xmath114 thus , close to the bolt , the periodicity of the compact coordinate @xmath115 is reduced by a factor @xmath116 which indeed points out to the existence of conical singularities whose angular deficit is @xmath117 . notice that these singularities are located on a sphere of radius @xmath118 , whose area is @xmath119 notice that the area takes discrete values depending on the integer numbers representing the number of branes and antibranes . even though the scalar curvature vanishes on the bolt , due to the existence of the conical singularities , one might wonder about higher order corrections to the eleven dimensional effective action close to the bolt . to clarify this issue , one can analyse the behaviour of the square of the riemann tensor . such corrections would be suppressed whenever @xmath120 working in the regime in which the number of branes is of the same order as the number of antibranes @xmath121 , the above constraint looks like @xmath122 therefore such corrections can be neglected when the size of the bolt is big in eleven dimensional planck units , or equivalently @xmath123 notice that in order to keep the string coupling constant small , the number of branes must be large . this is the approximation we would like to use . when the size of the bolt is big @xmath124 , the metric close to the bolt is a huge sphere times a cone . furthermore , in the regime , the effect of @xmath57 is negligible . vector bundle over a trivial @xmath125-space @xmath126 . that means that the @xmath125 is acting trivially on the sphere while rotating the fibre @xmath127 . the charge @xmath128 of the system specifies the first chern number as in the supersymmetric case . ] thus , locally , the four dimensional manifold @xmath12 looks like @xmath129 that such a description allows an orbifold singularity @xmath130 interpretation can be further checked by using in the regime , which ensures that l is an integer number . these orbifold singularities have always closed string tachyons in the twisted sectors . in the next section , we shall compare the annihilation of brane - antibrane pairs expected in the open string description , with the sequences of transitions for @xmath131 orbifolds discussed in @xcite , and we shall see that they are qualitatively the same . we shall now move to the vanishing charge sector , that is , the one with the same number of branes and antibranes , i.e. @xmath132 . in this case , the metric reduces to and depends on a single parameter @xmath33 , which can be written in terms of the number @xmath133 of d6-@xmath9 pairs as @xmath134 since @xmath57 vanishes , the surfaces r = constant are trivial fibrations @xmath135 . the assymptotic geometries are @xmath136 , whereas close to @xmath110 , one can check , proceeding in an analogous way to the previous discussion , that the bolt structure remains . in this case , the deficit in the periodicity is @xmath137 . that means that for an integer number of d6-branes the system has an orbifold interpretation as a @xmath138 orbifold . the scalar curvature vanishes everywhere , as it corresponds to a solution of einstein supergravity equations of motion with no matter , whereas the squared of the riemann tensor is given by @xmath139 which has a maximum at @xmath105 . once more , the gravity approximation is reliable in the large bolt limit . when one trivially reduces the previous m - theory configurations by adding an extra transverse compact circle , one finds a generically non supersymmetric purely gravitational ( geometrical ) type iia configuration . thus , the analysis of singularities discussed above still applies to this geometry . we are thus left with two different descriptions in type iia of a single m - theory configuration : first , the brane - antibrane system and on the other hand , geometrical configurations with conical singularities located on a sphere . furthermore , in the limit of big bolt , the geometry of the conical singularities is locally given by that of an orbifold type , @xmath140 . thus , it is clear that both systems contain tachyons ; the brane - antibrane system in the open string sector from strings stretching between a brane and antibrane , whereas in the orbifold side , there are closed string tachyons in the twisted sectors . these tachyons can be understood as localised on the bolt . some properties of this kind of closed string twisted sectors and their possible evolution have been analysed in @xcite . in the following , we shall show that the expected annihilation of brane - antibrane pairs in the open string side matches the reduction in the order of the non - supersymmetric orbifold observed in the previous cited references . we can consider the evolution of the system in the @xmath141 parameter space . in the d6-brane picture , we expect branes to annihilate the antibranes so that the total charge is preserved . the mass will decrease up to a supersymmetric system , @xmath142 , in which we are left either with all branes or all antibranes . this process is expected to be a discontinuous process : branes and antibranes are annihilated in pairs as closed string fields will be emitted to the bulk . we expect a sequence @xmath143 this process is represented schematically in figure [ brane ] . . ] when considered from the m - theory effective description in terms of a classical solution of the supergravity equations of motion , the latter depends on two continuous parameters : @xmath144 and @xmath128 . nevertheless , one can study the evolution in the geometry of the family of configurations by moving in such a two dimensional parameter space . indeed , we are interested in studying the decrease in the mass @xmath144 while keeping the charge @xmath128 fixed . it is clear that such a motion requires a decrease of @xmath33 while @xmath83 increases `` along the flow '' . heuristically , we can think of @xmath145 and @xmath146 as the starting point of the flow . the value of @xmath33 is thus determined to be @xmath147 the motion along the flow we are interested in , is described by decreasing the parameter @xmath148 , simulating the annihilation of a brane and antibrane . one can formally take the limit @xmath149 and get the bps configuration as expected . in the @xmath33 , @xmath83 parameter space this flow can be seen as a curve going to @xmath150 and @xmath89 ( see [ parameter ] ) . and @xmath83 parameter space representing the annihilation of brane antibrane - pairs . ] this flow has two effects : the radius of the bolt goes to zero and the conical singularity gets less conical with a factor @xmath151 . when the system arrives at the supersymmetric configuration , the bolt disappears into a nut and a supersymmetric orbifold singularity remains at the origin @xmath152 . see figure [ bolt ] . orbifolds have always tachyons in the closed string spectrum . by turning on some of this tachyons the cone expands till reaching flat space . ] one very interesting case is when the number of branes @xmath133 is exactly the same as the number of antibranes @xmath153 . in this case , the flow corresponds to a straight line at @xmath154 , and the decrease in @xmath33 is directly related to the decrease in @xmath155 . then close to the bolt there is an orbifold description as @xmath156 . the process of annihilating branes and antibranes takes @xmath157 . from the orbifold point of view that corresponds to a transition @xmath158 . notice that this process is very similar to the one found by @xcite where the orbifold singularity is desingularising till reaches the flat space by @xmath159 ( see figure [ orbi ] ) . notice that in the orbifold description in @xcite , the order of the orbifold is odd while in our case is even . however as we have already said , the correspondence between the two systems is expected to happen only at large n. notice that in both sides , brane - antibrane annihilation and the vev of the twisted field are discontinuous , so our approximation of continuous mass variation has no meaning between these points . when interpreted in terms of branes and antibranes , we have seen that the radius of the bolt takes discrete values as well as the @xmath83 parameter . notice that , as discussed in @xcite , the process of desingularising the cone is expected to be discontinuous . so one expects sudden changes in the volume of the bolt from both sides . for example , one can consider the emission of dilaton fields by the brane - antibrane annihilation into the bulk . that will correspond to a sudden change in the m - theory coordinate that looks like the cone change in the twisted orbifold side as described in @xcite . it will be very interesting to relate these two discontinuous processes in detail . from the m - theory point of view , we can see the bolt as emitting waves that change suddenly the shape of the cone till the bolt disappear to a point . it is important to notice that we are not mapping open string to closed string tachyons , we are just comparing the behaviour and evolution of two different systems related by an m - theory lift . if one naively tries to map one open to one closed string tachyon , one immediately realises that things are not working . for large number of pairs of branes and antibranes n the counting of open string tachyons goes like @xmath160 but the number of twisted closed string tachyons grows like n. also the perturbative masses of these states do not match . however , the number of steps driving the system to the supersymmetric configuration is the same , of order n. this is because when a pair brane - antibrane disappears there are also n open string tachyons that decouple from the spectrum . the relation among @xmath161 orbifolds and d6-@xmath9 systems in the large bolt limit leads us to consider a non - supersymmetric orbifold of type iia , perform its trivial lift to m - theory and reduce it afterwards along a circle inside the orbifold . the configuration thus obtained can not be trusted far away from the origin , but it must correspond to the local description of some d6-brane system . notice that this is exactly what happens for the supersymmetric orbifold @xmath162 : this produces the familiar supersymmetric @xmath99 orbifolds ( for review see @xcite ) , as reviewed at the beginning of section 3 , which upon reduction along the hopf fibre , gives rise to the local description of a system of n coincident d6-branes located at the fixed points of the @xmath163 along which we performed the reduction . we shall next consider some particular non - supersymmetric orbifold singularities of the form @xmath164 , where the action of each subgroup is defined in such a way that the complete orbifold breaks supersymmetry completely . we will see that after reduction along the hopf fibre , the type iia configuration has a line of conical singularities with some fractional d6-branes located at the origin whenever @xmath165 . it is important to stress , once more , that the forthcoming analysis is only reliable close to where the d - branes are located . let us define polar coordinates in @xmath19 by @xmath167 where the range of the different angular variables is @xmath168 , @xmath169 and @xmath170 . the action of the @xmath171 group on @xmath19 is of the form : @xmath172 and @xmath173 where @xmath174 are the generators of the group . the orbifold does not preserve supersymmetry because each subgroup preserves supersymmetries of different chirality . thus , whenever the order of both subgroups ( n , m ) is different from 1 , the total orbifold breaks supersymmetry completely . due to the identifications associated with the orbifold construction , there are now two cones associated with each of the subgroups @xmath175 one can work with angular variables satisfying the standard periodicity conditions by rescaling @xmath176 . in this way , the periods are manifest in the metric @xmath177\ ] ] there are many @xmath163 s along which one could reduce , but we shall take the usual hopf fibering , i.e. reducing on @xmath178 . using the kaluza - klein ansatz , the ten dimensional metric in the string frame looks like @xmath179 whereas the dilaton and rr one form are given by : @xmath180 orbifold . at a fix distance from the origin where the fractional d6-brane is located the @xmath126 presents two conical singularities that represents the intersection of the two dimensional sphere with a line of @xmath1 singularities . ] notice that if @xmath181 , the above configuration matches the local description of n coincident d6-branes close to the naked singularity ( r=0 ) , and half of the supersymmetry is preserved . whenever @xmath182 , the naked singularity remains but there is an additional line of conical singularities coming from a @xmath1 orbifold . indeed , after reducing along the hopf fibering , we are left with @xmath183 in the subspace transverse to the d6-branes , but with one angular coordinate of reduced period . , the metric is flat but @xmath184 is defined over @xmath185 . ] the set of fixed points of the orbifold which reduced the period of the angular variable is given by the line @xmath186 r. we can thus interpret the ten dimensional configuration as the local description close to r=0 of a set of d - branes on a @xmath1 orbifold carrying fractional charge n / m . notice that for the systems just discussed there is no open - closed string instability correspondence like in previous sections , since the analysis in both sides implies the existence of closed string tachyons in the twisted sectors . in this paper , we have analysed the geometry of the lift to m - theory of certain d6-@xmath9 systems . for any non - bps configuration , we find a bolt type singularity . the annihilation of d6-@xmath9 pairs in the open string description is realised , on the gravity side , by a reduction on the size of the bolt and a desingularization of the conical singularities on it . in the large bolt limit , the m - theory geometry is locally described by @xmath187 . this allowed us to qualitatively match the annihilation of d6-@xmath9 pairs with the sequences of transitions described in @xmath161 non - supersymmetric orbifolds . as we have already said , the process is discontinuous in both sides . it would be very interesting to analyse how the discrete evolution is produced . having realised this connection , we considered the non - supersymmetric orbifold @xmath188 and its relation with a local description of unstable branes , which turned out to be fractional d6-branes on a @xmath1 singularity . there are several natural questions related with the results reported here . due to the relation among d6-brane systems and @xmath161 orbifolds , it would be very interesting to investigate if there is any brane realisation for the sequences of transitions found in @xcite regarding non - supersymmetric @xmath189 orbifolds . we would also like to point out that the brane - antibrane system discussed in this paper can be interpreted as a particular case of a pair of d6-branes at generic angles , the one in which they have opposite orientations . these more general systems do generically break supersymmetry and in some regions of their moduli space , they are empty of tachyons . it would be interesting to understand the m - theory dynamics in these cases @xcite . other physical systems which have recently been given a lot of attention and do also have localised closed string tachyons are fluxbranes @xcite . it would be interesting to understand the stability and supersymmetry of some of them using similar local descriptions to the ones appearing in this work . on the other hand , the analysis in section 5 is just a local one , as can be seen from the fact that the dilaton ( string coupling ) increases as we move away from the origin . it would be nice to look for non - bps configurations whose validity of description goes beyond the region where the brane sits . we would like to thank p. barbn , r. emparan , y. oz and a. uranga for discussions , and especially r. emparan for pointing out the existence of reference @xcite and for his comments on the first version of this work . r.r . would like to thank the group in humboldt university in berlin for hospitality during the progress of this work . j.s . would like to thank the theory division at cern for hospitality during the initial stages of the present work . the research of j. s. has been supported by a marie curie fellowship of the european community programme `` improving the human research potential and the socio - economic knowledge base '' under the contract number hpmf - ct-2000 - 00480 . see among others ( and referencies therein ) : a sen , _ stable non - bps states in string theory _ , jhep * 9806 * ( 1998 ) 007 , hep - th/9803194 . + to3em__stable non - bps bound states of bps d - branes _ _ , jhep * 9808 * ( 1998 ) 010 , hep - th/9805019 . + to3em__tachyon condensation on the brane antibrane system _ _ , jhep * 9808 * ( 1998 ) 012 , hep - th/9805170 . + to3em__non - bps states and branes in string theory _ _ , hep - th/9904207 . a dabholkar , _ on condensation of closed - string tachyons _ , hep - th/0109019 . + to3em , _ tachyon condensation and black hole entropy _ , phys.rev.lett . 88 ( 2002 ) 091301 , hep - th/0111004 . + jlf barbon and e rabinovici , _ closed - string tachyons and the hagedorn transition in ads space _ , hep - th/0112173 . + s nam , sang - jin sin , _ condensation of localized tachyons and spacetime supersymmetry _ , hep - th/0201132 . sp de alwis , at flournoy , _ closed string tachyons and semi - classical instabilities _ , hep - th/0201185 . + a font , a hernandez , _ non - supersymmetric orbifolds _ , hep - th/0202057 . + sang - jin sin , _ tachyon mass , c - function and counting localized degrees of freedom _ , hep - th/0202097 . t sayama , _ closed string tachyons in non - supersymmetric heterotic theories _ , jhep * 08 * ( 2001 ) 037 ; hep - th/0106079 . + to3em__melvin background in heterotic theories _ _ , nucl . b621 * ( 2002 ) 235 - 256 ; hep - th/0107116 . + to3em__properties of string theory in kaluza - klein melvin background _ _ , hep - th/0110077 . + to3em__charged tachyons and gauge symmetry breaking _ _ , jhep * 02 * ( 2002 ) 033 ; hep - th/0112101 . m berkooz , mr douglas , rg leigh , _ branes intersecting at angles _ , nucl . * b480 * ( 1996 ) 265 [ hep - th/9606139 ] . + mm sheikh - jabbari , _ classification of different branes at angles _ , phys . * b420 * ( 1998 ) 279 [ hep - th/9710121 ] . + r blumenhagen , l goerlich , b kors , _ supersymmetric orientifolds in 6d with d - branes at angles _ , nucl . phys . * b569 * ( 2000 ) 209 [ hep - th/9908130 ] . + r blumenhagen , l goerlich , b kors , d lst , _ noncommutative compactifications of type i strings on tori with magnetic background flux _ , jhep * 0010 * ( 2000 ) 006 [ hep - th/0007024 ] . + g aldazabal , s franco , le ibaez , r rabadan , am uranga , _ d=4 chiral string compactifications from intersecting branes _ , hep - th/0011073 ; _ intersecting brane worlds _ , jhep 0102 ( 2001 ) 047 [ hep - ph/0011132 ] . + le ibaez , f. marchesano , r. rabadan , _ getting just the standard model at intersecting branes _ , hep - th/0105155 . + r rabadan , _ branes at angles , torons , stability and supersymmetry _ , hep - th/0107036 . + m cvetic , g shiu , am uranga , _ chiral four - dimensional n=1 supersymmetric type iia orientifolds from intersecting d6-branes _ , hep - th/0107166 ; _ three - family supersymmetric standard - like models from intersecting brane worlds _ , hep - th/0107143 . + these types of models can also be obtained as t - dual models of d - brane systems with fluxes . see , for instance , among others : + c. bachas , _ a way to break supersymmetry _ , hep - th/9503030 . + c. angelantonj , i. antoniadis , e. dudas , a. sagnotti , _ type - i strings on magnetised orbifolds and brane transmutation _ , phys . * b489 * ( 2000 ) 223 - 232 , hep - th/0007090 . tseytlin , magnetic backgrounds and tachyonic instabilities in closed string theory , hep - th/0108140 . russo , a.a . tseytlin , supersymmetric fluxbrane intersections and closed string tachyons , jhep * 0111 * ( 2001 ) 065 , hep - th/0110107 . + to3em__magnetic backgrounds and tachyonic instabilities in closed superstring theory and m - theory _ _ , nucl . * b611 * ( 2001 ) , 93124 , ` arxiv : hep - th/0104238 ` . + t. takayanagi and t.uesugi , _ orbifolds as melvin geometry _ , jhep * 0112 * ( 2001 ) 004 , hep - th/0110099 . + e dudas and j mourad , _ d - branes in string theory melvin backgrounds _ , nucl . * b622 * ( 2002 ) , 4672 , ` arxiv : hep - th/0110186 ` . + t takayanagi and t uesugi , _ d - branes in melvin background _ , j. high energy phys . * 11 * ( 2001 ) , 036 , ` arxiv : hep - th/0110200 ` . + ms costa and m gutperle , _ the kaluza klein melvin solution in m - theory _ , j. high energy phys . * 03 * ( 2001 ) , 027 , ` arxiv : hep - th/0012072 ` . + f dowker , jp gauntlett , gw gibbons , and gt horowitz , _ the decay of magnetic fields in kaluza klein theory _ , phys . rev . * d52 * ( 1995 ) , 69296940 , ` arxiv : hep - th/9507143 ` . + f dowker , jp gauntlett , sb giddings , and gt horowitz , _ on pair creation of extremal black holes and kaluza klein monopoles _ , phys . rev . * d50 * ( 1994 ) , 26622679 , ` arxiv : hep - th/9312172 ` . + r emparan , _ tubular branes in fluxbranes _ , nucl . * b610 * ( 2001 ) , 169189 , ` arxiv : hep - th/0105062 ` . + m gutperle and a strominger , _ fluxbranes in string theory _ , j. high energy phys . * 06 * ( 2001 ) , 035 , ` arxiv : hep - th/0104136 ` . + ms costa , car herdeiro , and l cornalba , _ fluxbranes and the dielectric effect in string theory _ , nucl . b619 * ( 2001 ) 155 - 190 , ` arxiv : hep - th/0105023 ` . + pm saffin , _ gravitating fluxbranes _ , phys . rev . * d64 * ( 2001 ) , 024014 , ` arxiv : gr - qc/0104014 ` . + r emparan and m gutperle , _ from p - branes to fluxbranes and back _ , jhep * 0112 * 023 ( 2001 ) [ arxiv : hep - th/0111177 ] . + jr david , m gutperle , m headrick and s minwalla , _ closed string tachyon condensation on twisted circles _ , jhep * 0202 * ( 2002 ) 041 [ arxiv : hep - th/0111212 ] . + j. m. figueroa - ofarrill and j. simn , _ generalised supersymmetric fluxbranes _ , jhep * 0112 * ( 2001 ) 011 , hep - th/0110170 .

we discuss the lift of certain d6-antid6-brane systems to m - theory . these are purely gravitational configurations with a bolt singularity . when reduced along a trivial circle , and for large bolt radius , the bolt is related to a non - supersymmetric orbifold type of singularity where some closed string tachyons are expected in the twisted sectors . this is a kind of open - closed string duality that relates open string tachyons on one side and localised tachyons in the other . we consider the evolution of the system of branes from the m - theory point of view . this evolution gives rise to a brane - antibrane annihilation on the brane side . on the gravity side , the evolution is related to a reduction of the order of the orbifold and to a contraction of the bolt to a nut or flat space if the system has non - vanishing or vanishing charge , respectively . we also consider the inverse process of reducing a non - supersymmetric orbifold to a d6-brane system . for @xmath0 , the reduced system is a fractional d6-brane at an orbifold singularity @xmath1 . cern/2002 - 066 + wis/12/02-mar - dpp + hep - th/0203243 + 0.3 cm 10.mm * ral rabadn @xmath2 and joan simn @xmath3 * + @xmath4 theory division cern , + ch-1211 genve 23 , switzerland + .5 cm @xmath5 department of particle physics , the weizmann institute of science + herzl street 2 , 76100 rehovot , israel + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ' '' '' width 5.cm 2.mm @xmath6 e - mail : raul.rabadan@cern.ch + @xmath7 e - mail : jsimon@weizmann.ac.il + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

molecular hydrogen is the most abundant molecule in the universe and the main constituent of regions where stars are forming . h@xmath2 plays an important role in the chemistry of the interstellar medium , and its formation governs the transformation of atomic diffuse clouds into molecular clouds . because of the inefficient gas phase routes to form h@xmath2 , dust grains have been recognized to be the favored habitat to form h@xmath2 molecules ( @xcite , @xcite ) . the sticking of h atoms onto surfaces has received considerable attention because this mechanism governs the formation of h@xmath2 , but also other molecules that contain h atoms . the sticking of h atoms onto dust grains can also be an important mechanism to cool interstellar gas ( @xcite ) . in the past decades , a plethora of laboratory experiments and theoretical models have been developed to understand how h@xmath2 forms . as h atoms arrive on dust surfaces , they can be weakly ( physisorbed ) or strongly ( chemisorbed ) bound to the surface . the sticking of h in the physisorbed state ( @xcite , @xcite , @xcite ; @xcite ) and in the chemisorbed state ( @xcite ; @xcite ; @xcite ) has been highlighted by several experiments on different types of surfaces ( amorphous carbon , silicates , graphite ) . in the ism , dust grains are mainly carbonaceous or silicate particles with various sizes and represent an important surface for the formation of h@xmath2 . however , a large part ( @xmath3 50@xmath4 ) of the available surface area for chemistry is in the form of very small grains or pahs ( @xcite ) . these pahs are predicted to have characteristics similar to graphite surfaces : however , once the first h atom is chemisorbed on the basal plane , subsequent adsorptions of h atoms in pairs appear to be barrierless for the para dimer and with a reduced barrier for the ortho dimer ( @xcite ) . h@xmath2 can then form by involving a pre - adsorbed h atom in monomer ( @xcite ; @xcite ; @xcite ; @xcite ) or in a para - dimer configuration ( @xcite ) . however , while these routes represent efficient paths to form h@xmath2 , the inefficient sticking of h atoms in monomers constitutes an important obstacle to enter the catalytic regime for h@xmath2 formation . this results in a very low h@xmath2 formation efficiency on graphitic / pah surfaces ( @xcite ) . the hydrogenation on the pah edges has been identified as an important route to form h@xmath2 in the ism ( @xcite ; @xcite ; @xcite ; @xcite ; @xcite ) . density functional theory calculations have shown that the first hydrogenation of neutral coronene is associated with a barrier ( @xmath360 mev ) but that subsequent hydrogenation barriers vanish ( @xcite ) . recently , coronene films exposed to h / d atoms at high temperature , were studied by means of ir spectroscopy ( @xcite ) and mass spectrometry ( @xcite ) . these measurements showed that neutral pahs , when highly hydrogenated , are efficient catalysts for the formation of h@xmath2 , and confirmed the high h@xmath2 formation rate attributed to pahs in pdrs ( @xcite ) . pah cations , which are usually present at lower extinction a@xmath5 , and therefore reside at the surfaces of pdrs , also represent an important route to form h@xmath2 ( @xcite ; @xcite ) . the addition of the first h atom is predicted to be barrierless . this reaction is exothermic but the product should be stabilized by ir emission . a second h atom can react with the already adsorbed h to form h@xmath2 without a barrier ( @xcite ; @xcite ) . in this letter , we study experimentally the hydrogenation of coronene cations in the gas phase through exposure to hydrogen atoms . by using mass spectrometry , we show that odd hydrogenation states of coronene cations predominantly populate the mass spectrum . our results highlight the fact that the further hydrogenation of pah cations is associated with a barrier if the number already attached h atoms is odd , and no barrier if this number is even . this alternanting barrier - no barrier occurence seems to remain with increasing hydrogenation . these results suggest that pah cations can also enjoy highly hydrogenated states in the interstellar medium , and acts as catalysts for h@xmath2 formation . in this pilot experiment we show the feasibility of studying the hydrogenation of pahs in the gas phase . for this purpose , we use a setup designed to study molecular ions in a radiofrequency ion trap . time - of - flight mass spectrometry of the trap content is used to identify the changes in mass of the coronene cations and therefore deduce their respective degrees of hydrogenation . the experiments have been performed using a home - built tandem - mass spectrometer shown schematically in figure [ fig : setup ] ( @xcite ) . a beam of singly charged coronene radical cations ( [ c@xmath6h@xmath7@xmath8 , m / z 300 ) was extracted from an electrospray ion source . the ions were phase - space compressed in an radiofrequency ( rf ) ion funnel and subsequently in an rf quadrupole ion guide . mass selection was accomplished by using an rf quadrupole mass filter . accumulation of the ions took place in a three dimensional rf ion trap ( paul trap ) . a he buffer gas at room temperature was used to collisionally cool the trapped cations . exposure to gas - phase atomic hydrogen for variable periods of time led to multiple hydrogen adsorption on the coronene cations . an electric extraction field was then applied between the trap end - caps to extract the trapped hydrogenated cations into a time - of - flight ( tof ) mass spectrometer with resolution m/@xmath9 m @xmath3 200 . to obtain mass spectra of sufficient statistics , typically a couple of hundred tof traces were accumulated . electrospray ionization allows to gently transfer ions from the liquid phase into the gas phase . inspired by the method of @xcite we have run the ion source with a solution consisting of 600 @xmath10l of saturated solution of coronene in methanol , 350 @xmath10l of hplc grade methanol and 50 @xmath10l of 10 mm solution of @xmath11 solution in methanol . in the liquid phase , electron transfer from a coronene molecule to a silver ion leads to formation of the required radical cation . the trapped ions are exposed to hydrogen atoms produced from h@xmath2 by a slevin type source which has been extensively used in crossed beam experiments ( @xcite,@xcite ) . while in the earlier work the dissociation fractions were determined by means of electron impact excitation or heii line emission , we now use charge removal ( captured ionization ) and dissociation induced by 40 kev he@xmath12 . for these processes the cross sections are well - known ( @xcite ) . in this way we determine a hydrogen dissociation fraction of @xmath13 . the temperature of the h beam is around room temperature ( @xmath325 mev ) . coronene ions are exposed to a constant flux of h atoms for different periods of time before their degree of hydrogenation is determined by means of mass spectrometry . the irradiation time is varied from 1.0 up to 30 s to study the time - dependence of coronene hydrogenation . the data obtained from our experiment are a series of mass spectra of hydrogenated coronene cations as a function of h exposure time . some of the spectra are shown in fig.[fig : rawdata ] . fig.[fig : rawdata](a ) shows the mass spectrum of the native m / z=300 coronene cations . a similar , thus unchanged , mass spectrum is obtained ( not shown in this article ) if we irradiate coronene cations with molecular hydrogen . this means that molecular hydrogen does not stick to coronene cations at room temperature . after turning on the hydrogen source and exposing the coronene cations to the atomic hydrogen beam for 1.0 s ( fig.[fig : rawdata ] , ( b ) ) , the peak at @xmath14 shifts to 301 , which means that the trap content main constituent is ( c@xmath6h@xmath15+h)@xmath8 . for increasing irradiation time ( fig.[fig : rawdata](c ) t= 2 s , ( d ) 3 s , ( e ) 4 s and ( f ) 4.75 s ) , the peak at @xmath16=301 disappears progressively while a peak at @xmath17 and then at @xmath18 ( for t = 4.75 s see fig.[fig : rawdata](f ) ) appears , which indicates the addition of 3 and 5 hydrogen atoms , respectively . at longer exposure time ( fig.[fig : longdata](a ) t @xmath315 s ) , the @xmath16=303 peak dominates the signal , and a peak at @xmath16=305 appears . at even longer irradiation times ( fig.[fig : longdata](b ) t @xmath330 s ) , the peak @xmath16=305 dominates and peaks at @xmath16=307 and 309 appear . these peaks clearly show the evolution of the hydrogenation states of coronene cations with h irradiation time . our results show that the most important peaks measured in the mass spectrum shift from lower masses to higher masses with increasing h exposure time . in order to follow the evolution of the first hydrogenated state of coronene cation ( c@xmath6h@xmath15+h)@xmath8 ( corh@xmath8 ) to the second ( c@xmath6h@xmath15 + 2h)@xmath8 ( corh@xmath19 ) , third ( corh@xmath20 ) and fourth ( corh@xmath21 ) hydrogenated states , we use a simple model that describes this evolution : @xmath22 @xmath23 @xmath24 @xmath25 hydrogenation of corh@xmath26@xmath8 follows an arrhenius expression where a@xmath27 is the prefactor and e@xmath27 is the barrier , while hydrogenation of corh@xmath28 follows the same expression with a prefactor a@xmath26 and no barrier . k@xmath29 is the boltzmann constant and t the temperature of the h beam ( t@xmath325 mev ) . in these equations we do not include abstraction , meaning that the time evolution of the contribution of each state is governed entirely by hydrogenation . this assumption is made in order to derive the first barriers of hydrogenation . abstraction can be neglected in the conditions of our experiments for low exposure times . this is supported by previous experiments where the cross section for addition of hydrogen to neutral coronene is predicted to be 20 times that for abstraction ( @xcite ) . further support is drawn from a kinetic chemical model we developed , which shows that abstraction has to be very low compared to hydrogenation to be able to mimic the experimental results ( boschman et al . in prep ) . however , for long h exposure time we expect the hydrogenation degree of the coronene cations to reach a steady state which will allow us to derive the contribution of abstraction relative to addition , and therefore derive the h@xmath0 formation rate due to pah cations . it should also be kept in mind that in the conditions of our experiments , the h atoms are at room temperature meaning that they cross the barriers for abstraction ( 10 mev , @xcite ) and addition ( 40 - 60 mev , @xcite ) with similar ease . under interstellar conditions , however , the abstraction will dominate by 8 orders of magnitude ( at 20 k ) because of the barrier differences . the first hydrogenation is expected to take place at the outer edge carbon atom ( @xcite ) . this state provides more conformational freedom to the four neighbouring outer edge carbon atoms , ensuring a preference for the second hydrogenation to take place at one of those four carbon atoms . the third hydrogenation will preferentially take place at the outer edge carbon next to the second h atom . again , the forth h atom can be bound to one of the four neighbouring outer edge carbon atoms , and the fifth sticks on the neighboring outer edge carbon . this scenario of h atoms sticking preferentially on outer edge carbons next to already adsorbed atoms is described in @xcite . the contribution of every peak is determined by fitting our data with gaussians with identical widths ( see fig.[fig : fit](a ) ) . the ratios between different hydrogenation states as function of time are reported in fig.[fig : fit](b ) . it appears that the ratio between the contribution of the first ( corh@xmath8 ) and the second ( corh@xmath19 ) hydrogenation state does not evolve with time for short time scales @xmath30 . also , the ratio between the third ( corh@xmath20 ) and the forth ( corh@xmath21 ) hydrogenation state shows identical behaviour after t@xmath31 2s @xmath32 . before this exposure time the n@xmath33 and n@xmath34 signals are very weak , and the ratio is uncertain . we can therefore assume that for these measurements @xmath35 and @xmath36 . the expression for the corh@xmath37 to corh@xmath38 as well as for the corh@xmath39 to corh@xmath40 energy barriers can then be written as : @xmath41 @xmath42 from these expressions we derive the energy barrier e@xmath2 as 72@xmath16 mev and e@xmath43 as 43@xmath18 mev , as shown in fig.[fig : fit](c ) . this shows that hydrogenation barriers are decreasing with increasing hydrogenation . however , our results also show that odd hydrogenated states dominate the mass spectrum even for high degrees of hydrogenation ( fig.[fig : longdata ] ) . this highlights the presence of a barrier - no barrier alternation from one hydrogenated state to another , up to high hydrogenation states . so our results indicate that even if the hydrogenation barriers decrease for the first hydrogenations , they do not vanish completely and remain at higher hydrogenation states . the barriers derived in our study are similar to the one calculated by @xcite for neutral coronene . this means that the first hydrogenations of coronene cations should be comparable to the hydrogenation of neutral coronene . however , for higher degree of hydrogenation we show that these barriers still exist , while the calculations from @xcite predict that these barriers vanish after a few hydrogenations . recent mass spectrometric measurements of coronene films exposed to h / d atoms do not show preferences for even or odd hydrogenation states of neutral coronene ( @xcite ) . however , these measurements are not very sensitive to barrier heights well bellow 100 mev , since the experiments were performed with atoms at beam temperature of 170 mev . in pdrs exposed to uv fields less than few hundreds g@xmath44 , the spatial distribution of h@xmath2 and pahs does correlate ( @xcite , @xcite , @xcite ) contrary to what is seen in the presence of strong uv fields ( @xcite , @xcite ) . the h@xmath2 formation rates have been derived for several pdrs exposed to various uv radiation fields . these rates can be explained by the contribution of pahs to the formation of h@xmath2 ( @xcite ) . depending on the uv intensity , the pahs observed can either be pah cations , that are present in regions at low visual exctinctions a@xmath5 , or neutral pahs , which are located at higher extinctions . work by @xcite and @xcite has shown that high - uv and high density pdrs ( n@xmath45@xmath3110@xmath46 @xmath47 and g@xmath44@xmath31100 , g@xmath48 ) can maintain a @xmath3 30@xmath4 cationic fraction upto a few mag in a@xmath5 . more relevant to this work , @xcite have studied low - uv pdrs ( g@xmath44@xmath49100 ) , and followed the pah charge balance for different densities , uv radiation fields and metallicities . they found that pah cations dominate over neutrals and anions for a@xmath5@xmath492 mag . the h@xmath2 formation rates observed in pdrs exposed to different uv fields can therefore be partly attributed to neutral and cationic pahs . our results show that the hydrogenation processes of neutral and cationic pahs are similar and should contribute similarly to the formation of h@xmath2 . further experimental investigations will allow us to derive the h@xmath2 formation rate for pah cations . we have investigated the addition of hydrogen atoms to coronene cations in the gas phase and observed increasing hydrogenation with h exposure time . our results show that odd hydrogenated states dominate the mass spectrum , which evidences the presence of a barrier for the further hydrogenation of odd hydrogenation states . the first hydrogen sticks to the coronene cations without a barrier ( @xcite , @xcite ) . the second and forth hydrogenations are associated with barriers of about 72 @xmath1 6 mev and 43 @xmath1 8 mev , while the third and fifth hydrogenation are barrierless . these barriers are similar to the one calculated for neutral coronene ( @xcite ) . our results indicate that superhydrogenated pah cations ( @xcite ) should also be found in the interstellar medium , and be important catalysts for the formation of h@xmath2 , as it is the case for their neutral counterparts . l. b. and s. c. are supported by the netherlands organization for scientific research ( nwo ) . g.r . recognizes the funding by the nwo dutch astrochemistry network . we would like to thank the anonymous referee for the helpful comments .

molecular hydrogen is the most abundant molecule in the universe . a large fraction of h@xmath0 forms by association of hydrogen atoms adsorbed on polycyclic aromatic hydrocarbons ( pahs ) , where formation rates depend crucially on the h sticking probability . we have experimentally studied pah hydrogenation by exposing coronene cations , confined in a radiofrequency ion trap , to gas phase atomic hydrogen . a systematic increase of the number of h atoms adsorbed on the coronene with the time of exposure is observed . odd coronene hydrogenation states dominate the mass spectrum up to 11 h atoms attached . this indicates the presence of a barrier preventing h attachment to these molecular systems . for the second and fourth hydrogenation , barrier heights of 72 @xmath1 6 mev and 40 @xmath1 10 mev , respectively are found which is in good agreement with theoretical predictions for the hydrogenation of neutral pahs . our experiments however prove that the barrier does not vanish for higher hydrogenation states . these results imply that pah cations , as their neutral counterparts , exist in highly hydrogenated forms in the interstellar medium . due to this catalytic activity , pah cations and neutrals seem to contribute similarly to the formation of h@xmath0 .

as is well known , the atom or atoms in the atomic clock are passive they do not `` tick''so the clock needs an active oscillator in addition to the atom(s ) . in designing an atomic clock to realize the second as a measurement unit in the international system of units ( si ) , one encounters two problems : ( a ) the resonance exhibited by the atom or atoms of the clock varies with the details of the clock s construction and the circumstances of its operation ; in particular the resonance shifts depending on the intensity of the radiation of the atoms by the oscillator . ( b ) the oscillator , controlled by , in effect , a knob , drifts in relation to the knob setting . problem ( a ) is dealt with by introducing a wave function parametrized by radiation intensity and whatever other factors one deems relevant . the si second is then `` defined '' by the resonance that `` would be found '' at absolute zero temperature ( implying zero radiation ) . for a clock using cesium 133 atoms , this imagined resonance is declared by the general conference of weights and measures to be 9 192 631 770 hz , so that the si second is that number of cycles of the radiation at that imagined resonance @xcite . to express the relation between a measured resonance and the imagined resonance at 0 k , a wave function is chosen . problem ( b ) is dealt with by computer - mediated feedback that turns the knob of the oscillator in response to detections of scattering of the oscillator s radiation by the atom(s ) of the clock , steering the oscillator toward an aiming point . a key point for this paper is that the wave function incorporated into the operation of an atomic clock can never be unconditionally known . the language of quantum theory reflects within itself a distinction between ` explanation ' and ` evidence ' . for explanations it offers the linear algebra of wave functions and operators , while for evidence it offers probabilities on a set of outcomes . outcomes are subject to quantum uncertainty , but uncertainty is only the tip of an iceberg : how can one `` know '' that a wave function describes an experimental situation ? the distinction within quantum theory between linear operators and probabilities implies a gap between any explanation and the evidence explained . @xcite : [ prop : one ] to choose a wave function to explain experimental evidence requires reaching beyond logic based on that evidence , and evidence acquired after the choice is made can call for a revision of the chosen wave function . because no wave function can be unconditionally known , not even probabilities of future evidence can be unconditionally foreseen . here we show implications of the unknowability of wave functions for the second as a unit of measurement in the international system ( si ) , implications that carry over to both digital communications and to the use of a spacetime with a metric tensor in explaining clock readings at the transmission and reception of logical symbols . clocks that generate universal coordinated time ( utc ) are steered toward aiming points that depend not only on a chosen wave function but also on an hypothesized metric tensor field of a curved spacetime . like the chosen wave function , the hypothesis of a metric tensor is constrained , but not determined , by measured data . guesses enter the operations of clocks through the computational machinery that steers them . taking incoming data , the machinery updates records that determine an aiming point , and so involves the writing and reading of records . the writing must take place at a phase of a cycle distinct from a phase of reading , with a separation between the writing and the reading needed to avoid a logical short circuit . in sec . [ sec : turing ] we picture an explanation used in the operation of a clock as a string of characters written on a tape divided into squares , one symbol per square . the tape is part of a turing machine modified to be stepped by a clock and to communicate with other such machines and with keyboards and displays . we call this modified turing machine an _ open machine_. the computations performed by an open machine are open to an inflow numbers and formulas incalculable prior to their entry . because a computer cycles through distinct phases of memory use , the most direct propagation of symbols from one computer to another requires a symbol from one computer to arrive during a suitable phase of the receiving computer s cycle . in sec . [ sec : phasing ] we elevate this phase dependence to a principle that defines the _ logical synchronization _ necessary to a _ channel _ that connects clock readings at transmission of symbols to clock readings at their reception recognizing the dependence of logic - bearing channels on an interaction between evidence and hypotheses about signal propagation engenders several types of questions , leading to a _ discipline of logical synchronization _ , outlined in sec . [ sec : patterns ] . the first type of question concerns patterns of channels that are possible aiming points , as determined in a blackboard calculation that assumes a theory of signal propagation . [ sec : typei ] addresses examples of constraints on patterns of channels under various hypotheses of spacetime curvature , leading to putting `` phase stripes '' in spacetime that constrain channels to or from a given open machine . an example of a freedom to guess an explanation within a constraint of evidence is characterized by a subgroup of a group of clock adjustments , and a bound on bit rate is shown to be imposed by variability in spacetime curvature . [ sec : adj ] briefly addresses the two other types of questions , pertaining not to _ hypothesizing _ possible aiming points ` on the blackboard ' , but to _ using _ hypothesized aiming points , copied into feedback - mediating computers , for the steering of drifting clocks . after discussing steering toward aiming points copied from the blackboard , we note occasions that invite revision of a hypothesized metric tensor and of patterns of channels chosen as aiming points . computer - mediated feedback , especially as used in an atomic clock , requires logic open to an inflow of inputs beyond the reach of calculation . to model the logic of a computer that communicates with the other devices in a feedback loop , we modify a turing machine to communicate with external devices , including other such machines . the turing machine makes a record on a tape marked into squares , each square holding one character of an alphabet . operating in a sequence of ` moments ' interspersed by ` moves ' , at any moment the machine scans one square of the tape , from which it can read , or onto which it can write , a single character . a move as defined in the mathematics of turing machines consists ( only ) of the logical relation between the machine at one moment and the machine at the next moment @xcite , thus expressing the logic of a computation , detached from its speed ; however , in a feedback loop , computational speed matters . let the moves of the modified turing machine be stepped by ticks of a clock . a step occurs once per a period of revolution of the clock hand . this period is adjustable , on the fly . we require that the cycle of the modified turing machine correspond to a unit interval of the readings of its clock . to express communication between open machines as models of computers , the modified turing machine can receive externally supplied signals and can transmit signals , with both the reception and the transmission geared to the cycle of the machine . in addition , the modified turing machine registers a count of moments at which signals are received and moments at which signals are transmitted . at a finer scale , _ the machine records a phase quantity in the cycle of its clock , relative to the center of the moment at which a signal carrying a character arrives . _ we call such a machine an _ open machine_. an open machine can receive detections and can command action , for instance the action of increasing or decreasing the frequency of the variable oscillator of an atomic clock . calculations performed on an open machine communicating with detectors and actuators proceed by moves made according to a rule that can be modified from outside the machine in the course of operation . these calculations respond to received influences , such as occurrences of outcomes underivable from the contents of the machine memory , when the open machine writes commands on a tape read by an external actuator . the wider physical world shows up in an open machine as both ( 1 ) unforeseeable messages from external devices and ( 2 ) commands to external devices . we picture a real - time computer in a feedback loop as writing records on the tape of an open machine . the segmentation into moments interspersed by moves is found not just in turing machines but in any digital computer , which implies the logical result of any computation is oblivious to variations in speed at which the clock steps the computer . + corollary 2.1 . _ no computer can sense directly any variation in its clock frequency . _ although it can not directly sense variation in the tick rate of its clock , the logic of open machine stepped by an atomic clock can still control the adjustment of the clock s oscillator by responding to variations in the detection rate written moment by moment onto its turing tape . a flow of unforeseeable detections feeds successive computations of results , each of which , promptly acted on , impacts probabilities of subsequent occurrences of outcomes , even though those subsequent outcomes remain unforeseeable . the computation that steers the oscillator depends not just on unforeseeable inputs , but also on a steering formula encoded in a program . * remarks * : 1 . to appreciate feedback , take note that a formula is distinct from what it expresses . for example a formula written along a stretch of a turing tape as a string of characters can contain a name @xmath0 for wave function as a function of time variable @xmath1 and space variables . the formula , containing @xmath0 , once written , just `` sits motionless , '' in contrast to the motion that the formula expresses . 2 . although unchanged over some cycles of a feedback loop , a feedback loop operates in a larger context , in which steering formulas are subject to evolution . sooner or later , the string that defines the action of an algorithm , invoking a formula , is apt to be overwritten by a string of characters expressing a new formula . occasions for rewriting steering formulas are routine in clock networks , including those employed in geodesy and astronomy . logical communication requires clocking . the reading of a clock of an open machine @xmath2an @xmath2-reading has the form @xmath3 where @xmath4 indicates the count of cycles and @xmath5 is the phase within the cycle , with the convention that @xmath6 . we define a channel from @xmath2 to @xmath7 , denoted @xmath8 , as a set of pairs , each pair of the form @xmath9 . the first member @xmath3 is an @xmath2-reading at which machine @xmath2 can transmit a signal and @xmath10 is a @xmath7-reading at which the clock of machine @xmath7 can register the reception of the signal . define a _ repeating channel _ to be a channel @xmath8 such that @xmath11)(\exists m , n , j , k ) ( m+\ell j.\phi_{a,\ell},n+\ell k.\phi_{b,\ell } ) ] \in \abr,\ ] ] for theoretical purposes , it is convenient to define an _ endlessly repeating channel _ for which @xmath12 ranges over all integers . again for theoretical purposes , on occasion we consider channels for which the phases are all zero , in which case we may omit writing the phases . because they are defined by local clocks without reference to any metric tensor , channels invoke no assumption about a metric or even a spacetime manifold . for this reason evidence from the operation of channels is independent of any explanatory assumptions involving a manifold with metric and , in particular , is independent of any global time coordinate , or any `` reference system '' @xcite . thus clock readings at the transmission and the reception of signals can prompt revisions of hypotheses about a metric tensor field . a record format for such evidence was illustrated in earlier work @xcite , along with the picturing of such records as _ occurrence graphs_. from the beating of a heart to the bucket brigade , life moves in phased rhythms . for a symbol carried by a signal from an open machine @xmath2 to be written into the memory of an open machine @xmath7 , the signal must be available at @xmath7 within a phase of the cycle of @xmath7 during which writing can take place , and the cycle must offer room for a distinct other phase . we elevate engineering commonplace to a principle pertaining to open machines as follows . [ prop : three ] a logical symbol can propagate from one open machine to another only if the symbol arrives within the writing phase of the receiving machine ; in particular , respect for phasing requires that for some positive @xmath13 any arrival phase @xmath14 satisfy the inequality @xmath15 prop . [ prop : three ] serves as a fixed point to hold onto while hypotheses about signal propagation in relation to channels are subject to revision . we call the phase constraint on a channel asserted by ( [ eq : main ] ) _ logical synchronization_. for simplicity and to allow comparing conditions for phasing with conditions for einstein synchronization , we take the engineering liberty of allowing transmission to occur at the same phase as reception , so that both occur during a phase interval satisfying ( [ eq : main ] ) . the alternative of demanding reception near values of @xmath16 can be carried out with little extra difficulty . + * remarks : * 1 . note that @xmath14 in the proposition is a phase of a cycle of a variable - rate clock that is _ not _ assumed to be in any fixed relation to a proper clock as conceived in general relativity . indeed , satisfying ( [ eq : main ] ) usually requires the operation of clocks at variable rates . the engineering of communications between computers commonly detaches the timing of a computer s receiver from that of the computer by buffering : after a reception , the receiver writes into a buffer that is later read by the computer@xcite . in analyzing open machines we do without buffering , confining ourselves to character - by - character phase meshing as asserted in prop . [ prop : three ] , which offers the most direct communication possible . given the definition of a channel and the condition ( [ eq : main ] ) essential to the communication of logical symbols , three types of questions arise : * type i : * what patterns of interrelated channels does one try for as aiming points ? * type ii : * how can the steering of open machines be arranged to approach given aiming points within acceptable phase tolerances ? * type iii : * how to respond to deviations from aiming points beyond tolerances ? such questions point the way to exploring what might be called a _ discipline of logical synchronization_. so far we notice two promising areas of application within this discipline : 1 . provide a theoretical basis for networks of logically synchronized repeating channels , highlighting 1 . possibilities for channels with null receptive phases as a limiting case of desirable behavior , and 2 . circumstances that force non - null phases . 2 . explore constraints on receptive phases imposed by gravitation , as a path to exploring and measuring gravitational curvature , including slower changes in curvature than those searched for by the laser gravitational wave observatory @xcite . answers to questions of the above types require hypotheses , if only provisional , about signal propagation . for this section we assume that propagation is described by null geodesics in a lorentzian 4-manifold @xmath17 with one or another metric tensor field @xmath18 , as in general relativity . following perlick @xcite we represent an open machine as a timelike worldline , meaning a smooth embedding @xmath19 from a real interval into @xmath17 , such that the tangent vector @xmath20 is everywhere timelike with respect to @xmath18 and future - pointing . we limit our attention to worldlines of open machines that allow for signal propagation between them to be expressed by null geodesics . to say this more carefully , we distinguish the _ image _ of a worldline as a submanifold of @xmath17 from the worldline as a mapping . consider an open region @xmath21 of @xmath17 containing a smaller open region @xmath22 , with @xmath21 containing the images of two open machines @xmath2 and @xmath7 , with the property that every point @xmath23 of the image of @xmath2 restricted to @xmath22 is reached uniquely by one future - pointing null geodesic from the image of @xmath7 in @xmath21 and by one past - pointing null geodesic from the image of @xmath7 in @xmath21 . we then say @xmath2 and @xmath7 are _ radar linkable _ in @xmath22 . we limit our attention to open machines that are radar linkable in some spacetime region @xmath22 . in addition we assume that the channels preserve order ( what is transmitted later arrives later ) . indeed , we mostly deal with open machines in a gently curved spacetime region , adequately described by fermi normal coordinates around a timelike geodesic . for simplicity and to allow comparing conditions for phasing with conditions for einstein synchronization , we take the liberty of allowing transmission to occur at the same phase as reception , so that both occur during a phase interval satisfying ( [ eq : main ] ) . the perhaps more realistic alternative of demanding reception near values of @xmath16 can be carried out with little difficulty . to develop the physics of channels , we need to introduce three concepts : \(1 ) we define a _ group of clock adjustments _ as transformations of the readings of the clock of an open machine . as it pertains to endlessly repeating channels , a group @xmath24 of clock adjustments consists of functions on the real numbers having continuous , positive first derivatives . group multiplication is the composition of such functions , which , being invertible , have inverses . to define the action of @xmath24 on clock readings , we speak ` original clock readings ' as distinct from adjusted readings an adjustment @xmath25 acts by changing every original reading @xmath26 of a clock @xmath2 to an adjusted reading @xmath27 . as we shall see , clock adjustments can affect echo counts . \(2 ) to hypothesize a relation between the @xmath2-clock and an accompanying proper clock , one has to assume one or another metric tensor field @xmath18 , relative to which to define proper time increments along @xmath2 s worldline ; then one can posit an adjustment @xmath28 such that @xmath29 where @xmath30 is the reading imagined for the accompanying proper clock when @xmath2 reads @xmath26 . \(3 ) we need to speak of positional relations between open machines . for this section we assume that when an open machine @xmath7 receives a signal from any other machine @xmath2 then @xmath7 echoes back a signal to @xmath2 right away , so the echo count @xmath31 defined in sec . [ sec : phasing ] involves no delay at @xmath7 . in this case , evidence in the form of an echo count becomes explained , under the assumption of a metric tensor field @xmath18 , as being just twice the radar distance @xcite from @xmath2 to the event of reception by @xmath7 . questions of type i concern constraints on channels imposed by the physics of signal propagation . here we specialize to constraints on channels imposed by spacetime metrics , constraints obtained from mathematical models that , while worked out so to speak on the blackboard , can be copied onto turing tapes as aiming points toward which to steer the behavior of the clocks of open machines . questions of types ii and iii are deferred to the sec . [ sec : adj ] . we begin by considering just two machines . assuming an hypothetical spacetime @xmath32 , suppose that machine @xmath2 is given as a worldline parametrized by its clock readings : what are the possibilities and constraints for an additional machine @xmath7 with two - way repeating channels @xmath8 and @xmath33 with a constant echo count ? we assume the idealized case of channels with null phases , which implies integer echo counts . for each @xmath2-tick there is a future light cone and a past light cone . with tick events indicated ; ( b ) light cones associated to ticks of @xmath2 ; ( c ) ticks of @xmath34 and @xmath35 at light cone intersections corresponding to @xmath36 . , width=470 ] the future light cone from an @xmath2-reading @xmath37 has an intersection with the past light cone for the returned echo received at @xmath38 . [ fig:3 ] illustrates the toy case of a single space dimension in a flat spacetime by showing the two possibilities for a machine @xmath7 linked to @xmath2 by two - way channels at a given constant echo count . in each solution , the clocking of @xmath7 is such that a tick of @xmath7 occurs at each of a sequence of intersections of outgoing and incoming light cones from and to ticks of @xmath2 . note that the image of @xmath7 , and not just its clock rate , depend on the clock rate of @xmath2 . determination of the tick events for @xmath7 leaves undetermined the @xmath7 trajectory between ticks , so there is a freedom of choice . one can exercise this freedom by requiring the image of @xmath7 to be consistent with additional channels of larger echo counts . a clock adjustment of @xmath2 of the form @xmath39 for @xmath40 a positive integer increases the density of the two - way channel by @xmath40 and inserts @xmath41 events between successive @xmath7-ticks , thus multiplying the echo count by @xmath40 . as @xmath40 increases without limit , @xmath7 becomes fully specified . turning to two space dimensions , the image of @xmath7 must lie in a tube around the image of @xmath2 , as viewed in a three - dimensional space ( vertical is time ) . so any timelike trajectory within the tube will do for the image of @xmath7 . for a full spacetime of 3 + 1 dimensions , the solutions for the image of @xmath7 fall in the corresponding `` hypertube . '' the argument does not depend on flatness and so works for a generic , gently curved spacetime in which the channels have the property of order preservation . and @xmath7 freely chosen ; ( b ) light signals lacing @xmath2 and @xmath7 define tick events ; ( c ) interpolated lacings of light signals added to make @xmath36.,width=470 ] a different situation for two machines arises in case only the image of @xmath2 s worldline is specified while its clocking left to be determined . in this case the image of @xmath7 can be freely chosen , after which the clocking of both @xmath2 and @xmath7 is constrained , as illustrated in fig . [ fig:4 ] for the toy case of flat spacetime with 1 space dimension . to illustrate the constraint on clocking , we define a `` lacing '' of light signals to be a pattern of light signals echoing back and forth between two open machines as illustrated in fig . [ fig:4 ] ( b ) . for any event chosen in the image of @xmath2 , there is a lacing that touches it . in addition to choosing this event , one can choose any positive integer @xmath40 to be @xmath31 , and choose @xmath41 events in the image of @xmath2 located after the chosen event and before the next @xmath2-event touched by the lacing of light signals . the addition of lacings that touch each of the @xmath41 intermediating events corresponds to a repeating channel @xmath8 with echo count @xmath42 , along with a repeating channel @xmath33 with the same echo count @xmath43 . this construction does not depend on the dimension of the spacetime nor on its flatness , and so works also for a curved spacetime having the property of order preservation . evidence of channels as patterns of clock readings leaves open a choice of worldlines for its explanation . in the preceding example of laced channels between open machines @xmath2 and @xmath7 , part of this openness can be reflected within analysis by the invariance of the channels under a subgroup of the group of clock adjustments that `` slides the lacings , '' as follows . suppose that transmissions of an open machine @xmath2 occur at given values of @xmath2-readings . we ask about clock adjustments that can change the events of a worldline that correspond to a given @xmath2-reading . if a clock adjustment @xmath28 takes original @xmath2-readings @xmath26 to a revised @xmath2-readings @xmath27 , transmission events triggered by the original clock readings become triggered when the re - adjusted clock exhibits the _ same readings_. as registered by original readings , the adjusted transmission occurs at @xmath44 . based on this relation we inquire into the action of subgroups of @xmath45 on the readings of the clocks of two open machines @xmath2 and @xmath7 . in particular , there is a subgroup @xmath46 that expresses possible revisions of explanations that leave invariant the repeating channels with constant echo count @xmath40 . an element @xmath47 is a pair of clock adjustments that leaves the channels invariant , and such a pair can be chosen within a certain freedom . for the adjustment @xmath28 one is free to : ( a ) assign an arbitrary value to @xmath48 ; and ( b ) , if @xmath49 , then for @xmath50 , choose the value of @xmath51 at will , subject to the constraints that @xmath52 and @xmath53 is less than the original clock reading for the re - adjusted first echo from @xmath54 . with these choices , @xmath55 is then constrained so that each lacing maps to another lacing . the condition ( a ) slides a lacing along the pair of machines ; the condition ( b ) nudges additional lacings that show up in the interval between a transmission and the receipt of its echo . in this way a freedom to guess within a constraint is expressed by @xmath56 . moving to more than two machines , we invoke the _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * definition : * an _ arrangement of open machines _ consists of open machines with the specification of some or all of the channels from one to another , augmented by proper periods of the clock of at least one of the machines . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ( without specifying some proper periods , the scale of separations of one machine from another is open , allowing the arrangement to shrink without limit , thus obscuring the effect of spacetime curvature . ) although gentle spacetime curvature has no effect on the possible channels linking two open machines , curvature does affect the possible channels and their echo counts in some arrangements of five or more machines , so that the possible arrangements are a measure spacetime curvature . the way that spacetime curvature affects the possible arrangements of channels is analogous to the way surface curvature in euclidean geometry affects the ratios of the lengths of the edges of embedded graphs . the effect on ratios shows up in mappings from graphs embedded in a plane to their images on a sphere . for example , a triangle can be mapped from a plane to a generic sphere , in such a way that each edge of the triangle is mapped to an arc of the same length along a great circle on the sphere . the same holds for two triangles that share an edge , as illustrated in fig . [ fig : sphere ] , panel ( a ) ; however , the gauss curvature of the sphere implies that the complete graph on 4 vertices generically embedded in the plane , shown in panel ( b ) , can not be mapped so as to preserve all edge lengths . the property that blocks the preservation of edge ratios is the presence of an edge in the plane figure that can not be slightly changed without changing the length of at least one other edge ; we speak of such an edge as `` frozen . '' in a static spacetime , which is all we have so far investigated , a generic arrangement of 4 open machines , is analogous to the triangle on the plane in that a map to any gently curved spacetime can preserve all the echo counts . [ prop : nine]assume four open machines in a static spacetime , with one machine stepped with a proper - time period @xmath57 , and let @xmath40 be any positive integer . then , independent of any gentle riemann curvature of the spacetime , the four open machines can be arranged , like vertices of a regular tetrahedron , to have six two - way channels with null phases , with all echo counts being @xmath58 . _ proof : _ assuming a static spacetime , choose a coordinate system with all the metric tensor components independent of the time coordinate , in such a way that it makes sense to speak of a time coordinate distinct from space coordinates ( for example , in a suitable region of a schwarzschild geometry ) . let@xmath59 denote the machine with specified proper period @xmath60 , and let @xmath61 , @xmath62 , and @xmath63 denote the other three machines . for @xmath64 , @xmath65 , we prove the possibility , independent of curvature , of the channels @xmath66 require that each of four machines be located at some fixed spatial coordinate . because the spacetime is static , the coordinate time difference between a transmission at @xmath59 and a reception at any other vertex @xmath67 ( a ) is independent of the value of the time coordinate at transmission and ( b ) is the same as the coordinate time difference between a transmission at @xmath67 and a reception at @xmath59 . for this reason any one - way repeating channel of the form ( [ eq : vs ] ) can be turned around to make a channel in the opposite direction , so that establishing a channel in one direction suffices . for transmissions from any vertex to any other vertex , the coordinate - time difference between events of transmission equals the coordinate time difference between receptions . a signal from a transmission event on @xmath59 propagates on an expanding light cone , while an echo propagates on a light cone contracting toward an event of reception on @xmath59 . under the constraint that the echo count is @xmath58 , ( so the proper duration from the transmission event to the reception event for the echo is @xmath68 ) , the echo event must be on a 2-dimensional submanifold a sphere , defined by constant radar distance @xmath69 of its points from @xmath59 with transmission at a particular ( but arbitrary ) tick of @xmath59 . in coordinates adapted to a static spacetime , this sphere may appear as a `` potatoid '' in the space coordinates , with different points on the potatoid possibly varying in their time coordinate . the potatoid shape corresponding to an echo count of @xmath58 remains constant under evolution of the time coordinate . channels from @xmath59 to the other three vertices involve putting the three vertices on this potatoid . put @xmath61 anywhere on the `` potatoid '' . put @xmath62 anywhere on the ring that is intersection of potatoid of echo count @xmath58 radiated from @xmath61 and that radiated from @xmath59 . put @xmath63 on an intersection of the potatoids radiating from the other three vertices . + q.e.d . according to prop [ prop : nine ] the channels , and in particular the echo counts possible for a complete graph of four open machines in flat spacetime are also possible for a spacetime of gentle static curvature , provided that three of the machines are allowed to set their periods not to a fixed proper duration but in such a way that all four machines have periods that are identical in coordinate time . the same holds if fewer channels among the four machines are specified . but for five machines , the number of channels connecting them matters . five open machines fixed to space coordinates in a static spacetime are analogous to the 4 vertices of a plane figure , in that an arrangement corresponding to an incomplete graph on five vertices can have echo counts independent of curvature , while a generic arrangement corresponding to a complete graph must have curvature - dependent relations among its echo counts . [ prop:9.5 ] assuming a static spacetime , consider an arrangement of five open machines obtained by starting with a tetrahedral arrangement of four open machines with all echo counts of @xmath58 as in prop . [ prop : nine ] , and then adding a fifth machine : independent of curvature , a fifth open machine can be located with two - way channels having echo counts of @xmath58 linking it to any three of the four machines of tetrahedral arrangement , resulting in nine two - way channels altogether . _ proof : _ the fifth machine can be located as was the machine @xmath63 , but on the side opposite to the cluster @xmath59 , @xmath61 , @xmath62 . + q.e.d . in contrast to the arrangement of 9 two - way channels , illustrated in fig . [ fig:5pt ] ( a ) consider an arrangement of 5 open machines corresponding to a complete graph on five vertices , with has ten two - way channels , as illustrated in fig . [ fig:5pt ] ( b ) . for five open machines in a generic spacetime , not all of the ten two - way channels can have the same echo counts . instead , channels in a flat spacetime as specified below can exist with about the simplest possible ratios of echo counts . label five open machines , @xmath70 , @xmath71 , @xmath72 , @xmath34 , and @xmath35 . take @xmath34 to be stepped by a clock ticking at a fixed proper period @xmath57 , letting the other machines tick at variable rates to be determined . let @xmath73 be any machine other than @xmath34 . for a flat spacetime it is consistent for the proper periods of all 5 machines to be @xmath57 , for the echo counts @xmath74 to be @xmath75 and for the echo counts @xmath76 to be @xmath77 , leading to twenty channels , conveniently viewed as in fig . [ fig:5pt ] ( b ) as consisting of ten two - way channels . [ prop : ten ] consider 5 open machines each fixed to space coordinates in a static curved spacetime in which the machines are all pairwise radar linkable , with 10 two - way channels connecting each machine to all the others ; then : 1 . allowing for the periods of the machines other than @xmath34 to vary , it is consistent with the curvature for all but one of the ten two - way channels to have null phases and echo counts as in a flat spacetime , but at least one two - way channel must have a different echo count that depends on the spacetime curvature . 2 . suppose @xmath4 of the 10 two - way links are allowed to have non - zero phases . if the spacetime does not admit all phases to be null , in generic cases the least possible maximum amplitude of a phase decreases as @xmath4 increases from 1 up to 10 . the periods of the clocks of the open machines can be taken to be the coordinate - time interval corresponding to the proper period @xmath57 at @xmath34 . _ proof : _ reasoning as in the proof of prop . [ prop : nine ] with its reference to a static spacetime shows that the same echo counts are possible as for flat spacetime _ with the exception _ that at least one of the two - way channels must be free to have a different echo count . for @xmath78 , similar reasoning shows that allowing @xmath79 machines vary in echo count allows reduction in the maximum variation from the echo counts in a flat spacetime , compared to the case in which only @xmath4 machines are allowed to vary in echo count . + q.e.d . adding the tenth two - way channel to an arrangement of five open machines effectively `` freezes '' all the echo counts . to define `` freezing '' as applied to echo counts , first take note an asymmetry in the dependence of echo counts on clock rates . consider any two machines @xmath2 and @xmath7 ; unlike echo count @xmath80 , which @xmath7 can change by changing it clock rate , the echo count @xmath31 is insensitive to @xmath7 s clock rate . an echo count @xmath31 will be said to be _ to _ @xmath7 and _ from _ @xmath2 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * definition : * an arrangement of open machines is _ frozen _ if it has an echo count to a machine @xmath7 that can not be changed slightly without changing the length of another echo count to @xmath7 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the property of being frozen is important because of the following . whether or not a frozen arrangement of open machines is consistent with an hypothesized spacetime depends on the weyl curvature of the spacetime . for example , think of the 5 open machines as carried by 5 space vehicles coasting along a radial geodesic in a schwarzschild geometry . in this example the variation of echo counts with curvature is small enough to be expressed by non - null phases of reception . in fermi normal coordinates centered midway between the radially moving open machine @xmath34 and @xmath35 one has the metric with a curvature parameter @xmath81 , where @xmath82 is the schwarzschild radial coordinate to the origin of the fermi normal coordinates , @xmath83 is the radial distance coordinate from from the center point between @xmath34 and @xmath35 , and @xmath84 and @xmath85 are transverse to the radial direction along which @xmath86 coasts@xcite . the speed of light is @xmath87 . we make the adiabatic approximation which ignores the time dependence of @xmath82 , so that in calculations to first order in curvature we take advantage of the ( adiabatically ) static spacetime by locating open machines at fixed values of @xmath88 . the metric is symmetric under rotation about the ( radially directed ) @xmath83-axis . let @xmath34 and @xmath35 be located symmetrically at positive and negative values , respectively , of the @xmath83-axis , and let @xmath89 , @xmath70 , and @xmath71 be located on a circle in the plane @xmath90 . with the five machines so located , the coordinate - time difference between transmissions is then the same as the coordinate - time difference between receptions , and the coordinate - time delay in one direction equals that in the opposite direction ( as stated in the proof of prop [ prop : nine ] ) . we construct seven two - way channels as above with null phases and show that the remaining 3 two - way channels can have the equal phases , but that this phase must be non - null with a curvature dependent amplitude @xmath91 . [ prop : eleven ] under the stated conditions , if @xmath92 is small enough so that + @xmath93 , then @xmath94 for a fixed separation @xmath95 between @xmath34 and @xmath35 , an adiabatic change in curvature imposes a constraint on bit rate possible for the channels , stemming from a lower bound on clock periods . suppose the cluster of 5 open machines is arranged so that the proper radar distance @xmath95 from @xmath34 to @xmath35 is 6,000 km , suppose the cluster descends from a great distance down to a radius of @xmath96 km from an earth - sized mass @xmath97 kg . for simplicity , assume that the positions and clock rates are continually adjusted to maintain null phases for all but the three channels @xmath98 . because @xmath99 , prop . [ prop : eleven ] implies @xmath100 , which with ( [ eq : main ] ) implies that @xmath101 . substituting the parameter values , one finds that for the phases for the channels @xmath98 to satisfy ( [ eq : main ] ) , it is necessary that @xmath102 s. if an alphabet conveys @xmath103 bits / character , the maximum bit rate for all the channels in the 5-machine cluster is @xmath104 bits / s . turning from type i to questions of type ii , we look at how the preceding `` blackboard modeling '' of clocks , expressed in the mathematical language of general relativity , get put to work when models are encoded into the open machines that manage their own logical synchronization . for questions of type ii ( and type iii ) both models that explain or predict evidence and the evidence itself , pertaining to physical clocks , come into play . models encoded into computers contribute to the steering of physical clocks in rate and relative position toward an aiming point , generating echo counts as evidence that , one acquired , can stimulate the guessing of new models that come closer to the aiming point . to express the effect of quantum uncertainty on logical synchronization , specifically on deviations from aiming points , one has to bring quantum uncertainty into cooperation with the representation of clocks by general - relativistic worldlines . this bringing together hinges on distinguishing evidence from its explanations . timelike worldlines and null geodesics in explanations , being mathematical , can have no _ mathematical _ connection to physical atomic clocks and physical signals . to make such a connection them one has to invoke the logical freedom to make a guess . within this freedom , one can resort to quantum theory to explain deviations of an atomic clock from an imagined proper clock , represented as a worldline , without logical conflict . because of quantum uncertainty and for other reasons , if an aiming point in terms of channels and a given frequency scale is to be reached , steering is required , in which evidence of deviations from the aiming point combine with hypotheses concerning how to steer @xcite . to keep things simple , consider a case of an aiming point with null phases , involving two open machines @xmath2 and @xmath7 , as in the example of sec . [ sec : typei ] , modeled by a given worldline @xmath2 with given clock readings @xmath26 , where @xmath7 aims to maintain two - way , null - phase channel of given @xmath105 . for this @xmath7 registers arriving phases of reception and adjusts its clock rate and its position more or less continually to keep those phases small . deviations in position that drive position corrections show up not directly at @xmath7 but as phases registered by @xmath2 , so the steering of machine @xmath7 requires information about receptive phases measured by @xmath2 . the knowledge of the deviation in position of @xmath7 at @xmath106 can not arrive at @xmath7 until its effect has shown up at @xmath2 and been echoed back as a report to @xmath7 , entailing a delay of at least @xmath80 , hence requiring that machine @xmath7 predict the error that guides for @xmath80 prior to receiving a report of the error . that is , steering deviations by one open machine are measured in part by their effect on receptive phases of other open machines , so that steering of one machine requires information about receptive phases measured by other machines , and the deviations from an aiming point must increase with increasing propagation delays that demand predicting further ahead . as is clear from the cluster of five machines discussed in sec . [ sec : typei ] , the aiming - point phases can not in general all be taken to be zero . for any particular aiming - point phase @xmath107 there will be a deviation of a measured phase quantity @xmath91 given by @xmath108 whatever the value of @xmath107 , adjustments to contain phases within tolerable bounds depends on phase changes happening only gradually , so that trends can be detected and responded to on the basis of adequate prediction ( aka guesswork ) . + * remarks : * 1 . unlike cycle counts of open machines , which we assume are free of uncertainty , measured phases and deviations of phases from aiming points are quantities subject to uncertainty . for logic to work in a network , transmission of logical symbols must preserve sharp distinctions among them ; yet the maintenance of sharp distinctions among transmitted symbols requires responses to fuzzy measurements . 2 . the acquisition of logical synchrony in digital communications involves an unforeseeable waiting time , like the time for a coin on edge to fall one way or the other @xcite . aiming points are not forever , and here we say a few words about questions of type iii , in which an aiming point based on a hypothesized metric tensor appears unreachable , and perhaps needs to be revised . we have so far looked at one or another manifold with metric @xmath32 as some given hypothesis , whether explored on the blackboard or coded into an open machine to serve in maintaining its logical synchronization . in this context we think of @xmath32 as `` given . '' but deviations of phases outside of tolerances present another context , calling for revising a metric tensor field . in this context one recognizes that a metric tensor field is hypothesized provisionally , to be revised as prompted by deviations outside allowed tolerances in implementing an aiming point . drawing on measured phases as evidence in order to adjust a hypothesis of a metric tensor is one way to view the operation of the laser interferometer gravitational - wave observatory ( ligo ) @xcite . while ligo sensitivity drops off severely below 45 hz , the arrangement of five open machines of prop . [ prop : ten ] has no low - frequency cutoff , and so has the potential to detect arbitrarily slow changes in curvature . 99 b. n. taylor and a. thompson , eds , _ the international system of units ( si ) _ , nist special publication 330 , 2008 edition , national institutes of science and technology . j. m. myers and f. h. madjid , `` a proof that measured data and equations of quantum mechanics can be linked only by guesswork , '' in s. j. lomonaco jr . and h.e . brandt ( eds . ) _ quantum computation and information _ , contemporary mathematics series , vol . 305 , american mathematical society , providence , 2002 , pp . . f. h. madjid and j. m. myers , `` matched detectors as definers of force , '' ann . physics * 319 * , 251273 ( 2005 ) . j. m. myers and f. h. madjid , `` ambiguity in quantum - theoretical descriptions of experiments , '' in k. mahdavi and d. koslover , eds . , _ advances in quantum computation _ , contemporary mathematics series , vol . 482 ( american mathematical society , providence , i , 2009 ) , pp . 107123 . j. m. myers and f. h. madjid , `` what probabilities tell about quantum systems , with application to entropy and entanglement , '' in a. bokulich and g. jaeger , eds . , _ quantum information and entanglement _ , cambridge university press , cambridge uk , pp . 127150 ( 2010 ) . a. m. turing , `` on computable numbers with an application to the entscheidungsproblem , '' proc . london math . soc . , series 2 , * 42 * , 230265 ( 193637 ) . m. soffel et al . , `` the iau resolutions for astrometry , celestial mechanics , and metrology in the relativistic framework : explanatory supplement , '' the astronomical journal , * 126 * , 26872706 ( 2003 ) . j. m. myers and f. h. madjid , `` rhythms essential to logical communication , '' in quantum information and computation ix , e. donkor , a. r. pirich , and h. e. brandt , eds , proceedings of the spie , * 8057 * , pp . 80570n112 ( 2011 ) . j. m. myers and f. h. madjid , `` rhythms of memory and bits on edge : symbol recognition as a physical phenomenon , '' arxiv:1106.1639 , 2011 . h. meyr and g. ascheid , _ synchronization in digital communications _ , wiley , new york , 1990 . the ligo scientific collaboration ( http://www.ligo.org ) `` ligo : the laser interferometer gravitational - wave observatory , '' rep . prog . phys . * 72 * , 076901 ( 2009 ) v. perlick , `` on the radar method in general - relativistic spacetimes , '' in h. dittus , c. lmmerzahl , and s. turyshev , eds . , _ lasers , clocks and drag - free control : expolation of relativistic gravity in space _ , ( springer , berlin , 2008 ) ; also arxiv:0708.0170v1 . f. k. manasse and c. w. misner , j. math phys . , `` fermi normal coordinates and some basic concepts in differential geometry , '' * 4 * , 735745 ( 1963 ) . t. e. parker , s. r. jefferts , and t. p. heavner , `` medium - term frequency stability of hydrogen masers as measured by a cesium fountain , '' 2010 ieee international frequency control symposium ( fcs ) , pp . 318323 ( ( available at http://tf.boulder.nist.gov/general/pdf/2467.pdf ) j. levine and t. parker , `` the algorithm used to realize utc(nist ) , '' 2002 ieee international frequency control symposium and pda exhibition , pp . 537542 ( 2002 )

a clock steps a computer through a cycle of phases . for the propagation of logical symbols from one computer to another , each computer must mesh its phases with arrivals of symbols from other computers . even the best atomic clocks drift unforeseeably in frequency and phase ; feedback steers them toward aiming points that depend on a chosen wave function and on hypotheses about signal propagation . a wave function , always under - determined by evidence , requires a guess . guessed wave functions are coded into computers that steer atomic clocks in frequency and position clocks that step computers through their phases of computations , as well as clocks , some on space vehicles , that supply evidence of the propagation of signals . recognizing the dependence of the phasing of symbol arrivals on guesses about signal propagation elevates ` logical synchronization . ' from its practice in computer engineering to a dicipline essential to physics . within this discipline we begin to explore questions invisible under any concept of time that fails to acknowledge the unforeseeable . in particular , variation of spacetime curvature is shown to limit the bit rate of logical communication .

within the framework of continuum mechanics there are surface and bulk material failure models . surface failure models are known by name of cohesive zone models ( czms ) . in the latter case , continuum is enriched with discontinuities along surfaces - cohesive zones - with additional traction - displacement - separation constitutive laws . these laws are built qualitatively as follows : traction increases up to a maximum and then goes down to zero via increasing separation ( barenblatt , 1959 ; needleman , 1987 ; rice and wang , 1989 , tvergaard and hutchinson , 1992 ; camacho and ortiz , 1996 ; de borst , 2001 ; xu and needleman , 1994 ; roe and siegmund , 2003 ; moes et al , 1999 ; park et al , 2009 ; gong et al , 2012 ) . if the location of the separation surface is known in advance ( e.g. fracture along weak interfaces ) then the use of czm is natural . otherwise , the insertion of cracks in the bulk in the form of the separation surfaces remains an open problem , which includes definition of the criteria for crack nucleation , orientation , branching and arrest . besides , the czm approach presumes the simultaneous use of two different constitutive models , one for the cohesive zone and another for the bulk , for the same real material . certainly , a correspondence between these two constitutive theories is desirable yet not promptly accessible . the issues concerning the czm approach have been discussed by needleman ( 2014 ) , the pioneer of the field . bulk failure models are known by name of continuum damage mechanics ( cdm ) . in the latter case , material failure or damage is described by constitutive laws including softening in the form of the falling stress - strain curves ( kachanov , 1958 ; gurson , 1977 ; simo , 1987 ; voyiadjis and kattan , 1992 ; gao and klein , 1998 ; klein and gao , 1998 ; menzel and steinmann , 2001 ; dorfmann and ogden , 2004 ; lemaitre and desmorat , 2005 ; volokh , 2004 , 2007 ; benzerga et al , 2016 ) . remarkably , damage nucleation , propagation , branching and arrest naturally come out of the constitutive laws . unfortunately , numerical simulations based on the the bulk failure laws show the so - called pathological mesh - sensitivity , which means that the finer meshes lead to the narrower damage localization areas . in the limit case , the energy dissipation in failure tends to zero with the diminishing size of the computational mesh . this physically unacceptable mesh - sensitivity is caused by the lack of a characteristic length in the traditional formulation of continuum mechanics . to surmount the latter pitfall gradient- or integral- type nonlocal continuum formulations are used where a characteristic length is incorporated to limit the size of the spatial failure localization ( pijaudier - cabot and bazant , 1987 ; lasry and belytschko , 1988 ; peerlings et al , 1996 ; de borst and van der giessen , 1998 ; francfort and marigo , 1998 ; silling , 2000 ; hofacker and miehe , 2012 ; borden et al , 2012 ) . the regularization strategy rooted in the nonlocal continua formulations is attractive because it is lucid mathematically . unluckily , the generalized nonlocal continua theories are based ( often tacitly ) on the physical assumption of long - range particle interactions while the actual particle interactions are short - range - on nanometer or angstrom scale . therefore , the physical basis for the nonlocal models appears disputable . a more physically - based treatment of the pathological mesh - sensitivity of the bulk failure simulations should likely include multi - physics coupling . such an attempt to couple mass flow ( sink ) and finite elastic deformation within the framework of brittle fracture is considered in the present work . cracks are often thought of as material discontinuities of zero thickness . such idealized point of view is probably applicable to nano - structures with perfect crystal organization . in the latter case fracture appears as a result of a separation - unzipping - of two adjacent atomic or molecular layers - fig . [ fig : schematic - cracks - of ] ( left ) . ] in the case of the bulk material with a sophisticated heterogeneous organization the crack appears as a result of the development of multiple micro - cracks triggered by the massive breakage of molecular or atomic bonds - fig . [ fig : schematic - cracks - of ] ( right ) . the bond breakage is not confined to two adjacent molecular layers and the process involves thousands layers within an area or volume with the representative characteristic size @xmath0 . it is in interesting that material failure does not require the breakage of all molecular or atomic bonds within a representative volume . only fraction of these bonds should be broken for the material disintegration . for example , in the case of natural rubber , roughly speaking , every third bond should be broken within a representative volume to create crack ( volokh , 2013a ) . the local bond failure leads to the highly localized loss of material . the latter , in our opinion , is the reason why even closed cracks are visible by a naked eye . thus , material flows out of the system during the fracture process . the system becomes open from the thermodynamic standpoint . however , cracks usually have very small thickness and the amount of the lost material is negligible as compared to the whole bulk . the latter observation prompts considering the system as the classical closed one . such approximation allows ignoring the additional supply of momenta and energy in the formulation of the initial boundary value problem described in the next sections . following the approach of continuum mechanics we replace the discrete molecular structure of materials by a continuously distributed set of material points which undergo mappings from the initial ( reference ) , @xmath1 , to current , @xmath2 , configuration : @xmath3 . the deformation in the vicinity of the material points is described by the deformation gradient @xmath4 . in what follows we use the lagrangean description with respect to the initial or reference configuration and define the local mass balance in the form @xmath5 where @xmath6 is the referential ( lagrangean ) mass density ; @xmath7 is the referential mass flux ; @xmath8 is the referential mass source ( sink ) ; and @xmath9 in cartesian coordinates . _ we further assume that failure and , consequently , mass flow are highly localized and the momenta and energy balance equations can be written in the standard form without adding momenta and energy due to the mass alterations . _ in view of the assumption above , we write momenta and energy balance equations in the following forms accordingly @xmath10 and @xmath11 where @xmath12 is the velocity of a material point ; @xmath13 is the body force per unit mass ; @xmath14 is the first piola - kirchhoff stress and @xmath15 ; @xmath16 is the specific internal energy per unit mass ; @xmath17 is the specific heat source per unit mass ; and @xmath18 is the referential heat flux . entropy inequality reads @xmath19 where @xmath20 is the absolute temperature . substitution of @xmath21 from ( [ eq : energy balance ] ) to ( [ eq : entropy inequality ] ) yields @xmath22 or , written in terms of the internal dissipation , @xmath23 we introduce the specific helmholtz free energy per unit mass @xmath24 and , consequently , we have @xmath25 substituting ( [ eq : specific internal energy ] ) in ( [ eq : dissipation 1 ] ) we get @xmath26 then , we calculate the helmholtz free energy increment @xmath27 and substitute it in ( [ eq : dissipation 2 ] ) as follows @xmath28 the coleman - noll procedure suggests the following choice of the constitutive laws @xmath29 and , consequently , the dissipation inequality reduces to @xmath30 _ we further note that the process of the bond breakage is very fast as compared to the dynamic deformation process and the mass density changes in time as a step function . so , strictly speaking , the density rate should be presented by the dirac delta in time . we will not consider the super fast transition to failure , which is of no interest on its own , and assume that the densities before and after failure are constants and , consequently , _ @xmath31 then , the dissipation inequality reduces to @xmath32 which is obeyed because the heat flows in the direction of the lower temperature . it remains to settle the boundary and initial conditions . natural boundary conditions for zero mass flux represent the mass balance on the boundary @xmath33 @xmath34 or @xmath35 where @xmath36 is the unit outward normal to the boundary in the reference configuration . natural boundary conditions for given traction @xmath37 represent the linear momentum balance on the boundary @xmath33 @xmath38 or , alternatively , the essential boundary conditions for placements can be prescribed on @xmath33 @xmath39 initial conditions in @xmath1 complete the formulation of the coupled mass - flow - elastic initial boundary value problem @xmath40 constitutive law for the lagrangean mass flux can be written by analogy with the fourier law for heat conduction @xmath41 where @xmath42 is a mass conductivity constant for the isotropic case . constitutive law for the mass source is the very heart of the successful formulation of the theory and the reader is welcome to make a proposal . we choose , for example , the following constitutive law , whose motivation is clarified below , @xmath43-\rho),\label{eq : mass source}\ ] ] where @xmath44 is a constant initial density ; @xmath45 is a material constant ; @xmath46 is the specific energy limiter per unit mass , which is calibrated in macroscopic experiments ; @xmath47 is a dimensionless material parameter , which controls the sharpness of the transition to material failure on the stress - strain curve ; and @xmath48 is a unit step function , i.e. @xmath49 if @xmath50 and @xmath51 otherwise . the switch parameter @xmath52 , which is necessary to prevent from material healing , will be explained below . substitution of ( [ eq : mass source ] ) and ( [ eq : mass flux ] ) in ( [ eq : mass flow ] ) yields @xmath53-\frac{\rho}{j\rho_{0}}=0,\label{eq : mass balance}\ ] ] where @xmath54 is the characteristic length . _ it is remarkable that we , actually , do not need to know @xmath55 and @xmath56 separately and the knowledge of the characteristic length is enough_. for example , the estimate of the characteristic length for rubber is @xmath57 ( volokh , 2011 ) and for concrete it is @xmath58 ( volokh , 2013b ) . to justify the choice of the constitutive equation ( [ eq : mass source ] ) for the mass source / sink we note that in the case of the homogeneous deformation and mass flow the first term on the left hand side of ( [ eq : mass balance ] ) vanishes and we obtain @xmath59.\ ] ] substituting this mass density in the hyperelastic constitutive law we have @xmath60\frac{\partial w}{\partial\mathbf{f}}=h(\zeta)\exp[-(w/\varphi)^{m}]\frac{\partial w}{\partial\mathbf{f}},\label{eq : stress - strain}\ ] ] where @xmath61 are the helmholtz free energy and energy limiter per unit referential volume accordingly . constitutive law ( [ eq : stress - strain ] ) presents the hyperelasticity with the energy limiters - see volokh ( 2007 , 2013a , 2016 ) for the general background . integrating ( [ eq : stress - strain ] ) with respect to the deformation gradient we introduce the following form of the strain energy function @xmath62 where @xmath63 here @xmath64 and @xmath65 designate the constant bulk failure energy and the elastic energy respectively ; @xmath66 is the upper incomplete gamma function . the switch parameter @xmath67 $ ] is defined by the evolution equation @xmath68 where @xmath69 is a dimensionless precision constant . the physical interpretation of ( [ eq : energy with limiter ] ) is straight : material is hyperelastic for the strain energy below the failure limit - @xmath64 . when the failure limit is reached , then the strain energy becomes constant for the rest of the deformation process precluding the material healing . parameter @xmath70 is _ not an internal variable_. it is a switch : @xmath71 for the reversible process ; and @xmath50 for the irreversibly failed material and dissipated strain energy . for illustration , we present the following specialization of the intact strain energy for a filled natural rubber ( nr ) ( volokh , 2010 ) @xmath72 where @xmath73 , @xmath74 , @xmath75 and the failure parameters are @xmath76 , and @xmath77 . the cauchy stress , defined by @xmath78 , versus stretch curve for the uniaxial tension is shown in fig . [ fig : cauchy - stress ] for both cases with and without the energy limiter . material failure takes place at the critical limit point in correspondence with tests conducted by hamdi et al ( 2006 ) . versus stretch . dashed line specifies the intact model ; solid line specifies the model with energy limiter.[fig : cauchy - stress ] ] for the implications and experimental comparisons of the elasticity with energy limiters the reader is advised to look through volokh ( 2013a ; 2016 ) , for example . we completely skip this part for the sake of brevity . thus , the proposed constitutive law for the mass source is motivated by the limit case of the coupled formulations in which the deformation is homogeneous . crack in a bulk material is not an ideal unzipping of two adjacent atomic layers . it is rather a massive breakage of atomic bonds diffused in a volume of characteristic size . the massive bond breakage is accompanied by the localized loss of material . thus , material sinks in the vicinity of the crack . evidently , the law of mass conservation should be replaced by the law of mass balance , accounting for the mass flow in the vicinity of the crack . the coupled mass - flow - elasticity problem should be set for analysis of crack propagation . in the present work , we formulated the coupled problem based on the thermodynamic reasoning . we assumed that the mass loss related to the crack development was small as compared to the mass of the whole body . in addition , we assumed that the process of the bond breakage was very fast and the mass density jumped from the intact to failed material abruptly allowing to ignore the transient process of the failure development . these physically reasonable assumptions helped us to formulate a simple coupled initial boundary value problem . _ in the absence of failure localization into cracks the theory is essentially the hyperelasticity with the energy limiters . however , when the failure starts localizing into cracks the diffusive material sink activates via the mass balance equation and it provides the regularization of numerical simulations . _ the latter regularization is due to the mass diffusion - first term on the left hand side of ( [ eq : mass balance ] ) . the attractiveness of the proposed framework as compared to the traditional continuum damage theories is that no internal parameters ( like damage variables , phase fields etc . ) are used while the regularization of the failure localization is provided by the physically sound law of mass balance . a numerical integration procedure for the formulated coupled initial boundary value problem is required and it will be considered elsewhere . the support from the israel science foundation ( isf-198/15 ) is gratefully acknowledged . barenblatt gi ( 1959 ) the formation of equilibrium cracks during brittle fracture . general ideas and hypotheses . axially - 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cracks are created by massive breakage of molecular or atomic bonds . the latter , in its turn , leads to the highly localized loss of material , which is the reason why even closed cracks are visible by a naked eye . thus , fracture can be interpreted as the local material sink . mass conservation is violated locally in the area of material failure . we consider a theoretical formulation of the coupled mass and momenta balance equations for a description of fracture . our focus is on brittle fracture and we propose a finite strain hyperelastic thermodynamic framework for the coupled mass - flow - elastic boundary value problem . the attractiveness of the proposed framework as compared to the traditional continuum damage theories is that no internal parameters ( like damage variables , phase fields etc . ) are used while the regularization of the failure localization is provided by the physically sound law of mass balance .

the anderson impurity model ( aim ) , in which a single , locally correlated orbital couples to a non - interacting metallic band of electrons , is a longstanding paradigm of strongly - correlated electron physics . conceived originally@xcite to explain the formation of localized magnetic moments on impurities in non - magnetic hosts , it has since formed the cornerstone of our understanding of the kondo effect@xcite and related many - body phenomena . interest in the area is currently particularly strong , both experimentally and theoretically , after the kondo effect was predicted@xcite and then directly confirmed@xcite to arise in mesoscopic quantum dot systems.@xcite after some 50 years of intense theoretical work , the spin-@xmath0 kondo effect as manifest in anderson s original model is naturally rather well understood@xcite . below some characteristic kondo temperature @xmath1 , a complex many - body state develops in which the impurity spin is completely screened by the host metal , leading at low energies to a ` local ' fermi - liquid and universal transport properties . being a low - energy phenomenon , the kondo effect is of course crucially dependent on both conduction band states near the fermi level and the low - energy spin degrees of freedom of the impurity . this has inspired much research into other quantum impurity models involving more complex impurities and/or host densities of states with the aim of identifying the various types of kondo effect that may arise , the conditions under which they do so , and the novel physics that results when kondo screening can not be achieved@xcite . here we consider the notionally simple problem of an anderson impurity in a gapped host , where the density of states vanishes over a finite range about the chemical potential , a model not only of relevance to anderson impurities in semiconductors but also@xcite to the topical issue of impurities in bcs superconductors@xcite . in removing the all - important low - lying states of the host , one would certainly expect the kondo effect to be precluded for large enough gaps : the question is , can the effect still arise for sufficiently - small gaps , or is it destroyed as soon as a gap is opened ? this question has indeed been the subject of a number of previous papers . poor man s scaling , the @xmath2 expansion and the non - crossing approximation predict @xcite that the kondo effect always arises whenever the gap is less than the kondo temperature in the absence of the gap , while for larger gaps the system undergoes a quantum phase transition to an ` local moment ' ( lm ) phase where the impurity spin remains unscreened as @xmath3 . in addition the problem has been studied numerically by the density - matrix renormalization group@xcite and quantum monte carlo@xcite , but with no general consensus reached regarding the nature of the quantum phase transition . the numerical renormalization group ( nrg)@xcite on the other hand has been used to argue that the fermi - liquid regime associated with the kondo effect exists only away from particle hole - symmetry , and then only below a certain critical gap . in the particle - hole symmetric limit it is found@xcite that the kondo effect _ never _ arises and the ground state is the doubly - degenerate lm phase for arbitrarily small gaps . in this paper we study the problem analytically , within a perturbative framework which includes both explicit second - order perturbation theory and self - consistent perturbation theory to all orders la luttinger @xcite . in addition to confirming the basic predictions of the nrg study @xcite , our analysis provides a number of exact results , including the analogue of the friedel sum rule , which serve as touchstones for approximate theories of the gapped aim ( gaim ) . in a subsequent paper@xcite , we present a local moment approach@xcite to the problem , the results of which agree very well with the conclusions of the present work . in standard notation the generic anderson hamiltonian@xcite is @xmath4 where @xmath5 is the number operator for @xmath6-spin electrons on ` site ' @xmath7 ( with @xmath8 referring to the impurity site and @xmath9 to the host band states ) . the first term in eqn . ( [ eq : h ] ) thus describes the non - interacting host band , the second and third terms describe the impurity with onsite coulomb interaction @xmath10 , and the fourth term hybridises the two . for a symmetric host band , the particle - hole symmetric limit corresponds to the special point @xmath11 ( where @xmath12 is invariant under a particle - hole transformation ) . the dynamics of the model will be obtained from the retarded green function @xmath13 @xmath14 differentiation of which leads straightforwardly to its equations - of - motion @xcite ; from which the impurity - diagonal green function in the non - interacting @xmath15 limit follows . its fourier transform , denoted by @xmath16 , is @xmath17 with @xmath18 the host - impurity hybridisation function @xmath19 and @xmath20 with @xmath21 a positive infinitesimal . the ` full ' and non - interacting green functions are related in the usual way by dyson s equation @xmath22 with @xmath23 the conventional ( single ) self - energy . it is convenient below to exploit the analytic structure of the impurity green functions and their constituent parts . let @xmath24 be analytic on the real axis and in the upper half plane , tending to zero as @xmath25 . then , with @xmath26 for real @xmath27 , one has the well known dispersion relation @xmath28 ( with @xmath29 denoting a principal value ) , and the spectral representation @xmath30 in particular , the full impurity green function can be determined entirely from its spectral function , @xmath31 the results above are valid for whatever form the host takes in eqn . ( [ eq : h ] ) : the details of the host bandstructure affect only the hybridisation function @xmath18 . assuming for simplicity that @xmath32 is fixed , and writing @xmath33 , eqn . ( [ eq : delta ] ) gives @xmath34 with @xmath35 the host density of states . hence , taking @xmath35 to be constant except for a gap of full width @xmath36 centred around @xmath37 , we write @xmath38 for the _ gapped _ anderson model as @xmath39 the corresponding @xmath40 ( @xmath41 ) follows from the hilbert transform eqn . ( [ eq : generalfhilbert ] ) as @xmath42 the logarithmic divergences of @xmath40 at @xmath43 arising from the gap are shown in fig . [ fig : delta ] . real part of the hybridization function , @xmath44 , _ vs. _ @xmath45 ( solid line ) . the dashed line illustrates the solution of eqn . ( [ eq : dpolepos ] ) as described in the text . ] the explicit form of the non - interacting spectrum @xmath46 is readily obtained from eqns . ( [ eq : giiw ] ) , ( [ eq : ditog ] ) , ( [ eq : deltagap ] ) and ( [ eq : deltargap ] ) . two distinct contributions arise : a continuum ( or ` band ' ) part @xmath47 arising when @xmath38 is non - zero @xmath48 ^ 2+{\delta_\mathrm{0}}^2 } , \label{eq : dband}\ ] ] and a single pole inside the gap . this pole arises for all @xmath49 when @xmath50 , occuring at a frequency @xmath51 determined by solution of @xmath52 that this equation always has one , and only one , solution is guaranteed by the monotonic decrease of @xmath40 from @xmath53 to @xmath54 for @xmath55 , as seen from the construction in fig . [ fig : delta ] . moreover , since @xmath56 , it is clear that @xmath57 for @xmath58 , respectively . in the particle - hole symmetric limit , where @xmath59 , the pole lies precisely at @xmath60 . we shall see that this zero - frequency pole is the basic reason for qualitatively distinct physics at particle - hole symmetry , compared to that elsewhere . the weight of the pole , @xmath61 , follows straightforwardly from a taylor expansion of @xmath40 as @xmath62^{-1 } , \label{eq : dpoleweight}\ ] ] and hence the total non - interacting spectrum is given by @xmath63 this non - interacting green function of course forms the basis for perturbation theory in @xmath10 . we begin by analysing the problem to second - order in @xmath10 , which is sufficient to highlight the essential features that lead to non - analyticities in the particle - hole symmetric limit . to second - order in @xmath10 , the self - energy @xmath64 ( @xmath65 ) is expanded diagramatically as shown in fig . [ fig:2pt ] . to simplify the notation , we henceforth drop the @xmath66 subscripts , since we focus on the local impurity green function . we also drop the spin index , as the green functions obtained are independent of @xmath6 . translating the diagrams in fig . [ fig:2pt ] using the feynman rules for the time - ordered green function gives @xmath67 where @xmath68 is the impurity charge in the non - interacting limit , @xmath69 and @xmath70 is the polarisation ` bubble ' @xmath71 using the spectral representation of the green function ( see eqn . ( [ eq : generalfspectral ] ) ) , @xmath72 is expressible entirely in terms of @xmath73 , _ viz . _ @xmath74 likewise , @xmath75 can be written as @xmath76 from eqns . ( [ eq : impinought ] ) and ( [ eq : sigmatwod ] ) , which hold also for the retarded green function under consideration here , the second - order self - energy @xmath77 can be constructed from the non - interacting spectrum @xmath73 alone . these integrals can of course be evaluated numerically for any given @xmath49 and @xmath78 to obtain the full frequency dependence of @xmath77 . this is not our aim here ; instead , we focus on the low - frequency behaviour ( which can be determined analytically ) , since it is this that contains the key physics of the problem . the generic behaviour away from the particle - hole symmetric limit can be examined by focussing on some given @xmath79 ; results for @xmath80 follow from a straightforward particle - hole transformation . given that the pole in @xmath73 lies at a frequency @xmath81 for @xmath82 ( see sec . [ sec : prelim ] ) , it is readily shown from eqns . ( [ eq : dbp ] ) and ( [ eq : impinought ] ) that @xmath83 since @xmath84 has no spectral weight for @xmath85 , the second term of eqn . ( [ eq : impinoughta ] ) is zero for @xmath86 . the additional contribution from the first term of eqn . ( [ eq : impinoughta ] ) also contains a gap for @xmath87 , and since @xmath88 we can write @xmath89 using this in eqn . ( [ eq : sigmatwod ] ) , similar arguments can be made for the low - frequency form of @xmath77 ; it too has a gap around @xmath37 , given by @xmath90 the gap in @xmath75 enables one to deduce the salient behaviour of @xmath91 . by writing @xmath92 , the hilbert transform eqn . ( [ eq : generalfhilbert ] ) gives @xmath93 and hence @xmath94{\sigma^\mathrm{i}}_2(\omega')\end{gathered}\ ] ] if both @xmath95 and @xmath96 are within the gap , the term in square brackets above is non - negative for all @xmath97 . since @xmath75 is necessarily non - negative , it follows that @xmath98 this monotonicity is important in arguments below . finally , the low - energy behaviour of @xmath77 can be inserted into the dyson equation , eqn . ( [ eq : dyson ] ) , to obtain the impurity green function to second - order in @xmath10 . for @xmath85 , @xmath99 and , by using similar arguments to those in the non - interacting limit , the monotonicity of @xmath91 in eqn . ( [ eq : sigrmono ] ) guarantees that @xmath100 has a single pole at a frequency @xmath101 given by @xmath102 inside the gap the second - order green function @xmath100 is thus seen to have the same basic structure as the non - interacting @xmath103 . in particular , the @xmath104 quasiparticle behaviour of the green function is obtained by expanding @xmath91 about @xmath37 , @xmath105 where @xmath106 is the renormalized level energy , and @xmath107^{-1}\ ] ] is the quasiparticle weight . that @xmath100 is a renormalized version of the non - interacting @xmath103 at low - frequencies is of course a direct reflection of adiabatic continuity to the non - interacting limit , in which sense the system is a generalized fermi liquid ( gfl ) . for some fixed @xmath49 and @xmath78 , on switching on a small @xmath10 , one expects the system to evolve smoothly into this perturbative state and as such to behave as a ( local ) fermi liquid . as we now show , however , such behaviour does _ not _ arise at the particle - hole symmetric limit of the model . that the behaviour at the particle - hole symmetric point is qualitatively different from that described above , arises because ( see sec . [ sec : prelim ] ) the pole in the non - interacting @xmath73 lies precisely at @xmath60 . on substituting eqn . ( [ eq : dbp ] ) into eqn . ( [ eq : impinought ] ) , @xmath72 itself contains a pole at zero frequency : @xmath108 physically , this pole describes zero - energy spin - flip excitations of the impurity at the non - interacting level : the @xmath109 pole in @xmath73 reflects a state which is equally likely to contain a single @xmath110- or @xmath111-spin electron . the second - order self energy term @xmath75 is obtained as before by substituting eqn . ( [ eq : impinoughtph ] ) into eqn . ( [ eq : sigmatwod ] ) . whereas away from particle - hole symmetry the key result was a _ gap _ in @xmath75 around the fermi level ( see eqn . ( [ eq : sigma2gap ] ) ) , here instead the pole in @xmath72 leads to a corresponding _ pole _ in the self - energy : @xmath112 such behaviour is strikingly different from the gfl physics described above . the real part of @xmath91 obtained from eqn . ( [ eq : sigma2iph ] ) diverges as @xmath104 , i.e. @xmath113 and the corresponding @xmath100 @xmath114^{-1}$ ] is thus of form @xmath115 ( noting that the renormalized level energy @xmath116 , given generally by @xmath117 , vanishes by symmetry at the particle - hole symmetric point @xmath118 ) . ( [ eq : g2ph ] ) clearly can not be written as a renormalized version of the non - interacting @xmath103 , indicating that the particle - hole symmetric point is not perturbatively connected in @xmath10 to the non - interacting limit . instead , it is readily shown by expanding the hybridisation function @xmath40 about @xmath37 that @xmath119 with a renormalized interaction @xmath120 ( and @xmath121^{-1}$ ] from eqn . ( [ eq : dpoleweight ] ) ) . the significance of this result is that it is a renormalized version of the _ atomic limit _ ( @xmath122 ) propagator , @xmath123^{-1}$ ] , indicative of the lm nature of the phase arising at particle - hole symmetry . instead of the single pole inside the gap seen away from particle - hole symmetry , _ two _ poles thus arise , at @xmath124 , and representing as such renormalized versions of the atomic limit hubbard satellites ( themselves occurring at @xmath125 ) . even at the basic level of second - order perturbation theory , one sees then that introducing a gap in the conduction band of the anderson model changes significantly its low - energy physics , with non - analyticities arising in the perturbative self - energies in the particle - hole symmetric limit . we now extend the treatment to arbitrary order in the interaction @xmath10 , bolstering the conclusions drawn above and in doing so obtaining some exact results for the problem . one can not of course derive perturbatively the full frequency dependence of the impurity green function to all orders , but its behaviour inside the gap can be ascertained exactly using some simple arguments . our main result is to determine the conditions under which perturbation theory in @xmath10 is valid , and the resulting low - frequency behaviour of the impurity green function . the argument follows that of the classic paper by luttinger@xcite , in which the skeleton expansion of the self - energy is used to obtain self - consistently the low - frequency behaviour of a wide class of interacting problems to all orders in the interaction . the basis of the approach is that if perturbation theory holds , the exact self - energy is equal to the sum of all skeleton self - energy diagrams constructed from the exact green function , as shown diagramatically in fig . [ fig : selfskel ] . by analysing the general low - frequency behaviour of all such skeleton diagrams , we show that this condition is satisfied only if the exact green function contains a _ single _ pole inside the gap at a non - zero frequency . a key insight of ref . is that the low - energy behaviour of the self - energy can be deduced perturbatively_via _ the skeleton expansion from the low - energy single - particle excitations of the exact @xmath126 . to be more precise , we refer to the positive frequency excitations of @xmath127 as ` particle energies ' and the negative frequency excitations as ` hole energies ' . then , it can be shown@xcite quite generally that @xmath128 has weight at positive frequencies corresponding to @xmath129 particle energies minus @xmath130 hole energies , or at negative frequencies corresponding to @xmath129 hole energies minus @xmath130 particle energies , with @xmath131 . combining this result with the dyson equation [ giving @xmath126 in terms of @xmath64 ] generates a pair of self - consistency equations for @xmath64 . in what follows , we show that any self - energy that contains poles inside the gap ( _ i.e . _ for @xmath85 ) is inconsistent with these requirements . consider the situation in which @xmath133 contains at least one pole in the range @xmath55 , and label by @xmath51 the frequency of the pole closest to @xmath37 . in the following we take @xmath134 ; similar reasoning can be applied if the closest pole is at a negative frequency . then define @xmath95 to be the least negative frequency at which @xmath133 has weight ( either in the form of another pole , or the upper edge of the negative - frequency continuum ) , and @xmath96 to be the next frequency above @xmath51 at which @xmath133 has weight . we sketch the resulting @xmath135 in fig . [ fig : poles](a ) , taking ( without loss of generality ) both @xmath95 and @xmath96 to correspond to discrete poles rather than the edges of the continua . sketch showing how poles in @xmath64 lead ( via the dyson equation ) to poles in @xmath126 . ( a ) solid lines represent the schematic form of @xmath135 with poles in the gap , dashed lines represent the corresponding @xmath136 from hilbert transformation , and the dotted lines show the constructions used for determing the poles in @xmath126 from dyson s equation ( [ eq : dyson ] ) . ( b ) the pole structure of @xmath126 obtained from case ( i ) of ( a ) . ( c ) the pole structure of @xmath126 obtained from case ( ii ) of ( a ) . ] in fig . [ fig : poles](a ) , we also sketch the @xmath136 corresponding to the @xmath135 described above . its form follows from the hilbert transform eqn . ( [ eq : generalfhilbert ] ) . within each frequency range in which @xmath133 is zero , an argument akin to that before eqn . ( [ eq : sigrmono ] ) shows that @xmath137 is monotonically decreasing . moreover , at frequencies where @xmath133 has poles , it is readily seen from eqn . ( [ eq : generalfhilbert ] ) that @xmath137 diverges as @xmath138 ( with @xmath139 the particular pole frequency ) . hence @xmath140 spans the full range from @xmath53 to @xmath54 between each excitation in @xmath141 . the real and imaginary parts of the schematic self - energy can then be inserted into the dyson equation to determine @xmath126 . since @xmath142 for @xmath143 , it follows from eqns . ( [ eq : giiw ] ) and ( [ eq : dyson ] ) that @xmath126 has poles only for @xmath85 , at frequencies given by the solution of @xmath144 . the resultant poles in @xmath126 thus lie at the frequencies where the dashed lines in fig . [ fig : poles ] intersect straight lines of unit slope , as shown by the dotted lines in fig . [ fig : poles](a ) . from the diagram , it is clear that within the region @xmath145 there will always be two such poles in @xmath126 . either there is one on each side of @xmath37 , which we denote by ` case ( i ) ' and show schematically in fig . [ fig : poles](b ) ; or both poles are at @xmath146 , denoted ` case ( ii ) ' ( see fig . [ fig : poles](c ) ) [ note that if @xmath60 , then the only possible case is ( i ) ] . now consider the @xmath146 behaviour of @xmath133 obtained by inserting this @xmath126 into the skeleton expansion . as discussed above , this is obtained @xcite by considering the energies of possible particle and hole excitations in @xmath126 . in case ( i ) , the lowest particle energy is at an @xmath147 , and the lowest hole energy is at an @xmath148 . since@xcite the lowest - energy pole in @xmath133 for @xmath146 is at an energy corresponding to two particle energies minus one hole energy , this lowest - energy pole is necessarily at a frequency @xmath149 _ and is therefore not consistent with the starting @xmath133 in fig . [ fig : poles](a)_. likewise in case ( ii ) , the lowest particle energy is at an @xmath150 , and the lowest hole energy is at an @xmath151 . it follows that any resulting pole in @xmath133 is at an @xmath152 with @xmath153 by definition . hence case ( ii ) is also not self - consistent . we have thus shown that there are no self - consistent perturbative solutions of the gaim with a pole in @xmath133 at an @xmath134 . similar arguments can be used to prove the same is true for a pole at @xmath154 . therefore , _ any perturbative ( fermi liquid ) solution can not contain poles in @xmath133 inside the gap_. finally we show that given the conclusion drawn above , particle - hole symmetry _ always _ leads to a breakdown of perturbation theory . to that end , imagine starting with a perturbative @xmath133 [ i.e. no poles in the gap ] in the particle - hole symmetric limit . its corresponding real part is monotonically decreasing inside the gap , and furthermore satisfies @xmath155 by particle - hole symmetry . therefore the green function obtained by substituting this self - energy into the dyson equation ( [ eq : dyson ] ) must have a pole at @xmath37 . but then the @xmath133 obtained by substituting this @xmath126 into the skeleton expansion would itself have a pole at zero frequency , breaking the perturbative self - consistency of the skeleton expansion . _ hence the particle - hole symmetric point can not be perturbatively connected to the non - interacting limit , and as such is a non fermi - liquid , for all @xmath156 . _ away from particle - hole symmetry by contrast , the pole in @xmath126 arising from a perturbative @xmath133 can ( and will ) be at a non - zero frequency . in this case the skeleton expansion would not lead to a pole inside the gap in @xmath133 , which is consistent with the form of the starting @xmath133 . hence such perturbative solutions can in principle exist ; and the results of sec . [ sec : sopt ] naturally lead us to expect them , at least for some non - vanishing radius of convergence in @xmath10 . by closing the skeleton self - energy diagrams in fig . [ fig : selfskel ] with an additional propagator line , one obtains the luttinger - ward functional @xmath157 $ ] @xcite , from which the exact self - energy follows from the functional derivative @xmath158}{\delta \mathcal{g}}\right)_{\mathcal{g}=g}. \label{eq : sigphi}\ ] ] using eqn . ( [ eq : sigphi ] ) , and that @xmath157 $ ] is invariant@xcite under a frequency shift of all its propagators @xmath159 , it can be shown that the ` luttinger integral ' @xmath160 must vanish : @xmath161 this can be used in turn to derive a generalized friedel sum rule@xcite for the gaim , which must be satisfied by any perturbative solution of the problem . consider the change in the density of states upon introducing the impurity , @xmath162 . this is given by @xmath163.\ ] ] the equations of motion for the host green functions generate @xcite @xmath164 and the dyson equation ( [ eq : dyson ] ) gives @xmath165 combining eqns . ( [ eq : lw1])([eq : lw3 ] ) thus leads to the general result @xcite @xmath166.\ ] ] the number of electrons introduced by the impurity , @xmath167 , is now obtained by integrating @xmath168 up to @xmath37 ( the factor of @xmath169 arising from a spin sum ) . from eqn . ( [ eq : lw3 ] ) , and using the luttinger integral condition eqn . ( [ eq : il ] ) , one has @xmath170 from equation ( [ eq : dyson ] ) it follows that @xmath171\ ] ] ( taking the principal range @xmath172 ) , and hence @xcite @xmath173.\ ] ] this friedel sum rule is quite general for the anderson impurity model . for the gaim in particular one has @xmath174 @xmath175 , and the results for the self - energy deduced in the previous section showed that @xmath176 necessarily vanishes if perturbation theory holds ( gfl phase ) . hence @xmath177 $ ] , from which @xmath178 with @xmath179 the renormalized level energy . for _ any _ @xmath180 , the additional number of electrons induced by the impurity is thus @xmath181 , while @xmath182 for any @xmath183 . this behaviour exclusively integral values of @xmath167is physically natural for a system with a gapped spectrum , although it is of course in marked contrast to the metallic ( gapless ) anderson model for which @xmath184 and hence ( from eqn . ( [ eq : wilber ] ) ) @xmath167 is a continuous function of @xmath185 . finally , we tie together the results of the previous sections . we have shown that self - consistent perturbative solutions of the problem must necessarily have a non - zero renormalized level , @xmath185 , since this leads to a single pole in @xmath126 at a non - zero frequency and hence a perturbatively - consistent @xmath64 from the skeleton expansion . if @xmath185 is negative the pole in @xmath126 is at a negative frequency and @xmath186 ; while if @xmath185 is positive the pole is at a positive frequency and @xmath187 . in sec . [ sec : inforder ] , we deduced that such perturbative solutions of the problem can only exist away from particle - hole symmetry : it is impossible to construct a @xmath126 that is both consistent with particle - hole symmetry _ and _ the perturbative skeleton expansion of the self - energy . hence the particle - hole symmetric point of the problem is a non - fermi - liquid lm phase , and any approximation to the gaim must therefore take this into account@xcite . away from particle - hole symmetry , there is nothing to prevent the existence of perturbative gfl solutions , and these are indeed found numerically in _ e.g. _ the nrg@xcite . finally we note that since the derivation of the generalized friedel sum rule , eqn . ( [ eq : nimplast ] ) above , has been obtained from a perturbative construction of the luttinger ward functional @xmath157 $ ] , it is not valid in the lm phase . at the particle - hole symmetric point , it is readily shown instead that @xmath188 by symmetry . in our forthcoming local moment approach to the model@xcite , we argue that @xmath188 is a general result that also holds for the lm phase away from particle - hole symmetry , and present results of this non - perturbative approach for the dynamics of the problem in the two phases .

we consider an anderson impurity model in which the locally correlated orbital is coupled to a host with a gapped density of states . single - particle dynamics are studied , within a perturbative framework that includes both explicit second - order perturbation theory and self - consistent perturbation theory to all orders in the interaction . away from particle - hole symmetry the system is shown to be a generalized fermi liquid ( gfl ) in the sense of being perturbatively connectable to the non - interacting limit ; and the exact friedel sum rule for the gfl phase is obtained . we show by contrast that the particle - hole symmetric point of the model is not perturbatively connected to the non - interacting limit , and as such is a non - fermi liquid for all non - zero gaps . our conclusions are in agreement with nrg studies of the problem .

despite the mounting evidence for the existence of dark matter ( dm ) in galaxies , clusters of galaxies and the universe at large scale , the nature and properties of the dark matter particle are still largely unconstrained by observations . in fact , viable dark matter models have been constructed with masses ranging between @xmath0ev and @xmath1 gev , and interaction cross sections ranging between @xmath2 pb and @xmath3 pb ( for a review , see @xcite ) . in this vast parameter space of dark matter models , weakly interacting massive particles ( wimps ) still stand as one of the most promising dark matter candidates , since for reasonable values of the model parameters , the freeze - out of dark matter wimps from the thermal plasma left a relic population with an abundance which reproduces qualitatively well the measured value of the dark matter density @xmath4 @xcite . there are presently three different approaches pursued in order to detect the non - gravitational effects of wimps with ordinary matter : direct detection , indirect detection and collider experiments . this decade is being especially prolific in experimental results in the three search strategies . indeed , various experiments currently in operation are setting strong limits on the wimp parameter space and ruling out regions where a dark matter signal could be expected , notably xenon100 @xcite and lux @xcite in direct searches , fermi - lat @xcite , ams-02 @xcite , h.e.s.s . @xcite , magic @xcite , icecube @xcite in indirect searches and the lhc in collider searches ( see e.g. @xcite ) . moreover , in the near future the @xmath5tev run of lhc , the xenon1 t @xcite and lz @xcite experiments , and the cerenkov telescope array @xcite will significantly improve the reach of collider , direct and indirect dark matter searches , respectively . these three different approaches constrain the parameter space of dark matter models in a complementary way , however , the synergy of the various search strategies is very model dependent . in this paper we focus on a simple scenario where the dark matter particle is a majorana fermion that couples to light quarks and a coloured scalar via a yukawa coupling . this scenario , despite its simplicity , offers a very rich phenomenology in direct detection @xcite , indirect detection @xcite and collider experiments @xcite . in particular , when the mediator mass is comparable to the dark matter mass , this model predicts a sharp and relatively intense gamma - ray spectral feature which , if observed , would constitute an unambiguous signal for dark matter annihilations @xcite . additionally , the collider phenomenology is distinct from the widely - used effective operator approach ( see e.g. @xcite ) , because the mediator can be directly produced in proton proton collisions . similar models , but with leptonic mediators , were studied in @xcite . in this paper we revisit the collider limits in this scenario . most analyses include only the production of coloured scalars via strong interactions , nevertheless , in this scenario the yukawa coupling can be sizeable and the production of coloured scalars via the exchange of a dark matter particle in the t - channel can become important or even dominant . this possibility has been discussed in @xcite . here we go beyond these analyses by performing a dedicated re - interpretation of collider searches which includes also jet matching , that is important when considering the quasi - degenerate mass spectrum . a similar analysis for the case of dirac dark matter has been recently presented in @xcite . we analyse the limits on the yukawa coupling from the atlas search for jets and missing transverse energy @xcite and investigate the complementarity of the collider limits with those from direct and indirect dark matter searches . furthermore we discuss various sources of experimental and theoretical uncertainties of collider limits and assess their impact on the exclusion power . finally , we consider an extension of the model by two coloured scalars coupling to the up - type quarks and we study the impact of extending the scalar sector on the dark matter searches in view of the stringent limits from flavour violation . the paper is organized as follows . in section [ sec : model ] , we introduce the simplified model and discuss its properties with respect to indirect , direct and collider searches . section [ sec : lhc ] explains some details of our collider analysis . our results are discussed and compared to direct and indirect detection constraints in section [ sec : results ] , and we conclude in section [ sec : conclusions ] . the appendix contains a brief discussion of flavour constraints . we assume the dark matter particle @xmath6 to be a majorana fermion which couples to the light quarks via a yukawa interaction with coloured scalars @xmath7 . the lagrangian of the model can be written as @xmath8 where @xmath9 denotes the standard model ( sm ) lagrangian while @xmath10 and @xmath11 are given by @xmath12 where @xmath13 denotes the covariant derivative . on the other hand , @xmath14 contains the interactions between the sm quarks and the dark sector , @xmath15 where @xmath16 is a yukawa coupling matrix , @xmath17 denote the right - handed quark fields and summation over flavours @xmath18 , @xmath19 is implied . this lagrangian generically leads to too large flavour changing neutral currents , hence some requirements must be imposed on the yukawa couplings to fulfil the stringent constraints from flavour observables . in the following we consider two scenarios : 1 . we consider a single scalar @xmath20 that couples exclusively to the right - handed up quarks , with coupling strength @xmath21 . this scenario corresponds to an alignment type set - up of the squark sector in the mssm and can be realized by appropriate flavour symmetries at a high scale @xcite . we consider a pair of mass degenerate scalars @xmath22 and @xmath23 which couple to right - handed up and charm quarks with a universal coupling @xmath24 . such a scenario is motivated by the paradigm of minimal flavour violation @xcite which requires flavour universality among quarks with the same gauge quantum numbers while allowing a separation of particles belonging to different multiplets . we show explicitly in appendix [ app : flavour ] that within these two scenarios the constraints from flavour observables are easily satisfied . one may also consider a coupling to down - type quarks , which is completely analogous and qualitatively very similar . in the following we concentrate on the above two scenarios for definiteness . the model is thus completely described by the two masses @xmath25 and @xmath26 of the dark matter and the mediator(s ) , respectively , and by the yukawa coupling @xmath21 . with this framework it is possible to calculate various dark matter observables , e.g. the relic density , the annihilation cross section , the dark matter - nucleon scattering cross section or event rates at the lhc , and compare their relative exclusion power . an interesting particularity of the model analyzed here is that the strongest experimental constraints can not be derived from a small set of effective operators , but require to consider higher order effects . concretely , for indirect detection the two - to - three and loop - induced annihilation channels play an important role , firstly because the leading order two - to - two channel is helicity and velocity suppressed , and second because the hard gamma - ray spectrum from @xmath27 and the loop induced processes @xmath28 generate a very distinct spectral signature @xcite . for direct detection , the lowest order operators mediating spin - independent interactions are suppressed for majorana dark matter with chiral interactions , such that higher order contributions and the spin - dependent scattering have to be also considered @xcite . lastly , the production at the lhc is governed not only by the strong processes , but also by processes mediated by the yukawa interaction with the dark matter particle @xcite . in the following , we summarize the relevant features of the model concerning the relic density , as well as the direct and indirect detection , and then discuss in detail the signatures at the lhc . probably the most compelling argument for wimp dark matter is that this class of particles is produced quite naturally in the early universe and can generate , after thermal freeze - out , the correct relic density @xmath4 @xcite as measured by the planck satellite . the lagrangian ( [ eq : l ] ) allows for tree level annihilations @xmath29 and in most of the parameter space the relic abundance is set by this process . however , for @xmath30 the scalar @xmath20 does not freeze - out before the dark matter particle @xmath6 and modifies the relic density @xcite . this process , which is known as coannihilation , can be approximately taken into account by first defining an effective cross section @xmath31 where @xmath32 corresponds to the freeze - out temperature while @xmath33 and @xmath34 correspond to the thermally averaged annihilation cross - section of a @xmath35 or an @xmath36 pair respectively , and then replacing the thermally averaged cross section by this effective cross section in the well - known solution to the boltzmann equation neglecting coannihilations . in our analysis , we use micromegas2.4 @xcite to calculate the relic density in a full numerical approach ( see also @xcite for a recent discussion of sommerfeld enhancement in a similar context ) . [ cols="^,^ " , ] the upper limits on the coupling @xmath21 can be translated into limits on the spin independent and spin dependent scattering cross section . the corresponding constraints are shown in fig.[fig : dd ] , together with upper limits from xenon100 @xcite and lux @xcite . for small mass splitting @xmath37 , the direct detection cross sections are resonantly enhanced , while the collider limits are weakened as discussed above . on the other hand , for @xmath38 , the collider search is very effective , while the direct detection cross section is suppressed for majorana dark matter with chiral couplings , as discussed in sec.[sec : dd ] . consequently , when converted into the direct detection cross section , the atlas limits can be stronger by one to several orders of magnitude than current bounds from direct detection experiments for masses in the range @xmath39gev . for masses around @xmath40gev and @xmath38 , the atlas constraint is strong enough to exclude even the qcd contribution to the production cross section at lhc . this translates into the dip in the constraint visible in the middle row of fig.[fig : dd ] . note , however , that while in general collider uncertainties only have a moderate impact , the upper limit is considerably affected in this range ( see blue dotted lines in fig.[fig : dd ] ) . for comparison , the atlas constraint for @xmath41 mediators is also shown by the blue dashed lines in fig.[fig : dd ] . the cross section expected for a thermal relic is also shown as black solid ( dashed ) line for @xmath42 ( @xmath41 ) mediators . comparison of constraints on the annihilation cross section obtained from searches for spectral features by the fermi - lat @xcite and h.e.s.s . @xcite ( _ cf_. @xcite ) , with constraints inferred from collider searches for jets and missing energy by atlas @xcite , as well as direct detection limits from xenon100 @xcite and lux @xcite . the black line corresponds to a thermal wimp , and the dotted lines indicate the uncertainty of the collider constraint , as discussed before . note that the results for @xmath41 mediator are very similar , and are therefore not shown . , title="fig:",scaledwidth=70.0% ] + comparison of constraints on the annihilation cross section obtained from searches for spectral features by the fermi - lat @xcite and h.e.s.s . @xcite ( _ cf_. @xcite ) , with constraints inferred from collider searches for jets and missing energy by atlas @xcite , as well as direct detection limits from xenon100 @xcite and lux @xcite . the black line corresponds to a thermal wimp , and the dotted lines indicate the uncertainty of the collider constraint , as discussed before . note that the results for @xmath41 mediator are very similar , and are therefore not shown . , title="fig:",scaledwidth=70.0% ] constraints on thermally produced wimp dark matter with a coloured mediator particle @xmath20 . the green region is excluded at @xmath43c.l . by the atlas search @xcite for jets and missing transverse energy . for comparison , the red shaded area is excluded by direct searches . the blue lines indicate the regions excluded by the search for an internal bremsstrahlung feature in the gamma - ray spectrum from the central galactic halo measured by h.e.s.s . @xcite , assuming a boost factor @xmath44 or @xmath40 , respectively @xcite . within the grey shaded region in the lower left corner , thermal production can not make up for the whole dark matter abundance due to efficient coannihilations . within the upper right corner , non - perturbatively large values of @xmath45 would be required . below the upper(lower ) gray line @xmath46 . the gray dashed line indicates the masses for which the coupling of the mediator equals the one of a squark.,scaledwidth=95.0% ] one of the most interesting features of the dark matter model discussed here is the presence of a sharp spectral feature in the dark matter annihilation spectrum . it arises mainly from internal bremsstrahlung @xmath27 for @xmath47 , while for @xmath48 , also the gamma - ray line resulting from the loop - induced process @xmath49 gives a significant contribution . in fig.[fig : mdmvssigv ] , we compare constraints on @xmath50 from gamma - ray observations of the central galactic halo by fermi - lat @xcite and h.e.s.s . @xcite ( blue shaded regions ) with those inferred from direct detection @xcite and from the atlas search @xcite considered here . as expected , for small mass splitting , the region excluded by the lhc search ( green region ) is less constraining than xenon100 and lux ( red region ) . however , for mass splitting of order one , the atlas search severely constrains the possibility to observe a spectral feature arising from dark matter with a coloured mediator below @xmath51tev energies . by requiring that thermal freeze - out yields a relic abundance that coincides with the value measured by planck @xcite , it is possible to fix the coupling @xmath52 between dark matter @xmath6 , the mediator @xmath20 and the sm quarks for each set of masses . under this assumption , the model has only two free parameters , which we take to be the dark matter mass @xmath25 and the mass splitting @xmath53 . the collider limits considered here can be translated into an exclusion region , which we show in fig . [ fig : mdmvssplitting_lhc_thermalcoupling ] ( green region ) . for mass splitting @xmath54 , it reaches up to @xmath55tev . however , for smaller or larger mass splitting , much lighter masses remain allowed . on the one hand , for much larger splitting , the mediator @xmath20 becomes too heavy to be produced effectively . for much smaller splitting , on the other hand , two effects play a role : first , the collider search becomes less effective in this regime . second , the coupling @xmath56 gets very small due to efficient coannihilations . the combination of these effects also leads to the relatively large uncertainties in the exclusion region ( green dotted lines ) , in particular as the thermal cross section and the lhc exclusion happen to exhibit a fairly similar dependence on the mass splitting in certain regions of parameter space , see e.g. fig . [ fig : splittingvscrosssection ] . for comparison , direct detection mostly probes a region with smaller mass splitting , due to the resonant enhancement of the nucleon scattering cross section for @xmath57 ( red regions ; note that for lux only limits on spin independent scattering are available at present ) . the limits from indirect detection are currently not sensitive to the flux expected for a thermal relic , if the standard einasto profile from @xcite is adopted . however , if the flux is enhanced by a boost factor of order @xmath58 , they probe the multi - tev region ( blue contour lines ) . for comparison , we also show a constraint inferred from mono - jet searches for nearly degenerate particle spectra @xcite , which is sensitive to very small splittings for low dark matter masses . the large hadron collider offers a unique environment to search for dark matter particles with masses below @xmath59 tev through their possible production in partonic collisions . to optimize the search it is convenient to identify simplified models that characterize the signals of a larger class of dark matter models . in this paper we have focused on a model with majorana dark matter particles that couple to the up - type quarks via one or several coloured mediators and which produces a signal consisting in two or more jets plus missing transverse energy , through the production and subsequent decay of the coloured scalar particles . we have carefully analysed the production of coloured scalar particles at the lhc , considering not only the production via the strong gauge interaction , but also via the exchange of a dark matter particle in the t - channel . the latter production channel can be relevant and even dominant in some regions of the parameter space leading to the observed dark matter abundance via the thermal freeze - out of dark matter particles in the early universe . more specifically , we have emphasized the importance of the partonic subprocess @xmath60 mediated by a majorana dark matter particle in the t - channel . due to the enhancement of the rate by the square of the dark matter mass and due to the unsuppressed parton distribution function of up - quarks inside the proton , this process is the dominant production channel in large regions of the parameter space . concretely , for large dark matter masses and a coloured scalar with comparable mass , we have found that the total production cross section of coloured scalars can be enhanced by more than two orders of magnitude compared to the production channels mediated by the strong interactions . we have then derived limits on the parameters of the model employing the atlas search @xcite for jets and missing transverse energy , based on @xmath61fb@xmath62 of data collected at a center of mass energy of @xmath63tev . to re - interpret the analysis for the model considered here , we have computed the appropriate efficiencies for the relevant production channels , taking jet matching with two additional hard jets into account , for all signal regions containing two to four jets . next , we have investigated the complementarity of the collider limits with those from direct detection and indirect detection experiments . we have found that , for some regions of the parameter space of the model , the atlas searches imposes the strongest limits and rules out choices of parameters leading to the observed dark matter abundance via thermal production . for small mass splitting between the dark matter and the mediator , the collider limits are comparable to bounds from direct detection . however , if the mass splitting is of the same order as the dark matter mass , the atlas limits are considerably stronger than the latest bounds from xenon100 and lux , reaching down to @xmath64@xmath65 for @xmath66gev@xmath67tev and @xmath68 . this is due to a relative suppression of the spin - independent scattering cross section for majorana dark matter with chiral couplings , and the enhancement of the production at lhc described above . searches for sharp spectral features at gamma - ray telescopes are fully complementary in the multi - tev region . however , for @xmath69tev direct detection and collider constraints in some cases even preclude the possibility of observing sharp spectral features at future gamma - ray telescopes for the standard choices of the astrophysical parameters . it is important to stress that these limits do not suffer from astrophysical uncertainties and are therefore very robust . we have estimated uncertainties arising from the determination of efficiencies and from higher - order contributions to the production cross section , which are typically @xmath70 but can be larger in particular cases . lastly , we have also considered an extension of the model by extra coloured scalars , inspired by the particle content of the minimal supersymmetric standard model . we have found that our main conclusions still remain for the scenarios in agreement with the flavour physics experiments . we are grateful to miguel pato for earlier collaborations on which part of this work was based , and to andreas weiler for helpful discussions and for cross - checking efficiencies . this work has been partially supported by the dfg cluster of excellence `` origin and structure of the universe '' and by the dfg collaborative research center 676 `` particles , strings and the early universe '' . s.v . also acknowledges support from the dfg graduiertenkolleg `` particle physics at the energy frontier of new phenomena '' . the interaction term of the dark matter particle and the coloured scalars with the right - handed quarks in general violates the @xmath71 flavour symmetry . therefore , it is necessary to check whether the stringent constraints arising from flavour physics are satisfied . in this appendix we discuss how two well - known possibilities to suppress flavour - changing neutral currents , namely degeneracy or alignment , can be realized within the toy - model considered in this work . consider first the possibility of a single coloured scalar @xmath20 , but allowing for arbitrary couplings @xmath73 to all right - handed quarks , @xmath74 where @xmath75 corresponds to @xmath76 . in this case the box diagram shown in fig . [ fig : ddbar ] gives a contribution to @xmath72 mixing , which is strongly constrained by the measured value of the @xmath77-meson mass splitting @xmath78 ( note that there is no contribution to cp violation in presence of a single species @xmath20 , such that constraints from @xmath79 do not apply ) . the box diagram gives a contribution to the operator @xmath80 given by @xmath81 where @xmath82 ( with @xmath83 ) . the functions @xmath84 and @xmath85 are given e.g. in @xcite . on the other hand , the experimental constraint inferred from measurements of @xmath78 is @xmath86 @xcite . for @xmath87 , this translates into an upper bound @xmath88 since thermal production requires typically @xmath89 , this means that @xmath20 has to couple nearly exclusively to the up - quark , with very suppressed coupling to charm ( or vice - versa ) . a possible exception are regions in parameter space with strong coannihilation for which @xmath90 . for a generic @xmath89 , the flavour - vector @xmath73 should be aligned with the mass eigenbasis of the quarks . this can be realized e.g. in the presence of a @xmath91 flavour symmetry under which @xmath92 and @xmath20 transform with equal charge , while all other states are uncharged . this symmetry is then broken only by the ckm mixing in the left - handed quark sector , and thus this breaking should lead to a misalignment suppressed by the quark masses as well as ckm mixing angles . more precisely , one may consider a situation where @xmath93 at some high scale @xmath94 . due to renormalization group running , the quark mass matrices @xmath95 and @xmath96 are scale - dependent . this leads to a running of the diagonalization matrices @xmath97 , with a similar expression for the down - type quarks . the left - handed rotations lead to the well - known running of the ckm matrix @xmath98 , while the right - handed rotations are unobservable in the standard model @xcite . however , in the present case they lead to a flavour - dependent running of the dark matter couplings , @xmath99 where we neglect flavour - insensitive contributions to the running and assume that @xmath100 . using the one - loop rges for the quark mass matrices from @xcite , one finds for the off - diagonal entry corresponding to @xmath101 and @xmath102 @xmath103 where @xmath104 etc . denotes the ckm matrix elements . thus , even for perfect alignment @xmath105 at the high scale , the coupling to the second generation induced by the running is approximately @xmath106 which is safely below the upper bound required from @xmath72 mixing . alternatively , one may consider a situation where three additional scalars @xmath7 are introduced , which are taken to transform under the @xmath71 flavour symmetry . then the allowed coupling is of the form @xmath107 and the @xmath7 are all mass - degenerate . one may consider a breaking of the symmetry in the scalar mass matrix , which induces non - degenerate mass eigenvalues of the @xmath7 , and singles out a preferred basis , namely the mass eigenbasis ( similar to the right - handed squarks in the mssm ) . after rotating into this basis ( as well as the mass basis for the quarks ) the interaction term has the generic form @xmath108 where @xmath109 is a unitary matrix , which can have large off - diagonal entries . the resulting contribution to the box diagram will be proportional to @xcite @xmath110 where @xmath111 is a function of the masses . in the limit of degenerate masses this expression goes to zero by virtue of the unitarity condition @xmath112 . lets assume for concreteness that the mixing with the third generation is negligible , similar as in the ckm matrix . in this case the box diagram gives a contribution @xmath113 with @xmath114 and @xmath115 . thus , the strong requirement of precise alignment @xmath116 found above can be considerably relaxed provided the masses are quasi - degenerate . for order one mixing , the upper bound on @xmath117 required from @xmath72 mixing then translates into an upper bound on the mass splitting @xmath118 thus , in both cases discussed above , the flavour constraints can be fulfilled in presence of an ( approximate ) flavour symmetry .

we investigate the signatures at the large hadron collider of a minimal model where the dark matter particle is a majorana fermion that couples to the standard model via one or several coloured mediators . we emphasize the importance of the production channel of coloured scalars through the exchange of a dark matter particle in the t - channel , and perform a dedicated analysis of searches for jets and missing energy for this model . we find that the collider constraints are highly competitive compared to direct detection , and can even be considerably stronger over a wide range of parameters . we also discuss the complementarity with searches for spectral features at gamma - ray telescopes and comment on the possibility of several coloured mediators , which is further constrained by flavour observables .

in this paper , we give canonical matrices of bilinear or sesquilinear forms @xmath12 where @xmath0 is a complex vector space and @xmath2 is its subspace . we use the following canonical matrices of bilinear or sesquilinear forms on a complex vector space given in @xcite ( see also @xcite ) . two square complex matrices @xmath13 and @xmath14 are said to be _ congruent _ or * _ congruent _ if there is a nonsingular @xmath15 such that @xmath16 or , respectively , @xmath17 , where @xmath18 denotes the complex conjugate transpose of @xmath15 . define the @xmath19-by-@xmath19 matrices @xmath20 @xmath21 [ @xcite ] [ bilin ] ( a ) every square complex matrix is congruent to a direct sum , determined uniquely up to permutation of summands , of matrices of the form @xmath22 in which @xmath23 , @xmath24 , and @xmath25 is determined up to replacement by @xmath26 . \(b ) every square complex matrix is congruent to a direct sum , determined uniquely up to permutation of summands , of matrices of the form @xmath27 in which @xmath28 and @xmath29 . alternatively , one may use the symmetric matrix @xmath30 instead of @xmath31 . @xmath32 a canonical form of a square matrix for congruence/*congruence over any field @xmath1 of characteristic different from 2 was given in @xcite up to classification of hermitian forms over finite extensions of @xmath1 . let us formulate the main result . for generality , we will consider matrices over any field or skew field @xmath1 with involution @xmath33 , that is , a bijection on @xmath1 such that @xmath34 for all @xmath35 . we denote the @xmath36-by-@xmath19 zero matrix by @xmath37 , or by @xmath38 if @xmath39 . it is agreed that there exists exactly one matrix of size @xmath40 and there exists exactly one matrix of size @xmath41 for every nonnegative integer @xmath19 ; they represent the linear mappings @xmath42 and @xmath43 and are considered as the zero matrices @xmath44 and @xmath45 . for every @xmath46 matrix @xmath47 we have @xmath48 and @xmath49 in particular , @xmath50 for each matrix @xmath51 $ ] over @xmath1 , we define its _ conjugate transpose _ @xmath52.\ ] ] if @xmath53 for some nonsingular matrix @xmath15 , then @xmath13 and @xmath14 are said to be * _ congruent _ ( or _ congruent _ if @xmath1 is a field and the involution on @xmath1 is the identity in what follows we consider congruence as a special case of * congruence ) . a _ sesquilinear form _ on right vector spaces @xmath2 and @xmath0 over @xmath1 is a map @xmath54 satisfying @xmath55 for all @xmath56 , and @xmath57 . if @xmath1 is a field and the involution on @xmath1 is the identity , then a sesquilinear form becomes bilinear we consider bilinear forms as a special case of sesquilinear forms . if @xmath58 and @xmath59 are bases of @xmath2 and @xmath0 , then @xmath60,\qquad \alpha_{ij}:={\cal g}(e_i , f_j),\ ] ] is the _ matrix of @xmath61 _ in these bases . its matrix in other bases @xmath62 and @xmath63 can be found by the formula @xmath64 where @xmath15 and @xmath65 are the change of basis matrices . for every @xmath66 and @xmath67 , @xmath68_e^*\,g_{ef}\,[v]_f,\ ] ] where @xmath69_e$ ] and @xmath70_f$ ] are the coordinate column - vectors of @xmath71 and @xmath72 . in this paper , we study sesquilinear forms @xmath73 in which @xmath2 is a subspace of @xmath0 , so we always consider their matrices in those bases of @xmath2 and @xmath0 that are concordant as follows . [ def0 ] let @xmath74 be one of sesquilinear forms , in which @xmath0 is a right space over @xmath1 , and @xmath2 is its subspace . choose a basis @xmath75 of @xmath0 such that @xmath76 by the _ matrix of @xmath61 in the basis @xmath75 _ , we mean the block matrix @xmath77=\left.\left [ \begin{matrix } \alpha_{11}&\dots&\alpha_{1m}\\ \vdots&\ddots&\vdots\\ \alpha_{m1}&\dots&\alpha_{mm } \end{matrix } \right| \begin{matrix } \alpha_{1,m+1}&\dots&\alpha_{1n}\\ \vdots&&\vdots\\ \alpha_{m , m+1}&\dots&\alpha_{mn } \end{matrix } \right],\ ] ] in which @xmath78 by the _ block - direct sum _ of block matrices @xmath79 $ ] and @xmath80 $ ] , we mean the block matrix @xmath81\uplus[a_2|b_2]:= \left.\left [ \begin{matrix } a_1&0\\0&a_2 \end{matrix } \right| \begin{matrix } b_1&0\\0&b_2\end{matrix } \right].\ ] ] in section [ s - pr ] we will prove the following theorem ( a stronger statement was proved in ( * ? ? ? * theorem 1 ) in the case @xmath82 ) . [ t0.01 ] let @xmath1 be a field or skew field with involution @xmath83possibly , the identity if @xmath84 is a field@xmath85 , @xmath0 be a right vector space over @xmath1 , and @xmath2 be its subspace . let @xmath74 be one of sesquilinear forms @xmath86 \(a ) there exists a basis @xmath75 of @xmath0 satisfying , in which the matrix of @xmath74 is a block - direct sum of a @xmath87-by-@xmath87 matrix @xmath88,\qquad \text{$k$ is nonsingular,}\ ] ] and matrices of the form @xmath89\ \ ( q{\geqslant}1 ) , \qquad [ j_q(0)|e_q]\ \ ( q{\geqslant}0),\ ] ] in which @xmath90 @xmath83the summands or may be absent@xmath85 . the block @xmath91 is determined by @xmath74 uniquely up to congruence , and the summands of the form are determined by @xmath74 uniquely up to permutation . \(b ) if @xmath8 , then one can replace in this direct sum the summand by @xmath92\uplus\dots\uplus [ k_s|0_{p_s0}],\ ] ] where @xmath93 is the canonical form of @xmath91 defined in theorem [ bilin ] and each @xmath94 is @xmath95-by-@xmath95 . the obtained block - direct sum is determined by @xmath74 uniquely up to permutation of summands , and so it is a canonical matrix of the sesquilinear @xmath83 in particular , bilinear _ _ ) _ _ form @xmath61 . let us formulate an analogous statement for matrices of linear mappings . [ defm ] let @xmath1 be a field or skew field , @xmath0 be a right vector space over @xmath1 , and @xmath2 be its subspace . let @xmath96 be one of linear mappings @xmath97 choose a basis @xmath75 of @xmath0 such that @xmath98 by the _ matrix @xmath99 of @xmath96 in the basis @xmath75 _ , we mean its matrix in the bases @xmath100 and @xmath75 of @xmath0 . we divide @xmath99 into two blocks @xmath101 , & \text{if $ u\to v$ or $ v / u\to v$ , } \\[5 mm ] [ a| b ] , & \text{if $ v\to u$ or $ v\to v / u$ , } \end{cases}\ ] ] where @xmath13 is @xmath36-by-@xmath36 . the following theorem will be proved in section [ s - pr ] . [ t.m ] let @xmath1 be a field or skew field , @xmath0 be a right vector space over @xmath1 , and @xmath2 be its subspace . let @xmath96 be one of linear mappings @xmath102 \(a ) there exists a basis @xmath75 of @xmath0 satisfying , in which for the matrix @xmath99 of @xmath103 we have : @xmath104 is a block - direct sum of a @xmath87-by-@xmath87 matrix @xmath105,\qquad \text{$k$ is nonsingular,}\ ] ] and matrices of the form @xmath106 , \qquad [ j_q(0)|e_q],\ ] ] where @xmath107 was defined in @xmath83the summands or may be absent@xmath85 . the block @xmath91 is determined by @xmath96 uniquely up to similarity , and the summands of the form are determined by @xmath96 uniquely up to permutation . \(b ) if @xmath8 , then one can replace the summand by a block - direct sum of square matrices of the form @xmath108.\ ] ] the obtained matrix is determined by @xmath96 uniquely up to permutation of summands , and so it is a canonical matrix of the linear mapping @xmath96 . we do not rate theorem [ t.m ] as new ; it is readily available from the canonical form problem solved in @xcite . we include it in our paper since the singular indecomposable summands of the canonical forms in theorems [ t0.01 ] and [ t.m ] coincide , and our proofs of theorems [ t0.01 ] and [ t.m ] are similar and are based on _ regularization algorithms _ that decompose the matrix of each form and each mapping into a block - direct sum of * its _ regular part _ @xmath109 $ ] with nonsingular @xmath91 ( see and ) , which is determined by or up to * congruence or similarity , and of * its _ singular summands _ of the form @xmath110 $ ] and @xmath111 $ ] ( see and ) , which are determined uniquely . if @xmath8 , then these algorithms can use only unitary transformations , which improves their numerical stability . these algorithms extend the regularization algorithm @xcite for a bilinear / sesquilinear form , which decomposes its matrix into a direct sum of a nonsingular matrix and several singular jordan blocks . an analogous regularization algorithm was given by van dooren @xcite for matrix pencils and was extended to matrices of cycles of linear mappings in @xcite . the canonical form problems for matrices of forms and mappings are special cases of the canonical form problem for block matrices , whose form resembles @xmath112 \ ] ] bangle.eps [ def1 ] by a _ bangle _ over @xmath113 we mean a matrix @xmath114\ ] ] over @xmath1 , partitioned into vertical strips , among which one strip @xmath115 is square and boxed . the number @xmath116 of rows of @xmath13 and the number @xmath117 of columns of each strip @xmath118 are nonnegative integers . let @xmath119\ ] ] be another bangle with the same sizes of strips and the same @xmath120 and @xmath121 . we say that the bangles @xmath13 and @xmath14 are * _ congruent _ or , respectively , _ similar _ and write @xmath122 if there exists a nonsingular upper block - triangular matrix @xmath123 over @xmath1 such that @xmath124 then @xmath125 this means that the boxed strips of * congruent / similar bangles are * congruent / similar . the following lemma is obvious . [ s - res1 ] two bangles are congruent / similar if and only if one reduces to the other by a sequence of the following transformations : * any transformation with rows of the whole matrix , and then the congruent / similar transformation with columns of the boxed strip @xmath83this transformation reduces to @xmath126\ ] ] or , respectively , @xmath127\ ] ] with a nonsingular @xmath128 . * any transformation with columns of an unboxed strip . * addition of a linear combination of columns of the @xmath129 strip to a column of the @xmath130 strip if @xmath131.@xmath32 note that the canonical form problem for matrices of forms and mappings is the canonical form problem for bangles with two strips . but applying our algorithm to bangles with two strips we can produce bangles with three strips ( see section [ sub - r2 ] ) ; so we consider bangles with an arbitrary number of strips . the paper is organized as follows . in section [ s - a ] we formulate our main theorem about the existence of a regularizing decomposition of a bangle . in sections [ s - red ] and [ s - redz ] we construct regularizing decompositions of bangles with respect to * congruence and similarity . in section [ s - pr ] we use these decompositions to prove the main theorem and theorems [ t0.01 ] and [ t.m ] . in this section , we formulate our main theorem , which reduces the canonical form problem for bangles up to * congruence / similarity to the canonical form problem for nonsingular matrices up to * congruence / similarity , and solves it for complex bangles . by the _ block - direct sum _ of two bangles and with the same number of strips and the same position of the boxed strip , we mean the bangle @xmath132.\ ] ] [ defin ] a _ regularizing decomposition _ of a bangle @xmath133\ ] ] over a field or skew field @xmath1 with respect to congruence / similarity is a bangle @xmath134 satisfying two conditions : * @xmath134 is congruent / similar to @xmath13 , and * @xmath134 is the block - direct sum of * * its _ regular part _ @xmath135 , \qquad \text{$k$ is nonsingular,}\ ] ] * * and its _ singular part _ being a block - direct sum of matrices of the form @xmath136 , \\ \label{e10.5c } \bigl[\dots|\,e_q\,|\dots \ , \boxed{\;j_q(0)\;}\ , \dots\bigr ] , \qquad \bigl[\dots\ , \boxed{\;j_q(0)\;}\ , \dots|\,e_q\,|\dots\bigr],\end{gathered}\ ] ] in which @xmath107 is defined in and the dots denote sequences of strips @xmath137 . + both the regular and the singular parts may have size @xmath138-by-@xmath138 . the following theorem generalizes theorems [ t0.01 ] and [ t.m ] . [ l10.1 ] ( a ) over a field or skew field @xmath1 , any bangle @xmath13 possesses regularizing decompositions for congruence and for similarity , their regular parts are determined by @xmath13 uniquely up to congruence and , respectively , similarity , and their singular parts are determined by @xmath13 uniquely up to permutation of summands . \(b ) if @xmath139 and @xmath134 is a regularizing decomposition of a bangle @xmath13 for to congruence , then its regular part is congruent to the block - direct sum @xmath140,\ ] ] in which @xmath141 is the canonical form of @xmath91 defined in theorem [ bilin ] and each @xmath94 is @xmath95-by-@xmath95 . replacing in @xmath134 the regular part by this block - direct sum , we obtain a canonical form of @xmath13 for congruence @xmath83 in particular , for congruence@xmath85 since the obtained bangle is congruent to @xmath13 and is determined by @xmath13 uniquely up to permutation of summands . \(c ) if @xmath139 and @xmath134 is a regularizing decomposition of a bangle @xmath13 for similarity , then its regular part is similar to a block - direct sum of matrices of the form @xmath142 , \qquad\lambda\ne 0.\ ] ] replacing in @xmath134 the regular part by this block - direct sum , we obtain a canonical form of @xmath13 for similarity since the obtained bangle is similar to @xmath13 and is determined by @xmath13 uniquely up to permutation of summands . note that for bangles with respect to similarity this theorem can be deduced from the canonical form problem solved in @xcite . we give an algorithm that for every bangle over a field or skew field @xmath1 constructs its regularizing decomposition for * congruence . if @xmath8 , then we can improve the numerical stability of this algorithm using only unitary transformations . the algorithm is the alternating sequence of left - hand and right - hand reductions , which we define in sections [ sub - r1 ] and [ sub - r2 ] . let @xmath143\ ] ] be a bangle over @xmath1 . producing * congruence transformations ( a)(c ) from lemma [ s - res1 ] with @xmath13 , we can reduce its submatrix @xmath144 $ ] by the following transformations : * arbitrary transformations of rows ; * arbitrary transformations of columns within any vertical strip @xmath118 ; * addition of a linear combination of columns of the @xmath129 strip to a column of the @xmath130 strip if @xmath131 . first we reduce @xmath144 $ ] to the form @xmath145\ ] ] using transformations ( b@xmath146 ) with @xmath147 and ( a@xmath146 ) , then make zero @xmath148 by transformations ( c@xmath146 ) . transforming analogously the submatrix @xmath149 $ ] , we reduce to the form @xmath150;\ ] ] and so on . repeat this process until obtain @xmath151,\qquad r_2{\geqslant}0,\ \dots,\ r_{k}{\geqslant}0,\ ] ] and extend the obtained partition into horizontal strips to the whole bangle . make zero all horizontal strips of the blocks @xmath152 except for the first strip and obtain @xmath153\ ] ] ( we have divided the boxed block @xmath115 into @xmath120 vertical strips conformally to its partition into horizontal strips ) for some @xmath154=:l(a).\ ] ] clearly , @xmath155 are uniquely determined by @xmath13 . we say that a bangle @xmath13 reduces to a bangle @xmath14 by _ admissible permutations _ and write @xmath156 if @xmath13 reduces to @xmath14 by a sequence of the following transformations : * permutation of rows of the whole matrix and then the same permutation of columns of the boxed strip , * permutation of columns in an unboxed strip . clearly , @xmath157 ( in the notation ) . [ lem_r1 ] ( a ) the equivalence @xmath158 holds for all @xmath159 , \qquad n=\bigl[\,\boxed{\;n_1\;}\ n_2\,|\dots|\,n_{t}\,\bigr],\ ] ] and each @xmath160 . \(b ) if @xmath161 , then for every bangle @xmath13 we can find using only unitary transformations . \(a ) the equivalence is trivial if @xmath162 . let @xmath163 . reasoning by induction on @xmath120 , we assume that @xmath164 and prove the equivalence as follows . * suppose @xmath165 , that is , @xmath166 for some nonsingular @xmath167 since both @xmath168 and @xmath169 have the same first vertical strip @xmath170 ( we join its zero horizontal strips ) , by we have @xmath171 and so @xmath172 has the form @xmath173 let @xmath174 be a submatrix of with @xmath172 of the form . due to , @xmath175 so @xmath176 , and by @xmath177 . * suppose @xmath178 . by , @xmath176 , this ensures @xmath179 for some nonsingular + @xmath180 + denote by @xmath181 and @xmath182 the strips of @xmath183 and @xmath184 : @xmath185,\ ] ] @xmath186.\ ] ] then @xmath187\ ] ] and by @xmath188\overset{*}{\sim } { \cal l}_k(n),\end{gathered}\ ] ] where @xmath189 are some matrices . this proves . let us give an alternative proof of using * congruence transformations ( a)(c ) from lemma [ s - res1 ] . due to that lemma , it suffices to show that those transformations ( a)(c ) with that preserve all of its blocks except for @xmath190 produce all transformations ( a)(c ) with . * we can add a column of @xmath191 to a column of @xmath192 if @xmath131 . indeed , in the case @xmath193 this is a column - transformation within the boxed block of @xmath194 , and so we must produce the * congruent row - transformation add the corresponding row of the @xmath195 horizontal strip of to the row of the @xmath196 horizontal strip . this spoils zero blocks of the @xmath196 horizontal strip , but they are repaired by additions of columns of @xmath197 . * we can also make arbitrary elementary transformations with columns of @xmath191 if @xmath198 : in the case @xmath199 these transformations spoil @xmath200 but it is restored by transformations with its columns . \(b ) let @xmath8 . we must prove that if @xmath133\ ] ] is reduced to by the algorithm from this section , then @xmath201 and @xmath202 can be found using only unitary transformations with @xmath13 . by unitary column - transformations within vertical strips @xmath203 of @xmath13 and by unitary row - transformations , we sequentially reduce its submatrix @xmath144 $ ] to the form @xmath204,\ ] ] where each @xmath205 is a nonsingular @xmath206-by-@xmath206 block and all @xmath207 s are unspecified blocks ( this reduction was studied thoroughly in ( * ? ? ? * section 4 ) ) . the matrix @xmath13 takes the form @xmath208,\ ] ] in which @xmath209 are @xmath210 matrices . replacing @xmath211 by the identity matrices of the same sizes and all @xmath207 s by the zero matrices , we obtain because can be reduced to by those transformations ( a)(c ) from lemma [ s - res1 ] that preserve @xmath201 and @xmath202 . let @xmath212\ ] ] be a bangle over a field or skew field @xmath1 . first we reduce @xmath13 by * congruence transformations @xmath213,\qquad \text{$s$ is nonsingular,}\ ] ] to the form @xmath214,\ ] ] in which the rows of @xmath215 $ ] are linearly independent and @xmath216 is square . then we make zero @xmath217 adding columns of @xmath218 and @xmath219 , and as in sequentially reduce @xmath220 $ ] to the form @xmath221,\ ] ] obtaining a partition of the first horizontal strip of into @xmath121 substrips . conformally divide the first vertical strip of the boxed block into @xmath121 substrips and obtain @xmath222\end{gathered}\ ] ] for some @xmath223=:r(a)\ ] ] with @xmath224 . [ lem_r2 ] ( a ) the equivalence @xmath225 holds for all @xmath226 , \qquad n=\bigl[\,n_1|\dots|n_{t}\ \boxed{\;n_{t+1}\;}\ \bigr].\ ] ] \(b ) if @xmath161 , then for every bangle @xmath13 of the form we can find using only unitary transformations . \(a ) let us prove the equivalence using * congruence transformations ( a)(c ) from lemma [ s - res1 ] ( alternatively , one could use induction on @xmath121 as in the proof of lemma [ lem_r1](a ) ) . due to lemma [ s - res1 ] , it suffices to show that those transformations ( a)(c ) with that preserve all of its blocks except for @xmath227 produce all transformations ( a)(c ) with . * we can add a column of @xmath191 to a column of @xmath192 if @xmath131 ; by the definition of * congruence transformations we must add the corresponding row of the @xmath195 horizontal strip of to the row of the @xmath196 horizontal strip ; although this spoils a zero block of the @xmath196 horizontal strip if @xmath198 , but it can be repaired by additions of columns of @xmath228 . * we can also make arbitrary elementary transformations with columns of @xmath191 if @xmath229 : these transformations spoil @xmath200 if @xmath198 , but it is restored by transformations with its columns . \(b ) let @xmath8 . first we reduce the bangle by transformations with unitary @xmath15 to the form , in which the rows of @xmath230 $ ] are linearly independent and @xmath216 is square . then we sequentially reduce @xmath231 $ ] by unitary column - transformations within vertical strips and by unitary row - transformations to the form @xmath232,\ ] ] where each @xmath205 is a nonsingular @xmath206-by-@xmath206 block and the @xmath207 s are unspecified blocks . the matrix @xmath13 takes the form @xmath233,\ ] ] where @xmath224 . replacing @xmath211 by the identity matrices of the same sizes and all @xmath207 s by the zero matrices , we obtain because can be reduced to by those transformations ( a)(c ) from lemma [ s - res1 ] that preserve @xmath234 and @xmath227 . for any bangle @xmath235\ ] ] over @xmath1 , its regularizing decomposition for * congruence can be constructed as follows . alternating the left - hand and the right - hand reductions for * congruence , we construct the sequence of bangles @xmath236 until obtain @xmath237 \quad\text{or}\quad a^{(n)}=\bigl[\,0_{p0}\,|\dots|\ , 0_{p0}\ \boxed{\ k\ } \ \bigr]\ ] ] with a nonsingular @xmath91 . producing this reduction , we in each step have deleted the reduced parts of @xmath13 ; say , in step 1 we reduced @xmath13 to the form and took only its unreduced part @xmath238 . let us repeat the reduction of preserving all the reduced parts of @xmath13 : * in step 1 we transform @xmath13 to @xmath239 of the form . * in step 2 we reduce its subbangle @xmath240 to @xmath241 preserving the other blocks of @xmath239 , and so on . after @xmath19 steps , instead of we obtain some bangle @xmath242 , which is * congruent to @xmath13 . due to the next theorem , @xmath243 is a regularizing decomposition of @xmath13 up to admissible permutations of rows and columns . [ tel.1 ] if @xmath13 is a bangle over a field or skew field @xmath1 , then @xmath243 reduces by admissible permutations of rows and columns to a regularizing decomposition of @xmath13 for congruence . we give a constructive proof of this theorem . by admissible permutations of rows and columns , @xmath243 reduces to a block - direct sum of the bangle in which @xmath91 is the same as in , and a bangle @xmath244 in which each row and each column contains at most one @xmath245 and its other entries are zero . we obtain a regularizing decomposition of @xmath13 for * congruence replacing @xmath244 in this block - direct sum by @xmath246 from the follows statement . @xmath247 let us prove . by admissible permutations of rows and columns of @xmath244 , we reduce its boxed strip @xmath248 to a direct sum of singular jordan blocks . then we rearrange columns in each unboxed strip such that if its @xmath249 and @xmath250 entries are @xmath245 and @xmath251 , then @xmath252 . it is easy to see that the obtained bangle @xmath246 is a block - direct sum of bangles of the form and : each singular jordan block @xmath253 in the decomposition of @xmath248 gives the summand if those row of @xmath244 that contains the last ( zero ) row of @xmath253 is zero , and the summand otherwise . the summands with @xmath254 give zero columns in unboxed strips of @xmath244 . we give an algorithm that for every bangle over a field or skew field @xmath1 constructs its regularizing decomposition for similarity . if @xmath8 , then we can improve the numerical stability of this algorithm using only unitary transformations . let @xmath255\ ] ] be a bangle over @xmath1 . using similarity transformations with @xmath13 , we can reduce its submatrix @xmath144 $ ] by transformations ( a@xmath146)(c@xmath146 ) from section [ sub - r1 ] . we reduce this submatrix to the form @xmath256,\qquad r_1{\geqslant}0,\ \dots,\ r_{k-1}{\geqslant}0,\ ] ] and obtain a partition of the bangle @xmath13 into @xmath120 horizontal strips . then we divide the boxed block @xmath115 into @xmath120 vertical substrips of the same sizes , make zero all horizontal strips in the blocks @xmath152 except for the last strip , and obtain @xmath257\end{gathered}\ ] ] for some @xmath258=:l(a).\ ] ] [ lem_r1z ] ( a ) the equivalence @xmath259 holds for all @xmath260,\\ n & = \bigl[\,n_1\,|\dots|\ , n_{k-1}\ \boxed{\ n_k\ } \ n_{k+1}\,|\dots|\,n_t\,\bigr].\end{aligned}\ ] ] \(b ) if @xmath161 , then for every bangle @xmath13 we can find using only unitary transformations . \(a ) this statement follows from lemma [ s - res1 ] since those transformations ( a)(c ) with that preserve all of its blocks except for @xmath190 produce all transformations ( a)(c ) with . for example , we can add a column of @xmath191 to a column of @xmath192 if @xmath131 : although in the case @xmath193 we must subtract the corresponding row of the @xmath196 horizontal strip of from the row of the @xmath129 horizontal strip , and this may spoil zero blocks of the @xmath129 horizontal strip , but they are repaired by additions of columns of @xmath261 . \(b ) let @xmath8 . by unitary column - transformations within vertical strips of @xmath13 and by unitary row - transformations , we sequentially reduce its submatrix @xmath144 $ ] to the form @xmath262,\ ] ] where each @xmath205 is a nonsingular @xmath206-by-@xmath206 block and all @xmath207 s are unspecified blocks . the matrix @xmath13 takes the form @xmath263,\ ] ] in which @xmath264 are @xmath265 matrices . replacing @xmath266 by the identity matrices of the same sizes and all @xmath207 s by the zero matrices , we obtain since reduces to by those transformations ( a)(c ) from lemma [ s - res1 ] that preserve @xmath267 , @xmath202 . let @xmath268\ ] ] be a bangle over @xmath1 . first we reduce @xmath13 by similarity transformations @xmath269,\qquad \text{$s$ is nonsingular,}\ ] ] to the form @xmath270,\ ] ] in which the rows of @xmath271 $ ] are linearly independent and @xmath272 is square . then we make zero @xmath273 adding columns of @xmath274 and @xmath275 , and sequentially reduce @xmath276 $ ] to the form @xmath277.\ ] ] the matrix @xmath13 transforms to @xmath278,\end{gathered}\ ] ] for some @xmath279=:r(a)\ ] ] with @xmath280 . [ lem_r2z ] ( a ) the equivalence @xmath281 holds for all @xmath282 , \qquad n=\bigl[\:\boxed{\ n_1\ } \ n_{2}\,|\dots|\,n_{t+1}\,\bigr].\ ] ] \(b ) if @xmath161 , then for every bangle @xmath13 of the form we can find using only unitary transformations . \(a ) it is easy to show that those transformations ( a)(c ) from lemma [ s - res1 ] with that preserve all of its blocks except for @xmath227 produce all transformations ( a)(c ) with . say , we can add a column of @xmath191 to a column of @xmath192 if @xmath131 : although we must subtract the corresponding row of the @xmath196 horizontal strip of from the row of the @xmath129 horizontal strip , and this spoils zero blocks of the @xmath129 horizontal strip if @xmath283 , but they are repaired by additions of columns of @xmath200 . \(b ) let @xmath8 . first we reduce @xmath13 by transformations with unitary @xmath15 to the form , in which the rows of @xmath284 $ ] are linearly independent and @xmath272 is square . then we sequentially reduce its submatrix @xmath285 $ ] by unitary column - transformations within vertical strips and by unitary row - transformations to the form @xmath286,\ ] ] where each @xmath205 is a nonsingular @xmath206-by-@xmath206 matrix . the matrix @xmath13 takes the form @xmath287.\ ] ] replacing @xmath288 by the identity matrices of the same sizes and all @xmath207 s by the zero matrices , we obtain since reduces to by those transformations ( a)(c ) from lemma [ s - res1 ] that preserve @xmath289 , @xmath227 . for any bangle @xmath255\ ] ] over @xmath1 , its regularizing decomposition for similarity can be constructed as follows . * first we apply subsequently the left - hand reduction for similarity to @xmath13 until obtain @xmath290,\ ] ] in which the first @xmath291 strips have no columns . * then we apply subsequently the right - hand reduction for similarity to @xmath292\ ] ] until obtain @xmath293\ ] ] with a nonsingular @xmath91 . producing this reduction , we in each step have deleted the reduced parts of @xmath13 . let us repeat the reduction preserving all the reduced parts of @xmath13 and denote the obtained bangle by @xmath294 . clearly , @xmath295 is similar to @xmath13 . due to the next theorem , @xmath294 is a regularizing decomposition of @xmath13 up to admissible permutations of rows and columns . [ tel.1z ] if @xmath13 is a bangle over a field or skew field @xmath1 , then @xmath294 reduces by admissible permutations of rows and columns to a regularizing decomposition of @xmath13 for similarity . we give a constructive proof of this theorem . by admissible permutations of rows and columns , @xmath294 is reduced to a block - direct sum of the bangle with @xmath91 from and a bangle @xmath244 in which each row and each column contains at most one @xmath245 and the other entries are zero . replacing @xmath244 in this block - direct sum by @xmath246 from , we obtain a regularizing decomposition of @xmath13 for similarity . \(a ) let us prove the statement ( a ) for * congruence ; its proof for similarity is analogous . let @xmath13 be a bangle over @xmath1 . in view of theorem [ tel.1 ] , @xmath13 possesses a regularizing decomposition for congruence , which is obtained from @xmath243 by admissible permutations of rows and columns . let @xmath296 and @xmath297 be two regularizing decompositions of @xmath13 . then @xmath298 . we need to prove that @xmath299 where @xmath300 and @xmath301 are the regular and the singular parts of @xmath302 ( @xmath303 ) . each row and each column of @xmath301 @xmath309 contains at most one @xmath245 , the other entries are zero . due to this property , the reduction of @xmath302 to @xmath310 of the form can be realized by admissible permutations : @xmath311 moreover , @xmath312 is a block - direct sum of a bangle of the form and a bangle , in which each row and each column contains at most one @xmath245 , the other entries are zero . by , @xmath312 reduces by admissible permutations of rows and columns to its regularizing decomposition , so we may take @xmath313 such that @xmath312 is a regularizing decomposition . since @xmath298 , we have @xmath314 , and so by and @xmath315 due to , the size of @xmath316 is less than the size of @xmath296 , reasoning by induction we may assume that holds for @xmath312 ; that is , @xmath317 then @xmath318 since @xmath319 and @xmath320 have the form . this proves due to . let @xmath74 be one of sesquilinear forms @xmath321 let us prove that the canonical form problem for its matrix @xmath322 $ ] ( defined in ) is the canonical form problem under congruence for the bangle @xmath323 \quad\text{or}\quad \bigl[b\ \boxed{\;a\;}\;\bigr],\ ] ] respectively , and so theorem [ t0.01 ] follows from theorem [ l10.1 ] . it suffices to prove that a change of the basis of @xmath0 reduces @xmath322 $ ] by transformations @xmath324\mapsto \begin{cases } s^*[a\ b ] \begin{bmatrix } s&p\\0&q \end{bmatrix } & \text{if $ { \cal g}\colon u\times v\to { \mathbb f}$ , } \\[6 mm ] s^*[a\ b ] \begin{bmatrix } s&0\\p&q \end{bmatrix } & \text{if $ { \cal g}\colon ( v / u)\times v\to { \mathbb f}$ } , \end{cases}\ ] ] in which @xmath15 and @xmath325 are nonsingular matrices and @xmath326 is arbitrary . _ case 1 : _ @xmath322 $ ] is the matrix of @xmath327 in a basis @xmath75 of @xmath0 satisfying . if @xmath328 is another basis of @xmath0 such that @xmath329 is a basis of @xmath2 , then the change matrix from @xmath75 to @xmath59 has the form @xmath330=\begin{bmatrix } s&p\\0&q \end{bmatrix},\ ] ] where @xmath15 is the change matrix from @xmath58 to @xmath329 in @xmath2 . due to , the matrix @xmath322 $ ] reduces by transformations . _ case 2 : _ @xmath322 $ ] is the matrix of @xmath331 in a basis @xmath75 of @xmath0 satisfying . if is another basis of @xmath0 such that @xmath332 is a basis of @xmath2 , then the change matrix from @xmath75 to @xmath59 has the form @xmath333=\begin{bmatrix } s&0\\p&q \end{bmatrix},\ ] ] where @xmath15 is the change matrix from @xmath334 to @xmath335 in @xmath336 . hence , the matrix @xmath322 $ ] reduces by transformations . let @xmath96 be one of linear mappings @xmath97 let us prove that the canonical form problem for its matrix @xmath337 & \text{if $ u\to v$ or $ v / u\to v$ , } \\[5 mm ] [ a| b ] & \text{if $ v\to u$ or $ v\to v / u$ , } \end{cases}\ ] ] ( see ) is the canonical form problem under similarity for the bangle @xmath338 , \quad \bigl[\,\boxed{\ ; a\ ; } \ b\,\bigr ] , \quad \bigl[\,\boxed{\ ; a^t\ ; } \ b^t\,\bigr ] , \quad\text{or}\quad \bigl[b\ \boxed{\;a\;}\;\bigr],\ ] ] respectively , and so theorem [ t.m ] follows from theorem [ l10.1 ] . it suffices to prove that a change of the basis of @xmath0 reduces @xmath99 by transformations @xmath339 \label{qqqq2 } [ a\ b]&\longmapsto s^{-1}[a\ b ] \begin{bmatrix } s & * \\0&q \end{bmatrix } \quad \text{if $ { \cal a}\colon v\to u$ , } \\[2 mm ] \label{qqqq3 } [ a\ b]&\longmapsto s^{-1}[a\ b ] \begin{bmatrix } s&0\\ * & q \end{bmatrix } \quad \text{if $ { \cal a}\colon v\to v / u$ } , \\[2 mm ] \label{qqqq4 } \begin{bmatrix } a \\ b \end{bmatrix } & \longmapsto \begin{bmatrix } s^{-1}&0\\ * & q^{-1 } \end{bmatrix } \begin{bmatrix } a \\ b \end{bmatrix } s\quad \text{if $ { \cal a}\colon v / u\to v$},\end{aligned}\ ] ] in which @xmath15 and @xmath325 are nonsingular matrices and the @xmath207 s denote arbitrary matrices . _ case 1 : _ @xmath99 is the matrix of @xmath340 in a basis @xmath75 of @xmath0 satisfying . if @xmath341 is another basis of @xmath0 such that @xmath329 is a basis of @xmath2 , then the change matrix from @xmath75 to @xmath59 has the form @xmath342=\begin{bmatrix } s&p\\0&q \end{bmatrix},\ ] ] where @xmath15 is the change matrix from @xmath58 to @xmath329 in @xmath2 . so the matrix @xmath99 reduces by transformations or . _ case 2 : _ @xmath99 is the matrix of @xmath343 in a basis @xmath75 of @xmath0 satisfying . if is another basis of @xmath0 such that @xmath332 is a basis of @xmath2 , then the change matrix from @xmath75 to @xmath59 has the form @xmath342=\begin{bmatrix } s&0\\p&q \end{bmatrix},\ ] ] where @xmath15 is the change matrix from @xmath334 to @xmath335 in @xmath336 . hence , the matrix @xmath99 reduces by transformations or . l. a. nazarova , a. v. roiter , v. v. sergeichuk , v. m. bondarenko , application of modules over a dyad for the classification of finite p - groups possessing an abelian subgroup of index p and of pairs of mutually annihilating operators . _ j. soviet math . 5 ) ( 1975 ) 636654 .

let @xmath0 be a vector space over a field or skew field @xmath1 , and let @xmath2 be its subspace . we study the canonical form problem for bilinear or sesquilinear forms @xmath3 and linear mappings @xmath4 @xmath5 @xmath6 @xmath7 we solve it over @xmath8 and reduce it over all @xmath1 to the canonical form problem for ordinary linear mappings @xmath9 and bilinear or sesquilinear forms @xmath10 . moreover , we give an algorithm that realizes this reduction . the algorithm uses only unitary transformations if @xmath11 , which improves its numerical stability . for linear mapping this algorithm can be derived from the algorithm by l. a. nazarova , a. v. roiter , v. v. sergeichuk , and v. m. bondarenko [ _ j . soviet math . _ 3 ( no . 5 ) ( 1975 ) 636654 ] . _ ams classification : _ 15a21 , 15a63 _ keywords : _ canonical matrices ; classification ; linear operators ; bilinear and sesquilinear forms [ theorem]lemma [ theorem]definition

the presence of large and rapidly varying electric and magnetic fields in relativistic heavy ion collisions results in charge - dependent effects , visible in a series of observables in the final state of the collision . these effects can be used as a new source of information on the space - time evolution of the non - perturbative process of particle production , and on the space - time properties of the system created in the heavy ion collision . to give one example , in 2007 we demonstrated that the distortion which the electromagnetic repulsion ( attraction ) of positive ( negative ) pions induced on charged pion ( @xmath1 ) ratios brought new information on the space - time scenario of fast pion production @xcite . in recent years , the general problematics of electromagnetically - induced effects in ultrarelativistic heavy ion reactions was subject of an important theoretical and experimental interest @xcite as it was connected to very interesting phenomena like the chiral magnetic effect ( cme @xcite ) . in the present paper we review our earlier studies of the electromagnetic distortion of charged pion spectra in the context of our more recent findings on the influence of spectator - induced @xmath4 and @xmath5 fields on the azimuthal anisotropies of charged pions . special attention is put on tracing the utility of both observables for studying the longitudinal evolution of the expanding matter created in the collision . a phenomenological model analysis is presented , aimed at explaining the space - time features of pion production which we deduced from the observed electromagnetic phenomena . of positively and negatively charged pions produced in peripheral pb+pb collisions at @xmath6 gev . the pion invariant density is drawn as a function of transverse momentum in fixed bins of @xmath7 as marked from top to bottom . the subsequent distributions are consecutively multiplied by 0.2 . the arrows point at the regions where the distortion induced by the spectator em - field is most visible . from @xcite.,title="fig:",scaledwidth=80.0% ] + the relatively moderate collision energy range available to the sps makes corresponding fixed - target experiments suitable for studying the electromagnetic influence of the spectator system on charged particle spectra in a large range of available rapidity . importantly , this includes the region of very low transverse momenta where the corresponding effects are expected to be largest . a detailed double - differential study of @xmath8 and @xmath9 densities as a function of longitudinal and transverse pion momentum is presented in fig . [ fig1a ] . the na49 experimental data cover , in the longitudinal direction expressed in terms of the c.m.s . feynman variable @xmath10 , the whole region from `` mid - rapidity '' ( @xmath11 ) up to @xmath12 which is about one unit above beam rapidity at lowest transverse momenta . the smooth exponential - like shape of the transverse momentum distribution gets visibly distorted in the region of low @xmath13 , where a dramatic decrease of invariant @xmath8 density and an accumulation of @xmath9 density is apparent as indicated by the arrows . this `` deformation '' is caused by the spectator system , which modifies the trajectories of charged pions by means of its space- and time - dependent @xmath4 and @xmath5 fields . the ratio of @xmath8 over @xmath9 density , fig . [ fig1](a ) , appears particularly sensitive to the spectator - induced electromagnetic field in the region of higher rapidity ( @xmath14 ) and lower transverse momenta . here , a deep two - dimensional `` valley '' is apparent with the @xmath1 ratio approaching zero in the region @xmath15 ( @xmath16 at low @xmath13 ) . note that with the pb nucleus composed of 39% protons over 61% neutrons , this implies breaking of isospin symmetry which unequivocally confirms the electromagnetic origin of the observed effect . quantitatively , this is confirmed in fig . [ fig1](b ) , where the observed distortion can be fairly well described by means of a simple two - spectator model with the two spectators assumed as lorentz - contracted homegenously charged spheres , and isospin effects being taken into account @xcite . it is important to underline that the unique free parameter in the model is the distance @xmath2 , in the longitudinal direction , between the pion emission point and the center of the spectator system . the reasonable agreement between data and model demonstrated in figs [ fig1](a),(b ) is obtained for values of @xmath2 in the range of 0.5 - 1 fm @xcite ; different values of @xmath2 lead to different detailed shapes of the distortion of @xmath1 ratios as described in @xcite . gev , ( b ) model simulation of this ratio as described in the text , ( c ) our monte carlo prediction for the ( pure ) electromagnetically - induced directed flow of positive pions , compared to the data from the wa98 experiment @xcite , ( d ) directed flow of charged pions in intermediate centrality au+au collisions @xcite , ( e ) , ( f ) electromagnetic component of @xmath8 and @xmath9 directed flow , extracted from star data @xcite and compared to our simulation made assuming @xmath17 fm . from : @xcite ( panels a , b ) , @xcite ( panel c ) , @xcite ( panels d , e , f).,title="fig:",scaledwidth=90.0% ] + in full analogy to charged pion ratios , the _ directed flow _ of charged pions emitted close to beam rapidity is also strongly affected by spectator - induced em effects . this is shown in fig . [ fig1](c ) where our prediction for a _ purely electromagnetic effect _ on the directed flow @xmath0 of positive pions is shown for three different values of the distance @xmath2 : 0 , 0.5 and 1 fm . as it can be seen in the figure , our monte carlo calculation shows that very large values of directed flow can be induced by the sole effect of electromagnetic repulsion of positive pions by the spectator system . our prediction is compared to the measurements provided by the wa98 collaboration at the same energy , @xmath6 gev @xcite . this comparison indicates that a very sizeable part of positive pion directed flow in the region close to beam / target rapidity can in fact come from the electromagnetic origin . at the same time , the wa98 experimental data apparently constrain the possible values of the distance @xmath2 , yielding the possible range of @xmath2 from 0 up to 1 fm . thus consistently from both observables ( @xmath1 ratios , fig . [ fig1](a ) and directed flow , fig . [ fig1](c ) ) , the longitudinal distance between the actual pion emission site and the center of the spectator system appears quite small , in the range below 1 fm . this small distance is to be viewed with respect to the longitudinal extent of the lorentz - contrated spectator system which is itself of the order of about 1 fm at this collision energy . the situation changes significantly when passing to pions produced close to _ central _ rather than _ beam _ rapidity . here experimental data on intermediate centrality au+au reactions exist from the star experiment at rhic @xcite at different collision energies ( from @xmath18 up to @xmath19 gev ) . the directed flow of positive and negative pions at the lowest available energy is presented in fig . [ fig1](d ) . a _ charge splitting _ is apparent between @xmath8 and @xmath9 . as shown in figs [ fig1](e),(f ) , the latter splitting can again be understood as a spectator - induced em effect , provided that a value of @xmath2 far larger than in the preceding case , @xmath17 fm , is assumed . gev . ( a ) subdivision of the nuclear matter distribution into longitudinal `` strips '' . ( b ) kinematical characteristics of the `` strips '' as a function of their position in the perpendicular plane ; the distance @xmath2 is indicated in the plot . ( c ) invariant mass of the `` strips '' projected in the perpendicular @xmath20 plane , where @xmath21 is the direction of the impact parameter vector . ( d ) longitudinal velocity @xmath22 of the `` strips '' as a function of their position . the `` hot '' participant and `` cold '' spectator regions are indicated in the plots.,title="fig:",scaledwidth=80.0% ] + this apparent sensitivity of the electromagnetic distortion of final state charged pion ratios and directed flow to the distance between the pion formation zone and the spectator system provides , in the opinion of the authors , a completely new and very welcome tool for studying the space - time evolution of charged particle production in the soft sector of ultrarelativistic heavy ion collisions . specifically , the elongation of the distance @xmath2 with decreasing pion rapidity is the reflection of the longitudinal evolution of the system created in the collision . summing up the findings from the precedent section , in our studies we obtained : * @xmath23 fm for pions moving at rapidities comparable to @xmath24 ( from our study based on na49 @xcite and wa98 @xcite data ) ; * @xmath17 fm for pions moving at central rapidities ( @xmath25 , from our study based on star data @xcite ) . while the mere fact that @xmath2 evolves with pion rapidity is simply the confirmation of the expansion of the system in the longitudinal direction , the latter is , especially at high pion rapidities , poorly known to hydrodynamical calculations due to the presence of a sizeable baryochemical potential @xcite , and difficult to access experimentally e.g. in lhc experiments ( in contrast to sps energies where the na49 and na61/shine experiments cover the whole region from @xmath26 to @xmath27 and above in the collision c.m.s . @xcite ) . in the present section we discuss this issue in the context of energy - momentum conservation in the initial state of the collision , in a model proposed by a.s . the spatial nuclear matter distribution in the volume of the two colliding nuclei is considered in a two - dimensional @xmath20 projection perpendicular to the collision axis ; peripheral pb+pb collisions at top sps energy are presented in fig . [ fig2](a ) . the resulting `` strips '' of highly excited nuclear ( or partonic ) matter , fig . [ fig2](b ) , define the kinematical properties of the longitudinal expansion of the system as a function of collision geometry . these are shown in figs [ fig2](c ) and ( d ) in the perpendicular @xmath20 plane . for the peripheral collision considered here , the overall energy available for particle production ( invariant mass of the `` strips '' as defined assuming local energy - momentum conservation ) has a well - defined `` hot '' peak at mid - distance between the centers of the two nuclei , and gradually decreases when approaching each of the two `` cold '' spectator systems . on the other hand , the longitudinal velocity @xmath22 of the `` strips '' depends strongly on their position in the @xmath20 plane . a careful comparison of figs [ fig2](c)-(d ) shows that significantly excited volume elements of the longitudinally expanding system can move at very large longitudinal velocities , comparable to that of the spectator system . assuming a given proper hadronization time of the different volume elements , a natural picture emerges . pions produced at high rapidity ( dominantly from `` strips '' moving at large values of the longitudinal velocity @xmath22 ) will emerge at a small distance from the `` cold '' spectator systems ; these originating from `` hot '' central `` strips '' , at smaller values of @xmath28 , will evidently show up at larger values of the distance @xmath2 . altogether , we conclude that a non - negligible amount of experimental data on charge - dependent effects in particle spectra and anisotropic flow exists , and much more can be obtained from existing fixed - target as well as collider experiments . these data can be used to trace the influence of the electric and magnetic fields in heavy ion collisions , which should be useful in future studies related to the chiral magnetic effect , the electromagnetic properties of the quark - gluon plasma , and others . our own studies demonstrate the sensitivity of the em - induced distortions of charged particle spectra and directed flow to the space - time scenario of particle production in heavy ion collisions , and allow us to trace the longitudinal evolution of the expanding matter created in the course of the collision . + + the authors , and especially a.r . , gratefully thank the organizers of the x workshop on particle correlations and femtoscopy ( wpcf 2014 ) , for their invitation and for the excellent organization of such a fruiful and interesting workshop . u. grsoy , d. kharzeev and k. rajagopal , phys . c * 89 * , 054905 ( 2014 ) [ arxiv:1401.3805 [ hep - ph ] ] . v. voronyuk , v. d. toneev , s. a. voloshin and w. cassing , phys . c * 90 * , no . 6 , 064903 ( 2014 ) [ arxiv:1410.1402 [ nucl - th ] ] . h. schlagheck ( wa98 collaboration ) , nucl . a * 663 * , 725 ( 2000 ) [ nucl - ex/9909005 ] . l. adamczyk _ et al . _ ( star collaboration ) , phys . * 112 * , 162301 ( 2014 ) [ arxiv:1401.3043 [ nucl - ex ] ] . a. rybicki and a. szczurek , phys . c * 87 * , 054909 ( 2013 ) [ arxiv:1303.7354 [ nucl - th ] ] , and references therein . a. rybicki , a. szczurek and m. klusek - gawenda , epj web conf . * 81 * , 05024 ( 2014 ) .

the large and rapidly varying electric and magnetic fields induced by the spectator systems moving at ultrarelativistic velocities induce a charge splitting of directed flow , @xmath0 , of positive and negative pions in the final state of the heavy ion collision . the same effect results in a very sizeable distortion of charged pion spectra as well as ratios of charged pions ( @xmath1 ) emitted at high values of rapidity . both phenomena are sensitive to the actual distance between the pion emission site and the spectator system . this distance @xmath2 appears to decrease with increasing rapidity of the pion , and comes below @xmath31 fm for pions emitted close to beam rapidity . in this paper we discuss how these findings can shed new light on the space - time evolution of pion production as a function of rapidity , and on the longitudinal evolution of the system created in heavy ion collisions .

existence of extra neutral gauge bosons has been predicted in many extensions of the standard model ( sm ) @xcite . string - inspired models and grand - unification ( gut ) models usually contain a number of extra @xmath7 symmetries , beyond the hypercharge @xmath8 of the sm . the exceptional group @xmath1 is one of the famous examples of this type @xcite . these extra @xmath7 s are broken at some intermediate energy scales between the gut and the electroweak scales . phenomenologically , the most interesting option is the breaking of these @xmath7 s at around tev scales , giving rise to extra neutral gauge bosons observable at the tevatron and the large hadron collider ( lhc ) . recent developments in model buildings also result in new models that contain extra gauge bosons . for example , little higgs models @xcite with additional gauge groups predict a number of new gauge bosons ; the sm gauge bosons propagating in the extra dimensions after compactification can give rise to kaluza - klein towers of gauge bosons @xcite ; stueckelberg @xmath0 model connecting a hidden sector to the visible sector in the context of dark matter @xcite , just to name a few . we will denote these neutral extra gauge bosons generically by @xmath0 for the general discussion in this introduction section . collider experiments such as cdf @xcite and d @xcite at the tevatron have been searching for the neutral extra gauge bosons @xmath0 , mainly through its leptonic decay modes . the leptonic mode is a very clean channel to probe for @xmath0 since it may give rise to a discernible peak above the drell - yan background right at the @xmath0 mass , provided that the size of the coupling strength to sm quarks and leptons is not too small . currently , the best limit comes from the negative search at the tevatron . the lower mass bound on @xmath0 is about @xmath9 gev for a number of @xmath0 bosons of the @xmath1 type and a stronger bound of almost @xmath10 gev for the sequential @xmath0 , which has exactly the same coupling strength and chiral couplings as the sm @xmath11 boson that can be served as a bench mark . the lhc with just an integrated luminosity of about @xmath12 pb@xmath4 has already set limits on @xmath0s @xcite almost as good as those from the tevatron . with more luminosity accumulated in the current lhc run the limit on @xmath0 will improve substantially in the near future . most of previous studies on @xmath0 bosons focused on the decays into sm fermions , and the corresponding limits were obtained based on the decay into leptons . this scenario is not necessarily a must , but just for simplicity and fewer choices of parameters . indeed , when the mass of @xmath0 is more than a tev or even larger it has chances of decaying into other exotic particles that must be included in the model for various theoretical reasons . for instance , in gut models or in little - higgs models there are other fermions needed to cancel the anomalies , the @xmath0 could decay into these exotic fermions if their masses are not too heavy . another example is the minimal supersymmetric standard model ( mssm ) with electroweak - scale susy partners and a @xmath0 which could also decay into sfermions and higgsinos . in a recently proposed @xmath0-mediated susy - breaking model @xcite , supersymmetry breaking in the hidden sector is communicated by a @xmath0 boson to the visible sector . the low - energy spectrum includes a @xmath0 boson of a few tev and light gauginos , such that the @xmath0 can also decay into susy particles other than the sm fermions . in this work , we consider a scenario of @xmath0 boson , which arised from @xmath7 symmetry breaking at around tev scale , in the context of weak - scale supersymmetry , in which all susy partners are relatively light ( a few hundred gev ) except for the squarks ( may of order @xmath13 tev ) . such a @xmath0 may come from breaking of one of the @xmath7 s in @xmath1 @xcite , @xmath2 @xcite , or @xmath14 @xcite etc . once we specify the @xmath7 charges for the matter superfields and higgs superfields , the couplings of @xmath0 to mssm particles are determined . we study the decays of @xmath0 and its production at the tevatron and the lhc . the decays of @xmath0 will be modified when the susy particle masses are only of order a few hundred gev , which will then affect the leptonic branching ratio of the @xmath0 . we investigate how much the limits from the tevatron will be affected , because of the reduction in the leptonic branching ratio . we also study how much the sensitivity at the lhc will be reduced when the susy decay modes are open for the @xmath0 boson . finally , we study the prospect of using the susy decay modes of the @xmath0 to search for the @xmath0 boson itself and investigate its properties . we note that some related works had appeared in literature , for example in refs . however substantial improvements over these previous works have been made in this work . these include 1 . ref . @xcite focused on how the presence of supersymmetric and other exotic particles of @xmath1 models in the @xmath0 decay can affect the @xmath0 boson discovery at the tevatron and the lhc . in our work , we used the most current updated limit on @xmath15 to put limits on the mass of various @xmath0 bosons . we have illustrated the case of decaying into sm particles only and the case of including both susy and sm particles . we have shown that the @xmath0 mass limits have to be relaxed by @xmath16 gev if including susy particles in the decay . @xcite was written in 2004 and certainly our paper used the newest 2011 data . for the lhc sensitivity we worked out the more realistic energy - luminosity combinations ( 7 tev , 10 tev , 14 tev ) . refs . @xcite did not know about the options of 7 and 10 tev at their time . @xcite focused on how the discovery potential of sleptons can be improved via the decay of @xmath0 into a slepton pair . they also studied various lightest supersymmetric particle ( lsp ) scenarios and used distributions to determine the masses of sleptons , gauginos , and the @xmath0 boson . in our work , in addition to the slepton - pair , we also studied the @xmath0 decays into a chargino pair and into a neutralino pair . we have shown clearly that the presence of @xmath0 is visible in the transverse - mass spectrum . one can therefore measure the transverse - mass spectrum and determine if there is a @xmath0 boson . this spectrum can also be utilized to estimate the mass differences , and couplings of @xmath0 to sleptons , neutralinos , and charginos . this can help us to understand the underlying supersymmetry breaking mechanism . 3 . in this work , we study @xmath1 models , @xmath2 , and the sequential @xmath0 model , while ref . @xcite studied only the @xmath1 models and ref . @xcite studied only the @xmath17 model . the organization of the paper is as follows . in the next section , we write down the interactions and briefly describe a few @xmath0 models and their extensions to include supersymmetry . in sec . iii , we calculate the branching ratios of the @xmath0 boson in various models . in sec . iv , we show the shift of the limits for the masses of the @xmath0 in various models due to opening of supersymmetric particles . we estimate the @xmath18 discovery reach at the lhc , including the susy decay modes in sec . v. we further discuss in sec . vi the susy decay modes of the @xmath0 boson . we conclude in sec . feynman rules that are related to the @xmath0 are collected in the appendix . following the notation of ref . @xcite , the lagrangian describing the neutral current gauge interactions of the standard electroweak @xmath19 and extra @xmath7 s is given by @xmath20 where @xmath21 is the sm @xmath11 boson and @xmath22 with @xmath23 are the extra @xmath11 bosons in the weak - eigenstate basis . for the present work we only consider one extra @xmath24 mixing with the sm @xmath21 boson . thus the second term of the lagrangian in eq . ( [ lag_nc ] ) can be rewritten as @xmath25 + g_2 z^0_{2\mu } \left [ \sum_f \bar \psi_f \gamma^\mu ( q'_{f_l } p_l + q'_{f_r } p_r ) \psi_f \right]\;,\ ] ] where for both quarks and leptons @xmath26 and @xmath27 are the chiral charges of fermion @xmath28 to @xmath29 and @xmath30 . here @xmath31 and @xmath32 are , respectively , the third component of the weak isospin and the electric charge of the fermion @xmath28 . the chiral charges of various @xmath0 models are listed in tables [ e6 ] and [ b - l ] . the overall coupling constant @xmath33 in eq . ( [ lag_nc ] ) is the sm coupling @xmath34 , while in grand unified theories ( gut ) @xmath35 is related to @xmath33 by @xmath36 where @xmath37 and @xmath38 is the weak mixing angle . the factor @xmath39 depends on the symmetry breaking pattern and the fermion sector of the theory , which is usually of order unity . the mixing of the weak eigenstates @xmath21 and @xmath29 to form mass eigenstates @xmath11 and @xmath0 are parametrized by a mixing angle @xmath40 : @xmath41 the mass of @xmath11 is @xmath42 gev . after substituting the interactions of the mass eigenstates @xmath11 and @xmath0 with fermions are @xmath43\,,\ ] ] where @xmath44 here the subscript `` s '' denotes the observed sm @xmath11 boson and `` n '' denoted the new heavy gauge boson @xmath0 . we have used the valid approximation @xmath45 and @xmath46 . in the following , we ignore the mixing ( @xmath47 ) such that the precision measurements for the sm @xmath11 boson are not affected , unless stated otherwise . the superpotential @xmath48 involving the matter and higgs superfields in a @xmath49 extended mssm can be written as @xmath50 \;,\ ] ] where @xmath51 , @xmath52 are family indices , and @xmath53 and @xmath54 represent the yukawa matrices for the up - type and down - type quarks respectively . here @xmath55 , and @xmath56 denote the mssm superfields for the quark doublet , lepton doublet , up - type quark singlet , down - type quark singlet , lepton singlet , up - type higgs doublet , and down - type higgs doublet respectively , and the @xmath57 is the singlet superfield . note that we have assumed other exotic fermions are very heavy . the @xmath49 charges of the fields @xmath58 and @xmath57 are related by @xmath59 such that @xmath60 is the only term allowed by the @xmath49 symmetry beyond the mssm . once the singlet scalar field @xmath57 develops a vev , it generates an effective @xmath61 parameter : @xmath62 . the case is very similar to nmssm , except we do not have the cubic term @xmath63 . the singlet field will give rise to a singlet scalar boson and a singlino , which will mix with other particles in the higgs sector and neutralino sector , respectively . in this work , we are contented with the assumption that the singlet scalar field and the singlino are heavy enough that it is out of reach at the lhc and the mixing effects are negligible . detailed phenomenological studies involving the singlet field will be presented in a future work . furthermore , we also take the superpartner , dubbed as @xmath0-ino , of the @xmath0 boson to be heavy . phenomenology involving the singlet scalar boson , singlino , and the @xmath0-ino of various singlet - extended mssm can be found in refs . @xcite . below the tev scale the particle content is the same as the mssm plus a @xmath0 boson . thus , the superpotential and the soft breaking terms are the same as in mssm . extra couplings of the @xmath0 boson with the mssm particles are coming from the gauge interactions of the extra @xmath7 and the corresponding supersymmetric vertices of yukawa interactions . the gauge interactions involving the fermionic and scalar components , denoted generically by @xmath64 and @xmath65 respectively , of each superfield are @xmath66 where the covariant derivative is given by @xmath67 here @xmath68 is the electromagnetic coupling constant , @xmath69 is the electric charge , @xmath70 are the rising and lowering operators on @xmath71 doublets and @xmath72 is the chiral charges of the @xmath7 associated with the @xmath0 boson . the interactions of @xmath0 with all mssm fields go through eqs . ( [ eq1 ] ) and ( [ eq2 ] ) . details and conventions are given in the appendix and the feynman rules that involve the @xmath0 boson are listed there as well . the sequential @xmath73 model is a reference model of extra @xmath11 bosons . it has exactly the same chiral charges as the sm @xmath11 boson but at a larger mass . the gauge coupling constant is also taken to be the same as the sm one , i.e. , @xmath74 . note , however , that when susy modes are open , the @xmath75 can also decay into sfermions , neutralinos , charginos , and higgs bosons . in general , experimental constraints on the sequential model are the strongest , because the other @xmath1 models , for example , have smaller gauge coupling constant , as in eq . ( [ g2g1 ] ) . two most studied @xmath7 subgroups in the symmetry breaking chain of @xmath1 occur in @xmath76 in @xmath1 each family of the left - handed fermions is promoted to a fundamental @xmath77-plet , which decomposes under @xmath78 as @xmath79 each @xmath77 contains the sm fermions , two additional singlets @xmath80 ( conjugate of the right - handed neutrino ) and @xmath57 , a @xmath81 and @xmath82 pair ( @xmath81 is the exotic color - triplet quark with charge @xmath83 and @xmath82 is the conjugate ) , and a pair of color - singlet su(2)-doublet exotics @xmath84 and @xmath56 with hypercharge @xmath85 . in the supersymmetric version of @xmath1 , the scalar components of one @xmath86 pair can be used as the two higgs doublets @xmath86 of the mssm . the chiral charges @xmath87 and @xmath88 for each member of the @xmath77 are listed in the third and fourth columns in table [ e6 ] . in general , the two @xmath87 and @xmath88 can mix to form @xmath89 where @xmath90 is the mixing angle . a commonly studied model is the @xmath91 model with @xmath92 which has @xmath93 . there are also the inert model with @xmath94 , the neutral @xmath95 model with @xmath96 , and the secluded sector model with @xmath97 . the chiral charges for each member of the @xmath77 are also listed in the last four columns in table [ e6 ] for these four variations of @xmath0 models within @xmath1 . here we take the assumption that all the exotic particles , other than the particle contents of the mssm , are very heavy and well beyond the reaches of all current and planned colliders . .[e6 ] the chiral charges of the left - handed fermions for various @xmath0 bosons arised in @xmath1 @xcite . note that @xmath98 since all the right - handed sm fermions are necessarily converted into left - handed charge - conjugated fermions in order to put them into the irreducible representation of @xmath77 of @xmath1 . [ cols="^,^,^,^,^,^,^,^",options="header " , ] identification of supersymmetric decay modes of the @xmath0 has it own interests , namely , to understand the role of @xmath0 in the susy breaking . moreover , if the @xmath0 boson decays frequently into susy particles , we can make use of the susy channels to probe for the @xmath0 . so far , in the models that we illustrate the branching ratio into charged leptons is not negligible , such that the best discovery mode is still the charged - lepton mode , which cleanly shows the peak in the invariant - mass distribution . nevertheless , there exist models , e.g , refs . @xcite , in which the charged - lepton decay mode is highly suppressed . one could also imagine that a @xmath0 does not couple to fermions or sfermions but only to the higgs sector , such that it couples solely to higgs bosons and higgsinos . in such an extreme the @xmath0 would substantially decay into higgsinos ( or the physical neutralinos and charginos after mixings ) . in other words , the supersymmetric decay modes of the @xmath0 boson could be sizable and useful for understanding the susy breaking . typically , the susy decay modes include ( i ) @xmath99 , ( ii ) @xmath100 , ( iii ) @xmath101 , etc . such leptonic modes give rise to a signature consisting of a charged - lepton pair and large missing energies . it is clean and we can construct the cluster transverse mass @xmath102 of the lepton pair and the missing energy . the transverse mass would indicate a broad peak structure , which is sensitive to the intermediate @xmath0 boson mass . here we only give a taste of what one can do to see the presence of the @xmath0 via the supersymmetric decays . detailed studies including various susy spectra and decay modes , and branching ratios will be given in a future publication . let us first investigate slepton - pair production in mssm and in mssm plus a @xmath0 . electroweak production of @xmath103 or @xmath104 goes through the @xmath105 exchanges . the differential cross section for the subprocess @xmath106 ( @xmath107 is given by @xmath108 where @xmath109 and @xmath110 . here the electric charge @xmath111 , the @xmath11 charge @xmath112 , and the @xmath0 charge @xmath113 . the subprocess cross section is then folded with parton distribution functions to obtain the total cross section . the so - produced @xmath114 will decay into the electron and positron and the lightest neutralinos under the normal hierarchy of susy masses .. ] thus , the final state consists of a charged - lepton pair and a missing energy . we can construct the cluster transverse mass given by @xmath115^{1/2 } \nonumber \\ & = & p_{te^+e^- } + \sqrt{p_{te^+ e^-}^2 + m_{e^+ e^-}^2 } \label{cluster } \;,\end{aligned}\ ] ] where the second equality is because @xmath116 . we show the distribution for this cluster transverse mass in fig . we have imposed a set of leptonic cuts before we construct the cluster transverse mass : @xmath117 the @xmath0 models shown in fig . [ mt ] are @xmath118 , @xmath91 , @xmath119 , and @xmath120 . the other @xmath0 models show similar features . the underneath curve is the mssm contribution only with @xmath121 and @xmath11 exchanges while the upper curve includes also the contribution from @xmath0 . the @xmath0 peak becomes broad because of the missing energies from the two neutralinos involved . nevertheless , the sharp edge of the peak is sensitive to the mass difference between the @xmath0 and the slepton masses . the differential cross section versus the cluster transverse mass @xmath102 defined in eq . ( [ cluster ] ) for @xmath122 . we have applied the leptonic cuts given in eq . ( [ leptoncut ] ) . both @xmath103 and @xmath104 are included . the underneath curve is without the @xmath0 boson while the upper curve includes the @xmath0 boson . the @xmath0 models shown here are @xmath123 and @xmath120 . the masses are @xmath124 tev , @xmath125 gev , and @xmath126 gev . , width=480 ] next , we study the production of a neutralino pair @xmath127 . assuming @xmath128 is the lsp , then @xmath129 can decay into @xmath130 via a virtual @xmath11 boson or a virtual slepton . the final state consists of a charged lepton pair and a missing energy . we can again construct the cluster transverse mass as in eq . ( [ cluster ] ) . electroweak production of @xmath131 goes through the @xmath11 and @xmath0 exchanges . we assume that the squarks are much heavier such that the @xmath132-channel squark exchanges are suppressed . the differential cross section for the subprocess @xmath133 is given by @xmath134 \nonumber \\ & & + \text{re}\,\big [ m_{rr}(\hat{s})\,m_{lr}^*(\hat{s } ) \big]\big ) \bigg\}\ ; , \end{aligned}\ ] ] where @xmath135 @xmath136 and @xmath137 ^ 2 - ( \ , 2\,m_{\tilde{\chi}_1 ^ 0}\,m_{\tilde{\chi}_2 ^ 0}/ \hat s \ , ) ^2 \big\ } ^{1/2}\ , .\ ] ] + here the chiral couplings @xmath138 of the @xmath11 boson and the chiral couplings @xmath139 of the @xmath0 boson to the neutralinos are , respectively , given by @xmath140 and @xmath141 where @xmath95 is the mixing matrix of the neutralinos defined in the appendix . numerically , with the choice of susy parameters of set ( a ) , we obtain the masses @xmath142 gev and @xmath143 gev , and the mixing parameters @xmath144 , @xmath145 , @xmath146 , and @xmath147 . we apply the same set of leptonic cuts as in eq . ( [ leptoncut ] ) and construct the cluster transverse mass . we show the cluster transverse - mass spectrum in fig . the @xmath0 models shown in fig . [ mt-2 ] are @xmath118 , @xmath119 , @xmath148 , and @xmath149 . the underneath curve is the mssm contribution only with the @xmath11 exchange while the upper curve includes also the contribution from @xmath0 . the @xmath0 peak becomes broad because of the missing energies from the two neutralinos involved . nevertheless , the edge of the peak is sensitive to the mass difference between the @xmath0 and the neutralinos . the differential cross section versus the cluster transverse mass @xmath102 defined in eq . ( [ cluster ] ) for @xmath150 followed by the leptonic decay of @xmath151 . we have applied the leptonic cuts given in eq . ( [ leptoncut ] ) and assumed the branching ratio @xmath152 . the underneath curve is without the @xmath0 boson while the upper curve includes the @xmath0 boson . the @xmath0 models shown here are @xmath118 , @xmath119 , @xmath148 , and @xmath149 . the masses are @xmath153 tev , @xmath142 gev , and @xmath143 gev.,width=480 ] lastly , we have electroweak production of @xmath154 that goes through the @xmath155 and @xmath0 exchanges . the differential cross section for the subprocess @xmath156 is given by @xmath157 \nonumber \\ & & + \text{re}\,\big [ m_{rr}(\hat{s})\,m_{lr}^*(\hat{s } ) \big]\big ) \bigg\}\ ; , \end{aligned}\ ] ] where @xmath158 @xmath136 , and @xmath159 . here the chiral couplings @xmath160 of the @xmath11 boson and the couplings @xmath161 of the @xmath0 boson to charginos are , respectively , @xmath162 and @xmath163 where @xmath164 and @xmath165 are the mixing matrices of the charginos . numerically , with the choice in set ( a ) for susy parameters we have @xmath166 , @xmath167 , @xmath168 , @xmath169 and @xmath170 gev . we only calculate the production of @xmath171 , because the second chargino is about twice as heavy as the first one . each of the charginos decays via a virtual @xmath48 , @xmath172 , or @xmath173 into a charged lepton , a neutrino , and the lightest neutralino ( if going through the virtual @xmath48 , light quarks are also possible ) . therefore , there will two charged leptons plus missing energies in the final state . just as the same as the case of slepton - pair or neutralino - pair production , we reconstruct the cluster transverse mass as in eq . ( [ cluster ] ) . we show the distribution of cluster transverse mass in fig . [ chargino ] for @xmath118 , @xmath91 , and @xmath149 . it is easy to see the bump due to the presence of the @xmath0 boson , though the bump is not as discernible as the previous two cases of slepton - pair and neutralino - pair production . the differential cross section versus the cluster transverse mass @xmath102 defined in eq . ( [ cluster ] ) for @xmath174 followed by the leptonic decay of @xmath175 . we have applied the leptonic cuts given in eq . ( [ leptoncut ] ) and assumed the branching ratio @xmath176 . the underneath curve is without the @xmath0 boson while the upper curve includes the @xmath0 boson . the @xmath0 models shown here are @xmath118 , @xmath91 , and @xmath149 . the masses are @xmath124 tev and @xmath170 gev.,width=480 ] in this work , we have studied the possible supersymmetric decay modes of an additional neutral gauge boson @xmath0 , which is currently limited to be at least 1 tev . grand unified theories have predicted one or more such @xmath0 bosons along the path through which the gut symmetry is broken down to the electroweak symmetry . when supersymmetry is included in the theory , such a @xmath0 boson can decay not only into the sm particles but also the supersymmetric partners . we have used @xmath1 , @xmath2 , and the sequential models to illustrate how the decays of the @xmath0 are affected . in particular , the golden search mode charged leptons for the @xmath0 will have a smaller branching ratio as the supersymmetric modes open . we have shown that the current limits obtained at the tevatron and the lhc will be reduced by a noticeable amount , of order 20 gev . we have also estimated the 5@xmath177 discovery sensitivities of the @xmath0 at the lhc , including the effect of supersymmetric decay modes . finally , we demonstrated that even though the @xmath0 decays into supersymmetric particles , giving rise to missing energies , one can still reconstruct the cluster transverse mass ( using the observable charged leptons ) to identify the existence of the @xmath0 . we believe further studies along this direction is worthwhile since it can help us to fully understand the role of the @xmath0 in the supersymmetry - breaking and the symmetry breaking pattern . the work was supported in parts by the national science council of taiwan under grant nos . 99 - 2112-m-007 - 005-my3 and 98 - 2112-m-001 - 014-my3 , and the wcu program through the kosef funded by the mest ( r31 - 2008 - 000 - 10057 - 0 ) . as explained in sec . iib , the extra term @xmath178 allowed by the @xmath49 symmetry will generate the effective @xmath61 term when the singlet scalar field @xmath57 develops a vev . with the assumption that the singlet scalar and singlino fields , @xmath0-ino , and other exotic fermions are very heavy , the low - energy particle content includes the @xmath0 boson and those of mssm . the effective superpotential @xmath179 involving the matter and higgs superfields is the same as the mssm s , and given by @xmath180 \;,\ ] ] where @xmath51 , @xmath52 are family indices , and @xmath53 and @xmath54 represent the yukawa matrices for the up - type and down - type quarks respectively . here @xmath55 , and @xmath56 denote the superfields for the quark doublet , lepton doublet , up - type quark singlet , down - type quark singlet , lepton singlet , up - type higgs doublet , and down - type higgs doublet respectively . the scalar interactions are obtained by calculating the @xmath181- and @xmath81-terms of the superpotential , and by including the following soft - susy - breaking terms @xmath182 \;,\end{aligned}\ ] ] where the @xmath183 represents the soft mass terms for the gauginos - ino . however , we will assume it is heavy and let it decouple from the low energy spectrum in the present work . ] and sfermions , and @xmath184 represents the trilinear @xmath185 terms . the gauge interactions for the fermionic and scalar components , denoted generically by @xmath64 and @xmath65 respectively , of each superfield mentioned above are given by @xmath186 where the covariant derivative is defined as usual @xmath187 here @xmath188 with @xmath189 the @xmath71 gauge coupling and @xmath38 the weak mixing angle , @xmath35 is the gauge coupling for the extra @xmath7 , @xmath70 and @xmath190 are the ladder operators and the third component of the @xmath71 generators , @xmath69 and @xmath72 are the charges of the two @xmath7s in unit of the electromagnetic charge @xmath68 and @xmath35 respectively , and finally , @xmath191 . since we only consider supersymmetric @xmath7 symmetry for the @xmath0 boson and due to the majorana nature of the @xmath0-ino , there is no coupling between the @xmath0-ino and the @xmath0 boson . simply from eqs . ( [ a1 ] ) and ( [ a2 ] ) we obtain the interactions of @xmath0 with fermions , sfermions , neutral and charged higgsinos ( which become the physical neutralinos and charginos after mixing effects are taken into account ) , and the higgs bosons . the rotation of neutral bino , wino , and higgsinos into the physical neutralinos is given by @xmath192 where @xmath95 is an orthogonal matrix . the rotation of the charged wino and higgsino into the physical charginos is via bi - unitary transformation @xmath193 where @xmath164 and @xmath165 are unitary matrices . the mixing angles involved in physical higgs bosons ( @xmath194 ) can be read off from the following decompositions of the higgs boson fields @xmath84 and @xmath56 : @xmath195 @xmath196 where the angle @xmath197 is the mixing of the neutral cp - even higgs bosons @xmath198 and @xmath199 , @xmath200 are the goldstone bosons , and @xmath201 gev . lastly , there are yukawa - type interactions between the gauginos and the scalar @xmath65 and fermionic @xmath64 components of the matter superfield of the following form @xmath202 where @xmath203 is the group index of the @xmath8 , @xmath71 , @xmath204 , or the extra @xmath205 . the interactions involving the @xmath0-ino are @xmath206 \;.\end{aligned}\ ] ] note that the @xmath0-ino will mix with the @xmath207 and @xmath208 higgsinos when the @xmath209 and @xmath210 take on vacuum expectation values . thus , we will have a @xmath211 neutralino mass matrix . however , we decouple the @xmath0-ino in this work by setting the @xmath0-ino mass heavy . we will come back to this in later work . in the following we list the feynman diagrams and the corresponding feynman rules involving the @xmath0 boson and the mssm particles . for each model , one should use the corresponding coupling strength and chiral charges . the coupling strength @xmath35 and the @xmath0 charges are given in tables [ e6 ] and [ b - l ] for the @xmath1 and @xmath2 models respectively . for the sequential @xmath0 model , replace the coupling strength @xmath35 by @xmath33 while the corresponding chiral charges are given by the sm values : @xmath212 . the chiral couplings of the charginos and neutralinos with the @xmath0 boson are , respectively , given by @xmath213 these coupling coefficients are the same for the three @xmath0 models that we have studied in this work , as long as we use the corresponding @xmath72 charges for the two higgs doublet fields . j. l. hewett and t. g. rizzo , phys . rept . * 183 * , 193 ( 1989 ) . for a review , see m. schmaltz and d. tucker - smith , ann . nucl . part . * 55 * , 229 ( 2005 ) [ arxiv : hep - ph/0502182 ] . for a review , see k. cheung , arxiv : hep - ph/0409028 . k. cheung and t. c. yuan , jhep * 0703 * , 120 ( 2007 ) [ arxiv : hep - ph/0701107 ] ; d. feldman , z. liu and p. nath , phys . d * 75 * , 115001 ( 2007 ) [ arxiv : hep - ph/0702123 ] ; d. feldman , b. kors and p. nath , phys . d * 75 * , 023503 ( 2007 ) [ arxiv : hep - ph/0610133 ] . t. aaltonen _ et al . _ [ cdf collaboration ] , phys . * 102 * , 091805 ( 2009 ) [ arxiv:0811.0053 [ hep - ex ] ] ; t. aaltonen _ et al . _ [ cdf collaboration ] , phys . rev . * 102 * , 031801 ( 2009 ) [ arxiv:0810.2059 [ hep - ex ] ] ; cdf coll . , `` search for high mass resonances decaying to muon pairs '' , cdf - note cdf / phys / exo / public/10165 . v. m. abazov _ et al . _ [ d0 collaboration ] , phys . b * 695 * , 88 ( 2011 ) [ arxiv:1008.2023 [ hep - ex ] ] . g. aad _ et al . _ [ atlas collaboration ] , arxiv:1103.6218 [ hep - ex ] . p. langacker , g. paz , l. -t . wang and i. yavin , phys . . lett . * 100 * , 041802 ( 2008 ) . 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the decay of the @xmath0 boson into supersymmetric particles is studied . we investigate how these supersymmetric modes affect the current limits from the tevatron and project the expected sensitivities at the lhc . employing three representative supersymmetric @xmath0 models , namely , @xmath1 , @xmath2 , and the sequential model , we show that the current limits of the @xmath0 mass from the tevatron could be reduced substantially due to the weakening of the branching ratio into leptonic pairs . the mass reach for the @xmath1 @xmath0 bosons is about @xmath3 tev at the lhc-7 ( 1 fb@xmath4 ) , about @xmath5 tev at the lhc-10 ( 10 fb@xmath4 ) , and about @xmath6 tev at the lhc-14 ( 100 fb@xmath4 ) . a similar mass reach for the @xmath2 @xmath0 is also obtained . we also examine the potential of identifying various supersymmetric decay modes of the @xmath0 boson because it may play a crucial role in the detailed dynamics of supersymmetry breaking .

collisionless shocks @xcite play an important role in energy transport and evolution of charged - particle distribution functions in space and astrophysical environments . although collisionless shocks in plasmas were first predicted in the 1950s @xcite and discovered in the 1960s @xcite , many questions relating to the microscopic physics of collisionless shock formation , evolution , and shock acceleration of particles to very high energies remain unanswered @xcite . laboratory studies of collisionless shocks have been conducted since the 1960s @xcite , but a recent renaissance of laboratory collisionless shock experiments @xcite stems from the fact that modern laboratory plasmas can satisfy key physics criteria for the shocks to have `` cosmic relevance '' @xcite . recently initiated experiments @xcite at los alamos national laboratory ( lanl ) aim to form and study astrophysically relevant collisionless shocks via the head - on merging of two supersonic plasma jets , each with order 10-cm spatial scale size . compared to most other modern collisionless shock experiments which use laser - produced or wire - array z - pinch @xcite plasmas , the lanl experiment has larger shock spatial size ( up to 30-cm wide and a few - cm thick ) and longer shock time duration ( order 10 @xmath6s ) but somewhat lower sonic and alfvn mach numbers . the lanl experiment plans to have the capability to apply magnetic fields of a few kg ( via coils ) that can be oriented either parallel or perpendicular to the direction of shock propagation . obtaining physical insights into and experimental data on collisionless shock structure , evolution , and their effects on particle dynamics are the primary reasons to conduct laboratory experiments on collisionless shocks . this paper reports results from particle - in - cell ( pic ) and hybrid - pic numerical simulations , using the lsp code @xcite , that informed the design of the lanl experiment and showed that collisionless shocks should appear with the expected plasma jet parameters . after a brief description of the lanl collisionless shock experiment , the remainder of the paper describes single - jet propagation and one- ( 1d ) and two - dimensional ( 2d ) pic head - on merging jet simulations . our 1d magnetized simulations , in which the jets are immersed in an applied magnetic field , are similar to those of shimada and hoshino @xcite who performed 1d pic simulations of magnetized shock formation using a reduced ion - to - electron mass ratio and a reflection boundary to model counter - propagating plasmas . we use the actual hydrogen mass ratio and the actual hydrogen plasma parameters expected in the lanl experiments , and we directly simulate both jets . this gives us the flexibility to independently vary the properties ( _ e.g. _ , the density profile ) of the two jets without assuming any symmetry . we have also performed 2d cartesian merging simulations of magnetized jets which allows us to consider the effects of the orientation of the magnetic field and plasma density gradients with respect to the jet propagation direction . these simulations demonstrate shock formation caused by the merging of magnetized jets with mach numbers as low as @xmath7 , where the mach number is defined as @xcite @xmath8 where @xmath9 is the pre - shock jet velocity in the shock frame , @xmath10 is the alfvn velocity ( in si units ) where @xmath11 is the pre - shock magnetic field strength @xmath11 , @xmath12 is the pre - shock ion density , @xmath13 is the ion mass , and @xmath14 where the @xmath15 and @xmath16 are the pre - shock electron and ion temperatures in energy units . in unmagnetized plasmas , collisionless shocks may also be formed by the weibel instability @xcite . simulations of this mechanism were described by kato and takabe @xcite , whose simulations were also performed at a reduced mass ratio and were restricted to relatively high velocities ( @xmath17 ) . when using the hydrogen mass ratio and a lower velocity ( @xmath18 km / s as expected in the experiment ) , we find no shock formation on relevant timescales ( a few @xmath6s ) . the outline of the paper is as follows . in sec . [ sec : setup - model ] we describe the simulation setup and numerical models used . in sec . [ sec : results ] , we present lsp simulation results of single hydrogen jet propagation ( sec . [ sec : single - h - jet ] and [ sec : fully - kinetic - single ] ) and 1d ( sec . [ sec:1d - magnetized - shock ] ) and 2d ( sec . [ sec:2d - simul - magn ] ) jet - merging with applied magnetic fields . conclusions are given in sec . [ sec : conclusions ] . the simulations described in this paper are based on the lanl collisionless shock experiment @xcite , which uses counter - propagating plasma jets formed and launched by plasma railguns @xcite mounted on opposite sides of a 2.74 m diameter spherical vacuum chamber ( fig . [ fg : exp ] ) . hydrogen , helium , and argon jets have been used in the experiments , but we focus exclusively on hydrogen in this paper due to its ability to better satisfy the physics criteria for cosmically relevant collisionless shocks @xcite . single - jet parameters and evolution have been characterized experimentally @xcite in preparation for using an array of thirty such jets to form spherically imploding plasma liners as a standoff compression driver for magneto - inertial fusion @xcite . for these collisionless shock studies , lower - density ( @xmath19@xmath1 @xmath2 ) and higher - velocity ( 100 km / s ) jets are desired ; this is accomplished primarily by reducing the injected mass for a given gun current . [ fg : exp ] the approach used in this numerical study is two - fold . we initially perform a large - scale simulation of a single jet propagating from the end of the plasma gun to the center of the vacuum chamber . the hydrogen jets emerge from the plasma gun with densities on the order of @xmath19@xmath1 @xmath2 and temperatures of a few ev . the jets emerging from the guns will be few centimeters in size ( on the order of the railgun aperture ) with masses of a few @xmath6 g , and will have a drift velocity @xmath18 km / s . but both must propagate on the order of 1 m before merging can begin , during which time the density , temperature , and equation - of - state ( eos ) of the jet can change . the single - jet propagation simulation models the time evolution of the initial jet as it propagates through the chamber . this 2d @xmath20@xmath21 simulation must be run for several @xmath6s . this requires using a fairly large timestep , for which @xmath22 at an initial plasma density @xmath23 @xmath2 , where @xmath24 is the electron plasma frequency . such simulations must be done with a hybrid - pic approach in which electron plasma oscillations do need not to be resolved . however , a fully kinetic approach is required to model the formation of shocks due to micro - instabilities induced by jet merging . as will be seen in sec . [ sec : single - h - jet ] , the hydrogen jets ejected from the plasma guns drop considerably in density as they propagate to the center of the chamber where the merging takes place . this density reduction during propagation allows us to perform fully kinetic explicit pic merging simulations in 1d and 2d cartesian coordinates , in which electron timescales are resolved ( @xmath25 ) . so these simulations , which are initialized with plasma parameters obtained from the propagation simulation , are intended to model only the merging process which occurs much later than the ejection of the jets from the guns . all of these simulations are performed using the hybrid - pic code lsp @xcite , which has been utilized widely and validated for applications in many areas of beam and plasma physics , including streaming instabilities @xcite and landau damping @xcite . in addition to the traditional pic paradigm , i.e. , a maxwell - vlasov solver for collisionless plasmas in which electron length and time scales must be resolved , lsp also contains algorithms for dense plasma simulation and includes physics modules for collisions ( among charged and neutral species ) , eos modeling , and radiation transport . this flexibility available in the code makes it useful for the two - fold simulation approach described above . all of the simulation results presented later in the paper have been checked to assure convergence of the relevant physics results with respect to numerical parameters such as cell size , timestep , and particle number per cell . to more fully focus on the physics results in the remaining sections of the paper , we provide in this section details on the models and numerical parameters used for the simulations . the jet propagation simulation is performed in lsp using a quasi - neutral hybrid - pic algorithm @xcite which has fewer constraints on the timestep . the ion macroparticles are kinetic . but there are no electron macroparticles , as the ions carry fluid information for the inertia - less electrons . the equation of motion for the composite ion - electron macroparticle is given by @xmath26 where @xmath27 is the macroparticle velocity , @xmath28 is the electron pressure , and @xmath29 is the full time derivative for the lagrangian macroparticle . the current is given by ohm s law : @xmath30 where @xmath31 is the electron density , @xmath32 is the conductivity , and @xmath33 is the drift velocity gathered at the grid nodes . the fields , current , densities , and electron pressure gradient are all calculated at the nodes and then interpolated to the macroparticle position when eq . ( [ eq : eom ] ) is applied . the full maxwell s equations are solved with the ohm s law term included . displacement current is not dropped . kinetic ions also undergo self - collisions ( ion - ion ) . it is for this reason that there is no ion pressure contribution to eq . ( [ eq : eom ] ) . coulomb collisions between electrons and ions species are included self - consistently through the spitzer conductivity in eq . ( [ eq : ohm ] ) . particle energies are advanced by the same method which is described in ref . the plasma eos ( plasma internal energy , charge state , @xmath34 , etc . ) and opacity tables for radiation transport are provided by the propaceos code @xcite . although lsp includes a full radiation transport algorithm @xcite , in this simulation we include only photon emission and neglect absorption . this allows radiation to be modeled as a simple energy sink on the fluid electron species . this approximation is justified in the optically thin regime , which is well satisfied for jets in the parameter regime under consideration . the single jet propagation simulation is carried out in 2d @xmath20@xmath21 cylindrical coordinates , as the jet is assumed to be azimuthally symmetric . this allows for full spatial hydrodynamic expansion of the jet in a 2d simulation . the simulation space is large enough to allow for propagation of the initial jet from the exit of the gun to the center of the vacuum chamber , and is bounded by perfectly conducting metal walls . the cell size is @xmath35 cm , and the timestep is given by @xmath36 cm . the use of the uniformly stable exact - implicit field solver allows the simulation to be run with @xmath37 the cell size @xcite . the initial plasma is characterized by @xmath38 ion macroparticles per cell . the results of the simulation are discussed in sec . [ sec : single - h - jet ] . the 1d cartesian fully kinetic propagation and merging simulations are initialized with input parameters based on results of the hybrid - pic single jet propagation simulation . as will be seen in sec . [ sec : single - h - jet ] , the jets at the chamber center have a much lower density ( @xmath39 @xmath2 ) than when ejected from the plasma guns . this allows us do explicit kinetic pic simulations . we can also afford better spatial and temporal resolution in 1d cartesian coordinates : @xmath40 cm . since the timestep is small , an explicit field solution is used rather than the exact - implicit solver . both electrons and ions are modeled kinetically . coulomb collisions are included for the electron species , as are ion collisions , which are found to have a negligible effect in this parameter regime . the cell size and time step given above resolve not only @xmath24 , but also the ion and electron cyclotron frequencies ( @xmath41 and @xmath42 , respectively ) , and ion and electron skin depths ( @xmath43 and @xmath44 , respectively ) . a cloud - in - cell @xcite particle model allows @xmath45 the debye length . the 1d simulations described below are all run with several thousand macroparticles per cell . we assume fully stripped hydrogen ions ( @xmath46 ) for simplicity . some justification for this assumption is given below . we also assume an ideal gas eos for the plasma jets and neglect radiation losses . we initially perform a few fully kinetic single - jet propagation simulations with and without magnetic fields to demonstrate the effect of applied fields on single jet propagation . these results are described in sec . [ sec : fully - kinetic - single ] . in sec . [ sec:1d - magnetized - shock ] we discuss the results of 1d fully kinetic simulations of two - jet merging with and without applied magnetic fields . we consider the effects of varying magnetic field strengths , as well as the effect of finite density gradients ( with scale length @xmath47 ) , and the effect of the initial spatial separation , or gap @xmath48 , between the two jets . the last group of simulations considered in this paper are in 2d cartesian coordinates . we simulate the 2d jets in the @xmath49@xmath50 plane , with the jets propagating in the @xmath49 direction . the total @xmath50 extent is many @xmath51 wide , and periodic boundaries are imposed at minimum and maximum values of @xmath50 . to maintain reasonable runtimes , 2d simulations performed at realistic length scales ( jet lengths @xmath52100 cm ) require coarser spatial resolution ( @xmath53 cm ) and a smaller number of particles per cell ( tens rather than hundreds ) than were possible in 1d . coulomb collisions are included , but we again assume an ideal eos and neglect radiation transport ( both photon emission and absorption ) . in sec . [ sec:2d - simul - magn ] we consider the results of 2d cartesian simulations of counter - propagating jets in perpendicular magnetic fields . as in 1d , the jet propagation remains in the @xmath49 direction , and we simulate the 2d jets in the @xmath49@xmath50 plane . the total @xmath50 extent is many @xmath51 wide , and periodic boundaries are imposed at minimum and maximum values of @xmath50 . in these simulations we find some slow numerical heating of the electrons at later times due to the coarse spatial resolution of the grid ( @xmath54 ) . however , if the simulation duration does not exceed @xmath55 @xmath6s , the energy conservation remains good to within a few percent . high fidelity simulations over longer time scales will require better spatial resolution and larger particle numbers . this will require the use of more processors than were available . hybrid - pic lsp simulations were performed of a single hydrogen plasma jet propagating from the railgun nozzle to the center of the chamber in order to connect the plasma jet parameters at the railgun exit with those in the region of head - on jet merging . details on the simulation setup and numerical methods are given in sec . [ sec : setup - single - jet ] . the initial ion density @xmath12 profile can be in seen in the upper left plot in fig . [ fg : ni_trans ] . at @xmath56 the single jet is assumed to have a peak density @xmath57 @xmath2 ( total mass @xmath58 @xmath6 g ) , electron and ion temperatures @xmath59 ev , and @xmath60 km / s ( in the @xmath61 direction ) . the initial jet parameters are also given in table [ tab_trans ] . the time evolution of the jet propagation can be seen in fig . [ fg : ni_trans ] , which shows @xmath12 contours at @xmath62 , @xmath63 , @xmath64 , and @xmath65 @xmath6s . figure [ fg : n_trans_lo ] shows @xmath12 line - outs at the same times . the approximate plasma parameters of the jet at the center of the chamber ( @xmath66 cm ) are also given in table [ tab_trans ] . so this simulation determines the approximate parameter regime of the individual jets after they have propagated to the center of the chamber and begun to merge : @xmath67 @xmath2 , @xmath68 km / s , @xmath69 ev , @xmath70 1 , and @xmath71 cm . we note that from spectroscopic data obtained from the lanl experiment the plasma density at the chamber center can be inferred to be below @xmath19 @xmath2 , which is consistent with the simulation result . the experimental jets are expected to be somewhat colder when emerging from the guns than the 5-ev value used in the simulations . but in the simulation results radiation cooling quickly causes the jet temperature to drop . nonetheless , we expect the amount of density decay seen in the simulation to be an upper bound on the experiments . based on the results of this simulation , we have chosen a set of simplified plasma parameters to be used for the fully kinetic simulations discussed in the following subsections . these values are given in table [ tab_merge ] . using these parameters we can estimate the coulomb collision frequency for the jets . to observe collisionless merging of the jets , it is necessary that the inter - jet ion collision time be much larger than the jet interaction time @xmath72 . for two counter - propagating ion beams ( @xmath73 , the ion thermal velocity ) , the spitzer collision frequency is proportional to @xmath74 @xcite . using the parameters above , we find @xmath75 s. the ion stopping time due to collisions with electrons in the opposing jet is @xmath76 s , while the jet interaction time @xmath77 s. so this simulation result demonstrates that these counter - propagating jets will indeed be in the collisionless regime when the jets merge at the center of the chamber , allowing the lanl facility to be used for the investigation of collisionless shock formation . [ fg : ni_trans ] [ fg : n_trans_lo ] .initial and approximate parameters at the center of the chamber ( @xmath78 ns ) in 2d @xmath20@xmath21 jet propagation simulation ( see also figs . [ fg : ni_trans ] and [ fg : n_trans_lo ] ) . [ cols="^,^,^",options="header " , ] [ tab_accel ] in this section we consider the results of 2d cartesian simulations of counter - propagating jets in perpendicular magnetic fields . the jet propagation remains in the @xmath49 direction , and we simulate the 2d jets in the @xmath49@xmath50 plane . in 2d , we now have to explicitly specify the direction of the perpendicular field in the @xmath50@xmath21 plane . details on the simulation setup were given in sec . [ sec : setup - explicit-2d ] . we consider initially a quasi-1d simulation with no variation of any physical quantity in the @xmath50 direction and periodic boundaries in @xmath50 . the initial density profiles can be seen in the top row of fig . [ fg : ni_q2d ] . the full initial conditions for all the 2d runs are given in table [ tab_merge ] . the total @xmath50 extent is many @xmath51 wide . the initial conditions are @xmath79 @xmath2 , @xmath80 ev . the magnetic field is in the @xmath81 direction ( analogous to the 1d simulations as the field is aligned in a virtual direction ) with @xmath82 g and @xmath83 km / s . we first note from the bottom row of fig . [ fg : ni_q2d ] , which shows @xmath12 and @xmath84 contours at @xmath85 ns , well after the shock formation , that there is no strongly evident structure in the @xmath50 direction in the bulk of the jets , although the @xmath50 extent is many @xmath51 wide . there is some small density variation along the @xmath86 line where the density is very low , but these variations are on the order of a cell size and are probably due to particle noise . due to the coarse spatial grid , we can no longer resolve the oscillatory structure along the @xmath49-direction in the shock transition region , which were seen in the highly resolved 1d simulations ( e.g. , see fig . [ fg : n_xpx_2jet_1 kg ] ) . but we obtain the same shock speed ( i.e. , @xmath87 ) and density discontinuity , @xmath88 , as the corresponding 1d simulation . [ fg : ni_q2d ] we now consider two uniform density disks with no applied magnetic field . the periodic boundaries in @xmath50 have been retained . ion density contours and @xmath49@xmath89 particle phase - space data are shown in fig . [ fg : ni_2dnob ] . the results are analogous to the unmagnetized 1d case , i.e. , there is very little interaction between the jets in this parameter regime in the absence of an applied magnetic field . there is certainly no evidence of weibel - induced unmagnetized shock waves on time scales @xmath90 @xmath6s . [ fg : ni_2dnob ] we repeat the previous simulation but now add a 350-g magnetic field in the virtual direction ( @xmath81 ) . the @xmath12 and @xmath84 contours are shown at various times in fig . [ fg : ni_2dwb ] . along the line @xmath91 cm , we find approximate values of @xmath92 , and @xmath93 , and a shock velocity of @xmath94 km / s . although the problem is now inherently 2d , we nonetheless estimate @xmath95 from the 1d formula of eq . ( [ eq : mach ] ) . we notice , however , that there is a conspicuous difference between this simulation and the 1d and quasi-1d ( fig . [ fg : ni_q2d ] ) analogs . namely , the density at the center - of - mass of the two jets , along the line at @xmath96 , remains large ( @xmath97 @xmath2 ) throughout the simulation , indicating at least some jet interpenetration at this point . in the 1d analogs , the density remained close to zero at the jet center of mass . in order to explain this qualitatively different behavior , we consider first the results for a 2d simulation in which the 350-g field is rotated from the virtual @xmath81 to the @xmath98 direction , which lies in the simulation plane . these results are shown in fig . [ fg : ni_2dwby ] . for this simulation we find , along @xmath99 cm , that @xmath100 and @xmath101 . for this magnetic field orientation , the results are qualitatively similar to the 1d case , as @xmath12 remains zero along @xmath96 . when the magnetic field is in the @xmath50 direction ( fig . [ fg : ni_2dwby ] ) , the currents which drive the magnetic field , @xmath102 , can flow in the virtual @xmath21 direction and remain localized near @xmath96 cm . this allows for large @xmath103 forces to reflect the incoming the jets . when the magnetic field is in the virtual @xmath21-direction ( fig . [ fg : ni_2dwb ] ) the magnetic field is supported by finite circular current paths around the perimeter of the bulk of the jet . the density gradient in @xmath50 acts to minimize the @xmath49 component of the @xmath104 force on the jets . [ fg : ni_2dwb ] [ fg : ni_2dwby ] as a final simulation , we consider a 2d simulation with more realistic density gradients . the 350-g field is in the @xmath81 direction . the initial @xmath12 contours can be seen in the top left of fig . [ fg : ni_2dgrad ] . from the @xmath12 and @xmath84 contours at later times , we note that it is difficult to clearly see the propagation front in the density contours as they are imposed on top of the initial gradient , but the magnetic field front of the shock is clearly seen in the field contours . the simulation results from figs . [ fg : n_xpx_2jet_varyn ] and [ fg : ni_2dwby ] demonstrate that shock structure can be observed in magnetized jets with realistic density gradients . [ fg : ni_2dgrad ] we have performed 1d and 2d pic simulations of hydrogen plasma jet propagation and head - on merging . in the parameter regime of the lanl experiment , unmagnetized collisionless shocks could not be detected in the simulation results . the simulations do demonstrate the formation of magnetized ( perpendicular ) collisionless shocks when @xmath105 ( @xmath106 ) . this requires that the jets be immersed in a field @xmath1071 kg . simulations predict @xmath1082 in this parameter range . non - shock jet interactions ( i.e. , the alfvn wave propagation discussed in sec . [ sec : effects - real - dens ] ) are also observed in the simulations as well as ion kinetic effects . these simulation confirm that the application of an appropriate magnetic field to the lanl experiment is required . some simple calculations as well as preliminary simulations show that an applied field can not fully diffuse into the oncoming jets on time scales @xmath109 @xmath6s . for this reason the jets need to be `` born '' in the magnetic field or penetrate into the field by some other mechanism than magnetic diffusion @xcite . since applied fields were shown to suppress ambipolar diffusion [ see fig . [ fg : n_xpx](c ) ] , it would be valuable to perform a large - scale 2d jet propagation simulation to see how jets would evolve when born or injected into applied fields . larger - scale 2d or 3d simulations require better spatial resolution to avoid numerical difficulties for longer simulation times ( @xmath110 @xmath6s ) . although not considered in this paper , we point out that the lanl experiment should also be able to observe unmagnetized _ collisional _ shocks with higher density merging jets and has observed such phenomena @xcite . this work was supported by the laboratory directed research and development ( ldrd ) program at lanl through u.s . dept . of energy contract de - ac52 - 06na25396 . the authors also acknowledge useful discussions with dr . d. v. rose and dr . n. l. bennett of voss scientific .

we describe numerical simulations , using the particle - in - cell ( pic ) and hybrid - pic code lsp [ t. p. hughes et al . , phys . rev . st accel . beams * 2 * , 110401 ( 1999 ) ] , of the head - on merging of two laboratory supersonic plasma jets . the goals of these experiments are to form and study astrophysically relevant collisionless shocks in the laboratory . using the plasma jet initial conditions ( density @xmath0@xmath1 @xmath2 , temperature @xmath3 few ev , and propagation speed @xmath4100 km / s ) , large - scale simulations of jet propagation demonstrate that interactions between the two jets are essentially collisionless at the merge region . in highly resolved one- and two - dimensional simulations , we show that collisionless shocks are generated by the merging jets when immersed in applied magnetic fields ( @xmath51 kg ) . at expected plasma jet speeds of up to 100 km / s , our simulations do not give rise to unmagnetized collisionless shocks , which require much higher velocities . the orientation of the magnetic field and the axial and transverse density gradients of the jets have a strong effect on the nature of the interaction . we compare some of our simulation results with those of previously published pic simulation studies of collisionless shock formation .

the anomalous magnetic moment of the muon has recently been measured to an accuracy of 0.54 ppm @xcite . the main source of uncertainty in the value predicted @xcite in the standard model is given by the hadronic contribution , @xmath7 , to the lowest order . this quantity is estimated with a dispersion integral of the hadronic cross section measurements . in particular , the pion form factor , @xmath8 , defined via @xmath9 , accounts for @xmath10 of the central value and for @xmath11 of the uncertainty in @xmath7 . the kloe experiment already published @xcite a measurement of @xmath8 with the method described below , using an integrated luminosity of 140 pb@xmath2 , taken in 2001 , henceforth referred to as kloe05 . -0.5 cm da@xmath0ne is an @xmath13 collider running at @xmath14 , the @xmath1 meson mass , which has provided an integrated luminosity of about 2.5 fb@xmath2 to the kloe experiment up to year 2006 . in addition , about 250 pb@xmath2 of data have been collected at @xmath15 gev , in 2006 . present results are based on 240 pb@xmath2 of data taken in 2002 . the kloe detector consists of a drift chamber @xcite with excellent momentum resolution ( @xmath16 for tracks with polar angle larger than @xmath17 ) and an electromagnetic calorimeter @xcite with good energy ( @xmath18}$ ] ) and precise time ( @xmath19}\oplus 100~\mathrm{ps}$ ] ) resolution . at da@xmath0ne , we measure the differential spectrum of the @xmath20 invariant mass , @xmath21 , from initial state radiation ( isr ) events , @xmath22 , and extract the total cross section @xmath23 using the following formula @xcite : @xmath24 where @xmath25 is the radiator function . this formula neglects final state radiation ( fsr ) terms . the cross section for isr photons has a divergence in the forward angle ( relative to the beam direction ) , such that it dominates over fsr photon production . the fiducial volume shown in fig . [ fig:1 ] is based on the following criteria : * two tracks with opposite charge within the polar angle range @xmath26 ; * small angle photon , @xmath27 , the photon is not explicitly detected and its direction is reconstructed from the track momenta in the @xmath28 center of mass system , @xmath29 . the above criteria result in events with good reconstructed tracks and enhance the probability of having an isr photon . furthermore , * fsr at the leading order is reduced to the @xmath30 level ; * the contamination from the resonant process @xmath31 where at least one of photons coming from the @xmath32 is lost is reduced to the level of @xmath33 . discrimination of @xmath34 -@xmath35 plane ; the selected area is shown.,title="fig:"]-0.5 cm from @xmath36 events is done via particle identification @xcite based on the time of flight , on the shape and the energy of the clusters associated to the tracks . in particular , electrons deposit most of their energy in the first planes of the calorimeter while minimum ionizing muons and pions release uniformly the same energy in each plane . an event is selected if at least one of the two tracks has not being identified as an electron . fig . [ fig:2 ] shows that contaminations from the processes @xmath37 and @xmath38 are rejected by cuts on the track mass variable , @xmath39 , defined by the four - momentum conservation , assuming a final state consisting of two particles with the same mass and one photon the analysis of data taken since 2002 benefits from cleaner and more stable running conditions of da@xmath0ne , resulting in less machine background and improved event filters than kloe05 . in particular , the following changes are implemented : * a new trigger level was added at the end of 2001 to eliminate the 30% loss from pions penetrating to the outer calorimeter plane and thus were misidentified as cosmic rays events . for the 2002 data , this inefficiency has decreased down to 0.2% , as evaluated from a control sample ; * the offline background filter , which contributed the largest experimental systematic uncertainty to the published work @xcite , has been improved . the filter efficiency increased from 95% to 98.5% , with negligible systematic uncertainty ; * the vertex requirement on the two tracks used in kloe05 is not applied , therefore eliminating the systematic uncertainty from this source . the absolute normalization of the data sample is measured using large angle bhabha scattering events , @xmath40 . invariant mass for the process @xmath41 , from an integrated luminosity of 240 pb@xmath2.,title="fig:"]-0.5 cm the integrated luminosity , @xmath42 , is obtained @xcite from the observed number of events , divided by the effective cross section evaluated from the monte carlo generator ` babayaga ` @xcite , including qed radiative corrections with the parton shower algorithm , inserted in the code simulating the kloe detector . an updated version of the generator , ` babayaga@nlo ` @xcite , decreased the predicted cross section by 0.7% , while the theoretical relative uncertainty improved from 0.5% to 0.1% . the experimental relative uncertainty on @xmath42 is 0.3% . the @xmath44 differential cross section is obtained from the observed spectrum , @xmath45 , after subtracting the residual background events , @xmath46 , and correcting for the selection efficiency , @xmath47 , and the luminosity : @xmath48 fig . [ fig:3 ] shows the differential cross section from the selected events . after unfolding , with the inversion of the resolution matrix obtained from monte carlo , & kloe05 & kloe08 + offline filter & 0.6 & negligible + background & 0.3 & 0.6 + @xmath39 cuts & 0.2 & 0.2 + @xmath49/e i d & 0.1 & 0.1 + vertex & 0.3 & not used + tracking & 0.3 & 0.3 + trigger & 0.3 & 0.1 + acceptance & 0.3 & 0.1 + fsr & 0.3 & 0.3 + luminosity & 0.6 & 0.3 + @xmath25 function eq.([eq:1 ] ) & 0.5 & 0.5 + vp & 0.2 & 0.1 + total & 1.3 & 1.0 + for events with both an initial and a final photon , the differential cross section is corrected using ` phokhara ` for shifting them from @xmath21 to the virtual photon mass , @xmath50 . then , it is divided by the radiator function ( ` phokhara ` setting the pion form factor @xmath51 ) to obtain the measured total cross section @xmath52 , of eq.([eq:1 ] ) . the pion form factor is evaluated subtracting the fsr term , @xmath53 @xcite , latexmath:[\[\sigma_{\pi\pi(\gamma)}~=~ \frac{\pi}{3}~ \frac{\alpha_{em}^2\,\beta_\pi^3}{m_{\gamma^*}^2}~ @xmath43 dispersion integral inclusive of fsr is obtained after removing vacuum polarization , vp , effects @xcite , @xmath55 ^ 2~.\ ] ] table [ tab:1 ] shows the list of relative systematic uncertainties in the evaluation of @xmath43 in the mass range [ 0.35,0.95 ] gev@xmath6 , for kloe05 and for the analysis of this new data set , kloe08 . published 05 & @xmath56 + updated 05 & @xmath57 + new data 08 & @xmath58 + + cmd-2 @xcite & @xmath59 + snd @xcite & @xmath60 + kloe08 & @xmath61 + the published analysis , updated for the new bhabha cross section and for a bias in the trigger correction @xcite , is compared with kloe08 , and also with the results obtained by the vepp2 m experiments @xcite , in the mass range @xmath62{\mathrm{\ mev}}$ ] . table [ tab:2 ] shows the good agreement amongst kloe results , and also with the published -0.5 cm cmd-2 and snd values . they agree with kloe08 within one standard deviation . the band of fig . [ fig:4 ] shows the kloe08 pion form factor smoothed accounting for both statistical and systematic errors and normalized to fix the 0 in the ordinate scale . cmd-2 and snd data points are interpolated and compared to this band , in the same panel . we obtained the @xmath3 contribution to @xmath4 in the mass range @xmath63{\mathrm{\ gev}}^2 $ ] integrating the @xmath44 differential cross section for the isr events @xmath22 , with photon emission at small angle : * measure @xmath64 using detected photons emitted at large angle , which would improve the knowledge of the fsr interference effects from kloe @xmath65 measurements @xcite ; * measure the pion form factor directly from the ratio , bin - by - bin , of @xmath34 to @xmath66 spectra @xcite ( see fig . [ fig:2 ] for the selection of @xmath67 events ) ; * extract the pion form factor from data taken at @xmath68 gev , off the @xmath1 resonance , where @xmath69 background is negligible . 99 g. w. bennett _ et al . _ [ muon g-2 collaboration ] , phys . d * 73 * ( 2006 ) 072003 f. jegerlehner , `` essentials of the muon g-2 '' , arxiv : hep - ph/0703125 a. aloisio _ et al . _ [ kloe collaboration ] , phys . b * 606 * ( 2005 ) 12 a. denig _ et al . _ [ kloe collaboration ] , kloe note 192 , july 2004 , ` www.lnf.infn.it/kloe/pub/knote/kn192.ps ` m. adinolfi _ et al . _ , [ kloe collaboration ] nucl . instrum . meth . a * 488 * ( 2002 ) 51 m. adinolfi _ et al . _ , [ kloe collaboration ] nucl . instrum . a * 482 * ( 2002 ) 364 s. binner , j. h. khn and k. melnikov , phys . b * 459 * ( 1999 ) 279 g. rodrigo , h. czy , j. h. khn and m. szopa , eur . j. c * 24 * ( 2002 ) 71 h. czy , a. grzelinska , j. h. khn and g. rodrigo , eur . j. c * 33 * ( 2004 ) 333 h. czy and e. nowak - kubat , phys . b * 634 * ( 2006 ) 493 f. ambrosino _ et al . _ [ kloe collaboration ] , eur . j. c * 47 * ( 2006 ) 589 c. m. carloni calame _ et al . _ , b * 584 * ( 2000 ) 459 g. balossini _ et al . _ , nucl . b * 758 * ( 2006 ) 227 f. jegerlehner , nucl . * 162 * ( 2006 ) 22 j. s. schwinger , `` particles , sources , and fields . vol . 3 '' , _ redwood city , usa : addison - wesley ( 1989 ) 318 p. ( advanced book classics series ) _ f. ambrosino _ et al . _ [ kloe collaboration ] , arxiv:0707.4078 [ hep - ex ] f. ignatov , these proceedings f. ambrosino _ et al . _ [ kloe collaboration ] , phys . b * 634 * ( 2006 ) 148 f. ambrosino _ et al . _ [ kloe collaboration ] , eur . j. c * 49 * ( 2007 ) 473 s. e. mller and f. nguyen _ et al . _ [ kloe collaboration ] , nucl . * 162 * ( 2006 ) 90

the kloe experiment at the da@xmath0ne @xmath1-factory has performed a new precise measurement of the pion form factor using initial state radiation events , with photons emitted at small polar angle . results based on an integrated luminosity of 240 pb@xmath2 and extraction of the @xmath3 contribution to @xmath4 in the mass range @xmath5 $ ] gev@xmath6 are presented , the systematic uncertainty is reduced with respect to the published kloe result .

one of the principal goals of the study of field theories on fuzzy spaces is to develop an alternative non - perturbative technique to the familiar lattice one @xcite . to date , this new approach in the case of four dimensional field theories has been limited to studies of euclidean field theory on @xmath0 @xcite , @xmath1 @xcite and @xmath2 @xcite . all but @xmath0 have additional complications . for example , @xmath1 is not spin but spin@xmath3 and @xmath2 is really a squashed @xmath4 and includes many unwanted massive kaluza - klein type modes . even @xmath0 is not ideal since it has curvature effects that drop off as power corrections rather than exponentially as in the case of toroidal geometries . the fuzzy approach does , however , have the advantage of preserving continuous symmetries such as the @xmath5 symmetry of a round @xmath6 and does not suffer from fermion doubling @xcite . the advantages are gained at the cost of introducing a non - locality associated with the non - commutativity of the fuzzy sphere . there is therefore a balance of advantages and disadvantages associated with the fuzzy approach . the final decision on whether the approach has real advantages over the standard lattice approach should be determined by doing genuine simulations . for this reason monte carlo simulations of the fuzzy approach are now in progress . in the lattice approach non - locality is also a problem when fermions are included . so our expectation is that as far as monte carlo simulations are concerned the fuzzy approach will not be competitive with the lattice one until fermions are included . the approach will gain further advantages in situations where symmetries are more important . it also extends naturally to allow for supersymmetry . ( see @xcite where a fuzzy supersphere was constructed ) . so we expect the true power of the approach to emerge when supersymmetry and chiral symmetry are present in a model . a radically different alternative to the euclidean monte carlo approach becomes available once one has a fuzzy three - dimensional space . such a space has the advantage that it allows one to develop very different non - perturbative methods , since now one can address the non - perturbative questions from a hamiltonian point of view . the purpose of this article is to introduce precisely such fuzzy three - dimensional spaces . we will begin by presenting a fuzzy version of the circle @xmath7 , from which one can obtain tori of arbitrary dimension . we will then present a fuzzy approximation to the three - sphere , @xmath8 . unfortunately , both of these spaces are still not ideal in that they involve many unwanted additional degrees of freedom which we suppress so that they do not contribute to the low energy physics . the presence of additional degrees of freedom is probably unavoidable as it seems to be the price one pays for the classical space not being a phase space . the three - sphere is also curved and hence the results obtained from studies of field theories on this space will approach those of a flat three - dimensional space with polynomial corrections . it has , however , the advantages of preserving the full @xmath9 symmetry of a round @xmath10 . from the construction it seems clear that both of these spaces will also be free of fermion doubling problems . we will restrict our focus here to scalar field theories and demonstrate how the unwanted degrees of freedom can be suppressed so that the limiting large matrix theory of a scalar field theory recovers field theory on the commutative spaces . we will argue that the data specifying the geometries can be cleanly specified by giving a suitable laplace - type operator for the scalar field , which together with the matrix algebra and its hilbert space structure gives a spectral triple . aside from our personal motivations , non - commutative geometry has recently become a very popular area of research from both the point of view of possible new physics in string theory and @xmath11-brane theory , @xcite , and as a new regularisation technique in ordinary quantum field theory , @xcite-@xcite and @xcite-@xcite . in both these endeavours `` fuzzy '' spaces play an important rle . roughly speaking a fuzzy space is a finite matrix approximation to the algebra of functions on a continuous manifold , the seminal example being the fuzzy two - sphere , @xcite . it has the important property of preserving the isometries of the space that it is approximating . as such the idea can serve as a source of examples related to matrix models in string theory and as a regularisation technique for ordinary quantum field theory . as a regularisation method it provides one that preserves the underlying space - time symmetries and is amenable to numerical computation . fuzzy spheres in dimensions other than two were analysed in @xcite-@xcite , but the construction there was incomplete . they also advocate projecting out the unwanted modes and working with a non - associative algebra which we consider unsatisfactory . also the case of odd spheres works very differently to that of even spheres . an alternative approach for the fuzzy four - sphere , @xmath12 , was given in @xcite , based on the fact that fuzzy @xmath4 and @xmath13 are well understood @xcite , and , in the continuum limit , @xmath4 is an @xmath6 bundle over @xmath2 . in this paper we show how the odd - dimensional fuzzy spheres @xmath7 and @xmath8 can be extracted from the matrix algebras associated with the fuzzy complex projective spaces @xmath14 and @xmath15 . an alternative approach to obtaining a finite approximation to @xmath16 , based on conformal field theory , was presented in @xcite , however , in this approach it is unclear how the unwanted modes are to be suppressed . our method uses a similar suppression mechanism to that used for @xmath12 in @xcite . although there is no closed finite dimensional matrix algebra for @xmath17 unless @xmath18 , the relevant degrees of freedom when @xmath19 and @xmath20 are contained in the matrix algebras for @xmath21 and @xmath4 respectively . one can therefore obtain functional integrals for field theories over @xmath7 and @xmath8 by starting with functional integrals over @xmath22 and @xmath15 and then suppressing the unwanted modes so that they do not contribute to the functional integral . because of the high degree of symmetry inherent in the construction , the unwanted modes can be suppressed simply by using appropriate quadratic casimirs in the laplacian . in this way we by - pass the problems associated with the fact that the algebra of matrices associated with functions on the sphere does not close on the sphere , but necessarily lifts into the enveloping complex projective space . in a similar fashion we expect that when a hamiltonian approach to field theory is developed using these spaces the unwanted modes will cause no difficulties since they can be made arbitrarily difficult to excite . the paper is organized as follows . in section [ geometry ] we summarize how a given geometry is captured in the fuzzy approach . section [ circle ] then gives our construction of a fuzzy circle , @xmath7 . section [ 4sphere ] summarises the construction of @xmath12 presented in @xcite and in section [ 3sphere ] we present our fuzzy three - sphere , @xmath8 . section [ 3sphere : v2 ] gives an alternative construction of @xmath8 which lends itself to a generalisation to @xmath17 for any @xmath23 @xcite . frhlich and gawdzki @xcite ( following connes , @xcite ) have demonstrated that the abstract triple @xmath24 , where @xmath25 is the hilbert space of square integrable functions on the manifold @xmath26 , with laplace - beltrami operator @xmath27 , @xmath28 being the metric , and @xmath29 is the algebra of smooth bounded functions on @xmath26 , captures a topological space together with its metrical geometry . in a similar fashion one can specify a fuzzy space , @xmath30 , as the sequence of triples @xmath31 parameterized by @xmath32 , where @xmath33 is the hilbert space acted on by the complete matrix algebra @xmath34 of dimension @xmath35 with inner product @xmath36 and @xmath37 is a suitable laplacian acting on matrices . one can readily extract information such as the dimension of the space from these data . the laplacian comes with a cutoff and so the dimension can be read from the growth of the number of eigenvalues . sub - leading corrections give such quantities as the euler characteristic and other information about the space . the data contained in the triple @xmath24 are precisely the data that go into the euclidean action for a scalar field theory on the space @xmath26 and hence specifying the scalar action is a convenient method of prescribing these data . in the fuzzy approach the algebra will always be a matrix algebra and we will retain the hilbert space inner product specified above so the only data from the triple , @xmath38 , remaining to be supplied are the permitted matrix dimensions , @xmath35 and a realization of the laplacian , @xmath37 . once this information is given the fuzzy geometry is specified . though it may be convenient to give a map to functions this is not necessary . once the laplacian is given its eigenmatrices and spectrum can be used to provide such a map if needed . suppose for example that the spectrum of @xmath37 is identical to that of @xmath39 up to some cutoff and a complete set of eigenmatrices is given by @xmath40 with the corresponding commutative eigenfunctions being @xmath41 , then the symmetric symbol - map @xmath42 given by @xmath43 provides a map to functions with @xmath44 the function corresponding to the matrix @xmath45 . by construction the map has no kernel and the symbol - map induces a @xmath46 product on functions given by @xmath47 which represents matrix multiplication in terms of an operation on the image functions . the @xmath46 product depends on @xmath42 , a different but equivalent one could be obtained by giving a nonzero weighting @xmath48 to the different terms in the sum ( [ symbol_map ] ) . in the case of @xmath49 a particular choice of the @xmath48 will give the diagonal coherent state prescription is referred to as the covariant symbol of the matrix @xmath45 and since the coefficients @xmath48 are not one it will differ from the corresponding contravariant symbol , see berezin @xcite . the symbol - map is referred to as symmetric when its covariant and contravariant symbols are equal and coincides with the case of @xmath50 . ] as discussed in @xcite . if the symbol - map ( [ symbol_map ] ) has the property that @xmath51 where @xmath39 is a natural laplacian for the space to be approximated , then the spectrum of the fuzzy space will be precisely a cutoff version of that of the commutative space @xmath26 . this is precisely what happens in the case of @xmath52 , see @xcite . however , it is convenient to extend the definition of fuzzy space to the case where the spectrum coincides for low - lying eigenvalues , but deviates for a family of eigenvalues that can be given arbitrarily high value and which correspond to degrees of freedom that have no counterpart in the commutative space @xmath26 . this allows us to obtain fuzzy approximations to additional spaces in particular , as we will see , to tori and the three sphere . if one takes the euclidean quantum field theory point of view then the desired geometry appears as that associated with the accessible configurations of the field theory and the deviations are suppressed in a probabilistic fashion . a successful method of suppressing the unwanted modes would be to add to the scalar action a term @xmath53 $ ] which is non - negative for any @xmath54 , zero only for matrices that correspond to functions on @xmath26 , and positive for those that do not . the modified action would therefore be of the form @xmath55+h s_i[\phi]$ ] . the parameter @xmath56 should be chosen to be large and positive . the probability of any given matrix configuration then takes the form @xmath57=\frac{{\rm e}^{-s[\phi]-hs_i[\phi]}}{z } \label{prob_of_config}\ ] ] where @xmath58 { \rm e}^{-s[\phi]-hs_i[\phi ] } \label{partition_fn}\ ] ] is the partition function of the model . if the prescription is to work for free field theories , then @xmath53 $ ] should be at most quadratic in @xmath54 . this can then be thought of as a modification of the laplacian in the triple ( [ fuzzy_triple ] ) . furthermore the problem of uv / ir mixing in scalar theories can be removed by including a higher derivative operator in the quadratic term of the field theory such that it renders all diagrams finite when the matrix size is sent to infinity . with such a prescription since each diagram has a limiting commutative value in the large matrix limit each diagram must take this value and hence no uv / ir mixing can occur . the prescription of sending the matrix size to infinity and sending the coefficient of the irrelevant higher derivative operator to zero do not commute . this prescription of adding an irrelevant operator to the action is simpler than the normal ordering prescription proposed in @xcite and works for any dimension . from the above discussion it should be clear that the entire problem of constructing a fuzzy approximation to a space is the problem of giving a suitable prescription for the matrix laplacian . consider the finite matrix algebra representation of the fuzzy sphere @xmath59 @xcite . the algebra of @xmath60 matrices , which will be denoted by @xmath61 , has the same dimension as the number of degrees of freedom in a spherical harmonic expansion of a function on @xmath6 , truncated at angular momentum @xmath32 , @xmath62 that is@xmath63 the precise identification between a matrix @xmath64 and a cut - off function @xmath65 , as discussed in the preceding section , is not unique , but the possible maps can be given in terms of coherent states or the symmetric symbol - map @xmath42 of ( [ symbol_map ] ) , and the resulting product of functions is non - commutative for finite @xmath32 . it is crucial to our construction that only maps for which the product of functions becomes commutative in the limit @xmath66 be considered . the symbol - map ( [ symbol_map ] ) associates orthonormal @xmath67 polarisation tensors @xmath68 with spherical harmonics @xmath69 . the conventions used here will be that @xmath70 where the polarisation tensors @xmath71 are those of @xcite . the @xmath72 symmetric laplacian , @xmath73 , on the fuzzy sphere acts on matrices @xmath54 and is represented by the second order casimir corresponding to the adjoint action of the angular momentum generators @xmath74 in the @xmath60 representation : @xmath75.\ ] ] hence the action can be taken to be @xmath76={1\over l+1 } tr\left ( { 1\over 2}\phi^\dagger \lap^2\phi + v(\phi)\right ) \label{fs2action}\ ] ] for some scalar potential @xmath77 , which is assumed to be bounded below . this action can then be used in a partition function which involves ordinary integration over @xmath78 degrees of freedom @xmath79}. \label{z}\ ] ] the probability distribution for field configurations is then @xmath57=\frac{{\rm e}^{-s[\phi]}}{z } \label{prob_of_s2}\ ] ] where @xmath55 $ ] given is by ( [ fs2action ] ) . this probability distribution is associated with the geometry @xmath80 which specifies a round fuzzy sphere . the field theory with quadratic potential , however , suffers from uv / ir mixing problems @xcite . if we add the term @xmath81 to the laplacian and use the triple @xmath82 the uv / ir mixing problem is removed and we recover a field theory on the commutative @xmath6 in the infinite matrix size limit . the parameter @xmath83 can finally be sent to zero with the result that the critical value of the mass parameter will be sent to infinity . the process of taking the large matrix limit and sending @xmath83 to zero do not commute . to obtain the commutative theory on the sphere the matrix size must be sent to infinity for non - zero @xmath83 . there is no finite matrix approximation to the algebra of functions on @xmath84 . nevertheless , the degrees of freedom relevant to a circle are certainly contained in @xmath61 . focusing on the top harmonic in ( [ ylm ] ) , with @xmath85 , the @xmath86 contain all @xmath87 and thus reproduce functions on the circle as @xmath88 . this implies that the partition function and correlation functions for a field theory on a circle can be extracted from that of the fuzzy sphere by suppressing all the modes with @xmath89 in ( [ z ] ) . one way of achieving this is to penalise modes with @xmath89 by giving them a large positive weight in the action . to this end we modify the action ( [ fs2action ] ) to @xmath90={1\over l+1 } tr\left\{{1\over 2 } \phi^\dagger [ l_3,[l_3,\phi]]+ { h\over 2}\phi^\dagger\left(l(l+1)-\lap^2\right)\phi + v(\phi)\right\}.\ ] ] all modes with @xmath89 now have the wrong sign for @xmath73 and , when @xmath56 is very large , are heavily penalised in the partition function ( [ z ] ) , contributing nothing as @xmath91 . in this limit only the modes with @xmath85 remain and these have the correct sign for their kinetic energy , because the term linear in @xmath56 vanishes on these and only these modes . the ` wrong sign ' for the @xmath73 contribution to the kinetic energy here is analogous to an anti - ferromagnetic coupling in a lattice theory and just as in the lattice theory with an anti - ferromagnetic coupling the action here is also bounded below . that the action remains bounded from below is intimately related to the fact that there is an ultraviolet cutoff in the model and therefore a maximum eigenvalue for the laplacian or equivalently a shortest wavelength . to see that the commutative algebra of functions on @xmath84 is recovered in the @xmath85 sector of the fuzzy sphere as @xmath66 , we first decompose the matrix @xmath54 using the basis of polarisation tensors : @xmath92 in our conventions ( [ tlm ] ) the commutator of the polarisation tensors is given by ( see e.g. @xcite page 191 , equation ( 46 ) ) @xmath93&= & \sqrt{(2l_1 + 1)(2l_2 + 1)\over l+1 } \sum_{l=0}^l(-1)^{l - l}\left\{1-(-1)^{l_1+l_2+l}\right\ } \nonumber \\ & & \hspace{50pt}\times \left\ { \matrix{l_1 & l_2 & l \cr l/2 & l/2 & l/2\cr}\right\ } c_{l_1m_1,l_2m_2}^{lm}\hat y_{lm},\nonumber\\\end{aligned}\ ] ] where @xmath94 are @xmath95-symbols and @xmath96 are clebsch - gordon co - efficients . now for large @xmath32 @xmath97 and @xmath98 when @xmath99 is odd . thus @xmath100\rightarrow 0\ ] ] and the algebra is commutative when @xmath101 as promised . in particular @xmath102\rightarrow 0\ ] ] and the top harmonic alone reproduces the commutative algebra of functions on @xmath84 in the continuum . to summarize we can encode the geometry specifying a fuzzy circle by the triple @xmath103 with @xmath104 . this picks out the fuzzy circle from the top angular momentum polarization tensor @xmath105 . one could equally pick it out from a lower one , @xmath106 by modifying the term proportional to @xmath56 to @xmath107 . this latter choice may have advantages for the suppression of uv / ir mixing effects in the fuzzy context . it roughly corresponds to a mixture of ` nearest neighbour ' and next nearest neighbour ferromagnetic and anti - ferromagnetic couplings . having constructed a fuzzy circle it is now clear that there is no obstacle to constructing fuzzy tori of arbitrary dimension , simply by taking products of fuzzy circles . this has the obvious advantage for numerical simulation of avoiding power - law curvature effects . we can use a similar procedure to approximate @xmath10 from a finite approximation to @xmath2 but first , in this section , we summarise the construction of the fuzzy @xmath2 from fuzzy @xmath4 given in @xcite . the construction utilises the fact that @xmath4 is an @xmath6 bundle over @xmath2 and there is a well - defined matrix approximation to @xmath108 . the harmonic expansion of a function on @xmath4 requires representations of @xmath109 that contain singlets of @xmath110 under @xmath111 : in terms of @xmath109 young tableaux the permitted representations are @xmath112 and are of dimension @xmath113 . all such representation , for @xmath114 , appear in the tensor product @xmath115 since the dimension of @xmath116 is @xmath117 the representations in ( [ matcp2 ] ) constitute a @xmath118 matrix and are thus in one - to - one correspondence with elements @xmath54 of @xmath34 . fuzzy @xmath4 is now identified with @xmath34 with an appropriate laplacian . the most natural laplacian on @xmath15 is the @xmath109 invariant one which is the the second order casimir corresponding to the adjoint action of the @xmath109 generators in the @xmath118 representation . for future convenience we shall use the fact that @xmath119 and denote the generators by @xmath120 , with @xmath121 and @xmath122 . the @xmath123 invariant laplacian on @xmath34 is then @xmath124 . \label{so6lap}\ ] ] as @xmath125 this corresponds to the continuum @xmath123 invariant laplacian on @xmath4 . in the notation of @xcite we shall label the @xmath123 irreducible representations by their highest weights @xmath126 , with @xmath127 either all integers or all half - integers and @xmath128 . the dimensions of these representations are @xmath129 and the eigenvalues of @xmath130 are @xmath131 the irreducible representations ( [ harmcp3 ] ) that appear in a harmonic expansion of function on @xmath4 are @xmath132 with @xmath133 an integer , so the quadratic casimir takes the value @xmath134 in this notation ( [ matcp2 ] ) translates to @xmath135 the extraction of a fuzzy @xmath2 from this algebra further relies on the curious fact that there is another possibility for the laplacian on @xmath136 that has a lower symmetry , @xmath137 , coming from the fact that it is also possible to represent @xmath4 as the coset space @xmath138 . in this representation the harmonic expansion of a function on @xmath4 requires all representations of @xmath139 that contain singlets of @xmath140 under the decomposition @xmath141 irreducible representations of @xmath139 can be labelled by two numbers @xmath142 , either both integers or both half - integers , and @xmath143 . they have dimension @xmath144 and second order casimirs@xmath145 from ( [ cp3embedding ] ) we see that the @xmath139 representations that contain singlets of @xmath140 are those with @xmath142 both integers these are all the tensor representations @xmath146 , with @xmath147 , and are therefore really representations of @xmath137 . the @xmath148 representations appearing in ( [ matcp2 ] ) decompose into @xmath137 representations as @xmath149 the fact that @xmath150 means that @xmath137 acts transitively on @xmath4 and functions on @xmath4 can be expanded in terms of @xmath137 irreducible representations @xmath151 with an @xmath137 invariant laplacian . as discussed in @xcite , there is no unique @xmath137 invariant laplacian on @xmath34 but rather any linear combination of the restrictions of ( [ so6lap ] ) to @xmath137 : _ i.e. _ any linear combination of @xmath152\ ] ] and @xmath153,\ ] ] with @xmath154 and where @xmath155 , can be used as a laplacian provided the combination has positive eigenvalues . the fuzzy @xmath2 can now be extracted from this by noting that the harmonic expansion of functions on @xmath156 require irreducible representations of @xmath137 that contain singlets of @xmath9 under the restriction of @xmath137 to @xmath9 . these are of course the symmetric tensor representations of @xmath137 , labelled by @xmath157 in the notation above . a laplacian whose low lying modes are those of @xmath158 can be constructed by penalizing the modes @xmath151 in ( [ so5mat ] ) with @xmath159 . from ( [ so6casimir ] ) and ( [ so5casimir ] ) we see that @xmath160 so the laplacian @xmath161 has eigenvalues @xmath162 and states with @xmath163 will be suppressed in a functional integral for large @xmath56 . the parameter @xmath56 here is acting like a `` squashing '' parameter , @xmath164 is the `` round '' @xmath148 invariant metric on @xmath4 , while @xmath165 breaks this symmetry down to @xmath137 . the lowest permitted value for @xmath56 is @xmath166 . we have @xmath167 , and we see that this laplacian is rather singular in the large @xmath32 limit as the representation @xmath168 for large @xmath133 develops a zero eigenvalue . the family of actions @xmath90={1\over d_l } tr\left\{\phi^\dagger \lap^2_h \phi + v(\phi)\right\ } \label{fuzzys4}\ ] ] gives a field theory on squashed @xmath136 for @xmath169 . furthermore as @xmath91 modes with @xmath170 are completely suppressed in a functional integral and ( [ fuzzys4 ] ) corresponds to a field theory on @xmath158 . note that it does not matter whether we use @xmath171 , @xmath172 or @xmath173 for the first term on the right - hand side of ( [ s4lap ] ) when the constraint @xmath174 is imposed all three become the same operator . we can build on the construction of the last section to get a laplacian whose low lying modes are those associated with a field on @xmath8 by using the same trick as in section [ circle ] to pick out the top mode @xmath175 of the @xmath158 . as an irreducible representation of @xmath137 this is the representation @xmath176 with dimension @xmath177 the harmonic expansion of a function on @xmath178 requires all irreducible representations of @xmath5 , both integral and half - integral , @xmath179 where @xmath180 are euler angles and @xmath181 are the wigner @xmath11-matrices . the key to extracting @xmath182 from @xmath158 is the observation that the total number of degrees of freedom in ( [ s3harmonic ] ) is @xmath183 which is the same as @xmath184 in ( [ matfs3 ] ) . this is because the top mode ( or indeed any mode @xmath157 ) of @xmath158 has the representation content of an @xmath182 . the top @xmath137 mode of @xmath158 can now be picked out by penalising the modes with @xmath185 with an ` anti - ferromagnetic ' kinetic - energy term . to this end we define the laplacian @xmath186 + h'\bigl ( 2l(l+3)-{\cal l}^2_{(6)}\bigr)\phi+ h\bigl(2\lap^2_{(5)}-\lap^2_{(6)}\bigr)\phi , \label{fs3def}\ ] ] with @xmath187 . for finite @xmath56 and @xmath188 , as both the last two terms are @xmath189 , this is a positive operator and contains all modes on @xmath15 . as @xmath190 modes not relevant to @xmath12 are sent to infinity and , finally , modes not relevant to @xmath182 are sent to infinity when @xmath191 . for very large @xmath56 and @xmath188 , the low lying eigenvalues are therefore precisely those of @xmath8 . in a functional integral for a scalar field based on this laplacian we recover scalar field theory on @xmath182 in the large @xmath56 and @xmath188 limit . field theory on @xmath182 therefore arises from the double limit @xmath192 in the action @xmath193= { 1\over d_l}tr\left\{\phi^\dagger \lap^2_{h , h'}\phi + v(\phi)\right\}\ ] ] with @xmath194 . the constructions described up till now have relied on matrix approximations to @xmath195 , specifically @xmath13 and @xmath4 . there is however another construction for @xmath182 based on the orthogonal grassmannian . this is a co - adjoint orbit and therefore a well - defined finite matrix approximation to the algebra of functions on this grassmannian exists . this space is not the same as @xmath4 : it arises from a different embedding of @xmath140 into @xmath139 , characterised by the decomposition @xmath196 in particular the @xmath197 does not give a singlet under this embedding and so must be excluded from the harmonic expansion on this space it is clearly not the same space as @xmath4 . the representation content here is such that @xmath198 which is known not to admit a spin structure @xcite . the expansion of a function on the orthogonal grassmannian @xmath199 can be obtained from that on @xmath200 simply by omitting all the odd rank tensors from the latter . in this way @xmath199 follows from moding out the @xmath137 representation of @xmath4 by the @xmath201 action @xmath202 on the @xmath197 , so @xmath203 as a side remark we note that @xmath204 is an @xmath6 bundle over the real projective space @xmath205 . in the notation of section [ 4sphere ] the even rank tensor representations of @xmath137 are @xmath151 with @xmath206 , and these have dimension @xmath207 so the total number of degrees of freedom for @xmath208 and @xmath209 ( @xmath32 even ) is @xmath210 ^ 2,\ ] ] which is the same as that of @xmath211 with @xmath212 thus for matrix dimensions @xmath213 we have fuzzy orthogonal grassmanians . the action on this fuzzy grassmannian looks essentially identical to that on a squashed @xmath15 , @xmath76={1\over d'_l}tr\left\ { \phi^\dagger \lap^2_{(5)}\phi + v(\phi ) \right\},\ ] ] except that the matrix algebras are restricted to those of size @xmath213 in ( [ dlprime ] ) containing only even rank tensor representations @xmath137 , @xmath214 and one has only one quadratic casimir , the @xmath137 one , at ones disposal . the harmonic expansion of a function on @xmath182 is contained in @xmath215 because the top representation for a given @xmath32 , @xmath216 with @xmath174 , has dimension ( [ matfs3 ] ) and , as observed in the previous section , this is a sum of the dimensions of the @xmath5 representations required for a harmonic expansion on @xmath10 , ( [ matfs2 ] ) . if we can penalise all modes with @xmath217 and @xmath159 for @xmath216 in a functional integral over @xmath204 then we will really be doing a functional integral over @xmath182 . this is easily achieved since @xmath216 and @xmath174 has the largest second order casimir , @xmath218 of all the @xmath137 representations in @xmath215 . in the now familiar manner the unwanted modes in the functional integral over @xmath204 can be suppressed by using the laplacian @xmath219 + h'\bigl(l(l+3 ) -\lap^2_{(5)}\bigr),\ ] ] which acts on fields @xmath220 and @xmath32 even . the unwanted modes are completely eliminated in the limit @xmath191 , giving @xmath182 truncated at level @xmath32 . the constraint that @xmath32 is even does not change the fact that we get the full continuum @xmath10 as @xmath66 . by starting with the known finite matrix algebras for @xmath4 and @xmath21 , the fuzzy @xmath15 and the fuzzy sphere @xmath221 , finite functional integrals for scalar field theories on @xmath8 and @xmath7 have been constructed . the geometry of a fuzzy space is specified by a triple @xmath38 and , although there is no known closed associated algebra giving a fuzzy @xmath84 as a triple directly , @xmath14 nevertheless contains the states required for a @xmath222 plus other unwanted states . the unwanted states are given large eigenvalues by modifying the laplacian on @xmath14 , as in equation ( [ s1fdef ] ) , leaving only the states of @xmath222 in the low energy spectrum of the laplacian . in a similar way @xmath136 contains the states necessary for a fuzzy description of @xmath10 ( via @xmath158 ) and the laplacian on @xmath15 can be modified , as in equation ( [ fs3def ] ) , so that states not related to @xmath182 are given large eigenvalues , leaving only @xmath182 states in the low energy spectrum . an alternative construction of @xmath182 , based on suppressing modes on a fuzzy version of the orthogonal grassmannian @xmath204 , has been presented in section [ 3sphere : v2 ] . this has the advantage of having a natural extension to @xmath223 for any @xmath23 , @xcite . thus @xmath8 can be obtained either in two steps , via the fuzzy @xmath12 constructed in @xcite , @xmath224 , or alternatively in a single step from the fuzzy version of @xmath225 $ ] as described in section [ 3sphere : v2 ] . the construction of the fuzzy circle allows fuzzy tori to be defined in an obvious way , by taking products of fuzzy circles , thus opening the way to numerical simulations on tori while preserving the full @xmath226 isometry group and avoiding the fermion doubling problem @xcite . one first writes down a finite functional integral for a field theory on @xmath227 , which contain the modes relevant for propagation @xmath228 in its spectrum , and then damps the unwanted modes . this can be done , in a manner that preserves the isometries of the torus , by introducing appropriate combinations of second order casimirs into the propagators . balachandran , t.r . govindarajan and b. ydri , _ mod . lett . _ * a15 * ( 2000 ) 1279 , hep - th/9911087 ; a.p . balachandran , t.r . govindarajan and b. ydri , _ fermion doubling problem and noncommutative geometry ii _ , hep - th/0006216 ; a.p . balachandran and g. immirzi , _ the fuzzy ginsparg - wilson algebra : a solution of the fermion doubling problem _ , hep - th/0301242 . j. frohlich and k. gawdzki , _ conformal field theory and geometry of strings _ , lectures given at mathematical quantum theory conference , vancouver , canada , 4 - 8 aug 1993 . published in vancouver 1993 , proceedings , mathematical quantum theory , * vol . 1 * 57 - 97 , hep - th/9310187 .

a fuzzy circle and a fuzzy 3-sphere are constructed as subspaces of fuzzy complex projective spaces , of complex dimension one and three , by modifying the laplacians on the latter so as to give unwanted states large eigenvalues . this leaves only states corresponding to fuzzy spheres in the low energy spectrum ( this allows the commutative algebra of functions on the continuous sphere to be approximated to any required degree of accuracy ) . the construction of a fuzzy circle opens the way to fuzzy tori of any dimension , thus circumventing the problem of power law corrections in possible numerical simulations on these spaces .

the recent discovery of 10 hypervelocity stars ( hvss ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) has raised many questions about their nature and origin . the most widely accepted ejection mechanism , proposed by @xcite , involves the encounter of a close binary with a supermassive black hole ( smbh ) . other possible mechanisms ejecting stars from the galactic center involve intermediate - mass black holes ( imbhs ; e.g. * ? ? ? * ; * ? ? ? * ) , a binary massive black hole ( bmbh ; e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , or a cluster of stellar mass black holes around the smbh @xcite . @xcite claimed that the rotational velocities of hvss should be lower than those measured for single stars of the same spectral type if they originated in binaries , because of tidal effects . he predicted that the rotational velocities of the known b - type hvss should be @xmath2 km s@xmath1 , based on values compiled by @xcite for b - stars in binaries . @xcite predicted high rotational velocities for hvss that were ejected by a very close encounter with an imbh in the galactic center , however such encounters are very unlikely . these predictions can not be tested with existing observations , as the low resolution of the discovery spectra of most hvss is not sufficient to determine projected rotational velocities ( @xmath3 ) . the only hvs with high resolution spectroscopy and a @xmath3 measurement is he 04375439 , found by @xcite . it has @xmath4 km s@xmath1 @xcite , in agreement with the prediction of @xcite . however , @xcite and @xcite measured half - solar metallicity for this early b - star , establishing its origin in the large magellanic cloud ( lmc ) . the possible ejection mechanisms for this star include an interaction with an imbh or a smbh , and a dynamical interaction of a single star in a dense cluster . this example demonstrates the importance of high resolution spectroscopy for understanding this newly discovered class of objects . of the remaining hvss , hvs2 ( or us708 ; * ) , is classified as an evolved sdo star and reasonably well understood . however there is some ambiguity in the nature of the late b - type hvss , since at their temperatures and gravities , the blue horizontal branch ( bhb ) crosses the main sequence . hot bhb stars generally have low rotational velocities and peculiar chemical abundances @xcite , thus high resolution spectroscopy of these faint hvss can determine their nature by measuring their atmospheric parameters , chemical abundances and @xmath3 . in addition , time series photometry can reveal pulsations and confirm their main sequence nature , as was done for hvs1 by @xcite . motivated by the lack of @xmath3 and stellar parameter measurements for most of the known hvss and the possibility of testing the nature of the smbh in the center of our galaxy , we performed high resolution spectroscopy of two hvss . in this letter we present our results . we collected spectra of hvs7 and hvs8 ( sdss j113312.12@xmath5010824.9 and j094214.04@xmath5200322.1 ) with the blue chip of the mike spectrograph @xcite installed at the 6.5-m magellan clay telescope at las campanas observatory ( chile ) , on two half nights on ut 2008 january 1819 . each star was observed twice , with individual exposure times between 900 and @xmath6 , using a @xmath7 slit and 3@xmath83 binning . the total exposure times were @xmath9 for hvs7 and @xmath10 for hvs8 . the resolution of the spectra is r = 32,000 at 4500 . the spectra were extracted using the mike reduction pipeline @xcite . the extracted spectra for each star were then averaged , normalized and merged . the wavelength coverage of the merged spectra is 3900 - 5050 , with an average s / n - ratio per pixel of 15 for hvs7 and 14 for hvs8 , based on the extracted continuum around 4500 . these s / n - ratios and our spectral resolution are sufficient to distinguish between high ( @xmath11 km s@xmath1 ; * ? ? ? * ) and low ( @xmath12 km s@xmath1 ; * ? ? ? * ) @xmath3 values for b - stars . next , we corrected the wavelength scale for doppler shift , to allow comparison of the spectra with models ( see 3 ) . we measured the heliocentric radial velocity of each star using the iraf cross - correlation package rvsao @xcite and the grid of models described in 3 . table [ tab : rv ] lists our results and the values previously reported by @xcite . 4 discusses the implications of our new radial velocity measurements . our high resolution spectra allow direct determination of the effective temperature @xmath13 , surface gravity @xmath14 , and @xmath3 of the stars by comparing synthetic model spectra to the observations . the s / n - ratio of the data is however too low to reliably measure abundances . we generated a grid of synthetic spectra using the lte atlas9 models and opacities developed by @xcite . the grid covers @xmath13 between 800015000 k in steps of 1000 k , and @xmath14 between 3.05.0 in steps of 0.25 dex . the metallicity was set to solar , assuming that the hvss are ejected from the galactic center , where abundances are solar or supersolar @xcite . for the macro- and micro - turbulence velocities we adopted 0 and 2 km s@xmath1 , which are typical for late b - stars @xcite . the models were broadened by 0.15 to match mike s instrumental profile and resampled to a dispersion of 0.03 / pix to match the dispersion of the stellar spectra . finally , we convolved each model with rotational profiles between 10350 km s@xmath1 in 10 km s@xmath1 velocity increments . simultaneous fits to @xmath13 , @xmath15 and @xmath3 were performed for each star by iteratively comparing each model to the data . the agreement between each model and the observed spectra is quantified by the spectroscopic quality - of - fit parameter , @xmath16 ( normalized @xmath17 ) , defined by @xcite and given by the equation @xmath18 where @xmath19 is the number of points in the spectrum , and @xmath20 and @xmath21 are the root mean squared deviation between each model and the stellar spectrum , and the smallest value of the rms found . @xmath16 = 0 gives the best model fit , and @xmath16 = 1 defines the statistical 1@xmath22 confidence interval of the result . the following subsections describe the derivation of @xmath13 , @xmath15 and @xmath3 for each target . the spectrum of hvs7 ( v=17.80 mag , s / n=15 ) includes four balmer lines ( @xmath23 @xmath24 ) from which @xmath13 , @xmath14 and @xmath3 can be estimated . we also detect , , and lines that can in principle be used to further constrain @xmath13 and @xmath3 , however the two main @xmath13 indicators ( and ) have anomalous line strengths and can not be used to constrain the @xmath13 . we are therefore left with only the balmer lines that are simultaneously sensitive to @xmath13 , @xmath14 and @xmath3 , but can still provide non - degenerate values of these parameters for late b - type stars ( see 4 of * ? ? ? we performed two tests to verify that balmer lines alone are sufficient to simultaneously derive the three parameters : a ) we applied our analysis to synthetic spectra with added random noise matching the s / n - ratios of the observations , and in all cases recovered the input values within errors , b ) we applied our analysis to a high s / n spectrum of a late b - type star ( hr7447 , b5 iii ) , kindly provided by l. lyubimkov . our analysis yielded @xmath13 = 14,000 @xmath0 1000 k , @xmath14 = 3.75 @xmath0 0.25 dex , and @xmath3 = 70 @xmath0 20 km s@xmath1 , in agreement with the parameters derived by ( * ? ? ? * ; * ? ? ? * @xmath13 = 13,400 k , @xmath14 = 3.64 dex , @xmath3 = 76 km s@xmath1 ) . we then proceeded to fit the spectrum of hvs7 for @xmath13 , @xmath14 and @xmath3 . we performed several tests to determine the stability of the best fit solutions . we ran fits to the entire spectrum , 100 windows centered on each balmer line ( to ensure that the wings and some continuum are included ) , portions of the spectrum outside the balmer lines , and 1020 windows centered on metal lines . in the last two cases we had to fix @xmath13 and @xmath14 to the values from the fits to the entire spectrum and the balmer lines and only fit for @xmath3 . all the tests give fully consistent results , with the following best fit parameters : @xmath13 = 12,000 @xmath0 1000 k , @xmath14 = 3.50 @xmath0 0.25 dex , and @xmath3 = 60 @xmath0 17 km s@xmath1 . the @xmath16 minimization results for the full spectrum are shown in figure [ fig : conts ] . we have adopted conservative errors for @xmath13 and @xmath14 equal to the grid step size , versus their smaller 1@xmath22 statistical errors . the statistical 1@xmath22 errors for @xmath3 ( horizontal dotted line in the @xmath16 vs. @xmath3 plot in figure [ fig : conts ] ) , are @xmath25 and @xmath26 km s@xmath1 , however visual comparison of the models to the observed spectrum show they are too large . instead we adopted the errors resulting from the fits to individual metal lines . the left panel in figure [ fig : hlines ] compares the best fit model to the balmer lines of hvs7 . figure [ fig : metals ] shows metal lines detected in hvs7 with @xmath3 = 40 , 60 , and 80 km s@xmath1 models overplotted . the purpose of this plot is two - fold ; the left - side panels show how the k and @xmath274233 lines ( @xmath3 is derived from these two and the @xmath284549 and 4583 lines ) give @xmath3 = 60 km s@xmath1 as the best model fit , and their depths agree with the solar abundance adopted in the models . the @xmath3 from these lines also agrees with the fits to the full spectrum and the balmer lines . the right - side panels show the behavior of the doublet , the and lines . none of the models in the grid can reproduce the depths of those lines . while and seem depleted in the atmosphere of hvs7 , the 4128/4130 lines seem strongly enhanced . the models can not reproduce either the depth nor the line strength ratio of the and lines . the enhancement of the 4128/4130 doublet is even more significant when taking into account that the kurucz lte atlas9 models overpredict the strengths of these lines . this problem persists even after including non - lte corrections @xcite . abundance peculiarities have been noticed before in bhb stars by several authors @xcite ; however , most of these are very slow rotators ( @xmath29 km s@xmath1 ; * ? ? ? the spectrum of hvs8 ( v=18.09 mag , s / n = 14 ) has a s / n similar to the hvs7 spectrum , however , inspection of the spectrum of hvs8 for metal lines gives null results . this can be explained by very low metal abundances , strong depletion , or highly broadened metal lines . the @xmath3 obtained below points towards the latter case . to derive @xmath13 , @xmath14 and @xmath3 we used only the spectrum above 4000 because of problems with the continuum normalization at shorter wavelengths . as with hvs7 , we simultaneously fit for @xmath13 , @xmath14 and @xmath3 by iteratively comparing the spectrum of hvs8 to our model grid . we ran fits to the entire spectrum and 160 windows centered on the balmer lines , which gave consistent parameters : @xmath13 = 11,000 @xmath0 1000 k , @xmath14 = 3.75 @xmath0 0.25 dex , and @xmath3 = 260 @xmath0 70 km s@xmath1 . the lack of metal lines in the spectrum of hvs8 is consistent with a high @xmath3 that results in strong line broadening . the error in @xmath3 in this case comes directly from the @xmath16 = 1 statistical 1@xmath22 result , as visual comparison of the spectrum to the models does not allow us to place a finer constraint . the right - side panel in figure [ fig : hlines ] compares the best fit model to the @xmath23 , @xmath30 and @xmath31 lines of hvs8 . the flat - bottomed cores of the balmer lines , which are the most sensitive regions to @xmath3 , clearly illustrate that hvs8 rotates faster than hvs7 . the new radial velocity observations in table [ tab : rv ] provide a third epoch for each star and allow to check for variations . our radial velocity measurement for hvs7 is identical to the values reported by @xcite and @xcite , within errors . such measurements provide clues to the nature of hvss . as pointed out by brown et al . , determining the nature of late b - type hvss is not straightforward because late - type main sequence b - stars and hot blue horizontal branch ( bhb ) stars have identical atmospheric parameters . bhb stars are less luminous and therefore closer in distance to us . establishing the evolutionary stage of hvs7 is critical because its radial velocity is marginally consistent with it being a bhb runaway star bound to our galaxy @xcite . the lack of significant radial velocity variations for hvs7 suggests it is not a binary , nor a pulsator . slowly pulsating main sequence b - type stars typically show radial velocity variations of @xmath32 20 km s@xmath1 in amplitude @xcite , while bhb stars appear to be stable , as they fall outside the rr lyrae instability strip @xcite . the long term radial velocity stability of hvs7 , combined with the metal abundance anomalies ( see [ sec : hvs7 ] ) , hint towards hvs7 being a bhb star . its @xmath3 ( 60@xmath33 km s@xmath1 ) is higher than typically found for bhb stars ( @xmath29 km s@xmath1 ) , although rotators with @xmath3 up to @xmath34 km s@xmath1 have been observed ( e.g. * ? ? ? * ; * ? ? ? * ) . the true nature of hvs7 as a bound bhb star will have to be disentangled by astrometry . for hvs8 we detect a radial velocity variation of 23 km s@xmath1 , consistent with a pulsating main sequence b - type star . we can not discard the possibility of hvs8 being a binary , although the system would have a very low mass - ratio , since there is no evidence of lines from a companion in the spectrum . the star is most likely a main sequence slow pulsator , like hvs1 @xcite . the low s / n of our spectrum does not allow to test for metal abundance anomalies in hvs8 , however , the high rotational velocity of this star will make its abundance analysis difficult , even with higher s / n spectra . we have derived @xmath3 , @xmath13 and @xmath14 for hvs7 and hvs8 , two of the ten currenty known hvss . their @xmath13 and @xmath14 are consistent with the stars being late b - type , as initially classified by @xcite using photometric color indexes . hvs7 has a projected rotational velocity @xmath3 = @xmath35 km s@xmath1 , while for hvs8 @xmath3 = @xmath36 km s@xmath1 . these measurements provide the first direct observational test to the prediction by @xcite , who suggests that hvss ejected via hills mechanism should rotate systematically slower ( @xmath12 km s@xmath1 ) than single stars of the same spectral type ( @xmath37 km s@xmath1 ; * ? ? ? if the hvss have fast rotational velocities typical of single b - type stars in the field , other ejection mechanisms , such as three - body encounters of single stars with @xmath32 @xmath38@xmath39 @xmath40 imbhs , with a binary mbh , or with @xmath32 10 @xmath40 stellar - mass black holes orbiting the galactic smbh , have to be invoked . the @xmath3 values of hvs7 and hvs8 are lower limits to their true rotational velocities , imposed by the inclination angle of the rotation axis of the stars . if the inclination of the rotation axis of hvs7 is low , its rotational velocity could in principle be much higher . in that case both targets are inconsistent with hansen s prediction for hills scenario . however , a sample of only two @xmath3 measurements is not enough to conclusively discern between the different scenarios proposed and more @xmath3 measurements are necessary . statistical tests performed by @xcite conclude that a sample of 25 or more hvss will be needed to distinguish between scenarios at a @xmath41 95@xmath42 confidence level . we also detect abnormal enhancement and depletion effects in the strength of some of the metal lines of hvs7 . these anomalies , together with the apparent lack of pulsations are consistent with hvs7 being a bhb star . its @xmath3 is also marginally consistent with it being a fast - rotating bhb star , if the inclination angle of the star s rotation axis is close to 90 degrees . however , confirmation that hvs7 is a bhb star will not be possible until its proper motion is accurately measured . finally , we find evidence of radial velocity variations in hvs8 consistent with a pulsating main - sequence b - type star nature . additional radial velocity measurements and time - series precision photometry will confirm this detection . minimization result for full spectrum of hvs7 . _ left : _ contour plot of @xmath14 vs. @xmath13 for a fixed @xmath3 of 60 km s@xmath1 ; contours correspond to @xmath43 ; crosses to the models in the kurucz grid . _ @xmath16 vs. @xmath3 for the best fit values @xmath13 = 12,000 k and @xmath14 = 3.50 dex . @xmath16 = 1 shows the statistical 1@xmath22 error of the fit ; the result is @xmath44 km s@xmath1 . a similar analysis was done for hvs8.,width=624 ]

we measure the projected rotational velocities of the late b - type hypervelocity stars hvs7 and hvs8 from high resolution spectroscopy to be 60 @xmath0 17 km s@xmath1 and 260 @xmath0 70 km s@xmath1 . the slow rotation of hvs7 is in principle consistent with having originated in a binary system , assuming a high inclination angle of the stellar rotation axis . however , the fast rotation of hvs8 is more typical of single b - type stars . hvs8 could have therefore been ejected by a mechanism other than that proposed by hills . we also estimate the effective temperatures and surface gravities for hvs7 and hvs8 and obtain an additional measurement of their radial velocities . we find evidence in support of a blue horizontal branch nature for hvs7 , and a main sequence nature for hvs8 .

generally , discussions of theory of dielectric response begin very formally and derive the lorentz - drude model by introducing a complex dielectric function that gives an out - of - phase damping term . in real space this corresponds to a spatially and temporally local damping term . often there is an appeal to the transfer functions of a localized driven damped oscillators as a strong analogy . however , the driving and damping are due to fields that are being changed by the motion of the charges and it is easy to get lost in the rather formal definitions of the `` macroscopic '' variables @xmath2 and @xmath3 . if we were to construct a complete basis of the system one might wonder how there can be any damping at all . the radiational degrees of freedom combined with the electron oscillations and core vibrations are all that exist in the theory . quantum statistical mechanics has never adequately reconciled this problem and the kubo formula is a formal approach to derive results @xcite . classical electrodynamics is the coherent limit of quantum electrodynamics . losses can take the form of a transition to fields and crystal and collective electronic oscillations that have no classical meaning . this suggests that the losses that we describe with the imaginary part of the dielectric constant have a purely quantum meaning ( in that they relate to incoherent motion with correlations outside of classical descriptions ) . there is a long history behind the differences between @xmath4 and @xmath3 and which are viewed as fundamental @xcite . originally , @xmath5 were considered fundamental because of our use of magnets to generate fields . now we consider @xmath6 as the fundamental microscopic fields and @xmath7 as some measure of their macroscopic response ( although more general mixing of linear responses than this are possible ) . we will confine ourselves to the electric case . in the case of electrostatics , we define the displacement vector @xmath8 where @xmath9 , the `` permeability of free space '' for vacuum and larger values for media . this quantity is chosen for the property that @xmath10 so that only the free charges act as sources . in general , solving for the electric field and polarization of the medium would require an iterative self - consistent approach of finding the polarization including the fields from the surface and other uncanceled fields from internal bound charges . the use of @xmath2 allows many highly symmetric problems to be quickly solved by boundary condition constraints and special functions . we can show that the internal energy density stored in the material is @xmath11 . beyond this , its meaning is unclear . it is certainly not the local spatial average of the electric field in a medium . it might best be thought of as an intermediary step to finding the polarization as @xmath12 which is a more physically meaningful quantity . when we seek a response to a time changing field , we generally elevate the dielectric constant to a function of frequency : @xmath13 . this implies that 1 . there has been a relaxation of the medium to a state where @xmath14 and @xmath15 obey a constitutive relation ( and there is only one such branch for a given @xmath16 ) and 2 . harmonic motion exists as solutions and linear combinations of these give general solutions . we know that electrostatics is not the low frequency limit of electrodynamics . ( note that e and b fields must both coexist in electromagnetic waves as @xmath17 . ) nonlinear effects at the edges of packets appear which are essential to any discussion of the fourier transformed fields and media response when it comes to momentum conservation . linear combinations are limited in their ability to capture this aspect of the physics . while these nonlinear effects can be locally made arbitrarily small by gentler packet gradients , the contributions are additive so can not be neglected this way . this suggests we will ultimately need to work with purely real space fields to answer such questions thus limiting the value of working with the eigenstate basis . the extension of the permittivity to complex values is done to consider linear responses that include dissipation . this could equivalently be done with a real response function that is just @xmath18 out of phase from the electric field . this distinction matters because extension to the nonlinear domain is not necessarily able to be done using complex fields where real parts are later taken . we wo nt be interested in such strong fields for this paper but when the nonlinearities are very small there are some simple workarounds @xcite . the kramers - kronig relations assume that the general response function is in this linear domain @xcite . the assumption of causality used in this derivation is not the relativistic one but a local one considering the polarization as response of the driving electric field and that this `` response '' temporally follows the driving . the motivation of this derivation seems to be the response function of a driven damped oscillator . such an oscillator is a spatially localized system where no space - time relativistic causality problems enter i.e. there is no evolving `` front '' to observe . radiation has this as an intrinsic feature and the response radiates out from each point . furthermore , these fields are constantly getting absorbed and reemitted by radiators to which the `` driver '' of the response , medium or field , is ambiguous . first , we will consider a dissipationless continuum model of an electromagnetic wave in a medium which makes no such distinctions and incorporates the full degrees of freedom available to the system then consider damping effects later . we now seek an exactly solvable model based on an idealized solid . realistic solids are composed of many atoms with essentially fixed cores and outer electronic shells that can oscillate . the actual displacements of these electrons at microwave frequencies and higher is very small for almost all typical radiation strengths . this is why the linear regime dominates . even the nonlinear regime generally exhibits relatively small signals and this is often treated by quantum optical methods where the equations again become linear . the problem that we run into in modeling the field in realistic solids is that there is already a standing electric field from the cores holding the electrons in place at equilibrium . this has a complicated structure even for a crystal . on top of this is the radiation field that will be passing through with possible evanescent local contributions . the clausius - mossatti relation gives an expression that relates the local atomic polarizability to the mean dielectric response of the medium . as the wavelengths in the medium become shorter and approach the interparticle separation , the derivation of it becomes less convincing . to get around these complications we introduce a model made of vertical charged plates so that we have a 2d translational symmetry in our solutions . as a first model we choose the arrangement in fig . [ plates ] . and charge @xmath19 . the negative plates are attached to springs with constant @xmath20 . the positive plates are fixed by constraint.[plates],width=480 ] this has a restoring force given by the springs with constant @xmath20 . there is some ambiguity as to the state of the static electric field between the plates . the plate pairs have alternately uniform field and zero field between them . this inhomogeneity does not seem very physical . closer inspection reveals that displacement of the plates gives strong fringe fields . these will contribute additional forces to restoring the plates . it is however causally problematic to have spring forces and fields playing a role at distances to the plates longer than @xmath21 . we are looking to mock up the role of restoring forces in a realistic solid . these are due to the deformation energy of deformed orbitals and the static electric fields of the cores on the electrons . for this reason we consider the modified plate arrangement as in fig . [ plates1 ] . this confines the fields between the plates so the large gap regions between them contain no field . we interpret the springs to be acting locally . the fringe fields that occur from displacements are spread among the many gaps in the material on a scale much less than @xmath21 of any to - be - considered radiation field . the restoring forces from displacements are due to the springs and the displacement fields . these we combine into a net effective force constant @xmath20 . note that we have not taken the dielectric constant @xmath22 or index of refraction @xmath23 as basic here . the mass and charge densities per area , @xmath24 , of the moving charges and the elastic response density per area @xmath25 are the primitive microscopic descriptors of the medium . note that the ratios of of surface densities per plate and corresponding ratios of volume densities , @xmath26 with @xmath27 the density of oscillators , will be the same regardless of @xmath28 , the plate separation since they each satisfy relations of the form @xmath29 . the advantages of this model are both its symmetry and its ability to let us separate the radiational and nonradiative internal fields in a convenient way . because of the symmetries of the system , the values of the fields are unambiguous in between the plates . the parcel averaged net field in a medium is not obviously of much value and one has to wonder if the fields will have different values in different media with the same macroscopic dielectric properties . if so , conservation laws may be the only universal way to describe such systems . the energy is a quadratic function of the fields so using the regional average of a strongly changing field can miss the correct energy by a large amount . ( we often talk about dielectric response as linear and , in the sense that the elastic response of the charges is in the linear regime it is , but it is not meant to imply that the fields are only slightly changed from the vacuum values . the changes can be quite large and vary rapidly over short distances . ) in our model , we never need to consider these restoring fields even when radiation is passing through the system . in this case , they are not static but restoring fields yet , assuming the vertical gaps and their separation between each other on the same plate is much smaller than the separation of the plate pairs , they still are clearly distinguishable from the radiation field in these gaps . in the case of a boosted medium the restoring field picks up magnetic components but the decomposition between these two types of fields is unchanged . this facilitates a simple consideration of the form of the final dielectric response functions under relativistic transformations . averaging methods have the problem that the medium is held up by a balance of electrostatic attraction and wavefunction curvature where the unshielded fields inside the electron shells become extremely large . the internal field of a disturbed solid has both radiative and velocity fields , only the former of which we tend to think of as radiation . however , when we try to compute how the energy in the system is stored and the response we do it in terms of the field strengths and it then becomes ambiguous if we should use the net or some nonradiationally subtracted local fields to average and if we should average over all of space or to subtract some interior region of the atoms . when it comes to the phase of the radiation it will get strongly distorted from a well defined plane wave on the scale of the atoms . in this model system , the symmetry of the problem allows a precise decomposition @xmath30 and we can give equations of motion for the @xmath31 henceforth to be called simply @xmath15 . the phase distortion is replaced by a discrete jump and the singular charge surfaces so that the wavelength and frequency can be well defined between the plates even when the separation @xmath32 ( for example , see fig . [ wavesteps ] ) . let us generally assume this plate separation is much smaller than the wavelength @xmath21 of a wave in this vacuum cavity between the plates . the displacement of the charged plates from equilibrium will be labelled @xmath33 so that the polarization density is @xmath34 . the current density is related to @xmath33 through @xmath35 . we assume that a sinusoidal wave is propagating in the x - direction with e - field polarized in the y - direction . the equations of motion are @xmath36 where @xmath37 , @xmath38 etc . for a plates of area @xmath39 . assuming a sinusoidal solution for the electric field at a given location @xmath40 ( so we can ignore any @xmath41 terms for now ) , @xmath42 and @xmath43 we find @xmath44 the current density @xmath45 is @xmath46 we can now apply maxwell s equations to obtain @xmath47 and using the full space and time dependent ansatz @xmath48 we find the dispersion relation ( see fig . [ dispersionrelation ] ) : @xmath49 gives a phase velocity of @xmath50 and , by the definition @xmath51 and index of refraction @xmath52 which is often written in terms of the plasma frequency : @xmath53 so that @xmath54 for completeness we should find the b - field and show consistency of the fields solution . note that since e and b are perpendicular to the direction of propagation , @xmath55 , and the plates are constrained to move in this plane , we do not need to worry about @xmath56 forces . these forces are periodic , so cancel @xcite , are additionally an order of @xmath57 smaller than those of @xmath15 but when we wish to consider coupling to phonon oscillations of the medium and damping we should include them . ( in the case of compact packets these magnetic forces will impart net end - of - packet impulses that distinguish between the abraham , total packet including electronic and core , and minkowskii , local internal electromagnetic , definitions of momentum . ) first we solve for the b - field : @xmath58 assuming @xmath59 we find @xmath60 with @xmath61 we get consistency with the equation of motion for @xmath62 and the original maxwell s equations . collecting our solutions we obtain : @xmath63 with the dispersion relation @xmath64 where @xmath23 is given above . [ dispersion ] gives a set of four branches of @xmath16 for each @xmath65 . two are the usual radiationally dominated modes that have @xmath66 as @xmath67 . there are two other modes that have @xmath17 as @xmath68 . these we might be inclined to call `` acoustic '' modes in analogy with similar results in condensed matter theory . in contrast with the first two modes these are `` elastically dominated '' ( to be made more clear later ) . if we let the charge density vanish , these modes degenerate to a set of independent oscillators with frequency @xmath69 . the phase and group velocity of these modes are all less than @xmath57 for the acoustic branch and no singularities occur . however we notice the singular denominators in @xmath33 and @xmath70 corresponding to @xmath71 the resonant cutoff of the acoustic modes where @xmath65 diverges . this means a packet would usually contain negligible amounts of such modes . these give a complete set of variables that allows us to specify any state of the field and medium . presumably we could expand any packet with vanishing fields , displacements and currents outside a compact set and observe the advance rate of the edge . before we move on to more subtle considerations , let us address the question of group velocity and the energy and longitudinal momentum transfer in the medium . our basis did not allow any longitudinal momentum as the plates were constrained in this direction . this momentum is shared with the elastic modes of the medium which are often very small . however the duration of a packet can be quite long and variations in its intensity can be on periods where acoustic oscillations can be excited . our basis requires we include these as an ad hoc modification that connects to these acoustic modes to conserve this momentum . it is this consideration that will lead to an understanding of the physical meaning of the abraham and minkowskii momenta for this model and will show a very nice built in self - consistency to the theory and a transparency of how momentum and energy are shared between medium and fields and a causal correction for nonlinear and varying damping effects . from this model we can now compute the local energy density and momentum directly for the progressive wave solutions . the energy has three sources 1 . ke of plates 2 . pe of springs 3 . energy of fields this gives the total energy density in eqns . [ edecomp]-[edecomp1 ] @xmath72 where @xmath73 it is related to the index of refraction by @xmath74 the charge and electromagnetic energy is decomposed here to show that one can view it as static and moving oscillatory components . this will be useful in later consideration of the phase velocity . we notice that the oscillatory components of the charge energy is exactly equal , and in phase with , the energy of the electromagnetic waves while the potential energy contribution is out of phase with them leading to a net constant energy density component . the time averaged energy is @xmath75 using the polarization vector defined by @xmath76 we have @xmath77 so that @xmath78 where @xmath79 and @xmath80 . @xmath81 and @xmath82 are contrived here to get the usual relations between energy and displacement . the poynting vector is given by @xmath83 where @xmath84 . the transverse work and longitudinal force on the plates are @xmath85 if we now consider time averages of these quantities we obtain : @xmath86 from the dispersion relation we can calculate the group velocity @xmath87 which is always less than c. one can show @xcite that center of mass and momentum conservation dictate that a packet must move at the group velocity through a uniform medium and carry momentum density @xmath88 . this is a universal relation for any disturbance that travels as a bound compact unit and leaves the medium undisturbed after it passes . this does not preclude stresses at the ends of the packet and complicated interactions with surfaces and momentum transfer at boundaries and through antireflective films as such a packet enters the medium . we can verify this theory in this case by noting that @xmath89 is the momentum density of the electromagnetic field . consider that the plates carry no net momentum . it is not just that they are laterally constrained . the forces in the x - direction average to zero over a cycle . this means that @xmath90 is all the momentum density there is to the packet . @xmath89 is the momentum density of the electromagnetic field and since there is no contribution from the electrons due the lateral constraints on the medium and the fact that these forces average to zero for a plane wave , this is the only momentum in the problem . this lets us immediately verify @xmath91 next we will consider details of the microscopic solution and how stress builds up in the medium and is shared between the charges and the fields . the abraham - minkowskii controversy revolves around the momentum of an electromagnetic wave in media . the use of momentum flux is common in treatments of elasticity and hydrodynamics but unlike mass flux ( i.e. momentum density ) , momentum is not a locally advected quantity ( see app . [ stressmomentum ] ) . the pressure plays the role of a source and sink and incompressibility introduces the ability to transport momentum over large distances apparently acausally . from this springs and endless list of sins and accidental successes from the use of pseudomomentum . for completeness , and to avoid such pitfalls , we seek to understand all the local forces and momentum densities in a system . these fall into several types : reflection impulses , pure electromagnetic momentum , longitudinal plate momentum ( that has been excluded from the degrees of freedom in our basis ) , radiative stress from `` hidden '' standing waves , forces from the static / velocity fields and bond distortion that we could term electrostriction , and , most subtle of all , conversion impulses due to momentum absorption of the material while it `` rings up '' to an equilibrium state with the wave . as mentioned before , a real understanding of the system and a universal theory of dielectrics will involve our ability to keep track of the conserved quantities as they propagate through the medium and exchange forces with it . we will see below that the stress tensor of the combination of medium and electromagnetic field together is not that illuminating but expressing them separately and including the local momentum transfers between them gives the observable impulse induced changes in the medium . the use of packets is essential in this process . infinite wave trains can provide exactly solvable solutions but , from the standpoint of conservation laws they can harbor hidden inconsistencies . two examples are the case of electromagnetic momentum of charged infinite plates in a magnetic field and that of the angular momentum of surface waves . in the first case , the net momentum is cancelled by fringe fields at infinity , which we might suspect since no composed collection of charges with no initial momentum can acquire any . in the second case , right moving progressive waves have a ccw angular momentum about the surface , despite cw particle motion but the value of this depends on where boundaries are periodically placed . hence the angular momentum density is sensitive to boundary conditions at infinity @xcite . in the case of an electromagnetic packet entering a block , [ fig : incident ] , we will see that , even given perfect ar coatings on the surfaces , the medium must acquire part of the momentum of the packet . there is additionally a stress that exists in the solid at the ends of a packet entirely within the block , fig . [ fig : enclosed ] . this is , however , different from the stresses that exist at the end of an infinite wavetrain , fig . [ fig : traversing ] , or long packet that traverses the block . we will compare this with the case of a partly in and partly out packet , fig . [ fig : halfin ] , to demonstrate how the outwards forces on the surface can be present while the medium picks up a net impulse . we will also investigate the effects of finite time duration of such impulses and how energy can get drained from the packet into acoustic impulses that traverse the block . the case of electromagnetic waves in media is complicated by the fact that there are two very distinct components : fields and charges . in acoustics and hydrodynamics , this is not the case . each has other complications that make them more complicated but in this one respect they are simpler . progressive surface gravity waves have a number of serious complications not the least of which that they carry mass hence have a nonzero momentum density . this is generally considered to be a higher order effect and , when it comes to ( the lowest nonrelativistic component of ) energy transport , it can be ignored @xcite . they will however shed some light on how to consider energy transport and the origins of group and phase velocity from the point of view of conservation laws . 0.4 0.4 0.4 0.4 the small amplitude solution of a deep water surface wave is given by the airy wave with surface profile @xmath92 and velocity potential function @xmath93 with the following dispersion relation : @xmath94 . from this we derive the phase and group velocities : @xmath95 , @xmath96 . the energy density per unit area is @xmath97 where the energy is equally divided among the kinetic and potential energy . the kinetic energy comes from the the small circular motion that penetrates down to a depth of @xmath98 . there is a small higher order drift that we ignore here . averaged over depth this is a uniform spatial distribution . the potential energy , however oscillates with position as @xmath99 . this gives a total energy density of @xmath100 . the energy density is thus of the form of a constant @xmath101 plus a positive advancing oscillatory function of height @xmath102 . it seems that the potential energy is transporting at velocity @xmath103 and the kinetic energy is fixed . the usual interpretation , due to rayleigh , in terms of packets indicates we should have a flux of @xmath104 which , interestingly equals @xmath105 . when the energy reaches the end of a packet , it produces new elevated regions which do not yet have corresponding kinetic motion to propagate them . these crests then drive the flows and an analogous effect happens at the back end of the packet . this gives us a picture of a packet as one where potential energy is transported at @xmath103 and the ends act as sources and sinks that convert half of this back into kinetic energy . the kinetic energy in the middle is essentially static ( except for a small stokes drift ) . packet spreading must arise at the ends due to the fact that the pressure created by the surface distribution will extend down and forwards @xmath106 from the packet s end so always leads to some stretching out of surface elevation of the packet . let us now apply this mode of thinking to our packets of electromagnetic energy in media . unfortunately , it is not exactly true that the kinetic and potential energy of the charges is static from eqn . [ oscillations ] . however , we do have a combination of charge and electromagnetic energy that is constant in eqn . [ oscillations1 ] . here we can decompose the net energy as @xmath107 . using @xmath108 we see that a similar interpretation applies here . it seems that there is a static component that is not moved by the traveling oscillatory crests . unpublished investigation into other systems by the author suggests this is a very general feature when no net mass flux is present . since @xmath103 can be greater than @xmath57 this is clearly an imperfect interpretation . closer inspection shows that there is a back and forth sloshing of the energy that contributes to the oscillating force on the charges . this oscillating local backwards moving flux of electromagnetic energy explains how the crests of energy above @xmath109 can advance at greater than @xmath57 while the total local energy flux never exceeds it . at this point one might wonder why we can not simply compute the electromagnetic energy between the plates @xmath110 and use that it advances at the vacuum group velocity , @xmath57 , so that @xmath111 rather than @xmath23 . the plates themselves can only transfer energy via the em fields so this should be all of the energy flux . as we will see in sec . [ stress ] below , the phase shifts at the plates necessitate a microscopic backwards flux of energy for a macroscopically simple progressive wave . this generates both a stress on the walls of the medium and a reduced effective electromagnetic energy that advances at @xmath57 . first let us consider the impulses due to the ar coating , specifically , what is the momentum ( not internal stress ) of these waves in media and what momentum transfer has it exerted on the medium . ( internal stress is not ignorable here and must be considered in any experimental result . in this model some of this force is hidden at the far off edges of the block in the fringe fields of our plates . realistic media will generate this from the local velocity fields and changes in the orbitals due to driving . below we just consider the fraction that arises from global conservation laws . ) we will do this two ways . first by an explicit calculation then by an analysis of the microscopic fields of a wavetrain traversing the medium . it is tempting to simply use the lorentz force to compute the forces on the surface using the continuum or constitutive model and be done with it . unfortunately , this will not always be sufficient for reasons already suggested . however , we can directly calculate this force of a steady beam at an ar coating . the details of the ar coating is not important , since the energy flux is constant throughout the system , the impulses at it will not be system dependent . this is in contrast with the stress on it which may be medium dependent due to other sources of electrostriction . the difficulties with deriving an exactly solvable calculation @xcite lies in the asymmetry between the two dynamic maxwell s equations . the current term @xmath112 exists in the @xmath113 equation . we can correct this asymmetry by imposing a magnetic monopole current @xmath114 . as long as there is no net work done by these current the time averaged impulse will be unchanged . the corresponding maxwell equations are @xcite @xmath115 we now match the two solutions with the same phase but discontinuously at @xmath40 . @xmath116 the discontinuities give the implied singular current densities . @xmath117 the power can be calculated by dotting the midpoint field at @xmath40 with the singular currents @xmath118 . these vanish since they give opposite contributions . the averaged ( net outwards ) pressure at the surface is given by @xmath119 this is consistent with the results of mansuripur @xcite derived with considerably more effort . we next investigate the stress in the medium by investigating the microscopic decomposition of the fields between the plates and show it is consistent with the above result . considering the case of fig . [ fig : traversing ] , the medium has completed any relaxation and momentum absorption from the advancing action of the packet edge . the fields in the gaps are made of plane waves with dispersion @xmath120 where these are presumably of a nearly monochromatic form with the same @xmath16 as the macroscopic frequency . we can decompose the fields here into right and left moving components by first noting that , for analytic waveforms with @xmath121 , we have a traveling wave and removing this component gives a wave moving in the opposite direction . we start by assuming we have primarily a progressive wave in the x - direction and a standard right handed coordinate system . in the following @xmath15 and @xmath122 are components of these vectors in the induced @xmath123 axes . this gives a decomposition in to right and left moving component waves . @xmath124 this decomposition eliminates the cross terms in the stress so that we can write @xmath125 in terms of the respective crossed right and left moving fields . the corresponding momentum densities are : @xmath126 where @xmath15 has been relabeled as @xmath127 , the field strength between the plates , a distinction we will need shortly . as a consistency check we see that @xmath128 . this is the net momentum density . the remainder of the momentum is traveling right and left and canceling but still generating a stress and force on the boundaries of the material . we can think of this microscopically as the momentum as having a free standing wave component with a free propagating wave between each plate . ( an illustration is provided in fig . [ wavesteps ] ) . the residual flux gets a magnitude of @xmath129 in each direction so an obstruction reflects with twice this magnitude . ( absorbtion instead of reflection gives half the following result ) . the stress is therefore : @xmath130 note that @xmath131 here is the field in the vacuum before the packet entered the medium and @xmath127 is the field strength inside the medium . one could alternately view the role of the plates as inducing a phase shift in the local @xmath15 and @xmath122 fields to the extent that they are well approximated as sections of plane waves of wavevector @xmath132 . defining the crest maxima as @xmath133 and @xmath134 we can define the phase shift @xmath135 mod@xmath136 . the shifts at each plate gives an infinitesimal shift in the energy and momentum of the waves . this gives an alternate perspective to the above point of view in terms of right and left traveling waves . we can then do a calculation of forces bases on the `` phase shift density . '' ( realistic media have no such singular sheets of charge and this phase shift must be replaced by a stretching of phase near the charges . ) now let us do a careful calculation of the motion of a packet moving into a dielectric slab with perfect ar coatings in the spirit of balazs @xcite where the packet begins as in fig . [ fig : incident ] and arrives on the interior as in fig . [ fig : enclosed ] . let the packet have length @xmath137 ( less than the length of the slab ) , area @xmath39 , and roughly monotonic frequency @xmath16 . its energy is @xmath138 . the packet moving into the medium advances at @xmath139 and is contracted by a factor of @xmath140 so , by the above relations has energy @xmath141 , where @xmath127 is the maximum field intensity in the medium vs. @xmath131 as the field intensity maximum in the vacuum . the yields the relation between the field in vacuum and the medium : @xmath142 . computing the net poynting vectors of the packet inside and outside we find they are identical : @xmath143 therefore the momentum densities are also equal @xmath144 . since the packet gets contracted by @xmath145 we see that there is a deficit of momentum @xmath146 . this force is inwards and is returned later when the packet leaves the medium . the resulting force depends on the the duration of the packet @xmath147 . the induced pressure is @xmath148 now consider the evolution of the packet midway entering the slab as in fig [ fig : halfin ] . there must be a `` ring - up '' time to build up the standing wave and impart energy to the oscillating charges during which time the associated electromagnetic momentum is absorbed by the medium . we can estimate the lag experienced by the front of the wave . the energy density of the wave is given by eqn . [ energy ] @xmath149 assuming no reflection from the edge of the advancing front , we find a front velocity that advances at @xmath150 which is always @xmath151 . we can now summarize what we know in terms of forces on the surfaces and packet ends in terms of @xmath152 and @xmath153 . as the packet enters the medium as in fig . [ fig : halfin ] , there is a backwards pressure on the wall @xmath154 and a forwards pressure on medium at the advancing front @xmath155 such that the net force is @xmath156 so that @xmath157 . the electromagnetic standing wave is sitting between the back wall and the front edge of the packet so that we can identify @xmath158 as the pressure from the ring - up of the medium . once the packet is entirely enclosed as in fig . [ fig : enclosed ] in the medium we have equal forces on the medium on each side due to the stress and ring - up and ring - down respectively : @xmath159 . in the case of the long traversing wave in fig . [ fig : traversing ] we have only the @xmath152 forces at the surface . this gives us a picture of the equilibration of a long beam as transferring a net impulse to the medium then introducing a net stress across the walls . for reflections at an immersed reflector , as in the ashkin - dziedzic experiment @xcite , the net impulse on it is from the momentum flux of the external beam with no @xmath153 forces at all . this explains why the minkowskii definition of the momentum works for this experiment . we have already discussed how damping in media must be an essentially quantum event since , otherwise , the associated elastic damping modes would simply enrich our band structure and never remove energy from the beam . however , for the case of modulated beams one can have losses into acoustic modes . in principle such modes simply give new branches of an otherwise lossless dispersion relation . when the relative occupancy of these acoustic modes are small one can view them as a kind of sink and this is the approach we take here . in practice , acoustic modes will be damped to quantum incoherent motions of the media and never recontribute to the beam . as the front of a half infinite beam advances across the medium it imparts a force at this rapidly advancing layer . the front surface of the block experiences a nearly uniform force @xmath160 . the medium , of length @xmath137 , will equilibrate to this on some time scale @xmath161 where @xmath162 is the speed of sound in the medium and @xmath33 is its young s modulus . let us now modulate our beam into a set of pulses with duration @xmath163 and spacing @xmath164 . the impulses create acoustic stress waves of length @xmath165 and amplitude @xmath166 . these move through the medium at @xmath162 so that if @xmath167 these packets are well separated . the mean averaged energy flux removed by acoustic means is then @xmath168 this gives an acoustic loss in the beam energy that is not present in the case of a uniform beam traversing the medium and reduces the power flux from @xmath169 to @xmath170 . one may wonder if there are acoustic losses in the medium during uniform propagation of a beam . at zero temperature there will be beam solutions that incorporate acoustic response into them but at finite temperature , thermal fluctuations will remove a fraction of energy @xcite . the classical fields correspond to coherent states of photons . this means that we do not expect a well defined number of photons to exist in the medium . however , we can consider the case of a single photon entering the medium and ask what is the expected result . we know that a fraction of the photon must now be absorbed into energy of the oscillators . this suggests that the situation is one of a superposition of a system in the zero photon and single photon sector . the waves between the plates are represented by right and left moving plane waves obeying @xmath120 and , although the get the phase shifts at each plane we need a broad distribution of frequencies , we can use that the energy density to momentum density in each right and left moving component is constant by the quantization conditions @xmath171 and @xmath172 . the occupancy of the single photon sector is therefore given by @xmath173 . the remaining momentum and energy is in some excited state of the medium in the zero photon sector being distributed among various phonon and electronic excitations . the extremely formal nature of quantum field theory and quantum optics generally preclude the kind of local detailed balancing analyses using conservation laws that classical systems allow . this seems to be a nice exception and hopefully a step towards a richer set of cross checks on problems with quantum response in media . the utility of the stress - energy tensor is in generating forces or , in the case of general relativity , providing sources for gravity . the bulk stress tensor in the absence of a medium is often of questionable value since when we consider gradients of stress to get forces there must be something to push against . this is the problem with idealized `` photon - gas '' hydrodynamic models in cosmology @xcite . the local photons just travel ballistically regardless of whether or not they are in thermal equilibrium . if there is a medium present then there must be sufficient time to equilibrate and transfer those forces to the material part of the medium and for the medium to reestablish equilibrium with it ( either equilibrium with the charges as in our continuum model or thermal equilibrium ) . the photon - photon interaction is so weak that we probe material changes , typically with other photons , to determine the response . the stress - energy tensor can be decomposed as @xmath174 where , for the progressive wave in the above coordinates , @xmath175 the force responses we usually desire involve the absorption of electromagnetic momentum by the medium itself at the boundaries and during propagation in the bulk where field s averaged behavior is changing . since the rate of transfer depends on both the opacity of the medium@xcite and response rate of the charges , it is hard to see how a collective stress - energy tensor for the combined system is of much value for this purpose . furthermore , there can be other sources of field induced stress in more general media due to changes in the mean electronic structure and nonradiative fields . in this sense our dielectric is `` minimal '' in that the only forces that arise come from the radiative field and conservation laws for it . nevertheless , in the local frame of the medium , each component is simple , well defined and local conservation laws hold explicitly . the dispersion relation @xmath176 gives us ( fig . [ dispersionrelation ] ) how the frequency @xmath16 relates to the macroscopic wavevectors @xmath65 . microscopically , however , the waves in the gaps are right and left moving combinations of wave with @xmath177 . the group velocity is the mean velocity of motion of a packet and is plotted in fig . we see from the dispersion relation eqn . [ dispersion ] that there are two frequencies where the macroscopic @xmath65 vectors diverge , @xmath178 and @xmath179 , however now we see that only the first second one contains arbitrarily short microscopic wavelengths . the second critical frequency , @xmath180 , has @xmath181 but these are again made of microscopic waves of @xmath182 . we can consider the internal stress of the medium in fig . [ stress ] and we see that it diverges at both these critical frequencies . this implies the internal standing waves are very large . at @xmath183 , the resonant frequency , we see that the internal energy diverges ( for finite field strength ) but @xmath184 also does so that we can say the system is `` elastically dominated . '' the other divergence seems to be an artifact of averaging fields since the microscopic @xmath65 goes to zero and both energies are finite . if we compute it in terms of the internal field strengths the stress is actually finite . in the case of metamaterials one can have a `` negative index of refraction '' so that the group and phase velocities move in opposite directions . this analysis reassures us that , microscopically , the waves are all primarily moving in the same direction as the groups . , width=288 ] , width=288 ] so far we have only studied a nondissipative case of a simple medium with one resonance . we could easily add other oscillators to get more complicated dispersion relations but they will always give singular behavior at the resonances . damping rounds these peaks and this is generally described by an imaginary part in the dielectric function . since we are seeking a description directly in terms of the microscopic motions instead of constitutive laws , we wo nt seek a direct analog of the dielectric function . we are also only interested in using real quantities since we want an easy generalization to nonlinear and time changing media . if one specified a general real dielectric function one can not even be sure that the imaginary part of the analytic extension will not include source terms ( `` negative damping '' ) . this is certainly problematic from the point of view of justifying causal evolution . one could simply give a set of frequencies and damping rates for every function @xmath185 and enforce a linear evolution however , on fourier transforming it , one generally finds that the damping function is not local . this locality condition is what enforces the kramers - kronig relations for linear solutions . causality arises from these relations and the various precursors arise @xcite as higher velocity components that the rest of the packet during the ring - up time . the usual treatments of causality involves some impressive complex analysis and one might wonder why this should be so . below we will discuss how conservation laws and positive definiteness of the energy density enforce this easily . there is an analogous causality problem in the case of heat conduction . since this has led to a number of erroneous attempts at modifying the heat equation including higher order modified or extended schemes that give hyperbolic solutions ( in violation of the fundamental degrees of freedom of the system ) we include a discussion in the appendix based on a similar microscopic analysis in app . [ discretew ] . a detailed description of the waves in the gaps showed there is a discontinuity in @xmath122 due to the currents at the plates but not in @xmath15 . ( the electric field of the right propagating part of the field is illustrated in fig . [ wavesteps ] . including the hidden standing wave contribution between the gaps joins the net fields smoothly . ) for a wave with a well defined @xmath16 the component waves in the gap region obey the relation @xmath120 exactly . the plates in this limit now just look like reflectors and energy storage devices . we saw that the distinction between traveling ( progressive ) and standing waves was that the relative phase of the magnetic field is different relative to the electric one . this suggested that a deviation in the magnetic field due to @xmath70 in the previous model is actually introducing a partial reflected wave in the plate gap region . if we initiate a pulse at one plate it causally travels to the next , partially reflects and transmits just as we expect from scattering theory . the continuum approach has too many advantages in its economy and analytic tools to abandon yet we must choose one that is not averaging away crucial information and retains some intuitive properties connecting it to the microscopic physics . in analogy with the heat equation , it would be nice if we could place some limits on the velocity of the energy flux through the form of the equations themselves . this would 1 . manifestly conserve energy and 2 . exploit that energy is a positive density function so its vanishing ensures all the other variables vanish at a packet edge . the electromagnetic part of the dynamics are fixed by maxwell s equations . since we have developed a model where @xmath31 and @xmath186 are nicely separable that worked so well in elucidating the role of conservation laws , we assume that such a decomposition is generally valid or that there is some kind of universality that allows such a model to be equivalently constructed to every realistic medium . the field evolution equations of @xmath187 and @xmath188 are then governed by the microscopic maxwell s equations . the interesting parts of the equations that involve ring - up , ring - down of the medium , damping , nonlinearity , time changing media response , etc . are then all in the details of the medium that manifest as the current and displacement functions . @xmath189 where we have included a magnetic current @xmath114 as a device to create magnetic polarization later without having to explicitly use dipoles or current loops . since @xmath190 we have the wave equation for @xmath15 ( and similarly for @xmath122 ) @xmath191 the fields then propagate at the speed of light with source and sink terms as long as @xmath114 is not a function of @xmath192 and @xmath193 is not a function of @xmath113 . this then preserves the characteristic structure of the equations . in a realistic medium there is an elastic constant for the electron distortions , @xmath194 , and one for the cores , @xmath195 . the electron distortion is measured relative to the cores so that these are coupled . rapid oscillations will tend to drive the electrons and leave the cores behind as in the case of internal black / gray body radiation . bulk elastic forces , as @xmath152 , that change slowly will transfer directly to the cores and create acoustic effects . however , the ability of the medium to store and deliver energy through the currents can be quite general and still maintain causality . we can let @xmath193 and @xmath114 be local functions of the fields but also of elastic and other local properties of the material and even of external driving forces . the simple linear case of a dielectric gives @xmath196 where @xmath193 itself is an independent variable so no constitutive law holds for it in terms of the fields . rapid damping of acoustic modes will tend to lead to relaxation to the optical modes in eqn . [ dispersionrelation ] so that an apparent constitutive law is observed . as long as the local medium dynamics are a function of the local medium effects and not derivatives of the fields , causality is manifestly preserved . the medium will tend to damp deviations from the optical modes through faster damping losses in the acoustic modes . these lead to the precursors that absorb electromagnetic energy as a packet evolves . the natural time scales associated with the medium that determine how fast energy is absorbed from a nearly monochromatic wave that is not yet in equilibrium with the medium are given by the rate of ring - up to an equilibrium value of @xmath197 . as an example , consider the case of the free electromagnetic wave as a monochromatic beam in a dielectric medium with no initial medium response . the source terms must conspire to give advance at the phase velocity so that @xmath198 deviations from this situation as in the case of a free space em wave inserted in the medium with no medium response yet present , allows the fields to advance at @xmath57 while the medium gradually steals energy and momentum from the beam . such a configuration can be expressed on the basis set using the four branches of the dispersion relation in eqns . [ b]-[solution1 ] . for an infinite wavetrain , the wavelengths do not change but the frequency is altered from the free space dispersion relation , the amplitude decreases , and backwards components are generated . the resulting four - wave mixing , @xmath199 gives an oscillatory change in the fraction of energy storage in the electric field . for an advancing packet , there is a broad spectral distribution of fourier components . the sommerfeld theory @xcite of precursors states that each component advances at approximately the group velocity of it . in real space , the local absorption of em energy from a gradually sloping monochromatic packet by the medium can be described with current such that @xmath200 gives the rhs of @xmath201 where @xmath202 is approximately antiparallel to @xmath113 . the most general medium is described by a set of variable @xmath203 with equations of motion @xmath204 . the displacement and currents @xmath205 are functions of the @xmath206 . electromagnetic restoring forces are hidden in the material variables so that electrostrictive effects can arise in these equations . additionally , external forces and sources of energy can be injected that can change the properties of the material , as in the case of electromagnetically induced transparency , or chemically driven medium changes . therefore @xmath207 can be explicit functions of time as well . the momentum effects on the medium can either be computed exactly through the local lorentz forces or by conservation laws utilizing that the form of the poynting vector in the medium is unchanged from that of the vacuum . we can now view eqn . [ law ] as a local real - space and manifestly causal set of equations of motion where the local evolution of @xmath193 and @xmath114 are functions of local medium conditions and the local values of @xmath15 and @xmath122 . we can think of this problem as a balance of energy in the equilibrium case where the input energy from the fields @xmath208 is balanced by the output @xmath209 for each charge layer . just as in the radiation reaction case of a point particle @xcite there are necessary nonlinearities here to give the right damping modification of the the driving force that arise from the relativistic acceleration . for larger field strengths these are unavoidable as the medium response gains nonlinear changes during ring - up and ring - down even if the elastic restoring forces stay in the linear regime . the extensions of linear response theory are typically rather formulaic . one imposes a structure based on a hydrodynamic framework and seeks relativistic or nonlinear corrections consistent with some physically important symmetries . the relativistic correction is not extremely important since it takes enormous fields to drive electrons at relativistic speeds and producing relativistic changes in medium velocity over field equilibration distances is similarly difficult . the whole constitutive approach has locality implicitly involved since it assumes the fields and medium reach an equilibrium over distances short compared to those of physical interest . a correction to medium response theory that seems more relevant is in how deformation and noninertial effects on the medium that may be small but potentially iterative many times over an optical path may build to produce large net effects . a medium undergoing acceleration will have the bound radiators move and reradiate without a time lag compared to the radiation that is momentarily unbound from it . such a treatment seems essentially nonlocal and to require the kind of explicit decomposition of fields and medium we have discussed here . there have been many attempts at determining the kinds of forces and stresses in dielectric media with some of the opinion there is a unique decomposition in terms of the electromagnetic and media response and others arguing that such a decomposition is meaningless and correct use of boundary conditions give equivalent results . the model introduced here , clearly gives a unique decomposition and leads to the hope that by tracking the flux of energy and momentum of a traversing radiation field and considering the absorption ring - up and damping of the medium one could arrive at a universal picture of dielectrics with a unique decompositions of field and medium energy and momentum . in the case of densely packed atoms in condensed matter , the delocalized electron wavefunctions give a momentum that is a linear combination of @xmath39 and @xmath210 , the phase gradient of the wavefunction , this is explicitly gauge dependent so this is problematic already . we can consider a medium made up of layers of independent dielectric blocks with ar coatings so that no standing wave exists between the layers as in fig . [ nonuniversal ] . these subblock layers will have internal stress but the walls of the net medium have none . this then gives a kind of metamaterial with nontrivial dielectric response but no stress which seems to kill our idea of such general universality . a second problem is the long range polarization forces that exist at the edge of beams or blocks of media . these fields can fall off rather slowly and can create local stretching and long range attraction that then couples to the elastic response of the medium . for long wavelengths these forces seem to involve the nonradiative fields of the medium . the injected radiative field is the only source of energy in the problem so must lose energy to fund such additional energy expenditures . the dispersion relation was shown to be a measure of how much energy is stored in the medium versus the field so must be impacted by such additional processes . even in the linear limit , one can have long range effects that are easy to neglect in a `` constitutive model . '' an example is the long range electrostatic forces on dc wires from the gradient of surface polarization to drive the internal small but nonzero electric field . it is unclear how much such considerations impact the dielectric response of more mundane materials that have not undergone some clever small scale engineering . however , it does suggest that universality does not exist generally for the stresses in dielectrics as function of the medium s dielectric and permeability functions . furthermore , even in the apparently linear regime for medium response one may need to do more careful detailed balancing of the internal fields and elastic medium response than can be obtained by use of a naive stress tensor as a function of internal field strengths . an alternate title for this article could have been `` electrodynamics in media without constitutive laws '' in that we provide a full dispersion relation so that fields and charge motion may be chosen with complete independent freedom . we have presented a simple dielectric model that is easily extendable to the case of multiple resonances and that does not depend on any averaging over localized oscillators and fields . the forces and impulses are now readily apparent by microscopic analysis in terms of the purely radiative field strengths that are not always clearly separable from the restoring fields in the averaging approaches . some nice byproducts of this have been a complete way to track momentum in the entire system , a measure of the photon number changes as a quanta enters and traverses the media , some understanding of how acoustic losses can be generated and an easy way to see how group velocity relates to the hidden microscopic phase velocity in metamaterials . nonlinearity and time delay are two possible routes to chaos and at higher intensities and frequencies , media response has both of these . as our ability to generate higher field intensities grow , more exotic dynamics are almost certain to arise . additionally , as our ability to measure smaller effects improve , the necessity to describe optical impulses and other competing small effects becomes more important . in addition to the case of large bond distortion , the abraham - lorentz - dirac force law ensures that nonlinearity will arise as a function of field gradients . the finite granularity scale of media gives a retardation for propagating changes to advance between separated radiators and to relax to the new mean frequency and field energy . in the case of nonlinear media one can seek a modification of the kramers - kronig relations . generally , a local damping law is assumed but phonons mediate most losses and these are nonlocal objects and we are nowadays frequently concerned with confined systems where discreteness in the phonon spectra is important . it therefore is advantageous to be able to discuss damping and dielectric response more generally than the linear response method and more intrinsically than a formal nonlinear extension of it . granular materials provide the canonical example of a system with no useful continuum limit . hydrodynamics does not work for granular flows . static packings are strongly history dependent and focus forces over many orders of magnitude at the scale of individual grains . frictional torques produce an `` indeterminacy '' whereby the packing and its boundary forces are inadequate to determine the internal stress . in the case of an electrodynamic field moving in media , one has to make the assumption that the fields are changing slowly on the scale of the separation of charges . for chirped and radically shaped pulses this is not so . x - ray and gamma ray shallow angle reflection will not satisfy this condition either . having an explicit model that can describe the fields on the scale where the wavelength becomes comparable or smaller than the granularity scale of the medium may give new ways to address such problems where continuum mechanical ideas may no longer be valid . precursors have been difficult to detect at a level of accuracy to test the sommerfeld - brillouin @xcite theory . for this reason people have been hunting for other systems e.g. surface gravity waves , for comparison @xcite . metamaterials are highly tunable . resistance and real time variations can be easily introduced into the shaped oscillators for microwave frequencies due to their larger scale . this would provide an excellent place to test for precursor shape and damping effects and introduce controls on the flux they generate . just as importantly , a clear understanding of the microscopic reality may help weed out some of the more fantastical theories and flawed analogs to validate them . there are other examples of systems with natural velocity limits that are not @xmath57 . the speed of ocean waves , the motion of oscillations on a spring or heat transport in a solid have natural limits in the speed of sound of the underlying medium . it is unclear if there is any universality to the resolution of the dynamics here . certainly some will seek a sweeping class of equations based on symmetry that ignore the underlying details of the dynamics . the results here and similar ones not included here are suggestive that this is a mistake . ultimately everything comes from microscopic physics and shortcuts that seem initially successful or are only successful in particular cases can contribute to long lasting confusion . the quantum use of quasiparticles is widespread yet the form of their dispersion relations imply that they must face such similar constraints . the best form for such corrections is an important challenge . high temperature superconductivity comes from a strongly interacting system utilizing very shallow hole filling bands . it seems that one can have extra energy carried with electrons in the interactions to give larger effective mass but how @xmath211 can arise seems mysterious . this is only possible due to the fact that the electron - electron interaction is greater at the brillouin zones so that excitations can actually reduce the net interaction . it would be interesting to see if such a composite model of electron waves modeled on the example given for electromagnetic waves here might lead to new insight on the forces and transport in conducting media . a current exciting topic in optics concerns the so called `` left handed '' or `` negative index '' materials @xcite . in this case one has a group velocity that is opposite to the phase velocity . the electric , magnetic and propagation vectors then become a left handed pair . such a state is clearly not possible for a simple dielectric . it it therefore necessary that the medium s dielectric and permeabilities are both nontrivial . in the case that they are both negative such a condition exists . it is therefore interesting to see how such a state can be made sense of microscopically in terms of the model of free radiation fields traveling between oscillators that act as temporary energy and momentum storage devices . such materials have been constructed in the microwave regime with `` split ring '' lattices to get both electric and magnetic responses that are coupled . such devices can be modeled with the more general case of electric and magnetic monopoles as oscillators as was used in sec . although such magnetic monopoles may not exist in nature they can model any current loop s action and do so in a parallel and simple fashion of elastically bound massive monopoles . the resulting equations of motion are @xmath212 where @xmath213 is the charge density of the electric oscillators , @xmath214 is the monopole density of the magnetic oscillators , etc . the dispersion relation is @xmath215 as an obvious extension of the results for the purely dielectric case in eqn . [ dispersion ] . a typical solution is shown in fig . [ lefthanded ] . comparing with the purely dielectric dispersion relation , we see that presence of two additional bands . the second from the bottom gives negative @xmath216 and positive @xmath139 consistent with the properties of left handed materials . interestingly , the energy flux between the plates must be advancing as free photons in the group velocity direction . the advancing phase direction is for the spatially averaged wave . in our model this can be represented as free waves with phase shifts at each oscillator layer as in fig . [ wavesteps ] . for a dielectric with no magnetic response , this can be viewed as the magnetic field intensity or the component of the electric field intensity that is purely propagating . for our negative index material , it should be viewed as the purely propagating component of either field . consider a 1d lattice of points separated by @xmath28 that each contain @xmath217 walkers . every discrete time increment @xmath218 the system is updated and half of the walkers move left and half right one step . ( we assume @xmath23 is always so large that problems posed by odd values do not make important contributions . ) there is no a priori reason @xmath217 should be quasicontinuous but , assuming it is , we can modify the finite difference equation into an approximate continuum one : @xmath219 where @xmath220 . in the last line we reformulated this into a form reminiscent of @xmath221 where @xmath222 is the current of the scalar quantity @xmath223 . first note that the discreteness of the system places bounds on the possible gradients of @xmath23 . since @xmath23 is a positive definite quantity @xmath224 . initial data should remain less than this and the evolved function should preserve this condition . from here we could attempt to improve the accuracy of the equation by using higher order terms in @xmath225 and @xmath226 derivatives . higher order terms in @xmath225 violate the sufficiency of @xmath227 as initial data . we could attempt to remove these terms by some iteration of lower order approximations to get self consistency ( as it done with the radiation reaction ( see ll ) ) . it is doubtful if any finite number of spatial derivatives would accomplish the goal of keeping the evolution of the edge bounded by the velocity @xmath228 , as is true for the finite difference equation , and any localized initial data . infinite order equations are unwieldy and can be argued to be disguised nonlocal equations . to look at this from a different perspective , consider the current @xmath222 and what it tells us about the velocity of the propagation . we can decompose this current in to right and left moving components @xmath229 . we know from above that the velocity of the flux in each direction is @xmath230 where @xmath231 is just the fraction of n from a site that moves right or left respectively . by the above physicality condition we have @xmath232 . if the solution approaches this we know we have moved into unphysical territory i.e. @xmath217 is `` very large '' but an adjacent value is now approaching zero this means the flux from one side is effectively terminated and increasing @xmath233 does not increase the flux beyond @xmath234 . we can best modify fick s law by introducing a scale dependent diffusion constant that drives the diffusion to zero as a function of @xmath235 . @xmath236 where @xmath237 . @xmath238 can be as simple as a step function @xmath239 or a smoothed version that still vanishes at @xmath240 . this introduces some essential nonanalyticity and nonlinearity in the problem . this is the price we have to pay to preserve the possibility of having a continuous and smooth set of descriptors to evolve the system with the correct dynamical degrees of freedom . in the case @xmath241 we can immediately see that causality is preserved because the evolution is just the heat equation until the local velocity reaches @xmath242 when it halts until the nearby function changes enough so that it can again evolve at a lower speed . in practice this state is never reached if @xmath243 is rounded over to a smooth function . as a final note , we see energy is conserved both globally and locally by gauss s law . it still remains to find a microscopic derivation of the function @xmath238 or even to see what the form of corrections to this model might be . this will have to remain to future work . to experimentally measure this quantity , using the edge of a packet of heat seems extraordinarily difficult . one could probe rapidly brought together surfaces with large temperature differences . for diffusion , one could look for rapidly moving outliers and large number averages . nonlinear equations follow as descriptions of truly linear phenomena when we suppress dimensionality e.g. the schrdinger equation and hartree - fock , density functional theory and the gross - pitaevskii equations . in the case of quantum field theory , nonlinearities show up in the running of the coupling constant as a result of suppressing scales that correspond to energies beyond what is physically relevant for the given problem . here we have shown that nonlinearities follows from neglected ( and apparently intractable to the analytic tools of smoothing and taking limits ) small scale and discrete physics when we need to work with a best fit continuum model . it is interesting to note that equations like the porous medium equation @xmath244 with @xmath245 give a set of weak solutions ( essentially continuous solutions with smoothness discontinuities ) that have finite velocity of front propagation . we can see that these also have a vanishing current where @xmath23 approaches zero . in contrast we have sought a solution where the velocity of propagation is bounded by a physically fundamental quantity . if we were to extend this to heat transport in a solid we note that the phonons are bounded by the velocity of sound . as long as we are at temperatures where the relevant phonons are from the linear part of the dispersion relation , we can expect such an equation to be relevant . the gas of heat flow in gases is more complicated . the particles can have a very broad distribution of velocities . this above result would have to use contribution over bounds from all these velocities and the sharp edge we see above would become blurred . even though sound speed is closely related to the thermal mean velocity the `` thermal edge '' of the heat distribution would creep out beyond it . the speed of light bound will be established once a relativistic distribution of particle speeds is used . the stress - energy tensor of a system @xmath246 is often described as the flux of @xmath247 momentum across the @xmath248th hypersurface composed of normals to the vector @xmath249 . this is a common approach in many texts on continuum mechanics . since we are interested in the microscopic interplay between material and radiation in a way that treats neither as the `` driver '' or `` responder , '' as is generally done in derivations of the kramers - kronig relations , it is good to pause and look at some specific cases from a microscopic point of view . in the case of a gas , the above is a good model . all the pressures and stresses in the medium are the results of kinetics . as such , these are the direct result of microscopic transfer of momentum . the case of solids and liquids are different . there is a kinetic component but there is also potential energy in the bonds between atoms that can be strained and do the work of transferring forces across the parcels . this may seem academic but when we look at some formulations of momentum conservation in continua , we can start to consider the naively implied `` momentum flux '' from such a @xmath246 as though it is a conserved quantity . specifically @xmath250 implies a conserved momentum ( since @xmath251 ) however the conservation law @xmath252 ( or @xmath253 ) is not generally true even when external forces are zero . ( note that for constant density fluids @xmath254 we can rewrite this as @xmath255 for @xmath256 . this is the correct eulerian momentum flux in the lab frame assuming any microscopic motions can be neglected . we generally refer to these fluids as `` incompressible '' but one can have fluids with varying density e.g. from varying solute concentration where each parcel is effectively incompressible so `` constant density '' is more accurate here . ) this is what we would expect for a conserved momentum . instead we see that @xmath257 plays the role of a source and sink term for it . momentum is exactly conserved and , absent any electromagnetic contribution , is the same as the mass density flux . the subtleties with momentum flux is that it is not locally conserved and that one often tries to write forces as momentum fluxes that may partially cancel on a given material parcel . during a microscopic evaluation of the sample one only sees bond strain and net motion . the net result is the same net force but momentum flux in this sense has no microscopic meaning . if we consider a gas we can locally keep track of the momentum of the system and how it shifts at each collision . in fact we can give a locally well defined right and left moving momentum flux . in the case of condensed matter , this is not the case and the elastic forces driven by the potential energy must be included in the microscopic description of the stress . inspecting a stressed zero temperature solid microscopically will indicate the strained bonds but no momentum is being transferred . when the fields exist in the presence of media we can lump the net local stress into a stress tensor . it is not clear how valuable this is since the waves might propagate through the medium for some distance before it relaxes . we generally try to use such stress tensors to take gradients to derive local forces . these must be the net forces on particles and fields . if this induces reflection and packets do not evolve as we expect from linear combination of the states in a dispersion law it is not clear how much each one responds to this . when electromagnetic fields are being included , we need to include its momentum . how to decouple these from media is not entirely obvious for realistic media . several respectable sources have promoted some confusion about the nature of the `` canonical '' momentum of a particle in an electromagnetic wave @xcite and this ends up playing a role in some discussions of the meaning of the abraham and minkowskii momentum in media @xcite . since it is easy to clarify we shall do it here . the canonical momentum of a particle is stated as @xmath258 . ( gaussian units are used in this paragraph . ) there are attempts to use this and other pseudomenta to give true forces at gradients or boundaries . some discussion has taken place as to when this is valid @xcite . let us first consider the gauge problem and this proposed definition . clearly it is not invariant . we know that these problems go away in the case of the schrdinger equation because the phase of the wave transforms to cancel any vector potential gauge change . in this case we have @xmath259 . the gauge transformation of @xmath260 is now compensated by a stretching in the phase , @xmath261 , of the wavefunction . by ehrenfest s theorem , a localized packet moving with with velocity @xmath262 continues along the lorentz force law path . to get the original canonical relation to give the true momentum we need to choose and maintain a gauge choice of @xmath263 at each particle . the wavefunction packet constitutes a model for the classical charged particle to the extent the ( in this case , irremovable ) self - forces can be neglected . since @xmath264 is a gauge invariant quantity , the classical canonical momentum really only coincides with this gauge restricted case . in this model , @xmath262 is a function of the phase gradient of the packet and the particular value of @xmath265 chosen for the given magnetic field . any modification of this vector potential by a gauge choice @xmath266 should leave @xmath267 unchanged and give no new forces on the particle . this implies that the only classical `` canonical '' transformation of the momentum that should be allowed obey the same local constraints i.e. @xmath268 at each charge . the author acknowledges helpful feedback from david aspnes . we properly should consider the elastic coupling of the electrons to the cores and the cores to each other as distinct . the time scale of these oscillating forces and responses determine how much momentum is taken up by electrons alone versus electrons and cores together . this creates a distinction between a plasmon - like and acoustic modes . these vibrations will create an elastic stress on the bonds , hence the medium , despite having no net longitudinal momentum contributions . the inclusion of such modes will increase the number of branches in the dispersion relation . the model we have here of uniform plates will be replaced by separated localized oscillators in realistic media . these are characterized not only by mass , charge and elastic response but cross section . this allows for longer distances to equilibrate changes in field strength and frequency of the propagating waves for a fixed average @xmath269 , @xmath270 and @xmath271 .

we present an exactly solvable model of a classical dielectric medium that gives an unambiguous local decomposition of field and charge motion and their contribution to the conserved quantities . the result is a set of four branches to the dispersion law that gives full independent freedom in the selection of initial data of the fields and charge motion , in contrast with constitutive laws . this is done with special care to the forces that exist at surfaces , coatings and the ends of packets . as a result the utility of a stress - tensor as a function of field strengths and dielectric response for deriving general forces is called into question . the abraham - minkowskii paradox is clarified from this point of view and the export of such notions to realistic media and metamaterials are discussed . one result of this model is a mathematically simpler and more intuitive understanding of causality in media than the brillouin and sommerfeld theories . necessary elastic medium response is estimated and some implications of this picture for quantum effects are included based on conservation laws . this model can be extended to manifestly maintain these features as general nonlinear and time and space dependent changes in medium response are introduced . the extent to which this can provide a universal description for all dielectrics is discussed . a microscopic treatment of negative index materials from this point of view is included as an illustration of the extreme economy and simplicity of these methods . the abraham - minkowskii paradox has led to a century long debate on the proper assignment of momentum to electromagnetic waves in media @xcite . minkowskii argued that the momentum of a photon should be @xmath0 and abraham as @xmath1 . some now argue that this is now resolved and both assignments are correct when boundary conditions are correctly applied @xcite . the purpose of this article is not so much to dispute that position but to view this problem and general dielectric response from a different and more intuitive point of view . this will generally support a unique decomposition of momentum into field and material components but also ultimately challenges the universality of such stresses as a function of the dielectric response . the physical questions we can ask of an electromagnetic wave interacting with a dielectric medium are 1 . what are the forces at the surfaces ? 2 . what stresses exist due to a steady state beam traversing the medium ? 3 . what are the impulses ( and transient losses ) associated with a wave packet traversing the medium ? 4 . how many photons are involved and how does this change at boundaries ? 5 . what are causal , damping and nonlinear ( and nonlocal ) effects on propagation ? to be able to answer such questions , hopefully easily and with physical intuition , is a sign that we truly do understand the system and are in a good position to extend the theory to more challenging configurations that can arise from nonlinearity , confined geometries and quantum effects . in this article , we will see that the energy and momentum in the photons in a particularly simple medium can be unambiguously specified and separated from that in the charge and core motions . internal stresses can be identified with phase shifts / time lags at the charges and induced standing waves in the medium . the phase velocity turns out to be the rate at which the spatially oscillating part of the energy density advances . the group velocity will be the velocity that the total energy of the packet , including the energy of the charges , propagates which will involve conversion of charge kinetic and potential energy at the ends . in the nondissipative case , the resonant limit gives diverging stress and ratio of charge energy to electromagnetic wave energy . elastic medium response can absorb large fractions of the electromagnetic momentum that is typically returned when the field leaves the medium but , for rapidly changing packets , intense fields or large media , the induced acoustic motions can drain energy from the waves in an unrecoverable fashion . this is just one sort of nonlinearity that media can display and will lead us to a general discussion of nonlinearity and causality in media from a physical point of view rather than the formal approaches used in the kramers - kronig relations . one might wonder why one should pursue such a reformulations of the established treatments of dielectric response . one reason is the frustrating and persistent problems that pseudomomentum and pseudostress introduce into physics @xcite . even those bent on resolving these conflicts often add new elements to the confusion . by providing a system that can be exactly parsed locally in terms of conserved quantities one can clarify exactly where the real momentum is , how it is transferred and what the possibilities are for real stresses . additionally , there is pedagogical value . one should not confuse agreement with experiment as a correct derivation . it is often joked that graduate students can derive whatever you ask them to by some means . however , none of us are beyond finding overly clever and hopeful derivations based on pleasant seeming abstraction to obtain a known or suspected result . to be able to attack a problem from multiple directions is a great comfort that our results are valid even when some arguments seem too clever and might belong more in the domain of mathematics than physics . many students express frustration that the treatment of dielectric response suddenly involves the tools of complex analysis , often their first introduction to it , and wonder why an explanation must require such a formal approach . physicists and mathematicians may differ on what the more `` fundamental '' sort of treatment means . most physics students would consider a microscopic description of the medium to be the most illuminating and natural since it has a clear connection to the fundamental laws of motion . however , the complexity of such treatments is generally out of reach of their time and ability and have their own approximations that leave room for doubt . furthermore , one has the feeling that this is a simple classical problem by nature and it should not require hard tools or wild leaps from the case of single driven oscillators to have an intuitive answer to medium response . in the following , we will introduce an exactly solvable model that shows that the energy transfer entirely consists of the electromagnetic flux where the ends of packets act as sources and sinks of this in terms of charge kinetic and elastic bond energy . this then gives a very nice consistency condition with conservation laws and the allowed dispersion relations that are related to a similar result for surface waves , a much more complicated system . delightfully , this approach is not just intuitive but also mathematically rudimentary , brief and leads to some new insight on the ways nonlinearity must essentially enter these systems . in this model , momentum effects can be exactly separated and classified as : reflection impulses , pure electromagnetic flux , longitudinal plate momentum , radiative stress , forces from the static / velocity fields and bond distortion and conversion impulses due to the `` rings up '' time of the medium to create an equilibrium state with the wave . during this ring - up , the medium is absorbing energy ( hence momentum ) from the propagating wave so that the forces on it are not a simple function of the instantaneous fields which means deriving the forces from a stress tensor built from them must fail in general . we will see that the rest of the internal stress in the medium is due to a hidden standing wave component between the layers due to a phase shift induced by the charges . the linearized theory of medium response gives a consistent causal theory even though some aspects of induced stress and conservation laws are obscured . how to handle nonlinearities as extra interactions on a basis built from a linearized theory is not always obvious . formal procedures exist in the form of perturbation theory and path integrals @xcite but the domain where this is valid and applicable is not always clear . as an example , the case of water surface waves is quite troublesome . a broad enough spectral distribution can lead to local fluctuations large enough to induce surface breaking which introduce vorticity and remove energy , momentum and angular momentum from the waves and convert it to heat , nonirrotational flow and angular momentum of the gravitational source . the end - of - packet contributions of optical signals in media will be seen to necessarily give nonlinear contributions as a function of the medium s elasticity independent of the extent to which the restoring forces are becoming nonlinear . the internal properties of a medium can easily be induced to change rapidly in real time so that no well defined basis of eigenstates , even including acoustic response , can span the solution . to this end , we will present a general local theory that includes damping and nonlinearity for which causality manifestly holds . ( nonlocality due to a sparse phonon spectrum will not be included since this will create a kind of incoherence among photons that is not describable by a classical theory . ) this allows the introduction of any local damping law and medium response restricted only by conservation laws that are clear from the initial form of the equations . the organization of the paper is as follows . in sec . [ comment ] we comment on the lorentz - drude model and the vagueness in it that makes extracting forces from it difficult . in sec . [ exact ] , we create an exactly solvable model that gives these results in the nondissipative limit in a fashion that allows an exact decomposition and tracking of energy and momentum as it moves through the medium . this allows us in sec . [ impulseandstress ] to explicitly describe the forces that must exist at boundaries , ends of packets and the time delays that must occur in energy transfer from radiative field to medium . additionally , we can estimate acoustic losses to the medium , photon number changes in the medium ( from purely classical considerations ! ) and forces on anti - reflective coatings . in the latter case , we exploit that the momentum flux conservation implies the result must be universal for any medium and show that this can be calculated with extraordinary economy by introducing magnetic monopole currents . in sec . [ damping ] , we show how this model can be extended to introduce damping and nonlinear effects which completely respect causality and then show that this model is not completely universal but is , in some sense , minimal in its account of energy and forces in the medium . an appendix covers the extension of the model to `` left handed '' materials , where group and phase velocities are opposite . the radiation field is shown to consist entirely of photon fields that move opposite to the phase velocity of the electric field vector that is smoothed over many radiator spacings . another appendix discuss causality in diffusive systems as another example of how correct small scale accounting fixes the apparent superluminal problems introduced by continuum mechanical limits . the final appendix discusses some subtle points on stress tensors and why they may be of limited value in describing the forces on an optical media in the presence of radiation .

a gaseous atomic target with very low momentum spread is an ideal starting point for atomic scattering experiments . this was demonstrated with great success by the invention of the coltrims ( cold target recoil ion momentum spectroscopy ) technique @xcite . while in coltrims experiments , the target is an atomic beam with low transverse momentum spread , the advent of laser cooling and trapping has provided a related platform . it is known as motrims ( magneto - optical trap recoil ion momentum spectroscopy ) @xcite , and uses an atomic cloud as target which is cooled in all three spatial dimensions with a magneto - optical trap . the achievable temperature of @xmath0 100@xmath1k corresponds to an energy spread of only 10nev . the above mentioned experiments focus on charged reaction products which can be detected with a position sensitive micro - channel plate . the inclusion of scattering processes involving neutral reaction products is possible if one looks , e.g. , at the temporal evolution of the target . this approach has the benefit that absolute cross - sections can be measured . in this context , the atom loss of a mot under electron bombardment has enabled the measurement of the total scattering cross - section and the total ionization cross - section for electrons on rubidium atoms at electron energies up to 500ev @xcite . in this work , we discuss the extension of this principle to a target of ultracold atoms which are held in an optical dipole trap . we give a first example of this technique measuring the total electron - rubidium scattering cross - section at energies between 1.7kev and 6kev . we assess the new possibilities of this experimental platform and the additional benefits compared to the preparation of the atoms in a mot . the measurement of absolute scattering cross - sections is of great importance for a quantitative comparison between experiment and theory . there are two different experimental strategies for their determination . in the first approach , the absolute density of target atoms has to be known . then , it is sufficient to measure the relative number of scattered projectiles . the second strategy is reversed and requires the knowledge of the flux density of the incident projectiles . then , the relative decay of the number of target atoms is sufficient to extract the total cross - section . this strategy can be used in crossed beam experiments or in experiments involving a gaseous target which is fixed in space . in both strategies , the spatial overlap integral between the projectiles and the target has to be determined as well . this task is simplified if the incoming flux density @xmath2 of projectiles is spatially homogeneous and if the target - which we assume to be fixed in space - is completely immersed in the incoming projectiles . then , the number of target atoms @xmath3 evolves according to @xmath4 here , @xmath5 is the total scattering cross - section and @xmath6 accounts for an additional loss channel which might be present in the experiment . the value of @xmath6 must be measured separately . eq.[eq:1 ] is valid when each scattering process leads to the loss of exactly one target atom . this requires that the trap which holds the target atoms is shallow enough to let every scattered target atom escape . furthermore , collisions in which a scattered target atom removes another atom on its way out of the target have to be negligible . the solution of eq.[eq:1 ] is an exponential decay of the number of target atoms . the total scattering cross - section @xmath5 is directly given by the decay constant @xmath7 devided by the flux density @xmath2 . this principle has been experimentally demonstrated with rubidium atoms in a mot which were exposed to an electron beam with energies up to 500ev@xcite . in an analogous approach , a light beam which intersects a cloud of trapped negative ions has recently been used to measure absolute photodetachment cross - sections @xcite . in our experiment , we extend this approach to an ultracold gaseous target which is prepared in an optical dipole trap . starting from mot , we load @xmath8 rubidium atoms in an optical dipole trap . the dipole trap is formed by a focussed co@xmath9 laser beam with a waist of 30@xmath10 m . after an additional stage of forced evaporation we obtain samples of @xmath11 rubidium atoms at a temperature between 50nk and 200nk . below 150nk the atoms form a bose - einstein condensate . this temperature range corresponds to a trap depth between 30 an 140 pev . the details of the experimental setup can be found in @xcite . ) . as the time scale of this decay is very long , the resulting correction to the determined scattering cross - section is small . ] the collisional system is completed by an incident electron beam originating from an electron column . as the experimental setup has been developed in the context of scanning electron microscopy of ultracold quantum gases @xcite , the electron beam can be focussed down to about 100 nm diameter and has an energy between 1.7kev and 6kev . typical beam currents vary between 10na and 1@xmath10a , depending on energy and beam diameter . the cloud of target atoms is cigar shaped with a radial extension of 10@xmath10 m and an axial extension of 100@xmath10 m . after the preparation stage we switch on the focussed electron beam and repeatedly scan an area @xmath12 which is about three times as large as the size of the cloud . each one of these frames takes 18ms and consists of 400 parallel lines which are oriented perpendicular to the long axis of the cloud ( see fig.[fig : working_principle ] ) . the scanning speed within each line and the propagation speed of the lines along the axial direction of the cloud is much faster than the motion of the atoms . therefore , the electron beam crosses an unperturbed cloud during one frame and the action of the electron beam can be integrated over the frame time . we make one hundred consecutive frames , resulting in a total exposure time of 1.8s . at the end of the exposure the cloud is depleted almost entirely . the total experimental cycle has a duration of 15s . when the diameter of the electron beam is much larger than the distance between two neighboring scan lines it is obvious that the integration of the current density over the frame time results in an effectively homogeneous current density . however , for a tightly focussed electron beam where the electron beam diameter is smaller than the distance between two neighboring scan lines , the current density after integration is strongly inhomogeneous . nevertheless , it can be considered homogeneous provided that ( i ) the target density is sufficiently constant over the distance between two neighboring lines and ( ii ) the dwell time at a certain position is short enough that only a small fraction of the target atoms is lost , i.e. , the number of scattered target atoms is linear in dwell time . both conditions are fulfilled for our experimental parameters . we measure the area over which the electron beam is scanned with help of a two - dimensional optical lattice which sets a regular structure with 600 nm period . imaging the atoms in the lattice @xcite allows us to calibrate the scan system of the electron column . a faraday cup measures the total beam current @xmath13 and we get the incoming flux density as @xmath14 where @xmath15 is the electron charge . during the exposure , electron impact ionization leads to a continuous production of ions which we detect with a channeltron . the number of produced ions is recorded and binned for each frame . as the motion of the electron beam is much faster than the atomic motion the binned signal is proportional to the total atom number at the beginning of each frame . we repeat the experiment cycle several hundred times and sum the signal over all runs . collisions with the background gas limit the lifetime of the target atoms in the optical dipole trap and constitute an additional decay as introduced in eq.[eq:1 ] . we measure the corresponding decay constant @xmath6 in a separate measurement ( see fig.[fig : trap_lifetime ] ) . . inset : the logarithmic plot reveals the pure exponential decay which sets in after 1s . ] a typical decay curve of the atom number is presented in fig.[fig : decay_curve ] . the data was taken at a beam energy of 6kev which corresponds to the standard working point of the electron column . according to eq.[eq:1 ] , the decay of the atom number should be exponential . we find that the exponential decay sets not in until a substantial fraction of the atoms is already lost . the deviation from eq.[eq:1 ] is due to secondary processes , where a scattered target atom or a produced ion can remove another atom from the target . these processes can be modelled with an additional decay term which is quadratic in the atom number @xmath16 the coefficient @xmath17 describes the strength of these processes . as can be seen from fig.[fig : decay_curve ] the agreement with the data is very good over the whole exposure time . this confirms the presence of secondary processes . we attribute them to cold ion - atom collisions , as only this collisonal system has a sufficient cross - section to explain the frequency of these processes @xcite . from the fit to eq.[eq:3 ] we extract the decay constant @xmath7 , and together with the previously measured flux density @xmath2 we deduce the absolute total scattering cross - section @xmath5 . we have performed the measurement for incident energies between 1.7kev and 6kev . the results are summarized in fig.[fig : cross_sections ] . the uncertainty of our measurement has been evaluated by averaging over several datasets taken at an energy of 6kev . we have varied the beam diameter , the size of the scanned area and have changed between condensed and thermal samples . at 6kev energy , we determine a cross section of @xmath18@xmath19 , corresponding to an uncertainty of @xmath20 . while there are no experimental data available in this energy range , we can compare our results to theoretical predictions . the total scattering cross - section has three contributions : * * electron impact excitation:*. the general expression for the differential cross - section in first born - approximation is given by @xcite + @xmath21 + here , @xmath22 is a prefactor that depends on the wave vectors @xmath23 and @xmath24 of the incoming and outgoing electron and the total number of target electrons @xmath25 , whose positions are denoted by @xmath26 . the momentum transfer is defined as @xmath27 and @xmath28 and @xmath29 are the initial and final state of the target . for vanishing momentum transfer ( @xmath30 ) , the matrix element approaches that for optical transitions @xmath31 . these kind of collisions constitute the dominant excitation channel and are referred to as dipole regime . for rubidium , excitation on the 5s - 5p resonance line is the most important contribution . in ref.@xcite an empirical formula for the cross - section of the 5s - 5p transition has been given , based on the first born - approximation . we estimate that the excitation to higher lying states ( 5s - np , n=6,7 , ... ) cumulate to about 10% of the cross - section of the strong resonance line . * * elastic scattering : * we employ an empirical formula , which is supposed to be applicable to all elements and all energies above 100ev @xcite . within our energy range , the elastic cross - section is about 10% of the impact excitation cross - section . in all elastic or exciting collisions , the momentum tranfer to the atom is substantially larger than the trap depth , and every scattering process leads to the loss of the atom . * * electron impact ionization : * in the context of scanning electron microscopy of ultracold quantum gases , electron impact ionization is the relevant scattering mechanism as it produces a detectable signal . the ionization process includes singly and multiply charged ions . a time of flight spectrum , presented in fig.[fig : tof_spectrum ] , shows the relative weights of these contributions : more than 80% of the ions are singly charged . while experimental data for the total ionization cross - section is availible only up to an energy of 500ev @xcite , theoretical calculations in plane - wave born - approximation have been made up to an energy of 10kev@xcite . . ] for a quantitative comparison we plot the three contributions together with their sum in fig.[fig : cross_sections ] . at all data points the theoretical prediction differs form the experimental result by an almost constant factor ( @xmath320.04 ) . this suggests that either the experimental data systematically overestimates the cross - section or the theoretical predictions miss a substantial amount of scattering processes . the measurement could in principle be influenced by the presence of the optical dipole trap . we have tested this in the following way : we direct the electron beam into the center of the cloud and record the ion signal for 100@xmath10s . we then repeat the experiment switching off the dipole trap during this time . within the first 100@xmath10s , the expansion of the cloud can be neglected and the electron beam interacts with a cloud of the same density distribution but without the presence of any trapping light . we find the same ion signal and can therefore exclude such an effect . varying the parameters of the electron beam and the size and properties of the scan area , we can also exclude any influence stemming from the specific realization of the experiment . we can furthermore exclude an inaccurate determination of the size of the scan area , as the two - dimensional optical lattice provides a perfectly periodic ruler . a potential source for a systematic error could be the determination of the electron beam current . we use a faraday cup which can be biased and which has an internal transverse magnetic field in order to prevent elastically backscattered electrons from escaping from the cup . we conclude that a systematic error originating from the faraday cup is unlikely , however we have currently no means to independently calibrate it as it is an integral part of the setup . the above presented theoretical predictions are based on simplifying assumptions such as the first born - approximation . this might lead to a systematic underestimation of the cross - sections . in addition , more complicated excitation channels , such as the excitation of inner shell electrons , the inclusion of optically forbidden transitions or the excitation of more than one electron , have not been accounted for in our simple model . our results might indicate that these channels also contribute significantly to the toal cross - section . apart from the normalization factor , the data trend shows very good agreement between experiment and theory . the good quality of the measurement is further confirmed by the small uncertainty . we therefore conclude that the presented approach is suitable for high precision measurements of absolute scattering cross - sections . compared to previously reported experiments using a magneto - optical trap @xcite several differences are apparent . in our approach , no switching of magnetic fields is necessary , as the dipole trap is extremely shallow . the experiment can be performed continously until the target is fully depleted . thus , a single experimental run is already sufficient to derive the cross - section . as only the relative atom number is important , individual experimental runs can be summed without normalization . finally , the atoms in the dipole trap can be polarized , which allows for spin - resolved scattering experiments . we have described a new experimental platform for the measurement of absolute scattering cross - sections based on optically trapped atoms . we have demonstrated the principle studying electron rubidium collisions at high incident energies . even though the measurement principle relies on the use of optical trapping fields , a surprisingly large variety of scattering scenarios is feasible . based on the actual status of cold atom physics , including the recent developments in non - optical cooling techniques and molecule formation , we can identify a number of interesting scattering scenarios and applications for the future : * * low energy electron - atom collisions : * the combination of laser cooling and subsequent photoionization is well suited to produce ions and electrons with extremely small initial energy spread @xcite . implementing two neighboring dipole traps , one of which is used as an electron source and the other is used as a target , allows to study low energy electron - atom collisions with unprecedented energy resolution . polarizing the atoms in both traps provides full control over the spins of the incoming electrons and the target atoms . * * low energy ion - atom collisions : * the same holds for the investigation of ultracold ion - atom collisions . as the recoil energy of the ion is negligible in photoionization , an even higher energy resolution should be feasible . * * electron - rydberg atom collisions : * recently , a new form of molecular binding mechanism has been identified for ultracold rydberg atoms @xcite . extending these studies by exciting atoms to rydberg states and exposing them to a low energy electron beam can give more insight in these phenomena . it also complements various studies of plasma physics with rydberg atoms @xcite . * * molecular targets : * feshbach resonances can be used to produce ground state molecular dimers from alkali atoms @xcite . the control over all internal and external degress of freedom offers the unique opportunity to produce a well defined molecular target , where fundamental electron - molecule or ion - molecule collisions can be studied . more complex molecules could be available soon combining stark deceleration and subsequent optical trapping . we gratefully acknowledge financial support from the dfg under grant no . ot 222/2 - 3 . 10 drner r _ et al _ 2000 _ phys . rep . _ * 330 * 95 van der poel m , nielson c v , gearba m a , and anderson n 2001 _ phys . lett . _ * 87 * 123201 flchard x , nguyen h , wells e , itzhak i b , and depaola b d 2001 _ phys lett . _ * 87 * 123203 turkstra j w _ et al _ 2001 _ phys . lett . _ * 87 * 123202 schappe r s , feng p , anderson l w , lin c c , and walker t 1995 _ europhys . * 29 * 439 schappe r s , walker t , anderson l w , and lin c c 1996 _ phys lett . _ * 76 * 4328 hlavenka p _ et al _ 2009 _ j. chem . phys . _ * 130 * 061105 gericke t , wrtz p , reitz d , langen t , and ott h 2008 _ nature physics _ * 4 * 949 wrtz p , langen t , gericke t , koglbauer a , and ott h 2009 _ phys . rev . lett . _ * 103 * 080404 ct r and dalgarno a 2000 _ phys . a _ _ 62 _ 012729 inokuti m 1971 _ rew . * 43 * 297 chen s t and gallagher a c 1978 _ phys . a _ * 17 * 551 browning r , li t z , chui b , ye j , and pease r f w 1994 _ j. appl . * 76 * 216 bartlett p l and stelbovics a t 2004 _ at . dat . tab . _ * 86 * 235 hanssen j l , hill s b , orloff j , and mcclelland j j 2008 _ nano lett . _ * 8 * 2844 bendkowsky v _ et al _ 2009 _ nature _ * 458 * 1005 robinson m p , tolra b l , noel m w , gallagher t f , and pillet p 2000 _ phys . lett . _ * 85 * 4466 ferlaino f , knoop s , and grimm r _ cold molecules : theory , experiment , applications edt . by r. v. krems , b. friedrich , and w. c. stwalley ( crc press , boca raton , 2009 ) ultracold feshbach molecules _ ; arxiv:0809.3920

we report on a new experimental platform for the measurement of absolute scattering cross - sections . the target atoms are trapped in an optical dipole trap and are exposed to an incident particle beam . the exponential decay of the atom number directly yields the absolute total scattering cross - section . the technique can be applied to any atomic or molecular species that can be prepared in an optical dipole trap and provides a large variety of possible scattering scenarios .

in this paper we consider the numerical integration of autonomous stochastic differential delay equations ( sddes ) in the it s sense @xmath5 with initial data @xmath6 $ ] . here @xmath7 is a delay term satisfying @xmath8 and @xmath9 , @xmath10 . we assume that the initial data is independent of the wiener measure driving the equations and @xmath11 is an @xmath12-dimensional wiener process defined on the complete probability space @xmath13 with a filtration @xmath14 satisfying the usual conditions ( that is , it is increasing and right continuous while @xmath15 contains all @xmath16-null sets ) . for a given constant stepsize @xmath17 , we propose a split - step backward euler ( ssbe ) method for sddes ( [ sddes1 ] ) as follows @xmath18 @xmath19 where @xmath20 and for @xmath21 @xmath22 for arbitrary stepsize @xmath23 , @xmath24 denotes the approximation of @xmath25 at time @xmath26 . we remark that @xmath27 in ( [ y*n ] ) depends on how memory values are handled on non - grid points . generally there are two ways , the first is to use piecewise constant interpolation , corresponding to @xmath28 , and the second to use piecewise linear interpolation . in later development , we prefer to assume @xmath29 to cover both cases . also , we mention that the scheme ( [ ssbe1])-([ssbe2 ] ) here is quite different from the ssbe method in @xcite , which will be explained at the end of this section . in ( [ ssbe1])-([ssbe2 ] ) , @xmath30 serves as an intermediate stage value , and in order to continue the process , we have to solve the implicit equation ( [ ssbe1 ] ) at every step to acquire @xmath30 . existence and uniqueness of solutions to the implicit equations ( [ ssbe1 ] ) will be discussed in section 4 . here , we always assume that numerical solution of ( [ ssbe1 ] ) exists uniquely . and one can easily check that @xmath31 is @xmath32-measurable . the key aim in this work is to propose a new ssbe method for sddes with variable delay and its convergence and stability in mean - square sense are investigated under a non - globally lipschitz condition . this situation has been investigated in @xcite for stochastic differential equations ( sdes ) without delay . for sdes with delay , most of previous work has been based on the more restrictive assumption that the coefficients @xmath33 satisfies global lipschitz and linear growth conditions , see , for example , @xcite . in @xcite , the authors showed that the numerical solution produced by euler - maruyama ( em ) method will converge to the true solution of the sddes under the local lipschitz condition . note that the proof of the convergence result in this paper is based on techniques used in @xcite . in @xcite , by interpreting the implicit method ssbe as the em applied to a modified sde the authors were able to get a strong convergence result . this paper , however , provides an alternative way to get the convergence result for ssbe . that is , by giving a direct continuous - time extension we accomplished the convergence proof for ssbe without considering the modified sddes . also , in deriving moment bounds of numerical solution , due to the delay term of our ssbe , i.e. , @xmath34 in ( [ ssbe1 ] ) , @xmath35 can not be explicitly dominated by @xmath24 as ( 3.25 ) in @xcite . starting with a recurrence of @xmath35 given by substituting ( [ ssbe2 ] ) into ( [ ssbe1 ] ) , we overcome this difficulty and obtained the desired moment bounds . note that a similar approach is adopted in the stability analysis . of course , the most important contribution of this work is to propose an improved ssbe method for sddes and to verify its excellent stability property . in @xcite , the authors proposed a ssbe method for a linear scalar sdde with constant lag and its convergence and stability are studied there . it is worth emphasizing that our proposed method is a modified version of ssbe in @xcite . the changes are in two aspects : firstly , we drop the stepsize restriction @xmath36 and allow for arbitrary stepsize @xmath17 ; secondly and most importantly , the scheme has been modified to a new one . to see this , the two methods are applied to a linear scalar sdde in section [ linear_ms ] . one can observe that the second terms of @xmath33 in the scheme in @xcite is the numerical solution @xmath37 ( see ( [ ssbez ] ) below ) . while the corresponding terms in our scheme is the intermediate stage value @xmath38 ( see ( [ ssbew ] ) below ) . note that the modifications of the method do not raise the strong order of the numerical solution , but they indeed improve the stability of the method greatly . in fact , it is shown below that our method can well replicate exponential mean - square stability of nonlinear test problem , including the linear test equation as a special case , without any restrictions on stepsize @xmath4 . the convergence and stability results of ssbe can be regarded as an extension of those in @xcite for sdes without delay to variable delay case . this unconditional stability property of ( [ ssbe1])-([ssbe2 ] ) demonstrates that the proposed method is promising and will definitely be effective in solving systems with stiffness in the drift term , where stability investigations are particularly important . this article is organized as follows . in next section , a general convergence result ( theorem [ ssbemain ] ) is established . in section 3 , a convergence result is derived under a one - sided lipschitz condition ( assumption [ olc ] ) . section 4 and 5 are devoted to exponential mean - square stability property of the method . numerical experiments are included in section 6 . throughout the paper , let @xmath39 denote both the euclidean norm in @xmath40 and the trace norm(f - norm ) in @xmath41 . as the standing hypotheses , we make the following assumption . [ lcmc ] the system ( [ sddes1 ] ) has a unique solution @xmath25 on @xmath42 $ ] . and the functions @xmath3 and @xmath0 are both locally lipschitz continuous in @xmath1 and @xmath2 , i.e. , there exists a constant @xmath43 such that @xmath44 for all @xmath45 with @xmath46 . moreover , we assume that @xcite [ init ] @xmath47 is hlder continuous in mean - square with exponent 1/2 , that is @xmath48 and @xmath7 is a continuous function satisfying @xmath49 in the following convergence analysis , we find it convenient to use continuous - time approximation solution . hence we define continuous version @xmath50 as follows @xmath51 where @xmath52 . for @xmath53 we can write it in integral form as follows @xmath54 where @xmath55 it is not hard to verify that @xmath56 , that is , @xmath50 coincides with the discrete solutions at the grid - points . in additional to the above two assumptions , we will need another one . [ anmb ] the exact solution @xmath25 and its continuous - time approximation solution @xmath50 have p - th moment bounds , that is , there exist constants @xmath57 such that @xmath58 \vee \mathbb{e}\left[\sup\limits_{0 \leq t \leq t}|\bar{y}(t)|^{p}\right ] \leq a \label{mb0}.\ ] ] now we state our convergence theorem here and give a sequence of lemmas that lead to a proof . [ ssbemain]under assumptions [ lcmc],[init],[anmb ] , if the implicit equation ( [ ssbe1 ] ) admits a unique solution , then the continuous - time approximate solution @xmath50 ( [ ce1 ] ) will converge to the true solution of ( [ sddes1 ] ) in the mean - square sense , i.e. , @xmath59 we need several lemmas to complete the proof of theorem [ ssbemain ] . first , we will define three stopping times @xmath60 where as usual @xmath61 is set as @xmath62 ( @xmath63 denotes the empty set ) . [ lem1 ] under assumption [ lcmc ] , [ init ] , there exist constants @xmath64 , @xmath65 such that for @xmath66 and @xmath67 @xmath68 _ proof . _ for @xmath66 , by definition of @xmath69 and @xmath70 , @xmath71 noticing that for @xmath72 @xmath73 with @xmath74 . using linear growth condition of @xmath75 and moment bounds in ( [ mb ] ) , we have appropriate constant @xmath64 so that @xmath76 as for estimate ( [ yd2 ] ) , there are four cases as to the location of @xmath77 and @xmath78 : @xmath79 1 ) @xmath80 , @xmath79 2 ) @xmath81 , @xmath79 3 ) @xmath82 , @xmath79 4 ) @xmath83 . + noticing that the delay @xmath84 satisfies lipschitz condition ( [ initial2 ] ) , one sees that @xmath85 in the case 1 ) , combining hlder continuity of initial data ( [ initial1 ] ) and ( [ tau ] ) gives the desired assertion . in the case 2 ) , without loss of generality , we assume @xmath86 , @xmath87 . thus we have from ( [ ssbe1 ] ) and ( [ y*n2 ] ) that @xmath88 \nonumber \\ & & + \mu \sum_{k = j+1}^{i-1 } \left[h f(y^*_{k+1},\tilde{y}^*_{k+1})+ g(y^*_{k } , \tilde{y}^*_{k})\delta w_{k } \right ] , \label{yd5}\end{aligned}\ ] ] where as usual we define the second summation equals zero when @xmath89 . noticing from ( [ tau ] ) that @xmath90 , and combining local linear growth bound ( [ lf ] ) for @xmath91 , global linear growth condition for @xmath75 and moment bounds ( [ mb ] ) , we can derive from ( [ yd5 ] ) that @xmath92 in the case 3 ) and 4 ) , using an elementary inequality gives @xmath93 then combining this with results obtained in case 1 ) and 2 ) gives the required result , with @xmath65 a universal constant independent of @xmath4 . [ lem2 ] under assumption [ lcmc ] , [ init ] , for stepsize @xmath94 , there exists a constant @xmath95 such that @xmath96 \leq c_r h,\ ] ] with @xmath95 dependent on @xmath97 , but independent of @xmath4 . _ for simplicity , denote @xmath98 from ( [ sddes1 ] ) and ( [ ce2 ] ) , we have @xmath99 = \mathbb{e } \left[\sup_{0 \leq s \leq t}\left|\bar{y}(s\wedge \sigma_r)-x(s\wedge \sigma_r)\right|^2 \right ] \nonumber \\ & = & \mathbb{e } \left[\sup_{0 \leq s \leq t}\left|\int_0^{s\wedge \sigma_r}f(y^*(r),\tilde{y}^*(r))-f(x(r),x(r-\tau(r)))\mbox{d}r \right.\right.\nonumber \\ & & \left.\left.+ \int_0^{s\wedge \sigma_r}g(y^*(r),\tilde{y}^*(r))-g(x(r),x(r-\tau(r)))\mbox{d}w(r)\right|^2\right ] \nonumber \\ & \leq & 2 t \mathbb{e } \int_0^{t\wedge \sigma_r } \left|f(y^*(s),\tilde{y}^*(s))-f(x(s),x(s-\tau(s)))\right|^2 \mbox{d}s \allowdisplaybreaks \nonumber \\ & & + 2\mathbb{e}\left[\sup_{0 \leq s \leq t } \left|\int_0^{s\wedge \sigma_r}g(y^*(r),\tilde{y}^*(r))-g(x(r),x(r-\tau(r)))\mbox{d}w(r)\right|^2\right ] \allowdisplaybreaks \nonumber \\ & \leq & 2(t+4)l_r \mathbb{e } \int_0^{t\wedge \sigma_r } \label{3.14}\end{aligned}\ ] ] where hlder s inequality and the burkholder - davis - gundy inequality were used again . using the elementary inequality @xmath100 , one computes from ( [ 3.14 ] ) that @xmath101 \nonumber \\ & \leq & 4(t+4)l_r \mathbb{e } \int_0^{t\wedge \sigma_r } & & + 4(t+4)l_r \mathbb{e } \int_0^{t\wedge \sigma_r } |\tilde{y}^*(s)- \bar{y}(s-\tau(s))|^2 + |\bar{y}(s-\tau(s))-x(s-\tau(s))|^2 \mbox{d}s \allowdisplaybreaks \nonumber \\ & \leq & 8(t+4)l_r \int_0^t \mathbb{e } [ \sup_{0 \leq r \leq s}|\bar{y}(r\wedge \sigma_r)-x(r\wedge \sigma_r)|^2 ] \mbox{d}s \nonumber \\ & & + 4(t+4)l_r \mathbb{e } \int_0^{t\wedge \sigma_r } \bar{y}(s)|^2 \mbox{d}s \nonumber \\ & & + 4(t+4)l_r \mathbb{e } \int_0^{t\wedge \sigma_r } |\tilde{y}^*(s)- \bar{y}(s-\tau(s))|^2 \mbox{d}s , \label{3.15}\end{aligned}\ ] ] where the fact was used that @xmath102 . by taking lemma [ lem1 ] into account , we derive from ( [ 3.15 ] ) that , with suitable constants @xmath103 @xmath104 & \leq & 8(t+4)l_r \int_0^t \mathbb{e } [ \sup_{0 \leq r \leq s}|\bar{y}(r\wedge \sigma_r)-x(r\wedge \sigma_r)|^2 ] \mbox{d}s \nonumber \\ & & + 4(t+4)tl_rc_1(r)h + 4(t+4)tl_rc_2(r)h \nonumber \\ & = & \tilde{c}_r\int_0^t \mathbb{e } [ \sup_{0 \leq r \leq s}|e(r \wedge \sigma_r)|^2 ] \mbox{d}s + \bar{c}_rh.\end{aligned}\ ] ] hence continuous gronwall inequality gives the assertion . _ proof of theorem [ ssbemain ] . _ armed with lemma [ lem2 ] and assumption [ anmb ] , the result may be proved using a similar approach to that in ( * ? ? ? * theorem 2.2 ) and ( * ? ? ? * theorem 2.1 ) , where under the local lipschitz condition they showed the strong convergence of the em method for the sodes and sddes , respectively . under the global lipschitz condition and linear growth condition ( cf @xcite ) , we can choose uniform constants @xmath64 , @xmath105 in previous lemma [ lem1],[lem2 ] to be independent of @xmath97 . accordingly we can recover the strong order of 1/2 by deriving @xmath106 \leq c h,\ ] ] where @xmath107 is independent of @xmath97 and @xmath4 . in this section , we will give some sufficient conditions on equations ( [ sddes1 ] ) to promise a unique global solution of sddes and a well - defined solution of the ssbe method . we make the following assumptions on the sddes . [ olc ] the functions @xmath3 are continuously differentiable in both @xmath1 and @xmath2 , and there exist constants @xmath108 , such that @xmath109 @xmath110 @xmath111 the inequalities ( [ olc1]),([olc2 ] ) indicate that the first argument @xmath1 of @xmath91 satisfies one - sided lipschitz condition and the second satisfies global lipschitz condition . it is worth noticing that conditions of the same type as ( [ olc1 ] ) and ( [ olc2 ] ) have been exploited successfully in the analysis of numerical methods for deterministic delay differential equations ( ddes)(see @xcite and references therein ) . as for sdes without delay , the conditions ( [ olc1 ] ) and ( [ olc3 ] ) has been used in @xcite . we compute from ( [ olc1])-([olc3 ] ) that @xmath112 @xmath113 on choosing the constant @xmath114 as @xmath115 the following condition holds @xmath116 in what follows we always assume that for @xmath117 the initial data satisfies @xmath118 [ eu ] assume that assumption [ olc ] is fulfilled . then there exists a unique global solution @xmath25 to system ( [ sddes1 ] ) . morever , for any @xmath119 , there exists constant @xmath120 @xmath58 \leq c(1+\mathbb{e}\|\psi\|^p).\ ] ] _ proof . _ see the appendix . [ mblem ] assume that @xmath33 satisfy the condition ( [ mc ] ) and @xmath67 is sufficiently small , then for @xmath121 the following moment bounds hold @xmath122 \vee \mathbb{e}\left[\sup\limits_{0 \leq t \leq t}|\tilde{y}^*(t)|^{2p}\right ] \vee \mathbb{e}\left[\sup\limits_{0 \leq t \leq t}|\bar{y}(t)|^{2p}\right ] \leq a \label{mb}.\ ] ] _ proof . _ inserting ( [ ssbe2 ] ) into ( [ ssbe1 ] ) gives @xmath123 hence @xmath124 expanding it and employing ( [ mc ] ) yields @xmath125 by definition of @xmath126 , one obtains @xmath127 . taking this inequality into consideration and letting @xmath128 , we have from ( [ 3.1 ] ) that @xmath129 denoting @xmath130 , one computes that @xmath131 by recursive calculation , we obtain @xmath132 raising both sides to the power @xmath133 gives @xmath134^p + \alpha^p n^{p-1 } \sum_{j=0}^{n-1 } thus @xmath135^p + \alpha^p m^{p-1}\mathbb{e}\sum_{j=0}^{m-1}|g(y^*_j,\tilde{y}^*_j)\delta w_j|^{2p}\right \}.\label{y*n3}\end{aligned}\ ] ] here @xmath136 , where @xmath137 is the largest integer number such that @xmath138 . now , using the burkholder - davis - gundy inequality ( theorem 1.7.3 in @xcite ) gives @xmath139^p \leq c_p\mathbb{e}\left [ \sum_{j=0}^{m-1}|y^*_j|^2|g(y^*_j,\tilde{y}^*_j)|^2h\right]^{p/2 } \allowdisplaybreaks\nonumber\\ & & \leq c_p ( kh)^{p/2}m^{p/2 - 1}\mathbb{e}\left[\sum_{j=0}^{m-1 } & & \leq \frac{1}{2}c_pk^{p/2}t^{p/2 - 1}h \mathbb{e}\left[\sum_{j=0}^{m-1 } \left(|y^*_j|^{2p } + 3^{p-1}(1+|y^*_j|^{2p}+|\tilde{y}^*_j|^{2p})\right)\right ] . \label{estimate1}\end{aligned}\ ] ] noticing that @xmath140 inserting it into ( [ estimate1 ] ) , we can find out appropriate constants @xmath141 such that @xmath142^p \nonumber \\ & \leq & \bar{c}h \sum_{j=0}^{m-1}\mathbb{e}\max_{0 \leq i \leq j } |y_i^*|^{2p } + \bar{c}(\mathbb{e}\|\psi\|^{2p}+1 ) . \label{estimate2}\end{aligned}\ ] ] at the same time , noting the fact @xmath143 and @xmath144 is independent of @xmath145 , one can compute that , with @xmath146 a constant that may change line by line @xmath147 \allowdisplaybreaks \nonumber \\ & \leq & \hat{c}h^{p-1}(\mathbb{e}\|\psi\|^{2p}+1 ) + \hat{c}h^p \sum_{j=0}^{m-1}\mathbb{e}\max_{0 \leq i \leq j } |y^*_i|^{2p}. \label{estimate3}\end{aligned}\ ] ] by definition ( [ ssbe1 ] ) , one sees that @xmath148 then using a similar approach used before , we can find out a constant @xmath149 to ensure that @xmath150 inserting ( [ estimate2]),([estimate3 ] ) into ( [ y*n3 ] ) and considering ( [ y*0 ] ) and @xmath67 , we have , with suitable constants @xmath151 @xmath152 thus using the discrete - type gronwall inequality , we derive from ( [ mbend ] ) that @xmath153 $ ] is bounded by a constant independent of @xmath137 . then by considering the elementary inequality @xmath154 , boundedness of @xmath155 $ ] is immediate . to bound @xmath156 $ ] , we shall first bound @xmath157 $ ] . from ( [ ssbe2 ] ) , we have @xmath158 & \leq & 2^{2p-1}\left\ { \mathbb{e}\left[\sup_{0 \leq nh \leq t}|y^*_n|^{2p}\right ] + \mathbb{e}\left[\sup_{0 \leq nh \leq t}|g(y^*_n,\tilde{y}_n^*)\delta w_n|^{2p}\right]\right\ } \nonumber \\ & \leq & 2^{2p-1}\left\ { \mathbb{e}\left[\sup_{0 \leq nh \leq t}|y^*_n|^{2p}\right ] + \mathbb{e}\sum_{j=0}^{n}|g(y^*_j,\tilde{y}^*_j)\delta w_j|^{2p}\right\}. \nonumber\end{aligned}\ ] ] now ( [ estimate3 ] ) and bound of @xmath159 $ ] gives the bound of @xmath157 $ ] . to bound @xmath156 $ ] , we denote by @xmath160 the integer for which @xmath161 . by definitions of ( [ ssbe1 ] ) and ( [ ce1 ] ) , for @xmath162 , @xmath163 where @xmath164 . thus @xmath165 & \leq & 2^{2p-1}\left\ { \gamma \mathbb{e}\left[\sup\limits_{0 \leq nh \leq t}|y^*_n|^{2p}\right]+(1-\gamma ) \mathbb{e}\left[\sup\limits_{0 \leq nh \leq t}|y_n|^{2p}\right]\right . \nonumber \\ & & \left.+\mathbb{e}\left[\sup\limits_{0 \leq t \leq t}|g(y^*_{n_t},\tilde{y}^*_{n_t})\delta w_{n_t}(t)|^{2p}\right ] \right\}.\label{mb1}\end{aligned}\ ] ] using doob s martingale inequality ( * ? ? ? * theorem 1.3.8 ) , we derive that @xmath166 \leq \sum_{n=0}^{n}\mathbb{e}\left[\sup\limits_{0 \leq s \leq h}|g(y^*_{n},\tilde{y}^*_{n})\delta w_{n}(s)|^{2p}\right ] \nonumber \\ \leq \left(\frac{2p}{2p-1}\right)^{2p}\sum_{n=0}^{n}\mathbb{e } \left[|g(y^*_{n},\tilde{y}^*_{n})\delta w_{n}(h)|^{2p}\right].\label{ge}\end{aligned}\ ] ] thus the last term in ( [ mb1 ] ) is bounded by considering ( [ estimate3 ] ) and bounds of @xmath167 $ ] , @xmath168 $ ] . now boundedness of @xmath169 $ ] follows immediately . [ lem6 ] under assumption [ olc ] , if @xmath170 , the implicit equation in ( [ ssbe1 ] ) admits a unique solution . _ let @xmath171 , then the implicit equation ( [ ssbe1 ] ) takes the form as @xmath172 where at each step , @xmath173 are known . observing that @xmath174 the assertion follows immediately from theorem 14.2 of @xcite . under assumption [ init],[olc ] , if @xmath175 , then the numerical solution produced by ( [ ssbe1])-([ssbe2 ] ) is well - defined and will converge to the true solution in the mean - square sense , i.e. , @xmath59 _ proof . _ noticing that assumption [ olc ] implies assumptions [ lcmc],[anmb ] by theorem [ eu ] and lemma [ mblem ] , and taking lemma [ lem6 ] into consideration , the result follows directly from theorem [ ssbemain ] . we remark that the problem class satisfying condition ( [ initial2 ] ) includes plenty of important models . in particular , stochastic pantograph differential equations ( see , e.g. , @xcite ) with @xmath176 and sddes with constant lag fall into this class and therefore corresponding convergence results follow immediately . in this section , we will investigate how ssbe shares exponential mean - square stability of general nonlinear systems . in deterministic case , nonlinear stability analysis of numerical methods are carried on under a one - sided lipschitz condition . this phenomenon has been well studied in the deterministic case ( @xcite and references therein ) and stochastic case without delay @xcite . in what follows , we choose the test problem satisfying conditions ( [ olc1])-([olc3 ] ) . moreover , we assume that variable delay is bounded , that is , there exists @xmath177 , for @xmath178 @xmath179 we remark that this assumption does not impose additional restrictions on the stepsize @xmath4 and admits arbitrary large @xmath4 on choosing @xmath180 and @xmath181 close to 1 . to begin with , we shall first give a sufficient condition for exponential mean - square stability of analytical solution to underlying problem . [ ems1 ] under the conditions ( [ olc1]),([olc2]),([olc3 ] ) and ( [ bd ] ) , and with @xmath182 obeying @xmath183 any two solutions @xmath184 and @xmath185 with @xmath186 and @xmath187 satisfy @xmath188 where @xmath189 $ ] is the zero of @xmath190 , with @xmath191 . _ by it formula , we have @xmath192 } \mathbb{e}|x(r)-y(r)|^2 \mbox{d}s.\end{aligned}\ ] ] letting @xmath193 and noticing that @xmath194 exists for @xmath195 and is continuous , we derive from ( [ 5.12 ] ) that @xmath196}u(s),\ ] ] where the upper dini derivative @xmath197 is defined as @xmath198 using theorem 7 in @xcite leads to the desired result . based on this stability result , we are going to investigate stability of the numerical method . [ ems3 ] under the conditions ( [ olc1]),([olc2]),([olc3 ] ) and ( [ bd ] ) , if @xmath199 , then for all @xmath23 , any two solutions @xmath200 produced by ssbe ( [ ssbe1])-([ssbe2 ] ) with @xmath186 and @xmath187 satisfy @xmath201 where @xmath202 is defined as @xmath203 _ proof . _ under @xmath199 , the first part is an immediate result from lemma [ lem6 ] . for the second part , in order to state conveniently , we introduce some notations @xmath204 from ( [ y*n2 ] ) , we have @xmath205 thus @xmath206 taking expectation and using ( [ olc3 ] ) yields @xmath207 now using the cauchy - schwarz inequality and conditions ( [ olc1])-([olc2 ] ) , we have @xmath208 inserting it into ( [ 5.3 ] ) gives @xmath209 here we have to consider which approach is chosen to treat memory values on non - grid points , piecewise constant interpolation ( @xmath210 ) or piecewise linear interpolation . in the latter case , let us consider two possible cases : @xmath79 if @xmath211 , then @xmath212 inserting ( [ 5.10 ] ) , we derive from ( [ 5.4 ] ) that @xmath213\mathbb{e}|x_n^*-y_n^*|^2 \\ & \leq(1 + h\gamma_3+\tilde{\mu } h \gamma_2)\mathbb{e}|x_{n-1}^*-y_{n-1}^*|^2 + h\gamma_4 \mathbb{e}|\tilde{x}_{n-1}^*-\tilde{y}_{n-1}^*|^2 . \end{split}\ ] ] hence using the fact @xmath199 in ( [ beta ] ) gives @xmath214 @xmath79 if @xmath215 , it follows from ( [ 5.4 ] ) and @xmath216 that @xmath217 therefore , it is always true that inequality ( [ 5.6 ] ) holds for piecewise linear interpolation case . obviously ( [ 5.6 ] ) also stands in piecewise constant interpolation case . further , from ( [ ssbe1 ] ) one sees @xmath218 using a similar approach as before , one can derive @xmath219 denote @xmath220 noticing that @xmath199 , one can readily derive @xmath221 , we can deduce from ( [ 5.6 ] ) and ( [ 5.11 ] ) that @xmath222 here @xmath223 denotes the greatest integer less than or equal to @xmath1 . + finally from ( [ ssbe2 ] ) , we have for large @xmath224 such that @xmath225 @xmath226 where @xmath227 is defined as in ( [ nuh2 ] ) . the stability result indicates that the method ( [ ssbe1])-([ssbe2 ] ) can well reproduce long - time stability of the continuous system satisfying conditions stated in theorem [ ems1 ] . note that the exponential mean - square stability under non - global lipschitz conditions has been studied in @xcite in the case of nonlinear sdes without delay . the preceding results can be regarded as an extension of those in @xcite to delay case . although the main focus of this work is on nonlinear sddes , in this section we show that the ssbe ( [ ssbe1])-([ssbe2 ] ) has a very desirable linear stability property . hence , we consider the scalar , linear test equation @xcite given by @xmath228 note that ( [ lineartest ] ) is a special case of ( [ sddes1 ] ) with @xmath229 , and satisfies conditions ( [ olc1])-([olc3 ] ) with @xmath230 by theorem [ ems1 ] , ( [ lineartest ] ) is mean - square stable if @xmath231 for constraint stepsize @xmath232 , i.e. , @xmath233 in ( [ bd ] ) , the ssbe proposed in our work applied to ( [ lineartest ] ) produces @xmath234 , \\ y_{n+1 } & = y_n^ * + [ cy_n^ * + d y_{n-\kappa}^*]\delta w_n . \end{array } \right.\end{aligned}\ ] ] in @xcite , the authors constructed a different ssbe for the linear test equation ( [ lineartest ] ) and their method applied to ( [ lineartest ] ) reads @xmath235 , \\ z_{n+1 } & = z_n^ * + [ cz_n^ * + d z_{n-\kappa+1}]\delta w_n . \end{array } \right.\end{aligned}\ ] ] the stability results there ( * ? ? ? * theorem 4.1 ) indicate that under ( [ linearms ] ) the method ( [ ssbez ] ) can only preserve mean - square stability of ( [ lineartest ] ) with stepsize restrictions , but the new scheme ( [ ssbew ] ) exhibits a better stability property . for the linear equation ( [ lineartest ] ) , if ( [ linearms ] ) holds , then the ssbe ( [ ssbew ] ) is mean - square stable for any stepsize @xmath236 . _ the assertion readily follows from theorem [ ems3 ] . apparently , the ssbe ( [ ssbew ] ) achieves an advantage over ( [ ssbez ] ) in stability property that the ssbe ( [ ssbew ] ) is able to inherit stability of ( [ lineartest ] ) for any stepsize @xmath236 . if one drops the stepsize restriction @xmath36 and allow for arbitrary stepsize @xmath17 , one can arrive at a sharper stability result from theorem [ ems3 ] . for the linear equation ( [ lineartest ] ) , if ( [ linearms ] ) holds , then the ssbe([ssbe1])-([ssbe2 ] ) is mean - square stable for any stepsize @xmath237 . in this section we give several numerical examples to illustrate intuitively the strong convergence and the mean - square stability obtained in previous sections . the first test equation is a linear it sdde @xmath238 . \end{array } \right.\end{aligned}\ ] ] denoting @xmath239 as the numerical approximation to @xmath240 at end point @xmath241 in the @xmath242-th simulation of all @xmath243 simulations , we approximate means of absolute errors @xmath244 as @xmath245 in our experiments , we use the ssbe ( [ ssbew ] ) to compute an `` exact solution '' with small stepsize @xmath246 and @xmath247 . we choose two sets of parameters as follows @xmath79 example i : @xmath248 @xmath79 example ii : @xmath249 @xmath79 example iii : @xmath250 with @xmath251 versus @xmath252 for example i ( left ) and example ii ( right).,title="fig:",width=240,height=172 ] with @xmath251 versus @xmath252 for example i ( left ) and example ii ( right).,title="fig:",width=240,height=172 ] 16cm@ccccccc + & & example ii & & & example iii + ( r)2 - 4 ( l)5 - 7 @xmath4 & em & ssbe ( [ ssbez ] ) & ssbe ( [ ssbew ] ) & em & ssbe ( [ ssbez ] ) & ssbe ( [ ssbew ] ) + + @xmath253 & 0.0008 & 0.0011 & 0.0008 & 0.0014 & 0.0020 & 0.0014 + @xmath254 & 0.0013 & 0.0016 & 0.0013&0.0025 & 0.0036 & 0.0023 + @xmath255 & 0.0021 & 0.0029 & 0.0019 & 0.0058 & 0.0070 & 0.0035 + @xmath256 & 0.0034 & 0.0058 & 0.0027 & 0.2744 & 0.0157 & 0.0053 + @xmath257 & 0.0086 & 0.0148 & 0.0038 & 6.1598e+010 & 0.0628 & 0.0078 + in figure [ 1 ] , computational errors @xmath244 versus stepsize @xmath4 on a log - log scale are plotted and dashed lines of slope one half are added . one can clearly see that ssbe ( [ ssbew ] ) for linear test equation ( [ lssde ] ) is convergent and has strong order of 1/2 . in table [ table1 ] , computational errors @xmath244 with @xmath258 are presented for the well - known euler - maruyama method @xcite , the ssbe method ( [ ssbez ] ) and the improved ssbe method ( [ ssbew ] ) in this paper . there one can find that the improved ssbe method ( [ ssbew ] ) has the best accuracy among the three methods . in particular , for example iii with stiffness in drift term ( i.e. , @xmath259 ) , when the moderate stepsize @xmath260 was used , the euler - maruyama method becomes unstable and the two ssbe methods still remain stable , but with the improved ssbe ( [ ssbew ] ) producing better result . to compare stability property of the improved ssbe and ssbe in @xcite , simulations by ssbe ( [ ssbew ] ) and ( [ ssbez ] ) are both depicted in figure [ 2 ] , [ 3 ] . there solutions produced by ( [ ssbew ] ) and ( [ ssbez ] ) are plotted in solid line and dashed line , respectively . as is shown in the figures , methods ( [ ssbew ] ) and ( [ ssbez ] ) exhibit different stability behavior . one can observe from figure [ 2 ] that ( [ ssbew ] ) for example ii is mean - square stable for @xmath261 . but ( [ ssbez ] ) is unstable for @xmath262 . for example iii , the improved ssbe ( [ ssbew ] ) is always stable for @xmath263 , but ( [ ssbez ] ) becomes stable when the stepsize @xmath4 decreases to @xmath264 . the numerical results demonstrate that the scheme ( [ ssbew ] ) has a greater advantage in mean - square stability than ( [ ssbez ] ) . ) with @xmath265 . upper left : @xmath266 , upper right : @xmath267 , lower left : @xmath268 , lower right : @xmath269.,title="fig:",width=249,height=144 ] ) with @xmath265 . upper left : @xmath266 , upper right : @xmath267 , lower left : @xmath268 , lower right : @xmath269.,title="fig:",width=249,height=144 ] ) with @xmath265 . upper left : @xmath266 , upper right : @xmath267 , lower left : @xmath268 , lower right : @xmath269.,title="fig:",width=249,height=144 ] ) with @xmath265 . upper left : @xmath266 , upper right : @xmath267 , lower left : @xmath268 , lower right : @xmath269.,title="fig:",width=249,height=144 ] ) with @xmath270 . upper left : @xmath266 , upper right : @xmath269 , lower left : @xmath271 , lower right : @xmath272.,title="fig:",width=249,height=144 ] ) with @xmath270 . upper left : @xmath266 , upper right : @xmath269 , lower left : @xmath271 , lower right : @xmath272.,title="fig:",width=249,height=144 ] ) with @xmath270 . upper left : @xmath266 , upper right : @xmath269 , lower left : @xmath271 , lower right : @xmath272.,title="fig:",width=249,height=144 ] ) with @xmath270 . upper left : @xmath266 , upper right : @xmath269 , lower left : @xmath271 , lower right : @xmath272.,title="fig:",width=249,height=144 ] consider a nonlinear sdde with a time - varying delay as follows @xmath273 dt + \left [ x(t)+x(t-\tau(t))\right ] dw(t ) , t > 0 , \\ x(t ) = 1 , \quad t \in [ -1,0 ] , \end{array } \right.\end{aligned}\ ] ] where @xmath274 . obviously , equation ( [ nlsdde ] ) satisfies conditions ( [ olc1])-([olc3 ] ) in assumption [ olc ] , with @xmath275 . thus @xmath276 and the problem is exponentially mean - square stable . as is shown in figure [ 4 ] , the ssbe ( [ ssbew ] ) can well reproduce stability for quite large stepsize @xmath277 . this is consistent with our result established in theorem [ ems3 ] . ) by ssbe ( [ ssbew ] ) using various stepsizes.,width=288,height=240 ] _ proof of theorem [ eu ] . _ since both @xmath91 and @xmath75 are locally lipschitz continuous , theorem 3.2.2 of @xcite shows that there is a unique maximal local solution @xmath25 on @xmath278 , where the stopping time @xmath279 . by it s formula we obtain that for @xmath280 @xmath281 where the condition ( [ mc ] ) was used . thus @xmath282 now , raising both sides of ( [ ito2 ] ) to the power @xmath283 and using hlder s inequality yield @xmath284 by the burkholder - davis - gundy inequality @xcite , one computes that , with @xmath285 , @xmath286 & & \leq c_1\left\{1+\mathbb{e}\|\psi\|^p + \int_0^t \mathbb{e}\sup_{0 \leq r \leq s}|x(r\wedge\rho_r)|^p\mbox{d}s\right . \nonumber \\ & & \left . + \mathbb{e}\left[\int_0^{t\wedge\rho_r } \label{ito5}\end{aligned}\ ] ] next , by an elementary inequality , @xmath287 + \frac{c_1}{2}t^{p/2 - 1}\mathbb{e}\int_0^{t\wedge\rho_r}|g(x(s),x(s-\tau(s)))|^p \mbox{d}s \nonumber \\ & \leq & \frac{1}{2c_1}\mathbb{e}\left[\sup\limits_{0 \leq s \leq t}|x(s\wedge\rho_r)|^{p}\right]+\frac{c_1}{2}(3t)^{p/2 - 1}k^{p/2}\int_0^t ( 1 + \mathbb{e}\sup_{0 \leq r \leq s}|x(r\wedge\rho_r)|^p+\mathbb{e}\|\psi\|^p ) \mbox{d}s . \nonumber\end{aligned}\ ] ] inserting it into ( [ ito5 ] ) , for proper constants @xmath288 we have that @xmath289 the gronwall inequality gives @xmath290 this implies @xmath291 letting @xmath292 leads to @xmath293 since @xmath294 is arbitrary , we must have @xmath295 a.s . and hence @xmath296 a.s . the existence and uniqueness of the global solution is justified . finally , the desired moment bound follows from ( [ ito6 ] ) by letting @xmath292 and setting @xmath297 . baker , e. buckwar , exponential stability in pth mean of solutions , and of convergent euler - type solutions , to stochastic delay differential equations , j. comput . , 184 ( 2 ) ( 2005 ) , pp.404 - 427 . k.burrage , p.m.burrage , t.tian , numerical methods for strong solutions of stochastic differential equations : an overview , proceedings : mathematical , physical and engineering , royal society of london 460(2004 ) , pp.373 - 402 . y.hu , semi - implicit euler - maruyama scheme for stiff stochastic equations , in stochastic analysis and related topics v : the silvri workshop , progr . 38 , h. koerezlioglu , ed . , birkhauser , boston , 1996 , pp.183 - 202 . a.jentzen , p.e.kloeden , a.neuenkirch , pathwise approximation of stochastic differential equations on domains : higher order convergence rates without global lipschitz coefficients , numer . 112 , 1 ( 2009 ) , 41 - 64 .

a new , improved split - step backward euler ( ssbe ) method is introduced and analyzed for stochastic differential delay equations(sddes ) with generic variable delay . the method is proved to be convergent in mean - square sense under conditions ( assumption [ olc ] ) that the diffusion coefficient @xmath0 is globally lipschitz in both @xmath1 and @xmath2 , but the drift coefficient @xmath3 satisfies one - sided lipschitz condition in @xmath1 and globally lipschitz in @xmath2 . further , exponential mean - square stability of the proposed method is investigated for sddes that have a negative one - sided lipschitz constant . our results show that the method has the unconditional stability property in the sense that it can well reproduce stability of underlying system , without any restrictions on stepsize @xmath4 . numerical experiments and comparisons with existing methods for sddes illustrate the computational efficiency of our method . + * ams subject classification : * 60h35,65c20,65l20 . + * key words : * split - step backward euler method , strong convergence , one - sided lipschitz condition , exponential mean - square stability , mean - square linear stability

it is an interesting fact that the liquid state has proven to be difficult to describe by theory throughout the history of condensed matter research @xcite . the problem extends beyond condensed matter and exists in other areas where strong interactions are combined with dynamical disorder such as field theory . in a weakly - interacting system such as a dense gas , the potential energy is much smaller than the kinetic energy . these systems are amenable to perturbation treatment giving corrections to the non - interacting case @xcite . perturbation approaches have been widely explored to calculate liquid thermodynamic properties but have not been able to agree with experiments . for example , the analysis of tractable models such as van der waals or hard - spheres systems returns the gas - like result for the liquid constant - volume specific heat @xmath0 @xcite . this is in contrast to experimental results showing that @xmath1 of monatomic liquids close to the melting point is nearly identical to the solid - like result , @xmath2 and decreases to about @xmath3 at high temperature @xcite . as expected on general grounds , the perturbation approach does not work for strongly - interacting systems . strong interactions are successfully treated in solids , crystals or glasses , where the harmonic model is a good starting point and gives the most of the vibrational energy . however , this approach requires fixed reference points around which the energy expansion can be made . with small vibrations around mean atomic positions , solids meet this requirement but liquids seemingly do not : liquid ability to flow implies that the reference lattice is non - existent . therefore , liquids seemingly have no simplifying features such as small interactions of gases or small displacements of solids @xcite . in other words , liquids have no small parameter . one might adopt a general approach not relying on approximations and seek to directly calculate the liquid energy for a model system where interactions and structure are known . this meets another challenge : because the interactions are both strong and system - dependent , the resulting energy and other thermodynamic functions will also be strongly system - dependent , precluding their calculation in general form and understanding using basic principles , in contrast to solids and gases @xcite . consistent with this somewhat pessimistic view , the discussion of liquid thermodynamic properties has remained scarce . indeed , physics textbooks have very little , if anything , to say about liquid specific heat , including textbooks dedicated to liquids @xcite . as recently reviewed @xcite , emerging evidence advances our understanding of the thermodynamics of the liquid state . the start point is the early theoretical idea of j frenkel @xcite who proposed that liquids can be considered as solids at times smaller than liquid relaxation time , @xmath4 , the average time between two particle rearrangements at one point in space . this implies that phonons in liquids will be similar to those in solids for frequencies above the frenkel frequency @xmath5 : @xmath6 the above argument predicts that liquids are capable of supporting shear modes , the property hitherto attributable to solids only , but only for frequencies above @xmath5 . we note that low - frequency modes in liquids , sound waves , are well - understood in the hydrodynamic regime @xmath7 @xcite , however eq . ( 1 ) denotes a distinct , solid - like elastic regime of wave propagation where @xmath8 . in essence , this suggests the existence of a cutoff frequency @xmath5 above which particles in the liquid can be described by the same equations of motion as in , for example , solid glass . therefore , liquid collective modes include both longitudinal and transverse modes with frequency above @xmath5 in the solid - like elastic regime and one longitudinal hydrodynamic mode with frequency below @xmath5 ( shear mode is non - propagating below frequency @xmath5 as discussed below ) . recall the earlier textbook assertion @xcite that a general thermodynamic theory of liquids can not be developed because liquids have no small parameter . how is this fundamental problem addressed here ? according to frenkel s idea , liquids behave like solids with small oscillating particle displacements serving as a small parameter . large - amplitude diffusive particle jumps continue to play an important role , but do not destroy the existence of the small parameter . instead , the jumps serve to modify the phonon spectrum : their frequency , @xmath5 , sets the minimal frequency above which the small - parameter description applies and solid - like modes propagate . it has taken a long time to verify this picture experimentally . the experimental evidence supporting the propagation of high - frequency modes in liquids currently includes inelastic x - ray , neutron and brillouin scattering experiments but most important evidence is recent and follows the deployment of powerful synchrotron sources of x - rays @xcite . early experiments detected the presence of high - frequency longitudinal acoustic propagating modes and mapped dispersion curves which were in striking resemblance to those in solids @xcite . these and similar results were generated at temperature just above the melting . the measurements were later extended to high temperatures considerably above the melting point , confirming the same result . it is now well established that liquids sustain propagating modes with wavelengths extending down towards interatomic separations , comparable to the wave vectors of phonons in crystals at the brillouin zone boundaries @xcite . more recently , the same result has been asserted for supercritical fluids @xcite . importantly , the propagating modes in liquids include acoustic transverse modes . these were first seen in highly viscous fluids ( see , e.g. , refs . @xcite ) , but were then studied in low - viscosity liquids on the basis of positive dispersion @xcite ( the presence of high - frequency transverse modes increases sound velocity from the hydrodynamic to the solid - like value ) . these studies included water @xcite , where it was found that the onset of transverse excitations coincides with the inverse of liquid relaxation time @xcite , as predicted by frenkel @xcite . more recently , high - frequency transverse modes in liquids were directly measured in the form of distinct dispersion branches and verified on the basis of computer modeling @xcite , and the striking similarity between dispersion curves in liquids and their crystalline ( poly - crystalline ) counterparts was noted . we note that the contribution of high - frequency modes is particularly important for liquid thermodynamics because these modes make the largest contribution to the energy due to quadratic density of states . the above discussion calls for an important question about liquid thermodynamics . in solids , collective modes , phonons , play a central role in the theory , including the theory of thermodynamic properties . can collective modes in liquids play the same role , in view of the earlier frenkel proposal and recent experimental evidence ? we have started exploring this question @xcite just before the high - frequency transverse modes were directly measured and subsequently developed it in a number of ways @xcite . this involves calculating the liquid energy as the phonon energy where transverse modes propagate above @xmath5 in eq . ( [ omega ] ) . the main aim of this paper is to provide direct computational evidence to the phonon theory of liquid thermodynamics and its predictions . we achieve this by calculating the liquid energy and @xmath5 in extensive molecular dynamics simulations . in the next chapter , we briefly discuss the main steps involved in calculating the liquid energy . we then proceed to calculating the liquid energy and frenkel frequency independently from molecular dynamics simulations using several methods which agree with each other . we do this for three systems chosen from different classes of liquids : noble , metallic and molecular , and find good agreement between predicted and calculated results in the wide range of temperature and pressure . the range includes both subcritical liquids and supercritical state below the frenkel line where transverse waves propagate . we calculate and analyze liquid energy and @xmath1 using several different methods . finally , we discuss how our results offer insights into inter - relationships between structure , dynamics and thermodynamics in liquids and supercritical fluids . we summarize the main result of calculation of the liquid energy on the basis of propagating modes . a detailed discussion can be found in a recent review @xcite . according to the previous discussion , the propagating modes in liquids include two transverse modes propagating in the solid - like elastic regime with frequency @xmath9 . the energy of these modes , together with the energy of the longitudinal mode gives the liquid vibrational energy . in addition to vibrations , particles in the liquids undergo diffusive jumps between quasi - equilibrium positions as discussed above . adding the energy of these jumps to the phonon energy in the debye model gives the total energy of thermal motion in the liquid @xcite : @xmath10 where @xmath11 is the number of particles and @xmath12 is transverse debye frequency and the subscript refers to thermal motion . here and below , @xmath13 . at low temperature , @xmath14 , where @xmath15 is the debye vibration period , or @xmath16 . in this case , ( [ harmo ] ) gives the specific heat @xmath17 close to 3 , the solid - like result . at high temperature when @xmath18 and @xmath19 , eq . ( [ harmo ] ) gives @xmath1 close to 2 . the decrease of @xmath1 from 3 to 2 with temperature is consistent with experimental results in monatomic liquids @xcite . the decrease of @xmath1 is also seen in complex liquids @xcite . ( [ harmo ] ) attributes the experimental decrease of @xmath1 with temperature to the reduction of the number of transverse modes above the frequency @xmath20 . the comparison of this effect with experiments can be more detailed if @xmath1 is compared in the entire temperature range where it decreases from @xmath21 to @xmath22 . this meets the challenge that @xmath5 in eq . ( [ harmo ] ) is not directly available in the cases of interest . @xmath5 ( @xmath4 ) is measured is dielectric relaxation or nmr experiments in systems responding to electric or magnetic fields only . these liquids are often complex and do not include simple model systems that are widely studied theoretically such as liquid ar . importantly , the range of measured @xmath5 does not extend to high frequency comparable to @xmath12 , and it is in this range where liquid @xmath1 undergoes an important change from 3 to 2 as discussed above . @xmath5 can be calculated from the maxwell relationship @xmath23 , where @xmath24 is the instantaneous shear modulus and @xmath25 is viscosity taken from a different experiment @xcite . more recently , it has been suggested @xcite that taking the shear modulus at a finite high frequency ( rather than infinite frequency ) agrees better with the modelling data . apart from rare estimations @xcite , @xmath24 is not available . in practice , the comparison of experimental @xmath1 and @xmath1 predicted as @xmath26 with @xmath27 given by eq . ( [ harmo ] ) is done by keeping @xmath24 as a free parameter , obtaining a good agreement between experimental and predicted @xmath1 and observing that @xmath24 lies in the range of several gpa typical for liquids @xcite . in the last few years , eq . ( [ harmo ] ) and its extensions to include the phonon anharmonicity and quantum effects of phonon excitations was shown to account for the experimental @xmath1 of over 20 different systems , including metallic , noble , molecular and network liquids @xcite . in view of the persisting problem of liquid thermodynamics , it is important to test eq . ( [ harmo ] ) directly by linking the liquid energy ( @xmath1 ) on one hand and @xmath5 on the other and testing the theory in a precise way . this , together with achieving consistency with other approaches to calculate the liquid energy , is one of the objectives of this study . importantly , this programme includes supercritical fluids as well as subcritical liquids , as discussed below . if the system is below the critical point ( see figure 1 ) , the temperature increase eventually results in boiling and the first - order transition , with @xmath1 discontinuously decreasing to about @xmath28 in the gas phase . the intervening phase transition excludes the state of the liquid where @xmath1 can gradually reduce to @xmath28 and where interesting physics operates . however , this becomes possible above the critical point . this brings us to the interesting discussion of the supercritical state of matter . theoretically , little is known about the supercritical state , apart from the general assertion that supercritical fluids can be thought of as high - density gases or high - temperature fluids whose properties change smoothly with temperature or pressure and without qualitative changes of properties . this assertion followed from the known absence of a phase transition above the critical point . we have recently proposed that this picture should be modified , and that a new line , the frenkel line ( fl ) , exists above the critical point and separates two states with distinct properties ( see figure [ frenline ] ) @xcite . physically , the fl is not related to the critical point and exists in systems where the critical point is absent . the main idea of the fl lies in considering how the particle dynamics change in response to pressure and temperature . recall that particle dynamics in the liquid can be separated into solid - like oscillatory and gas - like diffusive components . this separation applies equally to supercritical fluids as it does to subcritical liquids . indeed , increasing temperature reduces @xmath4 , and each particle spends less time oscillating and more time jumping ; increasing pressure reverses this and results in the increase of time spent oscillating relative to jumping . increasing temperature at constant pressure or density ( or decreasing pressure at constant temperature ) eventually results in the disappearance of the solid - like oscillatory motion of particles ; all that remains is the diffusive gas - like motion . this disappearance represents the qualitative change in particle dynamics and gives the point on the fl in figure [ frenline ] . most important system properties qualitatively change either on the line or in its vicinity @xcite . in a given system , the fl exists at arbitrarily high pressure and temperature , as does the melting line . quantitatively , the fl can be rigorously defined by pressure and temperature at which the minimum of the velocity autocorrelation function ( vaf ) disappears @xcite . above the line defined in such a way , velocities of a large number of particles stop changing their sign and particles lose the oscillatory component of motion . above the line , vaf is monotonically decaying as in a gas @xcite . for the purposes of this discussion , the significance of the fl is that the phonon approach to liquids and eq . ( [ harmo ] ) apply to supercritical fluids below the fl to the same extent as they apply to subcritical liquids . indeed , the presence of an oscillatory component of particle motion below the fl implies that @xmath4 is a well - defined parameter and that transverse modes propagate according to eq . ( [ omega ] ) . the ability of the supercritical system to sustain solid - like rigidity at frequency above @xmath5 suggested the term `` rigid '' liquid to differentiate it from the `` non - rigid '' gas - like fluid above the fl @xcite . therefore , the fl separates the supercritical state into two states where transverse modes can and can not propagate . this is supported by direct calculation of the current correlation functions @xcite showing that propagating and non - propagating transverse modes are separated by the frenkel line . interestingly , eq . ( [ harmo ] ) can serve as a thermodynamic definition of the fl : the loss of the oscillatory component of particle motion at the fl approximately corresponds to @xmath18 ( here , @xmath15 refers to debye period of transverse modes ) or @xmath19 . according to eq . ( [ harmo ] ) , this gives @xmath1 of about 2 . using the criterion @xmath29 gives the line that is in remarkably good coincidence with the line obtained from the vaf criterion above @xcite . we have considered liquids from three important system types : noble ar , molecular co@xmath30 and metallic fe . we have used the molecular dynamics ( md ) simulation package dl_poly @xcite and simulated systems with @xmath31 particles with periodic boundary conditions . the interatomic potential for ar is the pair lennard - jones potential @xcite , known to perform well at elevated pressure and temperature . for co@xmath30 and fe , we have used interatomic potentials optimized tested in the liquid state at high pressure and temperature . the potential for co@xmath30 is the rigid - body nonpolarizable potential based on a quantum chemistry calculation , with the partial charges derived using the distributed multipole analysis method @xcite . fe was simulated using the many - body embedded - atom potential @xcite . in the case of co@xmath30 , the electrostatic interactions were evaluated using the smooth particle mesh ewald method . the md systems were first equilibrated in the constant pressure and temperature ensemble at respective state points for 20 ps . system properties were subsequently simulated at different temperatures and averaged in the constant energy and volume ensemble for 30 ps . we are interested in properties of real dense strongly - interacting liquids with potential energy comparable to kinetic energy and hence have chosen fairly high densities : @xmath32 g/@xmath33 and @xmath34 g/@xmath33 for ar , @xmath35 g/@xmath33 and @xmath36 g/@xmath33 for fe and @xmath37 g/@xmath33 for co@xmath30 . the lowest temperature in each simulation was the melting temperature at the corresponding density , @xmath38 . the highest temperature significantly exceeded the temperature at the frenkel line at the corresponding density , @xmath39 , taken from the earlier calculation of the frenkel line in ar @xcite , fe @xcite and co@xmath30 @xcite . as discussed above , the temperature range between @xmath38 and @xmath39 corresponds to the regime where transverse modes progressively disappear and where eq . ( [ harmo ] ) applies . we have simulated @xmath40 temperature points at each pressure depending on the system . the number of temperature points was chosen to keep the temperature step close to 10 k. as discussed above , eq . ( [ harmo ] ) applies to subcritical liquids as well as to supercritical fluids below the frenkel line . our simulations include the temperature range both below and above the critical temperature . this will be discussed in more detail below . we have calculated @xmath5 in ( [ harmo ] ) from its definition in ( [ omega ] ) , as @xmath20 . @xmath4 can be calculated in a number of ways . most common methods calculate @xmath4 as decay time of the self - intermediate scattering or other functions by the factor of @xmath41 or as the time at which the mean - squared displacement crosses over from ballistic to diffusive regime @xcite . these methods give @xmath4 in agreement with a method employing the overlap function depending on the cutoff parameter @xmath42 provided @xmath43 , where @xmath44 is the inter - molecuar distance @xcite . we use the latter method and calculate @xmath4 at 13 - 20 temperature points at each density depending on the system . at each density , we fit @xmath4 to the commonly used vogel - fulcher - tammann dependence and use @xmath20 to calculate the liquid energy predicted from the theory . the predicted @xmath1 is calculated as @xmath17 where @xmath27 is given by eq . ( [ harmo ] ) : @xmath45 where @xmath11 is the number of atoms for ar and fe and the number of molecules for co@xmath30 . the first two terms in ( [ cv ] ) give @xmath29 when @xmath5 tends to its high - temperature limit of @xmath5 . the last term reduces @xmath1 below 2 by a small amount because @xmath46 is close to zero at high temperature @xcite . we now compare the calculated energy and @xmath1 with those directly computed in the md simulations . we note that the energy in eq . ( [ harmo ] ) is the energy of thermal phonon motion , @xmath47 , which contributes to the total liquid energy as @xmath48 where @xmath49 is liquid energy at zero temperature and represents temperature - independent background contribution due to the interaction energy . in comparing the calculated @xmath50 in eq . ( [ harmo ] ) with the energy from md simulations , we therefore subtract the constant term from the md energy . the comparison of @xmath17 is performed directly because the constant term does not contribute to @xmath1 . we have also calculated @xmath1 using the fluctuations formula for the kinetic energy @xmath51 in the constant energy ensemble : @xmath52 @xcite . both methods agree well , as follows from figures [ ar]a and [ ar]b . there is only one adjustable parameter in eq . ( [ harmo ] ) , @xmath12 , which is expected to be close to transverse debye frequency . @xmath5 is independently calculated from the md simulation as discussed above . in figures [ ar ] and [ fe ] we compare the energy and @xmath1 calculated on the basis of eqs . ( [ harmo ] ) and ( [ cv ] ) and compare them with those computed in md simulations . blue circle in each figure shows the critical temperature . we observe good agreement between predicted and calculated properties in a temperature range including both subcritical and supercritical temperature . this involved using @xmath53 ps ( @xmath35 g/@xmath33 ) and @xmath54 ps ( @xmath36 g/@xmath33 ) for fe , @xmath55 ps ( @xmath32 g/@xmath33 ) and @xmath56 ps ( @xmath34 g/@xmath33 ) for ar and @xmath57 ps for co@xmath30 , in reasonable order - of - magnitude agreement with experimental @xmath15 of respective crystalline systems as well as maximal frequencies seen in experimental liquid dispersion curves ( see , e.g. , @xcite ) . we note the expected trend of @xmath15 reducing with density . at high temperature where @xmath58 , eq . ( [ cv ] ) predicts @xmath1 close to 2 , noting that the last term gives only a small contribution to @xmath1 because @xmath5 becomes slowly varying at high temperature . consistent with this prediction , we observe the decrease of @xmath1 from 3 to 2 in figures [ ar ] and [ fe ] . the agreement between the predicted and calculated results supports the interpretation of the decrease of @xmath1 with temperature discussed in the introduction : @xmath5 decreases with temperature , and this causes the reduction of the number of transverse modes propagating above @xmath5 and hence the reduction of @xmath1 . for co@xmath30 , the same mechanism operates except we need to account for degrees of freedom in a molecular system . we first consider the case of solid co@xmath30 . the md interatomic potential treats co@xmath30 molecules as rigid linear units , contributing the kinetic term of 2.5 to the specific heat per molecule including 1 from the rotational degrees of freedom of the linear molecular and 1.5 from translations ( here , we have noted that co@xmath30 molecules librate and rotate in the solid at low and high temperature , respectively @xcite ) . noting the potential energy contributes the same term due to equipartition , the specific heat becomes 5 per molecule . this implies that for molecular co@xmath30 , eqs . ( [ harmo ] ) modifies as @xmath59 , where @xmath11 is the number of molecules and @xmath5 is related to the jump frequency of molecules and which gives @xmath60 in the solid state where @xmath5 is infinite . we use the modified equation to calculate the energy and @xmath1 and compare them to those computed from the md simulation in figure [ co2 ] . consistent with the above discussion , we observe that @xmath1 for co@xmath30 calculated directly from the md simulations is close to 5 at low temperature just above melting . at this temperature , @xmath16 , giving the solid - like value of @xmath1 as in the case of monatomic ar and fe . as temperature increases , two transverse modes of inter - molecular motion progressively disappear , resulting in the decrease of @xmath1 to the value of about @xmath61 , in agreement with @xmath1 calculated from the theoretical equation for @xmath50 . we note that the temperature range in which we compare the predicted and calculated properties is notably large ( e.g. , @xmath62 k for ar , and @xmath63 k for fe ) . this range is 10 - 100 times larger than those typically considered earlier @xcite . the higher temperatures for fe might appear as unusual , however we note that liquid iron as well as supercritical iron fluid remains an unmodified system up to very high temperature : the first ionization potential of fe is 7.9 ev , or over 90,000 k. hence the considered temperature range is below the temperature at which the system changes its structure and type of interactions . the very wide temperature range reported here is mostly related to the large part of the temperature interval in figures [ ar]-[co2 ] being above the critical point where no phase transition intervenes and where the liquid phase exists at high temperature , in contrast to subcritical liquids where the upper temperature is limited by the boiling line . the agreement between predicted and calculated properties in such a wide temperature range adds support to the phonon approach to liquid thermodynamics we propose . we make three points regarding the observed agreement between the calculated and predicted results . first , the collective modes contributing to the thermal energy in ( [ harmo ] ) are considered to be harmonic . the anharmonicity can be accounted for in the grneisen approximation , however this involves an additional parameter @xcite . we attempted to avoid introducing additional parameters and sought to test eq . ( [ harmo ] ) which contains only one parameter , @xmath12 . second , eq . ( [ harmo ] ) involves the debye model and quadratic density of states ( dos ) . this approximation is justified since the debye model is particularly relevant for disordered isotropic systems such as glasses @xcite , which are known to be nearly identical to liquids from the structural point of view . furthermore , the experimental dispersion curves in liquids are very similar to those in solids such as poly - crystals @xcite . therefore , the debye model can be used in liquids to the same extent as in solids . one important consequence of this is that the high - frequency range of the phonon spectrum makes the largest contribution to the energy , as it does in solids including disordered solids . we also note that liquid dos can be represented as the sum of solid - like and gas - like components in the two - phase thermodynamic model @xcite , and the solid - like component can be extracted from the liquid dos calculated in md simulations . this can provide more information about the dos beyond debye approximation . third , eq . ( [ harmo ] ) assumes a lower frequency cutoff for transverse waves , @xmath20 , as envisaged by frenkel in ( [ omega ] ) . our recent detailed analysis of the frenkel equations shows that the dispersion relationship for liquid transverse modes is @xmath64 , where @xmath65 is the shear speed of sound and @xmath66 is wavenumber @xcite . here , @xmath67 gradually crosses over from @xmath68 to its solid - like branch @xmath69 when @xmath70 . in this sense , using a lower frequency cutoff in ( [ harmo ] ) might be thought of as an approximation . however , we have recently shown @xcite that the square - root dependence of @xmath67 gives the liquid energy that is identical to ( [ harmo ] ) . the results in the previous sections support the picture in which the decrease of liquid @xmath1 from 3 to 2 is related to reduction of the energy of transverse modes propagating above @xmath5 as described by eq . ( [ cv ] ) . according to eq . ( [ cv ] ) , @xmath29 corresponds to complete disappearance of transverse modes at the fl when @xmath58 ( the disappearance is supported by the direct calculation of transverse modes on the basis of current correlation functions @xcite ) . importantly , @xmath29 marks the crossover of @xmath1 because the evolution of collective modes is qualitatively different below and above the fl @xcite . below the line , transverse modes disappear starting from the lowest frequency @xmath5 . above the line , the remaining longitudinal mode starts disappearing starting from the highest frequency @xmath71 , where @xmath72 is the particle mean free path ( no oscillations can take place at distance smaller than @xmath72 ) . this gives qualitatively different behavior of the energy and @xmath1 below and above the fl , resulting in their crossover at the fl @xcite . interestingly , the thermodynamic crossover at @xmath29 implies a structural crossover . indeed , the energy per particle in a system with pair - wise interactions is @xmath73 where @xmath74 is number density and @xmath75 is radial distribution function . according to eq . ( [ toten ] ) , the liquid energy is @xmath76 , where @xmath50 is given by eq . ( [ harmo ] ) . if the system energy undergoes the crossover at the fl where @xmath29 , eq . ( [ ene ] ) implies that @xmath75 should also undergo a crossover . therefore , the structural crossover in liquids can be predicted on the basis of the thermodynamic properties . we also expect the structural crossover at the fl to be related to the dynamical crossover on general grounds . as discussed above , below the fl particles oscillate around quasi - equilibrium positions and occasionally jump between them . the average time between jumps is given by liquid relaxation time , @xmath4 ( figure [ atoms ] schematically shows a local jump event from its surrounding `` cage '' . ) this means that a static structure exists during @xmath4 for a large number of particles below the fl , giving rise to the well - defined medium - range order comparable to that existing in disordered solids @xcite . on the other hand , the particles lose the oscillatory component of motion above the fl and start to move in a purely diffusive manner as in gases . this implies that the features of @xmath75 are expected to be gas - like . as a result , @xmath75 medium - range peaks are expected to have different temperature dependence below and above the fl . this behavior was observed in ar in md simulations in the short - range structure @xcite . more recently , the crossover in supercritical ne in the medium range at the fl was ascertained on the basis of x - ray scattering experiments @xcite . in figure [ rdf]a we plot pair distribution functions ( pdfs ) of ar at density @xmath34 g/@xmath33 in a wide temperature range . using the fl criterion @xmath29 gives the temperature at the fl , @xmath39 , of about 4000 k at that density , which we find to be consistent with the criterion of the disappearance of the minimum of the velocity autocorrelation function @xcite . the pdf was calculated with the distance step of @xmath77 , giving 600 pdf points . we observe pdf peaks in the medium range order up to about 20 at low temperature . the peaks reduce and broaden with temperature . to study this in more detail , we plot the peak heights vs temperature in figure [ rdf]b . we observe that the medium - range third and fourth peaks persist well above the critical temperature ( @xmath78 k for ar ) : the highest temperature simulated corresponds to @xmath79 . this interestingly differs from the traditional expectation that the structure of the matter so deep in the supercritical state has gas - like features only . at temperature above @xmath39 , the height of the fourth peak becomes comparable to its temperature fluctuations ( calculated as the standard deviation of the peak height over many structures separated in time by 1 ps at each temperature ) by order of magnitude . the fifth and higher - order peaks disappear before the highest temperature in the simulated range is reached . we plot the peak heights in figure [ rdf]b in the double - logarithmic plot because we expect to see an approximate power - law decay of the peak heights at low temperature . indeed , pdf in solids can be represented as a set of gaussian functions with peaks heights @xmath80 depending on temperature as @xmath81 where @xmath82 is a temperature - independent factor @xcite . this temperature dependence of @xmath80 was also quantified in md simulations @xcite . @xmath80 decrease mostly due to the factor @xmath83 whereas the effect of the exponential factor on @xmath80 is small and serves to reduce the rate at which @xmath80 decrease @xcite . this implies that in solids , @xmath84 approximately holds . in liquids , we expect the same relationship to hold below the fl where @xmath14 , corresponding to a particle oscillating many times before diffusively moving to the next quasi - equilibrium position . indeed , the ratio of the number of diffusing particles @xmath85 to the total number of particles @xmath11 in the equilibrium state is @xmath86 @xcite at any given moment of time . @xmath87 is small when @xmath14 below the fl and can be neglected . hence , @xmath84 applies to liquids at any given moment of time below the fl where @xmath14 . this also applies to longer observation times if @xmath80 is averaged over @xmath4 @xcite . we note that the above result , @xmath88 , involves the assumption that the energy of particle displacements is harmonic ( see , e.g. , ref . anharmonicity becomes appreciable at high temperature , however the anharmonic energy terms are generally small compared to the harmonic energy . this is witnessed by the closeness of high - temperature @xmath1 to its harmonic result for both solids and high - temperature liquids @xcite . we therefore expect that @xmath89 approximately holds in the low - temperature range below the fl as in solids but deviates from the linearity around the crossover at the fl where @xmath18 and where the dynamics becomes gas - like ( the calculated pdf in fig . [ rdf]a is normalized to 1 where no correlations are present at large distances ; hence we plot @xmath90 in order to compare it with the theoretical result @xmath88 which tends to zero when no correlations are present at high temperature ) . we note that the crossover is expected to be broad because @xmath14 applies well below the fl only . a substantial diffusive motion takes place in the vicinity of the line where @xmath87 can not be neglected , affecting the linear relationship . consistent with the above prediction , we observe the linear regime at low temperature in figure [ rdf]b , followed by the deviation from the straight lines taking place around 3000 k for the 2nd peak , 5000 k for the 3rd peak and 4000 k for the 4th peak , respectively . the smooth crossover in the 3000 - 5000 k range is centered around 4000 k , consistent with the temperature at the frenkel line discussed above . we also note that 4000 k corresponds to the specific heat @xmath29 in figure [ ar]b , in agreement with the earlier discussion . as discussed in the introduction , liquids have been viewed as inherently complicated systems lacking useful theoretical concepts such as a small parameter @xcite . together with recent experimental evidence and theory @xcite , the modelling data presented here and its quantitative agreement with predictions are beginning to change this traditional perspective . our extensive molecular dynamics simulations of liquid energy and specific heat provide direct evidence for the link between dynamical and thermodynamic properties of liquids . we have found this to be the case for several important types of liquids at both subcritical and supercritical conditions spanning thousands of kelvin . this supports an emerging picture that liquid thermodynamics can be understood on the basis of high - frequency collective modes . a more general implication is that , contrary to the prevailing view , liquids are emerging as systems amenable to theoretical understanding in a consistent picture as is the case in solid state theory . in addition to the link between dynamical and thermodynamic properties , we have discussed how these properties are related to liquid structure . this research utilised midplus computational facilities supported by qmul research - it and funded by the epsrc grant ep / k000128/1 . we acknowledge the support of the royal society , rfbr ( 15 - 52 - 10003 ) and csc .

we develop an approach to liquid thermodynamics based on collective modes . we perform extensive molecular dynamics simulations of noble , molecular and metallic liquids and provide the direct evidence that liquid energy and specific heat are well - described by the temperature dependence of the frenkel ( hopping ) frequency . the agreement between predicted and calculated thermodynamic properties is seen in the notably wide range of temperature spanning tens of thousands of kelvin . the range includes both subcritical liquids and supercritical fluids . we discuss the structural crossover and inter - relationships between structure , dynamics and thermodynamics of liquids and supercritical fluids .

ultracold quantum gases provide a very exciting branch of physics . besides the interesting physics that the gases offer by themselves , it has also been possible in the last few years to model with quantum gases systems from other branches of physics , and by doing so to provide answers to long - standing questions . the latter is mainly due to the amazing accuracy by which their properties can be tuned and manipulated . this involves the trapping potential , the dimensionality , the interaction between the atoms , and the statistics . by using a three - dimensional optical lattice the superfluid - mott insulator transition in the bose - hubbard model has been observed @xcite . bosonic atoms confined in one - dimensional tubes by means of a two - dimensional optical lattice where shown to realize the lieb - liniger gas @xcite . the unitarity regime of strong interactions was reached by using feshbach resonances to control the scattering length @xcite . to this shortlist of examples from condensed - matter theory , also examples from high - energy physics can be added . in a spinor bose - einstein condensate with ferromagnetic interactions skyrmion physics has been studied @xcite , whereas an antiferromagnetic spinor bose - einstein condensate allows for monopole or hedgehog solutions @xcite . there is also a proposal for studying charge fractionalization in one dimension @xcite , and for creating ( static ) non - abelian gauge fields @xcite . in recent work @xcite we have added another proposal to model a system from high - energy physics . by combining a vortex line in a one - dimensional optical lattice with a fermionic gas bound to the vortex core , it is possible to tune the laser parameters such that a nonrelativistic supersymmetric string is created . this we called the ultracold superstring . this proposal combines three topics that have attracted a lot of attention in the area of ultracold atomic gases . these topics are vortices @xcite , bose - fermi mixtures @xcite , and optical lattices @xcite . apart from its potential to experimentally probe certain aspects of superstring theory , this proposal is also very interesting because it brings supersymmetry within experimental reach . supersymmetry is a very special symmetry , that relates fermions and bosons with each other . it plays an important role in string theory , where supersymmetry is an essential ingredient to make a consistent theory without the so - called tachyon , i.e. , a particle that has a negative mass squared . in the physics of the minimally extended standard model , supersymmetry is used to remove quadratic divergences . this results in a super partner for each of the known particles of the standard model . however , supersymmetry is manifestly broken in our world and none of these superpartners have been observed . a third field where supersymmetry plays a role is in modeling disorder and chaos @xcite . here supersymmetry is introduced artificially to properly perform the average over disorder . finally , supersymmetry plays an important role in the field of supersymmetric quantum mechanics , where the formal structure of a supersymmetric theory is applied to derive exact results . in particular this means that a supersymmetry generator @xmath0 is defined , such that the hamiltonian can be written as @xmath1 , which is one of the basic relations in the relativistic superalgebra . it is important for our purposes to note , that this relation is no longer enforced by the superalgebra in the nonrelativistic limit . careful analysis @xcite shows that in this limit the hamiltonian is replaced by the number operator , i.e. , @xmath2 . it may sometimes be possible to write a nonrelativistic hamiltonian as the anticommutator of the supersymmetry generators , but this does not correspond to the nonrelativistic limit of a relativistic theory . in our proposal , a physical effect of supersymmetry is that the stability of the superstring against spiraling out of the gas is exceptionally large , because the damping of the center - of - mass motion is reduced by a destructive interference between processes that create two additional bosonic excitations of the superstring and processes that produce an additional particle - hole pair of fermions . moreover , this system allows for the study of a quantum phase transition that spontaneously breaks supersymmetry as we will show . another very interesting aspect of the ultracold superstring is the close relation with string - bit models @xcite . these are models that discretize the string in the spatial direction , either to perturbatively solve string theory , or , more radically , to reveal a more fundamental theory that underlies superstring theory . string - bit models describe the transverse degrees of freedom of the string in a very similar fashion as in our theory of the ultracold superstring . in this article we investigate in detail the physics of ultracold superstrings , expanding on our previous work @xcite . the article is organized as follows . in sec . ii we give the detailed derivation of the conditions for the ultracold superstring to be created . in particular , we pay attention to the presence of the fermionic bound state in the vortex core and the tuning of the lasers to reach supersymmetry . in sec . iii we investigate the experimental consequences of the supersymmetry . iv contains a detailed description of the supersymmetry by studying the superalgebra . in sec . v we make connection with string theory . finally , we end with our conclusions in sec . our proposal makes use of the fact that a vortex line through a bose - einstein condensate in a one - dimensional optical lattice can behave according to the laws of quantum mechanics @xcite . such an optical lattice consists of two identical counter - propagating laser beams and provides a periodic potential for atoms . when applied along the symmetry axis of a cigar - shaped condensate , which we call the @xmath3 axis from now on , the optical lattice divides the condensate into weakly - coupled pancake - shaped condensates . in the case of a red - detuned lattice , the gaussian profile of the laser beam provides also the desired trapping in the radial direction . rotation of the bose - einstein condensate along the @xmath3 axis creates a vortex line that passes through each pancake . quantum fluctuations of the vortex position are greatly enhanced in this configuration because of the small number of atoms @xmath4 in each pancake , which can be as low as @xmath5 , but is typically around @xmath6 . an added advantage of the stacked - pancake configuration , as opposed to the bulk situation , is that the dispersion of the vortex oscillations is particle like . this ultimately allows for supersymmetry with the fermionic atoms in the mixture . in the one - dimensional optical lattice the vortex line becomes a chain of so - called pancake vortices . this produces a setup which is pictured schematically in fig [ artimpl ] . there is a critical external rotation frequency @xmath7 above which a vortex in the center of the condensate is stable . for @xmath8 the vortex is unstable , but because of its euler dynamics , it takes a relatively long time before it spirals out of the gas @xcite . we analyze in detail the case of @xmath9 , i.e. , the situation in which the condensate is no longer rotated externally after a vortex is created . however , the physics is very similar for all @xmath8 , where supersymmetry is possible . the temperature is taken to be well below the bose - einstein condensation temperature , so that thermal fluctuations are strongly suppressed . we only consider the zero - temperature limit , because supersymmetry is formally broken for nonzero temperatures . a convenient choice for the boson - fermion mixture is @xmath10rb and @xmath11k , since such bose - fermi mixtures have recently been realized in the laboratory @xcite , and because the resonance lines in these two atomic species lie very nearby . the mostly used @xmath12 hyperfine spin states are @xmath13 and @xmath14 for @xmath11k , and @xmath15 , @xmath16 , and @xmath17 for @xmath10rb @xcite . they all have a negative interspecies scattering lenght @xmath18 , which is not desirable for our purposes as we show below . it could be possible to use other spin states , which have a positive interspecies scattering length . an other possiblity is to tune the scattering length , using one of the various broad feshbach resonances that can make the interaction repulsive while keeping the probability to create molecules negligible @xcite . in principle it is also possible to use other mixtures . another bose - fermi mixture that has been realized in the laboratory consists of @xmath19na and @xmath20li atoms @xcite . this mixture is less convenient because the resonance lines are widely separated , so that the two species feel very different optical potentials and it is hard to trap both with a single laser . in addition , @xmath21li is relatively hard to trap in an optical lattice because of its small mass . for these reasons , the @xmath19na-@xmath21li mixture can only be used in a very restricted parameter regime , as we will show lateron in fig . [ tuning1 ] . for the same reasons , the mixture @xmath10rb - @xmath21li @xcite does not work well either . the mixture @xmath22li-@xmath21li @xcite can not be used at all , because the resonance lines of the species are the same , so it is impossible to tune the physical properties of the mixture . because the excited states of the bosonic and fermionic atoms have different transition frequencies , the optical lattice produces for the two species a periodic potential with the same lattice spacing , but with a different height , as schematically shown in fig . [ optpotential ] . ( a ) is the radial distance in the @xmath23 plane . the pink and blue blobs represent the bosonic and fermionic densities , respectively . moreover , @xmath24 is the wavelength of the laser . the blue and red lines indicate the strength of the optical potential , respectively , for the bosons and fermions as a function of the @xmath3 coordinate . ( b ) schematic fine structure level scheme of the bosonic and fermionic atomic species . because we consider only sufficiently large detunings the hyperfine level structure is not resolved . ] ( b ) is the radial distance in the @xmath23 plane . the pink and blue blobs represent the bosonic and fermionic densities , respectively . moreover , @xmath24 is the wavelength of the laser . the blue and red lines indicate the strength of the optical potential , respectively , for the bosons and fermions as a function of the @xmath3 coordinate . ( b ) schematic fine structure level scheme of the bosonic and fermionic atomic species . because we consider only sufficiently large detunings the hyperfine level structure is not resolved . ] this is very crucial , because it allows to tune the optical lattice for the bosonic and fermionic atoms seperately , by careful adjustement of the wavelength and the rabi frequency , i.e. , the intensity of the laser . this is required to be able to tune the system to become supersymmetric lateron . for the @xmath25rb-@xmath11k mixture the rabi frequencies are in a good approximation the same . for other mixtures the rabi frequencies are different and we then take the bosonic rabi frequency as a reference . we take into account the fine - structure level scheme of the atoms , but , assuming that we are sufficiently far from resonance , we neglect the hyperfine structure . as a result , the optical potential is given by @xmath26 where the well depths obey @xmath27 , \nonumber\end{aligned}\ ] ] @xmath28 is the laser frequency , and @xmath29 and @xmath30 are the frequencies of the @xmath31 and @xmath32 resonance lines . here we neglected spontaneous emission of photons . this effect slightly modifies the trapping potential , but gives a finite lifetime to the atoms . using the rotating - wave approximation and neglecting the fine structure , the effective rate of photon absorption can for red - detuned laser light be estimed as @xmath33 where @xmath34 is the linewidth of the bosonic or fermionic excited state , respectively . for blue - detuned laser light , the atoms are trapped in the regions of low laser intensity and spontaneous emission is strongly reduced . the optical potential should be sufficiently deep to have a bound state for the bosonic and fermionic atoms . to make sure that that is the case we impose the condition @xmath35 where we have used the recoil - energy @xmath36 which is the energy associated with the absorption of a photon . on the other hand , the optical lattice should not be so strong to drive the system in the mott - insulator state @xcite . in one dimension with many atoms per site , this requires an exceptionally deep lattice , which only occurs if the laser frequency is very close to the resonance frequency of the atomic species . since we stay away from resonance , this situation does not occur in our calculations . the wavefunctions in the @xmath3 direction are assumed to be the groundstate wavefunctions of the harmonic oscillator associated with the optical lattice and thus given by @xmath37 where @xmath38 for the tunneling amplitude , we use the expression @xcite @xmath39,\ ] ] which becomes exact for a deep lattice . therefore , the atomic dispersions along the @xmath3 axis are given by @xmath40 lateron we need for the fermions the relation between the average number of particle per site and the chemical potential @xmath41 . from the above dispersion we derive at zero temperature that @xmath42 , \label{fillingfrac}\ ] ] where we neglect also interaction effects . the wavefunction in the ( axial ) @xmath3 direction is fully specified by the optical lattice and all the dynamics thus takes place in the radial direction , i.e. , in the @xmath23-plane . since the vortex - fluctuations form the lowest - lying modes , we restrict the dynamics to the vortex motion . we follow the derivations in earlier work @xcite , where a specific ansatz for the wavefunction was used , to achieve this . in this work the condensate density was described by a gaussian wavefunction with size @xmath43 and the vortex core was approximated by a step function . furthermore , it was assumed that the vortex is close to the center . the motion of the vortices results in kelvons , i.e. , quantized oscillations of the vortex , described by the creation and annihilation operators @xmath44 which obey @xmath45 . without the optical lattice kelvin waves have already been observed @xcite . the kelvons have the dispersion @xmath46 \right ) + \hbar \omega \label{dispersion } \\ & & + 2 j_k \lbrack 1-\cos(k \lambda/2)\rbrack , \nonumber \end{aligned}\ ] ] where @xmath47j_b,\ ] ] @xmath48 $ ] is the incomplete gamma function , @xmath43 is the thomas - fermi radius in the radial direction , @xmath49 is the bosonic harmonic length in the radial direction , and @xmath50 is the associated frequency . using another ansatz for the condensate wavefunction can slightly change the constant of proportionality in the definition of the kelvon operators and in the details of the dispersion , but the dispersion always stays tight - binding like . for the calculation of the bound state in the vortex core , we need to go beyond the description of the core by a step function . this change of the calculation could improve the value of @xmath51 , but not the functional form of the kelvon dispersion . since the corrections on the value of @xmath51 are small , we just use the result in eq . ( [ jk ] ) . besides the bandwidth @xmath51 we derive from eq . ( [ dispersion ] ) also the chemical potential for the kelvons , which gives @xmath52 -1 \right ) - \hbar \omega . \nonumber\end{aligned}\ ] ] note that the chemical potential is positive only for sufficiently small rotations , which is due to the fact that the vortex is in principle unstable for these values of the rotation and wants to spiral out of the center of the gas cloud . by treating the interaction between the bosonic and fermionic atoms in mean - field approximation , we have to solve the gross - pitaevskii equation for the condensate wavefunction @xmath53 coupled to the schrdinger equation for the fermion wavefunction @xmath54 @xmath55 which we investigate for the case that @xmath9 . the interaction paramaters are related to the scattering lengths according to @xmath56 with @xmath57 the boson - boson scattering length and @xmath18 the boson - fermion scattering length and @xmath58 the reduced mass @xmath59 . although it is very well possible to solve these equations numerically , we prefer an analytic treatment , to gain more insight into the problem . to proceed we make the approximation that the condensate wavefunction is not affected by the presence of the fermions . this is justified , because the contribution of the fermions is @xmath60 smaller than the contribution of the bosons , where @xmath61 is the average number of bosons and fermions at a lattice site . this ratio will be smaller than @xmath62 as it turns out . taking into account the interaction with the fermions leads to a slightly wider vortex core , which enhances the possiblity of a bound state . so we first solve the gross - pitaevskii equation for the condensate density neglecting the presence of the fermions and then use the condensate density as an effective potential for the fermions . since we only want to estimate when there is a bound state and we do not need the details of this bound state , we make the following approximations . first , we assume the vortex to be in the center such that the problem is rotationally symmetric and we only have to solve the radial equation . because of the quantum uncertainty the vortex position in principle fluctuates around the center of the trap , but these fluctuations are small . second , we assume the envelope condensate wavefunction to be thomas - fermi like , i.e. , @xmath63 third , we describe the vortex core by @xmath64 @xcite , such that the total bosonic density is given by @xmath65 if we take for the healing length @xmath66 the usual expression in the center of the trap , i.e. , @xmath67 we obtain the relation @xmath68 by expressing the energy in terms of @xmath69 we can write the schrdinger equation for the fermions as @xmath70 \psi(r ) = 0,\ ] ] where @xmath71 and the dimensionless parameter @xmath72 determines whether or not there is a bound state in the core of the vortex . if we assume that @xmath73 and @xmath74 , we can neglect the harmonic confinement and the thomas - fermi profile of the bose - einstein condensate . the effective potential for the fermionic atoms is then given by @xmath75 . this potential has a bound state for each value of @xmath76 , because for large distances from the core , it behaves as @xmath77 . however , the size of the wavefunction describing the bound state becomes extremely large for small values of @xmath76 . hence it is necessary to take into account the exact form of the potential to make a quantitative estimate of the existence of the bound state . the potential is determined by the values of the radial bosonic and fermionic harmonic length @xmath49 and @xmath78 . since @xmath78 determines the potential outside the condensate , it determines whether or not the fermions can tunnel out of the core to this region . for the existence of the bound state we can neglect this contribution , which is always justified , because it enhances the possibility of having a bound state . the radial bosonic harmonic length @xmath49 is fixed by the normalization of the condenstate wavefunction @xmath79 neglecting the presence of the core we find the usual expression for the thomas - fermi profile @xmath80^{1/4 } \ ! . \label{tfr}\ ] ] using that @xmath81\log \left(\frac{r_{\rm tf}^2 + 2 \xi^2}{2 \xi^2 } \right)\right ) , \end{aligned}\ ] ] we see that taking into account the core implies that we have to solve the equation @xmath82 where the last two terms come form the presence of the core . since the core is small in this approximation , this results in a radial harmonic length that is only slightly modified . the requirement that the wavefunction should vanish well within the condensate can then be quantified to yield the expression @xmath83 where @xmath84 is the radial size of the fermionic wavefunction . in this way we obtain that for typical densities there is a bound state for @xmath85 which means that @xmath86 . in contrast to the radial bosonic length @xmath49 , the fermionic radial harmonic length @xmath78 is not fixed . when the optical lattice is red - detuned , the lattice can be used to trap the atoms also in the radial direction . as a consequence , the total confining potential for the fermions is a multiple of the confining potential of the bosons , i.e. , @xmath87 this gives the relation @xmath88 from which we derive @xmath89 however , if the lattice is blue - detuned or if the radial trapping is tuned independently , this relation is not true . the radial trapping can be tuned by introducing a second running laser in the same direction as the optical lattice , as shown in fig [ extra_laser ] . the new laser beam has a constant intensity along the @xmath3 axis , and does not influence the one - dimensional potential wells , but it does change the radial confinement . in principle this second laser also introduces interference terms , but they are much faster than the atoms can follow for the frequencies of interest to us . therefore , the intensities of the two lasers can simply be added . in particular , as we show lateron , adjusting the radial trapping potentials is needed to get supersymmetric interaction terms . the condition imposed by this requirement is @xmath90- \frac{3}{2}\right),\ ] ] which gives the following expression for the fermionic radial harmonic length @xmath91 - \frac{3}{2}}.\ ] ] in this last case , the harmonic radial potential for the fermions is very small . in principle this allows the fermionic atoms to tunnel out of the vortex core , to the region where the condensate density vanishes . however , the tunneling is suppressed by increasing the parameter @xmath76 . a wkb estimate gives that for @xmath92 the lifetime of the fermions in the core is larger than a second . this means that @xmath93 . further increasing this ratio increases this lifetime dramatically . since adjusting the radial trapping potentials is only needed close to the center of the trap , it is also a possibility to use a second laser with a much smaller waist , such that higher - order contributions from the potential prevent the fermions from tunneling out of the core . for various situations , the effective potential for the fermions is shown in fig [ effpot ] . and energies in units of @xmath69 . ( a ) @xmath94 , @xmath95 : no bound state , since the potential is too small . ( b ) @xmath96 , @xmath97 : bound state in the core , but possibility to tunnel outside . ( c ) @xmath96 , @xmath98 : bound state in the core , no tunneling possible . ] in our superstring realization there are also boson - boson and boson - fermion interactions . the kelvons interact repulsively among each other when @xmath99 . for @xmath9 the kelvon - kelvon interaction coefficient is given by @xcite @xmath100- \frac{3}{2}\right).\ ] ] in addition , a repulsive interaction between the kelvons and the fermionic atoms is generated by the fact that physically the presence of a kelvon means that the vortex core is shifted off center , together with the fermions trapped in it . because of the radial confinement experienced by the trapped fermions , this increases the energy of the vortex . when the vortex core is shifted from @xmath101 to @xmath102 , the fermion hamiltonian is extended by a term @xmath103 where @xmath104 is the number operator for the fermions in the core . defining @xmath105 as the spring constants associated with the radial confinement of the bosonic and fermionic atoms , respectively , and using the definition of the kelvon operators , this translates into @xmath106 so the kelvon - fermion interaction coefficient is found to be @xmath107 to obtain a supersymmetric situation we have three requirements . in the first place the hopping amplitudes have to be the same @xmath108 this can be done by adjusting the laser parameters @xmath24 and @xmath109 , as shown in fig [ tuning1 ] . the freedom in choosing the wavelength of the laser can be used to minimize the atom loss . in fig . [ atomloss1 ] , we plot the atom loss as a function of the wavelength of the laser . rb-@xmath11k for 10000 ( solid line ) , 1000 ( dashed line ) and 500 ( dotted line ) bosonic atoms per site . note that for the blue - detuned part , i.e. , @xmath110 nm for the @xmath10rb-@xmath11k mixture extra radial trapping is needed , either magnetically , or by using an extra running laser as discussed in the text and shown in fig . [ extra_laser ] . in fig . [ tuning3 ] we display how to tune the running laser to obtain also supersymmetric interactions . in the inset we plotted the rabi frequency that is required for the @xmath19na-@xmath21li mixture to obtain supersymmetry , again for 10000 ( solid line ) , 1000 ( dashed line ) and 500 ( dotted line ) bosonic atoms per site . note that this can only be obtained in a very limited range of wavelength s . ] rb-@xmath11k mixture . ] secondly , the chemical potentials have to be the same @xmath111 this can be achieved by adjusting the fermion filling fraction @xmath112 , as shown in fig [ tuning2 ] . using the result from eq . ( [ fillingfrac ] ) and using the requirements for supersymmetry we obtain @xmath113 -1 \right ) } { 4 j_b \gamma\left[0 , \frac{l^4}{r_{\rm tf}^4 } \right ] } } \right ) \nonumber \\ & & = \ ! \frac{2}{\pi } \arcsin \ ! \left ( \ ! \frac{\ell}{\ell_b^z } \sqrt { \frac { \sqrt{\pi } \frac{\ell^2 } { r_{\rm tf}^2 } \left(\gamma\left[0 , \frac{l^4}{r_{\rm tf}^4 } \right]\!- \ ! 1\ ! \right ) e^{\sqrt{\tfrac{v_b}{e_b}}}}{16 ( v_b / e_b)^{1/4 } \gamma\left[0 , \frac{l^4}{r_{\rm tf}^4 } \right ] } } \right ) \ ! . \nonumber\end{aligned}\ ] ] the ratio @xmath114 is undetermined by supersymmetry constraints . in order for the thomas - fermi approximation to apply in the radial direction , versus the gaussian wavefunction in @xmath3 direction , this ratio needs to be sufficiently small . in the figure a ratio of @xmath115 is chosen . . this ratio should be sufficiently small to be radially in the thomas - fermi limit . for this plot a ratio of 1/5 is chosen . ] finally , the interaction terms have to be the same . this implies @xmath116 setting these coefficients equal to each other gives a condition on the radial trapping given by @xmath117- \frac{3}{2}\right).\ ] ] the radial trapping can be tuned by introducing a second running laser , as explained before . for the second laser , we can again independently choose both the wavelength and the rabi frequency as shown in fig . [ tuning3 ] . this can again be used to minimize the atom loss due to the red - detuned laser , but it turns out that atom loss is always quite small anyway for reasonable system parameters . only for very small detunings , the lifetime is less than a second . combining everything , our superstring is described by the supersymmetric hamiltonian @xmath118 here @xmath119 is the annihilation operator of a kelvon at site @xmath120 , @xmath121 is the annihilation operator of a fermion at site @xmath120 , @xmath122 means that the summation runs over neighbouring sites , and @xmath123 . we used the convention for the fourier transformation @xmath124 , where @xmath125 is the number of lattice sites . we define @xmath126 as the lattice spacing and @xmath127 as the length of the system . assuming that @xmath128 , such that @xmath129 , we can perform a continuum approximation to obtain for the hamiltonian @xmath130 where we introduced the effective mass @xmath131 . this continuum hamiltonian turns out to be exactly solvable @xcite by a straightforward generalization of the bethe - ansatz solution of the one - dimensional bose gas @xcite . however , the exact solutions spontaneously break supersymmetry and do not give much insight in the role of supersymmetry in the problem . using that the lagrangian is given by @xmath132,\ ] ] the action in the continuum limit is obtained as @xmath133 which now explicitly shows the supersymmetry of the problem , because it remains invariant when @xmath134 and @xmath135 are rotated into each other . if we neglect the interaction terms , which are rather small anyway , the fermions fill a fermi sea and the low - energy excitations are particle - hole excitations around the fermi surface . therefore , the low - energy part of the theory is properly described by linearizing the fermionic dispersion around the fermi level . to preserve supersymmetry we do the same for the bosons and obtain at the quadratic level the action @xmath136 where @xmath84 indicates whether the particles are right movers or left movers and @xmath137 is the fermi velocity . we used that @xmath138 . we identify the fermi velocity with the velocity of light @xmath135 and perform the transformation @xmath139 . we introduce the dirac spinor @xmath140 and @xmath141 , with @xmath142 . the other dirac matrices are @xmath143 and @xmath144 . the two bosonic fields can be captured in a single klein - gordon field @xmath145 , such that @xmath146 . this enables us to rewrite the linearized action as @xmath147 which is the action for the transverse modes of a free relativistic @xmath148 superstring in @xmath149 dimensions @xcite . in modern language , the lorentz invariance of this action appears here as an emergent phenomenon at long wavelenghts , because the underlying theory is not lorentz invariant . this is very similar with the way in which lorentz invariance appears in string - bit models @xcite . a second property of this action is , that the fermionic part has classically chiral symmetry , but quantum - mechaniclly suffers from a chiral anomaly . whereas in string theory this is an unwanted feature , in our case it has a physical origin , because it comes about from the fact that the underlying microscopic theory does not conserve the chiral current @xmath150 , and only conserves the current @xmath151 associated with the conservation of the total number of fermions . the presence of a kelvon implies that neighbouring vortex cores are slightly shifted with respect to each other . this effect decreases the fermionic hopping amplitude and results in a interaction term that couples fermions on neighbouring sites of the form @xmath152 since this term breaks supersymmetry , we want to investigate the system parameters for which it can be neglected , i.e. , for which @xmath153 . to do so we consider a kelvon with a certain wavenumber @xmath154 . the relative distance between neighboring cores kan then be estimated to be @xmath155 from eq . ( [ vortexcore ] ) we know that for small distances the vortex core can be modeled as a harmonic potential with width @xmath66 . hence , the fermionic wavefunctions are gaussians with the same width . so we have to compute @xmath156 where we used the relation from eq . ( [ tfr ] ) . from this same relation we see that @xmath157 scales with the number of bosonic atoms @xmath4 , such that @xmath158 is independent of @xmath4 . we can estimate @xmath159 to be of order unity , such that the requirement for @xmath158 to be small only depends on the wavenumber @xmath154 . if we identify this wavenumber with the fermi momentum , i.e. , @xmath160 we obtain a restriction on the fermionic filling fraction which can be estimated to be @xmath161 from fig . [ tuning2 ] we see that for most of the parameter space this condition is fullfilled . it is an important question how the supersymmetry can be observed . therefore we need to distinguish between the question whether the hamiltonian is tuned to be supersymmetric and whether the quantum ground state is supersymmetric , since it is possible that the ground state can spontaneously break supersymmetry . we are primarely interested in the situation that both the hamiltonian and the quantum ground state are supersymmetric . the two observables that are most easy to measure experimentally are the average number of fermions at a site @xmath112 and the average number of kelvons @xmath162 . the average fermion number can be determined by usual absorpsion measurements . the number of kelvons can be obtained from the mean - square displacement @xmath163 of the pancake vortices , which can be measured by imaging along the @xmath3 direction the size of the circle within which the vortex positions are concentrated @xcite . because @xmath164 this can directly be translated to the number of kelvons at a site . it is clear that in order to have a supersymmetric state , the kelvon and fermion modes should have the same average occupation number , i.e. , @xmath165 this allows us to devise an experimental measure for the proximity to the supersymmetric point , which can be directly measured , namely @xmath166 this quantity has an absolute minimum of zero at the supersymmetric point , so that it s magnitude is a measure of the deviation from supersymmetry . we can extend this to higher order correlation functions . the condition that @xmath167 can be used to prove that in order to have supersymmetry also the condition @xmath168 should hold . the quantiy @xmath169 can again be measured from the distribution of the measured vortex postions . , the left diagram is called @xmath170,title="fig : " ] , the left diagram is called @xmath170,title="fig : " ] . [ diagramsl ] another consequence of supersymmetry that can be directly measured is the reduced dissipation . dissipation in this context results in the vortex spiraling out of the gas . the dominant part of the dissipation is given by the coupling to the kelvon modes and the fermionic modes . the lowest order diagrams are given in fig.[diagramsl ] and denoted by @xmath171 for the coupling to the kelvon modes and @xmath170 when there is also coupling to the fermionic modes . the imaginary part of these diagrams measures the dissipation . in order to be able to know the dissipation away from the supersymmetric point we perform the calculation for unequal dispersions @xmath172 and unequal coupling constant @xmath173 and @xmath174 . we introduce the usual notation for the bose - einstein and fermi - dirac distribution functions @xmath175 the diagrams are then given by @xmath176 where the minus sign in front of the expression for @xmath177 comes from the presence of the fermion loop . note that due to the different combinatorial factors the diagram @xmath178 comes with an extra factor of two , which is lacking in the case of the diagram @xmath179 . as a result , the two diagrams do not cancel exactly at the supersymmetric point , as we claimed previously @xcite . instead , the dissipation is reduces by a factor 2 @xcite . the imaginary part of the diagrams gives the following expressions @xmath180 at zero temperature we have that @xmath181 , and @xmath182 . using this , we see that if there is supersymmetry , i.e. , if @xmath183 and @xmath184 , we have that @xmath185 and in particular that @xmath186 such that at zero temperature supersymmetry results in a dissipation rate that is only half as large as in the case of a ordinary vortex - line . using these expressions , it is also possible to calculate the quantum dissipation at nonzero temperature , or when supersymmetry is broken . in particular , when the interaction coefficients are tuned away from the supersymmetric point such that @xmath187 and supersymmetry is maintained at the quadratic level , the dissipation exactly vanishes and the superstring is extremely stable in the center of the condensate . when the hamiltonian is supersymmetric , the ground state still can break supersymmetry . this is the phenomenon of spontaneous supersymmetry breaking . for @xmath188 , the ultracold superstring is unstable against bose - einstein condensation of kelvons . this breaks supersymmetry , because the fermionic modes can not bose - einstein condense . bose - einstein condensation implies that the kelvon annihilation operator obtains an expectation value @xmath189 from the definition of the kelvon operator we conclude that as a consequence @xmath190 this means that the vortex moves out of the center of the trap . experimentally this is easy to measure . moreover , by monitoring the vortex position when it moves out of the center of the trap , this also allows for the experimental investigation of the dynamics of supersymmetry breaking . as a consequence of the breaking of the @xmath191 symmetry because of the bose - einstein condensation , the dispersion of the kelvon modes becomes gapless . the dispersion becomes the usual bogoliubov dispersion , which reads @xmath192 with @xmath193 . for long wavelengths this yields a linear behaviour . also the fermionic modes become gapless . this is a result of the breaking of supersymmetry and this mode is called the goldstino . because @xmath194 , the dispersion for the goldstino is given by @xmath195 which results in a quadratic dispersion . clearly the bosonic and fermionic dispersion in eqs . and are now different , which signals a nonsupersymmetric situation . in this section we review the algebra associated with supersymmetric field theories both in the relativistic ( super poincar algebra ) and the non - relativistic limit ( super galilei algebra ) . we give an explicit representation of the super galilei algebra in terms of the bosonic and fermionic operators . associated with a relatistic field theory in @xmath196 dimensions is the poincar algebra , whose generators consist of a vector @xmath197 , that generates translations and an antisymmetric tensor @xmath198 , that generates lorenz transormations . the greek indices run from @xmath101 to @xmath199 , such that @xmath200 should be identified with the hamiltonian @xmath201 , up to a constant . the algebra is then given by @xmath202 where @xmath203 is the flat space minkowski metric . when there is supersymmetry we can extend this to the super poincar algebra . for @xmath148 supersymmetry in @xmath204 dimensions this involves the two - component majorana spinor @xmath205 , @xmath206 , which is the generator of supersymmetry transformations . the algebra is then extended to include also @xmath207 where the @xmath208 are again the dirac matrices . we use conventions such that @xmath205 has two real components . to make connection with the supersymmetry in the ultracold superstring we combine these two components in one complex supersymmetry operator @xmath209 this decomposition breaks manifest lorentz symmetry , but since we are ultimately interested in the nonrelativistic limit , this is of no concern to us here . as a result we obtain the following algebra @xmath210 in particular , we see that the hamiltonian @xmath200 is fixed by the supersymmetry generator . this is a very peculiar restriction on the hamiltonian , which is only true for the relativistic theory . in the nonrelativistic limit , the supersymmetry decouples from the space - time translation symmetry as we show now . the galilei algebra can be derived as a limit of the poincar algebra by performing a inn - wigner contraction @xcite in the following way @xcite . we write @xmath211 where @xmath135 is the speed of light and @xmath212 denotes the mass , which is the same for the bosonic and fermionic degrees of freedom . we also defined a number operator @xmath213 , which counts all the particles in the system , and boost operators @xmath214 . furthermore , we still have the space translation generators @xmath215 and the hamiltonian @xmath216 . we can now take the limit @xmath217 to obtain the super galilei algebra . the galilei algebra obtained in this manner has nonvanishing commutators @xmath218 the part involving the supersymmetry becomes only @xmath219 this defines the algebra @xmath220 @xcite . as is clear , in this case the hamiltonian is decoupled form the supersymmetry . in @xmath204 and @xmath221 dimensions , it is sometimes possible to define an extended superalgebra @xmath222 , which again involves the hamiltonian @xcite . in @xmath223 this amounts to introducing an extra scalar supersymmetry generator @xmath224 with the algebra : @xmath225 the representation for the @xmath226 algebra in terms of the bosonic and fermionic operators @xmath134 and @xmath135 , can easily be found to be @xmath227 in addition , we can thus also define @xmath228 this produces @xmath229 which indeed is the kinetic energy part of the hamiltonian . the full quadratic part of the hamiltonian can be expressed as @xmath230 for completeness , we mention that we can also use superspace techniques to write the hamiltonian in a manifest supersymmetric way . this involves the introduction of a complex superfield @xmath231 where @xmath232 is a grassman variable such that @xmath233 and @xmath234 . the hamiltonian is in terms of the superfield given by @xmath235 in this formulation the spontaneous breaking of supersymmetry is particularly elegant , because the hamiltonian has the form of a standard landau theory of a second - order phase transition with @xmath236 as the order parameter . in this section , we discuss the similarities and differences with superstring theory . for some textbooks on the subject , we refer to refs . in string theory , one usually starts with the polyakov action @xcite , describing the coordinates @xmath237 , with @xmath238 , of the string propagating in a @xmath239-dimensional curved space - time with metric @xmath240 , @xmath241 here @xmath242 are coordinates on the worldsheet sweeped out by the string , @xmath243 is the worldsheet time , and @xmath84 runs longitudinally over the string . furthermore , @xmath244 is the string tension , and @xmath245 is a two - dimensional metric on the worldsheet with @xmath246 . in agreement wit the standard practice in high - energy physics we are momentarily using units such that @xmath247 . we restore units when we come to the precise connection with our ultracold superstring . in fully quantized string theory , one also performs a path integral over these metrics , and this leads to the string loop expansion where one sums over all two - dimensional surfaces containing an arbitrary number of holes . in our setup , the worldsheet of the string is completely fixed , and contains no holes , i.e. , it is just the two - dimensional plane . on the plane , we can then make use of the local symmetries of the polyakov action , that are the reparameterizations of the worldsheet coordinates and the weyl rescalings of the metric . doing so , we can make the gauge choice @xmath248 this gauge choice is referred to as the conformal gauge . the space - time in which the string propagates is coordinatized by @xmath249 . in quantized superstring theory one has that @xmath250 , but at the classical level one can have @xmath251 as well . we come back to this issue below . it is useful to introduce light - cone variables @xmath252 and @xmath253 . then @xmath254 and @xmath255 describe the longitudinal and transversal degrees of freedom of the string , respectively . string theory has the special feature that there are only transversal physical degrees of freedom . this is because string theory has an additional constraint that can be understood as the equation of motion of the worldsheet metric @xmath245 . defining @xmath256 , these constraints read in conformal gauge @xmath257 and are sometimes called the virasoro constraints . in practice , solving the constraints is difficult , but in the so - called light - cone gauge @xmath258 where @xmath259 and @xmath260 is the center - of - mass momentum in the @xmath261 direction , the longitudinal modes @xmath254 can be eliminated explicitly , at least for certain space - time metrics @xmath262 . the light - cone gauge can always be taken as a consequence of the residual gauge symmetry after the gauge choice of eq . has been imposed @xcite . the implementation of the constraints in eq . in the quantum theory leads to the critical dimension , namely @xmath263 for the bosonic string and @xmath250 for the superstring . in our condensed - matter setup , these constraints are not present . there are physical longitudinal degrees of freedom , so this makes it different from the superstring . however , the longitudinal modes are suppressed and at the energy scales we are looking at , it suffices to study only the transversal degrees . it is in this transversal sector that we connect to string theory . to make this connection , we have to specify the space - time metric @xmath262 . a class of backgrounds that has been intensely studied in the string literature is that of plane wave metrics @xcite . the simplest of these backgrounds , and also the one relevant for our case , is given by @xmath264 where @xmath265 is a function of the transverse coordinates only . in light - cone gauge , the lagrangian for the string propagating in this background now becomes @xmath266 - v(x^i)\ , \ ] ] where @xmath267 . to derive this result , we simply substitute the background in eq . into eq . , and use the light - cone gauge from eq . to produce the potential @xmath268 term in the lagrangian . furthermore , this produces a term proportional to @xmath269 that is decoupled from the @xmath255 . therefore this term can be dropped . in fact @xmath269 is fixed by the virasoro constraints in eq . , so we only need a lagrangian for the transverse degrees of freedom . one of the remarkable facts of string theory is that its conformal symmetry at the quantum level forces the metric to satisfy einstein s equations in general relativity . this is the way in which gravity emerges from string theory . when there are no other background fields present , as in our case , einstein s equations reduce to a single constraint on the function @xmath270 given by @xmath271 in other words , @xmath270 has to satisfy the laplace equation in the transverse space . this constraint has to be understood on an equal footing as the constraint on the space - time dimension . they both follow from a consistent implementation of the conformal symmetry at the quantum level . since we are not taking into account the virasoro constraints in our system , and hence the conformal symmetry , we therefore also ignore the constraint in eq .. doing so , we can work with arbitrary potentials @xmath268 . when we take @xmath251 , as we shall below , the scalar potential depends on two real fields . we now include the fermions , and discuss supersymmetry . to make a superstring we have to add additional terms to the lagrangian in eq . containing the fermions in such a way that there is supersymmetry . we can then impose the conformal or light - cone gauges to arrive at a supersymmetric generalization of the lagrangian in eq .. alternatively , we can directly study supersymmetric extensions of eq . as two - dimensional field theories . the general construction of supersymmetric two - dimensional field theories with scalar potentials @xmath268 was given in ref . not all potentials lead to lagrangians that can be supersymmetrized . for the case of minimal supersymmetry with two supercharges , sometimes denoted by ( 1,1 ) susy , the potential needs to be of the following type @xmath272 where @xmath273 is a real function , @xmath274 stands for the derivative with respect to @xmath255 , and the quantities @xmath275 satisfy @xmath276 together with @xmath277 . the supersymmetric lagrangian can then be written as @xmath278 with @xmath279 the supersymmetry variations are @xmath280 and leave the lagrangian invariant , up to a total derivative . here @xmath281 and @xmath282 are two - component majorana spinors , and in our model we thus have two majorana spinors . the @xmath283-matrices are related to the pauli matrices as @xmath284 and @xmath285 as before . for more details on the spinor conventions , see ref . @xcite . examples of supersymmetric models are given by @xmath286 where @xmath287 and @xmath288 are arbitrary parameters . plugging this into eq . leads to to @xmath273 . this leads , however , to terms in the potential @xmath289 with odd powers in @xmath224 , which is not what we are looking for . ] @xmath290 up to an irrelevant additive constant , the coefficients @xmath288 can be chosen such that the potential is as in our condensed - matter setup . furthermore , we have that @xmath291 which leads to mass terms for the fermions , and supersymmetry variations of the fermions of the form @xmath292 this term rotates the fermions into the bosons , just like for the ultracold superstring . if we compute @xmath293 to determine the interactions between bosons and fermions , it produces complicated interaction terms , @xmath294 as a result of the supersymmetry constraints . to connect to our condensed - matter setup , we have to take the nonrelativistic limit in which only particle excitations of the two - dimensional field theory survive , and the anti - particle excitations are absent . to illustrate this procedure , we start with the bosonic part of the lagrangian in eq . , based on two real scalar fields . in terms of the complex field @xmath295 the lagrangian reads @xmath296 where we have reinserted the speed of light @xmath135 in order to take the nonrelativistic limit @xmath297 below , and we further used that the potential is a function of @xmath224 only since this is the case of interest . using now the mass @xmath212 , we decompose the complex scalar field in terms of positive and negative frequency modes @xmath298 and call @xmath134 the particle field and @xmath299 the antiparticle field . both @xmath134 and @xmath299 are complex . we now substitute eq . into eq . and send @xmath297 . in this limit , the lagrangian becomes first order in time derivatives , and particles and antiparticles decouple from each other such that we can effectively set @xmath300 . the remaining terms in the nonrelativistic limit are @xmath301 where we have reinserted the various factors of @xmath302 . moreover , we have absorbed a mass term proportional to @xmath303 into the potential . remind that we have chosen a potential of the form given in eq . , so this mass term can easily be absorbed into a change of the coefficients @xmath304 or @xmath305 . notice that this lagrangian precisely coincides with the bosonic sector of lagrangian of the ultracold superstring given in eq . . the fermionic sector can be obtained in a similar way . in this paper we presented a detailed account of the conditions under which the ultracold superstring can be created . the requirements for the laser parameters and the atomic interactions were given . moreover we payed attention to the experimental signatures of supersymmetry . the supersymmetry in the problem was investigated by studying the appropriate super algebra . finally , a precise mathematical connection with string theory in @xmath149 dimensions was made . the discussions in this article were limited to the case of a single string . it is left for future investigation to extend the analysis to involve more strings . a complication in this case is that for parallel vortex lines , supersymmetry is not possible , because of the different way vortices and fermions interact with each other . a proposal to overcome this problem is to study the interaction of two superstrings that are both in the center of the condensate , but are seperated on the @xmath3 axis . this would correspond to merging and splitting of ultracold superstrings . the typical fermionic number of particles that is needed to obtain supersymmetry is typically around @xmath306 per site . this is rather low , both to control and to observe . however , the density is rather high and can be estimated to be at least @xmath307 . moreover , a disadvantage of a higher fermionic atomic density is that this makes also the contribution of the kelvon - fermion hopping interaction more important . it remains to be investigated , whether a change of the varous parameters can improve on this situation . apart from the other possibilities mentioned in this article , it is also possible to gain experimental insight in the system by coupling the vortex motion to resonant quadrupole modes @xcite . this gives the possibility to measure the kelvon dispersion directly . if the system is brought out of equilibrium by populating a high - lying kelvon mode , it also opens up the exciting possibility to study collapse and revival phenomena between the bosonic and fermionic modes . we are grateful for helpful discussions with masud haque , randy hulet , jan ambjrn and bernard de wit . this work is supported by the stichting voor fundamenteel onderzoek der materie ( fom ) and the nederlandse organisatie voor wetenschappelijk onderzoek ( nwo ) . d. amati and c. klimcik , phys . b * 210 * , 92 ( 1988 ) ; + g. t. horowitz and a. r. steif , phys . lett . * 64 * , 260 ( 1990 ) . j. g. russo and a. a. tseytlin , jhep * 0204 * , 021 ( 2002 ) . g. t. horowitz and a. r. steif , phys . d * 42 * , 1950 ( 1990 ) . l. alvarez - gaume and d. freedman , commun . phys . * 91 * , 87 ( 1983 ) .

the combination of a vortex line in a one - dimensional optical lattice with fermions bound to the vortex core makes up an ultracold superstring . we give a detailed derivation of the way to make this supersymmetric string in the laboratory . in particular , we discuss the presence of a fermionic bound state in the vortex core and the tuning of the laser beams needed to achieve supersymmetry . moreover , we discuss experimental consequences of supersymmetry and identify the precise supersymmetry in the problem . finally , we make the mathematical connection with string theory .

recently , there has been a lot of interest in understanding the scaling behavior in submonolayer island nucleation and growth.@xcite one reason for this is that the submonolayer growth regime plays an important role in determining the later stages of thin - film growth.@xcite of particular interest is the dependence of the total island - density @xmath0 and island - size distribution @xmath1 ( where @xmath2 is the density of islands of size @xmath3 at coverage @xmath4 and @xmath3 is the number of monomers in an island ) on deposition parameters such as the deposition flux @xmath36 and growth temperature @xmath37 . one concept that has proven especially useful in studies of submonolayer epitaxial growth is that of a critical island size,@xcite corresponding to one less than the size of the smallest stable " cluster . for example , if we assume that only monomers can diffuse , then in the case of submonolayer growth of 2d islands on a solid 2d substrate , standard nucleation theory@xcite predicts that the peak island density @xmath38 and the monomer density @xmath39 at fixed coverage satisfy , @xmath40 where @xmath41 is the monomer hopping rate , @xmath42 is the critical island size , @xmath43 and @xmath44 . we note that in the case of irreversible island growth ( @xmath45 ) this implies that @xmath46 and @xmath47 . in addition , it has been shown that in the absence of cluster - diffusion and in the pre - coalescence regime the island - size distribution ( isd ) satisfies the scaling form , @xcite @xmath48 where @xmath27 is the average island size , and the scaling function @xmath49 depends on the critical island size.@xcite however , in some cases ( such as in epitaxial growth on metal(111 ) surfaces ) it is also possible for significant _ small _ cluster diffusion to occur.@xcite in addition , several mechanisms for the diffusion of _ large _ clusters on solid surfaces have also been proposed . @xcite in each case , scaling arguments predict that the cluster diffusion coefficient @xmath5 decays as a power - law with island - size @xmath3 ( where @xmath3 is the number of particles in a cluster ) , i.e. @xmath50 . in particular , three different limiting cases have been considered@xcite - cluster diffusion due to uncorrelated evaporation - condensation ( @xmath7 ) , cluster diffusion due to correlated evaporation / condensation ( @xmath51 ) , and cluster diffusion due to periphery diffusion ( @xmath9 ) . we note that the case @xmath7 also corresponds to the brownian ( stokes - einstein ) diffusion of compact 2d clusters in two - dimensions . in order to understand the effects of island diffusion on the submonolayer scaling behavior a number of simulations have previously been carried out . for example , jensen et al@xcite have studied the effects of island - diffusion with @xmath51 on the percolation coverage for the case of irreversible growth without relaxation , corresponding to islands with fractal dimension @xmath52 . more recently , mulheran and robbie@xcite have used a similar model to study the dependence of the exponent @xmath13 on the cluster - diffusion exponent @xmath15 for values of @xmath15 ranging from @xmath53 to @xmath54 . they found that for small values of @xmath15 the value of the exponent ( @xmath55 ) is significantly larger than the value ( @xmath56 ) expected in the absence of cluster diffusion , although it decreases with increasing @xmath15 . however , the scaling of the isd was not studied.@xcite motivated in part by these simulations , krapivsky et al@xcite have carried out an analysis of the scaling behavior for the case of point - islands , based on the corresponding mean - field smoluchowski equations.@xcite their analysis suggests that due to the large amount of diffusion and coalescence in this case , for @xmath19 the total island density saturates ( corresponding to steady - state " behavior ) while the isd exhibits power - law behavior of the form , @xmath57 , where @xmath58 and the prefactor does not depend on coverage . has also been derived by cueille and sire@xcite and camacho.@xcite ] this power - law dependence for the isd is predicted to hold up to a critical island - size @xmath22 , where @xmath28 and @xmath59 . in contrast , for @xmath32 continuous island evolution is predicted , e.g. the total island density does not saturate , and as a result no simple power - law behavior is predicted for the isd . their analysis also indicates that for all values of @xmath15 , one has @xmath60 with logarithmic corrections . however , it should be noted that the point - island approximation is typically only valid at extremely low coverages . here we present the results of kinetic monte carlo simulations of irreversible island growth with cluster diffusion for the case of compact islands with fractal dimension @xmath61 . among the primary motivations for this work are recent experiments@xcite on the growth of ( compact ) colloidal nanoparticle islands at a liquid - air interface in which significant cluster diffusion has been observed . accordingly , in contrast to much of the previous work@xcite our model is an off - lattice model . however , our main goal here is not to explain these experiments but rather to obtain results which may be used as a reference for future work . as already noted , if cluster diffusion is due to 2d brownian motion ( as might be expected at a fluid - interface ) then the value of the exponent @xmath15 ( @xmath7 ) is the same as that expected for uncorrelated evaporation - condensation . however , we also present results for @xmath51 ( corresponding to cluster - diffusion due to correlated evaporation - condensation ) , @xmath9 ( corresponding to cluster - diffusion due to periphery diffusion ) as well as for higher values of @xmath15 ( @xmath62 and @xmath63 ) . this paper is organized as follows . in sec . ii , we describe our model in detail along with the parameters used in our simulations , while in sec . iii we discuss the methods we have used to enhance the simulation efficiency . in sec . iv we derive a generalized scaling form for the isd which is appropriate for the case of a power - law isd with @xmath64 , corresponding to @xmath19 . we then present our results for the scaling of the island - size distribution and island and monomer densities as a function of @xmath65 , coverage , and @xmath15 in sec . v. finally , in sec . vi we discuss our results . for simplicity we have studied a model of irreversible aggregation in which all islands are assumed to be circular and rapid island relaxation ( perhaps due to periphery diffusion ) is assumed . in particular , in our model each island or cluster of size @xmath3 ( where @xmath3 is the number of monomers in a cluster ) is represented by a circle with area @xmath66 and diameter @xmath67 , where @xmath68 is the monomer diameter . in addition , each cluster of size @xmath3 may diffuse with diffusion rate @xmath69 where @xmath70 is the monomer diffusion rate , @xmath41 is the monomer hopping rate " , and @xmath71 is the hopping length . similarly , we may write @xmath72 where @xmath73 is the hopping rate for a cluster of size @xmath3 . in order to take into account deposition , monomers are also randomly deposited onto the substrate with rate @xmath74 per unit time per unit area . since instantaneous coalesce and relaxation is assumed , whenever two clusters touch or overlap , a new island is formed whose area is equal to the sum of the areas of the original clusters , and whose center corresponds to the center - of - mass of both islands . we note that in some cases a coalescence event may lead to overlap of the resulting cluster with additional clusters . in this case , coalescence is allowed to proceed until there are no more overlaps . in addition , if a monomer lands on an existing cluster , then that monomer is automatically ` absorbed ' by the cluster . thus , at each step of our simulation either a monomer is deposited ( followed by a check for overlap with any clusters ) or a cluster is selected for diffusion . if a cluster is selected for diffusion , then the center of the cluster is displaced by a distance @xmath71 in a randomly selected direction . for computational efficiency , and also because it is the smallest length - scale in the problem , in most of the results presented here we have assumed @xmath75 . however , we have also carried out some simulations with smaller values ( @xmath76 and @xmath77 ) in order to approach the continuum limit . as discussed in more detail in sec . vi , our results indicate that the dependence of the island and monomer densities on the hopping distance @xmath71 is relatively weak . we note that besides the exponent @xmath15 describing the dependence of the cluster diffusion rate on cluster - size , the other key parameter in our simulations is the ratio @xmath78 of the monomer hopping rate to the monomer deposition rate ( scaled by the ratio of the hopping length to the monomer diameter ) e.g. , @xmath79 we note that this definition implies that the dimensionless ratio @xmath80 of the monomer diffusion coefficient @xmath81 to the deposition flux satisfies , @xmath82 our simulations were carried out assuming a 2d square substrate of size @xmath83 ( in units of the monomer diameter @xmath68 ) and periodic boundary conditions . in order to avoid finite - size effects , the value of @xmath83 used ( @xmath84 ) was relatively large , while our results were averaged over @xmath85 runs in order to obtain good statistics . in order to determine the asymptotic dependence of the island density on coverage and @xmath78 our simulations were carried out using values of @xmath86 ranging from @xmath87 up to a maximum coverage of @xmath88 monolayers ( ml ) . in order to study the dependence on @xmath15 , simulations were carried out for @xmath7 ( corresponding to brownian diffusion or uncorrelated evaporation - condensation ) , @xmath51 ( corresponding to correlated evaporation - condensation ) , and @xmath89 ( corresponding to periphery diffusion ) as well as for higher values ( @xmath10 and @xmath11 ) as well as the case @xmath12 corresponding to only monomer diffusion . in order to obtain a quantitative understanding of the submonolayer growth behavior , we have measured a variety of quantities including the monomer density @xmath90 ( where @xmath91 is the number of monomers in the system ) as a function of coverage @xmath4 , and the total island density @xmath92 ( where @xmath93 is the total number of islands including monomers in the system ) . in addition , we have also measured the island - size distribution @xmath1 where @xmath94 corresponds to the density of islands of size @xmath3 . we note that the factors of @xmath95 in the definitions above take into account the fact that the area of a monomer is @xmath96 , and as a result the densities defined above all correspond to area fractions . similarly , the coverage @xmath97 corresponds to the fraction of the total area covered by islands ( including monomers ) . while a simple monte carlo approach can be used@xcite to simulate the processes of monomer deposition and cluster diffusion such a method can be very inefficient for large values of @xmath78 and small values of @xmath15 , since the large range of island - sizes and diffusion rates can lead to a low acceptance ratio . accordingly , here we use a kinetic monte carlo approach . in particular , if we set the deposition rate @xmath36 per unit area @xmath98 equal to @xmath99 , then the total deposition rate in the system is @xmath100 while the hopping rate for a cluster of size @xmath3 is given by @xmath101 . as a result , the total diffusion rate for all clusters is given by @xmath102 ( where @xmath103 is the number of clusters of size @xmath3 ) while the total rate of deposition onto the substrate is @xmath100 . the probability @xmath104 of depositing a monomer is then given by , @xmath105 while the probability of cluster diffusion is @xmath106 . if cluster diffusion is selected , then a binary tree@xcite ( whose bottom leaves correspond to the total hopping rate @xmath107 for each size @xmath3 ) may be used to efficiently select with the correct probability which cluster will move as well as to efficiently update @xmath108 . however , for large @xmath78 and small @xmath15 the maximum cluster - size can be larger than @xmath109 and as a result the computational overhead associated with the binary tree can still be significant . accordingly , we have implemented a variation@xcite of the binary tree approach in which a range of cluster - sizes are clustered together into a single ` leaf ' or bin . in particular , to minimize the size of the binary tree , starting with island - size @xmath110 we have used variable bin - sizes such that each bin contains several different cluster sizes ranging from a starting value @xmath42 to a value approximately equal to @xmath111 . using this scheme allows us to use a binary tree with a maximum of @xmath112 leaves and a rejection probability of only @xmath113 . to further decrease the computational overhead , our binary tree grows dynamically from @xmath114 leaves to as many as needed . by properly selecting the rates in the binary tree and the corresponding acceptance probabilities , one can ensure that each diffusion event is selected with the proper rate . in particular , if we define the rate of bin @xmath42 as @xmath115 , where @xmath116 is the maximum cluster - diffusion rate in bin @xmath117 ( corresponding to the smallest cluster - size in the bin ) and @xmath118 is the number of islands in the bin , then the sum over all leaves may be written , @xmath119 the probability of attempting a diffusion event is then given by , @xmath120 while the probability of selecting bin @xmath42 is given by @xmath121 . once a bin is selected using the binary tree , a specific cluster is then selected randomly from the list of all the clusters in that bin . this implies that a cluster of size @xmath3 will be selected with probability @xmath122 . thus , by assuming an acceptance probability for the selected cluster - diffusion event given by @xmath123 each diffusion event will be selected with the proper rate . since our simulations are carried out off - lattice , one of the most time - consuming processes is the search for overlaps every time a cluster is moved . while the simplest way to carry out such a search is to check for overlaps with all other islands in the system , the search time scales as @xmath100 , and as a result it becomes very time - consuming for large systems . accordingly , we have used a neighbor look - up table@xcite which contains a list of all other islands within a buffer - distance of each island . the search for overlaps is then carried out only among the neighbors on this list rather than over all the islands in the system . the neighbor list is updated whenever the total displacement of any island since the last update is larger than half the buffer - distance . to speed - up the updates of the neighbor table , we have also used a `` grid '' method@xcite in which our system is divided into an @xmath124 by @xmath124 grid of boxes of size @xmath125 and each cluster can be rapidly assigned to a given box . using this method the search for neighbors only includes clusters within an island s box as well as the @xmath126 adjacent boxes . as a result , the table update time is reduced to @xmath127 instead of @xmath100 . to further optimize the speed of our simulations , the grid size is varied as the average island - size increases . as discussed in sec . i , in both simulations and experiments on submonolayer epitaxial growth , the island - size distribution ( isd ) is typically assumed to satisfy the scaling form given in eq . [ isdscal ] . however , this scaling form has been derived@xcite on the assumption that there is only one characteristic size - scale @xmath27 corresponding to the average island - size , and that the isd does not diverge for small @xmath128 . however , in our simulations of monomer deposition and cluster diffusion and aggregation with @xmath19 , we find that the isd exhibits a well - defined power - law behavior for small @xmath128 . in addition , the existence of a shoulder in the isd for large @xmath129 implies the existence of a second characteristic length - scale which scales as @xmath28 . we note that this corresponds to an island size - scale such that steady - state behavior breaks down , due to the existence of mass - conservation and a finite diffusion length . in general one would expect this to lead to a more complicated two - variable scaling of the form @xmath130 . however , if the power - law behavior for small @xmath128 is well - defined ( and @xmath64 ) then it is possible to derive a generalized scaling form involving only one variable . in particular , we assume that a scaling form for the island - size distribution may be written , @xmath131 in order to determine @xmath132 note that @xmath133 . converting to an integral this may be rewritten as @xmath134 where @xmath135 . if we now assume that @xmath136 for small @xmath137 and @xmath64 , then the small-@xmath137 part of the integral dominates and we obtain , @xmath138 . this leads to the generalized scaling form , @xmath139 we note that a similar scaling form ( corresponding to the special case @xmath140 ) has previously been derived in ref . for the case of the deposition of spherical droplets with dimension @xmath141 on a @xmath142-dimensional substrate . we also note that for @xmath140 and @xmath143 ( corresponding to the critical value of @xmath23 ) the standard scaling form eq . [ isdscal ] is obtained . we first consider the case @xmath31 corresponding to stokes - einstein diffusion . [ fig : dens05](a ) shows our results for the total cluster density @xmath0 ( including monomers ) as well as for the monomer density @xmath39 as a function of coverage for three different values of @xmath144 ranging from @xmath145 to @xmath146 . in good agreement with the theoretical prediction in refs . and of steady - state " behavior for @xmath19 , we find that both the monomer density @xmath39 and total island density @xmath0 reach an approximately constant value beyond a critical coverage @xmath17 . we note that this coverage decreases with increasing @xmath78 , while the peak island and monomer densities also decrease with increasing @xmath78 . the inset in fig . [ fig : dens05](b ) shows our results for the exponents @xmath13 ( @xmath147 ) and @xmath16 ( @xmath148 ) corresponding to the dependence of the peak island density @xmath149 and coverage @xmath17 on @xmath144 . in qualitative agreement with the results of mulheran et al@xcite for fractal islands , the value of @xmath13 obtained in our simulations is slightly lower but close to @xmath150 . this is also consistent with the prediction@xcite that for point - islands @xmath13 should be equal to @xmath150 with logarithmic corrections . [ fig : dens05](b ) shows the corresponding scaled island density @xmath151 as a function of the scaled coverage @xmath152 . as can be seen there is good scaling up to and even somewhat beyond the value ( @xmath153 ) corresponding to the peak in the island - density . in contrast , replacing the scaled coverage by @xmath154 as in ref . , leads to good scaling at @xmath155 , but the scaling is significantly worse for @xmath156 . also shown is the scaled monomer density @xmath157 ( where the peak monomer density scales as @xmath158 and the coverage corresponding to the peak monomer density scales as @xmath159 and @xmath160 ) as a function of the scaled coverage @xmath161 . as for the case of the island density , there is good scaling up to and even beyond the scaled coverage corresponding to the peak of the monomer density . we note that in contrast to the exponents @xmath13 and @xmath16 , the exponent @xmath162 does not appear to depend on @xmath15 . in particular , we find that for all the values of @xmath15 that we have studied , the value of @xmath162 ( @xmath163 ) is close to the value ( @xmath164 ) expected in the absence of cluster - diffusion . we now consider the scaled island - size distribution ( isd ) . in refs . and steady - state " power - law behavior of the form , @xmath165 where @xmath58 was predicted for @xmath19 for island - sizes @xmath166 where @xmath22 corresponds to the shoulder in the isd for large @xmath3 . similarly , the exponent @xmath26 characterizing the scaling of @xmath22 as a function of @xmath27 ( e.g. @xmath28 ) was predicted to satisfy the expression @xmath167 . we note that for @xmath168 these expressions imply that @xmath169 and @xmath170 . since @xmath171 and @xmath172 , one has @xmath173 . accordingly , eq . [ steadystate ] may be rewritten as , @xmath174 fig . [ fig : isd05](a ) shows the isd scaled using this form . as can be seen there is reasonably good scaling for @xmath175 , although the tail of the distribution does not scale . however , the measured value of the exponent @xmath23 ( @xmath24 ) is significantly higher than the predicted value . in addition , the measured value of @xmath26 ( @xmath176 ) is also significantly higher than the predicted value . [ fig : isd05](b ) shows the corresponding scaling results obtained using the generalized scaling form eq . [ ns2q ] and assuming @xmath177 and @xmath178 . we note that this implies that , @xmath179 as can be seen , in this case both the power - law region for small @xmath128 as well as the ` bump ' for large @xmath128 scale well using this form . we note however , that for the smallest clusters ( e.g. monomers and dimers ) there is poor scaling due to deviations from power - law behavior for small @xmath3 . and @xmath39 as a function of coverage @xmath4 for @xmath180 and @xmath7 . ( b ) scaled densities @xmath181 and @xmath182 as a function of scaled coverage ( @xmath183 and @xmath184 , respectively ) . inset shows dependence of peak island density @xmath149 and coverage @xmath17 on @xmath144.,width=283 ] obtained using steady - state scaling form eq . [ steadystate2 ] . ( b ) scaled isd obtained using generalized scaling form eq . [ ns2q ] with @xmath185 and @xmath177 . , width=283 ] we now consider the case @xmath186 which corresponds to cluster diffusion via correlated attachment - detachment . we note that this is the critical value for power - law behavior of the isd ( which is expected to occur for @xmath187 ) and as a result krapivsky et al@xcite have predicted nested " logarithmic behavior for the island - density . since the simulations are not as computationally demanding as for @xmath7 , in this case we have carried out simulations up to @xmath188 . [ fig : dens1](a ) shows our results for the total island density @xmath0 and monomer density @xmath39 as a function of coverage for @xmath189 . as can be seen , while there is a plateau in the island - density which appears to broaden and flatten somewhat with increasing @xmath190 , the plateau is not as flat as for the case @xmath7 , thus indicating deviations from steady - state behavior . as for the case @xmath191 , a plot of the scaled densities @xmath181 ( @xmath182 ) as a function of scaled coverage @xmath183 ( @xmath184 ) shows relatively good scaling up to the coverage corresponding to the peak island - density , although the value of @xmath13 ( @xmath55 ) is slightly lower than that obtained for @xmath7 . we now consider the island - size distribution . as shown in fig . [ fig : dens1](b ) , in this case the isd does not exhibit a well - defined power - law behavior . in particular , on a log - log plot the isd is curved with a slope @xmath192 for small @xmath3 and a smaller effective slope ( @xmath193 ) for large @xmath3 . similarly , while @xmath194 its effective value ranges from @xmath195 to @xmath196 depending on the value of @xmath144 and coverage . as a result , neither the standard scaling form eq . [ isdscal ] nor the generalized scaling form eq . [ ns2q ] can be used to scale the entire island - size distribution . however , using the generalized scaling form ( [ ns2q ] ) with @xmath194 and @xmath197 , we find good scaling for small @xmath128 ( see fig . [ fig : dens1](b ) ) , although the isd does not scale for large @xmath128 . on the other hand , if we use the standard scaling form ( [ isdscal ] ) ( which corresponds to the generalized scaling form with @xmath140 and @xmath143 , see inset of fig . [ fig : dens1](b ) ) then the isd scales for @xmath198 but not for small @xmath3 . we note that this lack of scaling is perhaps not surprising since for @xmath32 there are two characteristic size - scales @xmath27 and @xmath22 , but no well - defined power - law behavior . and @xmath39 as a function of coverage @xmath4 for @xmath180 and @xmath51 . ( b)scaled isd for @xmath51 using generalized scaling form ( [ ns2q ] ) with @xmath140 and @xmath197 . results correspond to coverages @xmath199 , @xmath180 and @xmath200 . inset shows corresponding scaling results obtained using the standard scaling form ( [ isdscal ] ) . , width=283 ] we now consider the case @xmath89 which corresponds to cluster diffusion via edge - diffusion . [ fig : dens32](a ) shows our results for the total island density @xmath0 and monomer density @xmath39 as a function of coverage for @xmath201 . as can be seen , while there is a plateau in the island - density which appears to broaden with increasing @xmath190 , it is not as flat as for the case @xmath51 , thus indicating deviations from steady - state behavior . as for the case @xmath200 , a plot of the scaled densities @xmath181 ( @xmath182 ) as a function of scaled coverage @xmath183 ( @xmath184 ) shows relatively good scaling up to the coverage corresponding to the peak island - density . we note that for @xmath202 , krapivsky et al@xcite have predicted that for point - islands there is a continuous logarithmic increase in the total island density of the form , @xmath203^{\mu/2 } \label{sinequ}\ ] ] however , we find that for @xmath9 and higher ( not shown ) scaling plots using this form ( e.g. @xmath204 as a function of @xmath205^{\mu/2}$ ] ) provide very poor scaling . in particular , since @xmath55 , the scaled peak island - density increases with @xmath35 while the peak position also shifts significantly to smaller values . we now consider the scaled isd for @xmath9 . again in this case , it is not possible to scale the entire isd using the average island - size @xmath27 since there are two characteristic size - scales but no well - defined power - law behavior . in particular , if we use the generalized scaling form eq . [ ns2q ] with @xmath197 and @xmath140 , then reasonable scaling is only obtained for the small-@xmath3 tail " corresponding to @xmath206 ( not shown ) . in addition , as shown in fig . [ fig : dens32](b ) , using the standard scaling form eq . [ isdscal ] neither the tail nor the peak scale . we note that the height and width of the power - law " portion of the isd decreases with increasing @xmath190 and coverage , while the peak near @xmath207 becomes higher and sharper . as a result , the power - law portion of the isd is significantly less important than for smaller values of @xmath15 . in particular , for @xmath208 and @xmath209 , it corresponds to only approximately @xmath113 of the area under the curve . and @xmath39 as a function of coverage @xmath4 for @xmath180 and @xmath9 . ( b ) scaled isd for @xmath9 using standard scaling form ( [ ns2q ] ) with @xmath140 and @xmath197([isdscal]).,width=283 ] fig . [ fig : picture ] shows pictures of the submonolayer morphology for @xmath210 and @xmath209 for @xmath211 , and @xmath212 . we note that the size - scale @xmath213 of each picture decreases with increasing @xmath15 so that approximately the same number of islands is visible . as can be seen , in qualitative agreement with our results , there is a very broad distribution of island - sizes for @xmath7 while the distribution becomes narrower with increasing @xmath15 . ) of the submonolayer morphology at coverage @xmath209 and @xmath208 for ( a ) @xmath7 ( m = 4096 ) ( b ) @xmath51 ( m = 709 ) ( c ) @xmath9 ( m = 624 ) ( d ) @xmath214 ( m = 485).,width=283 ] in order to obtain a better understanding of the dependence of the island density and isd on the mobility exponent @xmath15 , we have also carried out additional simulations for larger values of @xmath15 ( @xmath216 and @xmath11 ) as well as in the limit @xmath12 in which only monomers can diffuse . [ fig : isd2](a ) shows the corresponding results for the scaled isd for @xmath217 using the standard scaling form eq . [ isdscal ] for different values of the coverage @xmath4 and @xmath144 . as for the case @xmath9 the isd does not scale , although the power - law " portion for small @xmath128 is significantly reduced . instead the peak of the scaled isd increases with increasing coverage and @xmath144 . we also note that for @xmath210 and @xmath209 , the peak height is significantly higher than for @xmath218 while the peak position is closer to @xmath219 . similar results for the scaled isd for @xmath220 are shown in fig . [ fig : isd2](b ) , although in this case it tends to sharpen more rapidly with increasing @xmath144 and coverage . these results also suggest that , while the scaled isd _ may _ approach a well - defined form ( independent of coverage and @xmath144 ) in the asymptotic limit of large @xmath144 , the corresponding scaling function depends on @xmath15 . such a @xmath15-dependence is consistent with the dependence of the exponent @xmath13 and @xmath14 on @xmath15 ( see fig . [ fig : allchi ] ) . and coverage @xmath221 for ( a ) @xmath222 and ( b ) @xmath220 . , width=283 ] fig . [ fig : isd6 ] shows our results for the scaled isd for @xmath223 as well as in the limit @xmath224 in which only monomers can diffuse . somewhat surprisingly , we find that for @xmath223 the scaled isd is significantly broader than for @xmath225 and @xmath220 , although it is still more sharply - peaked than for @xmath12 . these results suggest that , at least for ( finite ) fixed @xmath144 , the peak - height depends non - monotonically on @xmath15 , e.g. it increases from @xmath9 to @xmath220 but then decreases for higher @xmath15 . this is also consistent with our results for @xmath12 ( see fig . [ fig : isd6](b ) ) for which good scaling is observed but with a peak height which is lower than for @xmath223 . and coverage @xmath221 for ( a ) @xmath226 and ( b ) @xmath12.,width=283 ] fig . [ fig : allchi](a ) shows a summary of our results for the monomer density @xmath39 and total island density @xmath0 as a function of coverage for @xmath227 , and @xmath63 for the case @xmath210 . as can be seen , up to the coverage @xmath228 corresponding to the peak monomer density both the island and monomer density are essentially independent of @xmath15 . [ fig : allchi](a ) also shows clearly that both the island - density and the coverage @xmath17 corresponding to the peak island - density increase with increasing @xmath15 , while the monomer density decreases with increasing @xmath15 . [ fig : allchi](b ) shows a summary of our results for the dependence of the exponents @xmath13 , @xmath14 , and @xmath162 on @xmath15 . as can be seen , the exponent @xmath13 depends continuously on @xmath15 , decreasing from a value close to @xmath150 for small @xmath15 ( @xmath7 ) and approaching a value close to @xmath229 for large @xmath15 . we note that these results are similar to previous results obtained for fractal islands with @xmath230 by mulheran and robbie.@xcite similarly , we find that the exponent @xmath14 describing the dependence of the monomer density at fixed coverage on @xmath144 also shows a continuous variation with increasing @xmath15 , starting at a value close to @xmath150 for @xmath7 and increasing to a value close to @xmath231 for large @xmath15 . in contrast , the exponent @xmath162 describing the flux - dependence of the peak monomer density is close to @xmath150 for all @xmath15 . motivated in part by recent experiments on colloidal nanoparticle island nucleation and growth during droplet evaporation,@xcite we have carried out simulations of a simplified model of irreversible growth of compact islands in the presence of monomer deposition and a power - law dependence ( @xmath50 ) of the island mobility @xmath5 on island - size @xmath3 . in particular , we have considered the cases @xmath7 ( corresponding to cluster - diffusion via brownian motion ) , @xmath51 ( corresponding to cluster - diffusion via correlated evaporation - condensation ) , and @xmath9 ( corresponding to cluster - diffusion via periphery diffusion ) . for comparison , we have also carried out simulations for higher values of @xmath15 including @xmath216 and @xmath11 as well as @xmath12 . in agreement with the predictions of ref . and ref . for point - islands , we find that for small values of @xmath15 the value of the exponent @xmath13 characterizing the dependence of the peak - island density on @xmath144 is close to but slightly lower than @xmath150 . however , we also find that @xmath13 decreases continuously with increasing @xmath15 , approaching the value @xmath229 for large @xmath15 . as already noted , these results are in good agreement with previous results obtained for fractal islands.@xcite similarly , the exponent @xmath14 characterizing the dependence of the peak monomer density on @xmath144 is also close to @xmath232 for small @xmath15 , but increases with increasing @xmath15 , approaching the value @xmath231 in the limit @xmath233 . in contrast , the exponent @xmath16 describing the dependence of the coverage @xmath17 ( corresponding to the peak - island density ) on @xmath144 is significantly smaller than @xmath150 for small @xmath15 and also decreases with @xmath15 , approaching zero in the limit of infinite @xmath15 . this is consistent with the fact that when only monomers are mobile ( @xmath12 ) the peak island - density occurs at a coverage which is independent of @xmath144 in the asymptotic limit of large @xmath144 . for comparison , we note that while the monomer density @xmath234 depends on @xmath144 it only depends on @xmath15 for coverages _ beyond _ the peak monomer density ( see fig . [ fig : allchi](a ) ) . as a result , the exponents @xmath162 and @xmath18 corresponding to the dependence of the peak monomer density ( and corresponding coverage @xmath228 ) on @xmath144 are close to @xmath150 for all @xmath15 . the similarity of our results for @xmath13 and @xmath16 to previous results@xcite for fractal islands suggests that these exponents ( along with the exponent @xmath14 ) depend primarily on the cluster - mobility exponent @xmath15 and substrate - dimension @xmath142 but not on the shape or fractal dimension of the islands . we note that such a result is not entirely surprising , since for the case in which only monomers can diffuse ( @xmath12 ) it has been found that the exponent @xmath13 depends only weakly on the island fractal dimension.@xcite in addition , we have found that the scaled island and monomer densities ( @xmath235 and @xmath236 ) lead to reasonably good scaling as a function of @xmath237 , up to and somewhat beyond the peak island - density . we note that this scaling form is somewhat different from that used in ref . in which the coverage is scaled by @xmath238 so that only the peak scales . in addition to the scaling of the island and monomer densities , we have also studied the dependence of the island - size distribution ( isd ) on the cluster - mobility exponent @xmath15 . in agreement with the prediction@xcite that for point - islands well - defined power - law behavior should be observed for @xmath19 , for the case @xmath7 we find a broad distribution of island - sizes with a well - defined power - law . however , in contrast to the point - island prediction that @xmath58 ( which implies @xmath169 for @xmath7 ) the value of @xmath23 obtained in our simulations ( @xmath24 ) is somewhat larger . similarly , the value of the exponent ( @xmath176 ) describing the dependence of the crossover island - size @xmath22 on @xmath27 for @xmath7 is also significantly larger than the point - island prediction @xmath240 . one possible explanation for this is that for compact islands the coalescence rate decreases more slowly with increasing island - size than for point - islands due to the increase in aggregation cross - section " with increasing island - radius . however , another possible explanation is the existence of correlations that are not included in the mean - field smoluchowski equations . in particular , we note that in previous work for the case of irreversible growth in the absence of cluster diffusion ( @xmath12 ) , it has been shown@xcite that there exist strong correlations between the size of an island and the surrounding capture - zone . ( not including monomers ) and monomer density @xmath39 for @xmath241 , @xmath242 , and continuum limit corresponding to @xmath243 . inset shows dependence of peak monomer density @xmath39 on @xmath71.,width=283 ] we note that in contrast to previously studied growth models with only limited cluster - diffusion,@xcite in which there is a single well - defined peak in the isd corresponding to the average island - size @xmath27 , in the presence of significant cluster mobility there are typically two different size - scales @xmath27 and @xmath22 . as a result , in general it is not possible to scale the isd using just the average island - size @xmath27 . however , for the case @xmath19 ( corresponding to well - defined power - law behavior up to a critical island - size @xmath22 ) our results confirm that for compact islands the isd exhibits steady - state behavior . as a result , the power - law region corresponding to @xmath175 can be scaled using eq . [ steadystate2 ] , although the large-@xmath3 tail " does not scale . accordingly , we have proposed a generalized scaling form for the isd , @xmath30 for the case @xmath19 . using this form , we have obtained excellent scaling for the case @xmath7 . in contrast for @xmath51 , there are still two competing size - scales @xmath27 and @xmath22 , but there is no well - defined power - law behavior . as a result , no single scaling form can be used to scale the entire isd . however , we find that the value of the exponent @xmath26 ( @xmath194 ) is close to that obtained using the point - island expression @xmath59 . in addition , for small @xmath128 the isd satisfies @xmath244 where @xmath245 . as a result , we find that the small @xmath128 tail " of the isd can be scaled using the generalized scaling form eq . [ ns2q ] with @xmath197 and @xmath140 , while the standard scaling form eq . [ isdscal ] leads to reasonably good scaling of the isd for @xmath198 . however , for @xmath202 there is no effective power - law behavior and as a result , neither the general scaling form eq . [ ns2q ] nor the standard scaling form eq . [ isdscal ] lead to good scaling of the isd for finite @xmath144 . instead we find that , using the standard scaling form eq . [ isdscal ] the fraction of islands corresponding to small @xmath128 decreases with increasing @xmath144 , and coverage , while the peak of the scaled isd increases in height and becomes sharper . as a result , the peak position shifts to the left with increasing @xmath144 and coverage and appears to approach @xmath99 for large @xmath144 . interestingly , this implies , as shown in figs . [ fig : dens32](b ) , [ fig : isd2 ] , and [ fig : isd6 ] , that for @xmath202 the peak of the scaled isd is even higher than for the case of irreversible growth without cluster diffusion ( @xmath12 ) . however , our results also suggest that , at least for fixed coverage and finite ( fixed ) @xmath144 , the peak - height of the scaled isd exhibits a non - monotonic dependence on @xmath15 , since it increases from @xmath9 to @xmath220 but is smaller for @xmath223 . it is also interesting to compare our results for @xmath202 with those obtained by kuipers and palmer@xcite who studied the scaled isd for the case of fractal islands , assuming an exponential dependence of the cluster mobility , e.g. @xmath246 where @xmath247 . because of the rapid decay of the mobility with increasing cluster - size assumed in their model , the resulting scaled island - size distributions ( using the standard scaling form eq . [ isdscal ] ) were much closer to those obtained for the case of irreversible growth with no cluster mobility ( e.g. @xmath12 ) than the results presented here . however , for values of @xmath248 which were not too small , they also found some evidence of a small island - size tail " , although it was much weaker than found here . it is also interesting to consider the applicability of the model studied here to recent experiments by bigioni et al@xcite for the case of colloidal nanoparticle cluster formation during drop - drying . we note that in this case , one expects that clusters will diffuse on the droplet surface via brownian motion which implies that @xmath7 . however , one also expects that , due to the relatively weak van der waals attraction between nanoparticles , in this case cluster formation may be reversible . accordingly , it would be interesting to carry out additional simulations for the case of reversible growth corresponding to a critical island - size @xmath249 . finally , we consider the continuum limit of our simulations . as already mentioned , while our simulations are off - lattice , in all of the results presented so far we have assumed a hopping length @xmath71 equal to the monomer diameter @xmath68 . we note that this makes our simulations similar to previous simulations@xcite with and without cluster mobility in which a lattice was assumed . however , it is also interesting to consider the continuum limit @xmath250 . in order to do so , we have carried out additional simulations with smaller values of @xmath71 ( @xmath251 and @xmath252 ) . in general , we find that both the monomer density @xmath39 , as well as the density @xmath253 of all clusters not including monomers exhibit a weak but linear dependence on the hopping length @xmath71 ( see inset of fig . [ fig : delta](b ) ) e.g. , @xmath254\ ] ] ( where @xmath255 corresponds either to the monomer or island density and @xmath256 corresponds to the continuum limit ) . accordingly , by performing a linear extrapolation we may obtain the corresponding densities in the continuum limit . as shown in fig . [ fig : delta ] , for @xmath20 and @xmath9 the island - density @xmath253 depends relatively weakly on the hopping length , and as a result there is very little difference between our results for @xmath75 and the continuum limit . in contrast , the monomer density exhibits a somewhat stronger dependence on the hopping length @xmath257 . however , in general we find @xmath258 while the value of @xmath259 decreases with increasing @xmath15 . in particular , in the limit @xmath12 in which only monomers can diffuse , we find @xmath260 ( @xmath261 ) for the island and monomer density respectively . these results indicate that in the continuum limit the island and monomer densities are only slightly lower than in our simulations . accordingly , we expect that in the continuum limit the scaling behavior will not be significantly different from the results presented here . this work was supported by air force research laboratory , space vehicles directorate ( contract no . fa9453 - 08-c-0172 ) as well as by nsf grant che-1012896 . we would also like to acknowledge a grant of computer time from the ohio supercomputer center .

the effects of cluster diffusion on the submonolayer island density @xmath0 and island - size distribution @xmath1 ( where @xmath2 is the density of islands of size @xmath3 at coverage @xmath4 ) are studied for the case of irreversible growth of compact islands on a 2d substrate . in our model , we assume instantaneous coalescence of circular islands , while the mobility @xmath5 of an island of size @xmath3 ( where @xmath3 is the number of particles in an island ) satisfies @xmath6 . results are presented for @xmath7 ( corresponding to brownian motion ) , @xmath8 ( corresponding to correlated evaporation - condensation ) , and @xmath9 ( corresponding to cluster diffusion via edge - diffusion ) , as well as for higher values including @xmath10 and @xmath11 . we also compare our results with those obtained in the limit of no cluster mobility ( @xmath12 ) . in general , we find that the exponents @xmath13 and @xmath14 describing the flux - dependence of the island and monomer densities respectively , vary continuously as a function of @xmath15 . similarly , the exponent @xmath16 describing the flux - dependence of the coverage @xmath17 corresponding to the peak island - density also depends continuously on @xmath15 , although the exponent @xmath18 describing the flux - dependence of the coverage corresponding to the peak monomer density does not . in agreement with theoretical predictions that for point - islands with @xmath19 power - law behavior of the island - size distribution ( isd ) is expected , for @xmath20 we find @xmath21 up to a cross - over island - size @xmath22 . however , the value of the exponent @xmath23 obtained in our simulations ( @xmath24 ) is higher than the point - island prediction @xmath25 . similarly , the measured value of the exponent @xmath26 corresponding to the dependence of @xmath22 on the average island - size @xmath27 ( e.g. @xmath28 ) is also significantly higher than the point - island prediction @xmath29 . for @xmath19 , a generalized scaling form for the isd , @xmath30 , is also proposed , and using this form excellent scaling of the entire distribution is found for @xmath31 . however , for finite @xmath32 we find that , due to the competition between two different size - scales , neither the generalized scaling form nor the standard scaling form @xmath33 lead to scaling of the entire isd for finite values of the ratio @xmath34 of the monomer diffusion rate to deposition flux . instead , we find that the scaled isd becomes more sharply peaked with increasing @xmath35 and coverage . this is in contrast to models of epitaxial growth with limited cluster mobility for which good scaling occurs over a wide range of coverages .

among the various possibilities to explain the stripping of matter ( gas and stars ) from galaxies in clusters , the principal actors can be classified in three groups : + 1 . tidal forces : + interaction with a companion , merger : in this case , a correlation between morphological type ( t ) and density ( @xmath0 ) should be expected , ( t-@xmath0 relation ) + interaction with the cluster ; then a correlation between type and radius in the cluster is expected ( t - r relation ) + harassment due to numerous interactions at high velocity and density + 2 . icm - ism interactions : + ram pressure stripping , but also thermal evaporation , turbulent , viscous stripping ; these are purely hydrodynamical mechanisms , and should affect only the diffuse gas . however , they are acting simultaneously with the others , and relative roles are hard to disentangle . since they are efficient only when the cluster is formed , and the icm gathered , have they enough time to act ? or have tides acted before ? outflows due to violent events : + starbursts and winds + agn jets and outflows + all these processes result in morphological type changes for galaxies , and stripping of their gas , therefore star formation quenching , or `` starvation '' as is observed in clusters . the delicate issue is that many mechanisms are able alone to account for the stripping / quenching , and very specific tests have to be found to disentangle what is happening . one of the clear evidence of tidal interactions and stripping is the existence of intra - cluster diffuse light ( icl ) : these intergalactic stars , stripped from their parent galaxies by tidal interactions , represent a large fraction of the total stellar mass of the cluster , between 10 - 40% ( cf figure 1 , feldmeier et al 2003 ) . cluster images at low luminosity levels show evidence of tidal debris in the form of plumes and arclike structures ( example of the centaurus cluster , calcneo - roldn et al 2000 ) . the quantity of icl does not appear to depend on cluster radius , but more on the surface density of galaxies ( @xmath0 ) , which favors the interactions between galaxies . although ccd images are now able to reveal icl in most clusters , a large sensitivity for this diffuse component is gained from planetary nebulae tracers , without the problems of flat fielding , etc , since they are detected by emission lines ( feldmeier et al 1998 , arnaboldi et al 2002 ) the intra - cluster stars have moderate metallicity ( durrell et al 2002 ) , which supports the scenario of their stripping from intermediate mass galaxies . these tidal debris and plumes are expected from simulations of galaxy clusters ( cf dubinski 1998 ) , even more prominent than what is observed . however , the background noise dilutes the weaker features , explaining the difficulty to observe them clearly ( e.g. mihos 2003 ) . it has been known for a long time that there exists in clusters a larger fraction of blue galaxies at increasing redshift ( butcher & oemler 1978 , 1984 ) . these blue galaxies indicate much more star formation in the recent past , and correspond to irregular shapes in the clusters . the existence in z=0.4 clusters of sign of tidal interaction / mergers also confirm that clusters have evolved very recently : in the last few gyrs , there was a much larger fraction of perturbed galaxies , late - types and starbursts , as if the cluster had relaxed only since then . rings of star formation were much more frequent than 2-arms spirals , contrary to what is found today ( oemler et al 1997 ) . these rings could be due to bars triggered in tidal interactions . part of them could also be due to fast encounters , expected in galaxy clusters , that lead to head - on collisions like the cartwheel . alternatively galaxies , through harassment , could be stripped at this epoch of their dark halos , de - stabilising disks . and triggering more violent star formation . these tidal interactions visible at z=0.4 , must have profoundly and rapidly modified the galaxy morphologies , since at z=0.2 , the evolution is almost terminated . milder effects are observed by balogh et al ( 1999 ) in an x - ray selected sample of clusters ( cnoc1 ) , who suggest a more gradual decline of star formation . in an h@xmath1 line study of 11000 galaxies in the 2df survey , over 17 galaxy clusters , lewis et al . ( 2002 ) find the star formation rate ( sfr ) increasing gradually from low values at the cluster centers , towards the field value at about 3 virial radii . they find a strong correlation between sfr and local projected density , as soon as the density is above 1 galaxy / mpc@xmath2 , independent of the size of the structure ( i.e. also valid in groups ) . gmez et al ( 2003 ) find also a strong sfr-@xmath0 relation with the early data release of the sdss , the sf - quenching effect being even more noticeable for strongly star - forming galaxies . the same break of the sfr-@xmath0 relation is observed at 1 galaxy / mpc@xmath2 . this relation is somewhat linked to the morphological type - density ( t-@xmath0 ) relation , but can not be reduced to it , since at any given type , the sfr-@xmath0 relation is still observed . this strong relation valid even outside cluster cores is a precious clue to derive the dominant mechanisms . from the morphological segregation in nearby clusters , drawn by dressler et al ( 1980 ) , it is now possible to see the evolution from about 5 gyrs ago , at z=0.4 ( dressler et al 1997 , figure 2 ) : at z=0 , there was the same t-@xmath0 correlation for relaxed or non - relaxed clusters , but it is no longer true at z=0.4 . as main lines of evolution , there is at z=0.4 the same fraction of ellipticals than at z=0 , but a much smaller fraction of s0s ; at z=0.5 , the fraction of lenticulars is 3 times lower than now . this suggests that ellipticals form early , before the cluster virialisation . in the hierarchical scenario , clusters form out of loose groups mergers , and it is likely that ellipticals are the result of mergers in groups , before the formation of the cluster . s0 s are transformed from spirals in virialised clusters , in a few gyrs time - scale . the study of stellar populations , and the spectral classes of cluster galaxies at z=0.4 reveals that star formation is quenched with respect to the field ( poggianti et al 1999 ) . at z=0.4 , passive and post - starburst ( e+a , or k+a ) spirals are much more frequent than in the field . it appears that the mechanism reponsible for that must act on shorter time - scales than the mechanism reponsible for the transformation into s0s . in cluster regions where the density is not centrally symmetric , it is possible to compare the morphology - radius ( t - r ) and morphology - density ( t-@xmath0 ) relations the latter ( t-@xmath0 ) appears always better than the t - r relation ( treu et al . 2003 , example of cl0024 + 16 at z=0.4 ) . galaxies are more aware of their local density than cluster location . the morphological segregation as a function of radius is quite significant for radii lower than 200kpc ( cf figure 3 ) . the fraction of early - type galaxies drop steadily until 1 mpc , or nearly the virial radius . the correlation with radius is then weak , while several over - dense regions have galaxies with morphology typical of their high density . it seems that gas depletion and morphological transformation are already well advanced in groups , before forming the cluster . the mild gradient in the morphological mix outside the virial radius could be due to harassment and starvation ( icm interactions are not operating there ) . it is only upon arrival in the central regions ( r @xmath3 200 kpc ) that substructures are erased , as indicated by the tight correlation between cluster radius and @xmath0 . solanes et al . ( 2001 ) have recently made a data compilation on 1900 galaxies , and conclude that about 2/3 of galaxy clusters are hi deficient in their centers . many observations demonstrate that the interaction with the hot gas ( icm - ism interactions ) might be responsible for this hi stripping in cluster galaxies : large deficiencies ( up to a factor 100 ) , deficiency as a function of distance from the central x - ray peak , radial orbits of the stripped galaxies , that allow them to explore the dense hot center . in her review , van gorkom ( 2003 ) describes convincing individual cases proving ram pressure stripping , like galaxies in the virgo cluster with perturbed and reduced - size hi disks while the stellar disk is normal . ram pressure stripping appears quite efficient and rapid , playing a role in only one cluster crossing - time . however , the correlation between hi deficiency and x - ray properties of the cluster is not observed ( l@xmath4 , t@xmath4 ) , which is surprising ( solanes et al 2001 ) . instead , there are correlations with galaxy properties , early - type and probably dwarf spirals are more easily stripped than the intermediate spiral types . that early - types are more stripped is not only due to their position in the cluster , but their deficiency is larger than from late spirals at each cluster radius , until 4 mpc from the center . tidal interactions in galaxies were thought marginal because of high velocities , and un - resonant interactions . however , the large number of interations can accumulate perturbations and truncation effects : this has been called `` harassment '' by moore et al . ( 1996 ) , i.e. frequent high - velocities close encounters . gnedin ( 2003 ) has recently simulated the formation of galaxy clusters in the frame of a hierarchical cosmological scenario , varying the cosmological parameters . tidal interactions determine the galaxy evolution , and are intensified by the density irregularities , either the presence of massive galaxies , or the infalling groups of galaxies , still not relaxed . these substructures favor the interaction , the typical frequency being estimated at 10 interactions at 10kpc impact parameter per galaxy . mergers occur essentially at the cluster formation , and are very rare today . this means that elliptical galaxies predate the cluster , as already found by merritt ( 1984 ) . later the tidal interactions can transform spirals to lenticulars , and explain their large fraction increase in the last gyrs . tidal interactions truncate massive dark matter halos and thicken stellar disks , increasing disk stability and quenching star formation . dwarf galaxies can be totally disrupted . the collision rate per galaxy strongly decreases with time , from 8 to 2 per gyr along the cluster life - time . the tidal field of the cluster itself has strong dynamical influence on galaxies , in particular in their extended halos . first the dark matter haloes are stripped , and form a common halo , but also the gas reservoirs that replenish the interstellar medium of field galaxies all along their lives , by accretion along gas filaments , is stripped also , and this could easily explain the starvation , and gradual decline in star formation of cluster galaxies . this phenomenon was first invoked by larson et al ( 1980 ) , and simulated by bekki et al ( 2001 ) . the latter assume an accretion rate of 1 m@xmath5/yr for a normal field galaxy , and show that the tidal field of the cluster efficiently removes the gas reservoir from a galaxy , and consequently its fueling of star formation . this tidal truncation does not depend very strongly on the orbit of the galaxy in the cluster , and the resulting sf - quenching is widespread through the cluster ( contrary to what is expected from icm interactions ) . once their gas reservoir is stripped , spiral galaxies will slowly be transformed into lenticulars by harrasment , thickening and shortening their stellar disk . a large variety of models have been simulated , since the phenomena associated with icm interactions depend on many physical assumptions about the small - scale structure of the gas , instabilities , equivalent viscosity , temperatures etc .. with an isothermal sph gas model , abadi et al ( 1999 ) show that hi gas is effectively stripped in the core of rich clusters , for disks oriented perpendicular to the wind . galaxies can lose 80% of their gas , the final disk being restricted to 4kpc radius , in a time - scale of 10@xmath6 yrs . however , in the outer parts of the cluster , or for inclined disks with respect to the wind , the process is much less efficient . with a finite - difference code , quilis et al . ( 2000 ) show that viscous coupling could favour the stripping ; they show that a hole in the center of the galaxy ( mimicking the frequent hi depletion in the central regions ) , could fragilize the gas disk , and enhance considerably the stripping efficiency . holes can have an influence at several scales . if star formation has already formed shells through supernovae and winds , ram pressure can enlarge the holes in the disk ( bureau & carignan 2002 ) . supernovae and winds alone are not efficient enough , except may be in small dwarfs ( dekel & silk 1986 , martin 1998 ) . vollmer et al ( 2001 ) use sticky particles for the gas , and follow galaxies on their orbit through the cluster : they show that the icm has only a significant action in the cluster core , and the stripped gas then falls back on the galaxy , once its orbit gets out of the core . schulz & struck ( 2001 ) show that the stripping is a multi - step process : the outer gas is quickly stripped , while the inner gas is compressed , and forms a ring . the compressed gas could give rise to triggered star formation in a small starburst . it is also possible that the gas reservoir required to replenish the ism of galaxies is hot and diffuse , as assumed by bekki et al ( 2002 ) , who show through ram pressure simulations , that the halo will be efficiently stripped , if its density is typically lower than 3 10@xmath7 @xmath8 . in that case , the global tidal stripping from the cluster and the ram pressure compete to strip the gas reservoirs , and contribute to form passive or anaemic spirals , that will slowly be transformed into s0s . enhanced stellar activity , when spiral galaxies infall into the cluster , is observed ( kenney & yale 2002 ) . if the galaxies have been stripped by the cluster global tide of their halo , this can favor the escape of the winds . in a wide sample of galaxies , kauffmann et al ( 2003 ) have shown that star formation efficiency is a strong function of surface density . low surface density dwarf objects ( lsb ) are unevolved objects , in which stellar feedback have prevented rapid star formation , by ejection of their gas . the energy of supernovae is enough to disperse the gas , when the mass of the galaxy has fallen below a threshold of 3 10@xmath9 m@xmath5 , the observed transition between low and high surface density galaxies ( dekel & woo 2003 ) . nuclear activity ( agn ) could also be invoked to provoke gas outflows , and remove gas from galaxies . agn have been found less frequent in cluster environment , by a factor 5 , with a frequency of only 1 percent , compared to 5 percent in the field ( dressler et al . recent observations show however that they might be more frequent in x - rays , suggesting an obscuration effect ( martini et al 2002 ) . the mechanical and heating energy of agn has a strong feedback effect to reverse and self - regulate the gas cooling in the centers of elliptical galaxies , groups and clusters ( e.g. ciotti & ostriker 2001 ) . agn feedback has been invoked to account for recent x - ray observations incompatible with the old quasi - state cooling flow model : absence of extremely cool gas ( @xmath3 1kev ) in the center of cooling flow clusters , presence of cold bubbles related to agn radio lobes , etc ... the amount of effectively cooling gas has been revised downwards . the cooling flow could be intermittent , with alternate periods of agn activity , inflow and outflow . a cold gas phase is observed , as shown in the abell 1795 cooling flow ( figure 4 ) . this cluster center is not yet relaxed , with a cooling wake triggered by the central cd motion ( fabian et al . the x - ray data from the inner 200kpc indicate a mass deposition rate of about 100 m@xmath5/yr ( ettori et al . to find the different roles of the various stripping mechanisms discussed above , it is essential to go back to the formation of the cluster , and the chronology of events . the dense icm required for efficient ram pressure is in place only after the formation of the cluster . also the phenomenon is proportional to the square of velocity dispersion , so virialisation should have occured , for all galaxies to acquire their high velocities . in the hierarchical scenario of structure formation , groups form first , with low velocity dispersion , and a large frequency of resonant tidal interactions and frequent mergers . giant ellipticals are formed at this stage by spiral galaxy mergers in groups . this is part of the morphological segregation , since the fraction of spirals decreases . progressively , the gas in galaxy haloes is stripped by tidal interactions and heated by shocks to the virial temperature of the growing structure , and the importance of the icm interactions will grow . the icm itself is formed through gravitational forces . a large fraction of the icm gas comes from already processed galaxy disks , since its metallicity is important ( 1/3 solar ) . at the present epoch , when clusters are virialised and still relaxing , the merger rate has fallen to zero ( at least major mergers ) , and ellipticals are only passively evolving . ram pressure and the global cluster tide are almost equally efficient to strip gas from infalling galaxies ( i.e. bekki et al . 2002 ) . the fact that the t-@xmath0 relation is even tighter than the t - r relation , combined to the extension of the t-@xmath0 relation from clusters down to groups ( ramella et al . 1999 , helsdon & ponman , 2003 ) suggest that two - body gravitational interactions are dominating . the transformation of spirals into s0s in the outer parts of clusters , depending on the local density , does not pleade in favor of the icm interactions as the main mechanism of the morphological segregation . low density galaxies ( dwarfs and lsb ) entering a cluster can be entirely disrupted by tidal shocks , they form streams of stars contributing to the intra - cluster light . tidal phenomena lead similarly to a morphology - density relation in loose groups , like that observed in the local group ( mayer et al 2001 ) . in rich clusters , the mass of the hot gas can be much larger than the mass of baryons in galaxies . most of the baryons are in the hot icm , since it represents nearly the baryon fraction of the matter in the universe @xmath10 . given its relatively high metallicity , it is then not surprising that most of the metals in a cluster comes from the icm : there is about twice more fe in the icm than in galaxies ( renzini 1997 , 2003 ) . since the metals are synthetized in galaxies , this means that either part of the hot gas comes directly from galaxies ( by stripping , or disruption ) , or that stellar winds and supernovae have enriched the icm . in fact , both sources of metals should be there , since metals expelled by sne would not be sufficient . figure 5 shows that the iron mass - to - light ratio is about constant as a function of the virial temperature of the structure ( below 2 kev , the estimations are less simple to derive , and there could be biases ) . these observations suggest that metals are ejected via winds ( sne or agn ) , not ram pressure , since there is no dependence on richness , or cluster velocity dispersion , but only the dispersion of individual galaxies ( renzini 2003 ) . there is the same m@xmath11/l@xmath12 in clusters and galaxies , implying the same processing in clusters than in the field . ellipticals in the field or in clusters have the same properties , confirming that these galaxies have been formed before the clusters . since the stellar mass is essentially in elliptical galaxies today ( 3/4 of the mass in spheroids , 1/4 in disks , less than 1% in irr , fukugita et al . 1998 , and es are dominating even more in clusters ) , most cluster stars are formed before cluster formation . stars and corresponding metals have likely been made at z@xmath13 2 - 3 , at the peak of stellar activity in the universe . part of the iron has been made through snii quickly , then by snia , 1gyr after the main episodes of star formation . clusters have not lost iron , nor accreted pristine material since the ratio between @xmath1 elements ( made in snii ) and iron is about solar in the ism of all clusters ( with no or little variation from cluster to cluster ) . this means that the metals come from normal stellar nucleosynthesis , with similar ratio of type ia to type ii sne , as well as the same global imf , etc .. also , at the present time snia continue to enrich in fe the medium essentially near the cd at the center ; it has been shown that clusters have almost negligible metallicity gradients , except those with cooling flows , and bright central galaxies ( bhringer , this symposium ) . the various clues brought by observations and simulations allow to draw the following conclusions , about the stripping mechanisms and the recycling of matter in clusters : + tidal forces are at play from the beginning of structure formation , and are the dominant factor in the t-@xmath0 relation ; they need long time - scales ( 1gyr ) in groups that will finally ( recently , z@xmath131 ) merge in a cluster + icm interactions have entered in action only recently ( after the virialisation of rich clusters ) ; they are very efficient in hi stripping from galaxy disks infalling today into the cluster in nearly radial orbits . the corresponding stripping time - scales are very short ( 10@xmath6 - 10@xmath14 yr ) . + starburst and agn winds have played a role continuously ( with a peak at z=2 - 3 ) , and especially in the metallicity distribution + e s have formed in groups before the cluster , s0 s have been transformed from spirals in rich clusters in the last 5 gyr ; their gas reservoir has been stripped through both the global tide of the cluster and the icm interaction , and their star formation quenched . harassment progressively truncates and thickens their disks . the difficulty to derive exactly the cluster evolution is that the mechanisms may act simultaneously , and also they show some duality : for instance , on one hand tidal interactions trigger some star formation , leading to the bo - 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there are several physical processes to remove gas from galaxies in clusters , with subsequent starvation and star formation quenching : tidal interactions between galaxies , or tidal stripping from the cluster potential itself , interactions with the hot intra - cluster medium ( icm ) through ram pressure , turbulent or viscous stripping , or also outflows from star formation of nuclear activity , we review the observational evidence for all processes , and numerical simulations of galaxies in clusters which support the respective mechanisms . this allows to compare their relative efficiencies , all along cluster formation .

a @xmath0 is a finite set @xmath1 paired with a finite set @xmath2 of unordered pairs @xmath3 with @xmath4 . a simple graph has no multiple connections and no self - loops : every @xmath5 appears only once and no @xmath6 is in @xmath2 . elements in @xmath1 are called , elements in @xmath2 are called . given a simple graph @xmath0 , denote by @xmath7 the of a vertex @xmath8 . it is a subgraph generated by the set of vertices directly connected to @xmath8 . denote by @xmath9 the set of complete @xmath10 subgraphs of @xmath11 . elements in @xmath9 are also called . the set @xmath12 for example is the set of all triangles in @xmath11 . of course , @xmath13 and @xmath14 . if the cardinality of @xmath9 is denoted by @xmath15 , the of @xmath11 is defined as @xmath16 , a finite sum . for example , if no tetrahedral subgraphs @xmath17 exist in @xmath11 , then @xmath18 , where @xmath19 is the , the number of vertices , @xmath20 is the , the number of edges and @xmath21 the number of @xmath22 . is defined inductively as @xmath23 with @xmath24 . cyclic graphs , trees or the dodecahedron are examples of graphs of dimension @xmath25 , a triangle @xmath22 , an octahedron or icosahedron has dimension @xmath26 . a tetrahedron has dimension @xmath27 . a complete graph @xmath10 on @xmath28 vertices has dimension @xmath29 . dimension is defined for any graph but can become a fraction . for a truncated cube @xmath11 for example , each unit sphere @xmath7 is a graph of @xmath27 vertices and one edge , a graph of dimension @xmath30 so that @xmath31 . the euler characteristic of this graph @xmath11 is @xmath32 . a is a function on @xmath9 which is antisymmetric in its @xmath33 arguments . the set @xmath34 of all @xmath29-forms is a vector space of dimension @xmath15 . the remaining sign ambiguity can be fixed by introducing an orientation on the graph : a @xmath10 subgraph is called a if it is not contained in a larger @xmath35 graph . an attaches a @xmath29-form @xmath36 to each maximal simplex with value @xmath25 . it induces forms on smaller dimensional faces . if @xmath36 cancels on intersections of maximal graphs , it is a `` volume form '' and @xmath11 is called . an icosahedron for example has triangles as maximal simplices . it is orientable . a wheel graph @xmath37 in which two opposite edges are identified models a mbius strip and is not orientable . a @xmath38-form is a function on @xmath39 and also called a . call @xmath40 the . it is defined as a @xmath25-form if @xmath11 has an orientation . without an orientation , we can still look at the @xmath41 if @xmath42 is an edge attached to @xmath43 . define the @xmath44 and the @xmath45 . a vertex @xmath8 is a if @xmath46 . if @xmath47 and @xmath11 is @xmath29-dimensional , a vertex @xmath8 is an if @xmath7 is a @xmath48-dimensional graph for which every point is an interior point within @xmath7 ; for @xmath49 we ask @xmath7 to be connected . the base induction assumption is that an interior point of a one - dimensional graph has two neighbors . a vertex @xmath8 of a @xmath29-dimensional graph @xmath11 is a if @xmath7 is a @xmath48-dimensional ( for @xmath49 connected ) graph in which every vertex is either a boundary or interior point and both are not empty . the seed assumption is that for @xmath50 , the graph @xmath7 has one vertex . a @xmath29-dimensional graph @xmath51 is a @xmath52 if every @xmath53 is an interior point or a boundary point . glue two copies of @xmath51 along the boundary gives a graph @xmath11 without boundary . a wheel graph @xmath54 is an example of a @xmath26-dimensional graph with boundary if @xmath55 . the boundary is the cyclic one dimensional graph @xmath56 . cut an octahedron in two gives @xmath57 . for an oriented graph @xmath11 , the @xmath58 is defined as @xmath59 , where @xmath60 denotes a variable taken away . for example @xmath61 is a function on triangles called the of a @xmath25-form @xmath62 . a form is if @xmath63 . it is if @xmath64 . the vector space @xmath65 of closed forms modulo exact forms is a of dimension @xmath66 , the . example : @xmath67 is the number of . the is @xmath68 . for a @xmath29-form define the @xmath69 . let @xmath70 be the number of @xmath10 subgraphs of @xmath71 . especially , @xmath72 is the @xmath73 of @xmath8 , the order of @xmath7 . the local quantity @xmath74 is called the of the graph at @xmath8 . the sum is of course finite . for a @xmath26-dimensional graph without boundary , where @xmath7 has the same order and size , it is @xmath75 . for a 1-dimensional graph with or without boundary and trees in particular , @xmath76 . for an arbitrary finite simple graph we have @xcite for an arbitrary finite simple graph and injective @xmath77 , we have @xcite assume @xmath62 is a @xmath48-form and @xmath11 is an oriented @xmath29-dimensional graph with boundary , then with boundary @xmath52 , the later remains a graph . in general it is only a , an element in the group of integer valued functions on @xmath78 usually written as @xmath79 . ] the * transfer equations * are . by definition of curvature , we have @xmath80 since the sums are finite , we can change the order of summation . using the transfer equations we get @xmath81 the number of @xmath29 simplices @xmath82 in the exit set @xmath83 and the number of @xmath29 simplices @xmath84 in the entrance set @xmath85 are complemented within @xmath7 by the number @xmath86 of @xmath29 simplices which contain both vertices from @xmath83 and @xmath85 . by definition , @xmath87 . the index @xmath88 is the same for all injective functions @xmath77 . the * intermediate equations * are . let @xmath89 . because replacing @xmath62 and @xmath90 switches @xmath91 with @xmath92 and the sum is the same , we can prove @xmath93 instead . the transfer equations and intermediate equations give @xmath94 = 2v_0 + \sum_{k=0}^{\infty } ( -1)^k 2 v_{k+1 } = 2 \chi(g ) \ ; .\end{aligned}\ ] ] denote a @xmath29-simplex graph @xmath95 by @xmath96 . from @xmath97 and algebraic boundary @xmath98 = \sum_k ( -1)^k ( x_0 , ... , ,x_n))$ ] , stokes theorem is obvious for a single simplex : @xmath99 gluing @xmath29-dimensional simplices cancels boundary . a @xmath29-dimensional graph with boundary is a union of @xmath29-dimensional simplices identified along @xmath48- dimensional simplices . a @xmath29-dimensional oriented graph with boundary can be built by gluing cliques as long as the orientation @xmath29-form can be extended . we also used that the * boundary as a graph * agrees with the * algebraic boundary * if differently oriented boundary pieces cancel . here are families of graphs , where the curvature is indicated at every vertex : for the history of the classical stokes theorem , see @xcite . the history of topology @xcite . the collection @xcite contains in particular an article on the history of graph theory . a story about euler characteristic and polyhedra is told in @xcite . for an introduction to gauss - bonnet with historical pointers to early discrete approaches see @xcite . for poincar - hopf , the first volume of @xcite or @xcite . for morse theory and reeb s theorem @xcite . poincar proved the index theorem in chapter viii of @xcite . hopf extended it to arbitrary dimensions in @xcite . gauss - bonnet in higher dimensions was proven first independently by allendoerfer @xcite and fenchel @xcite for surfaces in euclidean space and extended jointly by allendoerfer and weil @xcite to closed riemannian manifolds . chern gave the first intrinsic proof in @xcite . y. tong m. desbrun , e. kanso . discrete differential forms for computational modeling . in j. sullivan g. ziegler a. bobenko , p. schroeder , editor , _ discrete differential geometry _ , oberwohlfach seminars , 2008 .

by proving graph theoretical versions of green - stokes , gauss - bonnet and poincar - hopf , core ideas of undergraduate mathematics can be illustrated in a simple graph theoretical setting . in this pedagogical exposition we present the main proofs on a single page and add illustrations . while discrete stokes is at least 100 years old , the other two results for graphs were found only recently .

during past few years research in areas of wireless ad - hoc networks and wireless sensor networks ( wsns ) are escalated . ieee 802.15.4 is targeted for wireless body area networks ( wbans ) , which requires low power and low data rate applications . invasive computing is term used to describe future of computing and communications [ 1 - 3 ] . due to these concepts , personal and business domains are being densely populated with sensors . one area of increasing interest is the adaptation of technology to operate in and around human body . many other potential applications like medical sensing control , wearable computing and location identification are based on wireless body area networks ( wbans ) . main aim of ieee 802.15.4 standard is to provide a low - cost , low power and reliable protocol for wireless monitoring of patient s health . this standard defines physical layer and mac sub layer . three distinct frequencies bands are supported in this standard . however , 2.4 ghz band is more important . this frequency range is same as ieee 802.11b / g and bluetooth . ieee 802.15.4 network supports two types of topologies , star topology and peer to peer topology . standard supports two modes of operation , beacon enabled ( slotted ) and non - beacon enabled ( unslotted ) . medium access control ( mac ) protocols play an important role in overall performance of a network . in broad , they are defined in two categories contention - based and schedule - based mac protocols . in contention - based protocols like carrier sense multiple access with collision avoidance ( csma / ca ) , each node content to access the medium . if node finds medium busy , it reschedules transmission until medium is free . in schedule - based protocols like time division multiple access ( tdma ) , each node transmits data in its pre - allocated time slot . this paper focuses on analysis of ieee 802.15.4 standard with non - beacon enabled mode configure in a star topology . we also consider that sensor nodes are using csma / ca protocol . to access channel data . in literature , beacon enabled mode is used with slotted csma / ca for different network settings . in [ 1 ] , performance analysis of ieee 802.15.4 low power and low data rate wireless standard in wbans is done . authors consider a star topology at 2.4 ghz with up to 10 body implanted sensors . long - term power consumption of devices is the main aim of their analysis . however , authors do not analyze their study for different data rates . an analytical model for non - beacon enabled mode of ieee 802.15.4 medium access control ( mac ) protocol is provided in [ 2 ] . nodes use un - slotted csma / ca operation for channel access and packet transmission . two main variables that are needed for channel access algorithm are back - off exponent ( be ) and number of back - offs ( nb ) . authors perform mathematical modeling for the evaluation statistical distribution of traffic generated by nodes . this mathematical model allows evaluating an optimum size packet so that success probability of transmission is maximize . however , authors do not analyze different mac parameters with varying data rates . authors carry out an extensive analysis based on simulations and real measurements to investigate the unreliability in ieee 802.15.4 standard in [ 3 ] . authors find out that , with an appropriate parameter setting , it is possible to achieve desired level of reliability . unreliability in mac protocol is the basic aspect for evaluation of reliability for a sensor network . an extensive simulation analysis of csma / ca algorithm is performed by authors to regulate the channel access mechanism . a set of measurements on a real test bed is used to validate simulation results . a traffic - adaptive mac protocol ( tamac ) is introduced by using traffic information of sensor nodes in [ 4 ] . tamac protocol is supported by a wakeup radio , which is used to support emergency and on - demand events in a reliable manner . authors compare tamac with beacon - enabled ieee 802.15.4 mac , wireless sensor mac ( wisemac ) , and sensor mac ( smac ) protocols . important requirements for the design of a low - power mac protocol for wbans are discussed in [ 5 ] . authors present an overview to heartbeat driven mac ( h - mac ) , reservation - based dynamic tdma ( dtdma ) , preamble - based tdma ( pb - tdma ) , and body mac protocols , with focusing on their strengths and weaknesses . authors analyze different power efficient mechanism in context of wbans . at the end authors propose a novel low - power mac protocol based on tdma to satisfy traffic heterogeneity . authors in [ 6 ] , examine use of ieee 802.15.4 standard in ecg monitoring and study the effects of csma / ca mechanism . they analyze performance of network in terms of transmission delay , end - to - end delay , and packet delivery rate . for time critical applications , a payload size between 40 and 60 bytes is selected due to lower end - to - end delay and acceptable packet delivery rate . in [ 7 ] , authors state that ieee 802.15.4 standard is designed as a low power and low data rate protocol with high reliability . they analyze unslotted version of protocol with maximum throughput and minimum delay . the main purpose of ieee 802.15.4 standard is to provide low power , low cost and highly reliable protocol . physical layer specifies three different frequency ranges , 2.4 ghz band with 16 channels , 915 mhz with 10 channels and 868 mhz with 1 channel . calculations are done by considering only beacon enabled mode and with only one sender and receiver . however , it consumes high power . as number of sender increases , efficiency of 802.15.4 decreases . throughput of 802.15.4 declines and delay increases when multiple radios are used because of increase in number of collisions . a lot of work is done to improve the performance of ieee 802.15.4 and many improvements are made in improving this standard , where very little work is done to find out performance of this standard by varying data rates and also considering acknowledgement ( ack ) and no ack condition and how it affects delay , throughput , end - to - end delay and load . we get motivation to find out the performance of this standard with parameters load , throughput , delay and end to end delay at varying data rates . ieee 802.15.4 is proposed as standard for low data rate , low power wireless personal area networks ( wpans ) [ 1],[2 ] . in wpans , end nodes are connected to a central node called coordinator . management , in - network processing and coordination are some of key operations performed by coordinator . the super - frame structure in beacon enabled mode is divided into active and inactive period . active period is subdivided into three portions ; a beacon , contention access period ( cap ) and contention free period ( cfp ) . in cfp , end nodes communicate with central node ( coordinator ) in dedicated time slots . however , cap uses slotted csma / ca . in non - beacon enabled mode , ieee 802.15.4 uses unslotted csma / ca with clear channel assessment ( cca ) for channel access . in [ 2 ] , ieee 802.15.4 mac protocol non - beacon enabled mode is used . nodes use un - slotted csma / ca operation for channel access and packet transmission . two main variables that are needed for channel access algorithm are back off exponent ( be ) and number of back offs ( nb ) . nb is the number of times csma / ca algorithm was required to back off while attempting channel access and be is related to how many back off periods , node must wait before attempting channel access . operation of csma / ca algorithm is defined in steps below : @xmath0 nb and be initialization : first , nb and be are initialized , nb is initialized to 0 and be to macminbe which is by default equal to 3 . + @xmath1 random delay for collision avoidance : to avoid collision algorithm waits for a random amount of time randomly generated in range of @xmath2 , one back off unit period is equal to @xmath3 with @xmath4s + @xmath5 clear channel assessment : after this delay channel is sensed for the unit of time also called cca . if the channel is sensed to be busy , algorithm goes to step 4 if channel is idle algorithm goes to step 5 . + @xmath6 busy channel : if channel is sensed busy then mac sub layer will increment the values of be and nb , by checking that be is not larger than @xmath7 . if value of nb is less than or equal to @xmath8 , then csma / ca algorithm will move to step 2 . if value of nb is greater than @xmath8 , then csma / ca algorithm will move to step 5 `` packet drop '' , that shows the node does not succeed to access the channel . + @xmath9 idle channel : if channel is sensed to be idle then algorithm will move to step 4 that is `` packet sent '' , and data transmission will immediately start . fig . 1 illustrates aforementioned steps of csma / ca algorithm , starting with node has some data to send . csma / ca is a modification of carrier sense multiple access ( csma ) . collision avoidance is used to enhance performance of csma by not allowing node to send data if other nodes are transmitting . in normal csma nodes sense the medium if they find it free , then they transmits the packet without noticing that another node is already sending packet , this results in collision . csma / ca results in reduction of collision probability . it works with principle of node sensing medium , if it finds medium to be free , then it sends packet to receiver . if medium is busy then node goes to backoff time slot for a random period of time and wait for medium to get free . with improved csma / ca , request to send ( rts)/clear to send ( cts ) exchange technique , node sends rts to receiver after sensing the medium and finding it free . after sending rts , node waits for cts message from receiver . after message is received , it starts transmission of data , if node does not receive cts message then it goes to backoff time and wait for medium to get free . csma / ca is a layer 2 access method , used in 802.11 wireless local area network ( wlan ) and other wireless communication . one of the problems with wireless data communication is that it is not possible to listen while sending , therefore collision detection is not possible . csma / ca is largely based on the modulation technique of transmitting between nodes . csma / ca is combined with direct sequence spread spectrum ( dsss ) which helps in improvement of throughput . when network load becomes very heavy then frequency hopping spread spectrum ( fhss ) is used in congestion with csma / ca for higher throughput , however , when using fhss and dsss with csma / ca in real time applications then throughput remains considerably same for both . 2 shows the timing diagram of csma / ca . @xmath10 data transmission time @xmath11 , backoff slots time @xmath12 , acknowledgement time @xmath13 are given by equation 2 , 3 , and 4 respectively[2 ] . + [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^ " , ] [ tab : addlabel ] @xmath14 the following notations are used : + + @xmath15 @xmath16 @xmath17 + @xmath18 @xmath19 @xmath20 @xmath21 + @xmath22 @xmath23 @xmath19 + @xmath24 @xmath25 @xmath19 + @xmath26 @xmath27 @xmath28 + @xmath29 @xmath20 @xmath30 @xmath31 + @xmath32 @xmath20 @xmath33 @xmath31 + @xmath34 @xmath20 @xmath35 @xmath36 @xmath37 @xmath38 + @xmath39 @xmath20 @xmath33 @xmath40 + @xmath41 @xmath20 @xmath42 @xmath43 @xmath44 + @xmath45 @xmath46 @xmath47 @xmath42 @xmath43 @xmath48 + @xmath49 @xmath50 in csma / ca mechanism , packet may loss due to collision . collision occurs when two or more nodes transmits the data at the same time . if ack time is not taken in to account then there will be no retransmission of packet and it will be considered that each packet has been delivered successfully . the probability of end device successfully transmitting a packet is modeled as follows[3 ] . @xmath51 where , @xmath52 is the number of end devices that are connected to router or coordinator . be is the backoff exponent in our case it is 3 . @xmath53 is the probability of transmission success at a slot . @xmath54 is the probability of end device successfully allocated a wireless channel . general formula for @xmath55 is given by equation 8 . probability of time delay caused by csma / ca backoff exponent is estimated as in [ 7 ] . maximum number of backoff is 4 . value of be=3 has been used in following estimation and we estimate by applying summation from 3 to 5 . @xmath56 is the probability of time delay event . @xmath57 @xmath58 expectation of the time delay is obtained as from [ 7 ] . @xmath59 and @xmath60 are taken from equations 7 and 8 respectively . @xmath61=p({e_{a}|e_{b}})\nonumber\\ \nonumber\\ = \frac{\sum_{n=0}^{7 } n\frac{1}{2_{be}}p + \sum _ { n=8}^{15 } n\frac{1}{2_{be}}p + \sum _ { n=16}^{31 } n\frac{1}{2_{be}}p}{\sum_{n=0}^{2^{be-1 } } n\frac{1}{2_{be}}p{1-p}^{be-2}}\end{aligned}\ ] ] statistical data of throughput , load , end - to - end delay and delay of ieee 802.15.4 at varying data rates is shown in table i. it shows different values of delay , throughput , end - to - end delay and load recorded at different time . load at all data rates and at all time intervals remains same . start time for simulation is kept at 0 seconds and stop time is kept to infinity . load in all three data rates at different time intervals remains same as shown in table i. there is very small difference between delay and end - to - end delay . at 20 kbps maximum delay of 2145 seconds is recorded with maximum throughput of 4352 bits / sec at 60 min . at 40kbps maximum delay of 380 seconds and minimum delay 2.5 seconds is recorded . throughput of 8805 ( bits / sec ) is the highest throughput recorded on 60 min . in case of 250 kbps delay remains very small , near to negligible where as throughput matches load with 10388 ( bits / sec ) . beacon order & 6 + superframe order & 0 + maximum routers & 5 + maximum depth & 5 + beacon enabled network & disabled + route discovery time & 10(sec ) + network parameter are given in table ii . non beacon mode is selected in our analysis and beacon order is kept at 6 . due to non - beacon enabled mode superframe order is not selected . maximum routers or nodes that can take part in simulation is 5 , each having tree depth of 5 . discovery time that is needed by each router to discover route is 10 sec . minimum backoff exponent & 3 + maximum number of backoff & 5 + channel sensing duration & 0.1 ( sec ) + data rates & 20 , 40 , 250 kbps + packet reception power & -85 ( dbm ) + transmission band & 2.4 ( mhz ) + packet size & 114 bytes + packet interarrival time & 0.045(sec ) + transmit power & 0.05 ( w ) + ack wait duration & 0.05 ( sec ) + number of retransmissions & 5 + simulation parameters of 802.15.4 with its value are shown in table iii . minimum be is kept at 3 with maximum no . of back - off to 5 . default settings of 802.15.4 are used in this simulation . packet reception power is kept at -85 dbm with transmitting power of 0.5 watt(w ) . in ack enabled case , ack wait duration is kept at 0.05 sec with no of retransmissions to 5 . in no ack case these parameters are disabled . 114 bytes is the packet size with interarrival time of 0.045 sec . transmission band used in this simulation is 2.4 ghz . simulations have been performed at varying data rates of 20 , 40 , 250 kbps . simulations for both ack and non ack cases have also been performed . opnet modeler is the simulator used for simulations . simulations are executed for one hour with update interval of 5000 events . graphs are presented in overlaid statistics . overlaid means that , graphs of each scenario has been combined with each other . data of graphs are averaged over time for better results . personal area network identity ( pan i d ) is kept at default settings , coordinator automatically assigns pan i d to different personal area networks if they are attached . we consider non beacon mode for our simulations . using non - beacon enabled mode improves the performance and changing different parameters affects performance of 802.15.4 . csma / ca values are kept to default with minimum backoff exponent to 3 and having maximum backoff of 5 . changing these parameters does not affect its performance . we perform simulations with ack and non ack . in non ack there is only delay due to node waiting while sensing medium , there is no delay due to ack colliding with packets . in ack case there is collision for packets going towards receiver and ack packet coming from receiver at same time . delay in ack is more as compare to non ack case . we use standard structure of ieee 802.15.4 with parameters shown in table ii . in this section , performance of default mac parameters of ieee 802.15.4 standard non beacon enabled mode . simulations are performed considering 10 sensor nodes environment with coordinator collecting data from all nodes . fig 3 , 4 , 5 and 6 show graphical representation of performance parameters of 802.15.4 . delay represents the end - to - end delay of all the packets received by 802.15.4 macs of all wpans nodes in the network and forwarded to the higher layer . load represents the total load in ( bits / sec ) submitted to 802.15.4 mac by all higher layers in all wpans nodes of the network . load remains same for all the data rates . throughput represents the total number of bits in ( bits / sec ) forwarded from 802.15.4 mac to higher layers in all wpans nodes to the network . end - to - end delay is the total delay between creation and reception of an application packet . delay , load and throughput are plotted as function of time . as load is increasing , there is increase in throughput and delay . when load becomes constant , throughput also becomes constant , however , delay keeps on increasing . delay in 802.15.4 occurs due to collision of data packets or sometimes nodes keeps on sensing channel and does not find it free . when node senses medium and find it free , it sends packet . at same time , some other nodes are also sensing the medium and find it free , they also send data packets and thus results in collision . collision also occurs due to node sending data packet and at same time coordinator sending ack of successfully receiving packet and causing collision . when ack is disabled this type of collision will not occur . delay , throughput and load is analyzed at 40 kbps in fig 4 . with increase in load , there is increase in throughput and delay , however , it is less as compared to 20kbps , this is due to increase in data rate of 802.15.4 . increase in bit transfer rate from 20 to 40kbps causes decrease in delay and hence increases throughput . fig 5 shows behavior of 802.15.4 load , throughput and delay at 250kbps data rate . delay is negligible at this data rate , with throughput and load showing same behavior . delay approaching zero shows that , at 250 kbps data rate there are less chances of collision or channel access failure . ieee 802.15.4 performs best at this data rate compared to 20 and 40 kbps . at same time end - to - end delay of ieee 802.15.4 at varying data rates of 20 , 40 and 250 kbps are shown in fig 6 . this figure shows that end to end delay for 20 kbps data rate is higher than 40 kbps and 250 kbps . minimum end - to - end delay is found at 250 kbps data rate . at 250 kbps , more data can pass at same time with less collision probability hence having minimum delay and at 20 kbps , less data transfers at same time causing more end to end delay . statistical data of end - to - end delay is shown in table i , which shows end to end variation with change in time . fig 7 shows the delay , throughput , load and end - to - end delay of ieee 802.11.4 at 20 kbps data rate with and without ack . load remains same in both cases . there is no collision because of ack packets due to which packets once send are not sent again . there is decrease in delay and increase in ack due to less collision . end - to - end delay performs same as delay . ieee 802.15.4 performs better with non ack other than ack due to decrease in collision probability in no ack compared to ack case . delay , throughput , load and end - to - end delay with and without ack at 40 kbps are presented in fig 8 . there is considerable difference between the analysis in ack and without ack case . delay is reduced to negligible at low value of @xmath62 in no ack case due to reason that , at this data rate there is no collision therefor , delay is nearly zero . as there is no collision and channel sensing time is also low , this increase throughput and load in non ack case , as compared to ack . 9 shows analysis with ack and no ack cases of delay , throughput , load and end to end delay at 250 kbps , at this high data rates load and throughput in both cases becomes equal to each other and data is sent in first instant to coordinator by nodes . delay in both cases nearly equal to zero , which shows that , there is very less collision at this high data rates and channel sensing time is also very low . end to end delay slightly differs from delay in no ack case . in this paper , performance of ieee 802.15.4 standard with non - beacon enabled mode is analyzed at varying data rates . we have evaluated this standard in terms of load , delay , throughput and end - to - end delay with different mac parameters . we have also analyzed performance with ack enabled mode and no ack mode . we considered a full size mac packet with payload size of 114 bytes for data rates 20 kbps , 40 kbps and 250 kbps . it is shown that better performance in terms of throughput , delay , and end - to - end delay is achieved at higher data rate of 250kbps . ieee 802.15.4 performs worse at low data rates of 20kbps . performance of this standard improves with increase in data rate . in future research work , we will investigate the performance of ieee 802.15.4 in wbans by changing frequency bands on different data rates . we also intend to examine the effect of changing inner structure of mac layer in ieee 802.15.4 . 1 f. timmons , n @xmath63 @xmath64 . , `` analysis of the performance of ieee 802.15.4 for medical sensor body area networking '' , sensor and ad hoc communications and networks , 2004 . c. buratti and r. verdone @xmath63 @xmath64 . , `` performance analysis of ieee 802.15.4 non beacon - enabled mode '' , ieee transaction on vehicular technology , vol . 58 , no . 7 , september 2009 . anastasi , g @xmath63 @xmath64 . , `` the mac unreliability problem in ieee 802.15.4 wireless sensor networks '' , mswim09 proceedings of the 12th acm international conference on modeling , analysis and simulation of wireless and mobile systems , october 2009 . s. ullah , k. s. kwak @xmath63 @xmath64 . , `` an ultra - low power and traffic - adaptive medium access control protocol for wireless body area network '' , j med syst , doi 10.1007/s10916 - 010 - 9564 - 2 . s. ullah , b.shen , s.m.r . islam , p. khan , s. saleem and k.s . kwak @xmath63 @xmath64 . , `` a study of medium access control protocols for wireless body area networks '' . x. liang and i. balasingham @xmath63 @xmath64 . , `` performance analysis of the ieee 802.15.4 based ecg monitoring network '' . b. latre , p.d . mil , i. moerman , b. dhoedt and p. demeester @xmath63 @xmath64 . , `` throughput and delay analysis of unslotted ieee 802.15.4 '' , journal of networks , vol . 1 , no . 1 , may 2006 .

ieee 802.15.4 standard is designed for low power and low data rate applications with high reliability . it operates in beacon enable and non - beacon enable modes . in this work , we analyze delay , throughput , load , and end - to - end delay of non - beacon enable mode . analysis of these parameters are performed at varying data rates . evaluation of non beacon enabled mode is done in a 10 node network . we limit our analysis to non beacon or unslotted version because , it performs better than other . protocol performance is examined by changing different medium access control ( mac ) parameters . we consider a full size mac packet with payload size of 114 bytes . in this paper we show that maximum throughput and lowest delay is achieved at highest data rate . ieee 802.15.4 , throughput , delay , end - to - end , load

the hypothesis testing theory is a well developed branch of mathematical statistics @xcite . the asymptotic approach allows to find satisfactory solutions in many different situations . the simplest problems , like the testing of two simple hypotheses , have well known solutions . recall that if we fix the first type error and seek the test which maximizes the power , then we obtain immediately ( by neyman - pearson lemma ) the most powerful test based on the likelihood ratio statistic . the case of composite alternative is more difficult to treat and here the asymptotic solution is available in the regular case . it is possible , using , for example , the score function test ( sft ) , to construct the asymptotically ( locally ) most powerful test . moreover , the general likelihood ratio test ( glrt ) and the wald test ( wt ) based on the maximum likelihood estimator are asymptotically most powerful in the same sense . in the non regular cases the situation became much more complex . first of all , there are different non regular ( singular ) situations . moreover , in all these situations , the choice of the asymptotically best test is always an open question . this work is an attempt to study all these situations on the model of inhomogeneous poisson processes . this model is sufficiently simple to allow us to realize the construction of the well known tests ( sft , glrt , wt ) and to verify that these test are asymptotically most powerful also for this model , in the case when it is regular . in the next paper we study the behavior of these tests in the case when the model is singular . the `` evolution of the singularity '' of the intensity function is the following : regular case ( finite fisher information , this paper ) , continuous but not differentiable ( cusp - type singularity , @xcite ) , discontinuous ( jump - type singularity , @xcite ) . in all the three cases we describe the tests analytically . more precisely , we describe the test statistics , the choice of the thresholds and the behavior of the power functions for local alternatives . note that the notion of _ local alternatives _ is different following the type of regularity / singularity . suppose we want to test the simple hypothesis @xmath0 against the one - sided alternative @xmath1 . in the regular case , the local alternatives are usually given by , @xmath2 . in the case of a cusp - type singularity , the local alternatives are introduced by @xmath3 , @xmath2 . as to the case of a jump - type singularity , the local alternatives are @xmath4 , @xmath2 . in all these problems , the most interesting for us question is the comparison of the power functions of different tests . in singular cases , the comparison is done with the help of numerical simulations . the main results concern the limit likelihood ratios in the non - regular situations . let us note , that in many other models of observations ( i.i.d . , time series , diffusion processes etc . ) the likelihood ratios have the same limits as here ( see , for example , @xcite and @xcite ) . therefore , the results presented here are of more universal nature and are valid for any other ( non necessarily poissonian ) model having one of considered here limit likelihood ratios . we recall that @xmath5 is an inhomogeneous poisson process with intensity function @xmath6 , @xmath7 , if @xmath8 and the increments of @xmath9 on disjoint intervals are independent and distributed according to the poisson law @xmath10 in all statistical problems considered in this work , the intensity functions are periodic with some known period @xmath11 and depend on some one - dimensional parameter , that is , @xmath12 . the basic hypothesis and the alternative are always the same : @xmath0 and @xmath13 . the diversity of statements corresponds to different types of regularity / singularity of the function @xmath14 . the case of unknown period @xmath11 needs a special study . the hypothesis testing problems ( or closely related properties of the likelihood ratio ) for inhomogeneous poisson processes were studied by many authors ( see , for example , brown @xcite , kutoyants @xcite , lger and wolfson @xcite , liese and lorz @xcite , sung _ et al . _ @xcite , fazli and kutoyants @xcite , dachian and kutoyants @xcite and the references therein ) . note finally , that the results of this study will appear later in the work @xcite . for simplicity of exposition we consider the model of @xmath15 independent observations of an inhomogeneous poisson process : @xmath16 , where @xmath17 , @xmath18 , are poisson processes with intensity function @xmath14 , @xmath19 . here @xmath20 , @xmath21 , is a one - dimensional parameter . we have @xmath22 where @xmath23 is the mathematical expectation in the case when the true value is @xmath24 . note that this model is equivalent to the one , where we observe an inhomogeneous poisson process @xmath25 with periodic intensity @xmath26 , @xmath27 , and @xmath28 ( the period @xmath29 is supposed to be known ) . indeed , if we put @xmath30 , @xmath31 $ ] , @xmath18 , then the observation of one trajectory @xmath32 is equivalent to @xmath15 independent observations @xmath33 . the intensity function is supposed to be separated from zero on @xmath34 $ ] . the measures corresponding to poisson processes with different values of @xmath35 are equivalent . the likelihood function is defined by the equality ( see liese @xcite ) @xmath36{\rm d}t\right\}\end{aligned}\ ] ] and the likelihood ratio function is @xmath37 we have to test the following two hypotheses @xmath38 a test @xmath39 is defined as the probability to accept the hypothesis @xmath40 . its power function is @xmath41 , @xmath13 . denote @xmath42 the class of tests @xmath43 of asymptotic size @xmath44 $ ] : @xmath45 our goal is to construct tests which belong to this class and have some proprieties of asymptotic optimality . the comparison of tests can be done by comparison of their power functions . it is known that for any reasonable test and for any fixed alternative the power function tends to @xmath46 . to avoid this difficulty , as usual , we consider _ close _ or _ contiguous _ alternatives . we put @xmath47 , where @xmath48 , @xmath49 and @xmath50 . the rate of convergence @xmath50 must be chosen so that the normalized likelihood ratio @xmath51 has a non degenerate limit . in the regular case this rate is usually @xmath52 . then the initial problem of hypotheses testing can be rewritten as @xmath53 the power function of a test @xmath54 is now denoted as @xmath55 the asymptotic optimality of tests is introduced with the help of the following definition ( see @xcite ) . we call a test @xmath56 locally asymptotically uniformly most powerful ( laump ) in the class @xmath57 if its power function @xmath58 satisfies the relation : for any test @xmath59 and any @xmath60 we have @xmath61\geq 0.\ ] ] below we show that in the regular case many tests are laump . in the next paper @xcite , where we consider some singular situations , a `` reasonable '' definition of asymptotic optimality of tests is still an open question . that is why we use numerical simulations to compare the tests in @xcite . we assume that the following _ regularity conditions _ are satisfied . * smoothness . * _ the intensity function @xmath14 , @xmath62 , of the observed poisson process @xmath63 is two times continuously differentiable w.r.t . @xmath35 , is separated from zero uniformly on @xmath64 , and the fisher information is positive : @xmath65 here @xmath66 denotes the derivative of @xmath67 w.r.t . @xmath35 and , at the point @xmath68 , the derivative is from the right . _ * distinguishability . * _ for any @xmath69 , we have @xmath70 here @xmath71 _ in this case , the natural normalization function is @xmath72 and the change of variables is @xmath73 . the key propriety of statistical problems in the regular case is the _ local asymptotic normality _ ( lan ) of the family of measures of corresponding inhomogeneous poisson processes at the point @xmath74 . this means that the normalized likelihood ratio @xmath75 admits the representation @xmath76 where ( using the central limit theorem ) we have @xmath77\\ & \longrightarrow \widetilde{\delta } \sim \mathcal { n}\left(0,{\rm i}\left(\vartheta _ 1\right ) \right)\end{aligned}\ ] ] ( convergence in distribution under @xmath68 ) , and @xmath78 ( convergence in probability under @xmath68 ) . moreover , the last convergence is uniform on @xmath79 for any @xmath60 . let us now briefly recall how this representation was obtained in @xcite . denoting @xmath80 and @xmath81 , with the help of the taylor series expansion we can write @xmath82- n\int_{0}^{\tau } \left[\lambda _ u-\lambda _ 0-\lambda _ 0\ln \frac{\lambda _ u}{\lambda _ 0}\right ] { \rm d}t\\ & = \frac{u}{\sqrt{n}}\sum_{j=1}^{n}\int_{0}^{\tau } \frac{\dot \lambda _ 0}{\lambda _ 0}\left[{\rm d}x_j\left(t\right)-\lambda _ 0{\rm d}t\right]-\frac{u^2}{2}\int_{0}^{\tau } \frac{\dot \lambda _ 0 ^ 2}{\lambda _ 0}{\rm d}t+r_n\\ & = u\widetilde{\delta } _ n\left(\vartheta _ 1,x^n\right)-\frac{u^2}{2}{\rm i}\left(\vartheta _ 1\right)+r_n\longrightarrow \widetilde{\delta } -\frac{u^2}{2}{\rm i}\left(\vartheta _ 1\right).\end{aligned}\ ] ] in the sequel , we choose reparametrizations which lead to _ universal _ in some sense limits . for example , in the regular case , we put @xmath83 with such change of variables , we have @xmath84 where @xmath85 the lan families have many remarkable properties and some of them will be used below . let us remind here one general result which is valid in a more general situation . we suppose only that the normalized likelihood ratio @xmath86 converges to some limit @xmath87 in distribution . note that this is the case in all our regular and singular problems . the following property allows us to calculate the distribution under local alternative when we know the distribution under the null hypothesis . moreover , it gives an efficient algorithm for calculating power functions in numerical simulations . suppose that @xmath88 @xmath89 converges in distribution under @xmath68 : @xmath90 then , for any bounded continuous function @xmath91 , we have @xmath92\longrightarrow { \mathop{\mathbf{\kern 0pt e}}\nolimits}\left[z\left(u\right ) g\left(y\right)\right].\ ] ] for the proof see @xcite . in the regular case , the limit of @xmath93 is the random function @xmath94 so , for any fixed @xmath2 , we have the convergence @xmath95 according to this lemma , we can write the following relations for the characteristic function of @xmath96 : @xmath97 which yields the asymptotic distribution of the statistic @xmath98 under the alternative @xmath99 : @xmath100 all the tests considered in this paper are functionals of the normalized likelihood ratio @xmath101 . for each of them , we have to evaluate two quantities . the first one is the threshold , which guarantees the desired asymptotic size of the test , and the second one is the limit power function , which has to be calculated under alternative . our study is based on the weak convergence of the likelihood ratio @xmath102 under hypothesis ( to calculate the threshold ) and under alternative ( to calculate the limit power function ) . note that the test statistics of all the tests are continuous functionals of @xmath102 . that is why the weak convergence of @xmath102 allows us to obtain the limit distributions of these statistics . we denote @xmath103 the distribution that the observed inhomogeneous poisson processes @xmath63 induce on the measurable space of their realizations . the measures in the family @xmath104 are equivalent , and the normalized likelihood ratio is @xmath105{\rm d}t , \ ] ] where @xmath106 . we define @xmath86 to be linearly decreasing to zero on the interval @xmath107 $ ] and we put @xmath108 for @xmath109 . now the random function @xmath86 is defined on @xmath110 and belongs to the space @xmath111 of continuous on @xmath110 functions such that @xmath112 as @xmath113 . introduce the uniform metric in this space and denote @xmath114 the corresponding borel sigma - algebra . the next theorem describes the weak convergence under the alternative @xmath115 ( with fixed @xmath116 ) of the stochastic process @xmath117 to the process @xmath118 in the measurable space @xmath119 . note that in @xcite this theorem was proved for a fixed true value @xmath35 . in the hypothesis testing problems considered here , we need this convergence both under hypothesis @xmath120 , that is , for fixed true value @xmath0 ( @xmath121 ) , and under alternative @xmath122 with `` moving '' true value @xmath123 . [ t1 ] let us suppose that the regularity conditions are fulfilled . then , under alternative @xmath124 , we have the weak convergence of the stochastic process @xmath125 to @xmath126 . according to ( * ? ? ? * theorem 1.10.1 ) , to prove this theorem it is sufficient to verify the following three properties of the process @xmath102 . 1 . the finite - dimensional distributions of @xmath102 converge , under alternative @xmath124 , to the finite - dimensional distributions of @xmath127 . the inequality @xmath128 holds for every @xmath129 and some constant @xmath130 . there exists @xmath131 , such that for some @xmath132 and all @xmath133 we have the estimate @xmath134 let us rewrite the random function @xmath93 as follows : @xmath135 for the first term we have @xmath136 therefore we only need to check the conditions 23 for the term @xmath137 the finite - dimensional distributions of @xmath102 converge , under alternative @xmath124 , to the finite - dimensional distributions of @xmath127 . the limit process for @xmath138 is @xmath139 hence @xmath140 for the details see , for example , @xcite . let the regularity conditions be fulfilled . then there exists a constant @xmath130 , such that @xmath141 for all @xmath142 and sufficiently large values of @xmath15 . according to ( * ? ? ? * lemma 1.1.5 ) , we have : @xmath143 where @xmath144 is some intermediate point between @xmath145 and @xmath146 . let the regularity conditions be fulfilled . then there exists a constant @xmath131 , such that @xmath147 for all @xmath148 and sufficiently large value of @xmath15 . using the markov inequality , we get @xmath149 according to ( * ? ? ? * lemma 1.1.5 ) , we have @xmath150 using the taylor expansion we get @xmath151 where @xmath144 is some intermediate point between @xmath152 and @xmath153 . hence , for sufficiently large @xmath15 providing @xmath154 , we have the inequality @xmath155 , and we obtain @xmath156 by distinguishability condition , we can write @xmath157 and hence @xmath158 and @xmath159 so , putting @xmath160 the estimate follows from and . the weak convergence of @xmath93 now follows from ( * ? ? ? * theorem 1.10.1 ) . in this section , we construct the score function test , the general likelihood ratio test , the wald test and two bayes tests . for all these tests we describe the choice of the thresholds and evaluate the limit power functions for local alternatives . let us introduce the _ score function test _ ( sft ) @xmath161 where @xmath162 is the ( @xmath163)-quantile of the standard normal distribution @xmath164 and the statistic @xmath165 is @xmath166.\ ] ] the sft has the following well - known properties ( one can see , for example , ( * ? ? ? * theorem 13.3.3 ) for the case of i.i.d . observations ) . the test @xmath167 and is laump . for its power function the following convergence hold : @xmath168 the property @xmath167 follows immediately from the asymptotic normality ( under hypothesis @xmath120 ) @xmath169 further , we have ( under alternative @xmath170 ) the convergence @xmath171 this follows from the le cam s third lemma and can be shown directly as follows . suppose that the intensity of the observed poisson process is @xmath172 , then we can write @xmath173\\ & \quad + \frac{1}{\sqrt{n{\rm i}\left(\vartheta _ 1\right)}}\sum_{j=1}^{n}\int_{0}^{\tau } \frac{\dot\lambda \left(\vartheta _ 1,t\right)}{\lambda \left(\vartheta _ 1,t\right)}\left[\lambda \left(\vartheta _ n , t\right)-\lambda \left(\vartheta _ 1,t\right)\right]{\rm d}t \\ & = \delta _ n^*\left(\vartheta_1,x^n\right)+\frac{u_*}{{n{\rm i}\left(\vartheta _ 1\right)}}\sum_{j=1}^{n}\int_{0}^{\tau } \frac{\dot\lambda \left(\vartheta _ 1,t\right)^2}{\lambda \left(\vartheta _ 1,t\right)}{\rm d}t+o\left(1\right)\\ & = \delta _ n^*\left(\vartheta_1,x^n\right)+u_*+o\left({1}\right)\longrightarrow \delta + u_*.\end{aligned}\ ] ] to show that the sft is laump , it is sufficient to verify that the limit of its power function coincides ( for each fixed value @xmath116 ) with the limit of the power of the corresponding likelihood ratio ( neyman - person ) test ( n - pt ) @xmath174 . remind that the n - pt is the most powerful for each fixed ( simple ) alternative ( see , for example , theorem 13.3 in lehman and romano @xcite ) . of course , the n - pt is not a real test ( in our one - sided problem ) , since for its construction one needs to know the value @xmath152 of the parameter @xmath153 under alternative . the n - pt is defined by @xmath175 where the threshold @xmath176 and the probability @xmath177 are chosen from the condition @xmath178 , that is , @xmath179 of course , we can put @xmath180 because the limit random variable @xmath181 has continuous distribution function . the threshold @xmath176 can be found as follows . the lan of the family of measures at the point @xmath68 allows us to write @xmath182 hence , we have @xmath183 therefore the n - pt @xmath184 belongs to @xmath185 . for the power of the n - pt we have ( denoting as usually @xmath186 ) @xmath187 therefore the limits of the powers of the tests @xmath188 and @xmath189 coincide , that is , the score function test is asymptotically as good as the neyman - pearson optimal one . note that the limits are valid for any sequence of @xmath190 . so , for any @xmath60 , we can choose a sequence @xmath191 $ ] such that @xmath192 which represents the asymptotic coincidence of the two tests and concludes the proof . let us remind that the maximum likelihood estimator ( mle ) @xmath193 is defined by the equation : @xmath194 where the likelihood ratio function is @xmath195{\rm d}t\right\},\qquad \vartheta \in \left [ \vartheta _ 1,b\right ) . \ ] ] the glrt is @xmath196 where @xmath197 the wald s test is based on the mle @xmath198 and is defined as follows : @xmath199 the properties of these tests are given in the following proposition . the tests @xmath200 and @xmath201 belong to @xmath202 , their power functions @xmath203 and @xmath204 converge to @xmath205 , and therefore they are laump . let us put @xmath206 and denote @xmath207 . we have @xmath208 according to theorem [ t1 ] ( with @xmath121 ) , we have the weak convergence ( under @xmath74 ) of the measure of the stochastic processes @xmath209 to those of the process @xmath210 . this provides us the convergence of the distributions of all continuous in uniform metric functionals . hence @xmath211 which yields ( we suppose that @xmath212 ) @xmath213 using the same weak convergence we obtain the asymptotic normality of the mle ( see @xcite or @xcite ) : @xmath214 and hence @xmath215 . so both @xmath216 and @xmath217 belong to @xmath202 . now , let us fix some @xmath116 and study the limit behavior of the power functions of the tests . using the weak convergence of the likelihood ratio process under the alternative @xmath218 , we have @xmath219 hence ( we suppose again that @xmath212 ) , @xmath220 similarly we have @xmath221 therefore the tests are laump . * example 1 . * as the family of measures is lan and the problem is asymptotically equivalent to the corresponding hypothesis testing problem for a gaussian model , we propose here a similar test for gaussian observations . suppose that the random variable @xmath222 and we have to test the hypothesis @xmath223 against @xmath224 . then the sft @xmath225 is the uniformly most powerful in the class of tests of size @xmath226 . its power function is @xmath227 . the log - likelihood function is @xmath228 the one - sided mle @xmath229 is given by @xmath230 and it is easy to see that the test @xmath231 and the wald test @xmath232 have identical power functions . let us note , that the asymptotic equivalence to the sft and the optimality is a well known property of these tests in regular statistical experiments ( see , for example , @xcite and @xcite ) . we present these properties here in order to compare the asymptotics of these tests in regular and singular situations ( see @xcite ) . in particular , we will see that the asymptotic properties of these tests in singular situations will be essentially different . suppose now that the unknown parameter @xmath35 is a random variable with _ a priori _ density @xmath233 , @xmath234 . here @xmath235 is a known continuous function satisfying @xmath236 . we consider two approaches . the first one is based on the bayes estimator and the second one on the averaged likelihood ratio function . the first test , wich we call bt1 , is a wald type test but based on the bayes estimator ( be ) @xmath237 : @xmath238 remind that the be for quadratic loss function is @xmath239 after the change of variables @xmath240 in the integrals , we obtain the relation @xmath241 the properties of @xmath102 established in the proof of theorem [ t1 ] yield the following convergence in distribution under the hypothesis @xmath120 ( see @xcite or @xcite ) @xmath242 where @xmath243 and @xmath244 are the density and the distribution function of the standard normal gaussian random variable @xmath245 . hence , if we take @xmath246 to be solution of the equation @xmath247 then the bt1 @xmath248 belongs to @xmath42 . a similar calculation under the alternative @xmath249 allows us to evaluate the limit power function of the bt1 as follows : @xmath250 another possibility in bayesian approach is to define the test as the test with the minimal mean error of the second kind . for a test @xmath54 , let us denote @xmath251 the error of the second kind and introduce the mean error of the second kind : @xmath252 the bayes test @xmath253 is defined as the test which minimizes this mean error : @xmath254 we can rewrite the above integral as follows @xmath255 where we denoted @xmath256 the distribution of the sample @xmath63 and @xmath257 the averaged power @xmath258 is the same as if we have two simple hypothesis . under @xmath259 we observe a poisson process of intensity function @xmath260 , and under the alternative @xmath261 the observed point process has random intensity and its measure is @xmath262 . this process is a mixture @xmath263according to the density @xmath264 of inhomogeneous poisson processes with intensities @xmath265 , @xmath266 . this means that we have two simple hypotheses and the most powerful ( neyman - pearson ) test is of the form @xmath267 where the averaged likelihood ratio @xmath268 to study this test under hypothesis we change the variables : @xmath269 the limit of the last integral was already described above and this allow us to write @xmath270 where @xmath244 and @xmath243 are again the distribution function and the density of the standard gaussian random variable @xmath245 . hence , if we take @xmath271 to be solution of the equation @xmath272 then the test @xmath273 , which we call bt2 , belongs to @xmath42 and coincides with the test @xmath274 if we put @xmath275 . a similar calculation under the alternative @xmath249 allows us to evaluate the limit power function of the bt2 as follows : @xmath276 below we present the results of numerical simulations for the power functions of the tests . we observe @xmath15 independent realizations @xmath277\right)$ ] , @xmath278 , of inhomogeneous poisson process of intensity function @xmath279 where @xmath280 . the fisher information at the point @xmath74 is @xmath281 . recall that all our tests ( except bayes tests ) in regular case are laump . therefore they have the same limit power function . our goal is to study the power functions of different tests for finite @xmath15 . the normalized likelihood ratio @xmath282 is given by the expression @xmath283 where @xmath284 . the numerical simulation of the observations allows us to obtain the power functions presented in figures [ pf_regular_2 ] and [ pf_regular_glrt_wald ] . for example , the computation of the numerical values of the power function of the sft was done as follows . we define an increasing sequence of @xmath153 beginning at @xmath285 . then , for every @xmath153 , we simulate @xmath286 i.i.d . observations of n - tuples of inhomogeneous poisson processes @xmath287 , @xmath288 , with the intensity function @xmath289 and calculate the corresponding statistics @xmath290 , @xmath288 . the empirical frequency of acceptation of the alternative gives us an estimate of the power function : @xmath291 we repeat this procedure for different values of @xmath153 until the values of @xmath58 become close to @xmath46 . power functions of sft and bt1 ] power functions of glrt and wt ] in the computation of the power function of the bayes test bt1 , we take as _ a priori _ law the uniform distribution , that is , @xmath292)$ ] . the thresholds of the bt1 are obtained by simulating @xmath293 random variables @xmath294 , @xmath295 , calculating for each of them the quantity @xmath296 and taking the @xmath297-th greatest between them . some of the thresholds are presented in table [ thr_bt1 ] . .[thr_bt1]thresholds of bt1 [ cols="^,^,^,^,^,^,^,^",options="header " , ] note that for the small values of @xmath15 , under alternative , the power function of sft starts to decrease ( see figure [ pf_regular_glrt_wald ] ) . this interesting fact can be explained by the strongly non linear dependence of the likelihood ratio on the parameter . the test statistic @xmath298 can be rewritten as follows : @xmath299\\ & \qquad+\sqrt{\frac{n}{{\rm i}\left(\vartheta_1\right)}}\int_{0}^{t}\frac{\dot\lambda \left(\vartheta _ 1,t\right)}{\lambda \left(\vartheta_1,t\right)}\left[\lambda \left(\vartheta _ 1+u\varphi_n , t\right)-\lambda \left(\vartheta_1,t\right ) \right]{\rm d}t\\ & = -3\varphi_n\sum_{j=1}^n\int_{0}^{3}\frac{t\sin(6 t)}{3\cos^2(3\,t)+1}\left[{\rm d}x_j\left(t\right)-\left(3\cos^2\!\left(\left(3+u\varphi_n\right)t\right)\!+1\right){\rm d}t \right]\\ & \qquad+9\sqrt{\frac{n}{{\rm i}\left(\vartheta_1\right)}}\!\int_{0}^{3 } \!\frac{t\sin(6 t)}{3\cos^2(3\,t)+1}\times\left[\cos^2(3\,t)- \cos^2\left(\left(3+u\varphi_n\,\right)t\right)\right]{\rm d}t . \ ] ] the last integral becomes negative for some values of @xmath153 , which explains the loss of power of the sft ( for @xmath300 ) . this study was partially supported by russian science foundation ( research project no . 14 - 49 - 00079 ) . the authors thank the referee for helpful comments .

we consider the problem of hypothesis testing in the situation when the first hypothesis is simple and the second one is local one - sided composite . we describe the choice of the thresholds and the power functions of the score function test , of the general likelihood ratio test , of the wald test and of two bayes tests in the situation when the intensity function of the observed inhomogeneous poisson process is smooth with respect to the parameter . it is shown that almost all these tests are asymptotically uniformly most powerful . the results of numerical simulations are presented . msc 2010 classification : 62m02 , 62f03 , 62f05 . _ key words : _ hypothesis testing , inhomogeneous poisson processes , asymptotic theory , composite alternatives , regular situation .

the dimensionality of the 115 materials , cerhin@xmath1 , ceirin@xmath1 , and cecoin@xmath1 , appears to be related to their superconducting transition temperature . the material with the highest t@xmath2 , cecoin@xmath0 , has the most 2d - like fermi surface ( fs ) of the three . @xcite cerhin@xmath0 has a high t@xmath2 ( @xmath32.1 k ) , but only under a pressure of @xmath316 kbar . at ambient pressures , cerhin@xmath0 is an anti - ferromagnet . the fs of cerhin@xmath0 was the subject of one of our recent publications.@xcite in order to confirm the link between the superconducting state and fs dimensionality , the fs as a function of pressure in cerhin@xmath0 should be measured . if the fs becomes more 2d - like as the critical pressure is approached , then this will be evidence for making a connection . in these materials it seems that superconductivity does not appear until the overlap between the _ f _ electron wavefunctions is sufficient to allow band - like behavior . measurements of the fs as a function of pressure should show this increasing overlap as a change in topography . here we present measurements up to 7.9 kbar , about half the critical pressure for cerhin@xmath1 . we have designed and built small pressure cells , capable of running in a dilution refrigerator and in a rotator . measuring torque inside a pressure cell is impossible , so we have made small compensated pickup coils which fit into the cell . each coil has four to five thousand turns . the filling factor approaches unity because we are able to situate the coil along with the sample inside the cell . a small coil is wound on the exterior of the cell to provide an ac modulation of the applied field . we have measured the fs of cerhin@xmath0 under several pressures . at each pressure we measure fs frequencies and their amplitude dependence as a function of temperature . from this we can extract information about how the effective mass of the quasiparticles is changing as the pressure is increased . the figures show the fourier spectra of cerhin@xmath0 under @xmath37.9 kbar . the crystal was oriented so that the a - b axis plane is perpendicular to the applied field . at @xmath37.9 kbar and at ambient pressures ( measured in the pressure cell prior to pressurization ) reveals little that is suggestive of change . ] we show the 7.9 kbar data compared with two sets of data taken at ambient pressure . in fig . [ highfft ] the fs at 7.9 kbar is compared with the ambient data taken with a torque cantilever ( the same data reported in @xcite ) . because the modulation field for the ac measurements ( in the pressure cell ) was so small , the lowest frequencies can be ignored . notice that the 1411 t ( f@xmath4 , the designation given in ref . @xcite ) and 1845 t peaks are reproduced exactly in the ambient and the pressure data sets . the 1845 t peak was not included in ref . @xcite because of its small amplitude in ambient pressure torque measurements . the 3600 t ( f@xmath5 ) and 6120 t ( f@xmath6 ) peaks are present in both data sets ; however , the f@xmath5 appears to have split and the f@xmath6 appears to have shifted down in frequency . such changes could be explained as slight differences of sample alignment with respect to the applied field between the torque measurement and the pressure cell measurement . three other frequencies , 2076 t , 2710 t , and 4613 t , emerge in the pressure data which are close to to some reported in ref . @xcite to be observed only at the lowest temperatures ( 25 mk ) . all but the first of these frequencies are seen also in ambient pressure data taken with the sample in the pressure cell prior to pressurization as shown in fig . [ lowfft ] . thus , assuming the differences in frequency between the torque measurements and pressure cell measurements are due to differences in alignment , we can make frequency assignments that follow ref . @xcite ( also shown in fig . [ lowfft ] ) . the relative increase in amplitude with increasing pressure of these three peaks could be a result of the increase of the coupling factor between the sample and the coil as the two are compressed together . the lack of any clear differences in the fs up to 7.9 kbar suggests that if the fs changes , then such change is not a linear function of pressure . nor is there a compelling reason to think that it should be a linear function . possibly , at some pressure closer the the critical pressure , the transition to _ f _ electron itinerate behavior will take place leading to more noticable changes in the fs . the fs of cerhin@xmath1 appears to remain topographically stable under the application of pressure up to 7.9 kbar . additional measurements which approach the critical pressure ( @xmath316 kbar ) are of prime importance . this work was performed at the national high magnetic field laboratory , which is supported by nsf cooperative agreement no . dmr-9527035 and by the state of florida . work at los alamos was performed under the auspices of the u. s. dept . of energy . donavan hall , e.c . palm , t.p . murphy , s.w . tozer , eliza miller - ricci , lydia peabody , charis quay huei li , u. alver , r.g . goodrich , j.l . sarrao , p.g . pagliuso , j. m. wills , and z.fisk . b _ * 64 * , 064506 ( 2001 ) , cond - mat/0011395

measurements of the de haas - van alphen effect have been carried out on the heavy fermion anti - ferromagnet cerhin@xmath0 at temperatures between 25 mk and 500 mk under pressure . we present some preliminary results of our measurements to track the evolution of the fermi surface as the pressure induced superconducting transition is approached . , , , , , de haas - van alphen ; heavy fermions ; superconductivity ; high pressure

it all began with the idea of an intrinsic limit to hadron thermodynamics . during the past fifty years , different conceptual approaches had led to an ultimate temperature of strongly interacting matter . pomeranchuk @xcite first obtained it from the finite spatial extension of hadrons : a hadron can only have an independent existence if it has an independent volume . then hagedorn @xcite arrived at a limiting temperature by postulating a self - similar hadronic resonance composition : a resonance consists of resonances which consist of resonances , and so on . the resulting excitation spectrum was later also derived in the dual resonance model @xcite . with the advent of the quark infrastructure of hadrons and of quantum chromodynamics , it became clear that the ultimate temperature found in all these considerations was really a transition point to a new state of matter , to a plasma of deconfined quarks and gluons @xcite . statistical qcd , in particular in the finite temperature lattice formulation , has subsequently confirmed this hypothesis : at a now rather well determined temperature ( for vanishing baryon density , @xmath0 mev ) , strongly interacting matter undergoes a transition from a medium of color singlet hadronic constituents to one of deconfined colored quarks and gluons @xcite . the energy density at the transition point was found to be @xmath1 gev/@xmath2 . moreover , the transition turns a hadronic state of spontaneously broken chiral symmetry into a quark - gluon plasma in which this symmetry is restored : at @xmath3 , the effective constituent quark mass of some 0.3 gev vanishes , and we recover the bare quark mass of the qcd lagrangian . the obvious desire to test this fascinating phase structure of strongly interacting matter first led to the fixed target experiments at the ags in brookhaven ( with @xmath4 gev ) and at the cern - sps ( with @xmath5 gev ) . in 1986/87 , light ion beams on heavy ion targets started the program , and in 1994/95 , heavy ion beams followed . today , much of this program is concluded . so , what have we learned during the past fifteen years ? in this opening talk , i will address that question by asking : * what did we expect to find ? * what did we find ? * what does that tell us ? in my report , i will first recall briefly the expectations concerning signatures at the beginning of the experimental heavy ion program at the ags and sps in 1986 and then summarize what had really been found when it was ( in first generation experiments ) completed in 2000 . following this , i will try to indicate what conclusions can be drawn from these results , for the conditions reached , from the hard probes of the early stages and from the observed hadronisation pattern at freeze - out . the evolution of a high energy nucleus - nucleus collision was pictured in the form shown in fig . after a rather short equilibration time @xmath6 fm , the presence of a thermalized medium was assumed , and for sufficiently high initial energy densities , this medium would be in the quark - gluon plasma phase . -4 mm -8 mm -4 mm the initial energy density of the produced medium at the time of thermalization was ( and still is ) generally determined by the bjorken estimate @xcite = ( dn_h dy)_y=0 w_h r_a^2 _ 0 , [ 2.1 ] where @xmath7 specifies the number of hadronic secondaries emitted per unit rapidity at mid - rapidity and @xmath8 their average energy . the effective initial volume is determined in the transverse plane by the nuclear radius @xmath9 , and longitudinally by the formation time @xmath10 of the thermal medium . the temperature of the produced medium was assumed to be observable through the transverse mass spectrum of thermal dileptons and the momentum spectrum of thermal photons @xcite . the observation of thermal dilepton / photon spectra would also indicate that the medium was indeed in thermal equilibrium . the functional form of the spectra is the same for radiation from hadronic matter and from a qgp ; but the observed rates and temperatures were expected to differ in the two cases . it was clear from the beginning that these signals would be emitted during the entire thermal evolution of the system , making a separation of different phases of origin not very straight - forward . the determination of the nature of the hot initial phase required a signature sensitive to deconfinement . it was argued that in a deconfined medium the j/@xmath11 would melt through color screening @xcite and that , therefore , qgp production should lead to a suppression of j/@xmath11 production in nuclear collisions , compared to the rates extrapolated from @xmath12 data . similarly , the qgp was expected to result in a higher energy loss for a fast passing color charge than a hadronic medium , so that jet quenching @xcite should also signal deconfinement . the behavior of sufficiently short - lived resonances , in particular the dilepton decay of the @xmath13 , was considered as a viable tool to study the hadronic medium in its interacting stage and thus provide information on the approach to chiral symmetry restoration @xcite . the expansion of the hot medium was thought to be measurable through broadening and azimuthal anisotropies of hadronic transverse momentum spectra @xcite . the size and age of the source at freeze - out was assumed to be obtainable through hanbury - brown twiss ( hbt ) interferometry based on two - particle correlations @xcite . it was expected that increasing the collision energy would increase the density and hence the expansion of the produced medium , so that the hbt radii should grow with @xmath14 . the final interacting hadronic medium was discussed in terms of an ideal resonance gas , following an old suggestion @xcite brought to hadron physics by hagedorn @xcite : an interacting system of elementary constituents can be replaced by a non - interacting gas of resonances , provided the elementary interactions are resonance - dominated . this would provide the relative abundances of all hadron species in terms of just two parameters , the temperature and the baryon number density . one particularly interesting feature here was the fact that in elementary hadronic interactions , an overall reduction of strangeness production was observed . nuclear collisions , in particular if leading to a qgp as initial stage @xcite , were expected to remove this reduction and lead to strangenes production in accord with thermal predictions . the initial energy density , as specified by the bjorken estimate , eq . ( [ 2.1 ] ) , was measured in almost all sps experiments . in fig . [ e - dens ] we show @xmath15 as function of centrality , determined by the number of participant nucleons @xcite ; it covers the range from somewhat above 1 to almost 3.5 gev/@xmath2 . finite temperature lattice calculations , as already mentioned , give for the energy density at deconfinement , @xmath16 , values around or slightly below 1 gev/@xmath2 @xcite . however , also high energy @xmath17 collisions lead to energy density estimates well above 1 gev/@xmath2 . -8 mm [ cols="^,^ " , ] -8 mm -4 mm from percolation theory @xcite , it is known that the formation of large - scale clusters is a critical phenomenon ; it does not occur gradually as function of parton density . in the ` thermodynamic ' limit of large nuclei , the cluster size diverges at a specific critical parton density , and even for finite systems it suddenly increases in a very narrow band of density . thus there exists a statistical approach to critical behavior which does not pre - suppose thermalization and which can be applied to the pre - equilibrium stage of the nuclear collision evolution @xcite . in a central high energy nucleus - nucleus collision , the partons in the two nuclei lead in the transverse plane to an initial condition schematically illustrated in fig . [ pp ] . the transverse size of the partons is essentially determined by the intrinsic transverse momentum , @xmath18 . the number of partons contained in a nucleon is known from deep inelastic scattering experiments . it is parametrized by a parton distribution function depending on the fraction @xmath19 ( denoting parton and nucleon momenta by @xmath20 and @xmath21 , respectively ) and on the scale @xmath22 used to resolve the nucleonic parton structure . while in lepton - hadron scattering the scale is set by the virtual photon , in minimum bias nucleon - nucleon or @xmath23 collisions it is determined by the transverse momentum of the partons themselves . we denote the nuclear radius in fig . [ pp ] by @xmath24 , the average parton radius by @xmath25 , and then study the variation of the average cluster size as function of the parton density @xmath26 . in the limit @xmath27 , the cluster size diverges at the percolation threshold @xmath28 : s_cl ~(n_p -n)^- , [ pp1 ] with the critical exponent @xmath29 @xcite . percolation thus specifies the onset of connection as a critical phenomenon . for finite @xmath24 , there is a pronounced but finite peak at a slightly shifted density , as shown in fig . [ cluster ] . -6 mm -8 mm -6 mm to apply this formalism to nuclear collisions @xcite , we need the transverse parton size and the effective number of partons for a given collision configuration . the distribution of partons in a nucleon is determined in the analysis of deep inelastic scattering data ; per nucleon in a nucleon - nucleon collision , at mid - rapidity and for c.m.s . energy @xmath14 , one has n_parton(x , q^2 ) = ( dn_q dy)_y=0 = xg(x , q^2 ) + , [ pp2 ] where @xmath30 label the gluon , quark and antiquark distributions , respectively , and the sum runs over the quark species . at @xmath31 , the fractional parton momentum @xmath19 becomes @xmath32 . as shown in fig . [ pp ] , there is a distribution of partons of different transverse sizes up to the resolution scale @xmath33 ; we approximate this by the average value @xmath34 . in nucleus - nucleus collisions at sps energies , it is a good approximation to consider that the activated partons originate from wounded or participant nucleons . at higher energies , there will also exist collision - dependent contributions @xcite . we thus obtain for central @xmath23 collisions the percolation condition n_parton(x , q^2 ) = 1.13 q^-2 ; [ pp3 ] it determines the onset of color connection as function of @xmath35 and of the c.m . collision energy @xmath14 . at @xmath36 gev , we obtain @xcite : for @xmath37 , no color connection , for @xmath38 , a connected parton condensate . the parton condensate formed beyond the percolation point consists of overlapping and hence interacting partons from all involved nucleons . it thus constitutes a deconfined pre - thermal medium , which is a necessary precursor for qgp formation , since color connection is a prerequisite for thermalization . the parton condensate is characterized by a scale @xmath39 , where @xmath40 specifies the onset of percolation for central collisions at a given @xmath35 and @xmath41 . the average transverse momentum of partons in the condensate is determined by this scale , so that @xmath42 is in a sense a precursor of temperature . increasing @xmath35 or @xmath14 leads to higher @xmath42 , indicating something like a hotter " medium . we have here concentrated on the onset of parton condensation as a critical phenomenon determined by percolation of quarks and gluons as geometric entities in the transverse plane . the behavior of the dense parton condensate in the limit of large @xmath35 and @xmath14 , the color glass condensate , has been studied intensively over the past years in terms of classical color fields . it is becoming increasingly clear from these studies that the pre - equilibrium stage plays a much more decisive role in nuclear collisions than previously envisioned . in the last part of this section , we consider one interesting experimental consequence of the onset of parton condensation . to study this onset as function of centrality in a given @xmath43 collision , we have to replace @xmath44 in eq . ( [ pp3 ] ) by the density of wounded nucleons @xmath45 calculated for a given impact parameter @xmath46 , using the actual nuclear profile ( woods - saxon ) . for pb - pb collisions at @xmath47 gev , this leads to for @xmath48 , no color connection , for @xmath49 , a connected parton condensate . instead of the impact parameter , we have here used the number of wounded nucleons to specify the collision centrality , with @xmath50 for @xmath51 . note that at this threshold point , the bjorken estimate for the energy density gives @xmath52 gev/@xmath2 . this clearly shows that the onset of color connection only occurs at a much higher energy density than the value given by finite temperature lattice qcd for a thermalized medium . as mentioned above , charmonium suppression in nuclear collisons was one of the proposed signatures for quark - gluon plasma formation . in the light of present thinking , we have to consider the fate of the charmonium states already in the parton condensate phase . this can be addressed in different ways , invoking generalized color screening @xcite , dipole break - up in a random color field @xcite , etc . we shall here simply consider the intrinsic scale @xmath53 of a given charmonium state @xmath54 ( j/@xmath11 , @xmath55 , @xmath56 ) and assume that when it is exceeded by the condensate scale , q_s > q_i , [ pp4 ] the charmonium state is dissolved . we are thus assuming that @xmath42 plays in the pre - equilibrium state the role of the critical temperature or of the screening mass in a thermal medium . to show the consequences of this assumption , we must first recall that j/@xmath11 production in hadronic collisions occurs in part through feed - down from higher excited states : about 60% of the observed j/@xmath11 in proton - proton collisions are directly produced @xmath57 states , the remaining 40% coming from @xmath55 ( @xmath5830% ) and @xmath56 ( @xmath5810% ) decays . the intrinsic scales of these states are given by their radii , r_j/ ( 0.9 gev)^-1 , r _ ( 0.6 gev)^-1 , r_^ ( 0.6 gev)^-1 . [ pp5 ] -8 mm -8 mm -4 mm from the centrality dependence of @xmath42 shown in fig . [ q - s ] , we find that , at sps energies , charmonia suppression sets in at @xmath49 for the @xmath55 and @xmath56 contributions , @xmath59 for direct j/@xmath11 production , to be compared with the observed pattern shown in fig . [ na50 ] . obviously these results must be studied in more detail , but it appears quite likely that the onset of j/@xmath11 suppression indeed indicates an onset of deconfinement , while not implying any thermalization . -8 mm -8 mm -4 mm so let us summarize what we have learned in the first fifteen years of ultra - relativistic heavy ion collisions . we know that there is an intrinsic limit to hadronic matter , in accord with the phase diagram determined in statistical qcd . the confinement boundary is established by the sps / ags program and agrees , both qualitatively and quantitatively , with the predictions of lattice qcd . i believe that the rhic and lhc experiments can only reconfirm this . for small @xmath60 , we know the onset of hadronization , and we know that strangeness enhancement , transverse momentum broadening and hbt radii saturate at this onset . the early stage of the medium produced in nuclear collisions at the sps is partonic . with increasing @xmath35 , the partons begin to form connected clusters , and at a certain critical density , they form a color condensate consisting of interacting partons from many different nucleons . hence , at this point , color deconfinement begins : partons no longer have a clear origin or exist in well - defined numbers . the parton condensate is a necessary precursor of the qgp : it provides the interacting deconfined partons which , if given enough time for equilibration , can make a qgp . charmonium states of different binding energies and spatial sizes probe different parton scales of the produced initial state . the observed step - wise form of anomalous j/@xmath11 suppression appears to provide the first signal for color deconfinement through parton condensation ; this does not require any thermalization . the forthcoming j/@xmath11 production measurements at full sps energy but with lower @xmath35 , to be performed by na60 , should further clarify this point . in particular , the comparison between pb - pb and in - in j/@xmath11 suppression patterns should determine the critical densities and scales . pioneering results in physics always need to be reconfirmed . but if these limits of confinement survive the test of time , we will remember that they were first found in sps / ags experiments . so , at the beginning of the collision evolution at the sps , there appears to be color deconfinement ; at the end , thermalization and collective behavior . to produce a quark - gluon plasma medium , we need to have both at the early stage . let us see if the future data from experiments at the much higher rhic and lhc energies can achieve this . but no matter how well - defined the road for further exploration may seem to be , i am sure that the forthcoming studies will rediscover the one feature which has made this field so challenging and exciting . it is perhaps best summarized by the spanish poet antonio machado @xcite : _ caminante , son tus huellas el camino , y nada ms ; _ _ caminante , no hay camino , se hace camino al andar . _ traveller , the road is nothing more than your footprints ; _ _ traveller , there is no road , you make it as you go . it is a pleasure to acknowledge the help of many colleagues in the preparation of this survey ; special thanks go to p. braun - munzinger , s. digal , s. fortunato , f. karsch , d. kharzeev , m. nardi , p. petreczky , k. redlich , h. specht , u. wiedemann and , in particular , to c. loureno . i am grateful to l. vzquez for literary support .

the study of high energy nuclear collisions has entered a new stage with rhic ; it therefore seems a good time to ask what we have learned from the experimental results obtained up to now . i recall what we had expected to find when the sps and ags programs were started , summarize what actually was found , and then try to assess what we have learned from the results .

in theory , gravitational lenses can be used to address astrophysical problems such as the cosmological model , the structure and evolution of galaxies , and the structure of quasar accretion disks ( see the reviews by kochanek ( @xcite ) of strong lensing and wambsganss ( @xcite ) of microlensing ) . one of the main challenges in using lenses for any of these applications is discovering large numbers of lenses efficiently ( see the review of lens surveys in kochanek ( @xcite ) ) . most known lenses have been found either from optical imaging surveys of known quasars ( see pindor et al . @xcite for a recent study ) , radio imaging surveys of flat - spectrum radio sources ( see browne et al . @xcite ) , or searches for anomalous , higher redshift emission lines in galaxy spectra ( see bolton et al . imaging surveys of all radio sources ( burke @xcite ) proved difficult because of the confusing array of structures observed for steep spectrum radio sources . haarsma et al . ( @xcite ) proposed improving the efficiency of searches for lensed steep - spectrum sources by looking for radio lobes with optical counterparts , but the approach is limited by the resolution and sensitivity of existing all - sky radio surveys . none of these methods is easily applied to the next generation of large scale imaging surveys such as the sdss supernova survey ( sako et al . @xcite ) , the dark energy survey ( des , abbott et al . @xcite ) , pan - starrs ( kaiser @xcite ) and the large synoptic survey telescope ( lsst , tyson et al . @xcite ) . one possibility is to use a combination of color and morphology to identify quasar lens candidates ( morgan et al . this strategy can be effective as long as emission ( or absorption ) by the lens galaxy does not significantly change the color of the system from that of the quasars , which restricts its applicability to systems in which the quasar images are significantly brighter than the lens galaxy . a new feature of all these projects , however , is that they are synoptic surveys which obtain light curves for variable sources . pindor ( @xcite ) suggested that the synoptic data could be used to find lenses by cross - correlating the light curves of closely separated sources to search for the time delays present in the lensed systems . this approach may be problematic as a search method because it requires the automated extraction of light curves for the individual lensed images , some of which may also be distorted by the effects of microlensing . however , it will be an essential component of verifying lens candidates in the synoptic surveys . in this paper we introduce a far simpler strategy . unlike almost any other source , lensed quasars are `` extended '' variable sources because the variable flux is spread out over the scale of the image separations . as we discuss in 2 , restricting the search to extended variable sources is an extraordinarily powerful means of eliminating sources other than gravitational lenses . in 3 we demonstrate the method using data we have been acquiring to measure time delays and microlensing variability in known lensed quasars ( kochanek et al . we summarize our proposed search in 4 . the basic problem in lens searches is that they are intrinsically rare objects . we start with the problem that quasars are relatively rare . [ fig : starcount ] shows the surface density of quasars ( @xmath1 ) computed from the g - band 2slaq quasar luminosity functions ( richards et al . @xcite ) . for these models , the surface density at 23 mag is approximately @xmath2 deg@xmath3 . lensed quasars are rarer still , since a conservative estimate for the lensing probability of these faint quasars is @xmath4 ( see the review of lens statistics in kochanek @xcite ) . thus , while the number of faint , lensed quasars is almost two orders of magnitude greater than the number of lenses presently known , it is not a trivial problem to find the one lensed quasar in each 5 deg@xmath5 region given the @xmath6 other sources in the same area . the problem is further complicated by the increasing importance of the contribution of the lens galaxy flux to the total flux of the lens as we search for fainter lensed sources . the lens galaxy masks both the color and morphology of the lensed images , making traditional quasar selection methods useless . the key to our approach is to apply difference imaging ( alard & lupton @xcite , alard @xcite ) to the synoptic data from large imaging surveys . some version of difference imaging will be used in all these surveys as the basis for identifying variable sources and extracting light curves . difference imaging works by scaling , in both flux and psf substructure , a reference image to match the data obtained for each epoch and then subtracting the two to form a series of difference images @xmath7 . the difference image has flux only for objects that have varied between the reference image and the epoch under consideration , so it has the immediate advantage of eliminating all the galaxies . we focus on time variability because quasars are intrinsically variable sources . on two year time scales , roughly 50% of quasars vary by more than 0.1 mag ( e.g. cimatti et al . @xcite ) with general correlations that fainter quasars observed at bluer wavelengths show greater variability ( vanden berk et al . . the variability of lensed quasars will be still greater than that of unlensed quasars because they are also microlensed by the stars in the lens galaxy ( see wambsganss @xcite ) . we will conservatively assume that fraction @xmath8 of detected quasars will show 10% flux variations during the course of the survey . we can divide variable sources into three general categories : variable point sources ( stars , quasars , supernovae and other explosive events ) , moving solar system objects ( asteroids , kuiper belt objects ) , and gravitational lenses . let us consider what these three classes of objects look like in an image of the variable flux formed by computing the absolute value or root - mean - square ( rms ) of the differenced images for a field . the point sources are variable but do not move , so they will appear as point sources . solar system objects move rapidly across the field , appearing as a sparsely realized `` track . '' small separation gravitationally lensed quasars appear as extended , non - circular objects , while wider separation lensed quasars appear as very closely separated groupings of variable objects . we will refer to them as examples of `` extended '' variable objects . while resolved four - image lenses are virtually impossible to mimic with any other source , two - image lenses can be mimicked by several other sources . the least interesting backgrounds are variable star pairs or variable star / quasar pairs which occur simply because of chance superpositions . two astrophysically interesting backgrounds are binary quasars , whose abundance on separations @xmath9 is comparable to that of gravitational lenses ( see hennawi et al @xcite ) , and lensed supernovae . very crudely , if there is one detectable supernovae per @xmath10 mag galaxy per century , then the abundance of lensed supernovae is comparable to that of lensed quasars . both binary quasars and lensed supernovae can be easily distinguished from lensed quasars based on their light curves . this can be done rapidly for the case of supernovae , but may take @xmath11years for the case of a binary quasar . we focus on comparing the surface density of lenses to the surface density of variable star pairs , since they represent the least interesting background . star pairs that can be resolved in normal seeing are separated by large physical distances and should show no significant correlations in their activity . this makes it straight forward to estimate the background of uninteresting extended variable sources . while in general the fraction of stars that are variable will vary greatly depending on the stellar population observed , for a normal " stellar mix about 1 - 2% of stars vary by more than 1% ( e.g. hartman et al . moreover , studies of variable quasars located behind the smc showed that their main stellar contaminant in the color - variability space are massive be stars , which will be rare away from the galactic plane ( dobrzycki et al . @xcite ) . since we will be interested in regions well away from the galactic plane and the significantly higher long term variability amplitudes of quasars , we will assume that fraction @xmath12 of stars will have variability comparable to that of quasars . thus , if there are @xmath13 deg@xmath3 stars with v@xmath10 mag at high galactic latitude ( bahcall & soneira @xcite ) the surface density of variable stars is @xmath14 deg@xmath3 . the surface density of ( uncorrelated ) pairs of variable stars separated by @xmath15 should be of order @xmath16 for a surface density 1/300 deg@xmath5 at v@xmath10 with @xmath17 that is well below that of gravitational lenses . [ fig : starcount ] shows the expected surface density of stars , variable stars and variable stars pairs ( @xmath17 ) as a function of galactic latitude ( at @xmath18 ) and v - band magnitude in the bahcall & soneira ( @xcite ) star count models . the surface density of star pairs with @xmath17 is comparable to the surface density of variable stars . variable star pairs and variable star / quasar pairs are comparable in abundance , with the former being being more common for v@xmath19 mag and the latter more common at fainter magnitudes . comparing the surface density of variable star pairs and lensed quasars , we see that samples of extended variable objects should be dominated by gravitational lenses rather than chance superpositions . in order to demonstrate our idea , we analyzed our monitoring data for four lenses using the isis difference imaging package ( alard @xcite ) . each epoch consisted of a 15 min r - band exposure ( taken as three 5 min exposures ) with the smarts 1.3 m telescope using andicam ( depoy et al . we analyzed only the images with seeing fwhm @xmath20 , sky backgrounds @xmath21 adu / pixel , and flux calibrations relative to the reference image @xmath22 ( to eliminate data taken through clouds ) . the four lenses we considered were qj 01584325 ( morgan et al . @xcite ) , sdss 0924 + 0219 ( inada et al . @xcite ) rxj 11311231 ( sluse et al . @xcite ) , and q 2237 + 0305 ( huchra et al . rxj11311231 consists of a bright cusp triple spanning 23 separated by 32 from a much fainter fourth image . sdss 0924 + 0219 is a more compact four - image lens . it has an einstein ring diameter of 17 and the flux is dominated by the brightest image . qj 01584325 is a still more compact two - image lens with an image separation of @xmath23 . in our monitoring program we only include lenses with image separations greater than @xmath24 , so we can not provide examples with smaller separations . this is not a significant bias since the median separation of gravitational lenses is approximately @xmath25 ( e.g. browne et al . @xcite ) and the surveys should have better resolution than our smarts data .. we do , however , include q2237 + 0305 , a four image lens with a 18 einstein ring diameter buried in the bulge of a very bright ( b@xmath26 mag ) low redshift spiral galaxy . we used 71 , 37 , 57 and 26 epochs of data for qj 01584325 , rxj 11311231 , sdss 0924 + 0219 , and q 2237 + 0305 , respectively . for each lens we computed the average of the images @xmath27 , its estimated noise @xmath28 , and the rms of the difference images @xmath29 . in computing @xmath30 , we rejected the two epochs with maximum values of @xmath31 for each pixel in order to remove satellite trails and low level cosmic ray events which had not been found by earlier processing procedures . the overall signal - to - noise ratio in the combined images is very high , with a point source sensitivity of roughly r@xmath32 mag . we compute the significance of the variability using the ratio @xmath33 between the variance image and the noise in the average image , a ratio which should be unity in the absence of variability or systematic errors . in practice , our rms images @xmath30 are limited by systematics beyond the point ( roughly 10 images ) that the statistical errors approach 1% of the sky level . the limitation is presumably due to systematic problems associated with flat fielding , interpolation , and the difference imaging algorithms . figs . [ fig : lenses1 ] and [ fig : lenses2 ] show our four examples , comparing the average image @xmath34 to a map of the significance of the variability , @xmath33 . the first point to note is the complete vanishing of everything in the field other than the lens . the second point to note is that all four lenses are easily recognized as multi - component / extended sources without any need for further processing . the appearance of the four - image lenses ( sdss 0924 + 021 , rxj 11311231 and q 2237 + 0305 ) is particularly striking . to be fair , this is true for all these lenses but q 2237 + 0305 in the direct images as well . on the other hand , emission from the lens galaxy masks a steadily increasing fraction of lensed quasars fainter than 20 mag , so q 2237 + 0305 is more `` typical '' of the faint quasar lenses that will comprise the majority of the systems detectable in the deep synoptic surveys than the other three systems . in the full 30 arcmin@xmath5 fields , there were no other variable sources . thus , to provide a comparison to the behaviour of the lenses , we applied the same procedures to the field of the microlensing event ogle-2005-smc-001 using 62 5 min i - band observations obtained with andicam by the microfun collaboration ( r. pogge , private communication ) . [ fig : smc ] shows the average and variability images for a region with the same size as was used in figs . [ fig : lenses1 ] and [ fig : lenses2 ] . in this case , the field contains four strongly variable sources , the microlensing event and three long period variable ( lpv ) stars . all four variable sources are obviously stellar in the variability map and would not be flagged as lens candidates even though they are all blended with other sources in the direct image . almost all new , large scale imaging surveys will be synoptic surveys that monitor the time variability of sources in the survey area . by using difference imaging to search for extended variable sources , these surveys can easily identify lensed quasars because almost all other variable sources are either point sources or orbital tracks created by solar system objects . we estimate that gravitational lenses are the most common extended variable sources for galactic latitudes @xmath35 , with modest contamination from pairs of variable stars , variable star / quasar pairs and binary quasars . limiting the search to extended variable sources reduces the number of non - lens background objects by more than @xmath36 . thus , it should be relatively straight forward for sdss , pan - starrs , des or lsst to identify the lensed quasars in their respective variability survey areas . for lsst , this should amount to roughly @xmath37 lensed quasars to v@xmath10 mag . note that the criterion of being an extended variable source can be combined with other criteria to further reduce the rate of false positives based on other information available from the same survey . for example , the quasars will have slowly varying aperiodic light curves , while many stars will show more rapid variability or periodic variability . where the source is resolved , the light curves of lensed quasars should be similar and can be cross - correlated to measure a time delay and verify that the source is a lens . note , however , that our discovery method depends only on the existence of variability rather than the measureability of the delay . the colors of stars and quasars are different at most redshifts ( e.g. richards et al . @xcite ) , and the colors of lensed images should be similar , up to concerns about differential extinction in the lens ( e.g. falco et al . finally , in the average image it should be possible to detect a lens galaxy , potentially using difference imaging methods to accurately subtract the quasar contribution . in general , the background of non - lens sources can be so greatly suppressed that we suspect the only significant issue for candidate selection will be systematic errors in identifying extended variable sources that are presently difficult to quantify . kochanek , c.s . , 2004 , strong gravitational lensing , part 2 of gravitational lensing : strong weak & micro , proceedings of the 33rd saas - fe advanced course , g. meylan , p. jetzer & p. north , eds . , ( springer - verlag : berlin ) [ astro - ph/0407232 ] wamsganss , j. , , 2004 , gravitational microlensing , part 3 of gravitational lensing : strong weak & micro , proceedings of the 33rd saas - fe advanced course , g. meylan , p. jetzer & p. north , eds . , ( springer - verlag : berlin )

we demonstrate that gravitationally lensed quasars are easily recognized using image subtraction methods as time variable sources that are spatially extended . for galactic latitudes @xmath0 , lensed quasars dominate the population of spatially extended variable sources , although there is some contamination from variable star pairs , variable star - quasar pairs and binary quasars that can be easily controlled using other information in the survey such as the object light curves and colors . this will allow planned large - scale synoptic surveys to find lensed quasars almost down to their detection limits without the need for extensive follow - up observations .

primordial gravity waves are expected to be produced during cosmic inflation in addition to scalar perturbations . if indeed present , they would leave a characteristic footprint on the polarized anisotropies of the cosmic microwave background ( cmb ) , as they are considered to be essentially the sole source of the so - called primordial @xmath0-mode residing at the super - horizon scales at the time of the last scattering . a detection of the @xmath0-mode angular power spectrum at large angular scales would be then treated as a smoking gun of inflation , while a precise measurement of its amplitude would constrain the energy scale of inflation , or , geometrically speaking , the expansion rate of the universe during inflation @xcite . this amplitude is expressed by the tensor - to - scalar ratio , @xmath2 , defined as the relative power of primordial gravity waves with respect to that of the scalar perturbations at some pivot scale @xmath8 , chosen here to be equal to @xmath9 mpc@xmath10 . currently , the most stringent upper bound on @xmath2 using temperature anisotropies has been derived by the planck collaboration : @xmath11 at 95% cl @xcite , while a recent joint analysis of the planck and polarized data set an upper limit @xmath12 at 95% cl @xcite . the measurement of the tensor - to - scalar ratio @xmath2 could allow to discriminate between different inflationary models . in particular , if this upper bound @xmath13 is indeed realized in nature , this would imply a rather high amount of primordial gravity waves thus favoring large - fields inflationary models @xcite . at smaller angular scales , the @xmath0-mode is dominated by the lensing induced signal . this signal is generated by the gravitational lensing of the cmb photons due to the large scale structure @xcite . the lensing contribution is well - understood from a theoretical point of view and can be uniquely predicted given the primary @xmath1-modes anisotropies and the lensing deflection field @xcite . such predictions have been been recently confirmed by the sptpol @xcite and polarbear experiments @xcite , with also constraints on the cmb lensing @xmath0-mode power spectrum @xcite . the lensing @xmath0-mode signal does not depend on @xmath2 . it therefore acts as a source of an additional noise masking the primordial , @xmath2-dependent @xmath0-mode , and making its detection more difficult . striving for a detection of @xmath2 , one has to either try to remove this lensing signal @xcite or rely solely on the large angular scales . in this latter case two features of the primordial @xmath0-mode spectrum are of particular interest as they are anticipated to be particularly prominent . these are so - called reionization and recombination bumps peaking at @xmath14 and at @xmath15 , respectively . + measuring @xmath0-mode is made even more difficult by the fact that measurements as performed by the majority of current experiments , which scan the sky in order to produce its maps , are straightforwardly expressed only in terms of the stokes parameters , @xmath16 and @xmath17 . the @xmath1- and @xmath0-mode are mathematically related to the stokes parameters @xcite and can be therefore recovered from the observational data . this however is only simple , if full - sky data were available . in contrast , realistic cmb experiments provide maps of polarized anisotropies , which only cover a reduced fraction of the celestial sphere , ranging from @xmath18 for balloon - borne and ground - based experiments to @xmath19 for satellite missions . in the context of pseudospectrum estimation of the angular power spectra on an incomplete sky part of the @xmath1-mode signal is unavoidably mislabelled as @xmath0-modes and vice verse . though such leakages can be corrected on average , the leaked signal inevitably contributes to the sampling variance of the other reconstructed spectrum . this dramatically increases the uncertainties of the estimated @xmath0-mode spectrum since the cosmological @xmath1-mode is expected to be at least two orders of magnitude higher than the @xmath0-mode in terms of their power spectrum @xcite . the nature of the leakages and approaches to their removal were investigated in ref . @xcite and a relevant pseudospectrum estimator , referred to as the pure pseudospectrum estimator , was proposed subsequently in ref . this estimator has been thoroughly investigated and extended to include an optimization of the sky apodization @xcite , cross - spectrum approaches @xcite , and , @xmath20 and @xmath21 cross - correlations @xcite . alternative constructions of pseudospectrum estimators correcting for @xmath1-to-@xmath0 leakages have been also proposed @xcite . nevertheless , the pure pseudospectrum method has been found the most mature and efficient one , particularly due to its ability of optimizing the sky apodizations @xcite , making it a method of choice for many practical applications . it is worth pointing out that the leakages are indeed ubiquities and correcting for them is as mandatory for small - scale experiments , covering @xmath18 of the sky , as for satellite - like missions , with access to as much as @xmath19 of the sky @xcite . though the impact of the @xmath1-to-@xmath0 leakage on the variance of the @xmath0-mode power spectrum is generally acknowledged , it is rarely included in projecting performance of planned cmb experiments or instrumental concepts from the point of view of their setting constraints on the tensor - to - scalar ratio , @xmath2 . instead , the major body of work ( see @xcite for some recent examples ) in this area is based on simplified mode - counting arguments ( see , however e.g. , @xcite for some exceptions ) . this stemmed mostly from the practical reasons , as the impact of the leakage is neither calculable analytically nor analysis method independent . + the objective of this work is to fill this gap and present a more systematic study of the impact of the presence of the leakage on the performance forecasts of cmb b - mode experiments . the paper consists of two parts . in the first part , sec . [ sec : cap ] , we consider idealized observations of azimuthally symmetric sky areas with homogenous noise and study differences between performance forecasts derived applying three different approaches for different assumed sky area sizes . subsequently , from these three different perspectives we revisit the issue of the optimal sky area , which would permit setting the most stringent constraints on the scalar - to - tensor ratio , @xmath2 , given a fixed length and sensitivity of the experiment . in the second part , [ sec : realistic ] , we complete those considerations by discussing more realistic sky areas defined for three types of experiments : small - scale observations covering @xmath22% of the sky , an array of ground - based telescope covering @xmath23% and a satellite - like mission capable of delivering up to @xmath24% of the foreground clean sky . our conclusions are drawn out in sec . [ sec : conclu ] , where we also briefly sketch the implications for constraining inflationary models . throughout this work we neglect complications such as polarized diffuse foregrounds , e.g. , @xcite and account for resolved points sources only by appropriately tailoring the adopted mask . we also assume that no subtraction of the lensing @xmath0-mode has been attempted @xcite . we consider first the case of small - scale experiments in an idealized way . the observed part of the celestial sphere is assumed to be azimuthally symmetric , given by a spherical cap . we however let vary the sky coverage from 0.5% to 10% . the noise is an homogeneous , white noise , and its level is fixed at @xmath25k - arcminute for @xmath26 ( a typical level for ongoing small - scale experiments ) . for a fixed sensitivity and a fixed time of observation , the noise level ( in @xmath27k - arcminute ) scales as : @xmath28}{1\%}}\times n_p(1\%).\ ] ] the instrumental noise reprojected on the sky thus varies from @xmath29k - arcminute to @xmath30k - arcminute for an observed fraction of the sky of 0.5% and 10% , respectively . finally , the angular resolution is given by an azimuthally symmetric , gaussian beam with a width of 8 arminutes . we subsequently investigate the signal - to - noise ratio , @xmath31 , as a function of the sky coverage . this will be done considering four values of the tensor - to - scalar ratio : @xmath32 and @xmath33 . we note that the last two values are disfavored by the current data , nevertheless we include them in our considerations as they are useful in demonstrating some of the effects we describe hereafter . translating the uncertainties on the @xmath0-mode angular power spectrum reconstruction into error bars on the measured tensor - to - scalar ratio , @xmath34 , can be done using a fisher matrix formalism . for the rather small observed fractions of the celestial sphere here - considered , the @xmath0-mode angular power spectrum is reconstructed within bandpowers , labelled @xmath35 hereafter , with bandwidths @xmath36 . the binned power spectrum is given by @xmath37 , where the binning operator is defined as : @xmath38 ( our specific choice of the binning will be given in sec . [ ssec : powspec ] . ) the error bars on @xmath2 are then derived from the fisher matrix via : @xmath39 with @xmath40 , which stands for the covariance matrix of the reconstructed , binned angular power spectrum of the @xmath0-mode . ( note that @xmath41 denotes the _ estimator _ of the angular power spectrum , @xmath42 . ) the @xmath0-mode angular power spectrum as a function of @xmath2 is modeled as : @xmath43 with @xmath44 and @xmath45 two fiducial angular power spectra , which do not depend on @xmath2 . the former is just obtained as the contribution of primordial gravity waves for @xmath46 ( taking into account that the primordial @xmath0-mode is itself lensed ) . the latter corresponds to the contribution of primary @xmath1-mode transferred into @xmath0-mode because of the gravitational lensing of large scale structures . we do not consider here a potential _ delensing _ of the @xmath0-mode anisotropies , and such a contribution will be assumed to act as an additional gaussian noise for the measurement of @xmath2 . this is a simplifying assumption since the lensing - induced @xmath0-mode is non - gaussian leading to an additional , non - gaussian contribution to the covariance @xcite . gaussianity remains however a good approximation for bandpowers which are narrow enough ( @xmath47 ) @xcite , which is the case in our study . the covariance matrix @xmath48 is estimated using three different approaches , as described here . first , we rely on a nave mode - counting expression ( or so - called @xmath49-formula ) . in this case , the covariance on @xmath50 is approximated by : @xmath51 with @xmath52 the noise power spectrum , @xmath53 the beam of the telescope , and , @xmath49 the portion of the celestial sphere , which is observed ( or kept in the analysis ) . the noise power spectrum scales linearly with the sky coverage . the covariance matrix for the binned power spectrum is thus given by : @xmath54\delta_{b , b'}. \label{eq : mcountcov}\ ] ] this is essentially used as a benchmark as such an evaluation of the statistical error bars on the @xmath0-mode reconstruction underestimates the error bars coming from any numerical methods to be applied to the data . second we consider the error bars that could be incurred by using a minimum variance quadratic estimator @xcite . the estimator is defined as follows : @xmath55-\tilde{n}_{\ell'}\right\}. \label{eq : copt}\ ] ] in the above , @xmath56 is the covariance matrix of the maps of the stokes parameter , and @xmath57 is the column vector composed of @xmath58 ( the trace operation is across pixels ) . the quantity @xmath59 stands for the noise debias . finally , @xmath60 is the fisher information matrix given by : @xmath61 . \label{eq : fishopt}\ ] ] it is then shown that the covariance of the above estimator is given by the inverse of the fisher matrix , i.e. @xmath62 . we remind that this estimator is precisely built to be the quadratic estimator with the lowest variance . if the @xmath0-mode power spectrum is indeed estimated for each multipole , @xmath63 ( that is chosing @xmath64 ) , this directly gives the following expression for the error bars expected on @xmath2 : @xmath65,\ ] ] with @xmath66 the same covariance matrix _ but _ assuming that only @xmath67 does depend on @xmath2 , in line with our approach consisting in constraining the tensor - to - scalar ratio from the @xmath0-mode s measurements only given by : @xmath68,\ ] ] with @xmath69 the covariance matrix assuming that all the angular power spectra do depend on @xmath2 , the equation ( [ eq : fisher ] ) is therefore replaced by : @xmath70 in the above , the indices @xmath71 runs over @xmath72 and @xmath21 . the equation ( [ eq : fisher ] ) is finally obtained assuming that only @xmath67 in @xmath69 does depend on the tensor - to - scalar ratio . ] . in the case of azimuthally symmetric patches , the numerical computation of such fisher matrices ( either @xmath73 or @xmath74 ) , can be performed in a reasonable time using the expression found in the appendix f of ref . @xcite , and by using the s@xmath75hat package to perform spherical harmonic transforms @xcite . ( the use of this massively parallel package allows for a rapid computation of the covariance matrix for large sky coverages . ) . in the standard case , for brute force calculation the fisher matrix requires @xmath76 operations to be evaluated , but in this calculation , evaluating the spin harmonics by recursion in @xmath63 makes the computational cost as @xmath77 , where @xmath78 is the number of rings actually used . practically speaking , one should nonetheless include the impact of binning , done as follows . first one defines the so - called _ optimal pseudospectrum _ : @xmath79-\tilde{n}_{\ell}.\ ] ] one easily checks that @xmath80 ( where we also include the impact of an azimuthally symmetric beam ) . one then introduces the matrix : @xmath81 with the interpolation operator , @xmath82 : @xmath83 the binned estimator , @xmath84 , is finally defined as : @xmath85 with @xmath86 the binned , optimal pseudospectrum . from that last definition , and making use of eqs . ( [ eq : copt ] ) and ( [ eq : fishopt ] ) , it is straightforward to show that : @xmath87_{bb_1}\left[p_{b_1\ell_1}~\mathbf{f}_{\ell_1\ell'_1}~p_{b'_1\ell'_1}\right][(\tilde{\mathbf{f}}^{-1})^\dag]_{b'_1b ' } , \label{eq : qmlcov}\ ] ] where summations over repeated indices ( i.e. @xmath88 and @xmath89 ) is implicitly assumed , and @xmath90 means the transpose operation . + we note that this way of estimating the uncertainties on the power spectrum reconstruction is also relevant for maximum - likelihood approaches , see e.g. ref . @xcite . third , we make use of the x@xmath75pure code and monte - carlo simulations to estimate the covariance matrix expected for the pure pseudospectrum approach . details on the pure pseudospectrum estimator can be found in refs . @xcite . in practice , the power spectrum is estimated within bandpower and the covariance matrix reconstructed from the mc simulations is directly @xmath91 . the numerical cost of this scales as @xmath92 , allowing for rapid mc simulations . for each sky coverage and for each value of @xmath2 here - considered , we compute optimized sky apodizations to apply to the maps of @xmath16 and @xmath17 . those optimized sky apodizations are described in refs . @xcite and they allow for having the smallest error bars on the @xmath0-mode power spectrum reconstruction within the context of pure pseudospectrum techniques . those sky apodizations are a set of spin-0 , spin-1 and spin-2 window functions to be applied to the maps of the stokes parameter . they can be interpreted as the window functions , which make the pure pseudospectrum estimator as close as possible to the minimum - variance quadratic estimator @xcite . numerically speaking , computing those optimized sky apodizations may be long , especially for intricate shape of the observed region and/or low level of noise . using an iterative method , the numerical cost is @xmath93 , with @xmath94 a number of iterations ranging from few tens to few hundreds for simple patch geometry and noise level considered here ( see sec . iii in @xcite ) . we stress that for a given sky patch , those sky apodizations are to be optimized for each value of the tensor - to - scalar ratio and bin - by - bin . taking into account the number of bins ( see sec . [ ssec : powspec ] ) , the number of sky fractions and the number of values of @xmath2 , which are sampled in this study , this means that 1664 of such sky apodizations have to be computed . fortunately , in the case of homogeneous noise and patches with relatively simple contours ( which is obviously the case for a spherical cap ) , it was demonstrated in ref . @xcite that an approximated but numerically fast computation of those sky apodizations in the harmonic domain is possible , and indeed leads to error bars equal to those obtained thanks to a direct , pixel - based computation of the optimized sky apodizations . the numerical cost of this technique is reduced to @xmath92 which allows us to derive optimized sky apodizations for each value of the sky coverage , and for each value of the tensor - to - scalar ratio . our chosen bandpowers for reconstructing @xmath67 are the following . the first bin starts at @xmath95 and we use a constant bandwidth , @xmath96 . our last bin extends up to @xmath97 . the value of the maximum multipole is chosen in order to include all the relevant contributions from the primordial @xmath0-mode , that is until the lensing @xmath0-mode be the dominant contribution to the total @xmath0-mode power . in addition , for the experimental cases under consideration in this paper , we use beamwidths up to 8 arcminutes , corresponding to a cut - off of @xmath98 . the choice of the bandwidth is mainly motivated by the use of the pure pseudospectrum estimator . especially , it is mandatory for the numerical inversion of the mode - mixing matrices to be possible . we also note that the bandwidth is wide enough so that the correlations between different bins are nearly uncorrelated in the covariance of pure pseudospectrum estimator . we discuss the role of the bin width later on . we stress that the multipoles ranging from @xmath99 to @xmath95 ( corresponding to the reionization peak , and gathered in one bandpower ) are actually used in the pure pseudo-@xmath100 estimation of @xmath67 . however , given the limited sky coverages considered here such low multipoles are difficult to estimate and hardly constrained by the data . this bin is therefore not included in our analysis of the signal - to - noise ratio . the uncertainties on the estimated power spectrum of the @xmath0-mode as functions of the sky fraction , are shown in fig . [ fig : powunc1 ] where four selected values of the sky coverage are depicted : 1% , 3.5% , 7% and 10% . the tensor - to - scalar ratio chosen for this figure is @xmath101 . each panel corresponds to a different approach to derive the covariance matrix , @xmath91 : mode - counting , minimum variance quadratic estimator , and , pure pseudospectrum estimator ( from left to right ) . as expected , we do observe that the lowest error bars are the ones from the mode - counting estimation of the uncertainties , while the highest error bars are obtained from the pure pseudo-@xmath100 estimator . the error bars from the minimum variance , quadratic estimator lie between those two . at the largest accessible scales , @xmath102 , the error bars from the pure pseudospectrum estimator are @xmath4 greater than the optimistic mode - counting estimation . similarly , the error bars from the pure pseudospectrum estimators are at most @xmath103 higher than the ones derived from the minimum variance , quadratic estimators . at the smaller angular scales where lensing dominates , the three approaches lead to almost the same uncertainties . + the behaviour of the uncertainties as a function of the sky fraction is common to the three approaches . at the smaller angular scales first ( for multipoles greater than @xmath104 ) , the behavior is monotonic since the uncertainties systematically increase with the value of @xmath49 . this is because at these scales , the variance is dominated by the noise , which increases with the sky fraction . at larger scales however ( for multipoles smaller than @xmath104 ) , the uncertainties have a more intricate behaviour . first one notes that the uncertainties _ decrease _ from @xmath26 to @xmath105 . this is because the variance is dominated by sampling variance , which decreases for higher values of @xmath49 . second , one notes that uncertainties at @xmath106 then _ increases _ for a sky coverage ranging from @xmath107 to 10% . this means that for @xmath108 , the noise is now dominating the variance . once this transition value of @xmath109 is crossed , the noise contribution dominates the variance for all our considered angular scales , @xmath110 $ ] . therefore , the variance will monotonically increase with @xmath49 at all the relevant angular scales once @xmath108 . ( note that an identical behaviour is observed for the other values of @xmath2 , though the specific value of @xmath49 at which the transition occurs depends on the specific value of @xmath2 . ) the signal - to - noise ratio on @xmath2 is computing using eq . ( [ eq : fisher ] ) considering the three above - described methods to estimate the uncertainties on the @xmath0-mode reconstruction , @xmath91 . we remind that the summation in ( [ eq : fisher ] ) is performed over bandpowers with a bandwidth of @xmath96 and considering a range of multipoles from @xmath95 to @xmath97 . our numerical results on the signal - to - noise ratio for @xmath2 are gathered in fig . [ fig : snrfsky ] , showing @xmath111 as a function of the sky coverage . each panel corresponds to a given value of the tensor - to - scalar ratio , @xmath32 , and , @xmath33 ( from top to bottom ) . for each panel , the black , red , and blue crosses correspond to the signal - to - noise ratio derived by using the mode - counting , the minimum variance quadratic estimator , and , the pure pseudo-@xmath100 estimator , respectively . the horizontal , dashed line marks a @xmath112 detection . the sky fraction varies from 0.1% to 10% , what is wide enough to sample the maximal values of the signal - to - noise ratio . we note that the signal - to - noise ratio keeps decreasing for @xmath113 . this is because for the level of noise and values of @xmath2 here considered , the uncertainties on the reconstructed @xmath0-mode are noise dominated at all scales for @xmath113 . similarly , the ( s / n)@xmath114 keeps decreasing for @xmath115 , because the uncertainties on angular scales greater than a degree are dominated by the sampling variance for such low values of the sky fraction . + from @xmath0-mode polarization data shown as a function of the sky coverage . the uncertainties are computed using three different approaches : mode - counting ( black crosses ) , minimum variance quadratic estimator ( red crosses ) and pure pseudospectrum estimator ( blue crosses ) . each panel corresponds to a different fiducial value of the tensor - to - scalar ratio , @xmath32 and @xmath33 from top to bottom.,title="fig:"]-0.1truecm from @xmath0-mode polarization data shown as a function of the sky coverage . the uncertainties are computed using three different approaches : mode - counting ( black crosses ) , minimum variance quadratic estimator ( red crosses ) and pure pseudospectrum estimator ( blue crosses ) . each panel corresponds to a different fiducial value of the tensor - to - scalar ratio , @xmath32 and @xmath33 from top to bottom.,title="fig:"]-0.1truecm from @xmath0-mode polarization data shown as a function of the sky coverage . the uncertainties are computed using three different approaches : mode - counting ( black crosses ) , minimum variance quadratic estimator ( red crosses ) and pure pseudospectrum estimator ( blue crosses ) . each panel corresponds to a different fiducial value of the tensor - to - scalar ratio , @xmath32 and @xmath33 from top to bottom.,title="fig:"]-0.1truecm from @xmath0-mode polarization data shown as a function of the sky coverage . the uncertainties are computed using three different approaches : mode - counting ( black crosses ) , minimum variance quadratic estimator ( red crosses ) and pure pseudospectrum estimator ( blue crosses ) . each panel corresponds to a different fiducial value of the tensor - to - scalar ratio , @xmath32 and @xmath33 from top to bottom.,title="fig : " ] for the case of the mode - counting first , the signal - to - noise ratio is systematically greater than 3 for all the considered values of the sky coverage , and for all the considered values of the tensor - to - scalar ratio . considering then the case of the minimum variance , quadratic estimator , the signal - to - noise ratio on @xmath2 is systematically greater than 3 for @xmath116 and @xmath33 . for a tensor - to - scalr ratio of @xmath117 , the ( s / n)@xmath114 is greater or equal to three for @xmath118 . assuming finally a pure pseudospectrum reconstruction of the @xmath0-mode , the signal - to - noise ratio is systematically greater than 3 for @xmath119 and @xmath120 only . for a tensor - to - scalar ratio of @xmath101 , a measurement of it with a ( s / n)@xmath114 of at least 3 , is possible for @xmath121 . for a smaller value of @xmath117 , its measurement with ( s / n)@xmath122 is possible assuming @xmath123 . we note however here that for @xmath117 and @xmath101 , the signal - to - noise ratios remains greater than 2 assuming a pure pseudospectrum reconstruction of @xmath42 . + as expected from the error bars on the reconstructed @xmath0-mode , the highest and lowest ( s / n)@xmath114 s are respectively obtained from the mode - counting estimation , and the pure pseudospectrum estimator , while the ( s / n)@xmath114 from the minimum variance quadratic estimator lies between those two . this is the case for all the values of the tensor - to - scalar ratio we consider . at the peak , the signal - to - noise ratio from the pure pseudospectrum estimation of the @xmath42 is @xmath12415% ( @xmath120 ) to @xmath12420% ( @xmath117 ) smaller than the signal - to - noise ratio derived from the optimistic mode - counting . this means that the statistical significance on the measurement of @xmath2 by using the optimistic mode - counting is overestimated by a factor @xmath4 as compared to the more realistic case of the pure pseudoreconstruction of the @xmath0-mode . similarly , the ( s / n)@xmath114 from the pure pseudospectrum estimator is @xmath1241.5% ( @xmath120 ) to @xmath1248% ( @xmath117 ) smaller than the signal - to - noise ratio derived from the minimum variance , quadratic estimator . using the minimum variance , quadratic estimator to estimate the @xmath0-mode , as compared to the use of the pure pseudospectrum , thus translates into a gain in the statistical significance on the measurement of @xmath2 , of a factor 1.01 to 1.08 . this gain appears rather small but is larger for smaller values of the tensor - to - scalar ratio . as clearly shown in figs . [ fig : snrfsky ] , there exists a value of the sky coverage , which maximizes the signal - to - noise ratio on @xmath2 . this _ optimal _ value of @xmath49 was already observed in ref . @xcite , using only the mode - counting expression for the statistical error bars on the @xmath0-mode estimation though . we found that such an optimal value also exists using the minimum variance quadratic estimator or the pure pseudo-@xmath100 estimator . this is intuitively understood as follows . the _ statistical _ uncertainties on the angular power spectrum estimation have two sources , the sampling variance , which is dominant at the largest angular scales , and the noise variance dominating at the smallest angular scales . reducing the sampling variance is obtained by covering a large fraction of the sky . however , for a given sensitivity and a given time of observation , covering a large fraction of the sky inevitably translates into a higher level of noise per pixel . one should therefore find the good balance between sampling and noise variance so as to minimize the total error on given targetted parameters , which is @xmath2 here . the salient features of those results are summarized in the table [ tab : fsky ] . for each value of the tensor - to - scalar ratio and for each techniques used to compute uncertainties on the @xmath0-mode , we provide the values of the sky fraction maximizing the signal - to - noise ratio , @xmath125 . its associated ( thus maximal ) value of the signal - to - noise ratio , ( s / n)@xmath126 is also reported in this table . we stress that the position of the peak of ( s / n)@xmath114 is well defined for the mode - counting and the minimum variance , quadratic estimator . such a position of the peaking signal - to - noise ratio is however less pronounced for the case of the pure pseudo-@xmath100 estimation of the @xmath0-mode ( see e.g. the case @xmath127 for which a range of @xmath128 leads roughly to the same ( s / n)@xmath114 ) . this means that the values of @xmath125 reported in tab . [ tab : fsky ] for the case of the pure pseudospectrum approach are more indicative than a sharply defined value . + lm0.5cmm0.75cmm0.75cmm0.75cmm0.75 cm @xmath2 & & 0.07 & 0.1 & 0.15 & 0.2 + @xmath129 $ ] : & & & & & + mode - counting & & 2.0 & 3.0 & 4.0 & 5.0 + minimum - variance @xmath100 & & 2.5 & 2.5 & 3.5 & 5.0 + pure pseudo-@xmath100 & & 3.5 & 3.5 & 5.0 & 4.0 + ( s / n)@xmath126 : & & & & & + mode - counting & & 4.4 & 5.5 & 7.0 & 8.2 + minimum - variance @xmath100 & & 3.7 & 4.7 & 5.9 & 7.0 + pure pseudo-@xmath100 & & 3.4 & 4.4 & 5.8 & 6.9 + for all the approaches used to estimate the uncertainties on the @xmath0-mode , we observe that the optimal sky fraction increases with the value of the tensor - to - scalar ratio . this is because for higher values of @xmath2 , the signal in the @xmath0-mode is higher . one should therefore minimize first the sampling variance by increasing the observed part of the sky . + except for the case of @xmath120 , we note that the optimal sky coverage assuming a minimum variance , quadratic estimator slightly differs by 0.5% ( either higher or lower ) than the value of @xmath130 ( in % ) as inferred from the mode - counting . we also note that the optimal sky fraction obtained for the pure pseudo-@xmath100 reconstruction of the @xmath0-mode differs by 1% to 1.5% ( depending on the value of @xmath2 ) from the one inferred from the mode - counting estimation of the uncertainties on the @xmath0-mode . nevertheless , the values of the sky fraction for which the detection of the tensor - to - scalar ratio is peaking in the case of the mode - counting expression and the minimum - variance quadratic estimator fall in the range of optimized @xmath49 as derived from the pure pseudo-@xmath100 estimator . those numerical results therefore show that ( at least ) for the range of values of @xmath2 here - considered , the value of the sky coverage , which maximizes the measurement of the tensor - to - scalar ratio is rather independent on the adopted method for evaluating the statistical uncertainties on the @xmath0-mode reconstruction . this means that using the mode - counting expression , though underestimating the error bars , allows for a rapid and reliable search of the range of values of the optimized sky fraction . ( obviously , such an optimization of @xmath49 based on the mode - counting expression is reliable providing the final data set to be analyzed using either the minimum - variance quadratic estimator or the pure pseudospectrum estimator . ) for the two specific cases of the mode - counting uncertainties and the minimum variance , quadratic estimator , we note that an explicit reconstruction of the power spectrum is not mandatory to derive the ( s / n)@xmath114 in the fisher formalism . one can indeed directly plugged in eq . ( [ eq : fisher ] ) the formulas ( [ eq : mcountcov ] ) or ( [ eq : qmlcov ] ) . this allows for a study of the impact of binning on the signal - to - noise ratio , letting the bandwidth to vary from @xmath64 ( i.e. no binning ) to @xmath96 ( i.e. the binning imposed by the use of the pseudospectrum estimator in this analysis ) . as a function of the observed sky fraction , derived for three methods used to estimate the uncertainties on the @xmath0-mode reconstruction : mode - counting ( black area ) , minimum variance quadratic estimator ( red area ) , and , pure pseudospectrum estimator ( blue crosses ) . for the two first methods , we let the bandwidth of the bins to vary from @xmath64 ( highest ( s / n)@xmath114 ) to @xmath96 ( lowest ( s / n)@xmath114 ) . for the specific case of the pure pseudospectrum estimator , the reconstruction of the @xmath42 requires to use the bandwidth @xmath96 . ( we remind that the range of multipoles used to compute the signal - to - noise ratio is @xmath131 . ) ] the impact of binning is illustrated in fig . [ fig : snrbin ] showing the signal - to - noise ratio on @xmath101 as a function of the sky coverage . the grey ( red ) area corresponds to the ( s / n)@xmath114 using the mode - counting ( minimum variance quadratic estimator ) to estimate the uncertainties the angular power spectrum of the @xmath0-mode . for each shaded area , the highest signal - to - noise ratio is obtained for @xmath64 and the lowest for @xmath96 . as a reference , we also show the ( s / n)@xmath114 obtained with pure pseudospectrum reconstruction ( thus using a bandwidth of @xmath96 ) depicted by the blue crosses . the overall effect of increasing the width of the bandpower is to lower the signal - to - noise ratio . the decrease is however more pronounced for the case of the minimum variance , quadratic estimator than for the mode - counting estimation of the error bars on the reconstructed @xmath0-mode . this is due to the fact that correlations between multipoles ( or bandpowers ) are accounted for in the minimum variance , quadratic estimator , while those are supposed to be systematically vanishing for the mode - counting estimation of the covariance matrix . this additional piece of information contained in the correlations is therefore partially lost by averaging over bandpowers . we also checked that artificially imposing those off - diagonal correlations to be zero lowered the signal - to - noise ratio in the minimum variance method , although we note that once the bins are sufficiently wide the effect of the binwidth on the ( s / n)@xmath114 should be weak . the maximum values of the ( s / n)@xmath114 obtained using a bandwidth of @xmath64 , and a bandwidth of @xmath96 , are reported in tab . [ tab : bin ] , for each values of @xmath2 and for the mode - counting and the minimum variance quadratic estimator . for each cases , we also report the value of the sky coverage corresponding to that maximum . for the mode - counting approach , increasing the bandwidth from @xmath64 to @xmath96 , degrades the maximum ( s / n)@xmath114 by a factor @xmath132 for @xmath117 and @xmath133 , and , by a factor @xmath134 for @xmath119 and @xmath33 . this however only mildly affects the values of the sky fraction at which the maximum is achieved . the impact of binning is more marked for the minimum variance quadratic estimator however . increasing the bandwidth from @xmath64 to @xmath96 , here degrades the maximum ( s / n)@xmath114 by a factor @xmath135 for all the values of the tensor - to - scalar ratio considered in this study . similarly , the values of the sky fraction ( in % ) at which this maximum is achieved is systematically lowered ( except for the case @xmath127 ) , by 1% for @xmath101 and by 2% for @xmath120 . we note that despite these changes in the value of @xmath130 with the bandwidth , the optimized values of the sky fraction still fall in the range of optimized @xmath49 as derived from the pure pseudo-@xmath100 estimator . @xmath2 & & 0.07 & 0.1 & 0.15 & 0.2 + ( s / n)@xmath126 and @xmath125 : & & & & & + mode - counting : & & & & & + @xmath64 & & 4.6 ( 2.5% ) & 5.7 ( 3% ) & 7.2 ( 4% ) & 8.4 ( 5.5% ) + @xmath96 & & 4.4 ( 2.0% ) & 5.5 ( 3% ) & 7.0 ( 4% ) & 8.2 ( 5% ) + minimum - variance @xmath100 : & & & & & + @xmath64 & & 4.2 ( 2.5% ) & 5.3 ( 3.5% ) & 6.8 ( 5% ) & 8.0 ( 7% ) + @xmath96 & & 3.7 ( 2.5% ) & 4.7 ( 2.5% ) & 5.9 ( 3.5% ) & 7.0 ( 5% ) + we turn to the question of the detection of @xmath2 in more realistic cases . clearly , a spherical cap is ideal . the issue of leakages is strongly related to the detailed shape of the contours of the observed ( or kept - in - the - analysis ) portion of the sky ( see e.g. the figure 20 of ref . @xcite for the impact of the shape of the mask on the statistical error bars ) . a spherical cap then leads to the smallest amount of leakages for a given sky fraction since its contour has the smallest perimeter for that given sky fraction . to this end , we consider three archetypal cases , which capture the main characteristics of ongoing , or being - deployed , small - scale experiments ( ground - based or balloon - borne ) , a possible upgrade of those ground - based experiments to an array covering a rather large fraction of the sky ( @xmath136 ) , and , a possible satellite mission covering the entire celestial sphere . + [ cols="^,^,^ " , ] a couple of comments about the numerical computation of the pixel - based , minimum - variance sky apodizations are in order here . they are theoretically built to give the smallest uncertainties in the context of the pure pseudospectrum estimators . however , they are pratically computed from a preconditionned conjugate gradient ( pcg ) algorithm , which efficiency strongly depends on the experimental configurations , especially with respect to the noise level and its distribution over the patch , as well as with respect to the complexity of the contours of the mask . first , the number of iterations in our implemented pcg rapidly increases for lower levels of noise : at the largest angular scales ( @xmath137 ) the number of iterations ranges from @xmath104 for a noise level of 5.75@xmath27k - arcminute to @xmath138 for a noise level of 1@xmath27k - arminute ( the number of iterations being one order of magnitude smaller for smaller angular scales , @xmath139 ) . the @xmath0-mode angular power spectrum is estimated using the same binning as in the previous section , leading to @xmath140 , and we selected 6 values of @xmath2 . considering 3 experimental setups , this would translate into @xmath124500 optimized sky apodization to compute , which is numerically too costly . we therefore compute the optimized sky apodization for @xmath141 only but use them for all the here - considered values of @xmath2 , meaning that the signal - to - noise ratios obtained for @xmath142 may be suboptimal within the context of the pure pseudospectrum estimator - modes leaking into the @xmath0-mode , and poorly affected by the amplitude of the primordial @xmath0-mode . this means that the resulting sky apodizations may be mildly dependant on the assumed value of @xmath2 and that the derived signal - to - noise ratios are only slightly suboptimal for the case of small - scale experiments . ] . second , it is not guaranteed that the algorithm converges towards the optimal solution , especially for inhomogeneous noise ( see sec . iv c of ref . @xcite where it was shown that trimming out the external , noisiest pixels is required ) , or a very low level of noise ( see ref . @xcite mentionning that convergence is not reached for a noise level of @xmath143k - arminute , corresponding to the level of the array of telescopes case ) . this means that the performances of those sky apodizations have to be assessed using numerical simulations at the level of power spectrum reconstruction , comparing the resulting error bars on the estimated power spectra to the error bars obtained by using the other types of sky apodizations . the relative performances of the different sky apodizations are appraised at the level of power spectrum uncertainties . for each case we performed a series of 500 monte - carlo simulations to compute the statistical uncertainties on the reconstructed @xmath0-mode angular power spectra , assuming different kinds of sky apodizations . such performances have been exhaustively studied for the small - scale experiment case and the satellite mission case ( see refs . @xcite and ref . @xcite , resp . ) . on the contrary , the applicability of the pure pseudospectrum estimator for the case of an array of telescopes was hitherto not studied . we then performed numerical simulations using the different classes of sky apodizations to assess the efficiency of the pure pseudospectrum reconstruction of the @xmath0-mode , and subsequently select those sky apodizations , which lead to the smallest uncertainties . in this section , we only briefly review the major conclusions concerning the cases of a small - scale experiment and a satellite mission . then , we present the results of our numerical investigations for the case of an array of telescopes . for the small - scale experiment case , it was shown that the lowest uncertainties in the range @xmath144 $ ] were obtained using either the pixel - based optimized sky apodizations or analytic sky apodizations appropriately chosen to minimize the variance per each bin . however , the harmonic - based computation of the sky apodization fails in providing error bars comparable to the previous ones in the entire range of multipoles considered here , simply because the noise is inhomogeneous . ( we refer to fig . 24 of ref . @xcite and discussions therein . ) for the satellite mission case , the pixel - based computation of the minimum - variance sky apodization yields the smallest uncertainties for the range @xmath144 $ ] . similar performance is obtained by using the harmonic - based computation of these sky apodizations for @xmath145 . the error bars however drastically increase for larger angular scales , the reason for that being the intricate contours of the galactic mask and the point - sources mask , which require to relax the neuman and dirichlet boundary conditions to keep ( part of ) the informations about the @xmath0-mode contained in the ambiguous modes . for such a case , analytic sky apodizations fails in providing comparable error bars at the largest angular scales . ( we refer to fig . @xcite and discussions therein . ) for the case of an array of telescopes , we systematically search for the type of sky apodizations , which lead to the smallest uncertainties bin per bin and for each values of @xmath2 considered in this study . we first found that the harmonic - based , optimized sky apodizations yield to error bars higher than the analytic sky apodizations or the pixel - based , optimized sky apodizations . this is similar to what was observed for the case of a satellite mission and the inefficiency of the harmonic - based , optimized sky apodizations is due to the complexity of the contours of the mask . an example of the uncertainties for @xmath101 and using the pixel - based , optimized sky apodizations ( the dashed - red curve ) or the analytic ones for different values of the apodization length ( the dashed - blue curves ) is shown in fig . [ fig : clhalfsky ] . the solid - black curve stands for the input angular power spectrum and the dashed - black curve stands for the binned , mode - counting computation of the error bars . this first shows that the pixel - based , optimized sky apodizations perform the best at the largest angular scales . this is systematically so for the two first bins . for the third bin , the pixel - based , sky apodizations and the analytic ones perform the same for @xmath146 , while for @xmath147 , the analytic sky apodizations with an apodization length of 4 degrees lead to a smaller error bar than the pixel - based , optimized sky apodization . however at smaller scales , @xmath148 , the smallest error bars are systematically obtained by using an analytic sky apodizations with an apodization length of 2 degrees for intermediate scales , @xmath149 , and an apodization length of 1 degree for small scales , @xmath150 . we found this to be independant of the value of @xmath2 ( at least for the grid of values considered here ) . + we note that the apparent failure of the pixel - based , optimized sky apodizations here may be rather due to practical difficulties in computing such apodizations sufficiently accurately , rather than an indication of some fundamental problems . indeed , we have found that for the noise levels the iterative solver used to compute the apodizations converges extremely slowly ( as also observed in ref . @xcite ) potentially preventing us in practice from achieving sufficient precision . -mode for the case of an array of telescopes . the solid - black curve stands for the input angular power spectrum with a tensor - to - scalar ratio equal to @xmath101 . the dashed - black curve is for the binned , mode - counting uncertainties used as a benchmark . the error bars obtained by using the pixel - based , optimized sky apodizations ( called pcg in the figure legend ) is represented by the dashed - red curve . the different dashed - blue curves correspond to the error bars obtained by using analytic sky apodizations with an apodization length ranging from 1 degree to 6 degrees . ] as a summary , the smallest statistical uncertainties obtained for @xmath101 are shown in fig . [ fig : cellallexp ] in which the orange , red and burgundy curves stand for the small - scale experiment , an array of telescopes and a satellite mission respectively . for each experimental setup , we show the smallest error bars , which are attained for each bandpower . for the cases of a small - scale experiment and a satellite mission , this is obtained by using the pixel - based , optimized sky apodizations throughout the entire range of angular scales . for the case of an array of telescopes , the pixel - based , optimized sky apodizations are used for multipoles lower than 100 , while analytic sky apodizations with an apodization of 2 degrees and 1 degree are used in the range @xmath151 and in the range @xmath152 , respectively . + -mode s angular power spectra with the pure pseudospectrum estimator . the orange , red and burgundy lines stand for the small - scale experiment , an array of telescopes and a satellite mission respectively . the black line corresponds to the input angular power spectrum for @xmath101 . ] as expected , the higher uncertainties are the ones from a small - scale experiment due to the tiny fraction of the sky it covers , and the relatively high level of instrumental noise . we provide the uncertainties for the first bandpower , @xmath153 , for completeness . these scales are nonetheless unaccessible starting from a map covering 1% of the sky due to the high uncertainties , as already stated in sec . [ sec : cap ] . for angular scales going from @xmath99 to @xmath154 , the smallest error bars corresponds to a satellite mission . this is because at these angular scales , the uncertainties are dominated by sampling variance and a satellite mission , as compared to an array of telescopes , benefits from its larger sky fraction . for multipoles smaller than @xmath104 , the error bars from a satellite mission are roughly 1.5 to 2 times smaller than the error bars obtained from an array of telescopes , which is in line with the fact that the sky fraction for a satellite mission is 2 times higher than the sky fraction observed by an array of telescopes , thus reducing the error bars by a factor of @xmath155 as compared to the error bars from an array of telescope . nevertheless , at small scales , @xmath156 , smaller error bars are obtained from an observation by an array of telescopes . this is because in that regime , the uncertainties for the case of a satellite mission are dominated by the noise term , @xmath157 . since the noise for a satellite mission is four times higher ( in power spectrum ) than the noise for an array of telescopes , and the beam is more than two times higher , this increase of the variance largely overcome the decrease due to a larger sky coverage . this quantitatively explains why at those small angular scales , the lowest error bars on the @xmath0-mode reconstruction are obtained from an array of telescope . ( one can even notice that for the range of angular scales considered here , the uncertainties obtained for an array of telescopes in sampling variance dominated . ) the computation of the signal - to - noise ratio on the tensor - to - scalar ratio is done by using the same fisher matrix formalism as employed in the previous section , eq . ( [ eq : fisher ] ) . for each experimental configurations and each value of @xmath2 , we select the smallest error bars we obtained _ bin per bin_. this means that for the specific case of an array of telescope , the estimation of the @xmath0-mode angular power spectra is done by mixing different kind of sky apodizations . we use the same bandpowers as in sec . [ sec : cap ] and now add the largest angular scales , from @xmath99 to @xmath95 gathered in one single bandpower , in the summation in eq . ( [ eq : fisher ] ) . adding these scales is relevant for the case of an array of telescopes , and the case of a satellite mission . we will first add this bandpower at the largest scales for the three experimental setups , sec . [ sssec : numres ] . we will subsequently study its impact on the measurement of @xmath2 , sec . [ sec : firstbin ] . ( note that we use the binned covariance for both the modecounting and the pure pseudospectrum reconstruction of @xmath42 . ) our results on the signal - to - noise ratio for @xmath2 ranging from @xmath158 to @xmath33 are shown in fig . [ fig : snrallexp ] ( note that for the specific case of a satellite mission the value @xmath159 has been added in order to fall below the @xmath112 limit ) . the red and black crosses correspond to a covariance matrix computed using the mode - counting expression for error bars on @xmath67 , and the pure pseudospectrum error bars , respectively . the solid red line is the @xmath112 limit . the left panel corresponds to a small - scale experiment covering @xmath1241% of the celestial sphere with a highly inhomogeneous noise distribution . the middle panel corresponds to an array of telescopes covering @xmath12436% of the sky with a low level of ( homogeneous ) noise . finally , the right panel is for a satellite mission covering @xmath12471% of the sky with a low level of homogeneous noise . + for the case of a small - scale experiment , the signal - to - noise ratio on @xmath2 ranges from 0.06 for @xmath160 to 4 for @xmath120 assuming a pure pseudo-@xmath100 reconstruction of the @xmath0-mode power spectrum ( meaning that a `` measurement '' of @xmath160 would be consistent with @xmath161 ) . this has to be compared to what would be inferred from the idealized mode - counting evaluation of the uncertainties , for which the signal - to - noise ratio varies from 0.25 for @xmath160 to 6.7 for @xmath120 . for @xmath162 and @xmath133 , the ( s / n)@xmath114 derived from a mode - counting estimation of the uncertainties on the @xmath0-mode is overestimated by a factor @xmath163 , @xmath164 and @xmath165 , resp . , as compared to the ( s / n)@xmath114 obtained from a pure pseudo-@xmath100 reconstruction of the angular power spectrum . for the case of an array of telescopes and assuming the pure pseudosepctrum estimation of the @xmath0-mode , the signal - to - noise ratio varies from 0.67 to 41 with @xmath2 varying from 0.001 to 0.2 . values of @xmath166 and @xmath101 would be measured with a statistical significance of 5.75@xmath167 and @xmath168 , respectively . using instead the mode - counting estimation of the uncertainties on @xmath67 , the ( s / n)@xmath114 varies from 3 to 54 for @xmath2 ranging from 0.001 to 0.2 . for the three selected values of @xmath162 and 0.1 , the signal - to - noise ratio obtained from the mode - counting approach is respectively overestimated by a factor 4.5 , 2.5 and 1.3 , as compared to the realistic ( s / n)@xmath114 derived from the pure pseudo-@xmath100 estimation of @xmath67 . for the case of a satellite mission , the signal - to - noise ratio varies from 0.66 for @xmath169 to 59 for @xmath120 , and assuming the pure pseudospectrum estimation of @xmath67 . the values of @xmath160 , @xmath166 and @xmath101 would be detected with a statistical significance of 1.34 , 10.84 and 46.19 , respectively . if one instead makes use of the mode - counting estimation of the error bars on the @xmath0-mode reconstruction , the ( s / n)@xmath114 varies from 2 to 72 for values of the tensor - to - scalar ratio ranging from 0.0005 to 0.2 . for @xmath162 and @xmath133 , the mode - counting evaluation overestimates the signal - to - noise ratio , as compared to the pure pseudo-@xmath100 reconstruction , by a factor 2.2 , 1.59 and 1.22 . + from a qualitative viewpoint , the signal - to - noise ratios computed in the framework of the mode - counting expression are always higher compared to the signal - to - noise ratios assuming the pure pseudospectrum reconstruction of @xmath0-mode . ( this is obviously expected from the fact that the mode - counting approach is an idealized and _ underestimated _ computation of the uncertainties . ) we observe that the overestimation using the mode - counting expression ( as compared to the more realistic pure pseudospectrum reconstruction of @xmath67 ) is less marked for higher values of @xmath2 . this behavior is common to the three experimental configurations here - considered , though there are differences from a quantitative viewpoint . the reason is that for low values of @xmath2 , most of the information comes from the largest scales , which is precisely at those large scales that the underestimation of the @xmath0-mode reconstruction using the mode - counting formul is more marked . we also stress that in the case of mode - counting approach , the leakages are ignored . on the contrary , the pure pseudospectrum approach consistently includes them but correct them in the analysis . this explains why the mode counting approach overestimate the signal - to - noise ratio on @xmath2 . cmb observations covering a large fraction of the sky are automatically contaminated by various astrophysical foregrounds with complex physics involved among which the emission from our galaxy is the strongest . masks are used to remove from the analysis the portion of sky with the highest foreground level , but the foreground emission is present on the entire celestial sphere . usually techniques - such as parametric component separation @xcite used to determinate the spectral parameters or template fitting method , which deprojects the template of the foreground from the map @xcite - are used to minimize the impact of the foreground . the residual level of foreground contaminants depends on the technique actually chosen . however , the power spectrum of the galactic dust , polarized emission ( which is the major contaminant of cmb measurements at frequencies above @xmath104ghz ) behaves as @xmath170 , to be compared to @xmath171 for the cmb @xmath0-mode angular power spectrum at scales above a degree @xcite . the impact of such a galactic foreground is therefore expected to be more pronounced at the largest angular scales . here , we considered the worst case scenario where the foreground contamination could not be removed at all on the largest scale , meaning that the information from the reionization peak is no more taken into account in the computation of the signal - to - noise ratio . in practice , we discard the first bin ( @xmath172 ) from the analysis , which necessarily lowers the signal - to - noise ratio on @xmath2 . we define this relative decrease as : @xmath173 with ( s / n)@xmath114 the signal - to - noise on @xmath2 accounting for _ all _ the angular scales , and ( s / n)@xmath174 the signal - to - noise ratio obtained by _ discarding _ the first bandpower . this relative decrease can alternatively be interpreted as the relative contribution from the first bin to the signal - to - noise on @xmath2 since : @xmath175 with ( s / n)@xmath176 the signal - to - noise ratio on @xmath2 that would be obtained by using the first bandpower _ only_. + this relative decrease of ( s / n)@xmath114 is shown in fig . [ fig : degradesnrallexp ] . the red crosses correspond to the mode - counting estimation of the error bars on the reconstruction of @xmath67 while the black crosses correspond to the error bars from a pure pseudospectrum estimation of @xmath67 . the left , middle and right panels respectively stand for the case of a small - scale experiment , an array of telescopes and a satellite mission . the case of a small - scale experiment is poorly affected by the removal of the first bin using the pure peudospectrum reconstruction of @xmath67 , the relative decrease being systematically smaller than 0.1% . this is because in such a case the signal - to - noise ratio for the first bandpower , @xmath177 , is much smaller than unity for all the values of @xmath2 considered here . this bandpower therefore does not bring any significant amount of informations on @xmath2 . this drastically differs if one uses the mode - counting evaluation for which @xmath178 varies from 0.7% for @xmath120 to 32% for @xmath160 . this is because in this case , the signal - to - noise ratio in the first bandpower , @xmath179 with @xmath153 , becomes greater than unity though the sky coverage is only of @xmath18 . the fact that the relative decrease is more pronounced for small values of @xmath2 is understood as follows . for lower values of @xmath2 , the recombination bump at the degree scale , falls below the lensing part of the @xmath0-mode while the reionization bump in the first bandpower remains above the lensing signal . as a consequence , the reionization peak carries more information , relative to the informations carried by the recombination peak , for lower values of the tensor - to - scalar ratio . for the case of an array of telescopes , the relative decrease ranges from 0.4% for @xmath120 to roughly 3% for @xmath160 . we note that this relative decrease is now roughly constant from @xmath160 to @xmath166 and then decreases for higher values of the tensor - to - scalar ratio . this behavior of @xmath178 is explained by the very same reason explaining why @xmath178 decreases for higher values of @xmath2 if one makes use of the mode - counting estimation of the uncertainties on the estimated @xmath67 , and because for an array of telescopes , the angular power spectrum in the first bandpower can now be measured with a signal - to - noise ratio greater than unity . we note that the relative decrease using the mode - counting behaves the same as in the case of a small - scale experiment ( with minor quantitative differences at high values of @xmath2 ) . for the case of a satellite mission , the relative decrease varies from 0.35% for @xmath120 to 9% for @xmath169 , with @xmath180 for @xmath160 . this relative decrease now monotonically increases with lower values of @xmath2 ( though our results suggest that a plateau is reached for @xmath181 ) . this behavior is explained by the same reason explaining the decrease of @xmath178 for higher values of @xmath2 in the case of an array of telescopes . we also note that @xmath178 obtained from the mode - counting expression behaves the same as in the case of a small - scale experiment and an array of telescopes . + as is clear from fig . [ fig : degradesnrallexp ] , the shape of @xmath178 as derived using the mode - counting expression , is qualitatively the same for the three experimental configurations , though sky fractions and shapes of the masks drastically change . this is because the impact of the limited sky fraction is simply modelled as an overall renormalization of the error bars , equally applied at all angular scales ( see eq . ( [ eq : mcount ] ) ) . neglecting the noise contribution to the error bars on the @xmath0-mode reconstruction ( which is a relatively fair assumption here ) , it is easy to figure out that this overall @xmath182 does not enter in the final expression of @xmath178 . ( we note that minor differences are however expected because of the different noise level and beamwidth . ) at low values of @xmath2 , the relative decrease is much less marked in the context of the pure pseudospectrum reconstruction of @xmath67 , with respect to the mode - counting expression . this is because the different leakages have stronger impacts at large scales ( in term of increase of the error bars on the estimated @xmath67 ) , thus reducing the relative contribution of the first bandpower to the constraint that can be set on @xmath2 . the impact of leakages in terms of error bars on @xmath67 at large angular scales increases with smaller @xmath49 , which therefore reduces the relative contribution of the first bandpower to the constraints on @xmath2 . this is clearly seen in fig . [ fig : degradesnrallexp ] where @xmath178 is more important from the case of a small - experiment , to the case of an array of telescopes , to the case of a satellite mission . as a result , at a given @xmath2 and considering all the angular scales from 2 to 1020 , the value of the signal - to - noise ratio is the highest in the case of a satellite mission . as an example , a tensor - to - scalar ratio @xmath183 would be detected at a statistical significance of about @xmath184 . in the case of an array of telescope , the value of the signal - to - noise ratio remains high for a large range of values of the tensor - to - scalar ratio , showing a detection of 28@xmath167 at @xmath185 . finally , a small scale experiment would set mild constraints on low values of the tensor - to - scalar ratio , reaching 3@xmath167 at @xmath185 . + in the frame of the primordial @xmath0-mode detection prospects , the minimal value of the tensor - to - scalar ratio @xmath2 that could be detected regarding the experimental setups is a relevant result . the table [ tab : snr ] summarizes in this perspective the aforementioned results , considering a measurement of @xmath2 with at least a 3@xmath167 statistical significance as a threshold . the minimal accessible value of @xmath2 is shown with respect to the experimental setups and the estimation of the @xmath0-mode variance over all the range of multipoles ( referred to as _ case a _ in the table ) . as explained above , the mode counting estimation of the variance overestimates the forecasts made on the minimal accessible @xmath2 as compared to the realistic @xmath0-mode estimation . in the case of a potential satellite mission for instance , the lowest accessible @xmath2 we could realistically expect is 2.88 greater than the one estimated using the mode - counting estimation . these results therefore highlight the inaccuracy that an approximative estimation of the @xmath0-mode induces on the performed forecasts of the detectable @xmath2 values . thus considering the realistic forecasts performed thanks to the pure estimation of the @xmath0-mode power spectrum , we conclude that a satellite mission would give access to the largest range of @xmath2 , with a minimal @xmath2 value of @xmath186 . a typical small scale experiment is indeed expected to reach only @xmath2 higher than @xmath133 at 3@xmath167 ( note that @xmath141 is detectable at @xmath187 ) . between these two cases lies the one of an array of telescopes , which warrants a detection of the tensor - to - scalar ratio if it is higher than @xmath188 . as a result , each studied experiments widens the accessible range of the tensor - to - scalar ratio @xmath2 . in terms of minimal detectable vlaue of @xmath2 , one gains about a factor 20 from small - scale experiments to an array of telescopes , and about a factor 2 between the latter and a satellite mission . + p2.4cmm1.9cmm1.9cmm1.9 cm & small - scale exp . & telescopes array & satellite mission + mode - counting : & & & + case a & @xmath189 & @xmath190 & @xmath191 + case b & @xmath192 & @xmath193 & @xmath194 + pure pseudo-@xmath100 : & & & + case a & @xmath195 & @xmath196 & @xmath197 + case b & @xmath195 & @xmath198 & @xmath199 + furthermore , the table [ tab : snr ] also displays the minimum accessible @xmath2 obtained without the information from the first bin ( @xmath200 ) of the @xmath0-mode power spectrum ( referred to as the _ case b _ ) . this lack of information obviously leads to a smallest range of accessible @xmath2 than in the _ case a _ for a naive estimation of the @xmath0-mode variances , as explained in the previous subsection . in particular , while the accessible @xmath2 range is little affected by removing the first bin in the case of a small scale experiment , the minimal accessible @xmath2 is @xmath201 ( @xmath202 resp . ) greater for a satellite mission ( an array of telescopes resp . ) as the largest angular scales are relevant for these experimental setups . we note here that contrary to what one might expect , a large scale experiment would still succeed in detecting @xmath2 of at least @xmath203 . for @xmath63 between 20 and 90 , the amplitude of the primordial signal is roughly 10% of the lensing signal while the total ( mode - counting estimated ) error budget varies from few percents to 10% of the lensing signal . summing over the multipoles range thus enables a detection of @xmath204 with a @xmath112 statistical significance . nonetheless , in this _ case b _ , the orders of magnitude of the realistic forecasts remain unchanged if the @xmath0-mode power spectrum is reconstructed from the pure pseudo-@xmath100 approach . the values of @xmath2 that could be detected at @xmath112 increase by a factor of less than one percent for a small - scale experiment , a factor of few percents for an array of telescopes , and , a factor of ten percents for a satellite mission . ( this obvisouly reflects the values of @xmath178 found in the previous section , sec . [ sec : firstbin ] . ) this means that the pure pseudo-@xmath100 estimation of the reionization peak of the @xmath0-mode mildly constraints the tensor - to - scalar ratio . to take full advantages of the range @xmath205 ( so as to lower the minimal detectable value of @xmath2 and to enlarge the lever arm to constraint e.g. the spectral index ) , one should probably rely on more optimal techniques for reconstructing @xmath67 at those largest angular scales at large scales is also plagued by other sources of uncertainties such as the level of residual foregrounds and/or the impact of filtering of the maps . ] . we have investigated the detection of the tensor - to - scalar ratio , @xmath2 , from forthcoming and potential future measurements of the cmb polarized anisotropies . we considered the @xmath0-mode as the main source of information on @xmath2 and assumed the pure pseudospectrum reconstruction of its angular power spectra from the maps of stokes @xmath16 and stokes @xmath17 , previously shown to be a method of choice for analyzing coming data sets . we focused on realistic statistical uncertainties ( i.e. sampling and noise variance ) as incurred by such a numerical method , and we purposefully did not consider the potential gain thanks to delensing , nor the loss due to polarized foreground contamination and instrumental systematics . we emphasize that in this paper we consider only the @xmath1-to-@xmath0 leakage due to a cut sky . in cmb practice there are numerous other potential sources of such leakages . for instance , they can arise from instrument limitations , such as beam mismatch @xcite or polarimeter orientation uncertainty @xcite , or be generated by data processing , say , via time - domain filtering @xcite . such leakages would also have an effect on estimated @xmath0-mode power spectrum . the effect will in general depend on a specific method used for the estimation but also on the detailed nature of the leakage itself , and thus would have to be studied cased by case . in many situations , such leakages could be corrected for already on the map - making stage , leaving therefore the cut - sky as the only fundamental source of the leakage to contend with on the power spectrum estimation level as assumed in this work . in contrast , we include the effects of the gravitational lensing , i.e. , of the `` cosmological @xmath1-to-@xmath0 leakage '' , in the total uncertainty budget , in spite of the fact that map - making - level , delensing procedures , which could correct for part of this effect have been proposed @xcite . the improvements on the detection of @xmath2 those methods could give depend on the noise level and the resolution of the experiment , and , on the potential use of external datasets ( if delensing can not be done internally ) . by including the lensing - induced @xmath0-mode , we adopt a more conservative viewpoint as far as forecasts on the tensor - to - scalar ratio are concerned . + in this framework , we first consider the case of small - scale ( either ground - based or balloon - borne ) experiments in an idealized way , assuming the observed sky patch is azimuthally symmetric . we consider four values of @xmath32 and @xmath33 , and let the sky fraction to vary from 0.5% to 10% ( with a noise level of @xmath206k - arcminute at @xmath26 . we compare the signal - to - noise on @xmath2 as obtained from the pure pseudospectrum reconstruction of the @xmath0-mode to the signal - to - noise ratio that would obtained assuming either the mode - counting estimation of the uncertainties on the @xmath0-mode , or a minimum variance , quadratic estimator . we show that the statistical significance on the detection of @xmath2 using the mode - counting is overestimated by a factor @xmath4 as compared to the more realistic case of the pure pseudospectrum estimation . ( the mode - counting also overestimate this significance by a factor @xmath3 as compared to the minimum variance , quadratic estimator . ) similarly , the ( s / n)@xmath114 obtained from the pure pseudospectrum estimator is reduced by 1.5% ( at @xmath120 ) to 8% ( at @xmath117 ) as compared to the lossless minimum variance , quadratic estimator . for the case of small - scale experiment for which the reionization bump is not accessible , and in the limitation of azimuthally symmetric patches , the pure pseudospectrum approach for @xmath0-modes reconstruction is thus almost as accurate as the more computationally costly minimum variance , quadratic estimator ( the former scaling as @xmath92 and the latter as @xmath207 if the observed sky patch is not azimuthally symmetric ) . as shown in fig . 20 of @xcite , non - azimuthal symmetry basically does not change the uncertainties on the @xmath0-mode reconstruction with the pure pseudospectrum estimator ( except for unrealistic , highly squeezed shapes ) . we can thus except this conclusion to holds for more intricate shapes of the observed sky . our results ( summarized in tab . [ tab : fsky ] ) show that for a given sensitivity typical of forthcoming small - scale experiments , the value of the sky fraction maximizing the signal - to - noise ratio on @xmath2 is rather insensitive to the method adopted to compute the uncertainties on the reconstructed @xmath67 ( either the mode - counting expression , a minimum - variance quadratic estimator or the pure pseudo-@xmath100 approach ) . we also show that the choice of the bandwidth only mildly affect this optimized sky fraction in the case of the mode - counting approach to estimate the uncertainties on the @xmath0-mode reconstruction ( see tab . [ tab : bin ] ) . this means that using the mode - counting expression provide a rather reliable estimate of the optimized sky fraction from the viewpoint of statistical uncertainties though being underestimated . + second , we consider the detection of the tensor - to - scalar ratio for three selected examples , each of them mimicking three archetypal experimental configurations . realistic sky coverage ( with intricate contours ) and realistic noise distribution for the small - scale experimental setup are considered and the statistical uncertainties on the @xmath0-mode reconstruction are derived from the mode - counting expression first ( used as a benchmark ) , and second , from the pure pseudospectrum estimators using optimized sky apodizations . our results are summarized in the table [ tab : snr ] . for each experimental setups , it shows the minimal values of @xmath2 that could be measured with at least a statistical significance of @xmath112 . one gains more than one order of magnitude for the minimal detectable value of @xmath2 from the small - scale experiment to an array of telescopes , and another factor 2 from an array of telescopes to a satellite mission . this conclusion stands even if the largest angular scales ( @xmath208 ) can not be used in the analysis . let us briefly discuss the impact of those results in the context of single - field , slow - roll inflation . our purpose here is to give a rough translation of the potential measurement of the tensor - to - scalar ratio with the pure pseudo-@xmath100 estimation of the @xmath0-mode , into a potential discrimination between small fields and large fields models of inflation . ( a more detailed study of inflationary models can be found in ref . @xcite , though it is restricted to satellite missions and assumes a different evaluation of the error budget for the @xmath0-mode reconstruction . ) the tensor - to - scalar ratio is an valuable source of information for the physics of the primordial universe . first , it is a direct measure of the energy scale of inflation , @xmath209 with @xmath210 the value of the inflaton potential during inflation : @xmath211 this means that a measured value of @xmath212 corresponds to test a physical regime in the playground of grand unified theories . second , the tensor - to - scalar ratio is directly related to the number of e - folds , @xmath213 , and the excursion of the scalar field , @xmath214 , from the instant when cosmological fluctuations observed in the cmb are created during inflation , to the end of inflation @xcite : @xmath215 with @xmath216 the reduced planck mass . ( we note that @xmath213 can be determined from the knowledge of the inflaton potential . we however let it free in order not to assume a too specific shape of this potential . ) single field inflationary models can be roughly classified between large fields models and small fields models , whether the excursion of the scalar field is transplanckian or subplanckian , respectively . though the value @xmath217 should not be considered as a sharp and univoquely defined frontier between small fields and large fields models , a precise measure of @xmath2 then allows for discriminating between this two classes of models . for @xmath218 and considering zero runnings of the spectral index ( see @xcite for extensions of the lyth bound with runnings ) , values of @xmath2 greater than @xmath219 would correspond to large fields models of inflation ( see also ref . @xcite and references therein for examples of small fields models evading the lyth bound ) . , that could be observed with , at least , a @xmath112 significance , as functions of the number of e - folds during inflation . darker blue to lighter blue respectively stands for small - scale experiment , an array of telescopes and a satellite mission . the minimal , detectable value of @xmath2 at @xmath112 allowing for such a measurement is the one derived from the pure pseudo-@xmath100 estimation of the @xmath0-mode angular power spectra . ] the figure [ fig : lyth ] shows the ranges of @xmath220 as a function of @xmath213 accessible assuming that the tensor - to - scalar ratio has been measured with at least a @xmath112 statistical significance . blue areas correspond to the accessible range for each experimental configurations ( notice that the higher @xmath214 , the higher @xmath2 ) . the dark blue region is for the case of small - scale experiments , while the somewhat lighter blue and light blue regions corresponds to the case of the array of telescopes and of the satellite mission respectively . the minimal detectable value of @xmath2 with at least @xmath112 is the one derived from a pure pseudospectrum reconstruction of the angular power spectra of the @xmath0-mode and using the entire set of angular scales ( the _ case a _ of tab . [ tab : snr ] ) . this shows that a measurement of @xmath2 from the pure pseudo-@xmath100 reconstruction of the @xmath0-mode thanks to datasets coming from a small - scale experiment , is impossible if small fields models appear to be realized in the early universe . though a detection is possible in the large field models , there is still a range of such models for which the level of primordial gravity waves is still undetectable by a small - scale experiment . small fields models are only marginally accessible from the pure pseudo-@xmath100 estimation of the @xmath0-mode using datasets from an array of telescopes , as @xmath221 is accessible for @xmath213 smaller than @xmath222 . a detection of @xmath2 consistent with zero with a @xmath112 confidence level implies an excursion of the scalar field ( in planck units ) smaller than 0.8 to 1.8 for @xmath213 varying from 30 to 70 . finally , datasets coming from a satellite mission allows for a detection of primordial gravity waves in the small fields models with the pure pseudospectrum estimation of @xmath67 , providing that the number of e - folds is smaller than @xmath223 . on the range of e - folds considered here , a measurement of the tensor - to - scalar ratio consistent with zero then implies @xmath224 , meaning that a discrimination between large fields models and small fields models is possible for a wide range of values of @xmath213 . this research used resources of the national energy research scientific computing center , which is supported by the office of science of the u.s . department of energy under contract no . de - ac02 - 05ch11231 . some of the results in this paper have been derived using s@xmath75hat @xcite , healpix @xcite and class @xcite software packages . 999 u. seljak & m. zaldarriaga , * 78 * 2054 ( 1997 ) d. n. spergel & m. zaldarriaga , * 79 * 2180 ( 1997 ) collaboration : p. a. r. ade , n. aghanim _ et al . _ , arxiv:1502.0211 [ astro-ph.co ] ; 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@xmath0-mode of polarized anisotropies of the cosmic microwave background is a unique and nearly direct probe of primordial inflation , which can constrain the amplitude of the primordial gravity waves . however , its detection and precise measurement is made difficult by a minute amplitude of the signal , which has to be discerned from many contributions of non - cosmological origin and reliable estimated in the presence of numerous sources of statistical uncertainties . among these latter , the @xmath1-to-@xmath0 leakage , arising as a result of partial sky coverage , has been found to play a key and potentially fundamental role in determining the possible statistical significance with which the primordial @xmath0-mode signal can be detected . in this work we employ the pure - pseudo formalism devised to minimise the effects of the leakage on the variance of power spectrum estimates and discuss the limits on the tensor - to - scalar ratio , @xmath2 , that could be realistically set by current and forthcoming measurements of the @xmath0-mode angular power spectrum . we compare those with the results obtained using other approaches : nave mode - counting , minimum - variance quadratic estimators , and re - visit the question of optimizing the sky coverage of small - scale , suborbital experiments in order to maximize the statistical significance of the detection of @xmath2 . we show that the optimized sky coverage is largely insensitive to the adopted approach at least for reasonably compact sky patches . we find , however , that the mode - counting overestimates the detection significance by a factor @xmath3 as compared to the lossless maximum variance approach and by a factor @xmath4 as compared to the lossy pure pseudo - spectrum estimator . in a second time , we consider more realistic experimental configurations . with a pure pseudospectrum reconstruction of @xmath0-modes and considering only statistical uncertainties , we find that a detection of @xmath5 , @xmath6 and @xmath7 at 99% of confidence level is within the reach of current sub - orbital experiments , future arrays of ground - based telescopes and a satellite mission , respectively . this means that an array of telescopes could be sufficient to discriminate between large- and small - field models of inflation , even if the @xmath1-to-@xmath0 leakage is consistently included but accounted for in the analysis . however , a satellite mission will be required to distinguish between different small - field models depending on the number of e - folds .

the solution of the absolute value equation ( ave ) of the following form is considered : @xmath0 here , @xmath1 , @xmath2 and @xmath3 denotes the component - wise absolute value of vector @xmath4 , i.e. , @xmath5 . the ave ( [ eq:1 ] ) is a special case of the generalized absolute value equation ( gave ) of the type @xmath6 where @xmath7 and @xmath8 . the gave ( [ eq:1a ] ) was introduced in @xcite and investigated in a more general context in @xcite . recently , these problems have been investigated in the literature @xcite . the ave ( [ eq:1 ] ) arises in linear programs , quadratic programs , bimatrix games and other problems , which can all be reduced to a linear complementarity problem ( lcp ) @xcite , and the lcp is equivalent to the ave ( [ eq:1 ] ) . this implies that ave is np - hard in its general form @xcite . beside , if @xmath9 , then the generalized ave ( [ eq:1a ] ) reduces to a system of linear equations @xmath10 , which have many applications in scientific computation @xcite . the main research of ave includes two aspects : one is the theoretical analysis , which focuses on the theorem of alternatives , various equivalent reformulations , and the existence and nonexistence of solutions ; see @xcite . and the other is how to solve the ave . we mainly pay attention to the letter . in the last decade , based on the fact that the lcp is equivalent to the ave and the special structure of ave , a large variety of methods for solving ave ( [ eq:1 ] ) can be found in the literature ; see @xcite . these also include the following : a finite succession of linear programs ( slp ) is established in @xcite , which arise from a reformulation of the ave as the minimization of a piecewise - linear concave function on a polyhedral set and solving the latter by successive linearization ; a semi - smooth newton method is proposed , which largely shortens the computation time than the slp method in @xcite ; furthermore , a smoothing newton algorithm is presented in @xcite , which is proved to be globally convergent and the convergence rate is quadratic under the condition that the singular values of @xmath11 exceed 1 . this condition is weaker than the one used in @xcite . recently , the picard - hss iteration method is proposed to solve ave by salkuyeh in @xcite , which is originally designed to solve weakly nonlinear systems @xcite and its generalizations are also paid attention @xcite . 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 iteration steps are often problem - dependent and difficult to be determined in actual computations . moreover , the iteration vector can not be updated timely . in this paper , we present the nonlinear hss - like iteration method to overcome the defect mentioned above , which is designed originally for solving weakly nonlinear systems in @xcite . the rest of this paper is organized as follows . in section [ sec:2 ] the hss and picard - hss iteration methods are reviewed . in section [ sec:3 ] the nonlinear hss - like iteration method for solving ave ( [ eq:1 ] ) is described . numerical experiments are presented in section [ sec:4 ] , to shown the feasibility and effectiveness of the nonlinear hss - like method . finally , some conclusions and an open problem are drew in section [ sec:5 ] . in this section , the hss iteration method for solving the non - hermitian linear systems and the picard - hss iteration method for solving the ave ( [ eq:1 ] ) are reviewed . let @xmath12 be a non - hermitian positive definite matrix , @xmath13 be a zero matrix , the gave ( [ eq:1a ] ) reduced to the non - hermitian system of linear equations @xmath14 because any square matrix @xmath11 possesses a hermitian and skew - hermitian splitting ( hss ) @xmath15 the following hss iteration method is first introduced by bai , golub and ng in @xcite for the solution of the non - hermitian positive definite system of linear equations ( [ eq:5 ] ) . * the hss iteration method . * + given an initial guess @xmath16 , compute @xmath17 for @xmath18 using the following iteration scheme until @xmath19 converges , @xmath20 where @xmath21 is a positive constant and @xmath22 is the identity matrix . when the matrix @xmath23 is positive definite , i.e. its hermitian part @xmath24 is positive definite , bai et al . proved that the spectral radius of the hss iteration matrix is less than 1 for any positive parameters @xmath25 , i.e. , the hss iteration method is unconditionally convergent ; see @xcite . for the convenience of the subsequent discussion , the ave ( [ eq:1 ] ) can be rewritten as its equivalent form : @xmath26 recalling that the linear term @xmath27 and the nonlinear term @xmath28 are well separated and the picard iteration method is a fixed - point iteration , the picard iteration @xmath29 can be used to solve the ave ( [ eq:1 ] ) . when the matrix @xmath23 is large sparse and positive definite , the next iteration @xmath30 may be inexactly computed by hss iteration . this naturally lead to the following iteration method proposed in @xcite for solving the ave ( [ eq:1 ] ) . * the picard - hss iteration method . * + let @xmath12 be a sparse and positive definite matrix , @xmath24 and @xmath31 be its hermitian and skew - hermitian parts respectively . given an initial guess @xmath32 and a sequence @xmath33 of positive integers , compute @xmath30 for @xmath34 using the following iteration scheme until @xmath35 satisfies the stopping criterion : \(a ) set @xmath36 \(b ) for @xmath37 , solve the following linear systems to obtain @xmath38 : @xmath39 where @xmath21 is a given positive constant and @xmath22 is the identity matrix ; \(c ) set @xmath40 . the advantage of the picard - hss iteration method is obvious . first , the two linear sub - systems in all inner hss iterations have the same shifted hermitian coefficient matrix @xmath41 and shifted skew - hermitian coefficient matrix @xmath42 , which are constant with respect to the iteration index @xmath43 . second , as the coefficient matrix @xmath41 and @xmath42 are hermitian and skew - hermitian respectively , the first sub - system can be solved exactly by making use of the cholesky factorization and the second one by the lu factorization . the last , these two sub - systems can be solve approximately by the conjugate gradient method and a krylov subspace method like gmres , respectively ; see @xcite . in the picard - hss iteration , the numbers @xmath44 of the inner hss iteration steps are often problem - dependent and difficult to be determined in actual computations @xcite . moreover , the iteration vector can not be updated timely . thus , to avoid these defect and still preserve the advantages of the picard - hss iteration method , based on the hss ( [ eq:6 ] ) and the nonlinear fixed - point equations @xmath45 the following nonlinear hss - like iteration method is proposed to solve the ave ( [ eq:1 ] ) . * the nonlinear hss - like iteration method . * + let @xmath12 be a sparse and positive definite matrix , @xmath24 and @xmath31 be its hermitian and skew - hermitian parts respectively . given an initial guess @xmath32 , compute @xmath30 for @xmath34 using the following iteration scheme until @xmath35 satisfies the stopping criterion : @xmath46 where @xmath21 is a given positive constant and @xmath22 is the identity matrix . it is obvious that both @xmath4 and @xmath3 in the second step are updated in the nonlinear hss - like iteration , but only @xmath4 is updated in the picard - hss iteration . furthermore , the nonlinear hss - like iteration is a monolayer iteration scheme , but the picard - hss is an inner - outer double - layer iteration scheme . to obtain a one - step form of the nonlinear hss - like iteration , we define @xmath47 and @xmath48 then the nonlinear hss - like iteration scheme can be equivalently expressed as @xmath49 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 , bai et al . established the local convergence theory for the nonlinear hss - like iteration method in @xcite . however , these convergence theory has a strict requirement that @xmath28 must be @xmath50-differentiable at a point @xmath51 such that @xmath52 . obviously , the absolute value function @xmath3 is non - differentiable . thus , the convergence analysis of the nonlinear hss - like iteration method for solving weakly nonlinear linear systems is unsuitable for solving ave , and need further discuss . at the end of this section , we remark that the main steps in the nonlinear hss - like iteration method can be alternatively reformulated into residual - updating form as follows . * the hss - like iteration method ( residual - updating variant ) . * + given an initial guess @xmath53 , compute @xmath30 for @xmath34 using the following iterative procedure until @xmath35 satisfies the stopping criterion : \(1 ) set : @xmath54 , \(2 ) solve : @xmath55 , \(3 ) set : @xmath56 , @xmath57 , \(4 ) solve : @xmath58 , \(5 ) set : @xmath59 , + where @xmath21 is a given positive constant and @xmath22 is the identity matrix . in this section , the numerical properties of the picard , picard - hss and nonlinear hss - like methods are examined and compared experimentally by a suit of test problems . all the tests are performed in matlab r2013a on intel(r ) core(tm ) i5 - 3470 cpu 3.20 ghz and 8.00 gb of ram , with machine precision @xmath60 , and terminated when the current residual satisfies @xmath61 where @xmath17 is the computed solution by each of the methods at iteration @xmath43 , and a maximum number of the iterations 500 is used . in addition , the stopping criterion for the inner iterations of the picard - hss method is set to be @xmath62 where @xmath63 , @xmath64 , @xmath65 is the number of the inner iteration steps and @xmath66 is the prescribed tolerance for controlling the accuracy of the inner iterations at the @xmath43-th outer iteration . if @xmath66 is fixed for all @xmath43 , then it is simply denoted by @xmath67 . here , we take @xmath68 . the first subsystem with the hermitian positive definite coefficient matrix @xmath69 in ( [ eq : hsslike ] ) is solved by the cholesky factorization , and the second subsystem with the skew - hermitian coefficient matrix @xmath70 in ( [ eq : hsslike ] ) is solved by the lu factorization . the optimal parameters employed in the picard - hss and nonlinear hss - like iteration methods have been obtained experimentally . in fact , the experimentally found optimal parameters are the ones resulting in the least numbers of iterations and cpu times@xcite . as mentioned in @xcite the computation of the optimal parameter is often problem - dependent and generally difficult to be determined . we consider the two - dimensional convection - diffusion equation @xmath71 where @xmath72 , @xmath73 is its boundary , @xmath74 is a positive constant used to measure the magnitude of the diffusive term and @xmath75 is a real number . we use the five - point finite difference scheme to the diffusive terms and the central difference scheme to the convective terms . let @xmath76 and @xmath77 denote the equidistant step size and the mesh reynolds number , respectively . then we get a system of linear equations @xmath78 , where @xmath11 is a matrix of order @xmath79 of the form @xmath80 with @xmath81 where @xmath82 , @xmath83 and @xmath84 are the identity matrices of order @xmath85 and @xmath86 respectively , @xmath87 means the kronecker product . in our numerical experiments , the matrix @xmath11 in ave ( [ eq:1 ] ) is defined by ( [ eq : ex ] ) with different values of @xmath88 and different values of @xmath89 . it is easy to find that for every nonnegative number @xmath74 the matrix @xmath11 is in general non - symmetric positive definite@xcite . we use the zero vector as the initial guess , and the right - hand side vector @xmath90 of ave ( [ eq:1 ] ) is taken in such a way that the vector @xmath91 with @xmath92 is the exact solution , where @xmath93 denotes the imaginary unit . .the optimal parameters values @xmath25 for picard - hss and nonlinear hss - like methods ( p=0 ) . [ cols="<,<,<,>,>,>,>,>,>,>,>,>,>",options="header " , ] in this paper we have studied the nonlinear hss - like iteration method for solving the absolute value equation ( ave ) . this method is based on separable property of the linear term @xmath27 and nonlinear term @xmath94 and the hermitian and skew - hermitian splitting of the involved matrix @xmath11 . compared to that the picard - hss iteration scheme is an inner - outer double - layer iteration scheme , the nonlinear hss - like iteration is a monolayer and the iteration vector could be updated timely . numerical experiments have shown that the nonlinear hss - like method is feasible , robust and efficient nonlinear solver . the most important is it can outperform the picard - hss in actual implementation .

salkuyeh proposed the picard - hss iteration method to solve the absolute value equation ( ave ) , which is a class of non - differentiable np - hard problem . to further improve its performance , a nonlinear hss - like iteration method is proposed . compared to that the picard - hss method is an inner - outer double - layer iteration scheme , the hss - like iteration is only a monolayer and the iteration vector could be updated timely . some numerical experiments are used to demonstrate that the nonlinear hss - like method is feasible , robust and effective . absolute value equation , nonlinear hss - like iteration , fixed point iteration , positive definite 15a06,65f10,65h10

the gravitational deflection of light beams by large scale structures of the universe ( cosmological lensing ) amplifies and modifies the shape of distant galaxies and quasars . magnification produces correlation between the density of foreground lenses and the apparent luminosity of distant galaxies or quasars ( magnification bias ) , whereas distortion induces a correlation of ellipticity distribution of lensed galaxies ( cosmic shear ) . in both cases , the properties of cosmological lensing signals probe the matter content and the geometry of universe and how perturbations grew and clustered during the past gigayears . + albeit difficult to detect , the recent cosmic shear detections claimed by several groups demonstrate that it is no longer a technical challenge . it is therefore possible to study the universe through a new window which directly probes dark matter instead of light and allows cosmologists to measure cosmological parameters and dark matter power spectrum from weak gravitational distortion . let us assume that the shape of galaxies can be simply characterize by their surface brightness second moments @xmath0 , ( see @xcite , @xcite and references therein ) : @xmath1 because of gravitational lensing , a galaxy with intrinsic ellipticity @xmath2 is measured with an ellipticity @xmath3 , where @xmath4 is the gravitational distortion , @xmath5 @xmath6 and @xmath7 are respectively the gravitational convergence and shear . both depend on the second derivatives of the projected gravitational potential , @xmath8 : @xmath9 in the case of weak lensing , @xmath10 , @xmath11 and @xmath12 . since large - scale structures have very low density contrast , this linear relation is in particular valid on cosmological scales . + light propagation through an inhomogeneous universe accumulates weak lensing effects over gigaparsec distances . assuming structures formed from gravitational growth of gaussian fluctuations , cosmological weak lensing can be predicted from perturbation theory at large scale . to first order , the convergence @xmath13 at angular position @xmath14 is given by the line - of - sight integral @xmath15 { \rm d}\chi\ ] ] where @xmath16 is the radial distance out to redshift @xmath17 , @xmath18 the angular diameter distances , @xmath19 is the redshift distribution of the sources . @xmath20 is the mass density contrast responsible for the deflection at redshift @xmath17 . its amplitude at a given redshift depends on the properties of the power spectrum and its evolution with look - back - time . + the cumulative weak lensing effects of structures induce a shear field which is primarily related to the power spectrum of the projected mass density , @xmath21 . its statistical properties can be recovered by the shear top - hat variance @xcite , @xmath22 ^ 2 , \label{theovariance}\ ] ] the aperture mass variance @xcite @xmath23 ^ 2 , \label{theomap}\ ] ] and the shear correlation function @xcite : @xmath24 where @xmath25 is the bessel function of the first kind . higher order statistics , like the skewness of the convergence , @xmath26 , can also be computed . they probe non gaussian features in the projected mass density field , like massive clusters or compact groups of galaxies . ( see @xcite ; @xcite and references therein ) . the amplitude of cosmic shear signal and its sensitivity to cosmology can be illustrated in the fiducial case of a power law mass power spectrum with no cosmological constant and a background population at a single redshift @xmath17 . in that case @xmath27 and @xmath26 write : @xmath28 and @xmath29 where @xmath30 is the spectral index of the power spectrum of density fluctuations . therefore , in principle the degeneracy between @xmath31 and @xmath32 can be broken when both the variance and the skewness of the convergence are measured . ( [ eqvar ] ) shows that the amplitude of weak lensing signal is of the order of few percents , which is much smaller than the intrinsic dispersion of ellipticity distribution of galaxies . van waerbeke et al ( @xcite ) explored which strategy would be best suited to probe statistical properties of such a small signal . they have shown that the variance of @xmath6 can be measured with a survey covering about 1 @xmath33 , whereas for the skewness one needs at least 10 @xmath33 . furthermore , more than 100 @xmath33 must be observed in order to uncover information on @xmath34 or the shape of the power spectrum over scales larger than 1 degree . for @xmath35 and @xmath36 , the limiting shear amplitude can be simply expressed as follows @xmath37^{{1 \over 4 } } \times \left[{\sigma_{\epsilon_{gal } } \over 0.4 } \right ] \times \left[{n \over 20}\right]^{-{1 \over 2 } } \times \left[{\theta \over 10'}\right]^{{-{1 \over 2 } } } \ , \ ] ] where @xmath38 is the total sky coverage of the survey . the numbers given in the brackets correspond to a measurement at @xmath39 confidence level of the shear variance . ( [ survey ] ) contains the specifications of a cosmic shear survey . despite technical limitations discussed above , on scale significantly smaller than one degree , non - linear structures dominate and increase the amplitude of the lensing signal , making its measurement easier . few teams started such surveys during the past years and succeeded to get a significant signal . table [ tabcs ] lists some published results . since each group used different telescopes and adopted different observing strategy and data analysis techniques , one can figure out the reliability of the final results . .present status of cosmic shear surveys with published results . [ cols="<,^,^,^,<",options="header " , ] figure [ sheartop ] show that all these independent results are in very good agreement . this is a convincing demonstration that the expected correlation of ellipticities is real . the detection of coherent signal is not a demonstration of its very nature . even if a cosmological signal were expected , it could be contaminated by systematics , like optical and atmospheric distortions , which mix together with the gravitational shear . contrary to lensing effects , these contributions are visible also on stars and can be corrected ( using for example the ksb method , @xcite ) . however , stars often show strong anisotropic shape with elongation much larger than the expected amplitude of the gravitational distortion . the reliability of artificial anisotropy corrections is therefore a critical step of the weak lensing analysis ( see for example @xcite , @xcite @xcite and @xcite ) . an elegant way to check whether corrections are correctly done and to confirm the gravitational nature of the signal is to decompose the signal into e- and b- modes . the e - mode contains signal produced by gravity - induced distortion whereas the b - mode is a pure curl - component , so it only contains intrinsic ellipticity correlation or systematics residuals . both modes have been extracted using the aperture mass statistics by van waerbeke et al ( @xcite , @xcite ) and pen et al ( @xcite ) in the virmos - descart survey as well as by hoekstra et al ( @xcite ) in the red cluster sequence survey . in both samples , the e - mode dominates the signal , although a small residual is detected in the b - mode . this strongly supports the gravitational origin of the distortion . + an alternative to gravitational lensing effect could be an intrinsic correlations of ellipticities of galaxies produced by proximity effects . it could results from galaxy formation processes . several recent numerical and theoretical studies ( see for example @xcite ; @xcite ) show that intrinsic correlations are negligible on scales beyond one arc - minute , provided that the survey is deep enough . in that case , most lensed galaxies along a line of sight are spread over gigaparsec scales and have no physical relation with its apparent neighbors . hence , since most cosmic survey are deep , they are almost free of intrinsic correlations . we therefore are confident that the signal measured by all teams is a genuine cosmic shear signal . . for most points the errors are smaller than the stars . , width=340 ] a comparison of the top - hat variance of shear with some realistic cosmological models is ploted in figure [ sheartop ] . the amplitude of the shear has been scaled using photo-@xmath17 which gives @xmath40 . on this plot , we see that standard cobe - normalized cdm is ruled at a 10@xmath41 confidence level . however , the degeneracy between @xmath31 and @xmath32 discussed in the previous section still hampers a strong discrimination among most popular cosmological models . the present - day constraints resulting from independent analyses by maoli et al ( @xcite ) , rhodes et al ( @xcite ) , van waerbeke et al ( @xcite ) , hoekstra et al ( @xcite ) and rfrgier et al ( @xcite ) can be summarized by the following conservative boundaries ( 90% confidence level ) : @xmath42 and , in the case of a flat - universe with @xmath35 , they lead to @xmath43 . + and @xmath32 for the flat cosmologies . the confidence levels are @xmath44 $ ] from the brightest to the darkest area . the gray area and the dashed contours correspond to the computations with a full marginalisation over the default prior @xmath45 $ ] and @xmath46 $ ] . the thick solid line contours are obtained from the prior @xmath47 $ ] and @xmath48 $ ] ( which is a mean redshift @xmath49 $ ] ) . from van waerbeke et al . , width=340 ] and @xmath50 contributions to the two - point correlation functions are included . the dot - dashed line with error bars corresponds to measurements where the contribution of the @xmath50 mode has been subtracted out from the two - point correlation function . these measurements are compared to results obtained in @xmath51cdm , ocdm and @xmath52cdm simulations ( dashed , dotted and dot - dashed lines respectively ) . , width=377 ] the measurement of non - gaussian features needs informations on higher order statistics than variance . although the afore mentioned skewness of @xmath6 looks a promising quantity for this purpose , its measurements suffers from a number a practical difficulties which are not yet fixed . recently , bernardeau , van waerbeke & mellier ( @xcite ) have proposed an alternative method using some specific patterns in the shear three - point function . despite the complicated shape of the three - point correlation pattern , they uncovered it can be used for the measurement of non - gaussian features . their detection strategy based on their method has been tested on ray tracing simulations and turns out to be robust , usable in patchy catalogs , and quite insensitive to the topology of the survey . + bernardeau , mellier & van waerbeke ( @xcite ) used the analysis of the 3-point correlations function on the virmos - descart data . their results on figure [ xi3 ] show a 2.4@xmath54 signal over four independent angular bins , or equivalently , a 4.9-@xmath54 confidence level detection with respect to measurements errors on scale of about @xmath55 to @xmath56 arc - minutes . the amplitude and the shape of the signal are consistent with theoretical expectations obtained from ray - tracing simulations . this result supports the idea that the measure corresponds to a cosmological signal due to the gravitational instability dynamics . moreover , its properties could be used to put constraints on the cosmological parameters , in particular on the density parameter of the universe . although the errors are still large to permit secure conclusions , one clearly see that the amplitude and the shape of the 3-point correlations function match the most likely cosmological models . remarkably , the @xmath52cdm scenario perfectly fit the data points . + the bernardeau et al . ( @xcite ) result is the first detection of non - gaussian features in a cosmic shear survey and it opens the route to break the @xmath53 degeneracy . furthermore , this method is weakly dependent on other parameters , like the cosmological constant or the properties of the power spectrum . however , there are still some caveats which may be considered seriously . one difficulty is the source clustering which could significantly perturb high - order statistics ( hamana et al 2000 , @xcite ) . if so , multi - lens plane cosmic shear analysis will be necessary which implies a good knowledge of the redshift distribution . for very deep cosmic shear surveys , this could be could be a challenging issue . because on going surveys increase both in solid angle and in number of galaxies , they will quickly improve the accuracy of cosmic shear measurements , at a level where @xmath31 and @xmath32 will be known with a 10% accuracy . since it is based on gravitational deflection by intervening matter spread over cosmological scales , the shape of the distortion field also probes directly the shape of the projected power spectrum of the ( dark ) matter . pen et al ( @xcite ) already explored its properties measuring for the first time the @xmath57 of the dark matter ( see figure [ cl ] ) . we therefore know this is feasible with present data . of dark matter ever measured in cosmology.,width=283 ] however , we expect much more within the next decade . surveys covering hundreds of degrees , with multi - bands data in order to get redshift of sources and possibly detailed information of their clustering properties , are scheduled . the cfht legacy survey will cover 200 deg@xmath58 and is one of those next - generation cosmic shear survey . figures 1 and [ future1 ] shows it potential for cosmology . on figure 1 we simulated the expected signal to noise of the shear variance as function of angular scale for a @xmath52cdm cosmology . the error bars are considerably reduced as compared to present - day survey . on fig . [ future1 ] , we compare the expected signal to noise ratio of the cfht legacy survey with the expected amplitude of the angular power spectrum for several theoretical quintessence fields models . it shows that even with 200 deg@xmath58 which include multi - color informations in order to get redshift of sources , one can already obtain interesting constraints on cosmology beyond standard models . + the use of cosmic shear data can be much more efficient if they are used together with other surveys , like cmb ( boomerang , map , planck ) , snia surveys , or even galaxy surveys ( 2df , sdss ) . for example , sdss will soon provide the 100 , 000 quasars with redshifts . mnard & bartelmann ( @xcite ) have recently explored the interest of this survey in order to cross - correlate the foreground galaxy distribution with the quasar population . the expected magnification bias generated by dark matter associated with foreground structures as mapped by galaxies depends on @xmath31 and the biasing @xmath32 . in principle magnification bias in the sdss quasar sample can provide similar constrains as cosmic shear . of the cfht legacy survey . the lines shows various models discussed by benabed & bernardeau ( @xcite).,width=302 ] we thank m. bartelmann , k. benabed , d. bond , t. hamana , h. hoekstra , b. mnard , s. prunet and p. schneider for useful discussions . this work was supported by the tmr network `` gravitational lensing : new constraints on cosmology and the distribution of dark matter '' of the ec under contract no . erbfmrx - ct97 - 0172 . ym thanks the organizers of the meeting for financial support . 99 mellier , y. ; 1999 araa 37 , 127 . bartelmann , m. ; schneider , p. ; 2001 phys . 340 , 292 . bernardeau , f. ; van waerbeke , l. ; mellier , y. ; 1997 a&a 322 , 1 . jain , b. ; seljak , u. ; 1997 apj 484 , 560 . van waerbeke , l. ; bernardeau , f. ; mellier , y. ; 1999 a&a 342 , 15 . erben , t. ; van waerbeke , l. ; bertin , e. ; mellier , y. ; schneider , p. ; 2001 a&a 366 , 717 . van waerbeke , l. ; mellier , y. ; erben , t. ; et al . ; 2000 a&a 358 , 30 [ vwme+ ] . r. blandford , a. saust , t. brainerd , j. villumsen ; 1991 mnras 251 , 600 wittman , d. ; tyson , j.a . ; kirkman , d. ; dellantonio , i. ; bernstein , g. 2000a nature 405 , 143 [ wtk+ ] . bacon , d. ; rfrgier ; a. , ellis , r.s . ; 2000 mnras 318 , 625 [ bre ] . bacon , d. ; rfrgier ; a. , clowe , d. , ellis , r.s . ; 2000 mnras 325 , 1065 . kaiser , n. ; , wilson , g. ; , luppino , g. 2000 preprint , astro - ph/0003338 [ kwl ] . maoli , r. ; van waerbeke , l. ; mellier , y. ; et al . ; 2001 a&a 368 , 766 [ mvwm+ ] . rhodes , j. ; rfrgier , a. , groth , e.j . ; 2001 apj 536 , 79 . van waerbeke , l. ; mellier , y. ; radovich , m. ; et al . ; 2001 a&a 374 , 757 [ vwmr+ ] . bacon , d. , massey , r. , , rfrgier , a. , . ellis , r. astro - ph/0203134 j. miralda - escud ; 1991 apj380,1 pen , u - l . , van waerbeke , l. , mellier , y. ; 2002 apj 567 , 31 kaiser , n. 1992 apj 388 , 272 kaiser , n. , squires , g. , broadhurst , t. 1995 , apj 449 , 460 . kaiser , n. et al . , 1994 , in durret et al . , _ clusters of galaxies _ , eds frontires p. schneider , l. van waerbeke , b. jain , g. kruse ; 1998 apj333 , 767 . a. rfrgier , j. rhodes , e. groth , apjl , in press , astro - ph/0203131 l. van waerbeke , y. mellier , r. pello et al . , a & a , in press , astro - ph/0202503 hmmerle , h. ; miralles , j .- m . ; schneider , p. ; erben , t. ; fosbury , r.a.e . ; freudling , w. ; pirzkal ; n. , jain , b. ; white , s.d.m . ; 2002 a&a385 , 743 hoekstra , h. ; yee , h. ; , gladders , m.d . 2001 apj 558 , l11 h. hoekstra , h. yee , m. gladders , apj , in press , astro - ph/0204295 pen , u. ; van waerbeke , l. ; mellier , y. ; 2001 apj in press astro - ph/0109182 . pierpaoli , e. , scott , d. , white , m. 2001 mnras 325 , 77 . crittenden , r.g . ; natarajan , p. ; pen , u. ; theuns , t. 2001 apj 559 , 552 . mackey , j. ; white , m. ; kamionkowski , m. ; 2001 preprint , astro - ph/0106364 . hamana , t. et al . 2000 . preprint astro - ph/0012200 davis , m. ; newman , j. ; faber , s. ; phillips , a. ; 2000 proc . eso / ecf / esa on deep fields springer . le fvre , o. ; saisse , m. ; mancini , m. ; 2000 spie 4008 , 546 . schneider , p. 1998 apj 498 , 43 . van waerbeke , l. 1998 a&a 334 , 1 . bernardeau , f. , van waerbeke , l. , mellier , y. 2002 preprint astro - ph/0201029 . bernardeau , f. , mellier , y. , van waerbeke , l. 2002 a&a in press . preprint astro - ph/0201032 benabed , k. ; bernardeau , f. ; 2001 preprint , astro - ph/0104371 . mnard , b. , bartelmann , m. 2002 preprint astro - ph/0203163

we present the current status of cosmic shear studies and their implications on cosmological models . theoretical expectations and observational results are discussed in the framework of standard cosmology and cdm scenarios . the potentials of the next generation cosmic shear surveys are discussed . # 1

the presence of waves and oscillations in the solar corona is a well known feature that has been observed for long time . for an overview of the early observational background see @xcite . nowadays , because of the increasing spatial and temporal resolution of the euv instruments onboard trace , soho and hinode spacecraft , accurate observations of oscillations in different coronal structures are accomplished . many authors have reported observations of transversal coronal loop oscillations from both ground and space - based instruments @xcite . when these observations are compared with theoretical models @xcite , the possibility of inferring some plasma parameters , otherwise difficult to measure , and of improving the existing theoretical models is open ; see @xcite for a review . magnetohydrodynamics ( mhd ) is the underlying theory of coronal seismology and it is believed that all these observed oscillations and waves can be interpreted theoretically in terms of mhd modes of different coronal plasma structures . the theoretical study of these oscillations and waves can be done from several points of view . the first approach is to make a normal mode analysis of the linearized mhd equations , which allows to obtain the spatial distribution of the eigenmodes of the structure together with the dispersion relation @xmath1 . once the elementary building blocks of the mhd normal mode theory are described , the main properties of the resulting mhd waves can be outlined . many authors have explored the normal modes of coronal structures , beginning with very simple cases such as the straight and infinite cylinder @xcite . in the context of curved coronal magnetic structures , @xcite investigated the continuous spectrum of ideal mhd . @xcite and @xcite derived the spectrum of modes in potential and nonpotential arcades . more complex configurations , such as sheared magnetic arcades in the zero-@xmath0 plasma limit , have been studied by @xcite . other authors have studied eigenmodes in curved configurations with density enhancements that represent coronal loops ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? an alternative approach is to obtain the time dependent solution of the mhd equations . using this method , @xcite studied analytically the propagation of fast waves in a two - dimensional coronal arcade for a particular equilibrium , namely one with uniform alfvn speed . @xcite studied the effect of impulsively generated fast waves in the same coronal structure . @xcite studied the properties of alfvn waves in an arcade configuration , including the transition region between the photosphere and the corona . other studies have analyzed the effect of the loop structure on the properties of fast and slow waves in two - dimensional curved configurations ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , see @xcite for a review . the main aim of this paper is to analyze the effect of including three - dimensional propagation on the resulting mhd waves as a first step before considering more realistic situations like the one observed by @xcite , where the effect of three - dimensional propagation is clear . in our model there is no density enhancement like that of a loop and the zero-@xmath0 approximation is assumed , so only the fast and alfvn modes are present . we focus our attention on the mixed properties displayed by the generated mhd waves that arise due to the coupling when longitudinal propagation is allowed . the paper is arranged as follows . in [ equilibrium_conf ] we briefly describe the equilibrium configuration as well as some of the approximations made in this work . in [ linear ] we present our derivation of the linear ideal mhd wave equations with three - dimensional propagation of perturbations . in [ numerical_method_and_test ] the numerical code used in our study is described , together with several checks that have been performed by solving problems with known analytical or simple numerical solution . our main results are shown in [ numerical_res ] , where the linear wave propagation properties of coupled fast and alfvn waves in a two - dimensional coronal arcade , allowing three - dimensional propagation , are described . finally , in [ conclusions ] the conclusions are drawn . we model a solar coronal arcade by means of a two - dimensional potential configuration contained in the @xmath2-plane in a cartesian system of coordinates ( see * ? ? ? . for this @xmath3-invariant configuration the flux function is @xmath4 and the magnetic field components are given by @xmath5 @xmath6 in these expressions @xmath7 is the magnetic scale height , which is related to the lateral extent of the arcade , @xmath8 , by @xmath9 , and @xmath10 represents the magnetic field strength at the photospheric level ( @xmath11 ) . the overall shape of the arcade is shown in figure [ fig : arc ] . in this paper gravity is neglected and the @xmath12 approximation is used for simplicity . therefore , the equilibrium density can be chosen arbitrarily . we adopt the following one - dimensional profile @xmath13 where @xmath14 is the density scale height and @xmath15 is the density at the base of the corona . as shown by @xcite , the combination of magnetic field components given by equation ( [ eq : arccomp ] ) with the density profile given by equation ( [ eq : density ] ) leads to a one - dimensional alfvn speed distribution in the arcade that can be cast as @xmath16}. \label{eq : alfven1}\ ] ] here @xmath17 represents the ratio of the magnetic scale height to the density scale height and @xmath18 is the alfvn speed at the base of the corona . the @xmath19 parameter completely determines the behavior of the alfvn speed profile and hence the wave propagation properties . the case @xmath20 represents a uniform alfvn speed model , while @xmath21 corresponds to an exponentially decreasing alfvn speed in a uniform density configuration . other values of @xmath19 represent situations in which both the alfvn speed and density depend on height in a different manner . in order to study small amplitude oscillations in our potential arcade the previous equilibrium is perturbed . for linear and adiabatic mhd perturbations in the zero-@xmath0 approximation the relevant equations are @xmath22 @xmath23 where @xmath24 is the magnetic permeability of free space and the subscript `` @xmath25 '' is used to represent perturbed quantities . these equations are next particularized to our two - dimensional potential arcade equilibrium . as the equilibrium is invariant in the @xmath3-direction , we can fourier analyze all perturbed quantities in the @xmath3-direction by making them proportional to @xmath26 . in this way , three - dimensional propagation is allowed and each fourier component can be studied separately . as a result of this fourier analysis the perpendicular perturbed velocity and magnetic field components appear accompanied by the purely imaginary number @xmath27 . this is undesirable from a practical point of view , since equations ( [ eq : momentum ] ) and ( [ eq : induction ] ) will be solved numerically and the code is designed to handle real quantities only . nevertheless , by making the appropriate redefinitions , namely @xmath28 and @xmath29 , it turns out that our wave equations can be cast in the following form @xmath30,\label{eq : velocityx}\\ \frac{\partial \tilde{v}_{1y}}{\partial t}&=&\frac{1}{\mu_{0}\rho_{0}}\bigg[\left(b_{x}\frac{\partial\tilde{b}_{1y}}{\partial x}+b_{z}\frac{\partial\tilde{b}_{1y}}{\partial z}\right)+k_{y}\left(b_{1x}b_{x}+b_{1z}b_{z}\right)\bigg],\label{eq : velocityy}\\ \frac{\partial v_{1z}}{\partial t}&=&-\frac{1}{\mu_{0}\rho_{0}}\bigg[\left(\frac{\partial b_{1x}}{\partial z}-\frac{\partial b_{1z}}{\partial x}\right)b_{x}\bigg],\label{eq : velocityz}\\ \frac{\partial b_{1x}}{\partial t}&=&-k_{y}\tilde{v}_{1y}b_{x}-\frac{\partial}{\partial z}\left(v_{1z}b_{x}-v_{1x}b_{z}\right ) , \label{eq : fieldx}\\ \frac{\partial \tilde{b}_{1y}}{\partial t}&=&\frac{\partial}{\partial z}\left(\tilde{v}_{1y}b_{z}\right)+\frac{\partial}{\partial x}\left(\tilde{v}_{1y}b_{x}\right),\label{eq : fieldy}\\ \frac{\partial b_{1z}}{\partial t}&=&-k_{y}\tilde{v}_{1y}b_{z}+\frac{\partial}{\partial x}\left(v_{1z}b_{x}-v_{1x}b_{z}\right).\label{eq : fieldz}\\ \nonumber\end{aligned}\ ] ] these equations constitute a set of coupled partial differential equations with non - constant coefficients that describe the propagation of fast and alfvn waves . as the plasma @xmath12 , slow waves are excluded from the analysis . when @xmath31 , equations ( [ eq : velocityx])([eq : fieldz ] ) constitute two independent sets of equations . the two equations for @xmath32 and @xmath33 are associated to alfvn wave propagation . on the other hand , the four equations for the remaining variables , @xmath34 , @xmath35 , @xmath36 , @xmath37 , describe the fast wave propagation . the basic normal mode properties of fast and alfvn modes in a potential arcade with @xmath31 are described in @xcite , while the case @xmath38 has been considered by @xcite . the time dependent propagation for @xmath31 was analyzed by @xcite . when longitudinal propagation of perturbations is allowed ( @xmath38 ) , the six equations and their solutions are coupled so we may anticipate fast and alfvn wave propagation to display a mixed nature , in an analogous way to the mixed character of eigenmodes obtained by @xcite in their analysis of the normal modes of the present equilibrium with @xmath38 . in the following the tildes in @xmath32 and @xmath33 are dropped . the set of differential equations ( [ eq : velocityx])([eq : fieldz ] ) is too complicated to have analytical or simple numerical solutions except for simplified configurations and under particular assumptions . for this reason we solve them by using a numerical code , although comparisons with known wave properties have been carried out whenever possible . when considering a potential arcade as the equilibrium magnetic field , it is advantageous to use field - related components instead of cartesian components in order to characterize the directions of interest related to the polarization of each wave type . the unit vectors in the directions normal , perpendicular , and parallel to the equilibrium magnetic field are given by @xmath39 where @xmath40 is the flux function given in equation ( [ eq : flux ] ) . these unit vectors are related to the cartesian ones as follows @xmath41 with @xmath42 . in the absence of longitudinal propagation ( i.e. for @xmath31 ) , these three directions are associated with the three types of waves that can be excited , namely @xmath43 for fast waves , @xmath44 for alfvn waves , and @xmath45 for slow waves . since we want to model a coronal disturbance with a localized spatial distribution we have considered as the initial condition a two - dimensional gaussian profile given by @xmath46 , \label{eq : perturbation}\ ] ] where @xmath47 is the amplitude of the velocity perturbation , @xmath48 and @xmath49 are the coordinates of the perturbation s center , and @xmath50 is the width of the gaussian profile at half height . in the following we use @xmath51 to excite fast waves and @xmath52 to excite alfvn waves . when @xmath31 the fast mode produces plasma motions purely normal to the magnetic field , while the alfvn mode is characterized by a purely perpendicular velocity component . when propagation along the @xmath3-direction is considered , pure fast or alfvn modes do not exist and both produce motions in the normal velocity component as well as in the perpendicular velocity component @xcite . it must be noted that the numerical code solves the time - dependent equations in cartesian coordinates and so the solution has to be transformed following expressions ( [ eq : potentialunitvec ] ) to the field - related coordinates . the same applies to the initial perturbation , which must be transformed into the corresponding cartesian components . the numerical code ( see * ? ? ? * for details about the method ) uses the so - called method of lines for the discretization of the variables and the time and space variables are treated separately . for the temporal part , a fourth - order runge - kutta method is used . for the space discretization a finite - difference method with a fourth - order centered stencil is choosen . for a given spatial resolution , the time step is selected so as to satisfy the courant condition . as for the boundary conditions , as we computed the time evolution of two initial perturbations , two kinds of boundary conditions are used . first , when the initial perturbation in @xmath43 is the fundamental normal mode of the @xmath31 problem , for the @xmath43 component line - tying conditions are chosen at all boundaries , while for the @xmath53 component , flow - through conditions are selected except at @xmath11 where line - tying condition is used . on the other hand , when an initial perturbation like ( [ eq : perturbation ] ) is considered , the large photospheric inertia is accomplished by imposing line - tying boundary conditions at @xmath11 . in all other boundaries flow - through conditions are used so that perturbations are free to leave the system . in order to increase numerical stability , fourth - order artificial dissipation terms are included in the numerical scheme . in all the simulations the effects of this artificial dissipation have been checked to ensure that they do not affect the obtained solution , but just contribute to eliminate undesired high - frequency numerical modes . some preliminary tests have been performed in order to figure out the appropriate values of numerical parameters , such as the grid resolution or the numerical dissipation , on the obtained results for fast and alfvn waves . the first test we have conducted has been to run the code with no perturbation at all and to check that the structure remains stable . the results of this numerical run were completely satisfactory . then the propagation of linear fast and alfvn mhd waves in a potential coronal arcade has been considered . the temporal evolution of impulsively generated perturbations with rather similar conditions has been accomplished by several authors : @xcite obtained analytical expressions for the temporal evolution of perturbations when a coronal arcade is taken as the equilibrium state ; @xcite numerically computed such solutions when different initial perturbations are used ; and more recently @xcite showed the main properties of the time evolution of fast and alfvn waves in low-@xmath0 environments . these works facilitate the comparison of our numerical results with known results as well as with analytical ones . as shown by @xcite , when different resolutions are used time dependent results reveal that the grid resolution in our two - dimensional domain is not a critical factor for the proper computation of fast waves and that a good representation of the temporal evolution of perturbations can be achieved even with a rather modest resolution of @xmath54 grid points in the @xmath55-plane . as mentioned above , numerical dissipation is introduced in our code in order to ensure numerical stability . this dissipation is proportional to an adjustable parameter , or dissipation factor , @xmath56 . we have conducted numerical simulations for different values of the dissipation factor and it turns out that the temporal evolution of fast wave perturbations is not modified . the properties of alfvn continuum normal modes in a potential coronal arcade described by @xcite allow us to anticipate and identify possible sources of difficulties in the numerical computation of alfvn wave solutions . first of all , since they are oscillatory solutions strongly confined around given magnetic surfaces ( both when propagating or in their standing mode version ) , spatial scales quickly decrease with time and so we can expect a rather important dependence of the numerical solutions on the number of grid points used to cover the area in and around the excited magnetic surfaces . the situation becomes even worse if we take into account that computations in a cartesian grid do not allow us to locate all the grid points along magnetic surfaces . this fact affects the numerical results and adds a numerical damping . furthermore , when time - dependent simulations are considered the sampling rate is no more an independent parameter . when the spatial resolution of the grid is defined , the courant condition gives a maximum value for the temporal resolution which in turn sets the maximum frequency that can be resolved . we have first generated alfvn waves in our potential arcade model by considering an impulsive initial excitation of the @xmath53 component given by equation ( [ eq : perturbation ] ) with @xmath57 and @xmath58 . this implies that the initial disturbance is even about @xmath59 and so odd alfvn modes are not excited . as described by @xcite , the spatial resolution of the numerical mesh affects the obtained amplitude and frequency values . better resolution provides a closer value to the analytical frequency and less numerical damping . we have also checked the influence of numerical dissipation and the results show that only the amplitude , and therefore the damping time , decreases when the @xmath56 parameter is decreased . the spectral analysis of these oscillations at different heights in the structure is shown in figure [ fig : spectrumalfvenky0]a . the resulting power spectrum is compared to the alfvn continuum frequencies obtained by @xcite . the frequency associated with the generated alfvn waves coincides with the theoretical normal mode frequencies of the system , which gives us further confidence on the goodness of our code . alfvn waves stay confined to the vertical range of magnetic surfaces that were excited by the initial disturbance , since they can not propagate energy across magnetic surfaces . the initial perturbation is decomposed by the system in a linear combination of normal modes , but keeping the even parity of the initial disturbance with respect to @xmath59 , so energy is only found in the fundamental mode , the second harmonic , etc . in order to better isolate and show the possible numerical artifacts that the code introduces into the numerical solution we have considered a simpler case , the excitation of a particular alfvn mode around a magnetic surface . according to @xcite , alfvn normal mode solutions can be obtained analytically when @xmath21 . for this reason we now select @xmath21 . the initial excitation could now be given by @xmath60 , \label{eq : normalmode}\ ] ] where @xmath61 is the regular part of the solution , @xmath62 is the flux function defined by equation ( [ eq : flux ] ) , and @xmath63 gives the maximum height of the magnetic field line in which the normal mode is excited . it is important to note that the regular solution has a well - defined parity with respect to the @xmath64-direction depending on whether @xmath65 is chosen even or odd . however , since a delta function is difficult to handle from a numerical point of view , our normal mode - like excitation is performed by an initial perturbation of the form @xmath66}. \label{eq : normalpert}\ ] ] for the regular part , @xmath67 , the fundamental mode with one maximum along the field lines has been chosen . it should be noted that the width , @xmath50 , of the initial perturbation now causes the excitation of several alfvn modes in a set of neighboring magnetic surfaces . it is important to consider an initial velocity profile which is sufficiently localized in the direction transverse to magnetic surfaces so that only a few of them are excited . as we concentrate on the dynamics of a restricted number of field lines around a magnetic surface the consideration of other models , with different values of @xmath19 , would change quantitatively the generated frequencies , but not the overall qualitative conclusions shown here . figure [ fig : spectrumalfvenky0]b shows the temporal evolution of the excited @xmath53 component at a particular location as a function of time for three different values for the width of the initial disturbance . it is clear that three different solutions are obtained . the two corresponding to the largest widths are rather similar , but the one for the smaller width shows a strong damping . it must be said that the exact solution of this ideal system should display no time damping , hence we assert that this is a numerical effect , that can not be attributed to a real physical damping mechanism . this undesired effect is less important for larger widths of the initial perturbation since , for a given number of grid points , the initial condition is better resolved spatially . we next fix the width of the initial disturbance , @xmath50 , and vary the spatial resolution in our domain . figure [ fig : resolution ] shows several numerical simulations when an initial normal mode - like excitation ( equation [ [ eq : normalpert ] ] ) is made at different heights . it is clear that larger spatial resolution provides the more accurately the undamped oscillatory solution . also from this analysis we conclude that the spatial resolution is not a factor that should be taken into account in an isolated manner when considering the numerical description of alfvn waves on given magnetic surfaces . indeed , and because of the cartesian distribution of grid points in a system of curved magnetic field lines , low - lying magnetic lines are poorly resolved when compared to high - lying magnetic lines for a given grid resolution . this has implications that are worth to be taken into account as can be seen in figure [ fig : resolution ] . if we compare signals in figure [ fig : resolution ] , we can see that , all parameters being the same , closer results to the analytical solution are obtained for higher magnetic field lines . we can therefore assert that for the numerical simulation of alfvn wave properties the resolution of the grid is an important parameter and that it becomes more critical for low - lying magnetic field lines than for higher ones . it should be noted that the conclusions of these tests can also be applied to the case in which an impulsive excitation is set as the initial perturbation . in this section we present the main results from our numerical investigation . for simplicity , first , the temporal evolution of a normal mode - like fast disturbance is analyzed in order to show how and where resonant absorption , due to three - dimensional propagation of perturbations in a non - uniform medium , takes place . it turns out that previous results obtained for the normal modes of coupled fast and alfvn waves in a potential arcade by @xcite can guide us to understand the time evolution of the system and the energy transfer between resonantly coupled modes . then , a more complex situation is considered by analyzing the time evolution of the initial perturbation given by equation ( [ eq : perturbation ] ) . it should be noted that our first normal mode time evolution analysis has been proof very useful to further better understand the resulting coupling process between both velocity components when a localized impulsive disturbance is used . in order to gain some insight into the propagation properties of coupled fast and alfvn waves in our configuration , we first study the time evolution caused by an initial disturbance having the spatial structure of a fast normal mode for @xmath31 ( propagation in the @xmath2-plane ) . as shown by @xcite , pure fast modes in a potential arcade are characterized by a global spatial structure determined by the wavenumbers @xmath68 and @xmath69 , which give rise to smooth distributions with a given number of maxima in the @xmath64- and @xmath70-directions . this results in a discrete spectrum of frequencies . the frequencies and spatial structure of the fast modes with @xmath38 were computed by @xcite , who showed that perpendicular propagation produces the coupling of the fast normal modes to alfvn continuum solutions , resulting in modes with mixed properties . we have chosen as initial perturbation the velocity perturbation @xmath43 of the fundamental fast mode for @xmath31 , with one maximum in each direction in the @xmath2-plane . when @xmath31 this produces a standing harmonic oscillation of the system , as in an elastic membrane . when @xmath38 , this initial perturbation is not a normal mode of the system , but we expect that , the obtained temporal evolution will not differ very much from the actual normal mode of the coupled solution . figure [ fig : chap4nomarlkyn0 ] displays the results of such simulation . the first frame for @xmath43 shows the initial spatial distribution of the perturbation . initially , @xmath53 , the velocity component associated to alfvn waves , is zero . as time evolves , a non zero @xmath53 component appears because of the coupling introduced by the three - dimensional propagation . the panels for @xmath53 in figure [ fig : chap4nomarlkyn0 ] show that unlike @xmath43 the excited transversal perturbations are not globally distributed in the potential arcade , but only at preferred locations , around a few magnetic surfaces . when the @xmath43 and @xmath53 signals are measured at one of those locations , @xmath59 , @xmath71 , it is seen that the amplitude related to the fast - like perturbation decreases in time , while the amplitude of the alfvn - like component of the perturbation increases in time , see figure [ fig : compvnvy ] . this is an indication of the wave energy transfer due to the resonant coupling of the excited fast normal mode to the alfvnic solution around the excited magnetic surface . for long times a decrease in the amplitude of the velocity component , @xmath53 , can be appreciated and is attributed to numerical damping , for the reasons explained in [ alfven_wave ] . further confirmation of the resonant wave energy transfer occurring between the modes can be obtained by computing the time evolution of the total energy density in our system . this total wave energy can be computed as @xmath72 . \label{eq : totalenergy}\\ \nonumber\end{aligned}\ ] ] the right - hand side panels in figure [ fig : chap4nomarlkyn0 ] show the spatial distribution of this quantity as a function of time . the different frames clearly indicate that , initially , the energy is distributed globally around the center of the system , such as corresponds to the initial perturbation we have used . at later times , this energy is transferred to magnetic surfaces around the particular magnetic field line in the arcade where the signals in figure [ fig : compvnvy ] have been measured . the location of this energy deposition is not an arbitrary one . as previous theoretical works on the resonant energy transfer have shown , ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , global fast modes resonantly couple to localized alfvn continuum modes at the magnetic surfaces where the frequency of the fast mode matches that of the corresponding alfvn mode . in our case the spectral analysis of the wave energy densities associated to the normal and perpendicular components , plotted in figure [ fig : spectrumnormalky1 ] , allow us to confirm the resonant energy transfer at the location where the fundamental fundamental fast mode frequency crosses the alfvn continuum , that exactly corresponds to the magnetic surface where alfvnic oscillations are excited and energy transfer occurs , see figure [ fig : spectrumnormalky1]b . although the fast mode frequency crosses other alfvn continua , coupling can only occur if the parity of the fast and alfvn eigenfunctions along the field lines is the same , see further details in @xcite . this prevents the coupling with alfvn continuum modes with two extrema along field lines . even if the coupling with alfvn modes with three extrema along field lines is allowed , we find no signatures of this resonant coupling in the power spectrum analysis nor the wave energy density evolution . oscillations in coronal magnetic structures are believed to be generated by nearby disturbances , such as flares or filament eruptions . it is clear that such disturbances are far from being a normal mode of a particular structure as our potential arcade . therefore , we have next considered the impulsive excitation of perturbations by means of a localized disturbance , which is expected to be a better representation of the real phenomena that often trigger waves and oscillations in the solar corona . in particular a gaussian velocity perturbation is considered and the response of the system is expected to be different from the one described in [ normal_like ] , since now the initial perturbation is likely to be decomposed in a linear combination of normal modes with different frequencies that will constitute the resulting propagating wave . we have produced an impulsive excitation of the @xmath43 velocity component of the form given by equation ( [ eq : perturbation ] ) and have considered @xmath38 . time evolution of the velocity components and the total energy density are displayed in figure [ fig : gaussd1kyn0 ] , which shows that the generated wave has both normal and perpendicular velocity components . note that @xmath73 in the absence of @xmath74 ( see * ? ? ? it is clear in figure [ fig : gaussd1kyn0 ] that the perturbed normal velocity component evolution is similar to the one presented by @xcite , for the decreasing alfvn speed model with constant density and @xmath31 . for the normal velocity component , the shape of the wavefront is not circular , due to the fact that perturbations propagate faster toward the photosphere . for large times the front tends to be planar as the initial curvature of the wave packet is lost . as for the perpendicular velocity perturbation that is excited because of the three - dimensional character of the wave , its spatial distribution is highly anisotropic , with the signal concentrated around many magnetic surfaces . a wavefront with fast - like properties , similar to the one present in @xmath43 , can also be seen to propagate upwards producing @xmath53 perturbations until it leaves the system . at the end , a collection of alfvnic oscillations are generated in the arcade . by comparing with the results presented in the previous section we can think about them as being generated by the resonant coupling between the fast - like wavefront and several alfvn continuum solutions , instead of the single resonance case shown in [ normal_like ] . once excited , magnetic surfaces remain oscillating with their natural period and for large times they are phase - mixed because of the transverse non - uniformity . we have next analyzed in a quantitative way the effect of @xmath74 on the properties of the generated fast - like wavefront and the induced alfvnic oscillations . regarding the fast - like wavefront , figure [ fig : posmaxkyn0 ] ( top - panels ) shows different snapshots of the cut along @xmath59 of the @xmath43 component for different values of @xmath74 . these figures indicate that the larger the value of @xmath74 the faster the wavefront propagates . the propagation velocity can be measured by plotting the position of the wavefront maximum as a function of time ( see figure [ fig : posmaxkyn0 ] bottom - left ) . the time evolution of the wavefront is followed for @xmath75 and the initial position of the maximum is denoted by @xmath76 . for this relatively simple case , the numerical results can be compared with the analytical formula obtained by the integration of the local alfvn speed profile ( see equation [ @xmath77 in oliver et al . the resulting expression is @xmath78 , \label{eq : localalfven}\ ] ] where the @xmath79 and @xmath80 signs correspond to upward and downward propagation , respectively . figure [ fig : posmaxkyn0 ] ( bottom - panels ) shows a perfect correspondence between the numerically measured speed and the analytical expression when different models of the solar atmosphere are considered . the increase of the travel speed of fast - like wavefronts when @xmath38 is an important property to be taken into account in the three - dimensional problem . for the alfvnic oscillations the power spectrum is analyzed in a cut along @xmath59 , which allows us to study the power on different magnetic surfaces . figure [ fig : spectrumd1ky1 ] shows power at a large number of magnetic surfaces , not just around a selected group of field lines around a given magnetic surface , so a wide range of magnetic surfaces are excited because of the coupling . also , not just the fundamental mode is excited , but also several higher harmonics . all of them have even parity with respect to @xmath59 , such as corresponds to the parity of the @xmath43 perturbation and the parity rule for @xmath38 @xcite . when comparing the power spectrum obtained from the numerical solution with the analytical alfvn continuum frequencies given by @xcite for the case @xmath31 , we see that the signal coincides with the analytical curves for @xmath31 . such as expected , perpendicular propagation has no effect on the frequencies of alfvn waves generated on different magnetic surfaces in the arcade . this is a known result since @xmath81 . as with the normal mode case , a quantitative analysis of the time - evolution of the wave energy of the system helps to better understand the process of energy conversion between fast and alfvn waves . figure [ fig : gaussd1kyn0 ] ( right - hand side panels ) , shows the evolution of the total energy density as a function of time . at early stages this quantity shows a clear signature of a fast - like wavefront propagating through the domain . for long times the energy deposition is spatially distributed on the whole system , not only on a single magnetic surface , as in the previous section . although a large part of the energy leaves the system in the form of fast - like wavefronts , part of the energy remains trapped in the alfvnic oscillations that are resonantly excited in the arcade . the amount of energy trapped in the system can be calculated by the integration of the total energy density ( see equation [ [ eq : totalenergy ] ] ) in the whole domain as a function of time . the result is shown in figure [ fig : energiesdelta ] ( solid line ) . for short times the total energy remains almost constant , but when the fast front reaches the boundaries of the system a strong decrease of this quantity is seen . resonant wave conversion very quickly produces velocity perturbations in the @xmath3-direction and the energy associated to these alfvnic components grows up to its maximum value before the fast wavefront leaves the system . at later stages , a fraction of around @xmath82 of the initial total energy is retained in the system and the total energy remains almost constant in the subsequent time evolution . we must note that this energy is trapped even in the absence of any density enhancement or wave cavity . so far , we have used fixed values for the perpendicular propagation wavenumber , @xmath74 , and the ratio of magnetic to density scale heights , @xmath19 . we have next analyzed the influence of these parameters on the obtained results concerning the energy transfer between fast and alfvn waves . figure [ fig : energiesky ] shows the total energy time evolution for different values of @xmath74 . several conclusions can be extracted . first , the amount of energy that is trapped by the system in the form of alfvnic oscillations increases with @xmath74 and is above @xmath83 for the largest value of this parameter that we have considered . this can be understood in terms of stronger resonant coupling occurring for larger values of @xmath74 . the relation between the total energy and the energy associated to the @xmath3-direction also changes with @xmath74 , in such a way that , while for relatively small @xmath74 almost all the energy of the system is stored in oscillations in the @xmath3-direction , for larger values of @xmath74 there is a difference between the total energy and the alfvnic energy for large times . to understand this we need to mention that for @xmath38 alfvn waves have both perpendicular and normal velocity components and so alfvn wave energy is not only contained in the @xmath3-direction . although this effect is less visible in the simulations it can be measured , such as shown in figure [ fig : energiesky ] . note also that for large times the two energy densities decay . this is due to numerical damping , since when very small scales are created the spatial resolution used is not fine enough to handle the localized alfvnic oscillations that are phase - mixed for large times ( see [ alfven_wave ] ) . as for the @xmath19 parameter , it controls our model atmosphere , since it allows us to select different ratios of the magnetic scale height to the density scale height . by repeating the previous numerical experiments for two additional values of this parameter the following results are obtained ( see figure [ fig : posmaxkyn0 ] bottom - panels ) . depending on the value of @xmath19 , the alfvn speed profile in the vertical direction has a steeper or flatter profile . this means that the time that a fast - like perturbation needs to reach the boundaries of the system and leave it varies with @xmath19 . therefore , the time at which the sudden decrease of the total energy of the system occurs differs for different values of @xmath19 , see figure [ fig : energiesdelta ] . however , the fractional amount of wave energy that is transferred to alfvn waves and is trapped in the system does not depend on the model atmosphere we use . nevertheless , the rate at which the energy transfer occurs does depend in the model atmosphere , such as can be appreciated from the different slopes of the energy in figure [ fig : energiesdelta ] . in this paper we have studied the temporal evolution of coupled fast and alfvn waves in a potential coronal arcade when three - dimensional propagation is allowed . because of the inclusion of three - dimensional dependence on the perturbed quantities , fast and alfvn waves are coupled and the resulting solutions display a mixed fast / alfvn character . the non - uniform nature of the considered medium produces the coupling to be of resonant nature , in such a way that transfer of energy and wave damping occur in the system . first , the nature of resonant coupling between a fast normal mode of the system and alfvn continuum modes has been analyzed . it is seen that the fast mode with a global nature resonantly couples to localized alfvn waves around a given magnetic surface in the arcade , thus transferring its energy to the later . the position of the resonant surface perfectly agrees with the resonant frequency condition predicted by several authors in previous studies of this kind , and with the parity rules given by @xcite . next , the temporal evolution of a localized impulsive disturbance has been analyzed . the inclusion of perpendicular propagation produces an increase in the wave propagation speed for the fast - like wavefront when compared to the purely poloidal propagation case . as in the previous case , perpendicular propagation induces the excitation of alfvnic oscillations around magnetic surfaces , due to the resonant coupling between fast and alfvn waves . now these oscillations cover almost the whole domain in the arcade , so that the energy of the initial perturbation is spread into localized alfvnic waves . the frequency of the induced alfvnic oscillations is seen to be independent from the perpendicular wavenumber . as time progresses and the initial wavefront leaves the system part of the energy is stored in these alfvn waves which remain confined around magnetic surfaces . phase mixing then gives rise to smaller and smaller spatial scales , until the numerical code is unable to properly follow the subsequent time evolution . the energy trapping around magnetic surfaces occurs even in the absence of a density enhancement or a wave cavity structure , and is only due to the non - uniformity of the density profile and the magnetic structuring , which lead to a non - uniform alfvn speed distribution . finally , the efficiency of this wave energy transfer between large scale disturbances and small scale oscillations has been studied as a function of the perpendicular wavenumber and for different values of the ratio of the magnetic scale height to the density scale height . it is seen that the first factor strongly affects the amount of energy trapped by alfvn waves . the amount of energy trapped by the arcade increases for increasing value of the perpendicular wavenumber . the particular ratio of magnetic to density scale heights determines how fast the available fast wave energy leaves the system and , therefore , the rate at which energy can be transferred to alfvn waves , but not the final amount of energy stored by the arcade in the form of alfvnic oscillations . these @xmath84d simulations should be extended in the future to more realistic @xmath85d simulations in order to ascertain the applicability of our conclusions to the real wave dynamics observed in coronal structures . the authors acknowledge the spanish mcyt for the funding provided under project aya@xmath86 and d. fanning ( http://www.dfanning.com/ ) for his helpful advices about idl . s.r . also acknowledges mcyt for a fellowship . constant . these curves in the @xmath2-plane become arcade surfaces in three dimensions . in this model @xmath70 measures the upward distance from the base of the corona ( placed at @xmath11 ) . the three orthogonal unit vectors , @xmath87 , @xmath88 and @xmath89 , defining the normal , perpendicular and parallel directions respectively , are also shown at a particular point.,scaledwidth=70.0% ] at @xmath59 as a function of height , @xmath90 , and normalized frequency , @xmath91 . in this simulation the spatial grid is set to @xmath92 while the numerical dissipation is fixed to @xmath93 . solid lines are the theoretical frequency of the normal alfvn modes for @xmath21 , given by @xcite . from bottom to top they represent the fundamental mode and its harmonics . ( b ) temporal evolution of the @xmath53 component when the initial perturbation is an alfvn normal mode of the system located at @xmath59 and @xmath94 ( see equation [ [ eq : normalpert ] ] ) . here @xmath21 , the numerical grid has @xmath95 points , and @xmath93 . different solutions correspond to @xmath96 ( solid ) , @xmath97 ( dash - dotted ) and @xmath98 ( long - dashed ) . time is given in units of @xmath99.,title="fig:",scaledwidth=42.0% ] at @xmath59 as a function of height , @xmath90 , and normalized frequency , @xmath91 . in this simulation the spatial grid is set to @xmath92 while the numerical dissipation is fixed to @xmath93 . solid lines are the theoretical frequency of the normal alfvn modes for @xmath21 , given by @xcite . from bottom to top they represent the fundamental mode and its harmonics . ( b ) temporal evolution of the @xmath53 component when the initial perturbation is an alfvn normal mode of the system located at @xmath59 and @xmath94 ( see equation [ [ eq : normalpert ] ] ) . here @xmath21 , the numerical grid has @xmath95 points , and @xmath93 . different solutions correspond to @xmath96 ( solid ) , @xmath97 ( dash - dotted ) and @xmath98 ( long - dashed ) . time is given in units of @xmath99.,title="fig:",scaledwidth=49.0% ] velocity component at @xmath59 and @xmath94 for an initial perturbation given by equation ( [ eq : normalpert ] ) with @xmath98 and @xmath93 . ( b ) temporal evolution of the @xmath53 velocity component at @xmath59 and @xmath100 for an initial perturbation with @xmath98 and @xmath93 . in both panels solid , dash - dotted , and long - dashed lines represent a resolution of @xmath101 , @xmath95 , and @xmath92 points , respectively.,title="fig:",scaledwidth=46.0% ] velocity component at @xmath59 and @xmath94 for an initial perturbation given by equation ( [ eq : normalpert ] ) with @xmath98 and @xmath93 . ( b ) temporal evolution of the @xmath53 velocity component at @xmath59 and @xmath100 for an initial perturbation with @xmath98 and @xmath93 . in both panels solid , dash - dotted , and long - dashed lines represent a resolution of @xmath101 , @xmath95 , and @xmath92 points , respectively.,title="fig:",scaledwidth=46.0% ] and ( b ) @xmath102 as a function height , @xmath90 , at the symmetry plane @xmath59 , for the simulation shown in figure [ fig : chap4nomarlkyn0 ] . note that because of the quadratic nature of the wave energy density the curved lines have double the frequency of the alfvn continua given by @xcite and the horizontal lines have double the frequency of the fast normal mode.,title="fig:",scaledwidth=49.0% ] and ( b ) @xmath102 as a function height , @xmath90 , at the symmetry plane @xmath59 , for the simulation shown in figure [ fig : chap4nomarlkyn0 ] . note that because of the quadratic nature of the wave energy density the curved lines have double the frequency of the alfvn continua given by @xcite and the horizontal lines have double the frequency of the fast normal mode.,title="fig:",scaledwidth=49.0% ] . the values of the longitudinal wavenumber are @xmath31 ( solid line ) , @xmath103 ( dotted ) , @xmath104 ( dashed ) , and @xmath105 ( dash - dotted ) . * bottom - panels * position of the wavefront as a function of time when different values of the delta parameter , @xmath106 ( left ) , @xmath20 ( middle ) , and @xmath107 ( right ) , and longitudinal wavenumber , @xmath31 ( squares ) and @xmath105 ( triangles ) , are selected . the solid line shows the analytical solution for the wavefront position when longitudinal propagation is not allowed ( @xmath31 ) ; see equation ( [ eq : localalfven]).,title="fig:",scaledwidth=30.0% ] . the values of the longitudinal wavenumber are @xmath31 ( solid line ) , @xmath103 ( dotted ) , @xmath104 ( dashed ) , and @xmath105 ( dash - dotted ) . * bottom - panels * position of the wavefront as a function of time when different values of the delta parameter , @xmath106 ( left ) , @xmath20 ( middle ) , and @xmath107 ( right ) , and longitudinal wavenumber , @xmath31 ( squares ) and @xmath105 ( triangles ) , are selected . the solid line shows the analytical solution for the wavefront position when longitudinal propagation is not allowed ( @xmath31 ) ; see equation ( [ eq : localalfven]).,title="fig:",scaledwidth=30.0% ] . the values of the longitudinal wavenumber are @xmath31 ( solid line ) , @xmath103 ( dotted ) , @xmath104 ( dashed ) , and @xmath105 ( dash - dotted ) . * bottom - panels * position of the wavefront as a function of time when different values of the delta parameter , @xmath106 ( left ) , @xmath20 ( middle ) , and @xmath107 ( right ) , and longitudinal wavenumber , @xmath31 ( squares ) and @xmath105 ( triangles ) , are selected . the solid line shows the analytical solution for the wavefront position when longitudinal propagation is not allowed ( @xmath31 ) ; see equation ( [ eq : localalfven]).,title="fig:",scaledwidth=30.0% ] + . the values of the longitudinal wavenumber are @xmath31 ( solid line ) , @xmath103 ( dotted ) , @xmath104 ( dashed ) , and @xmath105 ( dash - dotted ) . * bottom - panels * position of the wavefront as a function of time when different values of the delta parameter , @xmath106 ( left ) , @xmath20 ( middle ) , and @xmath107 ( right ) , and longitudinal wavenumber , @xmath31 ( squares ) and @xmath105 ( triangles ) , are selected . the solid line shows the analytical solution for the wavefront position when longitudinal propagation is not allowed ( @xmath31 ) ; see equation ( [ eq : localalfven]).,title="fig:",scaledwidth=30.0% ] . the values of the longitudinal wavenumber are @xmath31 ( solid line ) , @xmath103 ( dotted ) , @xmath104 ( dashed ) , and @xmath105 ( dash - dotted ) . * bottom - panels * position of the wavefront as a function of time when different values of the delta parameter , @xmath106 ( left ) , @xmath20 ( middle ) , and @xmath107 ( right ) , and longitudinal wavenumber , @xmath31 ( squares ) and @xmath105 ( triangles ) , are selected . the solid line shows the analytical solution for the wavefront position when longitudinal propagation is not allowed ( @xmath31 ) ; see equation ( [ eq : localalfven]).,title="fig:",scaledwidth=30.0% ] . the values of the longitudinal wavenumber are @xmath31 ( solid line ) , @xmath103 ( dotted ) , @xmath104 ( dashed ) , and @xmath105 ( dash - dotted ) . * bottom - panels * position of the wavefront as a function of time when different values of the delta parameter , @xmath106 ( left ) , @xmath20 ( middle ) , and @xmath107 ( right ) , and longitudinal wavenumber , @xmath31 ( squares ) and @xmath105 ( triangles ) , are selected . the solid line shows the analytical solution for the wavefront position when longitudinal propagation is not allowed ( @xmath31 ) ; see equation ( [ eq : localalfven]).,title="fig:",scaledwidth=30.0% ] velocity component corresponding to the simulation show in figure [ fig : gaussd1kyn0 ] as a function of the maximum height of field lines , @xmath90 , and normalized frequency , @xmath91 . solid lines are the theoretical frequency of the alfvn normal mode obtained by @xcite . the frequency analysis is made at the symmetry plane , @xmath59 . ] ) , dotted ( @xmath20 ) , and dashed ( @xmath107 ) lines are the normalized total energy ( thick lines ) and the normalized total energy associated to the @xmath3-direction ( thin lines ) as a function of time for @xmath108 . ] -direction ( dotted line ) for @xmath106 and different values of the longitudinal wavenumber . ( a ) @xmath103 , ( b ) @xmath109 , ( c ) @xmath104 , and ( d ) @xmath110.,title="fig:",scaledwidth=46.0% ] -direction ( dotted line ) for @xmath106 and different values of the longitudinal wavenumber . ( a ) @xmath103 , ( b ) @xmath109 , ( c ) @xmath104 , and ( d ) @xmath110.,title="fig:",scaledwidth=46.0% ] -direction ( dotted line ) for @xmath106 and different values of the longitudinal wavenumber . ( a ) @xmath103 , ( b ) @xmath109 , ( c ) @xmath104 , and ( d ) @xmath110.,title="fig:",scaledwidth=46.0% ] -direction ( dotted line ) for @xmath106 and different values of the longitudinal wavenumber . ( a ) @xmath103 , ( b ) @xmath109 , ( c ) @xmath104 , and ( d ) @xmath110.,title="fig:",scaledwidth=46.0% ]

we numerically investigate the excitation and temporal evolution of oscillations in a two - dimensional coronal arcade by including the three - dimensional propagation of perturbations . the time evolution of impulsively generated perturbations is studied by solving the linear , ideal magnetohydrodynamic ( mhd ) equations in the zero-@xmath0 approximation . as we neglect gas pressure the slow mode is absent and therefore only coupled mhd fast and alfvn modes remain . two types of numerical experiments are performed . first , the resonant wave energy transfer between a fast normal mode of the system and local alfvn waves is analyzed . it is seen how , because of resonant coupling , the fast wave with global character transfers its energy to alfvnic oscillations localized around a particular magnetic surface within the arcade , thus producing the damping of the initial fast mhd mode . second , the time evolution of a localized impulsive excitation , trying to mimic a nearby coronal disturbance , is considered . in this case , the generated fast wavefront leaves its energy on several magnetic surfaces within the arcade . the system is therefore able to trap energy in the form of alfvnic oscillations , even in the absence of a density enhancement such as that of a coronal loop . these local oscillations are subsequently phase - mixed to smaller spatial scales . the amount of wave energy trapped by the system via wave energy conversion strongly depends on the wavelength of perturbations in the perpendicular direction , but is almost independent from the ratio of the magnetic to density scale heights .

humans have evolved large brains , in part to handle the cognitive demands of social relationships @xcite . the social structures resulting from these relationships confer numerous fitness advantages . scholars distinguish between two types of social relationships : those representing strong and weak ties . strong ties are characterized by high frequency of interaction and emotional intimacy that can be found in relationships between family members or close friends . people connected by strong ties share mutual friends @xcite , forming cohesive social bonds that are essential for providing emotional and material support @xcite and creating resilient communities @xcite . in contrast , weak ties represent more casual social relationships , characterized by less frequent , less intense interactions , such as those occurring between acquaintances . by bridging otherwise unconnected communities , weak ties expose individuals to novel and diverse information that leads to new job prospects @xcite and career opportunities @xcite . online social relationships provide similar benefits to those of the offline relationships , including emotional support and exposure to novel and diverse information @xcite . how and why do people form different social ties , whether online or offline ? of the few studies that addressed this question , shea et al . @xcite examined the relationship between emotions and cognitive social structures , i.e. , the mental representations individuals form of their social contacts @xcite . in a laboratory study , they demonstrated that subjects experiencing positive affect , e.g. , emotions such as happiness , were able to recall a larger number of more diverse and sparsely connected social contacts than those experiencing negative affect , e.g. , sadness . in other words , they found that positive affect was more closely associated with weak ties and negative affect with strong ties in cognitive social structures . this is consistent with findings that negative emotional experiences are shared more frequently through strong ties @xcite , not only to seek support but also as a means of strengthening the tie @xcite . in addition to psychological factors , social structures also depend on the participants socioeconomic and demographic characteristics . a study , which reconstructed a national - scale social network from the phone records of people living in the united kingdom , found that people living in more prosperous regions formed more diverse social networks , linking them to others living in distinct communities @xcite . on the other hand , people living in less prosperous communities formed less diverse , more cohesive social structures . the present paper examines how psychological and demographic factors affect the structure of online social interactions . we restrict our attention to interactions on the twitter microblogging platform . to study these interactions , we collected a large body of geo - referenced text messages , known as tweets , from a large us metropolitan area . further , we linked these tweets to us census tracts through their locations . census _ tracts _ are small regions , on a scale of city blocks , that are relatively homogeneous with respect to population characteristics , economic status , and living conditions . some of the tweets also contained explicit references to other users through the ` @ ' mention convention , which has been widely adopted on twitter for conversations . we used mentions to measure the strength of social ties of people tweeting from each tract . using these data we studied ( at tract level ) the relationship between social ties , the socioeoconomic characteristics of the tract , and the emotions expressed by people tweeting from that tract . in addition , people tweeting from one tract often tweeted from other tracts . since geography is a strong organizing principle , for both offline @xcite and online @xcite social relationships , we measured the spatial diversity of social relationships , and studied its dependence on socioeconomic , demographic , and psychological factors . our work complements previous studies of offline social networks and demonstrates a connection between the structure of online interactions in urban places and their socioeconomic characteristics . more importantly , it links the structure of online interactions to positive affect . people who express happier emotions interact with a more diverse set social contacts , which puts them in a position to access , and potentially take advantage of , novel information . as our social interactions increasingly move online , understanding , and being able to unobtrusively monitor , online social structures at a macroscopic level is important to ensuring equal access to the benefits of social relationships . in the rest of the paper , we first describe data collection and methods used to measure emotion and social structure . then , we present results of a statistical study of social ties and their relationships to emotions and demographic factors . the related works are addressed after this . although many important caveats exist about generalizing results of the study , especially to offline social interactions , our work highlights the value of linking social media data to traditional data sources , such as us census , to drive novel analysis of online behavior and online social structures . eagle et al . @xcite explored the link between socioeconomic factors and network structure using anonymized phone call records to reconstruct the national - level network of people living in the uk . measures of socioeconomic development were constructed from the uk government s index of multiple deprivation ( imd ) , a composite measure of prosperity based on income , employment , education , health , crime , housing of different regions within the country . they found that people living in more prosperous regions formed more diverse social networks , linking them to others living in distinct communities . on the other hand , people living in less prosperous communities formed less diverse , more cohesive social structures . quercia et al . @xcite found that sentiment expressed in tweets posted around 78 census areas of london correlated highly with community socioeconomic well being , as measured by the index of multiple deprivation ( i.e. , qualitative study of deprived areas in the uk local councils ) . in another study @xcite they found that happy places tend to interact with other happy places , although other indicators such as demographic data and human mobility were not used in their research @xcite . other researcher used demographic factors and associated them to sentiment analysis to measure happiness in different places . for instance , mitchell et al . @xcite generated taxonomies of us states and cities based on their similarities in word use and estimates the happiness levels of these states and cities . then , the authors correlated highly - resolved demographic characteristics with happiness levels and connected word choice and message length with urban characteristics such as education levels and obesity rates , showing that social media may potentially be used to estimate real - time levels and changes in population - scale measures , such as obesity rates . psychological and cognitive states affect the types of social connections people form and their ability to recall them @xcite . when people experience positive emotions , or affect , they broaden their cognitive scope , widening the array of thoughts and actions that come to mind @xcite . in contrast , experiencing negative emotions narrow attention to the basic actions necessary for survival . shea et al . @xcite tested these theories in a laboratory , examining the relationship between emotions and the structure of networks people were able to recall . they found that subjects experiencing positive affect were able to recall a larger number of more diverse and sparsely connected social contacts than those experiencing negative emotions . the study did not resolve the question of how many of the contacts people were able to recall that they proceeded to actively engage . a number of innovative research works attempted to better understand human emotion and mobility . some of these works focuses on geo - tagged location data extracted from foursquare and twitter . researchers reported @xcite that foursquare users usually check - in at venues they perceived as more interesting and express actions similar to other social media , such as facebook and twitter . foursquare check - ins are , in many cases , biased : while some users provide important feedback by checking - in at venues and share their engagement , others subvert the rules by deliberately creating unofficial duplicate and nonexistent venues @xcite . los angeles ( la ) county is the most populous county in the united states , with almost 10 million residents . it is extremely diverse both demographically and economically , making it an attractive subject for research . we collected a large body of tweets from la county over the course of 4 months , starting in july 2014 . our data collection strategy was as follows . first , we used twitter s location search api to collect tweets from an area that included los angeles county . we then used twitter4j api to collect all ( timeline ) tweets from users who tweeted from within this area during this time period . a portion of these tweets were geo - referenced , i.e. they had geographic coordinates attached to them . in all , we collected 6 m geo - tagged tweets made by 340k distinct users . we localized geo - tagged tweets to tracts from the 2012 us census . a tract is a geographic region that is defined for the purpose of taking a census of a population , containing about 4,000 residents on average , and is designed to be relatively homogeneous with respect to demographic characteristics of that population . we included only los angeles county tracts in the analysis . we used data from the us census to obtain demographic and socioeconomic characteristics of a tract , including the mean household income , median age of residents , percentage of residents with a bachelor s degree or above , as well as racial and ethnic composition of the tract . to measure emotions , we apply sentiment analysis @xcite , i.e. methods that process text to quantify subjective states of the author of the text . two recent independent benchmark studies evaluate a wide variety of sentiment analysis tools in various social media @xcite and twitter datasets @xcite . across social media , one of the best performing tools is sentistrength @xcite , which also was shown to be the best unsupervised tool for tweets in various contexts @xcite . sentistrength quantifies emotions expressed in short informal text by matching terms from a lexicon and applying intensifiers , negations , misspellings , idioms , and emoticons . we use the standard english version of sentistrength to each tweet in our dataset , quantifying positive sentiment @xmath0 and negative sentiment @xmath1 , consistently with the positive and negative affect schedule ( panas ) @xcite . sentistrength has been shown to perform very closely to human raters in validity tests @xcite and has been applied to measure emotions in product reviews @xcite , online chatrooms @xcite , yahoo answers @xcite , and youtube comments @xcite . in addition , sentistrength allows our approach to be applied in the future to other languages , like spanish @xcite , and to include contextual factors @xcite , like sarcasm @xcite . beyond positivity and negativity , meanings expressed through text can be captured through the application of the semantic differential @xcite , a dimensional approach that quantifies emotional meaning in terms of valence , arousal , and dominance @xcite . the dimension of _ valence _ quantifies the level of pleasure or evaluation expressed by a word , _ arousal _ measures the level of activity induced by the emotions associated with a word , and _ dominance _ quantifies the level of subjective power or potency experienced in relation to an emotional word . research in psychology suggests that a multidimensional approach is necessary to capture the variance of emotional experience @xcite , motivating our three - dimensional measurement beyond simple polarity approximations . the state of the art in the quantification of these three dimensions is the lexicon of warriner , kuperman , and brysbaert ( wkb ) @xcite . the wkb lexicon includes scores in the three dimensions for more than 13,000 english lemmas . we quantify these three dimensions in a tweet by first lemmatizing the words in the tweet , to then match the lexicon and compute mean values of the three dimensions as in @xcite . the large size of this lexicon allows us to match terms in in 82.39% of the tweets in our dataset , which we aggregate to produce multidimensional measures of emotions . [ cols="^,^ " , ] figure [ fig : mobility - demo ] shows the association between spatial diversity and demographic characteristics . income does not appear to significantly affect spatial diversity : only the top tertile of tracts by incomes has a significantly different spatial diversity ( @xmath2 ) from the other two tertiles . education , however , has a stronger dependence : tracts with better - educated residents also have significantly higher ( @xmath2 ) spatial diversity than tracts with fewer educated residents . in addition , ethnicity appears to be a factor . tracts with larger hispanic population have significantly lower spatial diversity ( @xmath3 ) than other tracts . the availability of large scale , near real - time data from social media sites such as twitter brings novel opportunities for studying online behavior and social interactions at an unprecedented spatial and temporal resolution . by combining twitter data with us census , we were able to study how the socioeconomic and demographic characteristics of residents of different census tracts are related to the structure of online interactions of users tweeting from these tracts . moreover , sentiment analysis of tweets originating from a tract revealed a link between emotions and sociability of twitter users . our findings are broadly consistent with results of previous studies carried out in an offline setting , and also give new insights into the structure of online social interactions . we find that at an aggregate level , areas with better educated , somewhat younger and higher - earning population are associated with weaker social ties and greater spatial diversity ( or inter - tract mobility ) . in addition , twitter users express happier , more positive emotions from these areas . conversely , areas that have more hispanic residents are associated with stronger social ties and lower spatial diversity . people also express less positive , sadder emotions in these areas . since weak ties are believed to play an important role in delivering strategic , novel information , our work identifies a social inequity , wherein the already privileged ones ( more affluent , better educated , happier ) are in network positions that potentially allow them greater access to novel information . some important considerations limit the interpretation of our findings . first , our methodology for identifying social interactions may not give a complete view of the social network of twitter users . our observations were limited to social interactions initiated by users who geo - reference their tweets . this may not be representative of all twitter users posting messages from a given tract , if systematic biases exist in what type of people elect to geo - reference their tweets . for demographic analysis , we did not resolve the home location of twitter users . instead , we assumed that characteristics of an area , i.e. , of residents of a tract , influence the tweets posted from that tract . other subtle selection biases could have affected our data and the conclusions we drew @xcite . it is conceivable that twitter users residing in more affluent areas are less likely to use the geo - referencing feature , making our sample of twitter users different from the population of la county residents . recognizing this limitation , we did not make any claims about the behavior of la residents ; rather , we focused on the associations between emotions and characteristics of a place and the behavior of twitter users , with an important caveat that those who turn on geo - referencing may differ from the general population of twitter users . for the analysis of emotions , we only considered english language tweets , although a significant fraction of tweets were in spanish . this may bias the average affect of tracts , especially for low - valence tracts , which have a larger number of hispanic residents . in the future , we plan to address this question by conducting sentiment analysis of spanish language tweets . ma was supported by the usc viterbi - india internship program . lg acknowledge support by the national counsel of technological and scientific development cnpq , brazil ( 201224/20143 ) , and usc - isi visiting researcher fellowship . this work was also partially supported by darpa , under contract w911nf-12 - 1 - 0034 . this support is gratefully acknowledged . cramer , h. ; rost , m. ; and holmquist , l. e. 2011 . performing a check - in : emerging practices , norms and conflicts in location - sharing using foursquare . in _ proc . 13th international conference on human computer interaction with mobile devices and services_. garcia , d. ; mendez , f. ; serdlt , u. ; and schweitzer , f. 2012 . political polarization and popularity in online participatory media : an integrated approach . in _ proc . first edition workshop on politics , elections and data_. mitchell , l. ; frank , m. r. ; harris , k. d. ; dodds , p. s. ; and danforth , c. m. 2013 . the geography of happiness : connecting twitter sentiment and expression , demographics , and objective characteristics of place . 8(5):e64417 . thelwall , m. ; buckley , k. ; paltoglou , g. ; skowron , m. ; garcia , d. ; gobron , s. ; ahn , j. ; kappas , a. ; kster , d. ; and holyst , j. a. 2013 . damping sentiment analysis in online communication : discussions , monologs and dialogs . in _ computational linguistics and intelligent text processing_. springer .

the social connections people form online affect the quality of information they receive and their online experience . although a host of socioeconomic and cognitive factors were implicated in the formation of offline social ties , few of them have been empirically validated , particularly in an online setting . in this study , we analyze a large corpus of geo - referenced messages , or tweets , posted by social media users from a major us metropolitan area . we linked these tweets to us census data through their locations . this allowed us to measure emotions expressed in the tweets posted from an area , the structure of social connections , and also use that area s socioeconomic characteristics in analysis . we find that at an aggregate level , places where social media users engage more deeply with less diverse social contacts are those where they express more negative emotions , like sadness and anger . demographics also has an impact : these places have residents with lower household income and education levels . conversely , places where people engage less frequently but with diverse contacts have happier , more positive messages posted from them and also have better educated , younger , more affluent residents . results suggest that cognitive factors and offline characteristics affect the quality of online interactions . our work highlights the value of linking social media data to traditional data sources , such as us census , to drive novel analysis of online behavior .

for many years the forest has been considered a different class of objects with respect to galaxies . the available sensitivity was too low to detect any sign of non primordial composition in the intergalactic gas clouds at high redshift . thanks to the advent of high resolution and signal to noise spectroscopy , the old idea on the majority of quasar absorption lines has been revisited and opened in the last few years a still pending debate on the connection between the forest and the galaxy formation of the early universe . the detection of ions different from civ in optically thin clouds is made complicated by harder observational conditions , whereas the still too poor knowledge of the ionisation mechanisms which determine the ion abundances in those clouds has often discouraged attempts of metal content estimations as a function of redshift and of hi column density . however abundance investigation of the clouds has fundamental implications in the understanding of the enrichment processes in the igm by pop iii stars in the @xmath3 universe . the sample of optically thin absorption lines with @xmath4 has been obtained by high resolution spectroscopy , mainly hiras / keck ( songaila 1997b ) but also by emmi / ntt for the @xmath5 systems ( savaglio et al . for all the systems civ and/or siiv and cii detections or upper limits are given in redshift coverage @xmath6 . the lower bound in @xmath7 is due to the very rare metal detection in lower column density systems . in this range even if the line can be saturated ( depending on the doppler width ) monte carlo simulations showed that fitting procedures of synthetic individual lines with similar resolution and s / n ratio of the observed spectra give hi column density errors which are less than a few tens of @xmath8 ( for @xmath9 , @xmath10 , fwhm = 12 and s / n = 20 this is typically 0.1 @xmath8 ) . the blending effect has a much more dramatic impact on column density uncertainties and for this reason , we consider in the case of complex structures as an individual cloud the total column densities of hi and of metal lines . estimating the heavy element content in the clouds is mostly complicated by the poor knowledge of the ionising sources . as a first simplification , we assume that this is dominated by photoionisation of the uv background and neglect any other mechanism . collisional ionisation is important when the gas temperature exceeds @xmath11 k. at that temperature , the doppler parameter for hi is 41 , well above the mean value typically found in clouds . the analysis of metal lines in clouds ( rauch et al . , 1997 ) shows that the mean `` doppler '' temperature in these clouds is @xmath12 k , making any evidence of collisional ionisation hard to justify . once the photoionisation equilibrium is assumed , we first consider the subsample of clouds which show both civ and siiv absorption . to calculate the metallicity we use cloudy and assume six different shapes for the uv background normalized to the value at the lyman limit ( @xmath13 erg s@xmath14 @xmath15 hz@xmath14 sr@xmath14 ) changing the parameter @xmath16 in the range @xmath17 . we varied the [ c / h ] and gas density in such a way to reproduce the observed civ . we also assume the relative silicon to carbon abundance to be between 0 and three times solar and consider the cloud size along the line of sight to be in the range 1 kpc @xmath18 kpc . given these assumptions , we obtain for this subsample a set of 18 [ c / h ] measurements shown in fig . carbon abundance in clouds with detected carbon and silicon has a large spread with mean values of [ c / h ] @xmath19 and no evidence of redshift evolution . we notice that this sample might consist of metal rich clouds since it has been selected because of the siiv detection and might not be representative of the whole population of clouds . in a recent work , songaila ( 1997a ) has estimated the total universal metallicity at @xmath20 ( assuming that at that time the baryonic matter of the universe mostly resides in the forest ) to be in the range 1/2000 and 1/630 relative to solar . in a different approach , we consider the whole sample and regard the global observed properties instead of the individual systems and compare with models . results of column density ratios on the @xmath21 and @xmath7 planes are shown in figs . [ f1 ] and [ f2 ] . in fig . 2 we investigate the redshift evolution of observed column densities in the case of @xmath22 and @xmath23 as reported . the discussed trend of siiv / civ ( cowie et al . , this conference proceedings ) can be reproduced by a redshift evolution of @xmath22 from 200 at @xmath24 to 3000 at @xmath25 . the same model can take into account other observed ion ratios . in fig . 3 we compare observations with cloudy models assuming that all the clouds of the sample are at the same mean redshift of @xmath26 with @xmath27 and the gas density proportional to the square root of @xmath7 , as given in the case of spherical clouds in photoionisation equilibrium with the uvb . in both figures the solid lines are obtained for metallicity [ c / h ] @xmath19 and [ si / c ] = [ o / c ] = 0.5 , [ n / c ] = 0 . models of photoionisation equilibrium can include the majority of metal detections ( also considering the metallicity spread ) but cii / hi which , as function of @xmath7 , looks to be steeper than calculated . additional observations of cii would probably cast further light on the discussion on the ionisation state and metal content in the clouds . in both figures , the numerous upper limits falling below the dashed curve [ c / h ] @xmath28 is an indication that in many clouds the metallicity is lower than the values found in the selected sample . the investigation of low and intermediate redshift ( @xmath2 ) observations of ovi and nv in @xmath29 clouds might succeed in answering the question of how efficient the mixing processes in the igm at high redshift has been . relative abundances can provide new hints on the study of metal production by pop iii stars . in particular nv since it has been predicted to be underproduced in massive stars with low initial metallicity ( arnett 1995 ) . more observations of the siiv / civ ratio for @xmath30 and @xmath31 are a challenging probe of the redshift evolution of the uvb , though this can be one of the many possible reasons for the observed siiv / civ trend ( another would be redshift evolution of the gas density being lower at lower redshift ) . more interesting conclusions await outcomes from new high quality data of keck observations . arnett d. , 1995 , ara&a , 33 , 115 rauch m. , sargent w.l.w . , womble d.s . , barlow t.a . , 1997 , apj , 467 , l5 savaglio s. , cristiani s. , dodorico s. , fontana a. , giallongo e. , molaro p. , 1997 , a&a , 318 , 347 songaila a. , 1997a , apjl , _ in press _ , ph/9709046 songaila a. , 1997b , _ in preparation _

we present a detailed analysis of the ionisation state and heavy element abundances in the intergalactic medium ( igm ) . the civ doublet is shown by 30 % of the 182 selected optically thin clouds in 10 qso lines of sight . direct metallicy calculations have been performed on individual systems with detected civ and siiv ( 10% of the sample ) varying the uv photoionising source , cloud density and size and silicon relative abundance . the best solutions for carbon content in this subsample ( redshift coverage @xmath0 ) span between 1/6 and 1/300 of the solar value with no evidence of redshift evolution in both the metallicity and the ionising source . global properties of the whole sample indicate that the metallicity in clouds with civ and siiv is not typical of the igm . the redshift evolution of the uvb is one of the possible sources of the observed siiv / civ trend presented by cowie and collaborators during this meeting . future detection of heavy elements in lower hi column density ( @xmath1 ) clouds relies on the presence of ovi and nv at @xmath2 .

recent evolution of mobile devices such as smart - phones and tablets has facilitated access to multi - media contents anytime and anywhere but such devices result in an explosive data traffic increase . the cisco expects by 2019 that these traffic demands will be grown up to 24.3 exabytes per month and the mobile video streaming traffic will occupy almost 72% of the entire data traffic @xcite . interestingly , numerous popular contents are asynchronously but repeatedly requested by many users and thus substantial amounts of data traffic have been redundantly generated over networks @xcite . motivated by this , caching or pre - fetching some popular video contents at the network edge such as mobile hand - held devices or small cells ( termed as _ local caching _ ) has been considered as a promising technique to alleviate the network traffic load . as the cache - enabled edge node plays a similar role as a local proxy server with a small cache memory size , the local wireless caching has the advantages of i ) reducing the burden of the backhaul by avoiding the repeated transmission of the same contents from the core network to end - users and ii ) reducing latency by shortening the communication distance . in recent years , there have been growing interests in wireless local caching . the related research has focused mainly on i ) femto - caching with cache - enabled small cells or access points ( called as caching helpers ) @xcite , ii ) device - to - device ( d2d ) caching with mobile terminals @xcite , and iii ) heterogeneous cache - enabled networks @xcite . for these local caching networks , varieties of content placements ( or caching placements ) were developed @xcite and for given fixed content placement , the performance of cache - enabled systems with different transmission or cache utilization techniques was investigated @xcite . specifically , content placement to minimize average downloading delay @xcite or average ber @xcite was proposed for fixed network topology . in a stochastic geometric framework , various content placements were also proposed either to minimize the average delay @xcite and average caching failure probability @xcite or to maximize total hit probability @xcite , offloading probability @xcite . however , these caching solutions were developed in limited environments ; they discarded wireless fading channels and interactions among multiple users , such as interference and loads at caching helpers . recently , the content placement on stochastic geometry modeling of caching was studied in @xcite . a tradeoff between content diversity and cooperative gain according to content placement was discovered well in @xcite but the caching probabilities were determined with numerical searches only . moreover , in @xcite , cache memory size is restricted to a single content size and loads at caching helpers are not addressed . the optimal geographical caching strategy to maximize the total hit probability was studied in cellular networks in @xcite . however , only hit probability whether the requested content is available or not among the covering base stations was investigated . none of the previous works successfully addressed the channel selection diversity and interactions among multiple users such as network interference and loads according to content placement . success of content delivery in wireless cache network depends mainly on two factors : i ) _ channel selection diversity gain _ and ii ) _ network interference_. for given realization of nodes in a network , these two factors dynamically vary according to what and how the nodes cache at their limited cache memory . specifically , if the more nodes store the same contents , they offer the shorter geometric communication distance as well as the better small - scale fading channel for the specific content request , which can be termed as channel selection diversity gain . on the contrary , if the nodes cache all contents uniformly , they can cope with all content requests but channel selection diversity gain can not help being small . moreover , according to content placement , the serving node for each content request dynamically changes , so the network interference from other nodes also dynamically varies . thus , it might be required to properly control the channel selection diversity gain and network interference for each content . recently , in @xcite , a tradeoff between content diversity and channel diversity was addressed in caching helper networks , where each caching helper is capable of storing only _ one content_. however , although pathloss and small - scale fading are inseparable in accurately modeling wireless channels , the channel diversity was characterized with only small - scale fading and the effects of pathloss and network interference depending on random network geometry were not well captured . in this context , we address the problem of content placement with a more generalized model considering pathloss , network interference according to random network topology based on stochastic geometry , small - scale channel fading , and arbitrary cache memory size . in this generalized framework , we develop an efficient content placement to desirably control cache - based channel selection diversity and network interference . the main contributions of this paper are summarized as follows . * we model the stochastic wireless caching helper networks , where randomly located caching helpers store contents independently and probabilistically in their finite cache memory and each user receives the content of interest from the caching helper with the largest instantaneous channel power gain . our framework generalizes the previous caching helper network models @xcite by simultaneously considering small - scale channel fading , pathloss , network interference , and arbitrary cache memory size . * with stochastic geometry , we characterize the channel selection diversity gain according to content placement of caching helpers by deriving the cumulative distribution function of the smallest reciprocal of the channel power gain in a noise - limited network . we derive the optimal caching probabilities for each file in closed form to maximize the average content delivery success probability for given finite cache memory size , and propose an efficient algorithm to find the optimal solution . * in interference - limited networks , we derive a lower bound of the average content delivery success probability in closed form . based on this lower bound with rayleigh fading , we derive near - optimal caching probabilities for each content in closed form to appropriately control the channel selection diversity and the network interference depending on content placement . * our numerical results demonstrate that the proposed content placement is superior to other content placement strategies because the proposed method efficiently balances channel selection diversity and network interference reduction for given content popularity and cache memory size . we also numerically investigate the effects of the various system parameters , such as the density of caching helpers , nakagami fading parameter , memory size , target bit rate , and user density , on the caching probability . the rest of this paper is organized as follows . in section ii , we describe the system model and performance metric considered in this paper . we analyze the average content delivery success probability and desirable content placement of caching helpers in a noise- and interference - limited network in sections iii and iv , respectively . numerical examples to validate the analytical results and to investigate the effects of the system parameters are provided in section v. finally , the conclusion of this paper is given in section vi . we consider a downlink wireless video service network , where the caching helpers are capable of caching some contents in their limited caching storage , as depicted in fig . [ fig : system_model ] . we assume that all contents have the same size normalized to one for analytic simplicity . the caching helpers are randomly located and modeled as @xmath0-d homogeneous poisson point process ( ppp ) with intensity @xmath1 . the caching helpers are equipped with a single antenna and their cache memory size is @xmath2 , so @xmath2 different contents can be cached at each helper since each content has unit size . the total number of contents is @xmath3 and the set ( library ) of content indices is denoted as @xmath4 . the contents have own popularity and their popularity distributions are assumed to follow the zipf distribution as in literature @xcite : @xmath5 where the parameter @xmath6 reflects the popularity distribution skewness . for example , if @xmath7 , the popularity of the contents is uniform . the lower indexed content has higher popularity , i.e. , @xmath8 if @xmath9 . note that our content popularity profile is not necessarily confined to the zipf distribution but can accommodate any discrete content popularity distribution . the users are also randomly located and modeled as @xmath0-d homogeneous poisson point process ( ppp ) with intensity @xmath10 . based on slivnyak s theorem @xcite that the statistics observed at a random point of a ppp @xmath11 is the same as those observed at the origin in the process @xmath12 , we can focus on a reference user located at the origin , called a _ typical user_. in this paper , we adopt _ random content placement _ where the caching helpers independently cache content @xmath13 with probability @xmath14 for all @xmath15 . according to the caching probabilities ( or policies ) @xmath16 , each caching helper randomly builds a list of up to @xmath2 contents to be cached by the probabilistic content caching method proposed in @xcite . 2 presents an example of the probabilistic caching method @xcite and illustrates how a caching helper randomly chooses @xmath2 contents to be cached among total @xmath17 contents according to the caching probability @xmath16 when the cache memory size is @xmath18 and total number of contents is @xmath19 . in this scheme , the cache memory of size @xmath2 is equally divided into @xmath20 ( @xmath21 ) blocks of unit size . then , starting from content 1 , each content sequentially fills the @xmath2 discontinuous memory blocks by the amount of @xmath22 from the first block . if a block is filled up in the filling process of content @xmath23 , the remaining portion of content @xmath23 continuously fills the next block . then , we select a random number within @xmath24 $ ] and the contents at the position specified by the random number in each block are selected . because one content is selected from each block by the selected random number , total @xmath20 ( @xmath21 ) contents can be selected in a probabilistic sense according to @xmath16 . in this way , in fig . [ rev : caching_explain ] , the contents @xmath25 are chosen to be cached . the contents selected in a probabilistic sense at each helper are cached in advance by either its request or overhearing . the caching helpers storing content @xmath13 can be modeled as independent ppp with intensity @xmath26 and the locations of the caching helpers storing content @xmath13 can be represented by @xmath27 where @xmath28 . the typical user requests one among @xmath17 contents according to the content popularity @xmath29 ; the content with a higher popularity is requested with higher likelihood . if the typical user requests content @xmath13 and selects a serving helper to maximize the instantaneous channel power gain among the helpers storing content @xmath30 , the received signal power becomes @xmath31 where @xmath32 is the transmit power of a caching helper , @xmath33 and @xmath34 denote the channel fading coefficient and the distance from the typical user to the caching helper located at @xmath35 , respectively , and @xmath36 is the path loss exponent . for each content @xmath13 , we denote a set of the reciprocals of the channel power gains from @xmath37 to the typical user in ascending order as @xmath38 , where @xmath39 . the notation @xmath40 and @xmath41 represent the distance and the channel fading coefficient from the typical user to the caching helper with the @xmath42-th smallest reciprocal channel power gain among the caching helpers storing content @xmath13 , respectively . note that the caching helper with the largest instantaneous channel power gain is equivalent to that with the smallest reciprocal of the channel power gain ( i.e. , @xmath43 ) . assuming gaussian signaling and time / frequency resource sharing among the users associated with the same caching helper , the mutual information between the typical user requesting content @xmath13 and its serving caching helper is @xmath44 where @xmath45 is the load of the serving caching helper , @xmath46 is the noise power variance , and @xmath47 is the interference received at the typical user , given by @xmath48 where @xmath49 is the set of caching helpers which do not cache content @xmath13 in their cache memory . the small - scale channel fading terms of the desired link and the interfering links follow the independent nakagami - m distributions with parameters @xmath50 and @xmath51 , respectively . similar to @xcite , we define the average content delivery success probability as a performance metric to properly account for the success events of content delivery as @xmath52,\label{def : ftsp}\end{aligned}\ ] ] where @xmath29 is the content requesting probability and @xmath53 is the target bit rate of content @xmath13 [ bits / s / hz ] to successfully support the real - time video streaming service of content @xmath13 without playback delay . in this section , in order to investigate how channel selection diversity affects the optimal caching solution , we first consider a noise - limited network ; when the number of active users is much smaller than the number of caching helpers , the impact of interference is negligible compared to the noise power and the typical user can be served without resource sharing with other users . in noise - limited networks , assuming gaussian signaling , the mutual information between the typical user requesting content @xmath13 and its serving helper is obtained as @xmath54 where @xmath55 is the signal - to - noise ratio ( snr ) . the power gain distribution of a nakagami-@xmath56 fading channel is given by @xmath57 where @xmath58 is the gamma function , @xmath59 , and @xmath60 is the fading parameter for link @xmath61 where @xmath62 represents either the desired link ( @xmath63 ) or the i.i.d . interfering links ( @xmath64 ) . if @xmath65 , the power gain distribution follows the exponential distribution corresponding to rayleigh fading . for @xmath66 , the channel is a deterministic channel . when the typical user receives content @xmath13 from the caching helper with the smallest reciprocal of the channel power gain ( i.e. , the largest channel power gain ) , the cumulative distribution function ( cdf ) of the smallest reciprocal of the channel power gain ( i.e. , @xmath43 ) is derived in lemma 1 . the cdf of the smallest reciprocal of the channel power gain , @xmath43 , in a nakagami-@xmath50 fading channel is given by @xmath67 where @xmath68 and @xmath69 . for @xmath30 , let @xmath70 be the path losses between the typical user and the caching helpers caching content @xmath13 . from the mapping theorem [ theorem 2.34 , @xcite ] , @xmath71 is a non - homogeneous ppp and its intensity function is given by @xmath72 where @xmath73 . note that @xmath74 are also mutually independent due to independence among @xmath75 . using the displacement theorem [ theorem 2.33 , @xcite ] , we can also derive the intensity function of @xmath76 for a general nakagami-@xmath50 fading channel as @xmath77 since the ppp of @xmath37 is transformed by the displacement and mapping theorems , @xmath78 is also a ppp @xcite . therefore , the cdf of @xmath43 is obtained as @xmath79 where @xmath80 denotes the number point of @xmath78 in a circle with a radius @xmath81 and @xmath68 . _ remark : _ as @xmath82 or @xmath83=\frac{\gamma(\delta+m_d)}{m_d^{\delta}\gamma(m_d)}$ ] increases , the cdf of @xmath43 grows faster to 1 because the intensity @xmath84 of ppp @xmath78 is proportional to them . in other words , as the number of caching helpers that are storing the content of interest and accessible by the typical user increases or the small - scale fading channel becomes more deterministic , the intensity of ppp @xmath78 representing the reciprocal channel power gains grows and thus the smallest reciprocal @xmath43 becomes smaller . especially , for given @xmath1 and @xmath50 , the largest channel power gain ( i.e. , @xmath85 ) grows as @xmath22 increases , which implies an increase of the _ channel selection diversity gain _ according to the content placement . [ fig : cdf ] validates the accuracy of lemma 1 for varying @xmath1 and @xmath50 . the cdf of @xmath43 increases faster to 1 as either @xmath1 or @xmath50 increases . however , the cdf of @xmath43 depends more on @xmath1 than on @xmath50 , so optimal caching probabilities are affected more by the density of caching helpers than channel fading . from lemma 1 , the average success probability for content delivery is derived in the following theorem . when the typical user receives content @xmath13 from the caching helper with the largest instantaneous channel power gain , the average success probability for content delivery @xmath86 in a nakagami-@xmath50 fading channel is obtained as @xmath87 where @xmath68 , @xmath69 , @xmath88 , and @xmath53 is the target bit rate of content @xmath13 . @xmath89 & = \mathbb{p}\left[\log_2\big(1 + \frac{\eta}{\xi_{i,1 } } \big)\geq \rho_i\right]\\ & = \mathbb{p}\left[\xi_{i,1 } \leq \frac{\eta}{2^{\rho_i } - 1}\right]\\ & = f_{\xi_{i,1}}\left(\frac{\eta}{2^{\rho_i } - 1}\right)\\ & = 1-e^{-\kappa p_i\left(\frac{\eta}{2^{\rho_i}-1}\right)^{\delta}},\label{eqn : success_prob}\end{aligned}\ ] ] where is obtained from lemma 1 . substituting into , we obtain . from lemma 1 , we know that the channel selection diversity gain for a specific content increases as the number of caching helpers storing the content increases , i.e. , @xmath22 increases . however , due to limited memory space @xmath2 , i.e. , the constraint @xmath90 , storing the same content at more caching helpers ( @xmath22 increases ) loses the chance of storing the other contents and the corresponding channel diversity gains . in this subsection , we derive the optimal solution of problem @xmath92 , the optimal caching probabilities , in closed form . for each @xmath13 , the function @xmath93 is convex with respect to @xmath22 since @xmath94 . since a weighted sum of convex functions is also convex function , problem @xmath92 is a constrained convex optimization problem and thus a unique optimal solution exists . the lagrangian function of problem @xmath92 is @xmath95 where @xmath96 is a constant , @xmath97 and @xmath98 are the nonnegative lagrangian multipliers for constraints and . after differentiating @xmath99 with respect to @xmath22 , we can obtain the necessary conditions for optimal caching probability , i.e. , _ karush - kuhn - tucker_(kkt ) condition as follows : @xmath100 from the constraint in , the optimal caching probabilities are given by @xmath101^{+}\\ & = \frac{1}{\kappa t_i}\left[\log\left(f_i\kappa t_i\right)-\log\left(\omega \!+\ ! \mu_i\right)\right]^{+ } , ~\forall i \!\in\ ! \mathcal{f},\label{eqn : opt}\end{aligned}\ ] ] where @xmath102^{+}=\max\{z,0\}$ ] . the caching probability of content @xmath13 grows as the content popularity @xmath29 becomes large , but is regulated by the term of @xmath103 . for the constraint in , @xmath97 is not necessarily zero because the optimal solution should always satisfy @xmath104 . based on the kkt conditions in - , lagrangian multipliers @xmath97 and @xmath98 range , according to @xmath22 , as , which is placed at the top of next page . @xmath105^{+ } & ~~\textrm{for}~~p_i=1,\\ f_i\kappa t_ie^{-\kappa t_i}<\omega < f_i\kappa t_i,&\mu_i=0~ & ~~\textrm{for}~~0<p_i<1,\\ \omega \geq f_i\kappa t_i,&\mu_i=0 ~ & ~~\textrm{for}~~p_i=0 . \label{multiplier_range } \end{array } \right.\end{aligned}\ ] ] reveals that the caching probability @xmath22 is determined according to lagrangian multiplier @xmath97 only since @xmath98 is a function of @xmath97 ; if @xmath106 where @xmath107 , then @xmath108 and thus @xmath109 . if @xmath110 where @xmath111 , then @xmath112 and thus @xmath113 . when @xmath114 , @xmath115 is bounded by @xmath116 since @xmath115 is decreasing with respect to @xmath97 . therefore , using the fact that @xmath117 for the optimal @xmath118 , one - dimensional bisection search can find the optimal @xmath118 and the corresponding @xmath119 given by @xmath120^{+},1\right),~\forall i\in\mathcal{f}.\label{opt_sol_noise}\end{aligned}\ ] ] the proposed algorithm to find the optimal caching probabilities @xmath119 is presented in algorithm 1 . consequently , the content delivery success probability maximized with @xmath119 becomes @xmath121.\end{aligned}\ ] ] [ cols="<",options="header " , ] in the previous section , the cache - based channel selection diversity gain for each content has been highlighted and the optimal caching probabilities to balance them were derived without consideration of interference . in this section , in the presence of network interference , we derive near - optimal content placement and analyze the effects of network interference on the content placement . we assume that the density of users is much higher than that of caching helpers , i.e. , @xmath122 , so the effect of noise is almost negligible relative to interference . when the typical user receives content @xmath13 from the caching helper with the smallest reciprocal of the instantaneous channel power among the caching helpers storing content @xmath13 , the other caching helpers interfere with the typical user because they are assumed to serve other users . then , the received signal - to - interference ratio ( sir ) at the typical user is represented as @xmath123 where @xmath47 is the interfering signal power and given by @xmath124 where @xmath49 is a set of the caching helpers which do not cache content @xmath13 and @xmath78 is a set of the reciprocals of the channel power gains from @xmath37 . note that the interfering signal power dynamically changes according to content placement of caching helpers since it is a function of @xmath43 and @xmath125 . therefore , optimal caching probabilities are expected to be obtained by optimally controlling channel selection diversity and network interference for given content popularity and cache memory size . in interference - limited networks , the average success probability of content delivery in is represented by @xmath126 , \label{eqn : avg_ps_inter}\end{aligned}\ ] ] where @xmath127 is a random load of the tagged caching helper when an arbitrary user receives content @xmath13 from the caching helper with the largest instantaneous channel power gain . to characterize , we require both the probability mass function ( pmf ) of the load at the tagged caching helper and the sir distribution when multiple contents are cached at each helper and the association is based on the instantaneous channel power gains . however , unfortunately , the exact statistics of the required information are unavailable because they are complicatedly determined by many interacting factors , such as multiple cached contents , locations of caching helpers and users , content request of users , instantaneous channel fading gains , etc . thus , the optimal caching probabilities to maximize have to be found by numerical searches of which complexity is prohibitively high for a huge number of contents . in this context , we propose near - optimal content placement to obtain some useful insights in interference - limited scenarios . to this end , we first approximate with the average load of the tagged caching helper @xcite as @xmath128,\label{rev:5_1}\end{aligned}\ ] ] where @xmath129 is the average load of the tagged caching helper when the user requests content @xmath13 to the caching helper with the largest instantaneous channel power gain . the validity of approximation is demonstrated in fig . [ fig : approx_check ] , where red star and blue circle represent the monte - carlo simulation and its approximation , respectively . this figure verifies that the approximation is quite tight to for arbitrary @xmath130 . moreover , a lowerbound of is obtained in the following theorem . when the typical user receives the requesting content from the caching helper with the smallest reciprocal of instantaneous channel power gain , the average success probability of content delivery is bounded below by @xmath131 where @xmath132 , @xmath133 is a constant independent of @xmath13 and makes the inequality hold for all ranges of @xmath16 , @xmath50 and @xmath51 are the nakagami fading parameters of the desired and interfering links , respectively , and @xmath134vdv\right.\nonumber\\ & \left.+~2\pi p_i\lambda\int_0^r \!\left[1 -\frac{m_i}{(spv^{-\alpha}+m_i)^{m_i}}\right]vdv\right),\\ f_{|x_i|}(r ) & = 2\pi p_i\lambda r\exp\left(-\pi p_i\lambda r^2\right).\end{aligned}\ ] ] see appendix [ appendixa ] . based on the lower - bounded average success probability of content delivery , we formulate an alternative optimization problem as @xmath135 although it is still non - trivial to obtain the solution of this alternative optimization problem , fortunately , when @xmath136 , i.e. , a rayleigh fading channel , the objective function ( i.e. , the lower bound of delivery success probability ) becomes more tractable and sheds light on intuitively understanding the impacts of network interference on content placement . therefore , in the following subsection , we focus on the case of @xmath136 ( i.e. , rayleigh fading ) . for rayleigh fading channels ( i.e. , @xmath136 ) , the lower - bound of delivery success probability in is simplified as @xmath137 where @xmath132 , @xmath138 , @xmath139 and @xmath140 is the gauss hypergeometric function . we omit the proof since it can be readily obtained by substituting @xmath136 in theorem 2 . with arbitrary cache memory size of @xmath2 at each helper , the alternative optimization problem * p2 * is rewritten as @xmath141 now we show that the objective function in * p3 * is concave and optimization problem * p3 * is also the constrained convex optimization problem . if we define @xmath142 as @xmath143 where @xmath144 and @xmath145 , its first derivative is @xmath146 ^ 2 } > 0 $ ] because @xmath147 always holds and @xmath148 for @xmath149 . note that @xmath150 for all @xmath13 because @xmath151 the second derivative of @xmath142 is @xmath152 ^ 3}\leq 0 $ ] and thus @xmath142 is a strictly increasing concave function . since a weighted sum of concave functions still satisfies concavity , optimization problem * p3 * is a constrained convex optimization problem . applying the same approach in section [ section : opt ] , we obtain the optimal caching probability of problem * p3 * as @xmath153^{+}\!\!\!\!,~\forall i\!\in\!\mathcal{f},\\ & = \frac{1}{1 \!-\ ! a_i}\left[-b_i+\sqrt{\frac{f_ib_i}{\omega^{\star}\!+\!\mu_i^{\star}}}~\right]^{+}\!\!\!,~\forall i\!\in\!\mathcal{f } , \label{eqn : opt2}\end{aligned}\ ] ] where lagrangian multipliers @xmath97 and @xmath98 range , according to @xmath22 , as , which is placed at the top of next page . replacing with and letting @xmath155 and @xmath156 in algorithm 1 , we can find the optimal @xmath118 and @xmath157 with one - dimensional bisection search and the corresponding near - optimal caching probabilities @xmath158 given by @xmath159^{+}\!\!,~1\right),~\forall i\!\in\!\mathcal{f}. \label{eqn : opt_sol_interlimited}\end{aligned}\ ] ] _ remark : _ unlike noise - limited networks , the solution of content placement obtained in is independent of the transmit power of caching helpers . the caching probability is a function of @xmath160 , @xmath29 and @xmath132 . in other words , the content placement is determined by the pathloss exponent , content popularity , and target bit rate . in this section , we evaluate the average success probability of content delivery to validate our analytical results in the previous sections . we also examine how various system parameters , such as @xmath161 , content popularity exponent ( @xmath162 ) , nakagami fading parameter ( @xmath50 and @xmath51 ) , pathloss exponent ( @xmath160 ) , density of caching helpers ( @xmath1 ) , user density ( @xmath10 ) , maximum target content bit rate ( @xmath163 ) , and cache memory size ( @xmath2 ) affect on caching probabilities . unless otherwise stated , the baseline setting of simulation environments is as follows : @xmath164 , @xmath165 , @xmath18 , @xmath136 , @xmath166 = 20 ( db ) , @xmath167 , @xmath168 ( units/@xmath169 ) , @xmath170 ( units/@xmath169 ) and @xmath171 ( bits / s / hz ) . the target bit rate for each content is uniformly generated as @xmath172 $ ] and all simulation results are averaged over 10,000 realizations . [ fig : caching_comparison ] compares the average success probabilities of content delivery in a noise - limited network for three different content placement strategies ; i ) caching the @xmath2 most popular contents ( mpc ) , ii ) caching the contents uniformly ( uc ) , and iii ) proposed content placement found by algorithm 1 ( proposed ) . this figure demonstrates that the proposed content placement in is superior to both uc and mpc in terms of average success probability of content delivery . mpc is closer to the proposed content placement than uc for high @xmath162 , and vice versa for low @xmath162 . for varying @xmath1 and @xmath50 , the optimal caching probability of each content @xmath13 in a noise - limited network is plotted in fig . [ fig : opt_sol_lambda_m ] , where the lower index indicates the higher popularity , i.e. , @xmath8 if @xmath173 . as @xmath1 or @xmath50 increases , the optimal caching probability becomes more uniform . it implies that it would be beneficial to increase hitting probability for all contents instead of focusing on channel selection diversity for a few specific contents . this is because channel power gains become higher as either the number of caching helpers increases or channels become more deterministic although channel selection diversity can be limited . this figure also exhibits that the optimal caching probability depends more on the geometric path loss than on small - scale fading , which matches the implication of fig . [ fig : cdf ] . fig . [ fig : opt_sol_target_bit_rate ] shows the optimal caching probability of each content @xmath13 in a noise - limited network for varying maximum target bit rate @xmath163 . as @xmath163 grows , the optimal caching probability becomes biased toward caching the most popular contents . if @xmath163 is large , increasing channel selection diversity gains of the most popular contents is more beneficial to improve success probability of content delivery . in fig . [ fig : opt_sol_m ] , the optimal caching probability of each content @xmath13 in a noise - limited network is plotted for varying cache memory size @xmath2 . the optimal caching probabilities scale with the cache memory size @xmath2 , but they become more uniform as @xmath2 increases . this is because less popular contents are accommodated in memory of larger size . [ fig : ps_with_opt_and_subopt ] compares the average success probabilities of content delivery with optimal @xmath174 obtained from by brute - force searches , with the proposed sub - optimal @xmath175 obtained from * p3 * , and the lower bound with the sub - optimal @xmath175 versus @xmath176 , when @xmath177 ( units/@xmath169 ) , @xmath164 , @xmath178 , and @xmath179 . for each @xmath180 and @xmath10 , the value of @xmath181 for a tighter lower bound is numerically found . since the content placement obtained from the lower bound is sub - optimal , the average content delivery success probability with the sub - optimal @xmath175 is bounded below that with optimal @xmath174 . although there is a large gap between the lower bound in and @xmath86 , the gap between the average content delivery success probabilities with the optimal @xmath174 and the proposed @xmath175 is small for an arbitrary target bit rate because and have quite similar shapes . consequently , the proposed sub - optimal caching probability is close to optimal caching probability although the sub - optimal caching probability is found from the lower bound in . [ fig : inter_comparision ] compares the average content delivery success probabilities among the proposed content placement schemes with numerically found @xmath181 yielding a tight lower bound and with @xmath182 , uc , and mpc versus the content popularity exponent @xmath162 . although the value of @xmath181 needs to be numerically found , any suboptimal solution even with the value @xmath181 which does not always satisfy the inequality in yields a lower average success probability of content delivery because of its suboptimality . from this fact , a suboptimal solution can be found by setting the value of @xmath181 to be the average load of a typical caching helper as @xmath183 for simplicity . [ fig : inter_comparision ] demonstrates that that both the proposed content placement schemes with numerically found @xmath181 and @xmath182 are superior to both uc and mpc in terms of average content delivery success probability for general @xmath162 . the average content delivery success probability with @xmath183 is quite similar to that with numerically found @xmath181 and outperforms uc and mpc . in an interference - limited network , for varying user density @xmath10 , the proposed caching probability of each content @xmath13 obtained by solving the convex optimization problem in * p3 * is plotted in fig . [ fig : inter_opt_sol_user ] , where the value of @xmath181 yielding a tight lower bound is numerically found . as the user density @xmath10 decreases , the optimal content placement tends to cache all contents with more uniform probabilities . we studied probabilistic content placement to desirably control cache - based channel selection diversity and network interference in a wireless caching helper network , with specific considerations of path loss , small - scale channel fading , network interference according to random network topology based on stochastic geometry , and arbitrary cache memory size . in a noise - limited case , we derived the optimal caching probabilities for each content in closed form in terms of the average success probability of content delivery and proposed a bisection based search algorithm to efficiently reach the optimal solution . in an interference - limited case , we derived a lower bound on the average success probability of content delivery . then , we found the near - optimal caching probabilities in closed form in rayleigh fading channels , which maximize the lower bound . our numerical results verified that the proposed content placement is superior to the conventional caching strategies because the proposed scheme efficiently controls the channel selection diversity gain and the interference reduction . we also numerically analyzed the effects of various system parameters , such as caching helper density , user density , nakagami @xmath184 fading parameter , memory size , target bit rate , and user density , on the content placement . since the pathloss dominates the small - scale fading effects according to lemma 1 , @xmath129 is approximated as the load of the tagged caching helper with the largest channel power gain averaged over fading ( i.e. , the load based on the association with long - term channel power gains ) , @xmath185 . moreover , the received sir with the association based on instantaneous channel power gains is larger than that with the association based on long - term channel power gains . accordingly , can be further bounded below as @xmath186,\label{rev:5_2}\end{aligned}\ ] ] where @xmath187 which is also validated in fig . [ fig : approx_check ] , where blue circle and green solid line represent and , respectively . in case of @xmath178 , a closed form expression of @xmath188 is available as @xmath189 @xcite , but with multiple contents ( @xmath190 ) analytic evaluation of is hard due to the complicated form of @xmath188 . to circumvent this difficulty , we again take a lower bound of as @xmath191 , \label{rev:5_3}\end{aligned}\ ] ] where @xmath133 is a constant independent of @xmath13 and makes the inequality hold for all ranges of @xmath16 , and @xmath132 . note that since is a decreasing function with respect to @xmath181 and bounded below by zero , there must exist a certain value of @xmath192 which makes the inequality hold . the value of @xmath181 yielding a tight lower bound can be numerically determined ; in general @xmath181 becomes larger as @xmath193 diminishes and @xmath162 grows . [ fig : approx_check ] validates , where green and black dotted lines represent and our lower bound in , respectively . it is verified that there exists a finite value of @xmath181 yielding a lower bound of regardless of @xmath16 . in our setting , the value of @xmath181 for a tighter lower bound is @xmath194 . although there exists a gap between and its lower bound , the shape of those two functions looks quite similar and thus the caching probabilities obtained from are close to the optimal caching probabilities . & \stackrel{(a)}{=}\sum_{i=1}^f f_i \int_0^{\infty}\!\mathbb{e}_{i_i}\!\left[\frac{\gamma(m_d , m_dp^{-1}\tau_i r^{\alpha}i_i)}{\gamma(m_d)}\right]\ ! f_{|x_i|}(r)dr , \label{lower_aftsp}\end{aligned}\ ] ] where @xmath132 , @xmath196 is the gamma function defined as @xmath197 , @xmath198 is the upper incomplete gamma function defined as @xmath199 , @xmath200 is the location of the nearest caching helper storing content @xmath13 , @xmath201 is the pdf of the distance to the nearest caching helper storing content @xmath13 , and @xmath202 the equality ( a ) is obtained from the nakagami-@xmath50 fading channel power gain . since @xmath203}{\gamma(m)}=e^{-my}\sum_{k=0}^{m-1}\frac{m^k}{k!}y^k$ ] , we have @xmath204\\ & = \sum_{k=0}^{m_d-1}\frac{1}{k!}\left(m_d p^{-1}\tau_i r^{\alpha } \right)^k\mathbb{e}_{i_i}\left[i_i^ke^{-m_dp^{-1}\tau_i r^{\alpha}i_i}\right]\\ & \stackrel{(b)}{=}\!\sum_{k=0}^{m_d-1}\!\frac{1}{k!}\left(-m_d p^{-1}\tau_i r^{\alpha}\right)^k \!\frac{d^k}{ds^k}\mathcal{l}_{i_i}(s)|_{s=\frac{m_d\tau_i r^{\alpha}}{p } } , \label{inner}\end{aligned}\ ] ] where ( b ) is from @xmath205 and @xmath206 is the laplace transform of @xmath207 given by @xmath208 = \mathbb{e}\left[e^{-s\sum_{y\in \phi\setminus x_i}p|h_y|^2|y|^{-\alpha}}\right]\\ & \stackrel{(c)}{=}\mathbb{e}\left[\prod_{y\in \phi\setminus x_i } \mathbb{e}_{|h_y|^2}\left[e^{-sp|h_y|^2|y|^{-\alpha}}\right]\right]\\ & \stackrel{(d)}{= } \exp\left(-2\pi p_i\lambda\int_{r}^{\infty}\left[1-\mathbb{e}_g\left[e^{-spgv^{-\alpha}}\right]\right]vdv\right)\nonumber\\ & ~~~\times\exp\!\left(\!-2\pi(1 \!-\ ! p_i ) \lambda\!\int_0^{\infty}\!\left[1\!-\!\mathbb{e}_g\!\left[e^{-spgv^{-\alpha}}\right]\right]vdv\!\right)\\ & \stackrel{(e)}{= } \exp\left(-2\pi p_i\lambda\int_{r}^{\infty}\frac{(spv^{-\alpha}+m_i)^{m_i}-m_i}{(spv^{-\alpha}+m_i)^{m_i}}vdv\right)\nonumber\\ & ~\times\exp\!\left(\!-2\pi(1\!-\!p_i ) \lambda\!\!\int_0^{\infty}\!\!\frac{(spv^{-\alpha}\!+\ ! m_i)^{m_i}}vdv\!\right)\\ & = \exp\left(-2\pi\lambda\int_0^{\infty}\frac{(spv^{-\alpha}+m_i)^{m_i}-m_i}{(spv^{-\alpha}+m_i)^{m_i}}vdv\right)\nonumber\\ & ~~~\times\exp\left(2\pi p_i\lambda\int_0^r\frac{(spv^{-\alpha}+m_i)^{m_i}-m_i}{(spv^{-\alpha}+m_i)^{m_i}}vdv\right ) , \label{laplace}\end{aligned}\ ] ] where ( c ) is due to independence of the channel ; ( d ) comes from the probability generating functional ( pgfl ) of ppp ; ( e ) is from the mogment generating function ( mgf ) of the nakagami-@xmath51 distribution . substituting into , we obtain @xmath209 where @xmath50 and @xmath51 are the nakagami fading parameters of the desired and interfering links , respectively , and @xmath210vdv\right.\nonumber\\ & \left.+2\pi p_i\lambda\!\int_0^r \left [ 1 - \frac{m_i}{(spv^{-\alpha}+m_i)^{m_i}}\right]vdv\right),\\ f_{|x_i|}(r ) & = 2\pi p_i\lambda r\exp\left(-\pi p_i\lambda r^2\right).\end{aligned}\ ] ] 1 cisco , `` cisco visual networking index : global mobile data traffic forecast update , 2014 - 2019 , '' available at http://www.cisco.com . n. golrezaei , a. f. molisch , a. g. dimakis , and g. caire , `` femtocaching and device - to - device collaboration : a new architecture for wireless video distribution , '' _ ieee commun . mag . 142 - 149 , apr . 2013 . k. shanmugam , n. golrezaei , a. g. dimakis , and a. f. molisch , and g. caire , `` femtocaching : wireless content delivery through distributed caching helpers , '' _ ieee trans . inform . theory _ 8402 - 8413 , dec . 2013 . j. song , h. song , and w. choi , `` optimal caching placement of caching system with helpers , '' in proc . _ ieee int . 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content delivery success in wireless caching helper networks depends mainly on cache - based channel selection diversity and network interference . for given channel fading and network geometry , both channel selection diversity and network interference dynamically vary according to what and how the caching helpers cache at their finite storage space . we study probabilistic content placement ( or caching placement ) to desirably control cache - based channel selection diversity and network interference in a stochastic wireless caching helper network , with sophisticated considerations of wireless fading channels , interactions among multiple users such as interference and loads at caching helpers , and arbitrary memory size . using stochastic geometry , we derive optimal caching probabilities in closed form to maximize the average success probability of content delivery and propose an efficient algorithm to find the solution in a noise - limited network . in an interference - limited network , based on a lower bound of the average success probability of content delivery , we find near - optimal caching probabilities in closed form to control the channel selection diversity and the network interference . we numerically verify that the proposed content placement is superior to other comparable content placement strategies . probabilistic content placement , caching probability , stochastic geometry , channel selection diversity

a coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus . coamoebae are cousins to amoebae , which are images of subvarieties under the coordinatewise logarithm map @xmath0 . amoebae were introduced by gelfand , kapranov , and zelevinsky in 1994 @xcite , and have subsequently been widely studied @xcite . coamoebae were introduced by passare in a talk in 2004 , and they appear to have many beautiful and interesting properties . for example , coamoebae of @xmath1-discriminants in dimension two are unions of two non - convex polyhedra @xcite , and a hypersurface coamoeba has an associated arrangement of codimension one tori contained in its closure @xcite . bergman @xcite introduced the logarithmic limit set @xmath2 of a subvariety @xmath3 of the torus as the set of limiting directions of points in its amoeba . bieri and groves @xcite showed that @xmath2 is a rational polyhedral complex in the sphere . logarithmic limit sets are now called tropical algebraic varieties @xcite . for a hypersurface @xmath4 , logarithmic limit set @xmath5 consists of the directions of non - maximal cones in the outer normal fan of the newton polytope of @xmath6 . we introduce a similar object for coamoebae and establish a structure theorem for coamoebae similar to that of bergman and of bieri and groves for amoebae . let be the coamoeba of a subvariety @xmath3 of @xmath7 with ideal @xmath8 . the of @xmath3 , , is the set of accumulation points of arguments of sequences in @xmath3 with unbounded logarithm . for @xmath9 , the initial variety @xmath10 is the variety of the initial ideal of @xmath8 . the fundamental theorem of tropical geometry asserts that @xmath11 exactly when the direction of @xmath12 lies in @xmath2 . we establish its analog for coamoebae . [ t : one ] the closure of @xmath13 is @xmath14 , and @xmath15 johansson @xcite used different methods to prove this when @xmath3 is a complete intersection . the cone over the logarithmic limit set admits the structure of a rational polyhedral fan @xmath16 in which all weights @xmath17 in the relative interior of a cone @xmath18 give the same initial scheme @xmath19 . thus the union in theorem [ t : one ] is finite and is indexed by the images of these cones @xmath20 in the logarithmic limit set of @xmath3 . the logarithmic limit set or tropical algebraic variety is a combinatorial shadow of @xmath3 encoding many properties of @xmath3 . while the coamoeba of @xmath3 is typically not purely combinatorial ( see the examples of lines in @xmath21 in section [ s : lines ] ) , the phase limit set does provide a combinatorial skeleton which we believe will be useful in the further study of coamoebae . we give definitions and background in section [ s : defs ] , and detailed examples of lines in three - dimensional space in section [ s : lines ] . these examples are reminiscent of the concrete descriptions of amoebae of lines in @xcite . we prove theorem [ t : one ] in section [ s : phase ] . as a real algebraic group , the set @xmath22 of invertible complex numbers is isomorphic to @xmath23 under the map @xmath24 . here , @xmath25 is the set of complex numbers of norm 1 which may be identified with @xmath26 . the inverse map is @xmath27 . let @xmath28 be a free abelian group of finite rank and @xmath29 its dual group . we use @xmath30 for the pairing between @xmath28 and @xmath31 . the group ring @xmath32 $ ] is the ring of laurent polynomials with exponents in @xmath28 . it is the coordinate ring of a torus @xmath33 which is identified with @xmath34 , the set of group homomorphisms @xmath35 . there are natural maps @xmath36 and @xmath37 , which are induced by the maps @xmath38 and @xmath39 . maps @xmath40 of free abelian groups induce corresponding maps @xmath41 of tori , and also of @xmath42 and @xmath43 . if @xmath44 is the rank of @xmath31 , we may identify @xmath31 with @xmath45 , which identifies @xmath33 with @xmath46 , @xmath43 with @xmath47 , and @xmath42 with @xmath48 . the of a subvariety @xmath49 is its image under the map @xmath50 , and the of @xmath3 is the image of @xmath3 under the argument map @xmath51 . an amoeba has a geometric - combinatorial structure at infinity encoded by the logarithmic limit set @xcite . coamoebae similarly have phase limit sets which have a related combinatorial structure that we define and study in section [ s : phase ] . if we identify @xmath52 with @xmath53 , then the map @xmath54 given by @xmath55 is a real algebraic map . thus , coamoebae , as they are the image of a real algebraic subset of the real algebraic variety @xmath33 under the real algebraic map @xmath56 , are semialgebraic subsets of @xmath43 @xcite . it would be very interesting to study them as semi - algebraic sets , in particular , what are the equations and inequalities satisfied by a coamoeba ? when @xmath3 is a grassmannian , such a description would generalize richter - gebert s five - point condition for phirotopes from rank two to arbitrary rank @xcite . similarly , we may replace the map @xmath57 in the definition of amoebae by the map @xmath58 to obtain the of @xmath3 , which is a subset of @xmath59 . the algebraic amoeba is a semi - algebraic subset of @xmath59 , and we also ask for its description as a semi - algebraic set . [ ex : linep2 ] let @xmath60 be defined by @xmath61 . the coamoeba @xmath62 is the set of points of @xmath63 of the form @xmath64 for @xmath65 . if @xmath66 is real , then these points are @xmath67 , @xmath68 , and @xmath69 if @xmath66 lies in the intervals @xmath70 , @xmath71 , and @xmath72 respectively . for other values , consider the picture below in the complex plane . @xmath73 for @xmath74 fixed , @xmath75 can take on any value strictly between @xmath76 ( for @xmath17 near @xmath77 ) and @xmath78 ( for @xmath66 near @xmath78 ) , and thus @xmath62 consists of the three points @xmath79 , @xmath80 , and @xmath81 and the interiors of the two triangles displayed below in the fundamental domain @xmath82 ^ 2\subset { { \mathbb r}}^2 $ ] of @xmath63 . this should be understood modulo @xmath83 , so that @xmath84 . @xmath85 the coamoeba is the complement of the region @xmath86 ^ 2\;:\ ; |\alpha-\beta|\ \leq\ \pi=\arg(-1)\}\,,\ ] ] together with the three images of real points @xmath67 , @xmath68 , and @xmath69 . given a general line @xmath87 with @xmath88 , we may replace @xmath66 by @xmath89 and @xmath90 by @xmath91 , to obtain the line @xmath92 , with coamoeba . this transformation rotates the coamoeba by @xmath93 horizontally and @xmath94 vertically . let @xmath95 $ ] be a polynomial with support @xmath96 , @xmath97 where we write @xmath98 for the element of @xmath32 $ ] corresponding to @xmath99 . given @xmath100 , let @xmath101 be the minimum of @xmath102 for @xmath103 . then the initial form @xmath104 of @xmath6 with respect to @xmath100 is the polynomial @xmath105 $ ] defined by @xmath106 given an ideal @xmath107 $ ] and @xmath100 , the with respect to @xmath17 is @xmath108\,.\ ] ] lastly , when @xmath8 is the ideal of a subvariety @xmath3 , the @xmath109 is defined by the initial ideal @xmath110 . the sphere @xmath111 is the set of directions in @xmath42 . write @xmath112 for the projection . the of a subvariety @xmath3 of @xmath33 is the set of accumulation points in @xmath113 of sequences @xmath114 where @xmath115 is an unbounded set . a sequence @xmath116 is unbounded if its sequence of logarithms @xmath117 is unbounded . a @xmath118 is the set of points @xmath100 which satisfy finitely many inequalities and equations of the form @xmath119 where @xmath120 . the of @xmath20 is the dimension of its linear span , and of @xmath20 are proper subsets of @xmath20 obtained by replacing some inequalities by equations . the relative interior of @xmath20 consists of its points not lying in any face . also , @xmath20 is determined by @xmath121 , which is a finitely generated subsemigroup of @xmath31 . a is a collection of rational polyhedral cones in @xmath42 in which every two cones of @xmath16 meet along a common face . [ t : fttg ] the cone in @xmath42 over the negative @xmath122 of the logarithmic limit set of @xmath3 is the set of @xmath100 such that @xmath123 . equivalently , it is the set of @xmath100 such that for every @xmath95 $ ] lying in the ideal @xmath8 of @xmath3 , @xmath104 is not a monomial . this cone over @xmath122 admits the structure of a rational polyhedral fan @xmath16 with the property that if @xmath124 lie in the relative interior of a cone @xmath20 of @xmath16 , then @xmath125 . it is important to take @xmath122 . this is correct as we use the tropical convention of minimum , which is forced by our use of toric varieties to prove theorem [ t : one ] in section [ s : tropicalcompact ] . we write @xmath126 for the initial ideal defined by points in the relative interior of a cone @xmath20 of @xmath16 . the fan structure @xmath16 is not canonical , for it depends upon an identification @xmath127 . moreover , it may be the case that @xmath128 , but @xmath129 . bergman @xcite defined the logarithmic limit set of a subvariety of the torus @xmath33 , and bieri and groves @xcite showed it was a finite union of convex polyhedral cones . the connection to initial ideals was made more explicit through work of kapranov @xcite and the above form is adapted from speyer and sturmfels @xcite . the logarithmic limit set of @xmath3 is now called the tropical algebraic variety of @xmath3 , and this latter work led to the field of tropical geometry . we consider coamoebae of lines in three - dimensional space . we will work in the torus @xmath130 of @xmath131 , which is the quotient of @xmath132 by the diagonal torus @xmath133 and similarly in @xmath134 , the quotient of @xmath135 by the diagonal @xmath136 . by in @xmath134 we mean the images in @xmath134 of lines and planes in @xmath135 parallel to some coordinate plane . let @xmath137 be a line in @xmath131 not lying in any coordinate plane . then @xmath137 has a parameterization @xmath138\ \longmapsto\ [ \ell_0(s , t):\ell_1(s , t):\ell_2(s , t):\ell_3(s , t)]\,,\ ] ] where @xmath139 are non - zero linear forms which do not all vanish at the same point . for @xmath140 , let @xmath141 be the zero of @xmath142 . the configuration of these zeroes determine the coamoeba of @xmath143 , which we will simply write as @xmath62 . suppose that two zeroes coincide , say @xmath144 . then @xmath145 for some @xmath146 , and so @xmath137 lies in the translated subtorus @xmath147 and its coamoeba @xmath62 lies in the coordinate subspace of @xmath148 defined by @xmath149 . in fact , @xmath62 is pulled back from the coamoeba of the projection of @xmath137 to the @xmath150 plane . it follows that if there are only two distinct roots among @xmath151 , then @xmath62 is a coordinate line of @xmath148 . if three of the roots are distinct , then ( up to a translation ) the projection of the coamoeba @xmath62 to the @xmath150 plane looks like so that @xmath62 consists of two triangles lying in a coordinate plane . for each @xmath140 define a function depending upon a point @xmath152\in{{\mathbb p}}^1 $ ] and @xmath153 by @xmath154 for each @xmath140 , let @xmath155 be the image in @xmath134 of @xmath25 under the map @xmath156\,.\ ] ] for each @xmath140 , @xmath155 is a coordinate line in @xmath134 that consists of accumulation points of @xmath62 . this follows from theorem [ t : one ] . for the main idea , note that @xmath157 for @xmath153 is a curve in @xmath134 whose hausdorff distance to the line @xmath155 approaches 0 as @xmath158 . the of @xmath137 is the union of these four lines . [ l : constant ] suppose that the zeroes @xmath159 are distinct . then @xmath160 is constant along each arc of the circle in @xmath161 through @xmath159 . after changing coordinates in @xmath161 and translating in @xmath162 ( rotating coordinates ) , we may assume that these roots are @xmath163 and so the circle becomes the real line . choosing affine coordinates , we may assume that @xmath164 , @xmath165 and @xmath166 , so that we are in the situation of example [ ex : linep2 ] . then the statement of the lemma is the computation there for @xmath66 real in which we obtained the coordinate points @xmath79 , @xmath80 , and @xmath81 . [ l : disjoint ] the phase limit lines @xmath167 , @xmath168 , @xmath169 , and @xmath170 are disjoint if and only if the roots @xmath151 do not all lie on a circle . suppose that two of the limit lines meet , say @xmath167 and @xmath168 . without loss of generality , we suppose that we have chosen coordinates on @xmath135 and @xmath161 so that @xmath171 and @xmath172 for @xmath140 . then there are points @xmath173 such that @xmath174 comparing the last two components , we obtain @xmath175 and so the zeroes @xmath151 have the configuration below . @xmath176{figures / elemgeom.eps } } \put(14,24){$\theta$ } \put(56,33.5){$\theta$ } \put(1,52){$\zeta_3 $ } \put(66,61){$\zeta_2 $ } \put(-12,0){$\zeta_0 $ } \put(94,0){$\zeta_1 $ } \end{picture}\ ] ] but then @xmath151 are cocircular . conversely , if @xmath151 lie on a circle @xmath177 , then by lemma [ l : constant ] the lines @xmath155 and @xmath178 meet only if @xmath179 and @xmath180 are the endpoints of an arc of @xmath181 . [ l : immersion ] if the roots @xmath151 do not all lie on a circle , then the map @xmath182 is an immersion . let @xmath183 , which we consider to be a real two - dimensional manifold . after possibly reordering the roots , the circle @xmath184 containing @xmath185 meets the circle @xmath186 containing @xmath187 transversally at @xmath66 . under the derivative of the map @xmath188 , tangent vectors at @xmath66 to @xmath184 and @xmath186 are taken to nonzero vectors @xmath189 and @xmath190 in the tangent space to @xmath135 . furthermore , as the four roots do not all lie on a circle , we can not have both @xmath191 and @xmath192 , and so this derivative has full rank two at @xmath66 , as a map from @xmath193 , which proves the lemma . by these lemmas , there is a fundamental difference between the coamoebae of lines when the roots of the lines @xmath142 are cocircular and when they are not . we examine each case in detail . first , choose coordinates so that @xmath194 . after dehomogenizing and separately rescaling each affine coordinate ( e.g. identifying @xmath134 with @xmath195 and applying phase shifts to each coordinate @xmath196 of @xmath195 ) , we may assume that the map parametrizing @xmath137 is @xmath197 suppose first that the four roots are cocircular . as @xmath198 , the other three lie on a real line in @xmath199 , which we may assume is @xmath200 . that is , if the four roots are cocircular , then up to coordinate change , we may assume that the line @xmath137 is real and the affine parametrization is also real . for this reason , we will call such lines @xmath137 . we first study the boundary of @xmath62 . suppose that @xmath66 lies on a contour @xmath177 in the upper half plane as in figure [ f : contour ] ( 223,65 ) ( 0,10 ) ( 55,0)@xmath201 ( 120,0)@xmath202 ( 165,0)@xmath203 ( 213,18)@xmath200 ( 76,47)@xmath177 that contains semicircles of radius @xmath204 centered at each root and a semicircle of radius @xmath205 centered at 0 , but otherwise lies along the real axis , for @xmath204 a sufficiently small positive number . then @xmath206 is constant on the four segments of @xmath177 lying along @xmath200 with respective values @xmath207 moving from left to right . on the semicircles around @xmath201 , @xmath202 , and @xmath203 , two of the coordinates are essentially constant ( but not quite equal to either 0 or @xmath208 ! ) , while the third decreases from @xmath208 to 0 . finally , on the large semicircle , the three coordinates are nearly equal and increase from @xmath209 to @xmath210 . the image @xmath211 can be made as close as we please to the quadrilateral in @xmath195 connecting the points of in cyclic order , when @xmath204 is sufficiently small . thus the image of the upper half plane under the map @xmath212 is a relatively open membrane in @xmath195 that spans the quadrilateral . it lies within the convex hull of this quadrilateral , which is computed using the affine structure induced from @xmath213 by the quotient @xmath214 . for this , observe that its projection in any of the four coordinate directions parallel to its edges is one of the triangles of the coamoeba of the projected line in @xmath215 of example [ ex : linep2 ] , and the convex hull of the quadrilateral is the intersection of the four preimages of these triangles . because @xmath137 is real , the image of the lower half plane is isomorphic to the image of the upper half plane , under the map @xmath216 and so the coamoeba is symmetric in the origin of @xmath195 and consists of two quadrilateral patches that meet at their vertices . here are two views of the coamoeba of the line where the roots are @xmath217 : @xmath218{figures / cocircular_line.1.eps } \qquad \includegraphics[height=1.9in]{figures / cocircular_line.2.eps}\ ] ] now suppose that the roots @xmath151 do not all lie on a circle . by lemma [ l : disjoint ] , the four phase limit lines @xmath219 are disjoint and the map from @xmath137 to the coamoeba is an immersion . figure [ f : symmetric ] shows two views of the coamoeba in a fundamental domain of @xmath134 when the roots are @xmath220 , where @xmath221 is a primitive third root of infinity . this and other pictures of coamoebae of lines are animated on the webpage @xcite . the projection of this coamoeba along a coordinate direction ( parallel to one of the phase limit lines @xmath155 ) gives a coamoeba of a line in @xmath222 , as we saw in example [ ex : linep2 ] . the line @xmath155 is mapped to the interior of one triangle and the vertices of the triangles are the images of line segments lying on the coamoeba . these three line segments come from the three arcs of the circle through the three roots other than @xmath179 , the root corresponding to @xmath155 . [ p : linesegments ] the interior of the coamoeba of a general line in @xmath130 contains @xmath223 line segments in triples parallel to each of the four coordinate directions . the symmetric coamoeba we show in figure [ f : symmetric ] has six additional line segments , two each coming from the three longitudinal circles through a third root of unity and @xmath224 . two such segments are visible as pinch points in the leftmost view in figure [ f : symmetric ] . we ask : what is the maximal number of line segments on a coamoeba of a line in @xmath130 ? the of a complex subvariety @xmath225 is the set of all accumulation points of sequences @xmath226 , where @xmath227 is an unbounded sequence . for @xmath228 , @xmath229 is the ( possibly empty ) initial scheme of @xmath3 , whose ideal is the initial ideal @xmath110 , where @xmath8 is the ideal of @xmath3 . our main result is that the phase limit set of @xmath3 is the union of the coamoebae of all its initial schemes . this is a finite union . by theorem [ t : fttg ] , @xmath230 is non - empty only when @xmath17 lies in the cone over the logarithmic set @xmath2 , which can be given the structure of a finite union of rational polyhedral cones such that any two points in the relative interior of the same cone @xmath20 have the same initial scheme . if we write @xmath231 for the initial scheme corresponding to a cone @xmath20 , then the torus @xmath232 acts on @xmath233 by translation ( see , e.g. corollary [ c : inigeometry ] ) . ( here , @xmath234 is the span of @xmath121 , a free abelian group of rank @xmath235 . ) this implies that @xmath236 is a union of orbits of @xmath237 , and thus that @xmath238 . we review the standard dictionary relating initial ideals to toric degenerations in the context of subvarieties of @xmath33 ( * ? ? ? let @xmath49 be a subvariety with ideal @xmath107 $ ] . we study @xmath110 and the initial schemes @xmath241 for @xmath228 . since @xmath242 , so that @xmath243 , we may assume that @xmath244 . as @xmath31 is the lattice of one - parameter subgroups of @xmath33 , @xmath17 corresponds to a one - parameter subgroup written as @xmath245 . define @xmath246 by @xmath247 the fiber of @xmath248 over a point @xmath249 is @xmath250 . let be the closure of @xmath248 in @xmath251 , and set to be the fiber of @xmath252 over @xmath253 . we first describe the ideal @xmath255 of @xmath248 . for @xmath99 , the element @xmath256 $ ] takes the value @xmath257 on the element @xmath258 , and so if @xmath259 , then @xmath98 takes the value @xmath260 on @xmath261 . given a polynomial @xmath95 $ ] of the form @xmath262 define the polynomial @xmath263[m]$ ] by @xmath264 then @xmath265 , so @xmath255 is generated by the polynomials @xmath266 , for @xmath267 . a general element of @xmath255 is a linear combination of translates @xmath268 of such polynomials , for @xmath269 . if we set @xmath101 to be the minimal exponent of @xmath270 occurring in @xmath266 , then @xmath271 and @xmath272 this shows that @xmath273[m]$ ] is generated by polynomials @xmath274 , where @xmath267 . since @xmath275 $ ] and the remaining terms are divisible by @xmath270 , we see that the ideal of @xmath276 is generated by @xmath277 , which completes the proof . we use proposition [ p : initial_scheme ] to prove one inclusion of theorem [ t : one ] , that @xmath278 fix @xmath279 , and let @xmath248 , @xmath252 , and @xmath280 be as in proposition [ p : initial_scheme ] , and let @xmath281 . we show that @xmath282 . since @xmath283 , there is an irreducible curve @xmath284 with @xmath285 . the projection of @xmath286 to @xmath52 is dominant , so there exists a sequence @xmath287 that converges to @xmath288 with each @xmath289 real and positive . then @xmath290 is the limit of the sequence @xmath291 . for each @xmath292 , set @xmath293 . since @xmath289 is positive and real , every component of @xmath294 is positive and real , and so @xmath295 . thus @xmath290 is the limit of the sequence @xmath296 . since @xmath297 converges to @xmath298 and @xmath289 converges to @xmath78 , the sequence @xmath299 is unbounded , which implies that @xmath290 lies in the phase limit set of @xmath3 . this proves . we complete the proof of theorem [ t : one ] by establishing the other inclusion , @xmath300 suppose that @xmath301 is an unbounded sequence . to study the accumulation points of the sequence @xmath302 , we use a compactification of @xmath3 that is adapted to its inclusion in @xmath33 . suitable compactifications are tevelev s tropical compactifications @xcite , for in these the boundary of @xmath3 is composed of initial schemes @xmath19 of @xmath3 in a manner we describe below . by theorem [ t : fttg ] , the cone over the logarithmic limit set @xmath2 of @xmath3 is the support of a rational polyhedral fan @xmath16 whose cones @xmath20 have the property that all initial ideals @xmath110 coincide for @xmath17 in the relative interior of @xmath20 . recall the construction of the toric variety @xmath303 associated to a fan @xmath16 @xcite , ( * ? ? ? * ch . 6 ) . for @xmath18 , set @xmath304 set @xmath305 $ ] and @xmath306 $ ] , which is naturally isomorphic to @xmath307 , where @xmath308 is the subgroup generated by @xmath121 . the map @xmath309 determines a comodule map @xmath310\to{{\mathbb c}}[\sigma^\vee]\otimes{{\mathbb c}}[m]$ ] , which induces the action of the torus @xmath33 on @xmath311 . its orbits correspond to faces of the cone @xmath20 with the smallest orbit @xmath312 corresponding to @xmath20 itself . the inclusion @xmath313 is split by the semigroup map @xmath314 which induces a map @xmath32\twoheadrightarrow{{\mathbb c}}[\sigma^\perp]$ ] , and thus we have the @xmath33-equivariant split inclusion @xmath315 on orbits @xmath316 in @xmath311 , the map @xmath317 is simply the quotient by @xmath318 . if @xmath319 with @xmath320 , then @xmath321 , so @xmath310\supset{{\mathbb c}}[\tau^\vee]$ ] , and so @xmath322 . since the quotient fields of @xmath310 $ ] and @xmath32 $ ] coincide , these are inclusions of open sets , and these varieties @xmath311 for @xmath18 glue together along these natural inclusions to give the toric variety @xmath303 . the torus @xmath33 acts on @xmath303 with an orbit @xmath312 for each cone @xmath20 of @xmath16 . since @xmath323 , @xmath303 contains @xmath33 as a dense subset , and thus @xmath3 is a ( non - closed ) subvariety . let @xmath324 be the closure of @xmath3 in @xmath303 . as the fan @xmath16 is supported on the cone over @xmath2 , @xmath324 will be a tropical compactification of @xmath3 and @xmath324 is complete ( * ? ? ? 2.3 ) . to understand the points of @xmath325 , we study the intersection @xmath326 , which is defined by @xmath327 $ ] , as well as the intersection @xmath328 , which is defined in @xmath329 $ ] by the image @xmath330 of @xmath327 $ ] under the map @xmath310\twoheadrightarrow{{\mathbb c}}[\sigma^\perp]$ ] induced by . let @xmath267 . since @xmath20 is a cone in @xmath16 , we have that @xmath332 for all @xmath17 in the relative interior of @xmath20 . thus for @xmath333 , the function @xmath334 on exponents of monomials of @xmath6 is minimized on ( a superset of ) the support of @xmath335 , and if @xmath17 lies in the relative interior of @xmath20 , then the minimizing set is the support of @xmath335 . multiplying @xmath6 if necessary by @xmath336 , where @xmath337 is some monomial of @xmath338 , we may assume that for every @xmath333 , the linear function @xmath334 is nonnegative on the support of @xmath6 , so that @xmath339 $ ] , and the function is zero on the support of @xmath335 . furthermore , if @xmath17 lies in the relative interior of @xmath20 , then it vanishes exactly on the support of @xmath335 . thus @xmath340 $ ] , which completes the proof . let @xmath343 be a point in the phase limit set of @xmath3 . then there exists an unbounded sequence @xmath301 with @xmath344 since @xmath324 is compact , the sequence @xmath345 has an accumulation point @xmath66 in @xmath324 . as the sequence is unbounded , @xmath346 , and so @xmath347 . thus @xmath66 is a point of @xmath328 for some cone @xmath348 of @xmath16 . replacing @xmath349 by a subsequence , we may assume that @xmath350 . because the map @xmath317 is continuous and is the identity on @xmath312 , we have that @xmath351 converges to @xmath352 , and thus @xmath353 corollary [ c : inigeometry ] implies that @xmath354 , as @xmath355 . recall that on @xmath356 , @xmath317 is the quotient by @xmath318 . thus we conclude from that @xmath357 which completes the proof of theorem [ t : one ] as @xmath358 for any @xmath17 in the relative interior of @xmath20 . in @xcite , the closure of a hypersurface coamoeba @xmath359 for @xmath95 $ ] was shown to contain a finite collection of . these are translates of codimension one subtori @xmath360 for @xmath20 a cone in the normal fan of the newton polytope of @xmath6 corresponding to an edge . by theorem [ t : one ] , these translated tori are that part of the phase limit set of @xmath3 corresponding to the cones @xmath20 dual to the edges , specifically @xmath236 . since @xmath20 has dimension @xmath361 , the torus @xmath362 acts with finitely many orbits on @xmath233 , which is therefore a union of finitely many translates of @xmath362 . thus @xmath236 is a union of finitely many translates of @xmath360 . the logarithmic limit set @xmath363 of a curve @xmath364 is a finite collection of points in @xmath113 . each point gives a ray in the cone over @xmath363 , and the components of @xmath365 corresponding to a ray @xmath20 are finitely many translations of the dimension one subtorus @xmath360 of @xmath43 , which we referred to as lines in section [ s : lines ] . these were the lines lying in the boundaries of the coamoebae @xmath62 of the lines @xmath137 in @xmath366 and @xmath367 . alexander below , vanessa krummeck , and jrgen richter - gebert , _ complex matroids phirotopes and their realizations in rank 2 _ , discrete and computational geometry , algorithms combin . , vol . 25 , springer , berlin , 2003 , pp . 203233 .

a coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus . we describe the structure of the boundary of the coamoeba of a variety , which we relate to its logarithmic limit set . detailed examples of lines in three - dimensional space illustrate and motivate these results .

tidal interaction between the lmc , the smc , and the galaxy have long been considered to play vital roles not only in dynamical and chemical evolution of the magellanic clouds ( mcs ) but also in the formation of the magellanic stream ( ms ) and bridge ( mb ) around the galaxy ( e.g. , westerland 1999 ; murai & fujimoto 1981 ; bekki & chiba 2005 , b05 ) . although previous theoretical and numerical studies on the lmc - smc - galaxy tidal interaction discussed extensively the origin of dynamical properties of the mb ( e.g. , gardiner & noguchi 1995 , g96 ) , they have not yet investigated so extensively the long - term formation histories of field stars and star clusters ( scs ) in the mcs . therefore , long - standing and remarkable problems related to the interplay between the lmc - smc - galaxy interaction and the formation histories of stars and scs remain unsolved ( see bekki et al . 2004a , b for the first attempts to challenge these problems ) . one of intriguing and unexplained observations on scs in the lmc is that an intermediate - age sc ( ngc 1718 ) with the estimated age of @xmath2 gyr has a distinctively low metallicity of [ fe / h]@xmath3 among intermediate - age scs ( geisler et al . 2003 , g03 ; grocholski et al . 2006 , g06 ) . santos & piatti ( 2004 , s04 ) investigated integrated spectrophotometric properties of old and young scs and found that several young scs with ages less than 200 myr have metallicities smaller than @xmath4 . three examples of these low - metallicity objects including rolleston et al . ( 1999 , r99 ) are listed in the table 1 . given the fact that the stellar metallicity of the present lmc is about @xmath5 in [ fe / h ] ( e.g. , van den bergh 2000 , v00 ; cole et al . 2005 ) , the above examples of low - metallicity , young scs are intriguing objects . no theoretical attempts however have been made to understand the origin of these intriguing objects in the lmc . the purpose of this letter is to show , for the first time , that the observed distinctively low metallicities in intermediate - age and young scs in the lmc can be possible evidences for accretion and infall of low - metallicity gas onto the lmc from the smc . based on dynamical simulations of the lmc - smc - galaxy interaction for the last 2.5 gyr , we investigate whether gas stripped from the smc as a result of the tidal interaction can pass through the central region of the lmc and consequently can play a role in the star formation history of the lmc . based on the results of the simulations , we discuss how the sporadic accretion / infall of metal - poor gas onto the lmc from the smc ( referred to as `` the magellanic squall '' ) can control recent star formation activities of the lmc . .examples of distinctively metal - poor stars and scs for the lmc and the inter - cloud region close to the lmc . [ cols="^,^,^,^ " , ] recent observations on stellar kinematics of old stars in the smc have suggested that the smc is _ not _ a dwarf irregular with a strongly rotating stellar disk but a dwarf spheroidal / elliptical with little rotation ( harris & zaritsky 2006 ) . the smc however has been modeled as a low - luminosity disk system in previous simulations ( g96 ) . considering the above observations , we model the smc s stellar component either as a dwarf elliptical ( de ) with a spherical shape and no rotation or as a dwarf irregular ( di ) with a disky shape and rotation in the present study . the smc s stellar ( gaseous ) component with the size of @xmath6 ( @xmath7 ) and the mass of @xmath8 ( @xmath9 ) is embedded by a massive dark matter halo with the total mass of @xmath10 set to be roughly equal to @xmath11 and the `` universal '' density distribution ( navarro , frenk & white 1996 ) . the projected density profile of the stellar component has an exponential profile with the scale length of @xmath12 for the de and the di models . @xmath6 is fixed at 1.88 kpc so that almost no stellar streams can be formed along the ms and the mb . many dwarfs are observed to have extended hi gas disks ( e.g. , ngc 6822 ; de blok & walater 2003 ) . the smc is therefore assumed to have an outer gas disk with an uniform radial distribution , @xmath13 ( @xmath14 ) , and @xmath15 ( @xmath16 ) being key parameters that determine the dynamical evolution of the gas . the rotating gas disk is represented by _ collisionless particles _ in the present simulations , firstly because we intend to understand purely tidal effects of the lmc - smc - galaxy interaction on the smc s evolution and secondly because we compare the present results with previous ones by g96 and connors et al . ( 2006 ) for which the `` gas '' was represented by collisionless particles . although we investigate models with different @xmath17 and @xmath18 , we show the results of the models with @xmath19 and 3 and @xmath20 and 4 for which the magellanic stream with a gas mass of @xmath21 can be reproduced reasonably well . the baryonic mass fraction ( @xmath22 ) thus changes according to the adopted @xmath17 . owing to the adopted @xmath20 and 4 , a very little amount of stars in the smc can be transferred into the lmc for the last 2.5 gyr . the initial spin of the smc s gas disk in a model is specified by two angles , @xmath23 and @xmath24 , where @xmath23 is the angle between the @xmath25-axis and the vector of the angular momentum of the disk and @xmath24 is the azimuthal angle measured from @xmath26-axis to the projection of the angular momentum vector of the disk onto the @xmath27 plane . although these @xmath23 and @xmath24 are also considered to be free parameters , models with limited ranges of these parameters can reproduce the ms and the mb ( e.g. , connors et al . the gas disk is assumed to have a _ negative _ metallicity gradient as the stellar components has ( e.g. , piatti et al . the gradient represented by @xmath28}_{\rm g}(r)$ ] ( dex kpc@xmath29 ) is given as ; @xmath30}_{\rm g}(r)= \alpha \times r + \beta,\ ] ] where @xmath31 ( in units of kpc ) is the distance from the center of the smc , @xmath32 , and @xmath33 . these values of @xmath34 and @xmath35 are chosen such that ( i ) the metallicity of the _ central _ region of the smc can be consistent with the observed one ( @xmath28 } \sim -0.6 $ ] ; v00 ) and ( ii ) the slope is well within the observed range of @xmath34 for very late - type , gas - rich galaxies ( zaritsky et al . if we adopt a stellar gradient ( i.e. , smaller @xmath34 ) in a model , gas particles stripped from the smc show a smaller mean metallicity . we investigate ( i ) the time ( @xmath36 ) when gas particles stripped from the smc pass through the lmc s central 7.5 kpc ( corresponding to the disk size with the scale length of 1.5 kpc , v00 ) and ( ii ) the metallicities ( [ fe / h ] ) of the particles for models with different morphological types ( de or di ) , @xmath37 , @xmath17 , @xmath18 , @xmath23 , and @xmath24 in the smc . such stripped smc s particles are referred to as `` accreted particles '' in the present study just for convenience . we also examine the mean metallicity and the mass fraction of the `` accreted particles '' ( @xmath28}_{\rm acc}$ ] and @xmath38 , respectively ) in each of the six models for which values of model parameters are shown in the table 2 . the present simulations with no gas dynamics , no star formation , and a point - mass particle for the lmc can not precisely predict how much fraction of the `` accreted particles '' can be really accreted onto the lmc s gas disk and consequently used for star formation . we however believe that the present models enables us to grasp essential ingredients of gas transfer between the mcs for the last few gyrs . we mainly show the results for the `` standard model '' ( i.e. , model 1 ) which shows typical behaviors of gas stripping in the smc . in the followings , the time @xmath39 is measured with respect to the present - time ( @xmath40 ) : for example , @xmath41 gyr means 1.5 gyr ago in the present study . fig.1 shows , for the standard model ( model 1 ) , the time evolution of the total gas mass ( stripped from the smc ) which reaches and is just located within the central 7.5 kpc of the lmc at each time step , @xmath42 . it is noted that @xmath42 is not an accumulated gas mass but is changeable with time as gas particles can pass through the lmc in the current collisionless simulation . it is clear that the @xmath42 evolution shows a number of peaks with the first peak about @xmath43 gyr ( @xmath44 ) , just after the first pericenter passage of the smc with respect to the galaxy in the 2.5 gyr evolution . the highest peak is seen at @xmath45 myr ( @xmath46 ) , when the lmc and the smc interact the most strongly . since the gas mass ( @xmath42 ) at its peak is not negligibly small compared with the present - day hi mass of the lmc ( @xmath47 ; v00 ) , accretion and infall of the gas onto the lmc s gas disk can increase local gas densities and consequently can possibly trigger star formation in the lmc . 2 demonstrates the epoch of the `` magellanic squall '' , when the stripped gas particles of the smc are falling onto the disk of the lmc . 3 shows the initial locations of the smc s gas particles ( with respect to the smc s center ) with @xmath48 myr @xmath49 and @xmath50 myr , where @xmath36 denotes the time when a particle passed through the central region of the lmc last time . the particles with @xmath51 myr are initially located in the outer part of the smc so that they can be stripped from the smc and consequently pass through the lmc earlier . owing to the small pericenter distance of the lmc - smc orbital evolution at @xmath45 myr , the smc is strongly disturbed to lose gas particles not only from its outer part but from its inner one . as a result of this , gas initially located throughout the gas disk of the smc can pass through the central region of the lmc at @xmath45 myr and thus show @xmath48 myr @xmath49 . the abovementioned differences in the initial spatial distributions between gas particles with @xmath48 myr @xmath49 and @xmath50 myr can cause the differences in metallicity distributions of the gas between the two populations , because the smc s gas disk is assumed to have a negative metallicity gradient . 4 shows that the gas particles with @xmath50 myr have a larger fraction of metal - poor gas with @xmath52[fe / h]@xmath53 and a mean metallicity of [ fe / h]@xmath54 . the particles with @xmath55 gyr has a mean metallicity of [ fe / h]@xmath56 , because they are initially located in the outermost part of the smc s gas disk . 4 also shows that the gas particles with @xmath48 myr @xmath49 have a peak around [ fe / h]@xmath57 with a mean metallicity of [ fe / h]@xmath58 . the particles with @xmath59 myr has a mean metallicity of [ fe / h]@xmath54 . these results clearly suggest that the lmc can replenish gas supplies through accretion and infall of _ metal - poor gas from the smc _ onto the lmc s disk . it should be stressed here that the metallicities of accreted gas from the smc at @xmath59 myr can be appreciably higher than the above , if we consider chemical evolution of the smc due to star formation for the last 2.5 gyr . 5 shows that relative velocities ( @xmath60 ) of the smc s gas particles within the central 7.5 kpc of the lmc with respect to the lmc velocity range from 40 to 150 km s@xmath29 at @xmath45 myr . this result indicates that if the particles can infall onto the lmc s disk , they can give strong dynamical impact on the hi gas of the lmc and possibly cause shock energy dissipation owing to @xmath60 much higher than the sound velocities of cold gas . previous numerical simulations showed that cloud - cloud collisions with moderately high relative velocities ( @xmath61 km s@xmath29 ) can trigger the formation of scs ( bekki et al . therefore the above result implies that some fraction of the particles passing through the lmc s central region can be responsible for the formation of new scs in the lmc . the parameter dependences of @xmath28}_{\rm acc}$ ] and @xmath38 are briefly summarized as follows . firstly @xmath28}_{\rm acc}$ ] and @xmath38 do not depend so strongly on baryonic fractions , gas mass fractions , and orbital configurations ( see the table 2 ) : @xmath28}_{\rm acc}$ ] ( @xmath38 ) ranges from @xmath62 ( 0.34 ) to @xmath63 ( 0.47 ) for a fixed size ratio of @xmath18 ( @xmath64 ) . secondly , @xmath28}_{\rm acc}$ ] and @xmath38 are _ both larger _ in the model with smaller @xmath18 ( model 6 ) for which a smaller amount of gas particles can be tidally stripped from the smc . the reason for the larger @xmath38 is that a significantly larger fraction of particles once stripped from the smc can pass through the lmc in model 6 . thirdly , the morphological type of the smc in the present study is not important for @xmath28}_{\rm acc}$ ] and @xmath38 . given the fact that only 0.2% of gas can be converted into strongly bound scs ( rather than into field stars ) in the evolution of the mcs ( b05 ) , these results imply that the maximum possible mass of scs formed from smc s gas in the lmc is roughly @xmath65 in the present models . owing to the very short time scale ( @xmath66 yr ) of sc formation from gas clouds during the tidal interaction ( bekki et al . 2004b ) , the stripped smc s gas clouds are highly likely to be accreted onto the lmc within the dynamical time scale of the lmc ( @xmath67 yr ) and then converted into scs within @xmath66 yr after the accretion . ngc 1718 with an estimated age of @xmath2 gyr has a low metallicity ( [ fe / h]@xmath68 ) about 0.3 dex smaller than those of other scs with similar ages in the lmc ( e.g. , g03 ; g06 ) . if the interstellar medium ( ism ) of the lmc about @xmath69 gyr ago was very inhomogeneous in terms of chemical abundances , some fraction of stars could be born from quite low - metallicity gas clouds with [ fe / h]@xmath68 . the distinctively low metallicity therefore could be due to the abundance inhomogeneity of the ism in the lmc about @xmath69 gyr ago . however , intermediate - age scs _ other than _ ngc 1718 have very similar metallicities of [ fe / h]@xmath70 and a small metallicity dispersion of only 0.09 dex in the lmc ( g06 ) . the observed low - metallicity of ngc 1718 thus seems to be unlikely to be due to the abundance inhomogeneity of the ism . we suggest that the origin of the ngc 1718 can be closely associated with the magellanic squall about @xmath69 gyr ago . since gaseous abundance patterns ( e.g. , [ mg / fe ] ) of the smc about a few gyr ago might well be very different from those of the lmc , ngc 1718 could have abundance patterns quite different from those of other gcs . it should be here stressed that the simulated peak of the squall ( @xmath71 gyr ) is not very consistent with the observed age of ngc 1718 ( @xmath72 gyr , g03 ) . s04 recently have reported that eight young scs with ages less than 200 myr have metallicities smaller than [ fe / h]@xmath73 that is a typical stellar metallicity of the lmc ( e.g. , v00 ) . although there could be some observational uncertainties in age and metallicity determination based solely on integrated spectrophotometric properties of scs ( s04 ) , their results imply that these scs could have been formed from metal - poor gas in the lmc quite recently . the present numerical results imply that ngc 1711 , ngc 1831 , ngc 1866 , and ngc 1984 , all of which are observed to have possible metallicities smaller than [ fe / h]@xmath74 , can be formed as a result of the magellanic squall . since the chemical abundances of the outer gas disk of the smc can be significantly different from those of the present lmc s gas disk , the detailed abundances ( e.g. , [ c / fe ] , [ n / fe ] , and [ mg / fe ] ) of the above four clusters can be significantly different from those of other young scs with `` normal '' metallicities with [ fe / h]@xmath75 in the lmc . the observed young , metal - poor stars ( [ fe / h]@xmath76 ) in the inter - cloud region close to the lmc ( r99 ) will be equally explained by the gas - transfer between the mcs ( see also bekki & chiba 2007 ) . the present study has first pointed out that the magellanic squall can also play a role in the relatively recent star formation history of the lmc . sporadic infall of metal - poor gas like the magellanic squall might well be also important for recent star formation histories in pairs of interacting galaxies . previous hydrodynamical simulations showed that high - velocity collisions of hi gas onto a galactic disk can create hi holes and shells ( e.g. , tenorio - tagle et al . the magellanic squall , which inevitably can cause high - velocity impact of the gas clouds stripped from the smc on the lmc , can thus be responsible for _ some _ of the observed hi holes in the lmc ( e.g. , staveley - smith et al . we plan to investigate how collisions between low - metallicity gas clouds from the smc and those initially in the lmc trigger the formation of stars and scs in the lmc s disk based on more sophisticated , high - resolution hydrodynamical simulations with pc - scale star formation processes . our future studies thus will enable us to understand more deeply how the magellanic squall influences pc - scale star formation processes in the lmc . we are grateful to the referee , daisuke kawata , for his valuable comments , which contribute to improve the present paper . k.b . acknowledges the large australian research council ( arc ) . numerical computations reported here were carried out on grape system kindly made available by the astronomical data analysis center ( adac ) of the national astronomical observatory of japan .

we first show that a large amount of metal - poor gas is stripped from the small magellanic cloud ( smc ) and fallen into the large magellanic cloud ( lmc ) during the tidal interaction between the smc , the lmc , and the galaxy over the last 2 gyrs . we propose that this metal - poor gas can closely be associated with the origin of lmc s young and intermediate - age stars and star clusters with distinctively low - metallicities with [ fe / h ] @xmath0 . we numerically investigate whether gas initially in the outer part of the smc s gas disk can be stripped during the lmc - smc - galaxy interaction and consequently can pass through the central region ( @xmath1 kpc ) of the lmc . we find that about 0.7 % and 18 % of the smc s gas can pass through the central region of the lmc about 1.3 gyr ago and 0.2 gyr ago , respectively . the possible mean metallicity of the replenished gas from the smc to lmc is about [ fe / h ] = -0.9 to -1.0 for the two interacting phases . these results imply that the lmc can temporarily replenish gas supplies through the sporadic accretion and infall of metal - poor gas from the smc . these furthermore imply that if these gas from the smc can collide with gas in the lmc to form new stars in the lmc , the metallicities of the stars can be significantly lower than those of stars formed from gas initially within the lmc . [ firstpage ] magellanic clouds galaxies : structure galaxies : kinematics and dynamics galaxies : halos galaxies : star clusters

neptune s triton is the only large planetary satellite to orbit retrograde relative to the planet s rotation . @xcite and later @xcite suggested that triton is a captured satellite , whose originally eccentric orbit was circularized due to tidal dissipation within triton . @xcite postulate that triton was captured from heliocentric orbit by a collision with a pre - existing satellite , and its initial high - eccentricity orbit then evolved due to tidal dissipation alone . they showed that the tidal evolution timescale is significantly shorter than the age of the solar system ( a few times @xmath2 years ) , even when the variations in triton s eccentricity with @xmath3-yr period are accounted for . however , @xcite noted that the criteria for non - disruptive capture are much stricter than @xcite calculated . if the original regular satellites of neptune and uranus were similar , a collision with the largest moons ( preferred due to their cross - sections ) would disrupt triton , with re - accretion on an orbit inclined to neptune s equator being impossible . @xcite suggested instead that triton was captured and its orbit was evolved by aerodynamic drag in neptune s primordial protosatellite nebula , and that after its orbit ewas circularized triton s gravity would be sufficient to clear a disk gap and thus halt further evolution . gas drag has been suspected as a capture mechanism for small and distant irregular satellites , but capture of triton would require unprecedented gas densities , requiring very close approaches to neptune . @xcite propose a three - body capture scenario for triton . they suggest pre - capture triton may have been a member of a binary , whose disruption during a neptune encounter led to triton s capture and its companion s escape . their work addresses only the capture itself , leaving the physics of post - capture evolution unchanged . any capture mechanism , be it collision , gas drag or 3-body interaction , is likely to introduce triton on a large , highly eccentric orbit . distant satellite orbits are perturbed primarily by the sun , inducing precession of angular variables and oscillations in eccentricity and inclination , with minor semimajor axis variations . the two important periodic perturbations are those associated with @xmath3 of the planet s orbital period ( `` evection '' ) and @xmath3 of the precession period of the argument of pericenter @xmath4 ( `` kozai behavior '' ) . for early triton , evection was first noted by @xcite ; conservation of orbital angular momentum during tidal evoution implies initial pericenter of 7 neptune radii ( @xmath5 ; triton s present orbit has @xmath6 ) , and evection - induced oscillations in @xmath7 produced minimum pericenters of @xmath8 . however , @xcite ignored kozai oscillations in @xmath7 , which must have been present if triton s inclination @xmath9 at closest approach was same as now ( @xmath10 , measured with respect to neptune s equator ) . kozai osillations require that when @xmath11 or @xmath12 , both @xmath7 and @xmath13 are at a maximum ( @xmath7 and @xmath9 oscillate in phase for retrograde satellites ) . since almost all tidal dissipation occurs during this high-@xmath7 phase of the kozai cycle ( when the pericenter distance is smallest ) , this @xmath14 inclination will be conserved as the maximum one for the kozai cycle , while the minimum one ( coinciding with @xmath15 and @xmath16 ) will be affected by dissipation . using more complete treatment of tides , @xcite show that tides raised on triton could not have led to a significant inclination change as long as it was in cassini state 1 . trapping in the much less likely cassini state 2 would have caused a rapid increase in its inclination ( i.e. , closer to @xmath16 ) , rather than a decrease @xcite , so triton s inclination relative to the local laplace plane was always @xmath17 . this assumes orbital evolution slower than the nodal precession , preserving inclination relative to the local laplace plane , which would initially be close to neptune s orbital plane but would subsequently move closer to neptune s equator . this condition is true for the tidal model @xcite but not the gas - drag model @xcite . [ peric ] presents two illustrative short orbital evolutions . using initial conditions of @xmath18 km@xmath19 , and varying inclinations , we re - create two possible post - capture orbits for triton . both evection ( @xmath20 yrs ) and kozai oscillations ( @xmath21 yrs ) are clearly visible in the evolution of the inclined orbit , while the @xmath22 case , shows only evection - related oscillations ( whose amplitudes are in agreement with results from @xcite fig . 2 for @xmath23 ) . however , if tidal dissipation alone evolved triton s orbit , only the inclined case can lead to present inclination of @xmath24 . this conclusion points to a paradox . @xcite modelled the tidal orbital evolution with the standard relation : @xmath25 where @xmath26 and @xmath27 are respectively the semimajor axis , time , pericenter distance , mean motion , tidal love number and tidal disspation factor , and @xmath28 km is triton s radius . we numerically averaged eq . [ goldreich ] over several full kozai periods , based on the output of the integration shown in fig . [ peric ] . using @xmath29 and @xmath30 for triton @xcite , the resulting timescale for @xmath31 reduction in @xmath32 is 3.5 gyr for the inclined orbit and 0.11 gyr for the near - coplanar orbit ( cf . goldreich et al . the requirement of an inclined orbit means that tides alone are not capable of circularizing triton s orbit . neptune s satellite system predating triton s capture was likely similar to that of uranus , as the planets are virtual twins in many important characteristics . uranus possesses five sizeable satellites at @xmath33 , with total mass @xmath3440% of triton s . it is likely that triton s capture led to destruction or ejection of any existing regular satellites outside @xmath8 . to explore the fate of these moons , we integrated five massless particles under the influence of triton on the @xmath13 orbit of fig . [ peric ] . the innermost satellite was put at @xmath35 , and the others on @xmath36 ; all the orbits were circular and in the plane of neptune s equator ( using current values of neptune s obliquity and @xmath37 ) . we found that the orbital crossing of the pairs 4 - 5 and 3 - 4 only a few centuries into the simulation . to calculate the collisonal timescale , we assigned our three outermost particles the radii of umbriel , titania and oberon , and we took the satellites to be uniformly distibuted over all radii @xmath38 , and latitudes @xmath39 . under these ( admittedly rough ) approximations , we estimated the collision probability for every output interval in our simulation and kept track of the cumulative collision probability for each pair . in both cases , the cumulative probability reaches unity at about 1000 years . while our approximations likely underestimate collision timescales , a more sophisticated method would not raise the result more than a factor of a few . due to its long orbital period ( about 7 yrs ) , most of it spent far from neptune , eccentric triton would be unlikely to collide with any of the satellites on such short timescales . the collision would likely destroy both moons and create a debris ring around neptune . this debris would rapidly disrupt the remaining regular satellites , grinding them down to small pieces incapable of destroying triton . the resulting massive disk out to about @xmath40 would interact strongly with triton , whose perturbations would prevent re - accretion of the disk particles from taking place ( cf . banfield & murray , 1992 ) . the mutual interaction of the retrograde triton and prograde disk , either through tidal torques or direct collisions , would cause decay of triton s angular momentum ; here we focus on the latter . to find the approximate evolution timescale of triton s orbit due to the debris disk , we modeled the disk as having uniform surface density and extending out to @xmath40 , and then integrated triton s orbit giving it a instantaneous `` kick '' every time it goes through neptune s equatorial plane within @xmath40 . like @xcite , we find that triton s lifetime depends largely on its inclination : inclined orbits survive much longer than coplanar ones . our timescales for the decay of orbits with @xmath13 at @xmath11 are on the order of a few times @xmath0 yrs , while the orbit with @xmath41 at @xmath11 would have an @xmath7-folding time of a few times @xmath42 yrs . while our disk model and resolution of the passages are crude , the robust conclusion is that the debris - drag timescale for orbital evolution is much shorter than the tidal one . although our disk is much less massive than that of @xcite ( who use a minimum mass nebula ) triton can still evolve quickly due to deeper penetration into the disk . finally , since the disk is less massive than triton , there is a natural end to this stage of the evolution , so there is no need for specific mechanisms ensuring triton s survival . if triton s evolution had been this fast ( @xmath43 yrs ) , then the assumption of conservation of inclination relative to local laplace plane is incorrect , as the nodal precession timescale ( at the boundary between an inner region dominated by neptune s oblatneness and an outer region dominated by the sun ) is longer than @xmath0 yrs ( the transition distance depends on @xmath37 as the fifth root , making the influence of the additional material from the debris disk modest ) . therefore , the pre - transition inclination of triton could have been anything from @xmath44 to @xmath45 if the evolution was fast . because passages through the disk would at that point ( @xmath46 ) happen for a wide range of values for @xmath4 ( and not only @xmath47 when @xmath7 is high ) , the constraints on the initial inclination are very weak . nereid and the other five now - known irregular satellites of neptune @xcite can help us constrain triton s post - capture orbit . during its early orbital history , triton s orbit would intersect those of the irregulars , making gravitational scattering possible . there is reason to think that neptune s irregulars had their orbits modified after capture : almost all irregular satellites of the other three giant planets have pericenters @xmath48 around 100 and 200 planetary radii , in case of direct and retrograde satellites , respectively . four outer neptunians do not approach the planet closer than about @xmath49 , while s/2002 n1 and nereid come within about @xmath50 ; the latter two objects will be addressed in the next section . can large-@xmath48 irregulars result from triton s passage ? [ lt ] shows six simulations using a symplectic integrator that includes triton evolving under the influence of the disk while interacting with a swarm of test particles representing `` original '' irregulars . the 2044 test particles were originally put at @xmath51 , @xmath52 , @xmath11 and with fixed @xmath53 ( for direct ) or @xmath54 ( retrograde orbits ) . triton s initial conditions were varied ( see fig . [ lt ] caption ) . the curves in fig . [ lt ] give the fraction of surviving bodies over the 1 myr integrations , normalized to the number of survivors when triton is not present ( some particles are inherently unstable ) . fig . [ lt ] shows that a significant number of irregulars survives triton s passage only if the duration of the interaction is below @xmath55 yrs . in such cases , triton s apocenter drops below the inner edge of the irregular population , and the depletion stops . in these models such rapid evolution occurs only for @xmath56 , for then triton s pericenter is almost continuously within the disk . an orbit with this maximum inclination will occur for a large fraction of @xmath57 post - capture orbits . also , requirement for @xmath58 yrs practically excludes tidal evolution _ independently _ from arguments in section 2 . [ aqplot ] compares the semimajor axes and pericenters of nereid and the five other irregulars with the surviving test particles in the simulation represented by the topmost curve in fig . the scattered particles bridge the gap between initial conditions corresponding to other planetary irregulars and neptune s . however , the mechanism leaves many satellites at @xmath59 . interestingly , nereid s apocenter distance seems to separate the bulk of the surviving particles and the observed moons . to test the importance of scattering by nereid on the irregulars , we ran two simulations similar to the ones already described , except that now non - evolving nereid was the scatterer and the simulations lasted @xmath60 and @xmath61 years , respectively . there was a noticeable effect on the stability of particles , and some particles in the center of the `` stable zone '' were observed to escape . nereid s escape speed ( @xmath62 at apocenter ) apparently makes it an efficient scatterer , perhaps allowing the moons on orbits permanently out of nereid s reach to dominate the surviving population . nereid orbits between 55 and 385@xmath5 , and has been previously suggested to be a regular satellite scattered to its current orbit by triton . we report preliminary findings on the secular orbital evolution of nereid - like objects just after the end of close encounters with triton . by `` nereid - like '' we mean that the precession is dominated by triton , rather than the sun . the radius of the zone within which the planetary oblateness dominates solar perturbations is @xmath63^{1/5}$ ] , where @xmath64 and @xmath65 are the masses of neptune and the sun , and @xmath66 is the planet s semimajor axis . we estimate the importance of triton by putting @xmath67 , where subscript @xmath68 refers to triton . if we take @xmath69 , we find @xmath70 km . therefore , `` nereid - type '' objects are those orbiting within about 7% of neptune s hill sphere ( for nereid , @xmath71 km@xmath72 ) . the most important secular interaction between two non - intersecting orbits is through the quadrupole term in the interaction hamiltonian ( i.e. one containing @xmath73 ; subscripts 1 and 2 refer to inner and outer body , respectively ) . the disturbing function for the outer body is @xcite : @xmath74.\ ] ] where @xmath75 is the outer body s semiminor axis , and @xmath76 and @xmath77 are those of triton . if we assume triton s orbit precesses much more slowly ( due to neptune s @xmath37 ) than the outer body s ( due to triton and the sun ) , we can replace @xmath77 with @xmath78 , the longitude of the ascending node of the outer body , measured from triton s pericenter , in triton s orbital plane ( the angles are geometrically identical ) . the expression above has no dependence on @xmath79 , so @xmath80 . using lagrange s eqations , expressions for the evolution of @xmath81 and @xmath78 obey : @xmath82,\ ] ] @xmath83 where @xmath84 . putting @xmath85 ( which puts triton close to intersection with nereid , assuming current pericenters ) , we numerically integrated eqs . [ domdt ] and [ didt ] for initial conditions of @xmath86 and @xmath87 ( fig . [ ioplot ] ) . suprisingly , objects with @xmath88 can have nodes librating around @xmath89 ( or @xmath12 , as the results for @xmath90 are symmetric ) , and their inclination ( relative to triton ) can change from direct to retrograde . we tested this model using a numerical integration , in which triton had its current @xmath9 and @xmath48 but @xmath85 . test particles were placed outside its apocenter , on @xmath91 orbits in neptune s equator plane . several particles show librating @xmath78 and repeatedly change their sense of revolution ( fig . [ ioplot ] ) . inclinations in fig . [ ioplot ] are relative to triton s orbit , which has @xmath13 relative to neptune s equator , which itself has an obliquity of @xmath92 relative to the planet s orbit . objects in libration island can end up with a wide range of inclinations relative to the sun . in particular , bodies starting within the equatorial plane of neptune can be shuffled through the libration island to both direct and retrograde orbits . one recently discovered irregular , s/2002 n1 , has @xmath93 km , @xmath94 km , and @xmath95 @xcite . @xcite report that n1 s colors are similar to that of nereid and suggest that they are members of a collisional family . this is implausible if n1 was produced at the present epoch , but is consistent with both objects being pieces of circumplanetary debris ejected by triton and mixed by this process . we propose that , after its initial capture by neptune ( via any process ) , triton strongly pertubed all of the pre - existing satellites , both regular and irregular . regular satellites rapidly collide with each other , creating a debris disk to be subsequently swept up swept up by triton . the debris drag evolved triton s orbit rapidly enough ( @xmath96 yrs ) to preserve some of the irregular satellites , scattering some to high @xmath48 . bodies in the inner part of the hill sphere later suffered triton s secular perturbations , which in some cases `` flipped '' their inclination between direct and retrograde . the closest irregular satellites were subsequently depleted through encounters with nereid , producing the distribution observed today . agnor , c. b. , & hamilton d. p. 2004 , , 36 , 1169 banfield , d , & murray , n. 1992 , , 99 , 390 benner , l. a. m. , & mckinnon , w. b. 1994 , , 114 , 1 chyba , c. f. , jankowski , d. g. , & p. d. nicholson 1989 , , 219 , l23 goldreich , p. , murray , n. , longaretti , p. y. , & banfield , d. 1989 , science , 245 , 500 grav , t. , holman , m. j. , & fraser , w. c. 2004 , , 613 , l77 holman , m. j. et al . 2004 , , 430 , 865 innanen , k. a. , zheng , j. q. , mikkola , s. , & valtonen , m. j. 1997 , , 113 , 1915 jankowski , d. g. , chyba , c. f. , & nicholson , p. d. 1989 , , 80 , 211 mccord , t. b. 1966 , , 71 , 585 mckinnon , w. b. 1984 , , 311 , 355 mckinnon , w. b. , & leith , a. c. 1995 , , 118 , 392 mckinnon , w. b. , lunine , j. i. , & banfield , d. 1995 , neptune and triton , d. p. cruikshank , tucson : u. of arizona press , 807

we present simulations of triton s post - capture orbit that confirm the importance of kozai - type oscillations in its orbital elements . in the context of the tidal orbital evolution model , these variations require average pericenter distances much higher than previously published , and the timescale for the tidal orbital evolution of triton becomes longer than the age of the solar system . recently - discovered irregular satellites present a new constraint on triton s orbital history . our numerical integrations of test particles indicate a timescale for triton s orbital evolution to be less than @xmath0 yrs for a reasonable number of distant satellites to survive triton s passage . this timescale is inconsistent with the exclusively tidal evolution ( time scale of @xmath1 yrs ) , but consistent with the interestion with the debris from satellite - satellite collisions . any major regular satellites will quickly collide among themselves after being perturbed by triton , and the resulting debris disk would eventually be swept up by triton ; given that the total mass of the uranian satellite system is 40% of that of triton , large scale evolution is possible . this scenario could have followed either collisional or the recently - discussed three - body - interaction - based capture .

at the end of the stellar evolution on the asymptotic giant branch ( agb ) stars loose copious amounts of mass , which build up a circumstellar dust and gas shell hiding the star from optical view almost completely . stars departing from the agb and evolving towards the planetary nebula ( pn ) phase are therefore difficult to observe optically . it was found that a number of proto - planetary nebulae ( cf . in crl 2688 ; @xcite @xcite ) show high velocity bipolar outflows which are connected to a fast , axially - symmetric wind , which is taking the place of the much slower , spherically - symmetric wind operating on the agb . the physical mechanism responsible for the change of the spherically - symmetric to an axially - symmetric , or in some cases point - symmetric wind is strongly debated . observations of masers in transition objects often reveal that this morphological change takes place at a very early stage in the post - agb phase ( @xcite @xcite ; @xcite @xcite ) , while the star is still heavily obscured in the optical range . non - variable oh / ir stars @xcite and iras selected infrared sources with extreme red colors @xcite are candidates for such hidden post - agb stars . their study has made progress only in the last decade due to the improved observation capabilities in the infrared at @xmath4 m by space - based observatories . in the mid - infrared the emission emerges from the circumstellar envelopes ( cse ) and their gas and dust composition has to be used to infer on the evolutionary state of the underlying star and the mass loss process . the spectroscopic observations with iso showed that strong changes occur in the infrared seds during total obscuration @xcite . in the case of the c - rich agb stars the molecular c@xmath1h@xmath1 absorption and the amorphous sic emission feature at 11.3@xmath5 m suddenly disappear and become substituted by a broad plateau of emission from 11 to 15@xmath5 m due to hydrogenated pahs . these are later replaced by de - hydrogenated , narrow pah features at 3.3 , 6.2 , 7.7 , 8.6 and 11.3@xmath5 m , which are also observed in more evolved c - rich pne . in o - rich agb stars the strong silicate absorption features at 9.7 and 18@xmath5 m disappear and are replaced by several prominent crystalline silicate emission features in the @xmath6 m wavelength range . a mixed chemistry is found also in a few sources , but it is not clear whether it is associated to late thermal pulses at the end of the agb phase and/or to the preservation of o - rich material in long - lived circumstellar disks . globally considered , there seems to be a continuous evolution from an amorphous ( aliphatic ) to crystalline ( aromatic ) organization of molecules in the dust grains both in the c - rich and the o - rich sequence , which is still unexplained @xcite . the akari satellite @xcite and the spitzer space telescope @xcite offered the possibility to extend the iso observations to larger and better selected samples of hidden post - agb stars . observations between 2 and 26@xmath5 m were possible with the infrared camera ( irc ) @xcite on board of akari , and in the range @xmath7 m with the infrared spectrograph ( irs ) @xcite on board of spitzer . a first sample studied consisted of obscured oh / ir sources with associated radio continuum emission . the spitzer spectra allowed a re - classification of the sources in agb stars and post - agb stars close to the formation of pns @xcite . the new samples observed , consisted of extremely red iras sources from the glmp catalog @xcite , and of oh / ir stars selected on the base of their appearance in the spitzer glimpse survey . the 2mass and glimpse surveys were used to identify oh / ir stars with near - infrared excesses indicative for a post - agb nature of these sources @xcite . the seds of obscured variable oh / ir stars peak in the wavelength range @xmath8 m and show a strong 10@xmath5 m and a weaker 18@xmath5 m absorption feature . these seds can be modeled in detail using cold dust opacity functions of amorphous silicates @xcite . this is confirmed by the results we obtained from the modeling of the akari spectra of the infrared sources classified as agb stars ( bunzel et al . , these proceedings ) . the carbon - rich cousins of oh / ir stars are the extreme carbon stars ( extreme in terms of infrared color ) . the dust features seen in their seds are usually weak , but they often show a molecular absorption line at @xmath9 m attributed to c@xmath1h@xmath1 . the extreme carbon stars are rarer than oh / ir stars and harder to classify because of the lack of prominent dust features and radio maser emission . before akari , the most extreme carbon stars were studied by @xcite , who modeled the seds successfully with amorphous carbon dust . they found the evolutionary status compatible with the end phase of agb evolution . the extreme carbon stars , we identified among the infrared sources observed with akari , are the reddest found so far . the spectra of all of them ( except iras15408 - 5657 ) could be modeled with amorphous carbon dust ( @xmath10 ) , with minor contributions of sic and silicates . because of a low iras variability index we suspect that part of them could have started post - agb evolution already ( garca - hernndez et al . , these proceedings ) . iras15408 - 5657 is a peculiar source , in the sense that its silicate absorption features are too weak for its red continuum . the model sed required a mixture of carbon and silicate dust in almost equal parts to obtain the appropriate strength of the silicate band . its low iras variability index makes it a post - agb candidate . it is unlikely that both dust species spatially coexist , because the underabundant atomic species ( c or o ) should be locked in co , and would not be available for dust formation @xcite . thus , the mixed chemistry may indicate the presence of two shells , an inner shell with c - rich dust and an outer one with o - rich dust . for several sources with silicate absorption features observed by bunzel et al . with akari we had indications for their post - agb nature beforehand . either due to the presence of bipolar high - velocity outflows traced by the h@xmath1o masers ( iras19134 + 2131 , @xcite @xcite ; oh31.0 + 0.0 = w43a , @xcite @xcite ) , or due to the presence of a near - infrared excess . as for iras15408 - 5657 the seds of these sources could not be modeled by pure amorphous silicate dust , but required a model , where the inner carbon - rich shell is viewed through an outer shell containing 20 - 40% silicate dust . the results for these post - agb stars and for iras15408 - 5657 indicate that for oxygen - rich agb stars the departure from the agb marks also a change in dust chemistry . carbon - rich dust forms in the inner shell , while the silicate - rich dust shell formed on the agb expands outwards . a preliminary evaluation of the spitzer spectra in the @xmath11 m range and the @xmath0 m seds of 88 iras sources from the glmp catalog confirm the akari results and show that the mid - ir spectra are even more diversified than expected . judged from the iras variability index almost all these sources have a chance of @xmath12% to be variable and are therefore currently post - agb stars . based on the spectra and the @xmath0 m seds they can be divided into several groups : * about 45% have strong silicate absorption features and are heavily obscured in the near - infrared . these are former o - rich agb stars , where the remnant agb shell dominate the mid - ir spectra . * another 15% show the combination of a very red continuum longward of @xmath13 m and a weak silicate absorption . about two - third of them have a near - infrared excess at @xmath14 m as it is exemplified in iras18355@xmath150712 ( fig . [ fig : sp18355 ] ) . these sources are similar to the akari observed post - agb stars , which required a mixed chemistry to model their seds . * three sources show evidence for the presence of crystalline silicate dust judged from sharp absorption features in the @xmath16 m region . all of them are oh / ir stars with a relatively blue continuum ( see fig . [ fig : sp18470]a ) . * another 20% show featureless spectra ( see iras11444 - 6150 in fig . [ fig : sp18470]b ) , spectra with c@xmath1h@xmath1 absorption at 13.7@xmath5 m or weak carbon dust features . almost half of them show a near - infrared excess as in iras18355@xmath150712 . these objects are probably post - agb carbon stars with varying degrees of optical depths of their remnant agb shells and where the objects with a near - infrared excess are the more evolved . * the remaining objects have ( in part strong ) near - infrared excesses and show a variety of carbon dust features in their spectra . an example is iras19176 + 1251 , which shows strong pah features and in addition a ne 12.8@xmath5 m emission line , coming probably from material shocked by the stellar wind ( fig . [ fig : sp19176 ] ) . these objects are considered as the most advanced in their post - agb evolution . post - agb evolution starts when almost the complete stellar envelope has been lost by the stellar wind , and the stars are still hidden by their circumstellar envelope . such stars do not show the agb typical long - period variability anymore . the akari and spitzer spectra of such hidden post - agb stars indicate that the inner part of the cses contains carbon - rich dust formed in the post - agb wind irrespective of the chemistry of the star on the agb . the silicate absorption features seen in many of the sources may originate from the outer cse , which is composed mainly by dust of the remnant agb shell . these conclusions can be probed in those stars , where the remnant agb shell has been diluted far enough , that observations of the warm dust near the star are possible . spectroscopy in the @xmath17 m range using the irc during the akari warm phase will therefore be made , to search for c - rich matter in the inner dust shell of those post - agb stars showing the @xmath16 m silicate absorption and a strong near - infrared excess . this research is based on observations with akari , a jaxa project with the participation of esa , and on observations with spitzer , a nasa s great observatories program . d. engels acknowledges travel support by the conference organizers . engels , d. 2002 , , 388 , 252 engels , d. 2007 , in _ asymmetrical planetary nebulae iv _ , eds . corradi et al . , ( in press ) garca - lario , p. 1992 , _ ph.d . thesis _ la laguna , spain garca - hernndez , d.a . , perea - caldern , j.v . , bobrowsky , m. , garca - lario , p. 2007 , , 666 , l33 garca - lario , p. , perea caldern , j.v . 2003 , in _ exploiting the iso data archive . infrared astronomy in the internet age _ , esasp-511 p. 97 garca - lario , p. 2006 , in _ planetary nebulae in our galaxy and beyond _ , iau - symp . 234 , p. 63 houck , j. , roellig , t. , van cleve , j. , et al . 2004 , , 154 , 19 imai , h. , obara , k. , diamond , p.j . , omodaka , t. sasao , t. 2002 , , 417 , 829 imai , h. , morris , m. , sahai , r. , hachisuka , k. , azzollini , f.j.r . 2004 , , 420 , 265 ivezi , z. , elitzur , m. 1995 , , 445 , 415 murakami , h. , baba , h. , barthel , p. , et al . 2007 , , 59 , s369 onaka , t. , matsuhara , h. , wada , t. , et al . 2007 , , 59 , s401 sahai , r. , trauger , j.t . , watson , a.m. , et al . 1998 , , 493 , 301 sahai , r. , bujarrabal , v. , zijlstra , a. 1999 , , 518 , l115 surez , o. , garca - lario , p. , manchado , a. , et al . 2006 , , 458 , 173 suh , k. , kim , h .- y . 2002 , , 391 , 665 volk , k. , xiong , g. , kwok , s. 2000 , , 530 , 408 werner , m. , roellig , t. , low , f. , et al . 2004 , , 154 , 1 zijlstra , a.a . , chapman , j.m . , te lintel hekkert , p. , et al . 2001 , , 322 , 280 * answer : * not yet . only with glimpse covering the @xmath18 m wavelength range it was possible to verify that the 2mass counterparts found for hidden post - agb stars at @xmath19 m are not field stars . objects with confirmed near - infrared excess are currently observed by akari between 2 and @xmath20 m .

the akari and spitzer satellites provided an unique opportunity to observe a variety of stars , which are considered as departing from the asymptotic giant branch ( agb ) and have started their post - agb evolution recently . most of these stars are absent optically and are bright in the mid - ir wavelength range . spectra of close to 200 objects have been obtained . for all of them the @xmath0 m spectral energy distribution has been constructed using photometric data from various surveys . we report here on the results of spitzer observations of 88 iras selected post - agb candidates and discuss them in comparison to the results of the akari observations of post - agb candidates reported elsewhere in these proceedings . the dust compositions can be divided broadly in oxygen- and carbon - rich types , but a variety of intermediate types have been found . among the oxygen - rich stars amorphous dust prevails , but a few sources show emission features from crystalline dust . the spectra from carbon - rich shells may be completely featureless , may show emission features from pahs or a molecular absorption line from c@xmath1h@xmath1 . we found also sources with a neon emission line at @xmath2 m . more than a third of all sources show a near - infrared excess at @xmath3 m and almost all of them show evidence of c - rich dust in their shells . we postulate that the emerging post - agb wind after the end of agb evolution contains always carbon - rich dust irrespective of the chemistry of the former agb star .

the aim of statistical mechanics is to obtain a qualitative understanding of natural phenomena of phase transitions by the study of simplified models , often built on a lattices . in general the hamiltonian of a model of statistical mechanics is left invariant by the lattice symmetries : a prototypical example being the ising model describing a ferromagnet . however , one might argue that materials which are found in nature are usually not completely homogeneous and for this reason , physicists where led to considering systems in which the interaction terms , for example the potentials between nearest neighbor spins , are chosen by sampling a random field which we call _ disorder _ with good ergodic properties , often even a field of independent identically distributed random variables . an important question which arises is thus whether the results concerning the phase transition obtained for a model with homogeneous interactions referred to as _ the pure system _ ( e.g. the onsager solution of the two dimensional ising model ) remain valid when a system where randomness of a very small amplitude is introduced . in @xcite a. b. harris , gave a strikingly simple heuristical argument , based on renormalization theory consideration , to predict the effect of the introduction of a small amount of the system : in substance harris criterion predict that if the phase transition of the pure system is sufficiently smooth , it will not be affected by small perturbation ( disorder is then said to be _ irrelevant _ ) , while in the other cases the behavior of the system is affected by an arbitrary small addition of randomness ( disorder is _ relevant _ ) . to be complete , let us mention also the existence of a boundary case for which the criterion yields no prediction ( the _ marginal disorder _ case ) . the criterion however does not give a precise prediction concerning the nature of the phase transition when the disorder is relevant . the mathematical verification of the harris criterion is a very challenging task in general . in the first place , it can only be considered for the few special models of statistical mechanics for which we have a rigorous understanding of the critical properties of the pure system . in the last twenty years this question has been addressed , first by theoretical physicists ( see e.g. @xcite and references therein ) and then by mathematicians @xcite ( see also @xcite for reviews ) , for a simple model of a 1-dimensional interface interacting with a substrate : for this model the interface is given by the graph of a random walk which takes random energy rewards when it touches a defect line . in this case , the pure system has the remarkable quality of being what physicists call _ exactly solvable _ , meaning that there exists an explicit expression for the free energy @xcite . this model under consideration in the present paper can be seen as a high - dimension generalization of the rw pinning model . the random walk is replaced by a random field @xmath10 , and the random energies are collected when the graph of the field is close to the hyper - plane @xmath11 . while the pure model is not exactly solvable in that case , it has been studied in details and the nature of the phase transition is well known @xcite . on the other hand , the study of the disordered version of the model is much more recent @xcite . in @xcite , we gave a close to complete description of the free energy diagram of the disordered model when @xmath8 : * we identified the value of the disordered critical point , which is shown to coincide with that of the associated annealed model , regardless of the amplitude of disorder . * we proved that for gaussian disorder , the behavior of the free energy close to @xmath2 is quadratic , in contrasts with the annealed model for which the transitition is of first order . * in case of general disorder , we proved that the quadratic upper - bound still holds , and found a polynomial lower bound with a different exponent . let us stress that the heuristic of our proof strongly suggests that the behavior of the free energy should be quadratic for a suitable large class of environments ( those who satisfy a second moment assumption similar to ) . in the present paper , we choose to attack the case @xmath12 , for which only limited results were obtained so far . we have seen in the proof of the main result @xcite that the critical behavior of the model is very much related to the extremal process of the field . the quadratic behavior of the free - energy in ( * ? ? ? * theorem 2.2 ) comes from the fact that high level sets of the gaussian free field for @xmath8 look like a uniformly random set with a fixed density ( see @xcite ) . in dimension @xmath13 however , the behavior of the extremal process is much more intricate , with a phenomenon of clustering in the level sets ( see @xcite or also @xcite for a similar phenomenon for branching brownian motion ) . this yields results of a very different nature . given @xmath14 be a finite subset of @xmath0 , we let @xmath15 denote the internal boundary of @xmath14 , @xmath16 the set of interior points of @xmath17 , and @xmath18 the set of point which are adjacent to the boundary , @xmath19 in general some of these sets could be empty , but throughout this work @xmath17 is going to be a large square . given @xmath20 , we define @xmath21 to be the law of the lattice gaussian free field @xmath22 with boundary condition @xmath23 on @xmath15 . the field @xmath24 is a random function from @xmath17 to @xmath25 . it is satisfies @xmath26 and the distribution of @xmath27 is given by @xmath28 where @xmath29 denotes the lebesgue measure on @xmath30 and @xmath31 ( one of the two @xmath32 factors is present to compensate that the edges are counted twice in the sum , the other one being the one usually present for gaussian densities ) . in what follows we consider the case @xmath33 for some @xmath34 . note that we have @xmath35 we also introduce the notation @xmath36 , and we simply write @xmath37 for @xmath38 . we drop @xmath23 from our notation in the case where we consider zero boundary condition @xmath39 . we let @xmath40 be the realization of a family of iid square integrable centered random variables ( of law @xmath41 ) . we assume that they have finite exponential moments , or more precisely , that there exist constants @xmath42 $ ] such that @xmath43\ , < \ , \infty\ \text { for every } { \beta}\in ( -{\beta}_0\ , 2{\overline{{\beta}}}]\ , .\ ] ] for @xmath44 set @xmath45}(\phi_x)$ ] . for @xmath46 and @xmath47 , we define a modified measure @xmath48 via the density @xmath49 where @xmath50.\ ] ] note that in the definition of @xmath51 , the sum @xmath52 can be replaced by either @xmath53 or @xmath54 as these changes affect only the partition function . in the case where @xmath39 , we drop the corresponding superscript it from the notation . in the special case where @xmath55 , we simply write @xmath56 and @xmath57 for the pinning measure and partition function ( as they do not depend on @xmath58 ) respectively . this case is referred to as the _ pure _ ( or homogeneous ) model . when @xmath46 , defines the pinning model with _ quenched _ disorder . the important properties of the system are given by the asymptotic behavior of the partition function , or more precisely by the free energy . the existence of quenched free energy for the disordered model has been proved in ( * ? ? ? * theorem 2.1 ) . we recall this result here together with some basic properties [ freen ] the free energy @xmath59 \stackrel{{{\ensuremath{\mathbb p } } } ( { \mathrm{d}}{\omega})-a.s.}{= } \lim_{n\to \infty } \frac{1}{n^d}\log z^{{\beta},{\omega}}_{n , h } \ , , \ ] ] exists ( and is self - averaging ) . it is a convex , nonnegative , nondecreasing function of @xmath1 . moreover there exists a @xmath60 which is such that @xmath61 let us briefly explain why @xmath6 marks a transition on the large scale behavior of @xmath24 under @xmath62 . a simple computation gives @xmath63.\ ] ] hence by convexity , we have @xmath64,\ ] ] for the @xmath1 for which @xmath65 differentiable ( for the hypothetical countable set where @xmath66 may not exist , we can replace @xmath67 by @xmath68 resp . @xmath69 , @xmath70 by @xmath71 resp . @xmath72 and consider the left- resp . right - derivative in the above equation ) . for @xmath73 , we have @xmath74 by convexity and thus the expected number of point in contact with the substrate is asymptotically of order @xmath75 . on the contrary when @xmath76 , the asymptotic expected contact fraction vanishes when @xmath77 tends to infinity . note that the whole model is perfectly defined for all @xmath78 . however , the case @xmath79 , which is a variant of the random walk pinning model which as mentioned in the introduction was the object of numerous studies in the literature . however , the effect of disorder in dimension @xmath80 being quite different , in the remainder of the paper , we prove results for the case @xmath12 and discuss how they compare with those obtained in the more related case @xmath8 @xcite . in the case @xmath55 , we simply write @xmath81 for @xmath82 . in that case the behavior of the free energy is known in details ( see ( * ? ? ? * fact 2.4 ) and also ( * ? ? ? * section 2.3 and remark 7.10 ) for a full proof for @xmath8 ) . we summarize it below . [ propure ] for all @xmath78 , we have @xmath83 and moreover * for @xmath12 @xmath84 * for @xmath8 @xmath85 where @xmath86 $ ] and @xmath87 is the standard deviation for the infinite volume free field in @xmath0 . to be more precise @xmath88 where @xmath89 is the green function defined in . the result in dimension @xmath13 is well known folklore to people in the fields , but as to our knowledge , no proof of it is available in the literature . for this reason we present a short one in appendix [ secpropure ] . using jensen s inequality , we can for every @xmath90 , compare the free energy to that of the annealed system , which is the one associated to the averaged partition function @xmath91 $ ] , @xmath92 \le \lim_{n\to \infty } \frac{1}{n^d}\log { { \ensuremath{\mathbb e } } } \left [ z^{{\beta},{\omega}}_{n , h}\right].\ ] ] our choice of parametrization implies @xmath93={{\ensuremath{\mathbf e } } } _ n\left [ { { \ensuremath{\mathbb e } } } \left[e^{\sum_{x\in { \widetilde}{\lambda}_n } ( { \omega}_x-{\lambda}({\beta})+h)\delta_x}\right ] \right]= { { \ensuremath{\mathbf e } } } _ n\left [ e^{\sum_{x\in { \widetilde}{\lambda}_n}h\delta_x}\right]= z_{n , h},\ ] ] and thus for this reason we have @xmath94 it is known that the inequality is strict : for @xmath95 , we have @xmath96 in all dimensions ( cf . . however we can ask ourselves if the behavior of the model with quenched disorder is similar to that of the annealed one in several other ways * is the critical point of the quenched model equal to that of the annealed model ( i.e. is @xmath97 ) ? * do we have a critical exponent for the free energy transition : do we have @xmath98 and is @xmath99 equal to one , like for the annealed model ( cf . proposition [ propure ] ) ? this question has been almost fully solved in the case @xmath8 . let us display the result here for @xmath8 , for every @xmath100 $ ] we have * @xmath97 for all values of @xmath46 . * if @xmath58 is gaussian , there exist positive constants @xmath101 such that for all @xmath102 . @xmath103 * in the case of general @xmath58 , for all there exist positive constants @xmath101 such that for all @xmath104 @xmath105 we strongly believe that the quadratic behavior holds for every @xmath58 as soon as @xmath106 , and the gaussian assumption is mostly technical . however , if @xmath107 , we believe that the model is in a different universality class and the critical exponent depends on the tail of the distribution of the variable @xmath108 . the aim of the paper is to provide answers in the case of dimension @xmath13 . we present now the main achievement of this paper . we prove that similarly to the @xmath8 case , the critical point @xmath6 coincides with the annealed one for every value of @xmath7 ( which is in contrast with the case @xmath79 where the critical points differs for every @xmath46 @xcite ) . however , we are able to prove also that the critical behavior of the free energy is not quadratic , @xmath65 is becomes smaller than any power of @xmath1 in a ( positive ) neighborhood of @xmath109 . this indicates that the phase transition is of infinite order . [ mainres ] when @xmath12 , for every @xmath110 $ ] the following holds * we have @xmath97 . * we have @xmath111 more precisely , there exists @xmath112 such that for all @xmath113 @xmath114 we do not believe that either bound in is sharp . however it seems to us that the strategy used for the lower - bound is closer to capture the behavior of the field . we believe that the true behavior of the free energy might be given by @xmath115 while a lower bound of this type might be achieved by optimizing the proof presented in the present paper ( but this would require some significant technical work ) , we do not know how to obtain a significant improvement on the upper - bound . like in @xcite , it worthwhile to notice that the proof of the results of the present paper can be adapted to a model for with a different localization mechanism . it is the analog of the model of a copolymer in the proximity of the interface between selective solvents , see @xcite and references therein . for this model given a realization of @xmath58 and two fixed parameters @xmath116 , the measure is defined via the following density @xmath117 where we assume @xmath118 . a natural interpretation of the model is that the graph of @xmath119 models a membrane lying between two solvents @xmath120 and @xmath121 which fill the upper and lower half - space respectively : for each point of the graph , the quantity @xmath122 describes the energetic preference for one solvent of the corresponding portion of the membrane ( a if @xmath123 and b if @xmath124 ) . as @xmath1 is positive and @xmath125 is centered , there is , on average , a preference for solvent a ( by symetry this causes no loss of generality ) . if @xmath126 $ ] , there is a non - trivial competition between energy and entropy : the interaction with the solvent gives an incentive for the field @xmath24 to stay close to the interface so that its sign can match as much as possible that of @xmath127 , but such a strategy might be valid only if the energetic rewards it brings is superior to the entropic cost of the localization . a more evident analogy with the pinning measure can be made by observing that we can write @xmath128 where @xmath129 , that is @xmath130 is the indicator function that @xmath131 is in the lower half plane . it is probably worth stressing that from to there is a non - trivial ( but rather simple ) change in energy . and in the form . in particular , the strict analog of proposition [ freen ] holds the free energy in this case is denoted by @xmath132 and , precisely like for the pinning case , one sees that @xmath133 . we then set @xmath134 . adapting the proof for the lower - bound in we can identify the value of @xmath135 . [ th : cop ] for @xmath136 , for any @xmath137 we have @xmath138 moreover , with @xmath65 replaced by @xmath139 , holds true . note that while pure co - membrane model ( i.e. with no disorder ) displays a first order phase transition in @xmath1 , the above result underlines that the transition becomes of infinite order in the presence of an arbitrary small quantity of disorder . note that this result differs both from the one obtained in dimension @xmath8 ( for which the transition is shown to be quadratic at least for gaussian environment ( * ? ? ? * theorem 2.5 ) ) , and that in dimension @xmath80 : for the copolymer model based on renewals presented in @xcite , @xmath140 is in most cases a strict upper - bound on @xmath135 ( see e.g. the results in @xcite ) . the proof of theorem [ th : cop ] is not given in the paper but it can be obtained with straightforward modification , from that of theorem [ mainres ] . the proof of the upper - bound and of the lower - bound on the free energy presented in equation are largely independent . however some general technical results concerning the covariance structure of the free field are useful in both proofs , and we present these in section [ toolbox ] . most of the proofs for results presented in this section are in appendix [ appendix ] . the proof of the upper - bound is developed in section [ seclower ] . the proof of the lower - bound is spreads from section [ finicrit ] to [ intelinside ] . in section [ finicrit ] we present an estimate on the free energy in terms of a finite system with `` stationary '' boundary condition . in section [ decompo ] , we give a detailed sketch of the proof of the lower - bound based on this finite volume criterion , divided into several steps . the details of these steps are covered in section [ liminouze ] and [ intelinside ] . for the proof of both the upper and the lower - bound , we need fine results on the structure of the free field . although these results or their proof can not directly be extracted from the existing literature , our proof ( especially the techniques developed in section [ intelinside ] ) is largely based on tools that were developed in the numerous study on extrema and extremal processes of the two dimensional free field @xcite and other @xmath141-correlated gaussian processes @xcite ( the list of references being far from being complete ) . in particular for the lower bound , we present an _ ad - hoc _ decomposition of the field in section [ decompo ] and then exploit decomposition to apply a conditioned second moment technique , similarly to what is done e.g. in @xcite . for the upper - bound , we also make use a change of measure machinery inspired by a similar techniques developed in the study of disordered pinning model @xcite and adapted successfully to the study of other models @xcite . throughout the paper , to avoid a painful enumeration , we use @xmath142 to denote an arbitrary constant which is not allowed to depend on the value of @xmath1 or @xmath77 nor on the realization of @xmath58 . its value may change from one equation to another . for the sake of clarity , we try to write @xmath143 when the constant may depend on @xmath7 . when a constant has to be chosen small enough rather than large enough , we may use @xmath144 instead of @xmath142 . for @xmath145 we let @xmath146 denote its @xmath147 norm . @xmath148 the notation @xmath149 is also used to denote the cardinal of a finite set as this should yield no confusion . if @xmath150 and @xmath151 we set @xmath152 we use double brackets to denote interval of integers , that for @xmath153 in @xmath154 @xmath155\cap { { \ensuremath{\mathbb z } } } = \{i , i+1,\dots , j\}.\ ] ] if @xmath156 is a finite family of events , we refer to the following inequality as _ the union bound_. @xmath157 we let @xmath158 denote continuous time simple random walk on @xmath0 whose generator @xmath159 is the lattice laplacian defined by @xmath160 and we let @xmath161 denote its law starting from @xmath162 . we let @xmath163 denote the associated heat - kernel @xmath164 if @xmath165 denote a probability measure on a space @xmath166 , and @xmath167 a measurable function on @xmath166 we denote the expectation of @xmath167 by @xmath168 with an exception where the probability measure is denoted by the letter @xmath169 , in that case we use @xmath170 for the expectation . if @xmath171 is a gaussian of standard deviation @xmath172 , it is well known that we have @xmath173 \le \frac{\sigma } { \sqrt{2\pi}u}e^{-\frac{u^2}{2\sigma^2}}.\ ] ] we refer to the gaussian tail bound when we use this inequality . in this section we quickly recall the the definition and some basic properties of the massive free field . given @xmath174 , and a set @xmath175 and a function @xmath23 , we define the law @xmath176 of the massive free field on @xmath17 with boundary condition @xmath23 and mass @xmath177 as follows : it is absolutely continuous w.r.t @xmath21 and @xmath178 } \exp\left(-m^2 \sum_{x\in \mathring { \lambda } } \phi_x^2\right).\ ] ] we let @xmath179 denote the law of the massive field on @xmath180 . ( in the special case @xmath181 , @xmath23 is omitted in the notation ) . we let @xmath182 denote the law of the centered infinite volume massive free field @xmath0 , which is the limit of @xmath183 when @xmath184 ( see section [ hkernel ] for a proper definition with the covariance function ) . we will in some cases have to choose the boundary condition @xmath23 itself to be random and distributed like an infinite volume centered massive free field ( independent @xmath24 ) , in which case we denote its law by @xmath185 instead of @xmath182 . note that the free field and its massive version satisfy a markov spatial property . in particular the law of @xmath186 under @xmath187 is the same as under the infinite volume measure @xmath188 . even if the definition of the free energy given in proposition [ freen ] is made in terms of the partition function with @xmath189 it turns out that our methods to obtain upper and lower bounds involve considering non - trivial boundary conditions ( cf . proposition [ scorpiorizing ] and proposition [ th : finitevol ] ) . however , it turns out to be more practical to work with a fixed law for the field and not one that depends on @xmath23 . fortunately , given a boundary condition @xmath23 the law of @xmath190 can simply be obtained by translating the field with @xmath191 boundary condition by a function that depends only on @xmath23 . this is a classical property of the free field but let us state it in details . as the covariance function of @xmath24 under @xmath190 and @xmath192 are the same , we have we have @xmath193={{\ensuremath{\mathbf p } } } ^{m}_n[\phi + h^{m,{\widehat}\phi}_n \in \cdot \ ] , \ ] ] where @xmath194.\ ] ] it is not difficult to check that @xmath195 must be a solution of the system ( recall ) @xmath196 we simply write @xmath197 when @xmath198 . the solution of is unique and @xmath195 has the following representation : consider @xmath199 the simple random walk on @xmath0 and for @xmath200 let @xmath201 denotes the first hitting of @xmath120 . we have @xmath202.\ ] ] given @xmath23 and @xmath203 , we introduce the notation @xmath204}(\phi_x+ h^{{\widehat}\phi}_n(x)).\ ] ] in view of an alternative way of writing the partition function is @xmath205.\ ] ] in some situation the above expression turns our to be handier than the definition . in this section we present some estimates on the covariance function of the free field and massive free field in dimension @xmath13 , which will be useful in the course of the proof . these are not new results , but rather variants of existing estimates in the literature ( see e.g ( * ? ? ? * lemma 2.1 ) ) . the covariance kernel of the infinite volume free field with mass @xmath174 in @xmath206 or @xmath207 in @xmath180 is given by the green function @xmath208 which is the inverse of @xmath209 ( this can in fact be taken as the definition of the infinite volume free field , requiring in addition that it is centered ) . the covariance function of the field under the measure @xmath210 is @xmath211 which is the inverse of @xmath209 with dirichlet boundary condition on @xmath212 . both of these functions can be represented as integral of the heat kernel , we have @xmath213&=\int_{0}^{\infty}e^{-m^2t}p_t(x , y){\mathrm{d}}t=:g^m(x , y),\\ { { \ensuremath{\mathbf e } } } ^m_n[\phi(x)\phi(y)]&=\int_{0}^{\infty}e^{-m^2t}p^*_t(x , y){\mathrm{d}}t=:g^{m,*}(x , y ) , \end{split}\ ] ] where @xmath214 is the heat kernel on @xmath180 with dirichlet boundary condition on @xmath212 , @xmath215.\ ] ] we simply write @xmath216 in the case @xmath198 . note that , because of the spatial markov property ( section [ secmass ] ) and of , when @xmath23 has law @xmath185 and @xmath24 has law @xmath210 , @xmath217 has the same law as the ( marginal in @xmath180 of the ) infinite volume field . hence as a consequence @xmath218=g^m(x , y)-g^{m,*}(x , y)=\int_{0}^{\infty}e^{-m^2t}(p_t(x , y)-p^*_t(x , y)){\mathrm{d}}t.\ ] ] before giving more involved estimates , let us mention first a quantitative version of the local central limit theorem ( * ? ? ? * theorem 2.1.1 ) for the heat kernel which we use as an essential building brick to obtain them . there exists a constant @xmath142 such that for all @xmath219 , @xmath220 let us recall the notation for the distance between a set and a point . the following two lemmas are proved in appendix [ appendix ] . [ greenesteem ] there exists a constant such that @xmath142 * for all @xmath221 , for any @xmath151 @xmath222 * for all @xmath221 , for any @xmath44 @xmath223 [ lem : kerestimate ] the following assertions hold * there exists a constant @xmath142 such that for all @xmath219 , @xmath224 , we have @xmath225 * there exist a constant @xmath142 such that for all @xmath219 and @xmath226 satifying @xmath227 we have @xmath228 and as a consequence @xmath229 * we have for all @xmath230 @xmath231 ^ 2}{t}.\ ] ] * we have for all @xmath230 @xmath232 finally we conclude this preliminary section with an estimate for the probability to remain above a line for gaussian random walks . the statement is not optimal and the term @xmath233 could be replaced by @xmath80 but as the rougher estimate is sufficient for our purpose we prefer to keep the proof simpler . we include the proof in the appendix [ appendix ] for the sake of completeness . [ lem : bridge ] let @xmath234 be arandom walk with independent centered gaussian increments , each of which with variance bounded above by @xmath13 and such that the total variance satisfies @xmath235 . then we have for all @xmath236 @xmath237\le \frac { c(x+(\log k))^2}{k}.\ ] ] let us briefly discuss the structure of the proof before going into more details . the main idea is presented in section [ changeofme ] : we introduce a function which penalizes some environments @xmath58 which are too favorable , and use it to get a bet annealed bound which penalizes the trajectories with clustered contact points in a small region ( proposition [ nonrandom ] ) . however , to perform the coarse - graining step of the proof , we need some kind of control on @xmath24 . for this reason , in section [ restrictou ] we start the proof by showing that restricting the partition function to a set of uniformly bounded trajectory does not affect a lot the free energy . in this section , we show that restricting the partition function by limiting the maximal height of the field @xmath24 does not affect too much the free energy . this statement is to be used to control the boundary condition of each cell when performing a coarse - graining argument in proposition [ scorpiorizing ] . let us set @xmath238 and write @xmath239.\ ] ] [ boundtheprob ] there exists a constant @xmath144 such for any @xmath240 and @xmath46 we have @xmath241\ge -\exp\left(-c |\log h|^2\right).\ ] ] as a consequence , we have @xmath242 for practical purposes we introduce the two following events @xmath243 we have @xmath244 . in order to obtain a bound on the probability of @xmath245 we need to use the fkg inequality for the gaussian free field which we present briefly ( we refer to ( * ? ? ? * section b.1 ) for more details ) . we denote by @xmath71 the natural order on the set of functions @xmath246 defined by @xmath247 an event @xmath120 is said to be increasing if for @xmath248 we have @xmath249 and decreasing if its complement is increasing . let us remark that all the events described in are either decreasing or increasing . a probability measure @xmath165 is said to satisfy the fkg inequality if for any pair of increasing events @xmath250 we have @xmath251 . note that this yields automatically similar inequalities for any pairs of monotonic events which we also call fkg inequalities . it is well know that @xmath210 satisfies the fkg inequality : it is sufficient to check that holley s criterion @xcite is satisfied by the hamiltonian in . the same argument yields that @xmath62 as well as the conditionned measures @xmath252 and @xmath253 also satisfy the fkg inequality . hence using the fkg inequality for @xmath62 , we have @xmath254 then , using the fkg inequality for @xmath255 and we have @xmath256 where we used symmetry to get the last equality . then we can conclude that @xmath257 ^ 2 \ge \left[{{\ensuremath{\mathbf p } } } _ { n}({{\ensuremath{\mathcal a } } } ^{h,1}_n \cap { { \ensuremath{\mathcal b } } } _ n ) \right]^2.\ ] ] we are left with estimating the last term . note that changing the boundary condition by a constant amount does not affect the leading order of the asymptotic thus to conclude it is sufficient to bound asymptotically the probability of the event @xmath258 which is a translated version of @xmath259 . more precisely we have for an adequate constant @xmath260 @xmath261 ^ 2\ ] ] to bound the probability of @xmath262 we use the following result , whose proof is postponed to the end of the section . [ controlgrid ] there exists a constant @xmath142 such that for any @xmath77 , and for any set @xmath263 which is such that @xmath264 is connected , we have @xmath265 \ge \exp(-c |{\gamma}|).\ ] ] we divide @xmath180 in cells of side - length @xmath266 for some small constant @xmath144 . we set @xmath267 we apply lemma [ controlgrid ] for the following set @xmath268 which is is a grid which splits @xmath180 in cells of side - length @xmath269 . we obtain that @xmath270 \ge \frac{2c}{n_0}= 2c \exp(-c|\log h|^2),\ ] ] where we used the inequality @xmath271 valid for all @xmath77 . to conclude we need to show that @xmath272 \ge - ( n_0)^{-2}.\ ] ] to prove it is sufficient to remark that conditioned to @xmath273 , the variance of the field @xmath274 is uniformly bounded by @xmath275 ( cf . for @xmath198 ) . thus , for any realization of @xmath24 satisfying @xmath276 , for any @xmath277 , using the gaussian tail bound we have for @xmath1 sufficiently small @xmath278 \le \exp\left ( -\frac { \pi |\log h|^4 } { 4 \log n_0 } \right)\le \exp\left(-\frac{\pi}{4c } |\log h|^2\right).\ ] ] now with this in mind we can apply union bound in @xmath279 and obtain @xmath280\\ \ge 1- ( n_0 - 1)^2\exp\left(-\frac{\pi}{4c } |\log h|^2\right ) \ge e^{-1/2}.\end{gathered}\ ] ] where the last inequality is valid provided the constant @xmath144 is chosen sufficiently small . as , conditioned to the realization of @xmath273 , the fields @xmath274 are independent for different values of @xmath281 , we prove that the inequality holds by multiplying for all distinct @xmath279 which fit ( at least partially ) in @xmath180 ( there are at most @xmath282 full boxes , to which one must add at most @xmath283 uncompleted boxes ) , and taking the expectation with respect to @xmath273 conditioned on the event @xmath276 . this ends the proof of proposition [ boundtheprob ] . we can prove it by induction on the cardinality of @xmath284 . assume that the result is valid for @xmath284 and let us prove it for @xmath285 . @xmath286 \ | \ \max_{x\in { \gamma } } |\phi_x|\le 1 \right]\ge \exp(-c).\ ] ] note that conditioned to @xmath287 , @xmath288 is a gaussian variable . its variance is given by @xmath289\le 1.\ ] ] the reason being that as by assumption @xmath290 , the walk @xmath291 is killed with rate one while it lies on @xmath292 . in addition , if @xmath293 , then necessarily @xmath294\in[-1,1].\ ] ] for this reason , the above inequality is valid if one chooses @xmath295 } \log p\left ( { { \ensuremath{\mathcal n } } } \in [ -1+u,1+u ] \right)= -\log p\left({{\ensuremath{\mathcal n } } } \in [ 0,2 ] \right),\ ] ] where @xmath296 is a standard normal . to bound the expectation of @xmath297 $ ] we use a `` change of measure '' argument . the underlying idea is that the annealed bound obtained by jensen s inequality is not sharp because some very atypical @xmath58 s ( a set of @xmath58 of small probability ) give the most important contribution to the annealed partition function . hence our idea is to identify these bad environments and to introduce a function @xmath298 that penalizes them . this idea originates from @xcite where it was used to prove the non - coincidence of critical point for a hierarchical variant of the pinning model and was then improved many times in the context of pinning @xcite and found application for other models like random - walk pinning , directed polymers , random walk in a random environment or self - avoiding walk in a random environment @xcite . in @xcite , we used the detailed knowledge that we have on the structure of the set of contact points , ( which is simply a renewal process ) in order to find the right penalization function @xmath298 . here we have a much less precise knowledge on the structure @xmath299 under @xmath210 ( especially because we have to consider possibly very wild boundart condition ) , but we know that one typical feature of the two - dimensional free field is that the level sets tend to have a clustered structure . we want to perform a change of measure that has the consequence of penalizing these clusters of contact points : we do so by looking at the empirical mean of @xmath58 in some small regions and by giving a penalty when it takes an atypically high value . let us be more precise about what we mean by penalizing with a function @xmath298 . using jensen inequality , we remark that @xmath300= 2{{\ensuremath{\mathbb e } } } \left [ \log \sqrt { z^{{\beta},{\omega}}_{n , h } ( { { \ensuremath{\mathcal a } } } ^h_n ) } \right]\le 2 \log { { \ensuremath{\mathbb e } } } \left [ \sqrt{z^{{\beta},{\omega}}_{n , h } ( { { \ensuremath{\mathcal a } } } ^h_n ) } \right]\ ] ] if we let @xmath298 be an arbitrary positive function of @xmath301 , we have by cauchy - schwartz inequality @xmath302 ^ 2 \le { { \ensuremath{\mathbb e } } } [ f({\omega})^{-1 } ] { { \ensuremath{\mathbb e } } } \left [ f({\omega})z^{{\beta},{\omega}}_{n , h } ( { { \ensuremath{\mathcal a } } } ^h_n ) \right ] , \ ] ] and hence @xmath303\le \frac{1}{n^2}\log { { \ensuremath{\mathbb e } } } [ f({\omega})^{-1}]+ \frac{1}{n^2}\log { { \ensuremath{\mathbb e } } } \left [ f({\omega})z^{{\beta},{\omega}}_{n , h } ( { { \ensuremath{\mathcal a } } } ^h_n ) \right ] .\ ] ] let us now present our choice of @xmath298 . our idea is to perform some kind of coarse - graining argument : we divide @xmath180 into cells of fixed side - length @xmath304 @xmath305 and perform a change of measure inside of each cell . we assume that @xmath304 is an even integer ( the free energy being monotone this causes no loss of generality ) , and that @xmath306 is a sufficiently large multiple of @xmath304 . given @xmath307 , we let @xmath308 denote the translation of the box @xmath309 which is ( approximately ) centered at @xmath310 ( see figure [ fig : structure ] ) @xmath311+{\widetilde}{\lambda}_{n_1}.\ ] ] in the case @xmath312 we simply write @xmath313 ( note that it is not identical to @xmath309 ) . we define the event @xmath314 which is simply denoted by @xmath315 in the case when @xmath312 . here @xmath316 denotes the derivative of @xmath317 defined in . finally we set @xmath318 the effect of @xmath298 is to give a penalty ( multiplication by @xmath319 ) for each cell in which one can find a region of @xmath58 with diameter @xmath320 and atypically high empirical mean . combining proposition [ boundtheprob ] and , we have ( provided that the limit exists ) @xmath321 + \liminf_{k\to \infty } \frac{1}{n}\log { { \ensuremath{\mathbb e } } } \left [ f({\omega})z^{{\beta},{\omega}}_{n , h}({{\ensuremath{\mathcal a } } } ^h_n)\right].\ ] ] we can conclude the proof with the two following results , which evaluate respectively the cost and the benefit of our change of measure procedure . [ cost ] there exists positive constants @xmath322 and @xmath112 and such that for all @xmath113 sufficiently small , for all @xmath323 @xmath324\le ( k-1)^2 e^{-c({\beta})(\log h)^2}.\ ] ] as a consequence we have @xmath325\le e^{-c({\beta})(\log h)^2}.\ ] ] [ benefit ] there exists @xmath326 such that for all @xmath113 @xmath327\le e^{-2|\log h|^{3/2}}.\ ] ] as a consequence of and of the two propositions above , we obtain that for @xmath328 we have @xmath329 the proof of proposition [ cost ] is simple and short and is presented below . the proof proposition [ benefit ] requires a significant amount of work . we decompose it in important steps in the next subsection . because of the product structure , we have @xmath330= \big ( { { \ensuremath{\mathbb e } } } \left [ \exp\left ( 2{{\ensuremath{\mathcal e } } } _ { n_1}\right ) \right]\big)^{(k-1)^2}.\ ] ] hence it is sufficient to obtain a bound on @xmath331\le ( e^2 - 1 ) { { \ensuremath{\mathbb p } } } [ { { \ensuremath{\mathcal e } } } _ { n_1}(0)].\ ] ] as an easy consequence of the proof of cramrs theorem ( see e.g. ( * ? ? ? * chapter 2 ) ) , there exists a constant @xmath322 that any @xmath332 @xmath333 \le e^{-c({\beta})(\log n_1)^2},\ ] ] and by union bound we obtain that @xmath334\le n^2_0\exp(-c(\log n_1)^2)$ ] , which in view of and is sufficient to conclude = 10.5 cm [ c][l]@xmath191 [ c][l ] @xmath335 [ c][l ] @xmath336 [ c][l ] @xmath337 [ c][l ] @xmath77 [ c][l]@xmath338 [ c][l]@xmath339 [ c][l]@xmath340 [ fig : structure ] the proof is split in three steps , whose details are performed in section [ proufin ] , [ proufeux ] and [ proufoix ] respectively . in the first one we show that our averaged partition function @xmath341 $ ] , can be bounded from above by the partition of an homogenous system where an extra term is added in the hamiltonian to penalize the presence of clustered contact in a small region ( here a region of diameter @xmath320 ) . we introduce the event @xmath342 which indicates the presence of such a cluster in @xmath308 , @xmath343 we simply write @xmath344 for the case @xmath312 . [ nonrandom ] we have @xmath345\\ \le { { \ensuremath{\mathbf e } } } _ n\left [ \exp\left ( h\sum_{x \in { \widetilde}{\lambda}_{n } } \delta_x - \sum_{y \in { \llbracket}1 , k-1 { \rrbracket}^2 } { \mathbf{1}}_{{{\ensuremath{\mathcal c } } } _ { n_1}(y ) } \right){\mathbf{1}}_{{{\ensuremath{\mathcal a } } } ^h_n } \right]= : { \widehat}z(n , n_1,h).\end{gathered}\ ] ] in the second step , we perform a factorization in order to reduce the estimate of @xmath346 to that of similar system with only one cell . let us set ( see figure [ fig : structure ] ) @xmath347 note that for every for @xmath348 we have @xmath349 and that @xmath350 . [ scorpiorizing ] we have @xmath351 \right)^{\frac{(k-1)^2}{4}}\ ] ] let us notice two important features in our factorization which are present to reduce possible nasty boundary effects : * there is a restriction on the boundary condition @xmath352 , which forbids wild behavior of the field . this restriction is directly inherited from the restriction to @xmath353 in the partition function and brings some light on the role of proposition [ boundtheprob ] in our proof . * the hamiltonian @xmath354 is a functional of @xmath355 i.e. of the field restricted to a region which is distant from the boundary of the box @xmath356 . the final step of the proof consists in evaluating the contribution of one single cell to the partition function . [ onecell ] there exists a constant @xmath144 such that for all @xmath1 sufficiently small for all @xmath23 satisfying @xmath352 we have @xmath357\le e^{-2(\log h)^{3/2}}.\ ] ] combining the three results presented above , we have @xmath358\le 2 n_1 n h+\frac{(k-1)^2}{4}e^{-2(\log h)^{3/2}},\ ] ] and this is sufficient to conclude the proof of proposition [ benefit ] . given a realization @xmath24 , we let @xmath359 be a probability law which is absolutely continuous with respect to @xmath41 and whose the density is given by @xmath360 under @xmath361 , the variables @xmath362 are still independent but they are not iid , as the law of the @xmath125s for which @xmath363 have been tilted . in particular it satisfies @xmath364= { \lambda}'({\beta})\delta_x \quad \text { and } { \mathrm{var}}_{{{{\ensuremath{\mathbb p } } } } ^{\phi}}[{\omega}_x]=1 + ( { \lambda}''({\beta})-1)\delta_x\ ] ] where @xmath316 and @xmath365 denote the two first derivatives of @xmath317 the function defined in . this notation gives us another way of writing the quantity that we must estimate @xmath366= { { \ensuremath{\mathbf e } } } _ n\left [ { { \ensuremath{\mathbb e } } } ^{\phi}[f({\omega } ) ] e^{h \sum_{x \in { \widetilde}{\lambda}_n } \delta_x}{\mathbf{1}}_{{{\ensuremath{\mathcal a } } } ^h_n}\right].\ ] ] to conclude it is sufficient to prove that @xmath367 \le \exp \left ( -\sum_{y \in { \llbracket}0 , k-1 { \rrbracket } } { \mathbf{1}}_{{{\ensuremath{\mathcal c } } } _ { n_1}(y ) } \right).\ ] ] note that because both @xmath368 and @xmath298 have a product structure , it is in fact sufficient to prove that for any @xmath369 we have @xmath370\le e^{-{\mathbf{1}}_{{{\ensuremath{\mathcal c } } } _ { n_1}(y)}}.\ ] ] with no loss of generality we assume that @xmath312 . the result is obvious when @xmath371 hence we can also assume @xmath372 . let @xmath373 be a vertex satisfying @xmath374 ( e.g. the smallest one for the lexicographical order ) . we have @xmath375&= { \lambda}'({\beta } ) \!\!\!\!\!\ ! \!\!\!\!\!\ ! \sum_{\ { z\in { \widetilde}{\lambda}'_{n_1 } \ : \ |z - x_0|\le ( \log n_1)^2\ } } \!\!\!\!\!\ ! \!\!\!\!\!\ ! { \delta}_z \ , \ge \ , { \lambda}'({\beta})(\log n_1)^3,\\ { \mathrm{var}}_{{{\ensuremath{\mathbb p } } } ^{\phi}}\left [ \sum_{\ { z\in { \widetilde}{\lambda}'_{n_1 } \ : \ |z - x_0|\le ( \log n_1)^2\ } } \!\!\!\!\!\ ! \!\ ! { \omega}_z \ \right ] & \le \left[2(\log n_1)^2 + 1\right]^2\max({\lambda}''({\beta}),1 ) . \end{split}\ ] ] hence in particular if @xmath304 is sufficiently large , chebychev s inequality gives @xmath376\le e^{-1}-e^{-2},\ ] ] which implies . we start by taking care of the contribution of the contact points located near the boundary @xmath212 , as they are not included in any @xmath308 . assuming that all these points are contact points we obtain the following crude bound @xmath377 + \sum_{y \in { \llbracket}1 , k-1 { \rrbracket}^2 } \sum_{x \in { \widetilde}{\lambda}_{n_1}(y ) } \delta_x.\ ] ] and the first term is smaller than @xmath378 . hence we have @xmath379.\ ] ] we partition the set of indices @xmath380 into @xmath381 subsets , according to the parity of the of the coordinates . if we let @xmath382 and @xmath383 denote the first and second diadic digits of @xmath384 . we set @xmath385 using hlder s inequality we have @xmath386^{4 } \\ \le \prod_{i=1}^4 { { \ensuremath{\mathbf e } } } _ n\left [ e^{4\sum_{y \in \xi(i ) } \left ( h\sum_{x \in { \widetilde}{\lambda}_{n_1}(y ) } \delta_x - { \mathbf{1}}_{{{\ensuremath{\mathcal c } } } _ { n_1}(y)}\right ) } { \mathbf{1}}_{{{\ensuremath{\mathcal a } } } ^h_n}\right].\end{gathered}\ ] ] for a fixed @xmath387 , the interiors of the boxes @xmath388 , @xmath389 are disjoint ( neighboring boxes overlap only on their boundary , we refer to figure [ fig : structure ] ) . this gives us a way to factorize the exponential : let us condition the expectation to the realization of @xmath390 where @xmath391 the spatial markov property implies that conditionally on @xmath390 , the restrictions @xmath392_{y \in \xi(i)}$ ] are independent . hence we can factorize the expectation and get @xmath393\\ \le \prod_{y\in \xi(i ) } { { \ensuremath{\mathbf e } } } _ n\left [ e^{4 \left ( h\sum_{x \in { \widetilde}{\lambda}_{n_1}(y ) } \delta_x - { \mathbf{1}}_{{{\ensuremath{\mathcal c } } } _ { n_1}(y)}\right ) } \ | \ ( \phi_x)_{x\in { \gamma}(i ) } \right].\end{gathered}\ ] ] on the event @xmath394 we have for any @xmath389 , by translation invariance , @xmath395\\ \le \max_{\ { { \widehat}\phi \ : \ \|{\widehat}\phi\|_{\infty}\le |\log h|^2 \ } } { { \ensuremath{\mathbf e } } } ^{{\widehat}\phi}_{2n}\left[e^{4h\left ( \sum_{x \in { \widetilde}{\lambda}'_{n_1 } } \delta_x\right ) - 4{\mathbf{1}}_{{{\ensuremath{\mathcal c } } } _ { n_1 } } } \right ] . \end{gathered}\ ] ] and hence we can conclude by taking the expectation of restricted to the event @xmath396 ( which includes @xmath353 ) . note that because of our choice of @xmath397 we always have @xmath398 which is small . hence for that reason , if @xmath1 is sufficiently small , the taylor expansion of the exponential gives @xmath399 \le \log { { \ensuremath{\mathbf e } } } ^{{\widehat}\phi}_{2n_1}\big[1 + 5h\sum_{x \in { \widetilde}{\lambda}'_{n_1 } } \delta_x- \frac{1}{2}{\mathbf{1}}_{{{\ensuremath{\mathcal c } } } _ { n_1 } } \big ] \\ \le 5h { { \ensuremath{\mathbf e } } } ^{{\widehat}\phi}_{2n_1}\big [ \sum_{x \in { \widetilde}{\lambda}'_{n_1 } } \delta_x \big]- \frac{1}{2}{{\ensuremath{\mathbf p } } } ^{{\widehat}\phi}_{2n_1}[{{\ensuremath{\mathcal c } } } _ { n_1}]\\ \le 5 n_1^{-2 } \max_{x\in { \widetilde}{\lambda}'_{n_1}}{{\ensuremath{\mathbf p } } } ^{{\widehat}\phi}_{2n_1}[\phi_x\in[-1,1]]-\frac{1}{2}{{\ensuremath{\mathbf p } } } ^{{\widehat}\phi}_{2n_1}[{{\ensuremath{\mathcal c } } } _ { n_1}].\end{gathered}\ ] ] we have to prove that the r.h.s . is small . before going into technical details let us quickly expose the main idea of the proof . for the r.h.s . of to be positive , we need @xmath400\big)}{{{\ensuremath{\mathbf p } } } ^{{\widehat}\phi}_{2n_1}[{{\ensuremath{\mathcal c } } } _ { n_1}]}\ge \frac{n_1 ^ 2}{10}.\ ] ] what we are going to show is that for this ratio to be large we need the boundary condition @xmath23 to be very high above the substrate ( or below by symmetry ) , but that in that case the quantity @xmath401\right)$ ] itself has to be very small and this should allow ourselves to conclude . to understand the phenomenon better we need to introduce quantitative estimates . let @xmath216 denote the green function in the box @xmath402 with @xmath191 boundary condition , and set @xmath403 we have from lemma [ greenesteem ] @xmath404 recall that from we have @xmath405 \ \big)\le { { \ensuremath{\mathbf p } } } _ { 2n_1}\left(\phi_x\in \left[-1-h^{{\widehat}\phi}_{2n_1}(x),1-h^{{\widehat}\phi}_{2n_1}(x)\right ] \ \right).\ ] ] with this in mind we fix @xmath406 hence using basic properties of the gaussian distribution , we obtain ( provided that @xmath1 is sufficiently small ) @xmath407\big ) \le e^{-\frac{(u-1)^2}{2v_{n_1}}}.\ ] ] it requires a bit more work to obtain a good lower bound for @xmath408 $ ] which is valid for all values of @xmath409 . fortunately we only need a rough estimate as the factor @xmath410 in gives us a significant margin in the computation . recall that @xmath411 denotes the two - dimensional heat - kernel with zero boundary condition on @xmath356 . let us set @xmath412 from the estimates in lemma [ lem : kerestimate ] , we can deduce that @xmath413 for instance we have @xmath414 for some appropriate @xmath142 ( the estimate is obtained using and ) so that the result can be deduced from the estimate in the green - function . [ boundforcluster ] for all @xmath1 sufficiently small , for all @xmath23 satisfying @xmath415 , and all @xmath416 we have @xmath417\ge c(\log n_1)^{-1 } e^{-\frac{u^2}{2v'_{n_1}}},\ ] ] combining the above result with and we have @xmath418\\\le \sup_{u>0}\left ( 5n_1^{-2 } e^{-\frac{(u-1)^2}{2v_{n_1}}}- c ( 2\log n_1)^{-1}e^{-\frac{u^2}{2v'_{n_1}}}{\mathbf{1}}_{\ { u \le ( \log n_1)^2\ } } \right ) \\ = \sup_{u>0 } \frac{5e^{-\frac{(u-1)^2}{2v_{n_1}}}}{n_1 ^ 2}\left[1- \frac{cn_1 ^ 2}{10(\log n_1 ) } e^{- \frac{u^2(v_{n_1}-v'_{n_1})}{2v'_{n_1}v_{n_1}}-\frac{2u-1}{2v_{n_1 } } } { \mathbf{1}}_{\ { u \le ( \log n_1)^2\}}\right].\end{gathered}\ ] ] now note that for the second factor to be positive , we need one of the terms in the exponential to be at least of order @xmath419 in absolute value . using the estimates we have for @xmath420 and @xmath421 , we realize that the exponential term is larger than @xmath422 and hence the expression is negative if @xmath423 for some small @xmath144 . for the other values of @xmath409 we can just consider the first factor which already gives a satisfying bound , and we can conclude that the l.h.s . of is smaller than @xmath424 we show here how to split the proof the proposition into three lemmas which we prove in the next subsection . set @xmath425 ( it is not necessarily unique but in the case it is not we choose one minimizer in a deterministic manner ) and @xmath426 we bound from below the probability of @xmath344 by only examining the possibility of having a cluster of contact around @xmath427 . using we have @xmath428\le { { \ensuremath{\mathbf p } } } ^{{\widehat}\phi}_{2n_1}\left [ \sum_{z\in { \widehat}{\lambda } } \delta_z \ge ( \log n_1)^3 \right]\\ = { { \ensuremath{\mathbf p } } } _ { 2n_1}\left [ \sum_{z \in { \widehat}{\lambda } } { \mathbf{1}}_{[-1,1]}\left(\phi_z+h^{{\widehat}\phi}_{2n_1}(x)\right)\ge ( \log n_1)^3 \right].\end{gathered}\ ] ] to estimate the last probability , we first remark that for @xmath429 , @xmath430 is very close to @xmath431 which we assume to be equal to @xmath432 for the rest of the proof ( the case @xmath433 is exactly similar ) . the factor @xmath419 in the estimate is not necessary , but it yields a much simpler proof . [ regular ] we have for all @xmath434 @xmath435 in particular if @xmath1 is sufficiently small , @xmath436 and @xmath352 , we have @xmath437 then to estimate the probability for @xmath24 to form a cluster of point close to height @xmath409 , we decompose the field @xmath438 into a rough field @xmath439 which is almost constant on the scale @xmath320 and an independent field @xmath440 which accounts for the local variations of @xmath24 . we set @xmath441 we let @xmath442 and @xmath443 denote two independent centered fields with respective covariance function @xmath444 and @xmath445 . by construction the law of @xmath446 has a law given by @xmath447 , and thus we set for the remainder of the proof @xmath448 and use @xmath447 to denote the law of @xmath449 . we have by standard properties of gaussian variables that for every @xmath450 , and for @xmath1 sufficiently small @xmath451\right]\ge \frac{1}{4 \sqrt { 2\pi v_{n_1 } } } e^{-\frac{u^2}{2v'_{n_1 } } } \ge \frac{1}{5 \sqrt{\log n_1}}e^{-\frac{u^2}{2v'_{n_1}}}.\ ] ] now we have to check that the field @xmath439 remains around level @xmath409 on the whole box @xmath452 . [ locareg ] there exists a constant @xmath144 such that for all @xmath1 sufficiently small we have @xmath453 \le e^{-c(\log n_1)^4}.\ ] ] finally we show that it is rather likely for @xmath440 to have a lot of points around level zero . [ lastep ] there exists a constant @xmath144 such that for all @xmath1 sufficiently small we have @xmath454\ge c(\log\log n_1)^{-1/2}.\ ] ] we can now combine all these ingredient into a proof according to lemma [ regular ] , if @xmath455 we have @xmath456 \right\ } \supset \left\ { \phi_z \in [ -3/4+u,3/4+u ] \right\ } \\ \supset \left\ { \ \left |\phi_1(x_{\min})-u \right| \le 1/4 \right\}\cap \left\ { |\phi_1(x_{\min})-\phi_1(z)| \le 1/4 \right\ } \cap \left\ { |\phi_2(z)| \le 1/4 \right\}.\end{gathered}\ ] ] thus we obtain as a consequence @xmath457}(\phi_z+h^{{\widehat}\phi}_{2n_1}(x ) ) \ge ( \log n_1)^3\right\ } \\ \subset \left\ { \ \left|\phi_1(x_{\min})-\phi_1(z ) \right| \le 1/4 \right\ } \cap \left\ { \forall z\in { \widehat}{\lambda } , \ |\phi_1(x_{\min})-\phi_1(z)| \le 1/4 \right\}\\ \cap \big\ { \sum_{z\in { \widehat}{\lambda } } { \mathbf{1}}_{\ { |\phi_2(z)|\le ( 1/4)\ } } \ge ( \log n_1)^3 \big\}.\end{gathered}\ ] ] using combined with lemmas [ locareg ] and [ lastep ] and the independence of @xmath439 and @xmath440 we conclude that @xmath458}(\phi_z+h^{{\widehat}\phi}_{2n_1}(x ) \ge ( \log n_1)^3 \right ] \\ \ge \left [ \frac{c}{\sqrt { \log n_1 } } e^{-\frac{u^2}{2v'_{n_1 } } } - e^{-c(\log n)^4 } \right]c(\log n_1)^{-1/2 } \ge \frac{c'}{(\log n_1)}e^{-\frac{u^2}{2v'_{n_1}}}.\end{gathered}\ ] ] where the last inequality is holds if @xmath459 and @xmath1 is sufficiently small . we can thus conclude using . given @xmath434 , let @xmath460 and @xmath461 be two simple random walk starting from @xmath462 and @xmath463 , and coupled as follows : the coupling is made as the product of two one - dimensional couplings , along each coordinate the walk are independent until the coordinate match , then they move together . let @xmath464 be the time where the two walks meet and @xmath465 be the time when @xmath460 hits the boundary . recalling we have @xmath466.\ ] ] we conclude by showing that @xmath467 < \frac{c|x - y| ( \log n_1 ) } { n_1}\ ] ] by union bound , we can reduce to the one dimensional case . let @xmath468 and @xmath469 denote the first coordinates of @xmath460 and @xmath461 . until the collision time , they are two independent one dimensional random walk in @xmath470 with initial condition @xmath471 and @xmath472 in @xmath473 . let @xmath474 and @xmath475 denote respectively their collision time and the first hitting time of @xmath476 for @xmath468 . we are going to show that @xmath477 < c \frac{|x_1-y_1|(\log n_1)}{n_1}\ ] ] note that before collision , @xmath478 is a nearest neighbor random - walk with jump rate equal to @xmath13 and for that reason we have for any @xmath479 @xmath480\le c |x_1-y_1| t^{-1/2}.\ ] ] on the other hand , we have for any @xmath481 @xmath482 \le 2 \exp\left(- \frac{cn_1 ^ 2}{t } \right).\ ] ] we can conclude choosing @xmath483 . we obtain the result simply by performing a union bound on @xmath484 . hence we only need to prove a bound on the variance @xmath485 \\ \le \int_{(\log n_1)^8}^{\infty } \left[p^*_t(x_{\min},x_{\min})- p^*_t(x_{\min},y)-p^*(y , y ) \right]{\mathrm{d}}t \end{gathered}\ ] ] using , we obtain that for any @xmath484 @xmath486\le c(\log n_1)^{-4},\ ] ] and thus that @xmath487\le |{\widehat}{\lambda}| e^{- c(\log n_1)^4 } , \ ] ] which allows to conclude we set @xmath488\}}.\ ] ] using the fact that the sum is deterministically bounded by @xmath489 , we have @xmath490}{2 } \right]\ge \frac{{{\ensuremath{\mathbf e } } } _ n[j]}{2c ( \log n_1)^4}.\ ] ] from , we have for small @xmath1 , @xmath491 then as @xmath492 are centered gaussians , we have @xmath493\ } } \right]\ge c ( \log n_1)^{4 } ( \log \log n_1)^{-1/2},\ ] ] which combined with allows to conclude . let us remark that it seems technically easier to get a lower bound for @xmath494 $ ] for a given @xmath77 than to prove one directly for the limit . however as there is no obvious sub - additivity property which allows to compare the two . in @xcite , for @xmath8 we introduced the idea of replacing the boundary condition by an infinite volume free field in order to recover sub - additivity . in dimension @xmath13 , the infinite volume free field does not exists as the variance diverges with the distance to the boundary of the domain . a way to bypass the problem it to artificially introduce mass and then to find a comparison between the free energy of the system with massive free field and the original one . this is the method that we adopted in our previous paper ( see ( * ? ? ? * proposition 7.1 and lemma 7.2 ) ) . however our previous results turn out out to be a bit two rough for our proof . we present here an improvement of it ( proposition [ th : finitevol ] ) on which we build the proof of theorem [ mainres ] . let us recall the comparison used in @xcite . even it is not sufficient for our purpose in this paper , it will help us to explain the improvement presented in section [ optfinit ] . given @xmath450 and @xmath174 , we introduce the notation @xmath495}(\phi_x)\ ] ] and set @xmath496.\ ] ] and @xmath497 the existence of the above limit is proved in @xcite . we can compare this free - energy to the original one using the following result . [ massivecompa ] we have for every @xmath409 and @xmath177 @xmath498 where @xmath499 ^ 2 } \log \left ( 1 + \frac{m^2}{4 \left[\sin^2(\pi x/2)+ \sin^2(\pi y/ 2)\right]}\right ) { \mathrm{d}}x { \mathrm{d}}y.\ ] ] there exists @xmath500 such that for every @xmath221 we have @xmath501 moreover for all @xmath77 we have @xmath502\right].\ ] ] the result is proved in @xcite ( as proposition 7.1 and lemma 7.2 ) but let us recall briefly how it is done . for the first point , we have to remark that changing the height of the substrate ( i.e. replacing @xmath503 by @xmath504 in ) for the original model does not change the value of the free energy , that is , @xmath505 heuristically this is because the free field hamiltonian is translation invariant but a proof is necessary to show that the boundary effect are indeed negligible ( see ( * ? ? ? * proposition 4.1 ) ) . note that for the massive free field , the limit really depends on @xmath409 because adding an harmonic confinement breaks this translation invariance . then we can compare the partition of the two free fields by noticing that the density of the massive field with respect to the original one ( recall ) satisfies @xmath506 } \le \frac{1}{{{\ensuremath{\mathbf e } } } _ n\left[\exp\left ( -\frac{m^2}{2}\sum_{x \in \mathring{\lambda}_n } \phi^2_x\right)\right]},\ ] ] and that @xmath507= : \lim_{n\to \infty}\frac{1}{n^2}\log w^m_n = -f(m).\ ] ] equation then follows from of a sub - additive argument ( see the proof of proposition 4.2 . in @xcite or that of below ) . note that proposition [ massivecompa ] gives a bound on @xmath65 which depends only on the partition function of a finite system . @xmath508\right]-f(m).\ ] ] in particular we can prove theorem [ mainres ] , if for any @xmath95 , @xmath509 we can find values for @xmath409 and @xmath177 and @xmath77 such that the l.h.s . is positive . however it turns out that with our techniques , we can not prove that the l.h.s is positive for very small @xmath1 . this is mostly because of the presence of a @xmath510 factor in the asymptotic behavior of @xmath511 around @xmath191 . therefore we need a better criterion in which the subtracted term is proportional to @xmath512 . to obtain a more efficient criterion , we want to restrict the partition function to a set of @xmath24 where @xmath513 is much smaller than @xmath514 . we define @xmath515 as a set where the density @xmath516 takes `` typical '' values ( see proposition [ thednproba ] ) . for some constant @xmath517 , we set @xmath518 recall that @xmath519 denotes the law of the infinite volume massive free field ( see section [ secmass ] ) for the boundary condition @xmath23 . [ th : finitevol ] for any value of @xmath77 , and @xmath520 and @xmath177 we have @xmath521 \right ] -km^2.\ ] ] with the idea of working with a measure that does not depend on the boundary condition , we set similarly to @xmath522}(\phi(x)+h^{m,{\widehat}\phi}_n(x)),\ ] ] and @xmath523 with this notation and in view of the considerations of section [ grbc ] the expected value in the l.h.s . in is equal to @xmath524\right].\ ] ] before giving a proof of proposition [ th : finitevol ] let us show how we are going to use it to prove our lower bound on the free energy . for the remainder of the proof we set @xmath525 where @xmath526 ( we find that the computations are easier to follows with the letter @xmath527 instead of a specific number , in fact any value in the interval @xmath528 would also work ) . with proposition [ th : finitevol ] , the proof of the lower bound in is reduced to the following statement , whose proof will be detailed in the next three sections . [ mainproposition ] for any @xmath529 , there exists @xmath112 such that for any @xmath113 @xmath530\right ] -k(m_hn_h)^2\ge 1.\ ] ] indeed the result directly implies that @xmath531 let us start by setting @xmath532\\ = { { \ensuremath{\mathbf e } } } ^{m,{\widehat}\phi}_n \left [ \exp\left(\sum_{x\in { \widetilde}{\lambda}_n } \left ( { \beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal d } } } ^0_n } \right].\end{gathered}\ ] ] a simple computation ( see below ) is sufficient to show that for any @xmath533 we have @xmath534\ge 4^k { \widehat}{{\ensuremath{\mathbf e } } } ^m { { \ensuremath{\mathbb e } } } \left[\log z'_{n}\right].\ ] ] hence that it is sufficient to prove with @xmath77 replaced by @xmath535 for an arbitrary integer @xmath323 , or by the limit when @xmath323 tends to infinity . let us prove . we divide the box @xmath536 into @xmath381 boxes , @xmath537 , @xmath538 . set @xmath539 where @xmath540 is the @xmath541-th digit of the dyadic development of @xmath384 . set @xmath542 we notice that @xmath543 we define @xmath544 if we condition on the realization on @xmath24 in @xmath545 , the partition functions of the system of size @xmath546 factorizes into @xmath381 partition functions of systems of size @xmath77 , whose boundary conditions are determined by @xmath23 and @xmath547 , and we obtain @xmath548 \\ = \prod_{i=1}^{4}{{\ensuremath{\mathbf e } } } ^{m,{\widehat}\phi}_{2n } \left [ \exp\left ( \sum_{x\in { \widetilde}{\lambda}^i_{n } } ( { \beta}{\omega}_x-{\lambda}({\beta})+h)\delta^u_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal d } } } ^{0,i}_{n } } \ \bigg| \ \phi|_{{\gamma}_n } \right ] = : \prod_{i=1}^{4 } { \widetilde}z^i({\widehat}\phi , \phi|_{{\gamma}_n } , { \omega}).\end{gathered}\ ] ] by the spatial markov property for the infinite volume field , each @xmath549 has the same distribution as @xmath550 ( if @xmath23 and @xmath547 have distribution @xmath551 and @xmath552 respectively and the @xmath125s are iid ) . using and jensen s inequality for @xmath553 $ ] we have @xmath554\ge \sum_{i=1}^{4 } { { \ensuremath{\mathbb e } } } { \widehat}{{\ensuremath{\mathbf e } } } ^m { { \ensuremath{\mathbf e } } } ^{m,{\widehat}\phi}_{2n}\left [ \log { \widetilde}z^i({\widehat}\phi , \phi|_{{\gamma}_n } \ ) \right ] = 4 { { \ensuremath{\mathbb e } } } { \widehat}{{\ensuremath{\mathbf e } } } ^m\left [ \log z'_{n}\right ] , \end{split}\ ] ] which ends the proof of . now we set @xmath555 with @xmath323 large . in the computation , we write sometimes @xmath556 for @xmath557 for simplicity . we remark that for @xmath558 we have @xmath559 + m^2 \left [ \sum_{x\in { \lambda}_m } \phi_x h(x)+ \frac{1}{2 } \sum_{x\in { \lambda}_m } h^2(x)\right]\\ \le m^2 m^2 k+ m^2 \left [ \sum_{x\in { \lambda}_m } \phi_x h(x)+ \frac{1}{2 } \sum_{x\in { \lambda}_m } h^2(x)\right],\end{gathered}\ ] ] where the first inequality follows from the definition of @xmath560 and the last one from and is valid provided @xmath520 is sufficiently large . from this inequality we deduce that @xmath561 } \right]\\ : = e^{m^2k m^2 } z''_m.\end{gathered}\ ] ] to conclude the proof , we must show that the r.h.s . is not affected , in the limit , by the presence of @xmath556 ( which produces the two last terms and enters in the definition of @xmath562 ) i.e. that @xmath563={\textsc{f}}({\beta},h).\ ] ] we can replace @xmath562 by @xmath504 at the cost of a girsanov - type term in the density . for computations , it is pratical to define @xmath564 the distribution @xmath565 under @xmath566 is absolutely continuous with respect to that of @xmath24 . the density of its distribution @xmath567 with respect to @xmath566 is given by @xmath568 where we used the notation @xmath569 to obtain the second line in we have used the summation by part formula ( which is valid without adding boundary terms since the functions we are integrating have zero boundary condition ) and to obtain @xmath570 the substitution of @xmath556 by @xmath571 produces the second term ( boundary effects ) . hence the expectation in is equal to ( assume @xmath572 ) @xmath573}({\widehat}\phi(x))\\ + m^2\sum_{x\in { \widetilde}{\lambda}_m } \frac{h(x)^2+h^0(x)^2}{2 } -\sum_{x\in \partial{\lambda}_m}{\sum_{\substack{y\in \partial^- { \lambda}_m \\ y\sim x } } } \frac{h(x)h(y)}{2 } \bigg)\\ \times { { \ensuremath{\mathbf e } } } ^0_m\left [ \exp\left(\sum_{x\in { \widetilde}{\lambda}_m } \left ( { \beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x- { \sum_{\substack{x\in \partial{\lambda}_m , y\in \partial^- { \lambda}_m \\ y\sim x } } } h(x)\phi(y ) \right)\right].\end{gathered}\ ] ] let us show first that the exponential term in front of the expectation in does not affect the limit of @xmath574 . we have @xmath575}({\widehat}\phi(x))\right| \\ \le \lim_{m\to \infty } \frac{1}{m^2 } { { \ensuremath{\mathbb e } } } \sum_{x\in \partial { \lambda}_m \cap { \widetilde}{\lambda}_m}| { \beta}{\omega}_x-{\lambda}({\beta})+h |=0.\end{gathered}\ ] ] for the other terms , set @xmath576 being a maximum over @xmath577 gaussian variables of finite variance , it is not difficult to check that for all @xmath578 sufficiently large , @xmath579\le ( \log m)^2.\ ] ] moreover from the definition of @xmath580 we gave for any @xmath581 we have @xmath582 this implies that the maximum of @xmath556 is attained on the boundary and that @xmath583 this implies that @xmath584 in particular we have @xmath585 hence from , and , equation holds provided we can show that @xmath586 = { \textsc{f}}({\beta},h),\ ] ] where we have used the notation @xmath587 this is extremely similar to the proof of ( * ? ? ? * proposition 4.2 ) but we include the main line of the computation for the sake of completeness . first we note that because of uniform integrability , holds if we prove the convergence in probability , @xmath588={\textsc{f}}({\beta},h).\ ] ] note that conditioned to @xmath23 , @xmath589 is a centered gaussian random variable . we show in fact @xmath590 almost sure convergence for rather than convergence of the expectation of , but since @xmath591 \right|\\ \le m^{-2}\sum_{x\in { \widetilde}{\lambda}_m } | { \beta}{\omega}_x-{\lambda}({\beta})+h|+ m^{-2 } \log { { \ensuremath{\mathbf e } } } _ m\left[e^{t({\widehat}\phi,\phi)}\right]\\ = m^{-2}\sum_{x\in { \widetilde}{\lambda}_m } | { \beta}{\omega}_x-{\lambda}({\beta})+h|+ m^{-2}{\mathrm{var}}_{{{\ensuremath{\mathbf p } } } _ m } \left(t({\widehat}\phi,\phi ) \right),\end{gathered}\ ] ] and the sequence is uniformly integrable ( cf . ) , almost sure convergence implies convergence in @xmath592 . now to prove , we set @xmath593 as the covariance function of @xmath24 is positive , we have @xmath594\le { { \ensuremath{\mathcal m } } } _ m ^2 { { \ensuremath{\mathbf e } } } _ m \left [ \left(\sum_{x\in \partial{\lambda}_m}{\sum_{\substack{y\in \partial^- { \lambda}_m \\ y\sim x}}}\phi(y)\right)^2\right ] = 4(m-1){{\ensuremath{\mathcal m } } } _ m ^2.\ ] ] we define @xmath595 combining our bound on the variance and standard gaussian estimates , we obtain @xmath596&\le e^{-c m^{5/2}},\\ { { \ensuremath{\mathbf e } } } _ m\left [ e^{t({\widehat}\phi , \phi ) } { \mathbf{1}}_{a_m}\right]&\le e^{-c m^{5/2}}. \end{split}\ ] ] combining the second line of with an annealed bound we obtain that @xmath597=-\infty,\ ] ] and hence is equivalent to @xmath598={\textsc{f}}({\beta},h).\ ] ] to prove , we first note using the first line of that implies that @xmath599={\textsc{f}}({\beta},h).\ ] ] by definition of @xmath600 we have @xmath601}{{{\ensuremath{\mathbf e } } } _ m\left [ e^{\sum_{x\in { \widetilde}{\lambda}_m } \left ( { \beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{u}_x } { \mathbf{1}}_{a_m } \right ] } \right| \le m^{-1/4 } { { \ensuremath{\mathcal m } } } _ m.\ ] ] hence to conclude we just need to show that @xmath602 this follows from the definition of @xmath603 and borel - cantelli s lemma . the overall idea for the proof is to restrict the partition function to a set of typical trajectories @xmath24 and to control the first two moments of the restricted partition function to get a good estimate for the expected @xmath141 . however the implementation of this simple idea requires a lot of care . we decompose the proof in three steps . in section [ sketch ] , we briefly present these steps and combine them to obtain the proof and in section [ propnineone ] we perform the first step of the proof , which is the simpler one . the two other steps need some detailed preparatory work which is only introduced in section [ liminouze ] . the first step is to show that @xmath604 is a typical event in order to ensure that our restriction to @xmath604 in the partition function does not cost much . [ thednproba ] we can choose @xmath520 in a way that for all @xmath221 sufficiently large , for all @xmath605 , and for all realization of @xmath23 @xmath606\le c(\log n)^{-1/2}.\ ] ] the result is not used directly in the proof of proposition [ mainproposition ] but is a crucial input for the proof of proposition [ prop : boundary ] below . the aim of the second step is to show that at a moderate cost one can restrict the zone of the interaction to a sub - box @xmath607 defined by @xmath608 ^ 2.\ ] ] the reason for which we want to make that restriction is that it is difficult to control the effect of the boundary condition ( i.e. of @xmath195 ) in @xmath609 . inside @xmath607 however , due to the choice of the relative values of @xmath177 and @xmath77 in , @xmath195 is very small and has almost no effect . [ prop : boundary ] there exists an event @xmath610 satisfying @xmath611\le c ( \log n)^{-1/16}\ ] ] and a constant @xmath143 such that @xmath612\\ \ge { { \ensuremath{\mathbf e } } } ^m_n \left [ \exp\left(\sum_{x\in { \lambda}'_n } \left ( { \beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi , u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal c } } } _ n } \right]- c({\beta } ) ( \log \log n)^4(\log n)^{\alpha-1/16}. \end{gathered}\ ] ] finally we have to show that the expected @xmath141 of the restricted partition function in the r.h.s . of is indeed sufficiently large to compensate for the second term . we actually only prove that this is the case for the set of good boundary conditions @xmath23 which have no significant influence in the bulk of the box @xmath613 and show that the contribution of bad boundary condition is irrelevant . we have chosen @xmath614 in a way such that the density of expected density contact is very scarce ( the total expected number of contact in the box is a power of @xmath615 , see below ) , but the unlikely event that @xmath24 has a lot of contact is sufficient to make the second moment of the partition very large . hence for our analysis to work , it is necessary to restrict the partition function to trajectories which have few contacts . we set @xmath616 [ prop : inside ] we have * for @xmath77 sufficiently large @xmath617 \le n^{-4}.\ ] ] * for any @xmath618 @xmath619\ge -n^2 { \lambda}({\beta})-\log 2.\ ] ] * there exists a constant @xmath620 such that for any @xmath621 @xmath622\ge c h ( \log n)^{\alpha}-\log 2.\ ] ] using proposition [ prop : inside ] , we have @xmath623 \\ \ge -{\widehat}{{\ensuremath{\mathbf p } } } ^m [ { \widehat}{{\ensuremath{\mathcal a } } } ^{{\complement } } _ n]\left(n^2 { \lambda}({\beta})+\log 2\right)\\ + { { \ensuremath{\mathbb e } } } { \widehat}{{\ensuremath{\mathbf e } } } ^{m}\left [ \log { { \ensuremath{\mathbf e } } } ^m_n \left [ \exp\left(\sum_{x\in { \lambda}'_n } \left ( { \beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi , u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal b } } } _ n}\right ] { \mathbf{1}}_{{\widehat}{{\ensuremath{\mathcal a } } } _ n}\right ] \\ \ge c h ( \log n)^{\alpha}-1 . \end{gathered}\ ] ] using proposition [ prop : boundary ] and recalling our choice of parameters , we have , for @xmath1 sufficiently small @xmath624- k(mn)^2\\ \ge c h ( \log n)^{\alpha}-c({\beta})(\log \log n)^4 ( \log n)^{\alpha-\frac{1}{16}}- k ( \log n)^{1/2}-1\\ \ge ( c/2 ) ( \log n)^{\alpha-\frac{1}{20}},\end{gathered}\ ] ] where in the last line we used that @xmath625 . this is sufficient to conclude . again in this proof simply write @xmath556 for @xmath195 the proof simply relies on computing the expectation and variance of @xmath626 we have @xmath627 ^ 2\right]= { { \ensuremath{\mathbf e } } } ^m_n \left [ \sum_{x\in { \lambda}_n } \phi(x)^2\right]+\sum_{x\in { \widetilde}{\lambda}_n } h(x)^2.\ ] ] from , for an appropriate choice of @xmath142 , the following holds @xmath628 \ge \frac{1}{2\pi } | \log m | -c\ge \frac{2f(m)}{m^2}-c.\ ] ] now let us estimate the variance . with the cancellation of odd moments of gaussians , the expansion of the products gives @xmath629-\left({{\ensuremath{\mathbf e } } } ^m_n \left [ \sum_{x\in { \widetilde}{\lambda}_n } ( \phi(x)+h(x))^2\right]\right)^2\\ = { { \ensuremath{\mathbf e } } } ^m_n \left [ \left ( \sum_{x\in { \lambda}_n } \phi(x)^2 \right)^2\right ] - \left({{\ensuremath{\mathbf e } } } ^m_n \left [ \sum_{x\in { \lambda}_n } \phi(x)^2\right]\right)^2\\ + 4{{\ensuremath{\mathbf e } } } ^m_n\left [ \sum_{x , y \in { \widetilde}{\lambda}_n } \phi(x)\phi(y)h(x)h(y ) \right].\end{gathered}\ ] ] we treat the last term separately and first concentrate on the two firsts which correspond to the zero boundary condition case . we have @xmath630- { { \ensuremath{\mathbf e } } } ^m_n\left[\phi(x)^2\right]{{\ensuremath{\mathbf e } } } ^m_n\left[\phi(y)^2\right]= 2 \left [ g^{m,*}(x , y)\right]^2,\ ] ] and hence from we can deduce that @xmath631-\left({{\ensuremath{\mathbf e } } } ^m_n \left [ \sum_{x\in { \lambda}_n } \phi(x)^2\right]\right)^2 \\ = 2 \sum_{x , y\in { \lambda}_n } g^{m,*}(x , y ) \le c n^2 m^{-2}. \end{gathered}\ ] ] concerning the last term in , we bound it as follows @xmath632= \sum_{x , y\in { \widetilde}{\lambda}_n } g^{m,*}(x , y)h(x)h(y)\\ \le \sum_{x\in{\widetilde}{\lambda}_n } h(x)^2 \sum_{y\in { \lambda}_n } g^{m,*}(x , y)\le c m^{-2}\sum_{x\in { \widetilde}{\lambda}_n } h(x)^2,\end{gathered}\ ] ] where in the last inequality we used . this gives @xmath633 hence , as long as @xmath520 is chosen sufficiently large , using and - we obtain @xmath634\\ \le \frac{{\mathrm{var}}_{{{\ensuremath{\mathbf p } } } ^m_n}\left ( \sum_{x\in { \widetilde}{\lambda}_n } ( \phi(x)+h(x))^2 \right)}{\left({{\ensuremath{\mathbf e } } } ^m_n\left[\sum_{x\in { \widetilde}{\lambda}_n } ( \phi(x)+h(x))^2 \right]-n^2 \left[\frac{2 f(m)}{m^2}-k\right ] \right)^2 } \\ \le \frac { c m^{-2}\left ( \sum_{x\in { \widetilde}{\lambda}_n } h(x)^2+n^{2 } \right)}{\left((k - c)n^2+\sum_{x\in { \widetilde}{\lambda}_n } h(x)^2 \right)^2 } \le c m^{-2 } n^{-2}. \end{gathered}\ ] ] the result thus follows for our choice for the range of @xmath77 . both proofs require a detailed knowledge on the distribution of the number of contact in @xmath635 and in @xmath607 . the highly correlated structure of the field makes this kind of information difficult to obtain . we have chosen @xmath409 quite high in order to obtain a very low empirical density of contact . for this reason our problem is quite related to that of the study of the maximum and of the extremal process of the @xmath13-dimensional free field , which has been the object of numerous studies in the past @xcite together with the related subject of branching random walk @xcite or brownian motion @xcite . we borrow two key ideas from this literature : * the gaussian free field can be written as a sum of independent fields whose correlation spread on different scales . this makes the process very similar to the branching random walk . * the number of point present at a height close to the expected the maximum of the field is typically much smaller than its expectation ( that is : by a factor @xmath615 ) but this @xmath141 factor disapears if one conditions to a typical event . these two points are respectively developed in section [ galefash ] and [ typic ] . let us decompose the massive free field into independent fields in order to separate the different scales in the correlation structure . the idea of decomposing the gff is not new was used a lot to study the extremum and there are several possible choices ( see @xcite where a coarser decomposition is introduced or more recently @xcite ) . our choice of decomposition is made in order to have a structure similar to that present in @xcite . there are several possible choices for the decomposition . the advantage of the one we present below is that the kernel of all the fields are expressed in terms of the heat - kernel , for which we have good estimates ( cf . section [ hkernel ] ) . set ( recall [ greenff ] ) @xmath636 ( it does not depend on @xmath230 as @xmath637 is translation invariant ) . we perform the decomposition of @xmath24 into a sum @xmath323 subfield , each of which having ( roughly ) unit variance . with this construction , @xmath638 is the final step of a centered gaussian random walk with @xmath323 steps . with this in mind we define a decreasing sequence of times @xmath639 , @xmath640 as follows @xmath641 this definition implies that @xmath642 from the local central limit theorem we can deduce that there exists a constant @xmath500 such that @xmath643 we define @xmath644 to be a sequence of centered gaussian fields ( we use @xmath645 to denote their joint law ) indexed by @xmath180 , each with covariance functions given by @xmath646 and set @xmath647 note that the covariance of @xmath648 is given by @xmath649 and for this reason we simply set @xmath650 and work from now on this extended probability space . for this reason we use simply @xmath645 instead of @xmath651 ( this should bring no confusion as @xmath177 and @xmath77 a now fixed by ) . note that the distribution of the field @xmath652 in the bulk of @xmath180 is `` almost '' translation invariant and its variance is very close to one . when @xmath230 is close to the boundary @xmath653 becomes smaller , and this effect starts at distance @xmath654 from the boundary . the distance @xmath655 is also the scale on which covariance function @xmath656 varies in the bulk . for this reason it is useful to set @xmath657 as a consequence of , of the definition of @xmath323 and that @xmath658 , we have @xmath659 , up to a @xmath660 correction : there exists a constant @xmath142 such that @xmath661- ( i - j(x))_+ \right | \le c\ ] ] indeed from lemma [ lem : kerestimate ] @xmath662 , we have @xmath663 as the variance of @xmath664 is bounded by @xmath80 ( or @xmath13 when @xmath665 ) this implies @xmath666\le c+(i - j(x))_+.\ ] ] finally we obtain the other bound using the fact that , as the increments have variance smaller than one ( ore two for the last one ) we have @xmath667-{{\ensuremath{\mathbf e } } } [ \phi^2_i(x ) ] \le k - i+1\ ] ] and we conclude using . now we are going to use the decomposition in order to obtain finer results on the structure of the field @xmath24 . the idea is to show that with high probability the trajectory of @xmath668 tend to stay below a given line , for all @xmath44 , and thus if @xmath638 reaches a value close to the maximum of the field , then conditioned to its final point , @xmath669 look more like a brownian excursion than like a brownian bridge , as it `` feels '' a constraint from above . if one restricts to the typical event described above , this constraint yields a loss of a factor @xmath323 ( hence @xmath615 ) in the probability of contact . note that for technical reasons , points near the boundary are a bit delicate to handle and thus we choose to prove a property in a sub - box @xmath670 which excludes only a few points of @xmath180 . we set @xmath671,\ ] ] and @xmath672 and define the event @xmath673 we show that this event is very typical . this is a crucial step to define the event @xmath674 and to estimate the probability of @xmath675 . [ th : condfirstmom ] we have @xmath676\ge 1-(\log n)^{-99},\ ] ] we define for @xmath677 @xmath678 \right).\ ] ] it is trivial to check that it is a martingale for the filtration @xmath679 integrating the second inequality in on the interval @xmath680 , we have for all @xmath681 @xmath682\le c |x - y|^2 e^{-4\pi(k - i)}.\ ] ] using a union bound , this implies that for @xmath77 sufficiently large @xmath683 \le \frac{1}{n}.\ ] ] on the complement of this event , if for a fixed @xmath684 we have@xmath685 then @xmath686 } .\ ] ] now as @xmath687 , we realize that in the range of @xmath281 which is considered @xmath688 and hence from we have @xmath689 \le i - j(x)+c+1.\ ] ] for this reason , if @xmath77 is sufficiently large , implies that . @xmath690 the last inequality is valid for @xmath77 sufficiently large , it is obtained by using the definition and the fact that @xmath684 ( which implies that @xmath691 ) . using and the fact that @xmath578 is a martingale , we conclude that @xmath692\le \frac{1}{n}+ { { \ensuremath{\mathbf p } } } \left [ \exists i , \ , m_i\ge ( \log n)^{100 } \right ] \le \frac{1}{n}+ ( \log n)^{-100}.\ ] ] to conclude this section , we note that conditioning on the event @xmath693 the probability of having a contact drops almost by a factor @xmath694 , in the bulk of the box . more precisely [ probacont ] there exists a constant @xmath142 such that * for all @xmath44 we have @xmath695\le cn^{-2}(\log n)^{1+\alpha}.\ ] ] * for all @xmath684 , we have @xmath696 \le c n^{-2 } ( \log n)^{\alpha } \left [ h(x)^2+(\log \log n)^2\right ] \exp\left(\gamma h(x)- \frac{\gamma^2}{2}j(x ) \right).\ ] ] in particular @xmath697\le c n^{-2 } ( \log n)^{\alpha } ( \log \log n)^2.\ ] ] for the first point we notice that under law @xmath698 , @xmath699 is distributed like an infinite volume free field and hence has covariance @xmath700 . for this reason if @xmath701 we have @xmath702= \int_{u-1}^{u+1}\frac{{\mathrm{d}}t}{\sqrt{2\pi g^m(x , x ) } } e^{-\frac{-t^2}{2g^{m}(x , x ) } } \le \frac{2}{\sqrt{2\pi g^m(x , x ) } } e^{-\frac{-(u-1)^2}{2g^{m}(x , x)}},\ ] ] and the result ( the upper bound , but the lower bound is proved similarly ) follows by replacing @xmath409 by its value , and @xmath703 by the asymptotic estimate @xmath704 . let us now focus on the second point . first we note that the result is completely obvious is @xmath705 ( the l.h.s . of is larger than one ) . hence we assume @xmath706 . then note that @xmath707\le { { \ensuremath{\mathbf p } } } \big [ \forall i\in { \llbracket}j(x),k { \rrbracket},\ \phi_i(x)\le \gamma ( i - j(x))+100(\log \log n)\ ; \\ \phi(x)+h(x)\in [ u-1,u+1 ] \big]\end{gathered}\ ] ] a first step is to show that @xmath708 \big]\le c n^{-2}(\log n)^{\alpha+1 } \exp\left(\gamma h(x)- \frac{\gamma^2}{2}i(x ) \right).\ ] ] using the gaussian tail estimate and we have @xmath709 \big]\le \frac{c\sqrt{k}}{u - h(x)}\exp\left(-\frac{\left(u-1-h(x)\right)^2}{2(k - j(x)+c ) } \right).\ ] ] note that the factor in front of the exponential is smaller than @xmath710 when @xmath706 . concerning the exponential term , notice that @xmath711 this yields . to conclude the proof we need to show that for all @xmath712 $ ] @xmath713\\ \le c(\log n)^{-1}\left ( h(x)^2+(\log \log n)^2\right).\end{gathered}\ ] ] we use lemma [ lem : bridge ] , for the re - centered walk @xmath714.\ ] ] let @xmath715 denote the variance of @xmath716 and @xmath717 that of @xmath638 . we have by standard properties of gaussian variables @xmath718=(v_i / v ) t.\ ] ] using the bound , for all the considered values of @xmath479 we have @xmath719 hence we have @xmath720\\ \le { { \ensuremath{\mathbf p } } } \big [ \ \forall i\in \ { 0,\dots , k\ } , \phi_i(x)\le 200 ( \log \log n)+ |h(x)| \ | \ \phi(x)=0 \ \big],\end{gathered}\ ] ] and we conclude using lemma [ lem : bridge ] . we are now ready to define the event @xmath674 . we set @xmath721 where @xmath722 \right\}.\ ] ] from markov s inequality , it is obvious that @xmath723 \le ( \log n)^{-1/16},\ ] ] and we can conclude ( provided that @xmath77 is large enough ) by using proposition [ th : condfirstmom ] , that holds . let us turn to the proof of . we want to get rid of the environment outside @xmath607 . the reader can check ( by computing the second derivative that can be expressed as a variance ) @xmath724 \right]\end{gathered}\ ] ] is convex in @xmath725 and has zero derivative at @xmath191 . hence reaches its minimum when @xmath725 equals zero , and @xmath726 \right]\\ \ge { { \ensuremath{\mathbb e } } } \left [ \log { { \ensuremath{\mathbf e } } } \left [ \exp\left ( \sum_{x\in { \lambda}'_n } ( { \beta}{\omega}_x+h-{\lambda}({\beta } ) ) \delta^{{\widehat}\phi , u}_x -{\lambda}({\beta } ) \sum_{x\in { \widetilde}{\lambda}_n\setminus { \lambda}'_n}\delta^{{\widehat}\phi , u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal d } } } _ n } \right ] \right]\\ \ge { { \ensuremath{\mathbb e } } } \left [ \log { { \ensuremath{\mathbf e } } } \left [ \exp\left ( \sum_{x\in { \lambda}'_n } ( { \beta}{\omega}_x+h-{\lambda}({\beta } ) ) \delta^{{\widehat}\phi , u}_x\right){\mathbf{1}}_{{{\ensuremath{\mathcal c } } } _ n } \right ] \right ] \\ - ( \log n)^{1/16}{\lambda}({\beta}){{\ensuremath{\mathbf e } } } \left[\sum_{x\in { \widetilde}{\lambda}_n\setminus { \lambda}'_n}\delta^{{\widehat}\phi , u}_x \ | \ { { \ensuremath{\mathcal a } } } _ n \right],\end{gathered}\ ] ] where the last line is obtained by restricting the expectation to @xmath674 in order to bound @xmath727 from below . finally , using lemma [ probacont ] and the definition of @xmath607 we obtain that @xmath728 \le c(\log \log n)^4(\log n)^{\alpha-1/8},\ ] ] which is sufficient to conclude . we start with the easy part of the proposition : showing that the probability of a bad boundary condition is scarce , and that for this reason , a quite rough bound is sufficient to bound their contribution to the total expectation . to prove , we use lemma [ lem : kerestimate ] . for a fixed @xmath729 , we set in the next equation @xmath730 . we have @xmath731= \int^{\infty}_0 e^{-m^2 t } [ p_t(x , x)-p^*_t(x , x ) ] { \mathrm{d}}t\\ \le \int^{\infty}_0 \frac{c}{t } e^{-m^2 t } \exp \left ( -c^{-1}\min\left(\frac{d^2}{t},d\log[(d / t)+1 ] \right)\right){\mathrm{d}}t\\ \le e^{-c'dm } \le \exp\left(-c ' ( \log n)^{1/8 } \right).\end{gathered}\ ] ] we have used in the last inequality that @xmath732 for @xmath729 . hence we have for any @xmath729 @xmath733\le \exp\left(-e^{c(\log n)^{1/8}}\right),\ ] ] and we can conclude using a union bound . to prove , we use jensen s inequality and obtain @xmath734\\ \ge { { \ensuremath{\mathbb e } } } { { \ensuremath{\mathbf e } } } \left[\sum_{x\in { \lambda}'_n } \left ( { \beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi , u}_x \ \big| \ { { \ensuremath{\mathcal c } } } _ n \right ] \ge -{\lambda}({\beta})n^2 . \end{gathered}\ ] ] hence the conclusion follows from @xmath735\ge 1/2 $ ] . proving that good boundary conditions give a good contribution to the expected @xmath141 partition function , is the most delicate point . we divide the proof in several steps . first we want to show that conditioned on the event @xmath675 , the expected @xmath141 partition function is close to the corresponding annealed bound ( obtained by moving the expectation w.r.t . @xmath58 inside the @xmath141 ) . this result is obtained by a control of the second moment of the restricted partition function . the second point is to show that @xmath738 $ ] is large . what makes this difficult is that @xmath739 typically does not behave like its expectation @xmath740 $ ] ( cf . lemma [ probacont ] ) we are going to prove that conditioned to @xmath693 , @xmath739 almost behaves like its expectation . to prove such a statement , we will impose a restriction to the trajectories which is slightly stronger than @xmath693 , as this makes computation easier . combining and we have for @xmath736 , @xmath742\\ \ge { { \ensuremath{\mathbb e } } } \log { { \ensuremath{\mathbf e } } } \left [ \exp\left(\sum_{x\in { \lambda}'_n } \left ( { \beta}{\omega}_x-{\lambda}({\beta})+h \right)\delta^{{\widehat}\phi , u}_x\right ) \ | \ { { \ensuremath{\mathcal b } } } _ n\right ] + \log { { \ensuremath{\mathbf p } } } \left [ { { \ensuremath{\mathcal b } } } _ n \right]\\ \ge h{{\ensuremath{\mathbf e } } } \left [ l_n \ | \ { { \ensuremath{\mathcal b } } } _ n \right ] -1 \ge c h ( \log n)^{\alpha}-1 . \end{gathered}\ ] ] first let us get a rough estimate on the probability of @xmath675 , valid for @xmath77 sufficiently large @xmath743\le c(\log n)^{-\frac{1-\alpha}{4}}.\ ] ] according to , for all @xmath736 @xmath744\le c(\log n)^{\alpha}(\log \log n)^2.\ ] ] hence using the markov inequality and the definition of @xmath675 , we have @xmath745 \ge { { \ensuremath{\mathbf p } } } [ l_n\ge ( \log n)^{\frac{1+\alpha}{2 } } \ | \ { { \ensuremath{\mathcal a } } } _ n ] \le c(\log n)^{-\frac{1-\alpha}{2}}(\log \log n)^2.\ ] ] we deduce from the above equation , using the fact that @xmath746 $ ] tends to one very fast ( proposition [ th : condfirstmom ] ) . we continue the proof by setting , @xmath747,\ ] ] and @xmath748 $ ] . we can bound the first term from below using jensen s inequality as follows @xmath749= \log { { \ensuremath{\mathbb e } } } \left [ y_n \right]+ { { \ensuremath{\mathbb e } } } \log[\zeta].\ ] ] we have @xmath750=\log { { \ensuremath{\mathbf e } } } \left [ \exp\left ( h l_{n}\right){\mathbf{1}}_{{{\ensuremath{\mathcal b } } } _ n } \right ] \ge h { { \ensuremath{\mathbf e } } } \left[l_{\alpha } \ | \ { { \ensuremath{\mathcal b } } } _ n \right]+ \log { { \ensuremath{\mathbf e } } } [ { { \ensuremath{\mathcal b } } } _ n].\ ] ] by , the second term is larger than @xmath751 . to estimate @xmath752 $ ] we simply compute the second moment of @xmath753 . we have @xmath754= { \widetilde}{{\ensuremath{\mathbf e } } } _ h^{\otimes 2}\left[\exp\left(\sum_{x\in { \lambda}'_n}\chi({\beta})\delta^{(1)}_x \delta^{(2)}_x \right ) \right],\ ] ] where @xmath755 and @xmath756}\exp\left ( hl_{n } \right ) { \mathbf{1}}_{{{\ensuremath{\mathcal b } } } _ n}.\ ] ] note that as a consequence of the definition of @xmath675 for @xmath77 sufficiently large , the density is bounded from above as follows @xmath757}\exp\left ( h ( \log n)^{\frac{1+\alpha}{2 } } \right)\le n^{1/4}.\ ] ] using the inequality @xmath758x\ ] ] valid for @xmath759 $ ] , we obtain @xmath760\le 1+e^{\chi({\beta})(\log n)^{\frac{1+\alpha}{2}}}{\widetilde}{{\ensuremath{\mathbf e } } } _ h^{\otimes 2}[\delta^{(1)}_x \delta^{(2)}_x]\\ \le 1+n^{1/2}e^{\chi({\beta})(\log n)^{\frac{1+\alpha}{2}}}\sum_{x\in { \lambda}'_n } ( { { \ensuremath{\mathbf e } } } [ \delta^{{\widehat}\phi}_x])^2\le 1+n^{3/4}\sum_{x\in { \lambda}'_n } ( { { \ensuremath{\mathbf e } } } [ \delta^{{\widehat}\phi}_x])^2.\end{gathered}\ ] ] note that from and our choice for @xmath177 , the variance of @xmath24 satisfies @xmath761 thus using our assumption on @xmath762 , and replacing @xmath409 by its value we obtain that for all @xmath729 @xmath763 ^ 2\le \left [ \frac{2}{\sqrt{2\pi g^{*,m}(x , x ) } } \exp\left(- \frac{(u-1-h(x))^2}{2g^{*,m}(x , x ) } \right ) \right]^2\le c n^{-4}(\log n)^{2(1+\alpha)}.\ ] ] thus we deduce from that @xmath764 - 1\le n^{-1}.\ ] ] this ensures that @xmath753 is close to one with a large probability . however to estimate @xmath765 $ ] , we also need some estimate on the right - tail distribution of @xmath766 . we use a rather rough one @xmath767 to conclude we note that for @xmath768 we have @xmath769 and hence that @xmath770={{\ensuremath{\mathbb e } } } [ \log(\zeta)+1-\zeta]\ge -{{\ensuremath{\mathbb e } } } [ ( \zeta-1)^2]+{{\ensuremath{\mathbb e } } } \left [ ( \log ( \zeta ) + 1-\zeta){\mathbf{1}}_{\ { \zeta\le 1/2\ } } \right].\ ] ] the first term in the r.h.s . can be controlled using . by cauchy - schwartz , the second term is smaller in absolute value than @xmath771)^{1/2}\left({{\ensuremath{\mathbb e } } } \left [ ( \log \zeta+1-\zeta)^2 { \mathbf{1}}_{\ { \zeta\le 1/2\ } } \right ] \right)^{1/2}\le ( { { \ensuremath{\mathbb p } } } [ \zeta\le 1/2])^{1/2}\left({{\ensuremath{\mathbb e } } } \left [ ( \log \zeta)^2\right ] \right)^{1/2 } .\ ] ] using chebychev inequality together with , we get that @xmath772\le 4n^{-1}.\ ] ] using and the fact that @xmath58 have exponential tails ( cf . assumption ) , we have @xmath773\le c ( \log n)^2.\ ] ] altogether we obtain that @xmath750\ge h { { \ensuremath{\mathbf e } } } \left[l_{n } \ | \ { { \ensuremath{\mathcal b } } } _ n \right]+ \log { { \ensuremath{\mathbf e } } } [ { { \ensuremath{\mathcal b } } } _ n]- cn^{-1/2}(\log n),\ ] ] and we can conclude using . instead of counting all the contacts , we decide to consider only a subset of them : those for which the trajectory @xmath774 stays below a given line . we choose the restriction to be a bit stronger than the one used in the definition of the event @xmath693 . we set @xmath775 , \ , \forall i\in { \llbracket}1,k{\rrbracket } , \ , \phi_i(x)\le \frac{u i}{k}+10\big \}},\\ l'_n&:= \sum_{x\in { \lambda}'_n } \delta'_x . \end{split}\ ] ] let us first show how to reduce the proof of lemma [ sdasda ] to a control on the two first moment of @xmath776 . we have @xmath777\ge { { \ensuremath{\mathbf e } } } \left[l'_{n } { \mathbf{1}}_{{{\ensuremath{\mathcal b } } } _ n}\right ] = { { \ensuremath{\mathbf e } } } \left[l'_{n}\right]- { { \ensuremath{\mathbf e } } } \left [ l'_{n } { \mathbf{1}}_{{{\ensuremath{\mathcal b } } } ^{\complement}_n}\right]\\ \ge { { \ensuremath{\mathbf e } } } \left[l'_{n}\right]- \sqrt { { { \ensuremath{\mathbf e } } } \left [ ( l'_{n})^2 \right ] } \sqrt { { { \ensuremath{\mathbf p } } } \left [ { { \ensuremath{\mathcal b } } } ^{\complement}_n \right ] } . \end{gathered}\ ] ] thus we can conclude provided that one can prove the two following bounds on the expectation and variance of @xmath776 @xmath778 & \ge c(\log n)^\alpha,\\ { { \ensuremath{\mathbf e } } } [ ( l'_n)^2]&\le c(\log n)^{2\alpha}(\log \log n)^{8}. \end{split}\ ] ] it is then sufficient to combine these results with and . hence we need to prove the two following results . before giving the details of these lemmas , let us prove . the bound on the expectation follows immediately from . concerning the bound on the variance , as for a fixed @xmath786 , we have @xmath787 and a trivial bound of @xmath788 for the case @xmath789 . hence we have @xmath790= \sum_{x , y\in { \lambda}'_n } { { \ensuremath{\mathbf e } } } [ \delta'_{x}\delta'_{y}]\\ \le c ( \log n)^{2\alpha+3}(\log \log n)^8 \left [ ( \log n)^{-3}+ ( \log n)^{-1/2 } \sum_{l=1}^k \frac{e^{-l \left(4\pi-\frac{u^2}{2k}\right)}}{(l+1)^{3/2}(k - l+1)^3 } \right ] . \end{gathered}\ ] ] we must then control the above sum . note that from and we deduce that @xmath791 and hence that @xmath792 this implies . if @xmath793 ( which is satisfied if @xmath1 is small enough because as @xmath794 we have @xmath762 ) , we obtain from the expression of the gaussian density @xmath795+u - h(x ) \big]\ge \frac{2}{\sqrt{2\pi g^{*,m}(x , x)}}\exp\left ( -\frac{(u - h(x)+1)^2}{2g^{*,m}(x , x ) } \right).\ ] ] using again that @xmath796 $ ] , we obtain , using @xmath797+u - h(x)\right]\ge cn^{-2}(\log n)^{1+\alpha}.\ ] ] now we can conclude provided we show that for all @xmath479 in the interval @xmath798 $ ] , we have @xmath799\ge \frac{c}{\log n}.\ ] ] let us recall the notation of section [ typic ] : @xmath715 denotes the variance of @xmath716 . for @xmath800 , we have @xmath801{\mathrm{d}}t.\ ] ] hence from we have @xmath802.\ ] ] we can check that and @xmath803 $ ] implies @xmath804 to prove , we use simply lemma [ lem : bridge ] @xmath805 for the re - centered process @xmath806 . we have @xmath807\\ \ge { { \ensuremath{\mathbf p } } } \left [ \forall i\in { \llbracket}1,k{\rrbracket } , \ , \phi_i(x)\le 1 \ | \ \phi(x)=t \right ] \ge \frac{c}{k}. \end{gathered}\ ] ] we replace @xmath784 and @xmath808 and their intricate correlation structure by a simplified picture . let @xmath809 , @xmath810 be two walks , with iid standard gaussian increments which are totally correlated until step @xmath541 and independent afterwards . more formally the covariance structure is given by @xmath811:= \min(i_1,i_2,j),\\ { { \ensuremath{\mathbf e } } } [ x^{(1)}_{i_1 } & x^{(1)}_{i_2}]={{\ensuremath{\mathbf e } } } [ x^{(2)}_{i_1 } x^{(2)}_{i_2}]:= \min(i_1,i_2 ) . \end{split}\ ] ] for @xmath812 we set @xmath813 . the simplified version of we are going to prove is the following @xmath814 \right ] \\ \le \frac{c(\log \log n)^8}{(j+1)^{3/2}(k - j+1)^3}\exp\left ( -\frac{(k+j)u^2}{2k^2 } \right).\end{gathered}\ ] ] note that we replaced the interval @xmath815 $ ] and @xmath816 $ ] by @xmath817 $ ] , and we also do so in the true proof of lemma [ covariancee ] . this is ok since we are looking for an upper bound and as @xmath621 , the latter inverval includes the other two . the strategy is to first evaluate the probability @xmath818 \right],\ ] ] and then compute the cost of the constraint @xmath819 using lemma [ lem : bridge ] and the fact that conditioned to @xmath820 , @xmath821 and @xmath822 , the processes @xmath823 , @xmath824 and @xmath825 are three independent brownian bridges . for the first step , notice that we have @xmath826 \\ = \frac{1}{(2\pi)^{3/2}(k - j)\sqrt{j } } \exp\left(-\frac{t^2}{2j}-\frac{(s_1-t)^2+(s_2-t)^2}{2(k - j)}\right ) { \mathrm{d}}t { \mathrm{d}}s_1 { \mathrm{d}}s_2.\end{gathered}\ ] ] with the constraint @xmath827 $ ] and @xmath828 , at the cost of loosing a constant factor we can replace @xmath829 and @xmath830 by @xmath831 . we obtain , after integrating over @xmath829 and @xmath830 , @xmath832 \right]\le \frac{c}{(k - j)\sqrt{j}}\exp\left(-\frac{t^2}{2j}-\frac{(u-2-t)^2}{k - j}\right ) { \mathrm{d}}t \\ \le \frac{c}{(k - j)\sqrt{j } } \exp\left ( -\frac{(2k - j)u^2}{2k^2}-\left(\frac{u}{k}-\frac{2}{k - j}\right ) \left(\frac{uj}{k}-t\right ) \right ) { \mathrm{d}}t.\end{gathered}\ ] ] note that due to our choice for @xmath409 and value of @xmath323 we have @xmath833 $ ] provided that @xmath1 is sufficiently small and @xmath834 is sufficiently large ( and hence the term can be replaced by @xmath835 at the cost of changing the value of @xmath142 ) . now using lemma [ lem : bridge ] ( after re - centering the process ) , we obtain that @xmath836\\ = { { \ensuremath{\mathbf p } } } \left [ x_i \le \left(\frac{ui}{k}+10\right)- \frac{i t}{j},\ \forall i\in{\llbracket}0,j{\rrbracket}\ | \ x_j=0 \right ] \\ \le c j^{-1}\left ( \left(\frac{uj}{k}-t\right)^2 + ( \log j)^2 \right),\end{gathered}\ ] ] where we have used that for @xmath837 and @xmath812 @xmath838 in the same manner we obtain that for @xmath839 @xmath840 \right]\\ \le c(k - j)^{-1}\left ( \left(\frac{uj}{k}-t\right)^2 + ( \log ( k - j))^2\right).\end{gathered}\ ] ] hence using -- and conditional independence we obtain that @xmath841\ ; \ x_j\in { \mathrm{d}}t \right ] \\ \le c ( k - j)^{-3 } j^{-3/2 } ( \log k)^6 \exp\left ( -\frac{(2k - j)u^2}{2k^2}- ( \gamma/2)\left(\frac{uj}{k}-t\right ) \right ) { \mathrm{d}}t,\end{gathered}\ ] ] which after integration over @xmath842 gives @xmath843 \right]\\ \le c ( k - j)^{-3 } j^{-3/2 } ( \log k)^6\exp\left ( -\frac { ( 2k - j)u^2}{2k^2 } \right).\end{gathered}\ ] ] now , we are ready to handle the case were @xmath844 and @xmath845 are replaced by @xmath716 and @xmath846 . some adaptation are needed since the increments of @xmath716 and @xmath846 have a less simple correlation structure but the method presented above is hopefully robust enough to endure such mild modifications . given @xmath230 and @xmath281 set @xmath847 ^ 2.\ ] ] let us prove that there exists a constant @xmath142 such that @xmath848 to see this it is sufficient to remark that @xmath849 { \mathrm{d}}t\\ = \frac{1}{2}\int_{t_i}^{\infty } e^{-m^2 t } \left [ p_t(x , x)+p_t(x , y)\right ] { \mathrm{d}}t- r_i(x , y).\end{gathered}\ ] ] where @xmath850 { \mathrm{d}}t.\ ] ] using , we see that @xmath214 can be replaced by @xmath163 at the cost of a small correction i.e. that @xmath851 is small . using the definition of @xmath639 we have for @xmath852 @xmath853 { \mathrm{d}}t= i- \int_{t_i}^{\infty } e^{-m^2t}\left [ p_t(x , x)-p_t(x , y)\right ] { \mathrm{d}}t,\ ] ] while for @xmath854 we have @xmath855 { \mathrm{d}}t\\ = \frac{i+j}{2}- \int_{t_j}^{\infty } e^{-m^2 t } \left [ p_t(x , x)-p_t(x , y ) \right ] { \mathrm{d}}t+ \int^{t_j}_{t_i } e^{-m^2 t } p_t(x , y ) { \mathrm{d}}t . \end{gathered}\ ] ] the kernel estimates and then allow to conclude that the integrals in the r.h.s of and are bounded by a constant and thus that hold . similarly to , we are first going to show that we have , for all @xmath842 , @xmath856 \big]\\ \le \frac{c}{(k - j)\sqrt{j } } \exp\left ( -\frac{(2k - j)u^2}{2k^2}- ( \gamma/2 ) \left(\frac{uj}{k}-t\right ) \right ) { \mathrm{d}}t.\end{gathered}\ ] ] using the independence of @xmath857 and @xmath858 and the fact that , up to correction of a constant order their respective variance are respectively equal to @xmath541 and @xmath859 ( cf - ) , we can obtain ( provided that @xmath860 is large enough ) , similary to that @xmath861 \right]\\ \le \frac{c}{\sqrt{j(k - j ) } } \exp\left ( -\frac{(2k - j)u^2}{2k^2}- ( \gamma/2 ) \left(\frac{uj}{k}-t\right ) \right ) { \mathrm{d}}t.\end{gathered}\ ] ] now , on top of that , we want to show that @xmath862 \ | \ z_j \in { \mathrm{d}}t , z_k \in [ u-2,u+2 ] \ \big]\le c(k - j)^{-1/2}.\ ] ] as @xmath863 is a gaussian we can prove by showing that @xmath864 } \left [ \phi(x)-\phi(y ) \right]\ge c(k - j),\ ] ] at least when @xmath860 is large : it implies that conditional density is bounded by @xmath865 and thus that holds . in fact we prove this bound for the variance conditioned to @xmath866 and @xmath867 ( which is smaller as the conditioning is stronger ) as it is easier to compute . if one sets @xmath868 one can remark , first using the fact that the increments of @xmath869 are independent and then using the usual formula for the conditional variance of gaussian variable , that @xmath870 } = { { \ensuremath{\mathbf e } } } [ ( z'_k - z'_j)^2]-\frac{\left({{\ensuremath{\mathbf e } } } \left[(z'_k - z'_j)(z_k - z_j)\right]\right)^2}{{{\ensuremath{\mathbf e } } } [ ( z_k - z_j)^2]}.\ ] ] using ( to replace @xmath214 by @xmath163 ) and ( to control the term @xmath871 ) we have @xmath872= \int^{t_j}_0 e^{-m^2 t } \left [ p^*_{t}(x , x)+p^*_t(y , y)-2p^*_t(x , y)\right]{\mathrm{d}}t \\ \ge \int^{t_j}_0 e^{-m^2 t } \left[p_{t}(x , x)+p_t(y , y)\right]{\mathrm{d}}t - c\ge 2(k - j)-c.\end{gathered}\ ] ] obviously @xmath873 $ ] is of the same order , and from again . @xmath874|= \left| \frac{1}{2}\int^{t_j}_0 e^{-m^2 t } ( p^*_{t}(x , x)-p^*_t(y , y ) ) { \mathrm{d}}t \right|\le 1.\ ] ] hence combining these inequalities in we obtain that holds . to conclude the proof we need to show that @xmath875 , z_i\le \frac{uj}{k}+10 \ | \ z_{j}=t \right ] \le c \left [ \left(\frac{uj}{k}-t\right)^2+(\log j)^2 \right ] j^{-1}\ ] ] and @xmath876 , \phi_i(x ) , \phi_i(y)\le \frac{uj}{k}+10 \ | \ z_{j}=t , \ \phi(x),\phi(y)\in[u-2,u+2 ] \ \right ] \\ \le \left [ \left(\frac{uj}{k}-t\right)^2+(\log j)^2 \right]^2 ( k - j)^{-2}.\end{gathered}\ ] ] indeed using conditional independence we can multiply the inequalities and with to obtain @xmath877 \\ \le \frac{c\left [ \left(\frac{uj}{k}-t\right)^2+(\log k)^2 \right]^3}{(k - j)^{3}j^{3/2 } } \exp\left ( -\frac{(2k - j)u^2}{2k^2}- ( \gamma/2 ) \left(\frac{uj}{k}-t\right ) \right ) { \mathrm{d}}t,\end{gathered}\ ] ] and conclude by integrating over @xmath479 . the proof of is quite similar to that of . @xmath878\\ = { { \ensuremath{\mathbf p } } } \left [ \forall i\le j,\ z_i\le \frac{u i}{k}+10- ( u_i / u_j)t \ | \ z_j=0 \right].\end{gathered}\ ] ] we use to obtain for all @xmath879 , @xmath880 and apply lemma [ lem : bridge ] , we obtain @xmath881\le c j^{-1}\left ( \left(\frac{uj}{k}-t\right)^2 + ( \log j)^2 \right).\ ] ] to prove we have to be more careful as the increments of @xmath638 and @xmath882 are correlated . it is more practical in the computation to condition to the constraint @xmath883 than to @xmath884 . to obtain a bound we then take the maximum over the constraint @xmath885 . we consider only the case @xmath886 in the conditioning as the others can be deduced by monotonicity ( which follows from positive correlations in the gaussian processes that are considered ) . we can consider without loss of generality that @xmath887 the upper bound is due to the conditioning , and if the lower - bound is violated , @xmath479 is so small that the r.h.s . of is larger than one . similarly to , using to control the value of @xmath888 we can prove @xmath889,\ \phi_i(x)\le \frac{u i}{k}+10 \ | \ \phi_j(x)=t_1,\ ; \ \phi(x)\in[u-2,u+2 ] \right ] \\ \le c ( k - j)^{-1}\left ( \left(\frac{uj}{k}-t_1\right)^2 + ( \log ( k- j))^2 \right).\end{gathered}\ ] ] now the challenge lies in estimating the cost of the constraint @xmath890 , on the segment @xmath891 , knowing @xmath882 , @xmath892 and @xmath716 , @xmath893 . after conditioning to @xmath892 and @xmath894 , note that @xmath895 is still a process with independent increments . hence we can apply lemma [ lem : bridge ] provided we get to know the expectation and variance of these increments . let @xmath888 denote the conditional variance of @xmath846 knowing @xmath896 . for a sequence @xmath897 ( random or deterministic ) indexed by the integers , we set @xmath898 let @xmath899 measure the correlation between @xmath900 and @xmath901 . we have @xmath902- { { \ensuremath{\mathbf e } } } [ \nabla \phi_i(x)\nabla\phi_i(y)],\\ t_i&:= \frac{{{\ensuremath{\mathbf e } } } [ \nabla \phi_i(x)\nabla\phi_i(y)]}{{{\ensuremath{\mathbf e } } } [ ( \nabla \phi_i(y))^2]}. \end{split}\ ] ] note that from we have @xmath903\ge 1/2 $ ] , and thus we deduce from that @xmath904 also using we obtain that for all @xmath905 @xmath906 the conditional expectation of @xmath846 , @xmath907 given @xmath892 and @xmath908 is given by @xmath909= \sum_{r = j+1}^k t_r \nabla \phi_r(x).\ ] ] in particular this is smaller ( in absolute value ) than @xmath910 on the event @xmath911 note that @xmath912 is a very likely event . we have , uniformly in @xmath913 satisfying @xmath914 \le \exp\left(-c ( \log ( k - j))^2 \right).\ ] ] indeed , after conditioning , the increments @xmath900 are gaussian variables of variance smaller than @xmath80 ( or @xmath13 for @xmath665 ) and their mean , equal to @xmath915 , is bounded by a uniform constant , due to the restriction . if one add the conditioning to @xmath882 and @xmath892 one obtains , for all @xmath916 @xmath917 \\ \ge t_2 + \left(\frac{v_i - v_j}{v_k - v_j}\right)(u-2-t_2 ) + \sum_{r = j+1}^k t_r \nabla \phi_r(x ) \\ \ge t_2 + \left(\frac{v_i - v_j}{v_k - v_j}\right ) \frac{(k - j)u}{k}- c ( \log(k - j)+1 ) \end{gathered}\ ] ] where to obtain the last inequality we used and and the definition of @xmath912 . we have @xmath918 hence , using lemma [ lem : bridge ] , after the necessary re - centering for the bridge conditioned to @xmath908 we obtain that if @xmath919 and is satisfied we have @xmath920 \\ \le c ( k - j)^{-1}\left [ \left(\frac{ju}{k}- t_2 \right)^2 + c \log(k - j ) \right]^2.\end{gathered}\ ] ] using and , we obtain that @xmath921\big]\\ \\ \le c ( k - j)^{-2}\left [ \left(\frac{ju}{k}- t_1 \right)^2 + c \log(k - j ) \right]\left [ \left(\frac{ju}{k}- t_2 \right)^2 + c \log(k - j)^2 \right]\\ + { { \ensuremath{\mathbf p } } } \left [ { { \ensuremath{\mathcal h } } } ^{{\complement } } \ | \ \phi_j(x)=t_1 , \phi(x)=u-2 \right ] . \end{gathered}\ ] ] the last term is negligible when compared to the first and taking the maximum over @xmath922 satisfying , this concludes the proof of . now , using jensen s inequality we have @xmath991\ge \frac{h}{n^2 } { { \ensuremath{\mathbf e } } } ^{m}_n \left [ \sum_{x\in { \widetilde}{\lambda}_n } \delta_x \right ] \ge h p [ { { \ensuremath{\mathcal n } } } ( \sigma_m ) \in [ -1,1 ] ] \ ] ] where @xmath992 denote the standard deviation of the infinite volume massive free field and @xmath993 is a centered normal variable with standard deviation @xmath994 . as the variance grows when @xmath177 tends to zero we obtain that for arbitrary @xmath995 for @xmath996 we have @xmath997 using the above inequality for @xmath998 , using to estimate @xmath994 and for @xmath511 we obtain that for any @xmath999 , for @xmath1000 sufficiently small we have @xmath1001 ] - f(m)\ge \frac{h}{\sqrt{(1/2)\log h}}(1-{\varepsilon}).\ ] ] concerning the upper - bound , we can show as in ( * ? ? ? * equation ( 2.20 ) ) that @xmath1002 is a sub - multiplicative function and thus that we have for every @xmath1003 we have @xmath1004 we use this inequality for @xmath1005 in that case , the taylor expansion of the exponential in the partition function gives @xmath1006\le e^{(\log h)^{-2 } } { { \ensuremath{\mathbf e } } } _ n\left [ \sum_{x\in { \widetilde}{\lambda}_n } \delta_x\right],\ ] ] where in the last inequality we used that the probability for a gaussian of a given variance to be in @xmath1007 $ ] is maximized if its mean is equal to zero . using then to estimate the probability , it is a simple exercise to check that for any @xmath995 and @xmath77 large enough , we have @xmath1008\le \frac{(1+{\varepsilon } ) 2n^2 } { \sqrt{\log n}}.\ ] ] combining all these inequality , we obtain that for @xmath1 sufficiently small @xmath1009 e. aidekon and z. shi , _ weak convergence for the minimal position in a branching random walk : a simple proof _ period . ( special issue in the honour of e. cski and p.rvsz ) ( 2010 ) * 61 * 43 - 54 . e. bolthausen and d. brydges , _ localization and decay of correlations for a pinned lattice free field in dimension two _ , in : state of the art in probability and statistics , festschrift for willem r. van zwet , ims lecture notes vol . 36 ( 2001 ) , 134 - 149 . f. caravenna , g. giacomin and f. l. toninelli _ copolymers at selective interfaces : settled issues and open problems _ , in probability in complex physical systems , volume 11 of springer proceedings in mathematics , pages 289 - 311 . springer berlin heidelberg , 2012 . j. ding and o. zeitouni , _ extreme values for two - d imensional discrete gaussian free field _ * 42 * ( 2014 ) , 1480 - 1515 . m. e. fisher , _ walks , walls , wetting , and melting _ , j. statist . phys . * 34 * ( 1984 ) , 667 - 729 . to estimate the green function of the massive field we use a bit of potential theory . we let @xmath923 denote the potential kernel of @xmath159 in @xmath206 i.e. @xmath924 from ( * ? ? ? * theorem 4.4.4 ) we have @xmath925 set @xmath926 . now recall that @xmath291 is a continuous time random - walk on @xmath206 with generator @xmath159 and that @xmath161 denote is law when the initial condition is @xmath151 , and @xmath927 denote the hitting time of @xmath120 . let @xmath928 be a poisson variable of mean @xmath929 which is independent of @xmath291 . by adapting the proof of ( * proposition 4.6.2(b ) ) we obtain that @xmath930-a(x , y),\\ g^{m}(x , y)=e^x\left [ a\left(x_{t_m},y\right)\right]-a(x , y ) . \end{split}\ ] ] considering the case @xmath931 and when there is no boundary , it is not difficult to see that @xmath932:= -\frac{1}{2\pi}\log m+o(1).\ ] ] in the case @xmath933 with boundary , this is more delicate . on one side it is easy to deduce from that for some appropriate @xmath500 , @xmath934 what remain to prove is that @xmath935 is an upper - bound ( which is a concern only if @xmath936 ) . note that the green function with dirichlet boundary condition is an increasing function of the domain and a decreasing function of @xmath177 . hence to obtain an upper - bound on @xmath211 , we can compare it with the the variance of the massless free field in the half plane @xmath937 at the point @xmath938 that is given by @xmath939\ge g^{m}(x , x ) .\ ] ] now note that @xmath940 is simply the hitting time of zero by one dimensional simple random walk starting from @xmath941 . hence @xmath942\le c d(x,\partial { \lambda}_n ) t^{-1/2}.\ ] ] as the second coordinate of @xmath943 is simply the value of an independent random walk evaluated at @xmath944 we get that for some constant @xmath945 all @xmath450 @xmath946\le c ( u / d).\ ] ] this tail estimate , together with is sufficient to conclude that @xmath939\le \frac{1}{2\pi}\log d(x,\partial { \lambda}_n)+c.\ ] ] for the second one , we notice that we can reduce the problem to proving that for any @xmath949 $ ] @xmath950 where @xmath951 is the heat - kernel associated with the simple random - walk on @xmath952 with dirichlet boundary condition . indeed if @xmath230 and @xmath281 differ by only one coordinate , say @xmath953 we can factorize the l.h.s of by the common coordinate and obtain @xmath954 \\ \le \frac{c}{\sqrt{t}}\left[p^*_t(x_2,x_2)+p^*_t(y_2,y_2)-2p^*_t(x_2,y_2)\right ] .\end{gathered}\ ] ] if the two coordinates of @xmath230 and @xmath281 differ , then if we let @xmath955 be a field with covariance function @xmath214 , the l.h.s of can be rewritten as @xmath956\le 2\left({{\ensuremath{\mathbb e } } } [ \left(\varphi_x-\varphi_z\right)^2]+{{\ensuremath{\mathbb e } } } [ \left(\varphi_y-\varphi_z\right)^2]\right)\ ] ] and we reduce to the first case by choosing @xmath957 . for @xmath805 we can just use large deviations estimates for @xmath962 with @xmath142 chosen sufficiently large , and use the local central limit theorem ( * ? ? ? * theorem 2.1.1 ) to cover the case @xmath963 . for @xmath662 we can compare to the half - plane case where @xmath964 ( recall that from the argument presented before this gives an upper bound ) . in that case we have where @xmath291 is the simple random walk on @xmath206 starting from zero . by a reflexion argument we have @xmath966,\ x_s < d]= p [ x_t=0]-p[x_t=2d]=p_t(0,0)-p_t(0,2d{{\mathbf e}}_1).\ ] ] the later quantity can be estimated with ( * ? ? ? * theorem 2.3.6 ) , and shown to be smaller than @xmath967 . for @xmath968 we have the right - hand side is smaller than @xmath970\le 4 p_t(2d{{\mathbf e}}_1)\ ] ] the later quantity can be estimated with the lclt for large @xmath479 ( * ? ? ? * theorem 2.3.6 ) , or with large deviation estimates for small @xmath479 . let @xmath888 denote the variance of @xmath971 ( without conditioning ) , @xmath972 and set @xmath973 ( @xmath974 $ ] ) . after conditioning to @xmath975 , the process @xmath976 remains gaussian and centered but the covariance structure is given by @xmath977= \frac{v_i(v - v_j)}{v } \quad \quad 0\le i\le j\le k.\ ] ] we denote by @xmath978 the law of the conditioned process . we can couple this process with a brownian motion conditioned to @xmath979 : a centered brownian bridge @xmath980}$ ] , by setting @xmath981 . note that we have ( by applying standard reflexion argument at the first hitting time of @xmath230 ) @xmath982 } b_t \ge x \right]= 1 - e^{-\frac{x^2}{2v}}.\ ] ] as the max of @xmath121 is larger than that of @xmath291 this gives the lower bound . to prove @xmath947 , by monotonicity , we can restrict the proof to the case @xmath983 . two estimate the difference between and the probability we have to estimate , we let let @xmath984 denote the brownian bridges formed by @xmath121 between the @xmath971 , @xmath985}:= b_s- \frac{(s - v_{i-1})b_{v_{i-1}}+(v_{i}-s)b_{v_{i}}}{v}.\ ] ] we have @xmath986\le { \widetilde}{{\ensuremath{\mathbf p } } } \left[\max_{t\in[0,v ] } b_t\le 2x\right]+ \sum_{i=1}^k { \widetilde}{{\ensuremath{\mathbf p } } } \left[\min_{s\in [ v_{i},v_{i+1}]}b^i_s\le -x\right]\\ = \left ( 1-e^{\frac{-2x^2}{v}}\right)+\sum_{i=1}^k \exp\left(-\frac{x^2}{2 \nabla v_i } \right)\end{gathered}\ ] ] where in the last line we used for @xmath121 and @xmath987 . this is smaller than @xmath988 for some well chosen @xmath142 .

this paper continues a study initiated in @xcite , on the localization transition of a lattice free field on @xmath0 interacting with a quenched disordered substrate that acts on the interface when its height is close to zero . the substrate has the tendency to localize or repel the interface at different sites . a transition takes place when the average pinning potential @xmath1 goes past a threshold @xmath2 : from a delocalized phase @xmath3 , where the field is macroscopically repelled by the substrate to a localized one @xmath4 where the field sticks to the substrate . our goal is to investigate the effect of the presence of disorder on this phase transition . we focus on the two dimensional case @xmath5 for which we had obtained so far only limited results . we prove that the value of @xmath6 is the same as for the annealed model , for all values of @xmath7 and that in a neighborhood of @xmath2 . moreover we prove that , in contrast with the case @xmath8 where the free energy has a quadratic behavior near the critical point , the phase transition is of infinite order @xmath9 + 2010 _ mathematics subject classification : 60k35 , 60k37 , 82b27 , 82b44 _ + _ keywords : lattice gaussian free field , disordered pinning model , localization transition , critical behavior , disorder relevance , co - membrane model _

owing to the gapless energy spectrum @xcite , graphene layers ( gls ) absorb electromagnetic radiation in a wide spectral range ( from the ultraviolet to terahertz ) due to the interband transitions @xcite . therefore , gls can be used in photodetectors , light sources , modulators , and mixers using the interband transitions @xcite . the performance of these devices can be enhanced by utilizing multiple - gl structures @xcite . for the infrared and visible spectral ranges , the interband absorption prevails over the intraband ( drude ) absorption . however , in the terahertz ( thz ) range , especially at low thz frequencies , the drude absorption can dominate . the intraband absorption in gls can also be used in different devices for thz modulation and detection . the thz detectors , including uncooled detectors , exploiting the effect of electron or hole heating ( hot - electron or hot - hole bolometers ) in two - dimensional electron ( hole ) heterostructures made of a@xmath0b@xmath1 , cdhgte , and other compound systems were realized previously @xcite . in this paper , we propose and analyze thz uncooled bolometric detectors based on gl structures . we demonstrate that such bolometers can exhibit fairly high responsivity , effectively operating at room temperatures and surpassing thz bolometers based on the traditional semiconductor heterostructures . the main advantages of gl - based room temperature bolometers are associated with the following three factors : ( i ) high electron and hole thz conductivities at room temperature @xcite and , hence , elevated drude absorption ; ( ii ) the dominant mechanism establishing the interband and intraband equilibrium is the interaction with optical phonons @xcite ; ( iii ) long time of the electron and hole energy relaxation via optical phonons due to their large energy @xmath2 mev @xcite ( this time is proportional to a factor @xmath3 and is very large for gls even at room temperature @xmath4 k ) . figures 1(a ) and 1(b ) show the proposed ngl - gnr - pgl bolometers . the bolometers consist of two gapless n - type and p - type gl absorbing regions connected by an undoped array of gnrs with sufficiently large energy gap @xmath5 ( serving as the barrier region ) . the gls can be doped chemically [ as in fig . 1(a ) ] or `` electrically '' ( using the conducting gates with the bias voltages , @xmath6 , of different polarity , as shown in fig . the gates which control the electron and hole densities can be made using gls @xcite . it is assumed that the gnr width , @xmath7 , is sufficiently small , so that the energy gap @xmath8 , ( where @xmath9 cm / s is the characteristic velocity of electrons and holes in gls ) is large enough to provide essential confinement of electrons in the n - gl and holes in the p - gl due to the formation of the barrier . the room temperature operation of field - effect transistors with sub 10 nm gnrs exhibiting fairly large energy gap was reported in ref . the energy barrier in such gnrs ensures a relatively strong dependence of the current on the effective temperature of electrons and holes enhancing the bolometer responsivity . ( wavy arrows correspond to intraband transitions due to absorption of photons in gls , smooth arrows indicate propagation of electrons and holes above the pertinent barriers in gnrs ) . , width=264 ] figure 1(c ) shows the resulting device band structure at sufficiently large bias voltage @xmath10 , where @xmath11 is the built - in voltage , @xmath12 is the fermi energy of electrons and holes in gls in equilibrium , and @xmath13 is the electron charge . in the following , we assume that the interband absorption is relatively weak in comparison with the intraband absorption . this occurs when the energy of photons , @xmath14 , of the incident thz radiation is relatively small ( corresponding to the frequency about few thz and lower ) . if @xmath15 , the interband transitions are forbidden due to the pauli blocking . we assume that due to relatively high electron and hole densities , the intercarrier scattering time is sufficiently short to provide fast maxwellization ( or fermization ) of the photoexcited electrons and holes . therefore , the electron and hole systems in gls are characterized by quasi - fermi energy @xmath16 and by the effective temperature @xmath17 . the heating of the electron and hole gases in the pertinent sections , i.e. , the deviation of the effective temperature @xmath17 from the lattice temperature @xmath18 leads to the deviation of the fermi energy @xmath16 from its equilibrium ( dark ) value @xmath12 . the quantities @xmath16 and @xmath17 are related by the following equation : @xmath19 @xmath20 in the case of chemical doping , the quantity @xmath21 is equal to the donor ( acceptor ) density . in the detectors with electric doping , @xmath21 is given by @xmath22 , so that @xmath23 , where @xmath24 and @xmath25 are the dielectric constant and the thickness of the layer separating gls and the gates and @xmath26 is the gate voltage [ see fig . 1(b ) ] . in the case under consideration , the electron and hole systems are sufficiently strongly degenerated ( @xmath27 ) , hence , the fermi energy is given by @xmath28 . considering the one - dimensional electron and hole transport in gnrs and the fermi distributions of electrons and holes in gls , in particular , near the gnr edges at @xmath29 , the sum of the electron and hole currents ( i.e. , the terminal current ) between the p- and n - regions through @xmath30 parallel gnrs is equal to @xmath31^{-1}\ ] ] @xmath32^{-1 } \biggr\}.\ ] ] here @xmath33 is the kinetic energy of electrons and holes in gnr . in the absence of illumination , i.e. , when @xmath34 and @xmath35 , eq . ( 2 ) yields the following expression for the dark current @xmath36 : @xmath37.\ ] ] setting @xmath38 mev , and @xmath39 , for @xmath4 k we obtain @xmath40a . this value is in a good agreement with experimental results @xcite . at relatively weak irradiation , @xmath41 and @xmath42 . considering this , the variation of the current through the gnr array , @xmath43 , i.e. , the photocurrent , can be presented in the following form : @xmath44}\ ] ] @xmath45.\ ] ] the first and the second terms in the right - hand side of eq . ( 4 ) describe the effect of variation of the effective temperature and the quasi - fermi energy due to heating by the thz radiation . however , as follows from eq . ( 1 ) , when @xmath27 , the variation of the quasi - fermi energy is relatively small , hence , the last term in the right - hand side of eq . ( 4 ) can be omitted . considering that the energy relaxation due to the processes governed by the interaction with optical phonons , the electron and hole effective temperature @xmath17 and the number of optical phonons @xmath46 obey the following equations : @xmath47 @xmath48 here , @xmath49 is the thz photon flux , @xmath50 , where @xmath51 , @xmath13 is the electron charge , @xmath52 is the speed of light , @xmath53 is the rate of the intraband transitions involving the emission and absorption of optical phonons , @xmath54 is the rate of optical phonon decay , and @xmath55 is proportional to the gl drude ac conductivity @xcite : @xmath56\ ] ] @xmath57 where @xmath58 is the momentum relaxation time of electrons and holes , which , generally , is depending on @xmath12 and @xmath18 . equations ( 5 ) and ( 6 ) govern the balance of the energy of the electron - hole system and the number optical phonons in gls explicitly accounting for all the energy received by the electron - hole - optical phonon system from thz radiation going eventually to the thermostat . since @xmath59 , the expression for the term @xmath60 can be simplified @xcite : @xmath61.\ ] ] here , @xmath62 is the time of the intraband phonon - assisted processes : the quantity @xmath63 plays the role of the effective energy relaxation time of electrons and holes . in equilibrium , ( 5 ) and ( 6 ) yield @xmath35 and @xmath64 . for the rate of optical phonons decay due to the unharmonic contributions to the interatomic potential , resulting in the phonon - phonon scattering , one can use the following simplified equation : @xmath65 where @xmath66 is the decay time of optical phonons and @xmath67 is the number of optical phonons in equilibrium . considering high heat conductivity of gls @xcite , the lattice temperature , i.e. the temperature of acoustic phonons , is assumed to be equal to the temperature of the contacts @xmath18 . using eqs . ( 4)-(9 ) , we obtain @xmath68 here we also have introduced the rate of the generation of the electron - hole pairs due to the absorption of equilibrium optical phonons @xmath69 and parameter @xmath70 , where @xmath71 is the time of the interband transitions . the difference between @xmath72 and @xmath72 is due to the features of the density of states in gls . at @xmath27 , one obtains @xcite @xmath73 . the quantity @xmath74 weakly decreases with increasing the majority carrier concentration ( if @xmath75 ) and strongly ( exponentially ) drops with decreasing temperature . at room temperature @xmath76 @xmath77s@xmath78 ( compare with @xcite ) . one can see from eq . ( 10 ) that the intraband absorption of thz radiation leads to an obvious increase of the effective temperature @xmath17 . . , width=264 ] . , width=264 ] substituting @xmath79 from eq . ( 10 ) into eq . ( 4 ) , we obtain @xmath80 here @xmath81}\ ] ] @xmath82 using eqs . ( 11 ) and ( 12 ) , for the bolometer current responsivity @xmath83 ( @xmath84 is the area of gls ) , we obtain @xmath85 for instance , considering a quasi - optic thz bolometer with a single gnr ( @xmath39 ) integrated with a spiral antenna , we can assume that @xmath86 s , @xmath87 , @xmath88m@xmath89 , ( about that in @xcite , @xmath4 k , and @xmath90 thz . setting @xmath91 mev , @xmath92 mev , and @xmath93 @xmath77s@xmath78 , we find @xmath94 a / w . if the applied bias voltage @xmath95 mv , setting @xmath96a , for the voltage responsivity @xmath97 we obtain @xmath98 v / w . the later values of the current and voltage responsivities significantly exceed those for uncooled hot - electron bolometers based on the heterostructures made of the standard semiconductor ( for example , cdhgte hot - electron bolometers @xcite ) . using eqs . ( 3 ) and ( 13 ) , one can calculate the bolometer dark current limited detectivity @xmath99 . since @xmath100 and @xmath101 , @xmath102 ( for fixed @xmath84 ) . at fixed value of @xmath12 , the detectivity achieves its maximum at @xmath103 . equation ( 12 ) shows that the heating of the optical phonon system due to the energy which this system receives from heated electrons and holes promotes an increase in the responsivity . the relative contribution of the optical phonon heating is determined by the factor @xmath104 . this implies that the bolometric effect in question is not purely a hot - electron or hot - hole effect . the bolometer spectral characteristic is determined by the frequency dependence of the ac drude conductivity , which , as seen from eq . ( 13 ) at @xmath105 , results in @xmath106 . if @xmath86 s , the responsivity roll - off occurs at @xmath107 thz . figures 2 and 3 show the dependences of the responsivity and detectivity , respectively , on the energy gap in gnr , @xmath5 , calculated for the thz bolometers with @xmath39 and different momentum relaxation times @xmath58 for @xmath90 thz . it is assumed that @xmath92 mev and @xmath108 @xmath77s@xmath78 . according to eq . ( 13 ) , the responsivity increases with increasing number , @xmath30 , of gnrs , if the gl area @xmath84 is fixed . however , an increase in @xmath30 may require the related increase in the width of gls and , consequently , in their area . similar bolometer can be based on n - gnr - n heterostructures . the results obtained above can also be applied to this device with small modifications : the dark current and responsivity given by eqs . ( 3 ) and ( 13 ) should be multiplied by a factor @xmath109 , because the terminal dark current and photocurrent are due to the electrons injected from only one gl . in conclusion , novel thz uncooled bolometers based on ng - gnr - pg heterostructures have been proposed . using the developed model , we calculated the bolometer dark current and responsivity and demonstrated that ngl - gnr - pgl can surpass the hot - electron bolometers based on traditional semiconductor heterostructures . this work was supported by the japan society for promotion of science and terano - nsf grant , usa . the work at rpi was supported by nsf and arl alliance cooperative research agreement program . r. r. nair , p. blake , a. n. grigorenko , k. s. novoselov , t. j. booth , t. stauber , n. m. r. peres , and a. k. geim , science * 320 * , 1308 ( 2008 ) . j. m. dawlaty , s. shivaraman , j. strait , p. george , m. chandrashekhar , f. rana , m. g. spencer , d. veksler , and y. chen , appl . . lett . * 93 * , 131905 ( 2008 ) . v. ryzhii , t. otsuji , m. ryzhii , and m. s. shur , j. phys . d * 45 * , 3201 ( 2012 ) . x. wang , y. ouyang , x. li , h. wang , j. guo , and h. dai , phy . lett . * 100 * , 206803 ( 2008 ) . l. a. falkovsky and a. a. varlamov , eur . j. b * 56 * , 281 ( 2007 ) .

we propose the concept of a terahertz ( thz ) uncooled bolometer based on n - type and p - type graphene layers ( gls ) , constituting the absorbing regions , connected by an array of undoped graphene nanoribbons ( gnrs ) . the gls absorb the thz radiation with the gnr array playing the role of the barrier region ( resulting in ngl - gnr - pgl bolometer ) . the absorption of the incident thz radiation in the gl n- and p- regions leads to variations of the effective temperature of electrons and holes and of their fermi energy resulting in the variation of the current through the gnrs . using the proposed device model , we calculate the dark current and the bolometer responsivity as functions of the gnr energy gap , applied voltage , and the thz frequency . we demonstrate that the proposed bolometer can surpass the hot - electron bolometers using traditional semiconductor heterostructures .

this work continues earlier third - order relativistic many - body perturbation theory ( rmbpt ) studies of energy levels of ions with one valence electron outside a closed core . in refs . @xcite third - order rmbpt was used to calculate energies of the three lowest states ( @xmath9 , @xmath10 , and @xmath11 ) in li- , na- , and cu - like ions along the respective isoelectronic sequences , while in the present work , third - order rmbpt is used to calculate energies of the eleven lowest levels , @xmath12 , @xmath13 , @xmath14 , @xmath2 , @xmath15 , and @xmath16 in ag - like ions . it should be noted that the @xmath17 cores of li- , na- , and cu - like ions are completely filled , by contrast with ag - like ions , where the @xmath18 core [ cu@xmath19@xmath20 is incomplete . third - order rmbpt calculations of @xmath21 transition amplitudes in ag - like ions up to @xmath8=60 were previously performed by @xcite . in the present paper , we extend the calculations of @xcite to obtain energies , reduced matrix elements , oscillator strengths , and transition rates for the 17 possible @xmath4 and @xmath5 e1 transitions . additionally , we evaluate lifetimes of excited states . most earlier theoretical studies of ag - like ions were devoted to oscillator strengths and lifetimes @xcite rather than energy levels ; an exception is the work of @xcite in which energies , oscillator strengths and lifetimes of levels in ag - like ions were calculated using relativistic dirac - fock ( df ) wave functions @xcite . in the present paper , we use rmbpt to determine energies and lifetimes of @xmath2 and @xmath0 levels in neutral ag and ag - like ions with @xmath3 . we compare our results with experimental data from refs . @xcite . [ cols="<,>,>,>,>,>,>,>,^ , > , > , > , > , > , > , > " , ] we solve the core rpa equations iteratively . in our calculations , we set the number of core iteration to 10 to save computation time ; for convergence to machine accuracy , about 50 iterations are needed at low @xmath8 . for example , for the @xmath22 transition in neutral ag , first - order length and velocity matrix elements are 4.30225 and 4.26308 , respectively . the values of the electric - dipole matrix elements are given in atomic units , @xmath23 . the atomic unit for the corresponding line strength is @xmath24 . the corresponding rpa values are 3.77755 and 3.96707 after one iteration ; they become 3.82599 and 3.82636 after 10 iterations . the final _ third - order _ gauge - independent results are 3.41726 and 3.41745 for this matrix element in length and velocity forms , respectively . llllllll & & & & & & & + + @xmath12&@xmath25 & 7.50 & 5.71 & 6.97 & 6.72@xmath260.03&3455 & 3282 + @xmath12&@xmath27 & 7.98 & 6.24 & 7.62 & 7.41@xmath260.04&3562 & 3384 + + @xmath28&@xmath29 & 5.82 & 5.12 & 5.57 & 6.7 @xmath260.2&5417 & 5380 + @xmath30&@xmath31 & 6.16 & 5.41 & 5.90 & 6.2 @xmath260.1&5372 & 5338 + @xmath12&@xmath25 & 2.32 & 2.42 & 2.60 & 2.77@xmath260.07&2170&2145 + @xmath12&@xmath27 & 2.68 & 2.88 & 3.09 & 3.11@xmath260.04&2291&2266 + @xmath25&@xmath28 & 1.75 & 1.44 & 1.67 & 1.85@xmath260.15&2364&2314 + @xmath27&@xmath30 & 1.95 & 1.60 & 1.86 & 1.79@xmath260.11&2243&2195 + + @xmath29&@xmath32&2.79 & 2.52 & 2.71 & 2.84@xmath260.30 & 4121&4072 + @xmath28&@xmath29&1.71 & 1.62 & 1.74 & 1.72@xmath260.07 & 3007&3009 + @xmath30&@xmath31&1.78 & 1.69 & 1.82 & 1.70@xmath260.07 & 2969&2983 + @xmath12&@xmath25&1.20 & 1.42 & 1.45 & 1.50@xmath260.15 & 1630&1625 + @xmath12&@xmath27&1.48 & 1.81 & 1.84 & 1.64@xmath260.06 & 1760&1749 + @xmath25&@xmath28&0.58 & 0.56 & 0.61 & 0.58@xmath260.05 & 1507&1488 + @xmath27&@xmath30&0.64 & 0.61 & 0.67 & 0.75@xmath260.06 & 1423&1403 + + @xmath28&@xmath29&1.20 & 1.27 & 1.38 & 1.30@xmath260.20&2266 & 2230 + @xmath30&@xmath31&0.98 & 1.04 & 1.13 & 1.25@xmath260.20&2224 & 2222 + @xmath12&@xmath25&0.75 & 0.95 & 0.95 & 0.81@xmath260.15&1320 & 1315 + @xmath12&@xmath27&0.97 & 1.27 & 1.26 & 1.29@xmath260.20&1444 & 1438 + @xmath25&@xmath28&0.29 & 0.31 & 0.32 & 0.45@xmath260.05&1117 & 1119 + @xmath27&@xmath30&0.31 & 0.33 & 0.34 & 0.34@xmath260.04&1050 & 1044 + + @xmath28&@xmath29&1.77 & 2.23 & 2.57 & 2.5@xmath260.4&2268 & 2279 + @xmath30&@xmath31&1.38 & 1.73 & 2.00 & 2.4@xmath260.3&2202 & 2217 + @xmath12&@xmath25&0.51 & 0.68 & 0.67 & 0.65@xmath260.12&1108 & 1104 + @xmath12&@xmath27&0.70 & 0.95 & 0.92 & 0.77@xmath260.10&1230 & 1226 + @xmath25&@xmath28&0.18 & 0.20 & 0.20 & & 892.1 & + @xmath27&@xmath30&0.18 & 0.21 & 0.21 & 0.191@xmath260.020&834.1 & 831 + + @xmath12&@xmath25 & 0.38 & 0.510 & 0.493 & 0.47@xmath260.03 & 952.9 & 951 + @xmath12&@xmath27 & 0.58 & 0.738 & 0.713 & 0.65@xmath260.04 & 1073 & 1071 + @xmath25&@xmath28 & 0.12 & 0.140 & 0.141 & 0.13@xmath260.03 & 745.3 & 743 + @xmath27&@xmath30 & 0.12 & 0.146 & 0.146 & 0.14@xmath260.04 & 693.0 & 691 + + @xmath12&@xmath25 & 0.29 & 0.39 & 0.38 & 0.35@xmath260.02&834.7 & + @xmath12&@xmath27 & 0.43 & 0.60 & 0.57 & 0.48@xmath260.03&954.0 & + @xmath25&@xmath28 & 0.087 & 0.106 & 0.105 & 0.107@xmath260.016&641.3&640 + @xmath27&@xmath30 & 0.090 & 0.108 & 0.107 & 0.120@xmath260.020&592.9&592 + + @xmath12&@xmath25 & 0.23 & 0.31 & 0.30 & 0.33@xmath260.03&741.0&740.4 + @xmath12&@xmath27 & 0.35 & 0.50 & 0.47 & 0.50@xmath260.05&858.6&859.2 the results of our third - order calculations are summarized in table [ tab - osc ] , where we list oscillator strengths for @xmath33 , @xmath34 , @xmath35 , and @xmath36 transitions in neutral ag and low-@xmath8 ag - like ions with @xmath37 . in table [ tab - s ] , we present line strengths for @xmath33 , @xmath34 , @xmath35 , and @xmath36 transitions in xe@xmath38 . the values calculated in length form in first , second , and third approximations are listed in columns @xmath39 , @xmath40 , and @xmath41 , respectively . the difference between second - order values @xmath40 and third - order values @xmath41 is much smaller than the difference between @xmath39 and @xmath40 . the second - order corrections change @xmath39 by 20 - 50 % . the addition of the third - order corrections modifies line strengths by 5 - 10 % . the first approximation is just the frozen - core df approximation and the first - order line strengths @xmath39 in table [ tab - s ] are very close to the earlier df calculations by @xcite . trends of the @xmath8 dependence of transition rates are shown in fig . the @xmath33 , @xmath34 , @xmath35 , and @xmath36 transition rates are shown in fig . [ fig2 ] a , b , c , d , respectively . all graphs are plotted using second - order data for consistency . the @xmath8 dependences of the transition rates for @xmath33 transitions shown in fig . [ fig2]a and two @xmath34 transitions shown in fig . [ fig2]b are smooth ; however , all other @xmath8 dependences shown in fig . [ fig2 ] contain sharp features . the sharp feature in the curve describing the @xmath42 transition rates ( fig . [ fig2]b ) is explained by irregularity in the curve describing the @xmath28 energy shown in fig . this irregularity in the energy @xmath8 dependence was already discussed in the previous section . the sharp minima in the region @xmath43 in the curves describing the @xmath35 transition rates shown in fig . [ fig2]c are due to inversion of the order of @xmath44 and @xmath45 energy levels . in the region @xmath43 the @xmath35 transition energies become very small resulting in the small transition rates . the second sharp feature in the curves describing the @xmath35 transition rates shown in fig . [ fig2]c occurs in the region @xmath8 = 72 - 73 and results from the irregularity in the second - order correction to the @xmath35 dipole matrix elements . below , we describe some details of the calculation to clarify this issue . a typical contribution from one of the second - order rpa corrections to dipole matrix element ( @xmath46 ) has the form @xcite @xmath47\propto \sum_{nb } \sum_{k } \frac{d_{nb}x_{k}(vnv'b)}{\epsilon _ { n}+\epsilon _ { v}-\epsilon _ { v'}-\epsilon _ { b}}\ , .\ ] ] here , the index @xmath48 designates a core state and index @xmath49 designates an excited state . the numerator is a product of a dipole matrix element @xmath50 and a coulomb matrix element @xmath51 . for the special case of the @xmath52 transition , the energy denominator for the term in the sum with @xmath53 and @xmath54 is @xmath55 again , as in the case of the second - order @xmath56 energy , there is a nearly zero denominator when the lowest - order energies of the @xmath30 and @xmath57 states are close . the cause of this irregularity is once again traced to the near degeneracy of a single - particle state and a two - particle one - hole state . the remaining irregularities for @xmath7 in the curves presented in fig . [ fig2 ] have similar origins . we calculate lifetimes of @xmath0 and @xmath2 levels in neutral ag and in ag - like ions with @xmath37 using third - order mbpt results for dipole matrix elements and energies . in table [ tab - life ] , we compare our lifetime data with available experimental measurements . this set of data includes results for a limited number of levels in low-@xmath8 ions ( up to @xmath58 ) . we give a more complete comparison of the transition rates and wavelengths for the eleven transitions between @xmath59 and @xmath44 states in ag - like ions with @xmath60 including the third - order contribution in table iii of the accompanying epaps document @xcite . in table [ tab - life ] , we present our lifetime data @xmath61 calculated in the lowest- , second- , and third - order approximations . these results are listed in columns labeled @xmath62 , @xmath63 , and @xmath64 , respectively . the largest difference between the calculations in different approximations occurs for @xmath30 and @xmath28 levels for @xmath8 = 51 and 52 when @xmath65 transition energies become very small and contributions from the second and third orders become very important . it should be noted that for some levels of neutral ag and ag - like ions with @xmath8 = 48 and 49 , @xmath64 agrees better with @xmath62 than with @xmath63 . the accuracy of lifetime measurements is not very high for ag - like ions , and in some cases the lowest - order results @xmath62 , which are equivalent to the dirac - fock results of @xcite were enough to predict the lifetimes . the more sophisticated theoretical studies published recently in refs . @xcite were restricted to @xmath66 transitions and did not include wavelength data . in two last columns of table [ tab - life ] , we compare our theoretical wavelengths , @xmath67 with experimental measurements , @xmath68 . in the cases where more than one transition is allowed , the wavelength of the dominant transition is given . we find good agreement , 0.01 - 1% , of our wavelength results with available experimental data for ions with @xmath69 . in summary , a systematic rmbpt study of the energies of @xmath12 , @xmath27 , @xmath25 , @xmath30 , @xmath28 , @xmath31 , @xmath29 , @xmath70 , @xmath71 , @xmath72 , and @xmath32 states in ag - like ions is presented . these energy calculations are found to be in good agreement with existing experimental energy data and provide a theoretical reference database for the line identification . a systematic relativistic rmbpt study of reduced matrix elements , line strengths , oscillator strengths , and transition rates for the 17 possible @xmath33 , @xmath34 , @xmath35 , and @xmath36 electric - dipole transitions in ag - like ions throughout the isoelectronic sequence up to @xmath73 is conducted . both length and velocity forms of matrix elements are evaluated . small differences between length and velocity - form calculations , caused by the nonlocality of the df potential , are found in second order . however , including third - order corrections with full rpa leads to complete agreement between the length- and velocity - form results . we believe that our energies and transition rates will be useful in analyzing existing experimental data and planning new experiments . there remains a paucity of experimental data for many of the higher ionized members of this sequence , both for term energies and for transition probabilities and lifetimes . the work of w. r. j. and i. m. s. was supported in part by national science foundation grant no . phy-01 - 39928 . u.i.s . acknowledges partial support by grant no . b516165 from lawrence livermore national laboratory .

energies of @xmath0 ( @xmath1 ) and @xmath2 states in neutral ag and ag - like ions with nuclear charges @xmath3 are calculated using relativistic many - body perturbation theory . reduced matrix elements , oscillator strengths , transition rates and lifetimes are calculated for the 17 possible @xmath4 and @xmath5 electric - dipole transitions . third - order corrections to energies and dipole matrix elements are included for neutral ag and for ions with @xmath6 . second - order corrections are included for @xmath7 . comparisons are made with available experimental data for transition energies and lifetimes . correlation energies and transition rates are shown graphically as functions of nuclear charge @xmath8 for selected cases . these calculations provide a theoretical benchmark for comparison with experiment and theory .

microlensing is one of the most powerful methods that can be used to search for extrasolar planets @xcite . recently , two robust microlensing detections of exoplanets were reported by @xcite and @xcite . the signal of a planetary companion to microlens stars is a short - duration perturbation to the smooth standard light curve of the primary - induced lensing event occurring on a background source star . the planetary perturbation occurs when the source star passes close to the caustic . the caustic represents the set of source positions at which the magnification of a point source becomes infinite . studies of the properties of the caustic are important because the characteristics of the planetary perturbations in the microlensing light curve depend critically on the properties of the caustic . for example , the location of the perturbation on the lensing light curve depends on the location of the caustic . the duration of the perturbation and the probability of detecting the perturbation are proportional to the caustic size . in addition , the pattern of the perturbation is closely related to the caustic shape . therefore , it is essential to understand the properties of caustics for the interpretation of the planetary lensing signals . although some of the properties of the caustics in planetary microlensing have been known , our knowledges of them are mostly from scattered information based on numerical approaches . the problem of the numerical approach is that the dependence of the planetary lensing behavior on the planet parameters of the star - planet separation @xmath0 ( normalized by the einstein ring radius @xmath1 ) and the planet / star mass ratio @xmath2 is not clear . there have been several attempts to overcome this ambiguity using analytic methods . by treating the planet - induced deviation as a perturbation , @xcite and @xcite derived analytic expressions for the locations of the _ central _ caustic , which is one of the two sets of caustics of the star - planet lens system located close to the primary star . based on a similar perturbative approach , @xcite provides analytic expressions for the locations of the lensing images . @xcite derived analytic expressions for the locations of not only the central caustic but also the _ planetary _ caustic , the other set of caustics , which are located away from the central star . however , there has been no analytic work on the detailed properties of the caustics such as the location , size , and shape , except the very recent work of @xcite ( hereafter paper i ) on the central caustics . following paper i , we conduct a comprehensive and analytic analysis on the properties of the planetary caustics . under the perturbative approximation , we derive analytic expressions for the location , size , and shape of the planetary caustics as a function of @xmath0 and @xmath2 . based on these expressions combined with those for the central caustics derived in paper i , we compare the similarities and differences between the planetary and central caustics . we provide an expression for the size ratio between the two types of caustics . we also derive an expression for the condition of the merging of the two types of caustics . we finally discuss the validity of the perturbative approximation . a planetary lensing is described by the formalism of a binary lens with a very low - mass companion . because of the very small mass ratio , planetary lensing behavior is well described by that of a single lens of the primary star for most of the event duration . however , a short - duration perturbation can occur when the source star passes the region around the caustics , which are important features of binary lensing . the caustics of binary lensing form a single or multiple closed figures where each of which is composed of concave curves ( fold caustics ) that meet at cusps . for a planetary case , there exist two sets of disconnected caustics . one ` central caustic ' is located close to the host star . the other ` planetary caustic ' is located away from the host star and its number is one or two depending on whether the planet lies outside ( @xmath3 ) or inside ( @xmath4 ) the einstein ring . the size of the caustic , which is directly proportional to the planet detection efficiency , is maximized when the planet is located in the ` lensing zone ' , which represents the range of the star - planet separation of @xmath5 . the planetary caustic is always bigger than the central caustic . we start from the formula of @xcite for the position of the planetary caustics ( eqs . [ 49 ] and [ 50 ] of his paper ) . keeping up to the first order term , the formula are expressed as @xmath6 @xmath7 where @xmath8 is a variable and @xmath9^{1/2}. \label{eq3}\ ] ] in these expressions , the coordinates are centered at the position on the star - planet axis with a separation vector from the position of the star of @xmath10 where @xmath11 is the position vector of the planet from the star normalized by @xmath1 ( see figs . [ fig : one ] and [ fig : two ] ) . the origin of the coordinates corresponds to the center of the planetary caustic . for the pair of the planets with separations @xmath0 and @xmath12 , the centers of the caustics are separated from the star by the same distance ( because @xmath13 ) but directed toward opposite directions ( because @xmath14\neq { \rm sign } [ { \bold r}(1/s)]$ ] ) . therefore , the center of the caustic is located on the same and opposite sides of the planet with respect to the position of the star for the planets with @xmath3 and @xmath4 , respectively . if one defines the lensing zone as the range of the planetary separation for which the planetary caustic is located within the einstein ring , the exact range of the lensing zone is @xmath15 to the first - order approximation , the size of the planetary caustic is proportional to @xmath16 as shown in equations ( [ eq1 ] ) and ( [ eq2 ] ) . we will discuss the deviation of the approximation from the exact value in 4.3 . in this case , between the two values of @xmath17 in equation ( [ eq3 ] ) only the one with ` + ' sign is valid because the other one with ` @xmath18 ' sign results in @xmath19 . as a result , there exists only a single set of caustics for planets with @xmath3 as shown in figure [ fig : one ] . the planetary caustic of the planet with @xmath3 is composed of four cusps , with two of them are located on the @xmath20 axis and the other two are located on the @xmath21 axis ( see fig . [ fig : one ] ) . the positions of the individual cusps , @xmath22 , corresponds to the cases of @xmath23 ( for the two cusps on the @xmath20 axis ) and @xmath24 ( for the other two cusps on the @xmath21 axis ) . then , the positions of the cusps on the @xmath20 and @xmath21 axes are expressed respectively as @xmath25 @xmath26 if we define the horizontal and vertical widths of the planetary caustic as the separations between the cusps on the individual axes ( see fig . [ fig : one ] ) , the widths are expressed respectively as @xmath27 @xmath28 where the expressions after the arrow are those evaluated to the first non - vanishing order in @xmath0 in the limiting case of @xmath29 . then , the vertical / horizontal width ratio is expressed as @xmath30 in the limiting case of @xmath29 , @xmath31 and @xmath32 , i.e. the caustic size decreases as @xmath33 and the shape becomes less elongated as the star - planet separation increases . in this case , @xmath17 in equation ( [ eq3 ] ) is valid only in the following range of @xmath8 @xmath34 for @xmath8 within these ranges , there are two possible values of @xmath17 corresponding to the signs . as a result , there exist two sets of caustics for planets with @xmath4 ; one above and the other below the star - planet axis ( see fig . [ fig : two ] ) . each of the caustics for the planet with @xmath4 is composed of three cusps . one of them is located on the @xmath21 axis but the other two are not located on either of the axes . the caustic meets the @xmath21 axis at @xmath35 $ ] and @xmath36 when @xmath24 ( see fig . [ fig : two ] ) . among these two positions , the former corresponds to the cusp , and thus the location of the on - axis cusp is @xmath37 where the sign ` @xmath38 ' is for the cusps located above and below the star - planet axis , respectively . if we define the vertical width of the caustic as the separation between the two crossing points at @xmath39 and @xmath40 , the width is expressed as @xmath41 where the factor ` 1/2 ' is included into consideration that there exist two planetary caustics for planets with @xmath4 and the expression after the arrow is that evaluated to the first non - vanishing order in @xmath0 in the case of @xmath42 . by defining the center of _ each _ caustic as the midpoint between the two crossing points ( see fig . [ fig : two ] ) , its position is expressed as @xmath43 the other two cusps occurs when @xmath44 ( or @xmath45 ) . this condition is satisfied when @xmath46 ( or @xmath47 ) . then , combined with the possible range in equation ( [ eq11 ] ) , the values of @xmath8 corresponding to the off - axis cusps are found to be @xmath48 with this value combined with equations ( [ eq1 ] ) and ( [ eq2 ] ) , the positions of the off - axis cusps are expressed as @xmath49,\cr } \label{eq16}\ ] ] where @xmath50 . in the limiting case of @xmath42 , equation ( [ eq16 ] ) is approximated as @xmath51 because @xmath52 , @xmath53 , and @xmath54 . by defining the horizontal width as the separation between the two off - axis cusps , the width is expressed as @xmath55 once again , the factor ` 1/2 ' is included into consideration that there are two planetary caustics . from equations ( [ eq13 ] ) and ( [ eq18 ] ) , the vertical / horizontal width ratio is expressed as @xmath56 in the limiting case of @xmath42 , each caustic shrinks as @xmath57 , c.f . @xmath58 for planets with @xmath3 , and @xmath59 , c.f . @xmath32 for planets with @xmath3 . based on the analytic expressions derived in the previous section , we now investigate how the properties of the planetary caustics such as the location , size , and shape vary depending on @xmath0 and @xmath2 . we also compare the properties of the planetary caustics with those of the central caustics . in the upper panel of figure [ fig : three ] , we present example planetary caustics of several planetary systems with different @xmath0 and @xmath2 . in the lower panel , we present the separation of the caustic from the planet - hosting star as a function of @xmath0 . in figure [ fig : four ] , we also present the variation of the caustic size ( as measured by the horizontal and vertical widths ) and the shape ( as measured by the vertical / horizontal width ratio ) as a function of @xmath0 . the properties of the planetary caustics found from the figures and the dependence of these properties on the planet parameters are as follows . 1 . for @xmath3 , the location of the caustic center depends on @xmath0 but not on @xmath2 . on the other hand , for planets with @xmath4 , the caustic location depends on both @xmath0 and @xmath2 . in this case , the caustic is located farther away from the star - planet axis as @xmath2 increases ( see eq . [ [ eq14 ] ] ) . 2 . although the caustic size depends on the mass ratio as @xmath60 , the shape of the caustic does not depend on @xmath2 and solely dependent on @xmath0 ( see eqs . [ [ eq10 ] ] and [ [ eq19 ] ] ) . 3 . the rate of decrease of the caustic size with the increase of @xmath61 are different for planets with @xmath3 and @xmath4 . compared to the caustic of the planet with @xmath3 , the rate of decrease is steeper for the planet with @xmath4 . in the limiting cases of @xmath29 and @xmath42 , the caustic sizes decrease as @xmath58 and @xmath62 for planets with @xmath3 and @xmath4 , respectively ( see eqs . [ [ eq8 ] ] , [ [ eq9 ] ] , [ [ eq13 ] ] , and [ [ eq18 ] ] ) . @xcite presented the analytic expressions for the location , cusp positions , width , and shape of the central caustics analogous to those presented in the previous section for the planetary caustics . the expressions for the location of the central caustic , analogous to equations ( [ eq1 ] ) and ( [ eq2 ] ) for the planetary caustic , are @xmath63 @xmath64 where @xmath65 is a variable and the coordinates are centered at the position of the host star . there exists a single central caustic regardless of @xmath0 and it has an elongated asteroid shape with four cusps , of which two are located on the @xmath20 axis and the other two are off the axis . the analytic expressions for the positions of the individual cusps , which are analogous to equations ( [ eq6 ] ) and ( [ eq7 ] ) for the planetary caustic with @xmath3 and to equations ( [ eq12 ] ) and ( [ eq16 ] ) for the planetary caustic with @xmath4 , are @xmath66 , \label{eq22}\ ] ] @xmath67 , \label{eq23}\ ] ] where @xmath68^{1/2 } \}$ ] . the horizontal and vertical widths of the central caustic defined as the separations between the cusps on and off the star - planet axis are expressed respectively as @xmath69 @xmath70 which are analogous to those in equations ( [ eq8 ] ) and ( [ eq9 ] ) for the planetary caustic with @xmath3 and to equations ( [ eq13 ] ) and ( [ eq18 ] ) for the planetary caustic with @xmath4 . then , the width ratio of the central caustic is @xmath71 which is analogous to those in equations ( [ eq10 ] ) and ( [ eq19 ] ) for the planetary caustics with @xmath3 and @xmath4 , respectively . in the limiting cases of @xmath29 and @xmath42 , the size of the central caustic decreases respectively as @xmath72 the planetary and central caustics have the following similarities and differences . 1 . unlike the planetary caustic , the pair of the central caustics with separations @xmath0 and @xmath12 are identical as demonstrated by the fact that the inversion @xmath73 in equations ( [ eq20 ] ) and ( [ eq21 ] ) results in the same expressions . 2 . while the dependence of the size of the planetary caustic on the planet / star mass ratio is @xmath60 , the dependence of the central caustic is @xmath74 . therefore , the planetary caustic shrinks much more slowly with the decrease of the planet mass than the central caustic . 3 . for planets with @xmath3 , the rate of decrease of the size of the central caustic with the increase of @xmath61 is similar to that of the planetary caustic with @xmath3 , i.e. @xmath75 ( see eqs . [ [ eq8 ] ] and [ [ eq27 ] ] ) , but smaller than that of the planetary caustic with @xmath4 , which shrinks as @xmath57 ( see eq . [ [ eq18 ] ] ) . then , what is the size ratio between the planetary and central caustics . if we use the horizontal width as a representative quantity for the caustic size , the size ratio between the two types of the caustics is found from equations ( [ eq8 ] ) , ( [ eq18 ] ) , and ( [ eq24 ] ) and expressed as @xmath76 & for $ s<1 $ , \cr } \label{eq28}\ ] ] where the additional subscripts ` p ' and ` c ' denote the planetary and central caustics , respectively . in figure [ fig : five ] , we present the size ratio as a function of @xmath0 and @xmath2 . since @xmath77 while @xmath78 , the dependence of the size ratio on the mass ratio is @xmath79 . for a given mass ratio , the size ratio is maximized at around @xmath80 and decreases rapidly with the increase of @xmath61 . , the change rate of the size ratio is reversed as @xmath61 further increases beyond a critical value ( @xmath81 or @xmath82 ) . however , this reversal occurs at the separation beyond the lensing zone . ] as @xmath83 , the location of the planetary caustic , i.e. @xmath84 , approaches the position of the central star , around which the central caustic is located . then the two types of the caustics eventually merge together , resulting in gradual loss of distinction between the two types of caustics . the condition for the merging of the two caustics is that the separation between the two caustics is smaller than the half of the sum of the individual caustic widths , i.e.@xmath85 by using the analytic expressions for @xmath86 ( eqs . [ [ eq8 ] ] and [ [ eq18 ] ] ) and @xmath87 ( eq . [ [ eq24 ] ] ) , we compute the region of the caustic merging in the parameter space of @xmath88 and presented in figure [ fig : five ] ( the region enclosed by thick dashed lines ) . the region is confined in a small region around @xmath89 , but the width of the region increases as @xmath2 increases because the caustic size increases with the increase of @xmath2 . are the presented analytic expressions based on perturbative approximation good enough for the description of the caustics in planetary microlensing ? we answer this question by comparing the two sets of caustics constructed based on analytic and numerical computations . in figure [ fig : six ] , we present some pairs of the planetary caustics with different values of the planet parameters @xmath0 and @xmath2 . in each panel of the figure , the blue caustic is drawn by using the analytic expressions while the red caustic is the exact one based on numerical computations . for reference , we note that the mass ratios of the planets with masses equivalent to the jupiter , saturn , neptune , and earth around a host star with @xmath90 of the most probable galactic lensing event are @xmath91 , @xmath92 , @xmath93 , and @xmath94 , respectively . from the figure , we find that although the deviation increases with the increase of the planet / star mass ratio , the analytic approximation well describes the planetary caustic in most mass regime of planets ( @xmath9510^{-3}$ ] ) . for the earth - mass planet , we find that the two caustics are eventually indistinguishable . we derived analytic expressions for the location , size , and shape of the planetary caustic as a function of the star - planet separation and the planet / star mass ratio under perturbative approximation . based on these expressions , we conducted comprehensive analysis on the properties of the planetary caustics . combined with the analogous expressions for the central caustics derived in paper i , we compared the similarities and differences between the planetary and central caustics . we also presented the expressions for the size ratio between the two types of caustics and for the condition of the merging of the two types of caustics . these analytic expressions will be useful in understanding the dependence of the planetary lensing behavior on the planet parameters and thus in interpreting the planetary lensing signals . we would like to thank j. h. an and a. gould for making helpful comments . work by c.h . was supported by the astrophysical research center for the structure and evolution of the cosmos ( arcsec ) of korea science and engineering foundation ( kosef ) through science research center ( src ) program .

although some of the properties of the caustics in planetary microlensing have been known , our understanding of them is mostly from scattered information based on numerical approaches . in this paper , we conduct a comprehensive and analytic analysis of the properties of the planetary caustics , which are one of the two sets of caustics in planetary microlensing , those located away from the central star . under the perturbative approximation , we derive analytic expressions for the location , size , and shape of the planetary caustic as a function of the star - planet separation and the planet / star mass ratio . based on these expressions combined with those for the central caustic , which is the other set of caustics located close to the central star , we compare the similarities and differences between the planetary and central caustics . we also present the expressions for the size ratio between the two types of caustics and for the condition of the merging of the two types of caustics . these analytic expressions will be useful in understanding the dependence of the planetary lensing behavior on the planet parameters and thus in interpreting the planetary lensing signals .

gamma - ray binaries are systems composed of a massive star and a compact object and from which persistent gev and/or tev gamma - ray emission is detected and dominates the overall non - thermal spectrum . they emit across the electromagnetic spectrum from the radio to tev gamma ray ( see * ? ? ? * for a review ) . there are only five gamma - ray binaries known to date @xcite , and only for one source has the compact object been identified ( psr b1259@xmath063 ; * ? ? ? since most of the energy output of a gamma - ray binary is in the gamma - ray band , current theoretical studies focus on explaining the high energy emission properties . the gamma - ray emission models can be categorized into two classes : microquasar models ( e.g. , * ? ? ? * ; * ? ? ? * ) and pulsar models ( e.g. , * ? ? ? * ; * ? ? ? . in the microquasar model , relativistic electrons in a jet generated close to the compact object compton - upscatter the synchrotron emission of the jet itself and/or the stellar uv photons ( e.g. , * ? ? ? * ; * ? ? ? * ) , or relativistic hadrons collide with background nuclei creating pions that decay ( e.g. , * ? ? ? * ) , producing gamma rays . in the pulsar model , pulsar wind particles are accelerated in the pulsar wind / stellar wind shock , and compton - upscatter stellar photons to produce the observed gamma rays ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? non - thermal x - ray emission in gamma - ray binaries is thought to be produced by the electrons which are accelerated in the pulsar wind / stellar wind shock ( e.g. * ? ? ? * ; * ? ? ? * ) or in relativistic jets formed close to the compact object ( e.g. , * ? ? ? * ) . the models predict varying x - ray fluxes and spectra depending on the properties of the shock , which are determined by the thrust of the winds and the orbital geometry of the binary system ( e.g. , * ? ? ? * ) , or on the jet dynamics and cooling timescale ( e.g. , * ? ? ? * ; * ? ? ? hence , x - ray measurements can be used for constraining the orbital parameters and understanding the nature of the physical processes in gamma - ray binaries ( see also * ? ? ? * ; * ? ? ? * ; * ? ? ? the gamma - ray binary 1fgl j1018.6@xmath05856 was discovered with _ fermi _ in 2011 . @xcite found modulation in the radio to gamma - ray bands with a period of @xmath3days , identifying the source as a gamma - ray binary . they further identified the companion star to be an o6v((f ) ) star . soon after the discovery , subsequent broadband studies were carried out @xcite in order to better characterize the source properties , but in no case were they able to identify the nature of the compact object . x - ray properties of the gamma - ray binary 1fgl j1018.6@xmath05856 were measured in detail with _ swift_. @xcite showed that the x - ray flux peak seen at phase 0 ( gamma - ray maximum ) by @xcite seems not to be a persistent feature and instead shows a relatively large orbit - to - orbit variation . furthermore , @xcite found evidence of a correlation between flux and spectral hardness in the x - ray band . recently , @xcite refined the gamma - ray period using _ fermi _ observations with a longer baseline , and found the period to be @xmath4days . since this is slightly different from the value ( @xmath5days ) used for the previous x - ray study carried out by @xcite , the x - ray results need to be refined using the new gamma - ray period . the baseline of the x - ray observations is long ( 5 years ) , and thus phases of later observations may change significantly . important questions to be addressed for gamma - ray binaries are : what is the nature of the compact object ( known only for psr b1259@xmath063 , * ? ? ? * ) , and what is the physical emission mechanism . if the source is powered by accretion , a complex continuum spectrum is expected whether the compact object is a neutron star or a black hole . hence , accurate measurement of the spectrum will help us identify the compact object . furthermore , searching for a spectral turn - over in the hard x - ray band ( e.g. , * ? ? ? * ; * ? ? ? * ) and/or spectral lines often seen in high - mass x - ray binaries ( hmxbs ) may also provide clues about the emission mechanism of the source . in this paper , we measure x - ray properties of the gamma - ray binary 1fgl j1018.6@xmath05856 more accurately than before using new observations taken with _ nustar _ , _ swift _ and with archival _ xmm - newton _ observations . in section [ sec : sec1 ] , we describe the observations we used in this paper . we show data analysis and the results in section [ sec : ana ] . we then discuss our findings in section [ sec : disc ] , and conclude in section [ sec : concl ] . @xmath6 absorption - corrected flux . + @xmath7 for mos1,2/pn . fw : full window . sw : small window . + @xmath8 @xmath9 was frozen for the _ swift _ and _ nustar _ data fit . + we observed the gamma - ray binary 1fgl j1018.6@xmath05856 with _ nustar _ @xcite four times between 2014 june 4 and december 1 with exposures of @xmath220ks for each observation . the total exposure was 90 ks . soft x - ray band below and overlapping with the _ nustar _ band ( 379kev ) was covered with _ swift _ observations and two archival _ xmm - newton _ observations ( see table [ ta : ta1 ] ) . the total exposure of the 71 _ swift _ observations was 169ks , and each exposure was relatively short . the _ nustar _ observations were processed with the standard pipeline tools nupipeline and nuproducts of nustardas 1.4.1 integrated in heasoft 6.16 . we used _ nustar _ caldb version 20140414 and applied the standard filters . in order to process the _ swift _ data , we used the xrtpipeline tool along with heasarc remote caldb and standard filters @xcite . note that the source was not clearly detected in some _ swift _ observations , and that the _ swift _ observations taken until mjd 55984 were reported previously @xcite . the _ xmm - newton _ data were processed with epproc and emproc in science analysis system ( sas ) 14.0.0 using standard filters . detection of pulsations in gamma - ray binaries can be difficult for several reasons , such as the possibilities of an unfavorable emission geometry , absorption of soft x - rays by the wind , or a large background due to non - thermal unpulsed emission . even in a favorable situation where the above effects are minimal , the doppler effect due to binary motion can blur the pulse signal if the orbit is tight . for 1fgl j1018.6@xmath05856 , @xcite showed that the doppler broadening is not a concern for a 20-ks observation assuming a circular orbit with an inclination of 30@xmath10 . we therefore attempt to search for the pulsation . event arrival times measured at the spacecraft were transformed into those at the solar system barycenter with barycorr for the _ nustar _ and barycen for the _ xmm - newton _ data . we did not search the _ swift _ data because of the paucity of counts in individual _ swift _ observations . for the _ nustar _ data , we produced an event list for each observation in the 320kev band using a circular aperture with @xmath11 . we performed the timing analysis with the data from each _ nustar _ focal plane module as well as with the combined dataset . above 20kev background dominates , and hence we adopt that as the high end of our band . note that the results below do not depend strongly on the exact energy range or the aperture size we folded the event time series to test periods between @xmath12 s , and calculated @xmath13 @xcite . we find that @xmath13 is fairly large for some test periods . however , we find that the large @xmath13 seen in one observation is not reproduced in the others . we further verified that the large @xmath13 values are not significant . note that the measured @xmath13 distribution does not follow a @xmath14 distribution , but has a long tail , and thus we used a functional distribution obtained by fitting the measured @xmath13 distribution in order to estimate the significance . we performed the same study for the _ xmm - newton_/pn data in the 0.52kev and 0.510kev bands , and did not find any significant pulsations . assuming the pulse profile is a sine function with a period in the range of 0.11s , we estimate the 90% upper limit for the pulse fractions to be 47% and 6% in the 320 kev and 0.510 kev bands , respectively . next , we refine the x - ray measurement of the orbital period by using a longer baseline using the _ swift _ data over a longer time period than the previous work . note that we did not use the data taken with _ xmm - newton _ or _ nustar _ because their count rate measurements can not be directly compared to those of _ swift_. as was done by @xcite , we use epoch folding @xcite because of the unequal exposures of the observations . in the _ swift _ observations , we extracted source and background events in the 0.510kev band within a @xmath15 radius circle , and an annular region with inner radius 50@xmath16 and outer radius 100@xmath16 , respectively . we then folded the event time series at test periods around @xmath17days @xcite , producing a light curve with 16 bins . we used the same epoch for phase 0 as that used in the previous studies @xcite . we calculated @xmath14 of the light curve for each trial period , and followed the fitting technique as described in @xcite . note , however , that we modeled the underlying continuum using a power - law function instead of the constant model employed by @xcite , because the @xmath14 of the folded light curve is rising towards short periods ( see figure [ fig : fig1 ] ) . the best - fit continuum model is @xmath18 ; the exact value of the power - law index varies between 0.9 and 1.1 depending on the fit range . we find that the best - fit orbital period varies between 16.538 and 16.55 days , depending on the number of bins , and the search step or search range . we varied the number of bins between 8 and 18 to ensure that the light curve is resolved and the @xmath14 statistic is applicable , and the search step between 0.001 and 0.005 days , smaller than the uncertainty in the @xmath19 measurement ( @xmath20days ) . the search range was varied between @xmath21day and @xmath22 days . we find that the variations are within 1@xmath23 of the measurements . the resulting period is @xmath24 days . we show the folded light curve in figure [ fig : fig2]a . [ cols="^,^,^ " , ] we also used the second harmonic to measure the orbital period , since it can be measured with better precision , and found that the measurement is more stable , varying only @xmath25days as a function of the number of bins , search step or the search range . the result is @xmath26 days . our result is consistent with the _ fermi _ measured value ( @xmath27 , * ? ? ? * ) with a null hypothesis probability @xmath28 . cc & + although @xmath9 towards the source has been measured previously , the uncertainty was relatively large . since there is one more _ xmm - newton _ observation ( obs . i d 0694390101 , table [ ta : ta1 ] ) taken after the previous x - ray study @xcite , we can determine @xmath9 more precisely using the _ xmm - newton _ observations . we extracted the source spectrum from a circle with @xmath29 ( obs . i d 0604700101 ) or @xmath30 ( obs . i d 0694390101 ) , and background spectra from a circle with @xmath31 in a source - free region @xmath2200@xmath16 vertically upwards along the detector column from the source . note that we used different source extraction regions because of differences in exposure times . since it has been suggested that the spectral hardness varies orbitally , we used different spectral slopes for observations taken at different phases ( figure [ fig : fig2 ] ) . thus , we fit the two _ xmm - newton _ spectra separately allowing all the fit parameters to vary . we grouped the spectra to have 20 counts per bin , and fit them with an absorbed power - law model with the angr abundance in xspec @xcite using @xmath14 statistics or _ l _ statistics @xcite . the two methods yield consistent results . the best - fit @xmath9 values for the observations are statistically consistent with each other ( table [ ta : ta1 ] ) . best - fit @xmath9 values obtained with a different abundance model ( wilm in xspec ; * ? ? ? * ) for the two spectra are still consistent with each other . therefore , we use a common @xmath9 value and find that a power - law model successfully explains the data ( figure [ fig : fig3 ] , left ) . the best - fit value is @xmath32 , and we use this value throughout this paper . note that using @xmath32 does not change the other spectral parameters in table [ ta : ta1 ] significantly . we find that using the wilm abundance model changes the best - fit @xmath9 values ( to @xmath33 , @xmath34 , and @xmath35 for obs . ids 0604700101 , 0694390101 and combined , respectively ) , but the other spectral parameters do not change significantly . we note that the source count rates were less than 0.030.08 cps for mos1/2 , and 0.10.3 cps for pn , and hence pile - up is not a concern . for the _ nustar _ data , we extracted source and background events from circular regions with @xmath11 and @xmath36 , respectively . backgrounds were extracted in the same detector chip as the source , offset @xmath24@xmath37 from the source region . the source was detected above the background up to 2030kev . we grouped the spectra to have a minimum of 20 counts per spectral bin , and used @xmath14 statistics and _ l _ statistics ; they provide consistent results . we jointly fit the data with a power - law model having different photon indexes for different orbital phases , and found that the best - fit parameters are @xmath38 and @xmath39 for @xmath40 , and @xmath41 and @xmath42 for @xmath43 . we find that a power - law model with a constant photon index across orbital phases can also explain the data if we let the flux vary between phases . however , a model with separate spectral indices for different phases provides a significantly better fit than one with constant phase - independent index throughout the phases , having an f - test probability that the improvement is just due to statistical chance of @xmath44 . using separate power - law indexes for the three observations taken at @xmath40 does not improve the fit . furthermore , individual fits of the observations suggest that the photon index is statistically the same among the observations taken at @xmath40 and that the photon index at phase 0 is different from that at phase 0.2 . we show the _ nustar _ spectra in figure [ fig : fig3 ] and the fit results in figures [ fig : fig2 ] and [ fig : fig4 ] . for the _ swift _ data , we extract the spectra using the same regions as for the timing analysis ( section [ timingana ] ) . the center of the source extraction circle was determined for each observation separately . since the source spectral properties vary with orbital phase @xcite , we performed phase - resolved spectroscopy . we folded the observations using the new timing solution we found in section [ timingana ] and merged the data in each phase bin , for a total of twelve phase bins . we further produced two spectra for phase 0 , one for the high - flux state and another for the rest of the observations taken at that phase , and eleven spectra for the other phases hence producing a total of thirteen spectra . we grouped the data to have 1 count per energy bin because of the paucity of counts in some phase bins , and used _ l _ statistics . for the phases that have enough counts , we also tried to fit the spectra using @xmath14 statistics , after grouping to have more than 20 counts per energy bin , and found that the results are consistent with those obtained using _ l _ statistics . we then fit all 13 spectra jointly with an absorbed power - law model with a common @xmath9 ( frozen at the _ xmm - newton_-measured value of @xmath45 ) throughout the observations but a separate photon index and flux for each spectrum . we find that the power - law model explains the data with photon indices of @xmath46 and 310kev fluxes of @xmath47@xmath48 , where the maximum flux was for the high - flux state ( see figures [ fig : fig2 ] b and c ) . we also tried to determine @xmath9 at each orbital phase using the _ swift _ data . however , we were able to constrain the fit parameters reasonably only for phase 0 ( without the high - flux state ) . we fit the spectrum for phase 0 ( excluding the high - flux state ) with a power - law model , and find that the best - fit parameters are @xmath49 , @xmath50 , and @xmath51 . this @xmath9 value at phase 0 is consistent with those obtained using the _ xmm - newton _ data for phases 0.3 and 0.6 above , suggesting that @xmath9 does not strongly vary as a function of orbital phase . although we can not clearly rule out orbital variation of @xmath9 , a @xmath210% variation of @xmath9 does not significantly change the _ swift _ results . accretion - powered neutron star hmxbs often show spectral features such as emission lines or an exponential cutoff in the x - ray band @xcite . we find that the x - ray spectrum of 1fgl j1018.6@xmath05856 is well described with a power - law model without requiring any additional features ( e.g. , figure [ fig : fig3 ] ) . for example , fitting the spectrum for phase 0 with a cutoff power - law model ( pow*highecut in xspec ) does not improve the fit , and the best - fit parameters are not constrained . we further changed the spectral grouping in order to have the spectra cover a broader energy range , and to see if a cutoff is required at higher energy . specifically , we grouped the _ nustar _ spectra to have more than 15 counts per energy bin , covering the 370kev band . we fit the spectra with a power - law model and a cutoff power - law model , and found the same results as above ; no cutoff is required in the fit . we performed additional analysis to determine the lower limit for the cutoff energy ( @xmath52 of the highecut model ) . however , it is not possible to set a meaningful lower limit for @xmath52 without constraining the e - folding energy ( @xmath53 of the highecut model ) . we therefore limit @xmath53 bewteen 6kev and 12kev , values obtained for a sample of accretion - powered neutron star hmxbs @xcite , and found that the 90% lower limit for @xmath52 is 39kev and 34kev for @xmath53 of 6kev and 12kev , respectively . note that some accretion - powered black hole binaries are known to have the cutoff energy above 70kev and that our data are not sensitive to such high energy cutoff . the spectral hardness varies with orbital phase ( figure [ fig : fig2]c ) and flux ( figure [ fig : fig4 ] ) . figure [ fig : fig4 ] shows an apparent correlation between flux and spectral hardness . we fit the apparent correlation with a constant function and found that it does not provide an acceptable fit ( @xmath54dof=55/18 ) . we therefore added a linear slope to the constant function and find that the linear fit explains the data well ( @xmath14/dof=17/17 ) . the measured slope is @xmath55 ( per @xmath56 ) , consistent with that reported by @xcite . the best - fit function is shown in figure [ fig : fig4 ] . @xcite suggested that there is evidence for a correlation between x - ray flux and spectral hardness . with the new and larger dataset , it is clear that the two quantities are correlated . for example , we find that the spearman s rank order correlation coefficient is @xmath57 and the significance is @xmath58 . we further verified that the x - ray flux and the photon index vary orbitally using the @xmath14 test , which resulted in @xmath59 and @xmath60 , respectively . note that whether or not we include the high - flux data point in the correlation calculation does not significantly change the result . although the significance for the correlation is high , uncertainties in the measurements are significant ( see figure [ fig : fig4 ] ) and need to be considered for the significance calculation . in order to do so , we performed simulations . note that the photon index and the flux are correlated in the spectral fit , and one needs to take into account the covariance . we do this by using the covariance matrices in the simulation as was done by @xcite . in 100,000 simulations , a non - negative correlation occurred 316 times , which suggests that the significance of the negative correlation is @xmath299.7% . we also carried out simulations for the linear correlation , and measured the confidence level of the negative linear correlation to be @xmath299.9% . we also checked for short - term variability ( @xmath210ks ) using the longer _ xmm - newton _ observation ( obs . i d 0694390101 ) because it has the best statistics . we calculate the count rate and hardness ratio ( ratio of count rates in two energy bands ; e.g , @xmath61 ) on various time scales and energy bands . we find variabilities of @xmath220% and @xmath210% for the count rate and hardness ratio , respectively , but no correlation between them . our analysis of the _ nustar _ , _ xmm - newton _ , and _ swift _ data is largely consistent with the x - ray results reported by @xcite , but the current work provides improvements and refinements . first , using longer _ swift _ observations , we find that the orbital period of 1fgl j1018.6@xmath05856 is @xmath1days , consistent with the gamma - ray measurement ( @xmath4days ; * ? ? ? * ) . when folded on the new period , the light curve shows two distinct features ; a spike at phase 0 and a broad sinusoidal hump ( figure [ fig : fig2 ] ) , similar to those reported previously @xcite . with the new period measurement , however , we find that the spike at phase 0 is a persistent feature and shows less orbit - to - orbit variability than was suggested by @xcite . second , we clearly see the correlation between flux and spectral hardness for which @xcite found only marginal evidence . this is possible thanks to more sensitive observations made with _ nustar _ , _ swift _ and _ xmm - newton_. note that we combined all the _ swift _ observations taken over a period of @xmath22000days for the spectral analysis . if there is long - term ( @xmath62 ) and/or short - term ( 10100ks ) variability , the combined results may be incorrect . this is a concern because there are only a few observations per orbital phase bin , and individual exposure of the observations is only @xmath2ks . furthermore , if the orbital period is not accurate or varies with time , phases of later observations will change , introducing an additional error to the analysis . however , the agreement of the _ swift _ measurements with the _ nustar _ and _ xmm - newton _ results suggests that the errors may not be large compared to the statistical uncertainties , having no significant impact on the results . for example , the significance of the correlation between flux and photon index is still @xmath6399% when adding 10% systematic uncertainty to the _ swift _ measurements . the broadband x - ray spectra of 1fgl j1018.6@xmath05856 at phases 0 and 0.2 are well described with a power - law model in the 0.540 kev band . recently , @xcite find that , based on parameter space consistent with radial velocity measurements , a neutron star model is preferred over a typical stellar mass black hole , although both classes are still allowed for 1fgl j1018.6@xmath05856 . we check to see if the source shows any evidence for accretion , such as line features or an exponential cutoff in the x - ray spectrum , as is often seen in neutron star hmxbs , and find none ( figure [ fig : fig3 ] ) . furthermore , we find no clear evidence for an exponential cutoff at @xmath64kev ( for spectrum at phase 0 ) . we , therefore , set the 90% lower limit for @xmath52 to be 3439kev for e - folding energies of 612kev ( @xmath53 ; see * ? ? ? * for the range of @xmath53 of neutron star hmxbs ) . this lower limit is large for a neutron star hmxb ( typical @xmath65kev ; e.g. , see * ? ? ? note that the x - ray pulsar x per ( also known as 4u 0352@xmath66309 ) for which @xcite did not find a clear spectral cutoff turned out to have a cutoff at 69kev @xcite , which is comparable to the energy under which we did not find any evidence for a spectral cutoff in 1fgl j1018.6@xmath05856 . also , high cutoff energies @xmath6370kev have been seen in black hole binaries @xcite . therefore , we can not clearly rule out the possibility that 1fgl j1018.6@xmath05856 is a black hole binary or a neutron star bianry with unusually high cutoff energy based only on the spectral cutoff . nevertheless , the continuum spectrum of x per or other x - ray binaries is very complex ( e.g. , * ? ? ? * ) while we see a simple power - law spectrum for 1fgl j1018.6@xmath05856 . this suggests that 1fgl j1018.6@xmath05856 may be a non - accreting neutron star system , which has also been suggested for another gamma - ray binary ls 5039 ( e.g. , * ? ? ? * ) . in analogy to ls 5039 , we may identify the location of the sinusoidal x - ray peak at @xmath67 ( figure [ fig : fig2 ] ) as inferior conjunction , and the gamma - ray peak at @xmath68 @xcite as superior conjunction @xcite . then , the phase difference of @xmath69 between the two conjunctions implies that the orbit is eccentric . we note , however , that it is not clear whether the x - ray and gamma - ray peaks are physically related to the conjunctions or the apastron / periastron passages , and that alignment of the x - ray and gamma - ray peaks with inferior and superior conjunctions may not be precise . therefore , more observations and detailed modeling are required in order to draw a firm conclusion . we find that the x - ray spectral properties of 1fgl j1018.6@xmath05856 clearly show orbital modulation . pulsar models for gamma - ray binaries often attribute such orbital modulation with orbital variation in the adiabatic cooling timescale @xcite , the electron injection spectrum , or the location and the shape of the wind nebula @xcite . the pulsar models have been applied to the similar system ls 5039 ( e.g. , spectral variability and recurring x - ray flares ; * ? ? ? * ; * ? ? ? * ) , and have reproduced the overall spectral energy distribution ( e.g. , * ? ? ? . however , whether or not these models can explain the spiky feature at phase 0 we see in 1fgl j1018.6@xmath05856 needs to be investigated . we note that the high - flux state observed with _ swift _ at phase 0 ( figure [ fig : fig2]a and b ) is not reproduced in other observations taken at the same phase . it may be because the two observations in the high - flux state were made in a very narrow phase interval and the later observations did not cover that phase interval . in order to see if this is the case , we first verified that the high - flux state was not produced by short timescale variability ( @xmath2ks ) ; it lasted for the full duration of the exposures of the observations ( 24 ks at mjd 55585.7 and 7 ks at mjd 55618.7 for obs . ids 00031912004 and 00031912011 , respectively ) which cover a phase interval of @xmath70 ( at @xmath71 for @xmath72days ) . we then measured the phases of the other observations . we find that there are four observations made at the high - flux phase interval , and none of them was in the high - flux state . since the phase of an observation can change significantly for a different orbital period , we further varied @xmath19 within the measurement uncertainty of @xmath73day , and find the same result . this suggests that there is orbit - to - orbit flux variability at phase 0 . we find that the duration of the high - flux state is longer than 24ks and shorter than 1.8days . the minimum duration is set to be 24ks because the high - flux states last during the observation ( see above ) . the maximum duration is set to be the interval between a high - flux state and the next non - high - flux observation , which is 1.8days for both high - flux states . as noted by @xcite , the observational properties of the flare such as duration and orbital repeatability look more like that of ls 5039 ( e.g. , * ? ? ? * ) than those of ls i @xmath6661@xmath10303 ( e.g. , * ? ? ? * ; * ? ? ? this may support the idea that the flares are produced by clumpiness of the stellar wind ( e.g. , * ? ? ? * ) since the stellar companion ( be star ) of ls i @xmath6661@xmath10303 is different from those ( o stars ) of 1fgl j1018.6@xmath05856 and ls 5039 as the flare properties do . however , how the clumpiness produces flares at one orbital phase for 1fgl j1018.6@xmath05856 or ls 5039 but not at random orbital phases needs to be further investigated . we present results of _ nustar _ , _ swift _ , and _ xmm - newton _ observations of the gamma - ray binary 1fgl j1018.6@xmath05856 . using the _ swift _ data , we measured the orbital period of the source to be 16.544@xmath740.008 days , in agreement with the refined gamma - ray measurement of @xcite . the new period is only slightly different from that used in our previous x - ray study , and hence our spectral and temporal analysis results agree well with the previous x - ray measurements . we find that the flux enhancement at phase 0 occurs more regularly in time than was suggested previously based on _ swift _ data . the new _ nustar _ and _ xmm - newton _ data allow us to show clearly the correlation between x - ray flux and spectral hardness of 1fgl j1018.6@xmath05856 . finally , the broadband x - ray spectrum of 1fgl j1018.6@xmath05856 suggests that it may not be an accretion - powered system . + we thank r. w. romani for useful discussions . this work was supported under nasa contract no . nng08fd60c , and made use of data from the _ nustar _ mission , a project led by the california institute of technology , managed by the jet propulsion laboratory , and funded by the national aeronautics and space administration . we thank the _ nustar _ operations , software and calibration teams for support with the execution and analysis of these observations . this research has made use of the _ nustar _ data analysis software ( nustardas ) jointly developed by the asi science data center ( asdc , italy ) and the california institute of technology ( usa ) . this research has made use of data obtained from the high energy astrophysics science archive research center ( heasarc ) , provided by nasa s goddard space flight center . acknowledges supports provided by the nasa sponsored fermi contract nas5 - 00147 and by kavli institute for particle astrophysics and cosmology ( kipac ) . ln wishes to acknowledge the italian space agency ( asi ) for financial support by asi / inaf grant i/037/12/0 - 011/13 .

we report on _ nustar _ , _ xmm - newton _ and _ swift _ observations of the gamma - ray binary 1fgl j1018.6@xmath05856 . we measure the orbital period to be @xmath1 days using _ swift _ data spanning 1900 days . the orbital period is different from the 2011 gamma - ray measurement which was used in the previous x - ray study of @xcite using @xmath2400days of _ swift _ data , but is consistent with a new gamma - ray solution reported in 2014 . the light curve folded on the new period is qualitatively similar to that reported previously , having a spike at phase 0 and broad sinusoidal modulation . the x - ray flux enhancement at phase 0 occurs more regularly in time than was previously suggested . a spiky structure at this phase seems to be a persistent feature , although there is some variability . furthermore , we find that the source flux clearly correlates with the spectral hardness throughout all orbital phases , and that the broadband x - ray spectra measured with _ nustar _ , _ xmm - newton _ , and _ swift _ are well fit with an unbroken power - law model . this spectrum suggests that the system may not be accretion - powered .

several theoretical predictions and scenarios have been proposed for the existence of ultra high energy ( uhe ) cosmic rays ( uhecr ) and uhe neutrinos @xmath5 detection of the uhecr and particularly uhe neutrinos would be of great importance for understanding the energy of powerful agns , gamma ray bursts and possible existence of massive particles predicted by the gut theories . for detecting uhecr and uhen several ambitious terrestrial experiments are being carried out and also planned with very large collecting areas @xmath6 1 @xmath7 and volumes @xmath6 1 @xmath8 @xcite . askaryan noted in 1960s @xcite , that electromagnetic cascades in dense medium by the uhe particles will develop an excess of negative charge giving rise to incoherent erenkov radiation . later , dagkesamanski and zheleznykh @xcite noted that uhe particles impinging on the lunar regolith at @xmath9 10 m-20 m deep layers of the moon will give rise to radio pulses of nanosecond ( ns ) durations . the large surface area of the moon effectively provides a large surface area for detection of the rare uhe particles . observations have been made towards the moon at 1.4 ghz using the parkes 64 m diameter radio telescope @xcite , and at 2.2 ghz using the jpl / nasa 70 m and 30 m antennas ( glue experiment ) @xcite and using a single 64 m telescope at kalyazin radio astronomical observatory @xcite . these have put upper limits on the existence of uhe particles but these are appreciably higher than the predictions by waxman and bahcall @xcite . askaryan effect has been tested using different media in a series of accelerator experiments . one of such experiment is done in silica sand which resembles composition of lunar regolith @xcite . as shown by alvarez - muniz et al . @xcite , the angular distribution of the electric field emitted by 10 tev shower in ice , salt and the lunar regolith is much wider at 0.1 ghz than at 1 ghz . scholten et al . @xcite have calculated differential detection probability for cosmic rays of energy @xmath10 ev and neutrinos of energy @xmath11 ev hitting the moon as a function of apparent distance from the centre of the moon for different detection frequencies . it is shown that the radio emission at higher frequencies arises mostly from uhe particles impinging near the rim of the moon but at lower frequencies from a major part of the moon , indicating the advantage of making observations at lower frequencies using already existing or planned radio telescopes of large collecting areas in the frequency range of about 30 to 300 mhz . for detecting uhecr and uhe neutrinos , observations are currently being carried out by radio astronomers in netherlands using the westerbork radio telescope ( wsrt ) @xcite at @xmath2 140 mhz . observations are also planned with the lofar @xcite under construction . in section ii , we summarize equations giving the expected value of the electric field and flux density for uhe particles as well as 25 times rms detection threshold of a radio telescope of collecting area @xmath12 panda _ et al _ @xcite have recently considered prospects of using the giant metrewave radio telescope ( gmrt ) @xcite for observing radio pulse emission arising from the uhe particles interacting with the surface of the moon . in section iii , we describe appropriate parameters of the gmrt for searching the lunar erenkov emission and also summarize expected values of the signal strength as a function of energy of uhe particles and the receiver noise threshold . in section iv , we propose observations of the erenkov radiation from the lunar regolith using the large ooty radio telescope ( ort ) @xcite that has an effective collecting area , @xmath13 = 8000 @xmath1 and is operating at 325 mhz . at present ort provides a bandwidth of only 4 mhz but its receiver system has been modified to provide @xmath14 mhz @xcite and is being extended to 15 mhz . in contrast to the gmrt providing dual polarizations at several frequency bands , the ort provides only a single polarization but it would be possible to get observing time of @xmath15 hours , as it is being used mostly for day time interplanetary scintillations . as discussed in sections iv and v , search for uhe particles will also allow simultaneous observations of lunar occultation of radio sources in the path of the moon and also variation of brightness temperature of the moon with the lunar phase , the latter yielding parameters such as dielectric constant and electrical conductivity of the lunar regolith upto depths of 30 m to 100 m. in section vi we discuss model independent limits for detection of uhecr and uhe neutrinos for several current and planned experiments , including lofar , wsrt , gmrt and ort . discussions and conclusions are given in section vii . the electric field of radio waves on earth , @xmath16 from a erenkov shower in the lunar regolith due to uhe neutrinos , with energy @xmath17 has been parameterized based on accelerator measurements and monte carlo simulations @xcite ( neglecting angular dependence ) giving @xmath18}\right)\ , . \label{field}\ ] ] where r is the distance between the emission point on the moon s surface to the telescope , @xmath19 is the radio frequency of observations and @xmath20 ghz for the lunar regolith material . the power flux density at earth , @xmath21 is given by @xmath22 where free space impedance , @xmath23 = 377 ohms , receiver bandwidth , @xmath24 is in units of 100 mhz and 1 _ jy _ = @xmath25 substituting from eq . [ field ] , we get @xmath26}\right)^2\ , ( \delta \nu / 100 mhz ) jy.\ ] ] panda et al . @xcite has given the following value of the power flux density @xmath27}\right)^2 \ , \frac{\delta \nu}{100\,\mathrm{mhz}}\,\,\ , \mathrm{jy}. \label{f}\ ] ] furthermore there is an angular dependence given by @xmath28 with @xmath29 and @xmath30 here we used gaussian approximation for our calculation where the forward - suppression factor @xmath31 in ( [ f ] ) is ignored . for high frequencies this has no effect . for low frequencies , the differences at small angles only plays a role for showers nearly parallel to the surface normal , while the effects of changing the normalization near the erenkov angle is important also for more horizontal showers . a measure of the effective angular spread @xmath32 of the emission around the erenkov angle @xmath33 is given in terms of @xmath34 it is seen from above that the value of @xmath21 as given by panda et al . is about 3 times lower than that given by eq.(2 ) . we find that the value of f by panda _ @xcite is 0.92 that given by scholten _ et al _ @xcite . we have used here eq.[f ] as per panda _ et al _ @xcite . by equating the power , @xmath35 received by a radio telescope , due to the incident input threshold threshold power flux density , @xmath36 with the minimum detectable receiver noise power , @xmath37 we have @xmath38 where the factor @xmath39 is due to the reception of a single polarization , @xmath13 the effective area of the telescope , @xmath40 the bandwidth and the receiver rms noise , @xmath41 @xmath42 being the system temperature , @xmath43 boltzmann s constant and @xmath44 the integration time . hence @xmath45 is given by @xmath46 for detection of a narrow pulse with width @xmath44 using an optimum bandwidth @xmath24 @xmath47 and hence rms noise @xmath45 is given by @xmath48 in tables [ table1 ] , [ table2 ] , [ table3 ] we list the system temperatures at the different observation frequencies and the corresponding noise levels of two different configurations of gmrt and ort . using equation ( [ f ] ) , we can solve for @xmath49 at the threshold required for measurement with the radio telescope ( obtained for @xmath50 and @xmath51 ) . if we take a required signal - to - noise ratio @xmath52 the threshold shower energies @xmath53 which can be measured at the different observation frequencies at the gmrt and the ort are given in tables [ table1 ] , [ table2 ] , [ table3 ] . the gmrt is a synthesis radio telescope consisting of 30 nos . of fully steerable parabolic dish antennas each of 45 m diameter . fourteen antennas are located in a somewhat random array within an area of about 1 @xmath54 and other sixteen antennas along 3 y - shaped arrays with a length of each @xmath9 14 km . the gmrt is currently operating in 5 frequency bands ranging from about 130 mhz to 1430 mhz . the receiver system provides output at two orthogonal polarizations from each of the 30 antennas with a maximum bandwidth of 16 mhz for each polarization , being sampled at 32 ns each . the @xmath13 of each antenna is nearly 950 @xmath1 in the frequency range of 130 to 630 mhz and only 600 @xmath1 at 1430 mhz . panda et al . have made estimates of the sensitivity of the gmrt for observations of uhe cr and uhe neutrinos . they have considered the @xmath13 of the gmrt @xmath55 at 150 , 235 , 325 and 610 mhz and 18,000 @xmath1 at 1390 mhz . however , we may note that the gmrt provides the above area only when the voltage outputs of all the 30 antennas are added in phase resulting in antenna beam of @xmath9 2 arcsec at the highest frequency and @xmath9 15 arcsec at 150 mhz and therefore covering only a small part of the moon . however the receiver correlator allows incoherent addition of the outputs of the 30 antennas , covering the entire front surface of the moon and resulting in @xmath56 @xmath57 at the lower 4 frequency bands and @xmath58 at 1390 mhz . insetad if we measure coincidences of the power outputs of the 30 antennas , the effective area will also be 5203 @xmath1 at the lower frequency bands but would have the advantage of discrimination between the lunar cerenkov emission and any terrestrial radio frequency interference(rfi ) as the gmrt antennas are located in an array of @xmath9 25 km extent . an alternative strategy will be more effective if we use the recently installed software correlator at the gmrt for cross multiplications of the voltage outputs of the 30 gmrt dishes with @xmath59 mhz . it allows 32 ns sampling of the voltage outputs of each of the 30 antennas . by combining these voltage outputs for the central 14 antennas of the gmrt with appropriate phase values , it would be possible to form 25 phased beams covering the moon , each beam having a resolution of about 6 arcmin at 140 mhz . the effective area for each of the 25 beams will be 14250 @xmath1 at the lower frequency bands and 9000 @xmath1 at @xmath9 1 ghz , providing a competitive radio telescope for searching for uhe neutrinos . contributions by the moon s temperature to the system temperature of the gmrt receiver is negligible at 140 mhz but is appreciable at higher frequencies . using the system parameters of the gmrt as given in tables [ table1 ] and [ table2 ] , we have estimated sensitivity of uhe cr and uhe neutrinos fluxes as given in figs [ crlimit100 ] , [ crblimit100 ] , [ nlimit100 ] , [ nblimit100 ] and [ fluxgmrt ] . .gmrt parameters , sensitivity and threshold sensitivity at different frequencies for an incoherent array . @xmath60 is the full width half maximum(fwhm ) beam of the 45 m dishes , @xmath61 is the temperature of the moon at frequency @xmath62 @xmath45 is the expected threshold flux density ( noise intensity ) of the gmrt and @xmath63 the corresponding electric field . the threshold energy @xmath53 is given in the last column . [ cols="<,<,<,<,<,<,<,<,<,>",options="header " , ] the ort consists of a 530 m long and 30 m wide parabolic cylinder that is placed in the north south direction on a hill with the same slope as the latitude of the station @xcite . thus it becomes possible to track the moon for 9.5 hours on a given day by rotating the parabolic cylinder along it s long axis . the ort operates only at 325 mhz and has effective collecting area of @xmath64 a phased array of 1056 dipoles is placed along the focal line of the parabolic cylinder . each dipole is connected to an rf amplifier followed by a 4 bit phase shifter . signals received by 48 dipole units are connected to a common amplifier branching network @xcite . the 22 outputs of the phased array are brought to a central receiver room . an analogue system that was originally built for lunar occultation observations @xcite provided 12 beams to cover the moon ; each beam is 6 arcmin in the north side direction and 126 arcmin ( @xmath9 2 deg . ) in the east west direction . recently a digital system has been installed by the raman research institute ( rri ) , bangalore and the radio astronomy centre of ncra / tifr , at ooty allowing formation of phased array beams with collecting area = 8000 @xmath1 and a bandwidth of 10 mhz with @xmath9 40 ns sampling @xcite . it is possible to form 6 beams covering the moon and 7th beam far away for discrimination of any terrestrial rfi . the proposed upgrade of the 12 beam analogue system will provide a bandwidth of 15 mhz . the measured receiver temperature of the ort is 140 k + a contribution by moon of @xmath2 ( 31.5/126 ) x 230 k = 57 k. thus @xmath65 of ort for lunar observations at 327 mhz is about 200 k. as discussed in the next section , observations of the moon for 1000 hrs using the ort at 325 mhz will provide appreciably higher sensitivity than the past searches made by various workers and also compared to a search being made currently in netherlands using the westerbork synthesis radio telescope ( wsrt ) at 140 mhz . using the ort , it may be possible to reach sensitivity to test the predictions of the waxmann - bachall model based on theoretical arguments . proposed observations , particularly with the ort will also provide arcsec resolution for galactic and extragalactic radio sources occulted by the moon , and may also search for any transient celestial sources in the antenna beam outside the disc of the moon . it would be quite valuable to make passive radio maps of the moon using the gmrt at decimetre and metre wavelengths . the suface temperature of the moon is about 130 k in its night time and @xmath9 330 k in its day time . since moon s surface consists of lossy dielectric material , the radio waves emitted by its thermal properties arise from few cm at microwaves to more than 100 m deep at wavelength of several m. therefore , the observed values of brightness temperature of the moon varies by tens of degrees at microwaves to less than a degree at radio wavelengths . the gmrt provides a resolution of about 2 arcsec at @xmath9 1420 mhz and @xmath9 15 arcsec at 150 mhz . polarization observations are also possible with the gmrt . therefore , maps of radio emission of the moon for its night and day with the gmrt will provide estimates of the dielectric constant and electric conductivity of the lunar regolith . the data will be complimentary to the radar measurements @xcite . in this section we calculate model independent limits for detection of uhe cr and uhe neutrinos for gmrt and ort using the procedure given in panda _ et al _ @xcite . scholten et al . @xcite have considered dividing the wsrt antennas into 3 groups for the proposed search for neutrinos , whence a is likely to be @xmath66 they give a value of @xmath45 = 600 jy for wsrt and @xmath67 jy . the system temperature for the ort , @xmath65 = 200k , at 327 mhz including contribution by the moon and @xmath68 . hence @xmath69 and @xmath70 = 1687.5 jy , which are much lower than for the wsrt . the event rate that would be expected at the telescope can be related to an isotropic flux @xmath71 of uhe particles on the moon through @xmath72 where @xmath73 denote the type of primary particle and @xmath74 is an aperture function corresponding to the effective detector area . the aperture can be further decomposed into an angular aperture @xmath75 and a geometric area factor for the moon @xmath76 with @xmath77 km . to evaluate the aperture , we use the analytical methods described in @xcite . for the case of strongly interacting cosmic rays which can mainly interact on the surface of the moon , the angular aperture is given by @xmath78 \times \theta(\cos\beta ) \mathrm{d}\alpha\mathrm{d}\cos\beta , \label{angle}\ ] ] where @xmath79 and @xmath80 are the polar and azimuthal coordinates of the ray normal to the moon s surface in a system where the shower direction defines the @xmath81 axis . the full geometry and the different angles are described in fig . [ geometry ] . when the uhe primary is instead a neutrino , it can produce showers deep below the surface of the moon and there will be considerable attenuation of the radio waves which travel distances longer than @xmath82 below the surface . for the neutrino induced showers , the aperture is defined in the same way as for the cr , but the angular aperture is now given @xcite by @xmath83-{\cal e}_{th}\bigr\ } \times \exp[-l(z,\beta)/\lambda_{\nu } ] \times \mathrm{d}\alpha \mathrm{d}\cos\beta , \label{omega - nu}\ ] ] where @xmath84 is the distance the neutrino travels inside the material to reach the interaction point at a distance @xmath81 below the surface . in performing this integration we allow @xmath81 to go below the known depth of the regolith . despite the attenuation , the aperture therefore picks up contributions coming from deep showers , especially for the lower frequencies . numerically we find for the worst case ( when @xmath85 mhz ) , that imposing a sharp cutoff at a depth of @xmath86 m would reduce the aperture by nearly an order of magnitude , similarly to what was discussed in @xcite . as for the cosmic rays , the total aperture is obtained by substituting ( [ omega - nu ] ) into ( [ acr ] ) and integrating over the polar angle @xmath87 . to estimate the sensitivity of gmrt and ort to cosmic ray and neutrino events we have evaluated the angular apertures by employing this technique and performing numerical integrations for the different parameters given in tables [ table1],[table2 ] , [ table3 ] . in the next section we will discuss these results further in the context of prospectiveflux limits . if no events are observed at gmrt and ort over a time @xmath88 then an upper limit can be derived on uhe cr and neutrino fluxes at the moon . the conventional model - independent limit @xcite is given by @xmath89 where still @xmath90 , @xmath91 and @xmath92 . the poisson factor @xmath93 for a limit at @xmath94 confidence level . in fig.[ortcrlimit ] , are shown prospective limits on the flux of the uhe crs for t=100 , 1000 , 8760 hours ( one year ) of the observation time with ort . plots for wsrt and lofar for t=100 hours of the observation time are also shown . in figs [ crlimit100 ] and [ crblimit100 ] are given model independent limits on uhe cr flux at different frequencies of the gmrt for an incoherent array and 25 beams case respectively for 100 hours of observations . similarly for the uhe neutrinos , prospective limits on their flux for t=100 , 1000 and 8760 hours of observation with ort are given in fig . [ ortnulimit ] . figs [ nlimit100 ] and [ nblimit100 ] give limits on the uhe neutrinos at different frequencies of the gmrt for an incoherent array and 25 beams case respectively for 100 hours of observations . for all our calculations we take @xmath95 @xcite . it is clear from the plots that that low frequency observations give more stringent limits on the flux at the expense of a higher threshold . this is due to the well - known increase in the aperture @xcite from radiation spreading at lower frequencies . since many radio experiments exist for uhe neutrino detection , we have compiled a comparison in fig.[fluxgmrt ] . this figure contains , the predicted thresholds of the ort at 325 mhz for 1 year of observation time , of the gmrtb ( 25 beams case ) at 140 mhz for 100 hrs and 30 days of observation time and the already existing limits from rice @xcite , glue @xcite , forte @xcite and anita - lite @xcite . also we have indicated the prospective future limits that has been calculated for anita @xcite , lofar @xcite or lord @xcite . james and protheroe @xcite have recently calculated sensitivity of the next generation of lunar cerenkov observations for uhe cr and neutrinos . in addition to search for uhe cr and neutrinos , simultaneous observations with the full array of the gmrt will provide radio maps of the moon as a function of the lunar phase , giving information about the average thermal and electrical conductivity of the moon s regolith up to depth of @xmath9 30 m to 100 m. therefore , for the two experiments to be carried out simultaneously , it may be possible to get 2@xmath96 50 hours of observations in two gmrt time allocation cycles . also , observations with the ort at the same time will allow discrimination against man made rfi transients . it will be prudent to use both the ort and the gmrt for searching for the uhe neutrinos . the new software correlator being installed at the gmrt will allow forming @xmath9 25 beams at 140 mhz to cover the moon at @xmath9 140 mhz providing 2 bands of 16 mhz and @xmath97 14250 @xmath3 one may also conveniently use the incoherent mode of the gmrt with @xmath97 5203 @xmath3 although ort with @xmath988000 @xmath1 operates only at 325 mhz , it is well suited to track the moon for hundreds of hours . the rfi is also much lower at ooty than at the gmrt site . by using the new digital system installed recently at ooty by prabhu _ @xcite of the raman research institute , in conjunction with the 12 beams of the analogue system ort and also it s upgrade , it should be possible to reach adequate sensitivity to test the waxman bahcall limit proposed on theoretical arguments on the uhe particle flux . proposed observations , particularly with the ort will also provide arcsec resolutions for celestial radio sources occulted by the moon , and may also detect any transient celestial sources present in the antenna beam outside the disc of the moon . search for uhe neutrinos will also allow simultaneous observations for making radio maps of the moon as a function of the lunar phase ( full moon , 5 and 15 days earlier and later ) , providing information about the average thermal and electrical conductivity of the moon s regolith up to a depth of @xmath9 30 m. the existence of uhe neutrinos of @xmath99 ev is implied by the detection of for @xmath100 ev . the extremely high luminosity of the star burst galaxies , agns , gamma ray burst are likely to accelerate protons to very high energies that get scattered by the cmbr photons producing a flux of uhe neutrinos . there are also predictions of their occurrence by more exotic sources in the early universe . as may be seen from fig . [ fluxgmrt ] , observations with the ort and gmrt will provide a threshold sensitivity of @xmath101 being comparable to the current searches being made by other investigators . detection of the uhe cr and neutrinos of @xmath99 ev would be of great importance for testing theories of high energy physics and for understanding several phenomena of cosmological and astrophysical importance . * acknowledgement * we thank t. prabhu of the raman research institute , p.k . manoharan and a.j . selvanayagam of the radio astronomy centre ooty and s. sirothia of ncra , pune for many valuable discussions . the work of s.p . was supported by the ministerio de educacion y ciencia under proyecto nacional fpa2006 - 01105 , and also by the comunidad de madrid under proyecto hephacos , ayuda de i+d s-0505/esp-0346 . t. h. hankins , r. d. ekers and j. d. osullivan , mnras * 283 * , 1027 ( 1996 ) ; c. w. james , r. m. crocker , r. d. ekers , t. h. hankins , j. d. osullivan , r. j. protheroe , mnras * 379 * , 1037 ( 2007)[astro - ph/0702619 ] .

searching for the ultra high energy cosmic rays and neutrinos of @xmath0 is of great cosmological importance . a powerful technique is to search for the erenkov radio emission caused by uhecr or uhe neutrinos impinging on the lunar regolith . we examine in this paper feasibility of detecting these events by observing with the giant metrewave radio telescope ( gmrt ) which has a large collecting area and operates over a wide frequency range with an orthogonal polarisation capability . we discuss here prospects of observations of the erenkov radio emission with the gmrt at 140 mhz with 32 mhz bandwidth using the incoherent array and also forming 25 beams of the central array ( effective collecting area of 14250 @xmath1 ) to cover the moon . we also consider using the ooty radio telescope ( ort ) which was specially designed in 1970 for tracking the moon . the ort consists of a 530 m long and 30 m wide parabolic cylinder that is placed in the north south direction on a hill with the same slope as the latitude of the station . thus it becomes possible to track the moon for 9.5 hours on a given day by a simple rotation along the long axis of the parabolic cylinder . ort operates at 325 mhz and has an effective collecting area of @xmath2 8000 @xmath3 recently a digital system has been installed by scientists of the raman research institute ( rri ) , bangalore and the radio astronomy centre ( rac ) of ncra / tifr , at ooty allowing a bandwidth of 10 mhz with @xmath2 40 ns sampling . it is possible to form 6 beams covering the moon and 7th beam far away for discrimination of any terrestrial rfi . increasing the bandwidth of the existing 12 beam analogue system of the ort from 4 mhz to 15 mhz to be sampled digitally is planned . it is shown that by observing the moon for @xmath4 1000 hrs using the ort it will provide appreciably higher sensitivity than past searches made elsewhere and also compared to the search being made currently in netherlands using the westerbork synthesis radio telescope ( wsrt ) at 140 mhz . using the gmrt and ort , it may be possible to reach sensitivity to test the waxman - bachall limit based on theoretical arguments on the uhe particle flux .

the idea that our universe is an element in a vast set of universes , the multiverse , has been argued to be an interesting way to address the cosmological constant problem in the context of string theory @xcite . of course , this scenario raises many questions . how is the vacuum of our world chosen ? through anthropic arguments @xcite ? through quantum cosmology arguments @xcite ? is the string landscape scenario compatible with predictability @xcite ? do the universes of the multiverse interact @xcite ( see also ref . @xcite ) ? does the multiverse exhibit collective behavior @xcite ? the multiverse also arises in the context of the so - called many world interpretation of quantum mechanics @xcite and in the eternal inflationary model @xcite . actually , it has been recently proposed that the multiverse of eternal inflation and the many - worlds interpretation of quantum mechanics can be identified , yielding a new view on the measure and measurement problems . however , it has been argued that a non - linear evolution of observables in the quantum multiverse would be an obstacle for such a description as these non - linearities are expected from quite general arguments @xcite . in this paper we shall study the process of vacuum decay in the context of an interacting multiverse @xcite . the consideration of an interacting multiverse entails a new and richer structure for the whole set of universes . the aim of this paper is to analize the influence of this enriched structure in the process of the vacuum decay of a single universe . first , we shall consider the wheeler - de witt equation for the wave function of the space - time . for many cases of interest the space - time is described by a homogeneous and isotropic geometry whose spatial section volumes scale as @xmath0 , where the scale factor @xmath1 is a function of the cosmic time @xmath2 of a given multiverse . in this case the wave function of the universe , @xmath3 , simplifies and it only depends on the values of the scale factor and the matter fields , i.e. @xmath4 , with @xmath5 being a set of scalar fields . the se can t hus be considered as a field that propagates in the space spanned by the variables @xmath6 . following the usual prescriptions of quantum mechanics , a second quantization procedure can be applied to the field @xmath7 , which can be described in terms of quantum oscillators with their corresponding creation and annihilation operators . these operators would represent , in an appropriate representation , the creation and annihilation of pieces of the space - time with a given geometry . this description allows for representing the fluctuations of the space - time in terms of baby universes @xcite , i.e. small particle - like portions of space - time that pop up and branch off from the parent space - time and propagate therein . similarly , for a super - observer the field @xmath7 can be described in terms of particle - like pieces of space - time that we call universes . the aim of this work is to examine if a supra - universal structure can influence the properties of a single causally isolated region of the space - time . whatever the definition of a universe is , it can be associate to some notion of causal closure , i.e. a region of the space - time manifold where all causally related events are self - contained . in other words , something that may cause or may be caused by any effect on any observed part of the universe should be included as being part of the universe . thus , although it seems meaningless to consider _ external _ elements of the universe , we shall see that this is not the case . the classical and local notion of causal closure does not exclude the possibility that non - local interactions among different regions of the space - time may determine some of the global properties of single universes . in fact , it has already been shown @xcite that the interaction between two or more universes could determine the effective values of the cosmological constant of the universes . despite that , light ray cones and local causal relations and properties within each single universe still obey the usual relations and remain causal . however , the value of a global property like its the cosmological constant can be affected by the interaction among the universes . this cosmological picture is then completely different than the one single universe picture . interactions and collective behaviour might then occur among the universes of the multiverse . actually , this collective behaviour is fairly general and is at the very heart of quantum theory , which is a non - local theory and within which all the physical elements are fundamentally coupled to their environment and individual properties arise out of a result of some decoherence process . thus , the true quantum state of the space - time must account for the states of all the universes , if they exist . the aim of this paper is to examine whether some of these collective processes may have an observable influence on the properties of our universe . irrespective of the consideration of a multiverse and its implications , it seems therefore interesting to analyze the influence , if any , that different distant regions of the space - time may have on the properties of the observable part of our local universe ( see also ref . @xcite ) with a two - fold aim : i ) to analyze whether they might help to solve some of the open questions posed by the latest planck data @xcite , and ii ) to look for distinguishable imprints of other universes in , for instance , the properties of the cosmic microwave brackground ( cmb ) spectrum @xcite . this paper is organized as follows : in section ii , we discuss the hamiltonian quantum cosmology model of an interacting multiverse . in section iii , we consider the bubble formation , that is , the nucleation of universes in a parent space - time and specifically address this nucleation in a setting where the universes are interacting . finally , in section iv , we present a discussion of our results . let us consider a simply connected piece of a homogeneous and isotropic space - time manifold endowed with a scalar field @xmath8 that represents the matter content . more general topologies can also be considered by splitting the whole manifold into simply connected pieces of space - time @xcite , each of which is quantum mechanically described by a wave function @xmath9 that is the solution of the wheeler - dewitt equation @xcite @xmath10 where the scalar field has been rescaled according to ref . @xcite , @xmath11 , where @xmath12 is the planck mass . in eq . ( [ wdw01 ] ) the dots represent derivatives with respect to the scale factor and the prime denotes derivative with respect to the scalar field . the function @xmath13 contains the potential terms of the wheeler - de witt equation . in the case of a closed space - time it is given by @xmath14 where @xmath15 and @xmath16 is the hubble function . the frequency @xmath17 has units of mass or , equivalently , units of the inverse of time or length . we shall consider two contributions to the hubble function , i.e. , @xmath18 . the first one is due to the existence of a cosmological constant , @xmath19 , which is assumed to be very small . the second contribution is due to the potential of the scalar field , @xmath20 . let us now develop a quantum field theory for the wave function @xmath3 in the curved minisuperspace spanned by @xmath21 with a minisuperspace metric given by @xmath22 where @xmath23 stands for @xmath24 . the _ line _ element of the minisuperspace metric is therefore @xmath25 the scale factor , @xmath26 , formally plays the role of the time variable and the matter field the role of the spatial variable in the two dimensional lorentzian minisuperspace metric ( [ msm ] ) ( @xmath1 can actually be seen as a time reparametrization ) . we can now follow the usual procedure of a quantum field theory for the scalar field @xmath27 by considering the following action @xmath28 where the lagrangian density is given , as usual , by @xmath29 where @xmath30 . then , the corresponding euler - lagrange equation @xcite @xmath31 turns out to be the wheeler - de witt equation , eq . ( [ wdw01 ] ) . the hamiltonian density that corresponds to the lagrangian density , eq . ( [ l01 ] ) , is given by @xmath32 where @xmath33 is the momentum conjugated to the field @xmath3 . we can now pose an interaction scheme @xcite among a set of @xmath34 universes by considering a total hamiltonian density given by @xcite @xmath35 where @xmath36 is the unperturbed hamiltonian density of the @xmath37-universe , given by eq . ( [ h01 ] ) , and @xmath38 is the hamiltonian density of the interaction for the @xmath37-universe , that here we consider as the simple quadratic interaction between next neighbour universes , @xmath39 where @xmath40 is a coupling function that can depend on the value of the scale factor and we use periodic boundary conditions so that , @xmath41 . we consider that the hamiltonian density , eq . ( [ h02 ] ) , represents the evolution of a set of universes that are interacting to each other , where each internal observer do not see any interaction but only its own hamiltonian density . we can take into account , for the sake of simplicity , a new representation given in terms of the normal modes by means of the fourier transformation of @xmath3 and @xmath42 @xmath43 the hamiltonian density , eq . ( [ h02 ] ) , becomes @xmath44 the new quantum states oscillate now with a frequency given by @xmath45 this is formally the typical quantum description of a collective system in terms of normal modes . these represent a collective behaviour in which the wave function can oscillate . for a single mode @xmath46 the oscillation of the wave function @xmath47 is given by the equation @xmath48 which is the effective wheeler - de witt equation of the wave function of the @xmath46-universe in the @xmath49 representation that appears as an isolated , non - interacting universe . we shall assume that , although we work in this single multiverse wave function formalism , all the results can be decomposed in terms of the previous formalism of individual universes . the effective value of the potential term of the scalar field in the @xmath50-universe has been modified as a result of the interaction with other universes . let us notice that eq . ( [ frq02 ] ) can be written as @xmath51 where @xmath52 and @xmath53 , with @xmath54 let us now analyse the influence of the last term in eq . ( [ np ] ) in the terms of the @xmath46-universe . we restrict our interest to the regime where the wave function of the @xmath46-universe can be approximately described by the semiclassical wave funtion @xmath55 where @xmath56 is the action of the gravitational part alone with no interaction , that is : @xmath57 where the positive and negative signs in eq . ( [ wfsc01 ] ) correspond to the contracting and the expanding branches of @xmath58 , respectively . the wave function @xmath59 satisfies then , at first order in @xmath60 , the following wave equation @xcite @xmath61 let us notice that in the absence of any interaction scheme eq . ( [ sceq01 ] ) is the schrdinger equation for the scalar field @xmath8 with a hamiltonian given by @xmath62 the field equation for the scalar field is given by @xmath63 where the dots stands for derivative with respect to the friedmann time @xmath2 , i.e. @xmath64 . the last term in the potential @xmath65 , given by eq . ( [ np ] ) , has no influence upon eq . ( [ sf01 ] ) , so the classical behaviour of the scalar field remains unaltered with respect the usual description . quantum mechanically , however , the extra term in the potential introduces a modification in the vacuum state that has to be accounted for any vacuum decay process of the universe . this has a major influence in the process of bubble formation and in the global structure of the space - time . the quantum interactions among distant regions of the whole space - time manifold can modify the vacuum state of the matter fields . we have seen that for a chain of interacting universes , where the potential of the scalar field for a normal mode of the wave function of the universe depends on the value @xmath50 of the mode . different universes may remain in different mode states and observers therein feel their universes to be filled with a scalar field whose vacuum state is different for each mode state . the universes may then suffer a process of vacuum decay between two different values of their vacua . the key point for the vacuum decay is the existence of different local minima in the potential of the matter fields . the minimum of those local minima is called the true vacuum and the remaining are the false vacua . in the late 1970s coleman has generalized the quantum mechanical tunnelling effect of transition from the false " vacuum ( excited state ) into the true " vacuum ( ground state ) in field theory @xcite . subsequently , quantum radiative corrections were introduced @xcite and finally the effect of gravity was considered @xcite . the issue of adopting a set of consistent boundary conditions for the decay process was examined in ref . @xcite . in the framework developed by coleman and collaborators , a field in a false vacuum state can then decay to the state of true vacuum with a probability of occurrence per unit time per unit volume , @xmath66 , given in the semiclassical approximation by @xmath67 where @xmath68 and @xmath69 are two quantities to be determined . the result is the materialization of a bubble of true vacuum separated by a thin wall from the surrounding false vacuum . the global picture is then a vast space - time in the false vacuum splattered by bubbles of true vacuum that are continuously forming and expanding until they finally collide , merge and fill the whole space - time . let us now consider the process of vacuum decay in the context of a theory @xcite with a potential @xmath70 that has two minima , @xmath71 and @xmath72 , with values of their vacuum energy given by @xmath73 and @xmath74 , respectively ( see fig . [ fig1 ] ) . the thin wall approximation , that is going to be applied , it is satisfied when @xmath75 hence , a region of the space - time in the false vacuum may decay into the true vacuum by nucleating a bubble of true vacuum within the surrounding false vacuum that rapidly grows and expands when it is energetically favourable . let us now analyse the same process in the context of the interacting multiverse described in sect . ii . for a given value of the scale factor , the extra term of the potential of the @xmath46-universe is a constant that depends on the value @xmath46 of the normal mode of a given universe . the global effective value of the potential is then given by a set of curves separated by @xmath46 units ( see fig . [ fig2 ] ) , with @xmath76 . the global picture presents then a landscape structure with @xmath34 different vacua : @xmath77 false vacua states and a true vacuum state @xmath78 . we can consider , on one hand , the vacuum decay in a universe as a consequence of the multiverse interaction . on the other hand , we can also consider a global picture of vacuum in the multiverse and study the vacuum decay into a real vacuum that corresponds to a single universe . the vacuum decay process follows the description of ref . however , the process of bubble formation and the global structure of a single universe is now much richer . it can be envisaged as follows : small baby universes are created from quantum fluctuations of the space - time . at small values of the scale factor ( i.e. values close to the planck scale ) the fluctuations of the scalar field and the effects of the interaction among the universes are dominant , so the newborn universes are expected to remain in normal modes with a high value of @xmath46 . in some universes , the effective value of the potential would be high enough to trigger inflation even if we assume the new limit that is suggested by the most recent planck data @xcite . as the universe expands in the @xmath46-false vacuum , different processes of vacuum decay are expected to occur generating new bubbles of smaller and smaller false vacua until the bubbles are created in the true vacuum @xmath79 . ho wever , the process does not stop there . the quantum fluctuations of the space - time of large regions with false or true vacuum state would supply new baby universes where the process of vacuum decay and bubble formation would take place continuously in a self - contained eternal process . , which is suggested by the most recent planck data @xcite , the energy supplied by the quantum vacuum state could be large enough to trigger inflation in universes with a high mode state , i.e. @xmath80 , for high values of @xmath46 . a process of vacuum decay could then occur afterwards.,width=302 ] for the sake of concreteness , let us consider the scalar field with a quartic potential @xmath81 where @xmath82 is the mass and @xmath83 is the self - coupling of the scalar field . it has two minima located at @xmath84 , both with the same value of the potential given by @xmath85 . let us notice that in the case studied by coleman and de luccia the value of the potential at the two different minima differs by a small amount of energy @xmath86 , otherwise the process of vacuum decay would not be possible . in the case under consideration , this condition is not necessary because the process of vacuum decay will take place between the vacuum states of different modes that could correspond to different universes or to a single universe whose vacuums had been modified by the interaction with other universes this is schematically represented in fig . thus , instead of having eq . ( [ tw01 ] ) defining the thin wall approximation , here we have @xmath87 that is @xmath88~.\ ] ] the probability for a vacuum decay from the mode @xmath46 to the mode @xmath89 is given by @xcite @xmath90 where @xmath91 and @xmath92}$ ] . @xmath93 is a function chosen such that @xmath94 and that @xmath95 thus , @xmath96 given in eq . ( [ qv ] ) . we choose @xmath97 as the point at which @xmath8 is the average of its two extreme values , @xmath98.\end{aligned}\ ] ] furhermore , @xmath97 is assumed to be large compared to the length scale on which @xmath8 varies significantly . then , it is possible to write @xmath8 in terms of @xmath99 @xcite : @xmath100^{-1/2}}\leftrightarrow\nonumber\\ \leftrightarrow \phi(\rho)&\!\!\!=\!\!\!&{m\over \lambda_{\varphi } } + \tanh\left[{{m\over 2\sqrt{2}}(\rho-\bar{\rho})+\tanh^{-1}{{\lambda_{\varphi}\over m}\phi_{1/2}}}\right]~.\nonumber \\\end{aligned}\ ] ] hence , following ref . @xcite we now can evaluate @xmath101 in the thin wall approximation , @xmath102}={2\sqrt{2}\over3}{m^3\over \lambda_{\varphi}^2}~,\ ] ] and consequently the probability factor @xmath69 for a vacuum decay from the mode @xmath46 to the mode @xmath89 is @xmath103 ^ 3}}~.\ ] ] it is worth noticing that eq . ( [ b2 ] ) restricts the values of coupling function @xmath40 that suppress the vacuum decay at large values of the scale factor . for instance , with a polynomial value @xmath104 , @xmath37 must satisfy @xmath105 in order to fulfill the condition that the vacuum decay can not grow with the scale factor . we can thus analyze some plausible cases . let us first consider the value @xmath107 where the constants has been chosen for later convenience . then , eq . ( [ frq03 ] ) can be re - written as @xmath108 where @xmath109 , and @xmath110 with @xmath111 thus , eq . ( [ wdw02 ] ) would represent the quantum state of a universe with an effective value of the cosmological constant of the background space - time given by eq . ( [ effl ] ) . for a positive value of @xmath112 , @xmath113 $ ] . if we assume a small value of @xmath114 ( included the value @xmath115 ) and a value @xmath116 , then , at the onset of the universe where it is supposed to remain at a large value for the @xmath50 mode , the effective value of the cosmological constant would be large enough to trigger inflation . afterwards , as the universe decays into lower modes , the effective value of the cosmological constant would be getting smaller and smaller until it would reach the value @xmath114 that would be the currently observed value of the cosmological constant . another way to obtain a small effective value of the cosmological constant is to suppose a negative value for @xmath112 . then , @xmath117 $ ] provided that @xmath112 is of the same order of @xmath114 . however , it implies a strong fine tuning and it would mean that our universe is now in a state with a high value @xmath50 of the mode . this does not seems to be consistent . let us now consider the coupling function @xmath119 where @xmath120 is a constant parameter . now @xmath121 with @xmath122 because the factor @xmath123 in eq . ( [ frq05 ] ) , eq . ( [ wdw02 ] ) with eq . ( [ frq05 ] ) would not actually represent the quantum state of a closed universe . however , we can perform a scale factor transformation @xmath124 in terms of which eq . ( [ wdw02 ] ) turns out to be @xmath125 where now the dots stand for the derivative with respect to the transformed scale factor @xmath126 , and @xmath127 with @xmath128 eq . ( [ wdw02 ] ) with eq . ( [ frq06 ] ) does actually represent the quantum state of a closed universe with an effective value of the cosmological constant given by @xmath129 . for a value @xmath130 , the effective value of the cosmological constant satisfies , , for @xmath131 . it would imply that for large values of the mode @xmath46 , at the onset of the universe , the effective value of the cosmological constant would be large enough to trigger inflation . however , as the state of the universe is decaying the effective value of the cosmological constant is decreasing . the current state of the universe would then correspond to a very small value of the mode @xmath46 . in this case , eq . ( [ wdw02 ] ) would represent the quantum state of a universe for which @xmath133 with @xmath134 eq . ( [ frq07 ] ) is the frequency that arises in the third quantized model of a universe filled with a minimally coupled scalar field with mass @xmath82 , like the field @xmath8 considered in this paper , and another massless scalar field which is conformally coupled to the background space - time ( see refs . ) . the conformally coupled masses scalar field can effectively mimic a radiation like field with an energy given by @xmath135 . therefore , the result of the interaction between universes would imply in the present case the appearance of a radiation like content of the universe that would be of order @xmath136 for large values of the mode @xmath46 , and it would be decaying to the value @xmath137 for small values of the mode ( @xmath138 for @xmath139 ) . this effective content would imply the existence of a pre - inflationary stage of the universe that should have observable effects in the power spectrum of the cmb . let us us consider the flat branch of a de sitter ( or quasi - de sitter , i.e. @xmath140 ) space - time . then eq . ( [ frq07 ] ) simplifies as , @xmath141 with @xmath142 . the friedmann equation turns out to be then @xmath143 with solutions given by @xmath144 where @xmath145 , with @xmath146 some constant to fit with the boundary conditions . at late times , the scale factor ( [ sf03 ] ) grows in an exponential way approaching to an exact de sitter expansion . however , at the earliest epoch it shows a deviation from desitter evolution that would have a strong influence in the lowest modes of the cmb power spectrum @xcite . in refs . @xcite , it is analyzed the effects that a preinflationary phase dominated by an energy density inspired by the generalized chaplygin gas @xcite ( see also ref . @xcite ) @xmath147 has in the power spectrum of the cmb . let us notice that eq . ( [ nfdw ] ) can reproduce a radiation dominated preinflationary stage of the universe , like the one studied in this paper , with a suitable choice of parameters : @xmath148 , @xmath149 , @xmath150 , and @xmath151 . then , the same procedure used in ref . @xcite can be applied here . the results ( see also ref . @xcite ) indicate that a radiation dominated preinflationary stage of the universe may alleviate the quadrupole anomaly of the cmb in a better way than ia matter dominated preinflationary stage . however , it is concluded that a greater departure from de sitter space - time is needed during the preinflationary stage in order to better fit with the observational data . let us now analyze the case where @xmath153 . this case is particularly interesting for at least two reasons : i ) it shows also a preinflationary stage whose effects on the power spectrum of the cmb are expected to be stronger that those caused by a radiation dominated preinflationary stage @xcite , and ii ) the quantum effect of the interacting multiverse have no classical analogue so it can be considered a distinguishable imprint of the multiverse on the cosmic observational data . let us first point out that a term proportional to @xmath154 in the frequency in eq . ( [ frq01 ] ) arises also in the decomposition in partial waves of the wave function of a de sitter space - time @xcite . such a decomposition is equivalent to the interacting scheme presented here with a coupling function given by @xmath152 . it is therefore a pure quantum effect having no classical analogue . in both cases , it implies the appearance of a term proportional to @xmath155 in the effective equation of the energy density . in contrast to the term @xmath156 caused by the radiation dominated preinflationary stage , the departure from de sitter space - time caused by the interacting multiverse is thus stronger . the effective friedmann equation for the flat branch of the de sitter universe turns out now to be given by @xmath157 where @xmath158 is @xmath159 with @xmath160 is some constant parameter , and solutions @xmath161 and @xmath162 , being @xmath163 and @xmath164 constants of integration . the departure form the de sitter evolution lead by the term proportional to @xmath154 in the friedmann equation ( [ sf04 ] ) is stronger than the one produced by a radiation term ( proportional to @xmath165 ) ( see fig . [ fig5 ] ) . therefore , it is expected that its effects on the lowest modes of the power spectrum of the cmb would provide a better fit with the observed data @xcite . it would provide an observational support to the model presented in this paper and to the whole multiverse proposal . thus , it provides with distinguishable predictions that can be compared with observational data , making therefore falseable the whole multiverse theory . the interacting picture developed in sec . ii opens up the possibility of new and interesting processes of vacuum decay such as the one depicted in fig . [ fig4 ] . the decay between the vacuum state of the mode @xmath50 and those of the mode @xmath89 might occur through an intermediate vacuum decay into a metastable state given by the value @xmath166 of the mode @xmath89 . this state would rapidly decay into the vacuum states of modes in a process that parallels those occuring in quantum optics where a two - photon state is generated through a metastable state ( see , for instance , ref . @xcite ) , where the radiation field turns out to be described in terms of pairs of entangled photons ( see fig . [ fig4 ] ) . in the case of the vacuum decay of the space - time , it would result into the generation of two bubbles of true vacuum ( of the lower false vacua ) whose quantum states would be entangled , @xmath167 where @xmath168 are the expanding and contracting branches of eq . ( [ wfsc01 ] ) , with @xmath169 . the properties of the space - time inside the two entangled bubbles would be the same at large scales for observers inhabiting therein . for instance , the effective value of the cosmological constant would be the same , given by @xmath114 and the effective mass scale of the scalar field would be in both bubbles given by @xmath170 , so the inner part of the two bubbles would be very similar at large scales . if the two entangled bubbles would come out from a double instanton , like the one studied in ref . @xcite , then , the quantum state of one of the bubbles would be given by the reduced density matrix that is obtained by tracing out the degrees of freedom of the partner bubble of the entangled pair , with a probability given by @xcite @xmath171 with @xmath172 , \ ] ] where @xmath173 and @xmath174 are the complete elliptic integrals of first and second kind . we could follow ref . @xcite to obtain the probability for the double euclidean instanton given by the euclidean solutions of eq . ( [ wdw01 ] ) with the quartic potential ( [ qv ] ) . in general ( see also ref . @xcite ) , the resulted state is a thermal state that is indistinguishable from a classical mixture so that observers inhabiting the bubbles would see the scalar field of their universes in a thermal state , being completely unaware of the entanglement properties of their bubbles . there is also a time reversal symmetry between the time variables of the entangled bubbles @xcite so the regions inside the bubbles would present opposite symmetry assignments such as , for instance , baryon asymmetry or other discrete symmetries ( see also ref . ) that would be the consequence of the global symmetries of the entangled pair of bubbles . for an observer external to the bubbles , the symmetry assignments and asymmetries would disappear when he / she would consider the properties of the whole entangled pair . the question then is whether this would fix some of the apparent asymmetries of our universe . in this work we have examined the implications of an interacting multiverse and the issue of bubble nucleation of true " vacua in the universes filled by the false " vacuum at its genesis or later . the existence of a multiverse where the universes may interact opens up the door to a new scenario with important implications on the global structure of the universes . as a result of the interactions it appears a landscape structure where the universes are created with different effective values of the cosmological constant . thus , there exist the possibility of quantum tunneling transitions between different universal states giving rise to new bubbles with the corresponding value of their vacuum states . the interaction between universes and the generation of new bubbles is expected to be dominant only for small length scales of the parent space - time , where quantum effects of the space - time are significant . however , these newborn bubbles may expand and generate new bubbles in a self - reproducing process . the vacuum decay between the quantum states of two or more universes is expected to be cut off for large values of their scale factors . this condition imposes a restriction on the possible values of the coupling function in the interaction scheme presented in this paper . the possible cases of interest have been analysed and they all would have observable consequences in the global properties of each single universe . they could explain the very small value of the cosmological constant of our universe . however , a fine - tunning is always required making the proposal , in this respect , no better than others cosmological scenarios @xcite . in the case that the coupling function would be a constant , it would appear the existence of a pre - inflationary stage in the evolution of the single universes . a pre - inflationary stage would have observable implications in the power spectrum of the cmb of the universe @xcite . furthermore , it has been claimed that it would fi t some o f the anomalies that most models of inflation present with respect to the latest observational data provided by planck . the interacting multiverse with a coupling function proportional to @xmath175 provides us with a preinflationary stage of the universe whose departure from de sitter expansion is stronger than the one caused by a radiation dominated preinflationary stage . this would be a pure quantum effect having no classical analogue so : i ) it might hopefully alliviate the quadrupole anomaly of the cmb and , ii ) it could provide distinguishable predictions of the multiverse that can not be explained by any other known effect . thus , the model of the multiverse presented in this paper entails distinguishable predictions that can be compared with current observational data being therefore falsifiable leading to implications that have not been considered so far . was partially supported by the marsden grant _ topics in mathematical general relativity and theoretical cosmology " _ administered by the royal society of new zealand . the work of cb is supported by the fct ( portugal ) grant sfrh / bpd/62861/2009 .

we examine a new multiverse scenario in which the component universes interact . we focus our attention to the process of true " vacuum nucleation in the false vacuum within one single element of the multiverse . it is shown that the interactions lead to a collective behaviour that might lead , under specific conditions , to a pre - inflationary phase and ensued distinguishable imprints in the comic microwave background radiation .

understanding the wetting behavior of liquids on solid substrates @xcite is a prerequisite for making use of a myriad of biological and technological applications such as eye irrigation , cell adhesion , tertiary oil recovery , coating , lubrication , paper industry , micro - mechanical devices , and the production of integrated circuits . generically , the solid surfaces in the above mentioned examples are not ideal in the sense that they are neither smooth nor homogeneous . most surfaces are topographically or chemically heterogeneous . such heterogeneities may substantially change the wetting behavior of these surfaces @xcite , which is not necessarily detrimental with respect to envisaged applications . certain topographically structured surfaces are superhydrophobic or superhydrophilic . in the first case droplets roll off these substrates ( instead of flowing ) , such that these surfaces are self - cleaning @xcite . in the second case the surface topography leads to a complete spreading of droplets @xcite . tailored topographic surface structures can induce particular dewetting processes which in turn can be exploited to pattern substrates on the micron scale @xcite . microfluidics is another strong driving force for the research on the dynamics of fluids on structured substrates . shrinking standard laboratory setups to a lab - on - a - chip promises huge cost reduction and speed - up @xcite . open microfluidic systems , i.e. , with free liquid - vapor or liquid - liquid interfaces , may provide various advantages such as reduced friction , better accessibility of the reactants , and reduced risk of clogging by solute particles @xcite . in open microfluidic devices fluids are guided along chemical channels @xcite or in grooves @xcite , which can be chemically patterned in oder to provide additional functionality @xcite . wetting phenomena on topographically structured substrates have attracted substantial research efforts @xcite with , however , the main focus on equilibrium phenomena . in view of the aforementioned applications , dynamical aspects are of particular interest . in spite of this demand , theoretical work on the dynamics of liquid films and droplets on topographically structured substrates has started only recently . in most of these studies the dynamics of the fluids is assumed to be well described by macroscopic hydrodynamic equations , which are solved either directly @xcite , by a lattice boltzmann method @xcite , or in the thin film ( lubrication ) regime @xcite . the applicability of this latter method is limited because the inherent long - wavelength approximation does not keep track of many relevant microscopic features @xcite . on the nanoscale , macroscopic hydrodynamic equations turn out to be inadequate for describing the dynamics of fluids . overcoming this deficit is the focus of a new research area called nanofluidics @xcite . wetting phenomena in particular reveal these deviations ; for a recent review of these issues see ref . however , hydrodynamic equations can be augmented to include hydrodynamic slip , the finite range of intermolecular interactions , and thermal fluctuations . the resulting mesoscopic hydrodynamic equations have been rather successful in analyzing , e.g. , the dynamics of dewetting on homogeneous substrates @xcite . the presence of intermolecular interactions can be summarized into the so - called disjoining pressure ( djp ) , @xmath0 where the effective interface potential @xmath1 is the cost of free energy to maintain a homogeneous wetting film of prescribed thickness @xmath2 . on a homogeneous substrate @xmath1 is independent of lateral coordinates parallel to the substrate surface and the equilibrium wetting film thickness @xmath3 minimizes @xmath4 . however , on chemically or topographically inhomogeneous substrates ( structured , rough , or dirty ) the generalized disjoining pressure does depend in addition on these lateral coordinates . in most studies , the lateral variations of the disjoining pressure have been modelled rather crudely , i.e. , the substrate is assumed to be locally homogeneous and lateral interferences of heterogeneities are neglected : e.g. , a step is typically modelled by an abrupt change of the disjoining pressure @xcite . recently we have demonstrated , that the actually smooth variation of the lateral action of surface heterogeneities can change the behavior of droplets in the vicinity of chemical steps @xcite or topographical features ( edges and wedges ) @xcite even qualitatively . in the present study we extend these results to the case of an isolated straight topographic step in an otherwise homogeneous substrate ( as shown in fig . [ dpstep ] ) and we recover the previously studied case of isolated wedges and edges in the limit of infinite step height @xmath5 . we should emphasize that our investigation provides only a first but nonetheless essential step towards understanding the dynamics of droplets on arbitrarily structured substrates . although more refined than previously used models the present one is still rather simple . we only consider additive lennard - jones type intermolecular interactions , i.e. , we do not take into account electrostatic interactions which would be very important for polar fluids . we assume the fluid to be newtonian , non - volatile , and incompressible ( which is compatible with the frequently used so - called sharp - kink approximation of classical equilibrium density functional theory ( see , e.g. , ref . we also assume a no - slip boundary condition at the solid surface @xcite and neglect the influence of thermal fluctuations @xcite . for numerical reasons we restrict our investigation to two - dimensional ( 2d ) droplets , corresponding to three - dimensional ( 3d ) liquid ridges ( or rivulets ) which are translationally invariant in the direction parallel to the step ; nonetheless we expect our results to hold qualitatively also for 3d droplets . ( on the top side and bottom side of the step ) are exposed to the vertically and laterally varying disjoining pressure @xmath6 , the contour plot of which is shown . the topographic step and the drops are taken to be translationally invariant along the @xmath7 axis ( i.e. , orthogonal to the image plane ) . in ( a ) the substrate is chosen to correspond to the minus case with ( @xmath8 , @xmath9 ) and in ( b ) the substrate corresponds to the plus case ( @xmath10 , @xmath9 ) ( see eq . ( [ djpfar ] ) for definitions ) . lengths ( @xmath11 , @xmath2 , @xmath5 ) and the disjoining pressure @xmath6 are measured in units of @xmath12 and @xmath13 , respectively ( see the main text for definitions ) . ] we study the dynamics of non - volatile and newtonian nanodroplets ( corresponding to three - dimensional ridges which are translationally invariant in one lateral direction ) on topographically stepped surfaces within the framework of mesoscopic hydrodynamics , i.e. , by solving the augmented stokes equation presented in sec . [ mesohydrosec ] with the numerical method described in [ numersec ] . we consider in particular the effects due to the long range of lennard - jones type intermolecular interactions which enter the theoretical description in terms of the disjoining pressure ( djp ) as illustrated in fig . [ dpstep ] . we assume the substrate to be chemically homogeneous in the lateral directions and the surface to be covered by a thin layer of a different material . as detailded in sec . [ modeling ] this leads to two adjustable parameters @xmath14 and @xmath15 which enter into the djp and characterize the wetting properties of the substrate , i.e. , the equilibrium contact angle @xmath16 and the wetting film thickness @xmath3 ( see fig . [ djpshapes ] ) . as shown in fig . [ bc ] both for positive and for negative hamaker constants one can find a one - parameter family of pairs @xmath17 leading to the same @xmath16 on a flat substrate ( i.e. , without a step ) . as shown in fig . [ flat ] nanodroplets on substrates with the same @xmath16 but with different values of @xmath14 and @xmath15 assume shapes which differ mainly in the vicinity of the three - phase contact line with the apex region is almost unaffected by the substrate potential . the results of the numerical solution of the mesoscopic hydrodynamic equations are presented in sec . [ results ] . in contrast to macroscopic expectations based on a capillary model ( i.e. , taking into account only interface energies and neglecting the long range of the intermolecular interactions ) , topographic steps do influence droplets in their vicinity : on substrates with a positive hamaker constant ( figs . [ effstepplus ] and [ wedgediffhplus ] ) , droplets move in uphill direction while on substrates with a negative hamaker constant ( figs . [ effstepp ] and [ wedgediffhminus ] ) the droplets move in the opposite direction . as expected the forces on the droplets and their resulting velocity increase with the step height , but also with the absolute value of the hamaker constant . this is the case if the contact angle is varied ( see , e.g. , figs . [ edgeeffc ] and [ wedgediffcminus ] ) and even if the contact angle is fixed by varying the hamaker constant and the properties of the coating layer together ( see figs . [ edgediffb ] , [ edgeplus ] , [ wedgediffb ] , and [ wedgediffbplus ] ) . the speed of the droplets increases with their size as demonstrated in fig . [ wedgedropsize ] . as detailed in subsec . [ direction ] , the influence of the step on a droplet can be phrased in terms of an effective wettability gradient , i.e. , a spatially varying equilibrium contact angle . the driving force on droplets on such substrates increases linearly with the droplet size because the difference in equilibrium contact angle at the two contact lines of the liquid ridges increases roughly linearly with the distance from the steps . the velocity of droplets driven away from the step decreases rapidly with the distance from the step as shown in sec . [ discuss ] . but droplets moving towards the step ( either on the top side or on the bottom side of the step ) stop with their leading contact line close to the step edge or wedge , respectively . therefore they do not cross the step ( see figs . [ effstepp ] , [ comppisigma ] , [ edgeeffc ] , [ edgediffb ] , [ wedgediffhplus ] , and [ wedgediffbplus ] ) . accordingly , edges , wedges , and steps act as barriers for migrating droplets ( which is also true macroscopically ) because droplets sitting right at the tip of an edge are in an free - energetically unfavorable state ( see fig . [ overedge ] ) while droplets located in the corner of a wedge are in a state corresponding to a local minimum of the free energy ( see fig . [ wedgeforce ] ) . therefore , an external force is required to push droplets over edges ( see fig . [ edgeforce ] ) or to pull them out of wedges ( see fig . [ wedgeforce ] ) . in both cases , the total ( i.e. , integrated over the droplet volume ) force required to accomplish this increases slightly with the droplet volume , but less than linearly . this means , that if the force is applied via a body force density acting per unit volume ( e.g. , gravity ) larger droplets experience a larger force and therefore overcome steps more easily . in addition , the lateral action of intermolecular forces can also pin droplets at edges and near wedges . however , droplets which initially span a topographic step always end up filling the wedge at the step base , either with the upper contact line pinned at the step edge or , if the droplet volume is too small , with the upper contact line on the vertical part of the step , as shown in fig . [ overwedge ] . a deeper understanding of the dynamics of droplets in the vicinity of edges and wedges can be reached by analyzing the forces acting on the droplet surface , i.e. , the disjoining pressure and surface tension ( see eqs . ( [ dpforce ] ) and ( [ sigmaforce ] ) , respectively ) . as demonstrated in fig . [ comppisigma ] , if the droplets move under the influence of the topographic step only , the main contribution to the driving force stems from the disjoining pressure . as shown in figs . [ dropsize ] and [ stepsize ] , the numerically observed features of the dynamics of droplets can be understood in terms of the disjoining pressure induced force density on the droplets calculated for droplets of simple parabolic shapes used as initial conditions for the numerical solution of the hydrodynamic equations . as shown in fig . [ profsize ] the actual relaxed droplet shape is different but the calculated forces depend only weakly on the deviation of the actual shape from its parabolic approximation . in the limit of large distances from the step the force can be calculated analytically ( see subsec . [ direction ] ) : far from the step the total force per unit ridge length @xmath18 ( with the cross - sectional area @xmath19 ) essentially depends on the ratio of the step height @xmath5 and the distance from the step @xmath20 as well as on the ratio of the apex height @xmath21 and @xmath20 . the corresponding asymptotic results are summarized in fig . [ asymptofig ] . in all cases the force density varies according to a power law @xmath22 with @xmath23 . for finite sized droplet and steps of finite height we obtain the fastest decay and for almost macroscopic droplets in the vicinity of finite sized steps as well as for nanodroplets near isolated edges and wedges we get @xmath24 . while our present analysis can not be applied to the case of an almost macroscopic droplet in a wedge , for large drops ( @xmath25 ) next to an isolated edge we get the weakest decay with @xmath26 . in any case , the total force per unit length @xmath27 is proportional to the hamaker constant as observed in the numerical solution of the mesoscopic stokes dynamics as well as in the force analysis presented in subec . [ force ] . the dynamics of large drops ( @xmath25 ) is equivalent to the dynamics of macroscopic drops on a surface with an effective chemical wettability gradient ( i.e. , a spatially varying `` equilibrium contact angle '' @xmath28 ) @xcite . at low reynolds numbers the mean field dynamics of an incompressible newtonian fluid of viscosity @xmath29 is given by the navier - stokes equation for the local pressure @xmath30 and the flow field @xmath31 : @xmath32 with the stress tensor @xmath33 . in this study , we neglect the influence of the vapor phase or air above the film . therefore the tangential components of the component of the stress tensor @xmath34 normal to the liquid - vapor surface @xmath35 ( with outward pointing normal vector @xmath36 ) is zero . the normal component of @xmath34 , i.e. , the normal forces acting on the liquid surface , are given by the sum of the laplace pressure and of the disjoining pressure : @xmath37 with the surface tension coefficient @xmath38 and the local mean curvature @xmath39 of the liquid surface ; @xmath40 is the strength of a spatially constant external body force density pointing in the @xmath11-direction ( with @xmath41 as the corresponding potential ) which we introduce in order to study the strength of barriers to the lateral motion of droplets . alternatively , for incompressible fluids one can define a new pressure @xmath42 such that the external body force density @xmath40 enters into the stokes equation eq . ( [ eq : stokes ] ) rather than the boundary condition in eq . ( [ eq : surfacebc ] ) : @xmath43 . although this approach might be more intuitive , the equivalent form used here is more convenient for implementing the boundary element method used here to numerically solve these equations ( see [ numersec ] ) . the dynamics of the free liquid surface is determined by mass conservation together with the incompressibility condition : the local normal velocity is identical to the normal component of the local flow field . we neglect hydrodynamic slip at the liquid - substrate surface @xmath44 and we only consider impermeable substrates . since we assume the substrate to be stationary this results in the following boundary condition for the flow field : @xmath45 in order to avoid strong initial shape relaxation of the droplets ( in response to placing them on the substrate with a certain shape ) which can lead to significant lateral displacements @xcite , we choose a parabolic initial profile which is smoothly connected to a precursor film of thickness @xmath3 : @xmath46^{|x-\bar{x}|^m+1},\ ] ] such that @xmath47 is the droplet height at the center and half the base width . accordingly the distance of the droplet edge from the step at @xmath48 is given by @xmath49 with @xmath20 the position of the center of the droplet in the @xmath11-direction . the parameter @xmath50 specifies the smoothness of the transition region from the drop to the wetting layer . in this study we choose @xmath50 to be @xmath51 . we investigate the droplet dynamics for two different situations . in the first one we position the droplet on the top side of the step of height @xmath5 with the three - phase contact line @xmath52 at a distance @xmath53 with @xmath54 from the step edge at @xmath48 . in the second situation we place the droplet on the bottom side of the step with the three - phase contact line @xmath55 at a distance @xmath56 with @xmath57 from the wedge at the base of the step . in equilibrium ( [ eq : surfacebc ] ) reduces to the euler - lagrange equation of the effective interface hamiltonian of a fluid film on a substrate as derived , e.g. , in ref . this means that we approximate the normal forces on the liquid surface due to the intermolecular interactions by the disjoining pressure derived for equilibrium systems . in a non - equilibrium situation , the unbalanced forces acting on the fluid surface add up to a resulting net force on the liquid body . we separately consider the two contributions @xmath58 and @xmath59 from the disjoining pressure and from the laplace pressure , respectively , both normalized by the droplet volume @xmath60 and given by the following integrals over the liquid - vapor surface @xmath61 of the droplets : @xmath62 for a liquid ridge translationally invariant in @xmath7-direction both integrals as well as @xmath63 are proportional to the macroscopic ridge length @xmath64 , so that the latter drops out of the expressions for the force densities ( in units of @xmath65 ) @xmath58 and @xmath59 . in three dimensions @xmath66 is a two - dimensional surface area element . @xmath19 is the two - dimensional cross - sectional area of the liquid ridge . in the following we calculate the disjoining pressure for a fluid film or droplet near a topographic step as displayed in fig . [ dpstep ] . apart from a very thin coating layer of thickness @xmath67 we assume the substrate material to be homogeneous , disregarding its discrete molecular structure . many substrates used in experiments are coated , e.g. , by a native oxide layer or by a polymer brush which is used to modify the wetting properties of the substrate . however , a more refined analysis of the djp , which takes the molecular structure of the substrate and of the fluid into account , yields terms of a form similar to those generated by a coating layer @xcite . in general , i.e. , far from the critical point of the fluid , the vapor or gas phase covering the system has a negligible density which we neglect completely . assuming pairwise additivity of the intermolecular interactions , i.e. , the fluid particles as well as the fluid and the substrate particles are taken to interact with each other via pair potentials @xmath68 where @xmath69 and @xmath70 relate to liquid ( @xmath71 ) , substrate ( @xmath72 ) , or coating ( @xmath73 ) particles and @xmath74 is the interatomic distance , one can show that the disjoining pressure ( djp ) of the system is given by @xcite @xmath75}\ , \,d^3r,\ ] ] with @xmath76 and @xmath77 and @xmath78 as the number densities of the liquid and substrate , respectively . @xmath79 is the actual substrate volume . in order to facilitate the calculation of the disjoining pressure of the step we decompose it into contributions from quarter spaces ( edges ) forming building blocks which can be calculated analytically . we first consider an @xmath80dge occupying the lower left quarter space @xmath81 , which in the following we denote by @xmath82 . for lennard - jones type pair potentials @xmath83 , where @xmath84 and @xmath85 are material parameters , the djp in the vicinity of a non - coated edge occupying @xmath86 is given by @xmath87 where @xmath88 and @xmath89 . the first term dominates close to the surface of the edge and the second term at large distances from the substrate . all integrals in eq . ( [ eqpie ] ) can be calculated analytically and one obtains the djp as the corresponding difference @xmath90 of two contributions with @xmath91\nonumber\\\end{aligned}\ ] ] and @xmath92\cdot\end{aligned}\ ] ] the contributions to the disjoining pressure of a thin @xmath73oating layer of thickness @xmath67 on the @xmath93pper side of the edge occupying @xmath94 , the @xmath74ight part of the edge occupying @xmath95 , and the thin rod which fills the @xmath96ip area of the edge @xmath97 can be calculated analogously : @xmath98 with @xmath99 and @xmath100 ; @xmath101 stands for @xmath93 ( @xmath93pper ) , @xmath74 ( @xmath74ight ) , or @xmath96 ( @xmath96ip ) . actual coating layers have a more complicated structure , in particular in the direct vicinity of edges and wedges , which depends on the specific combination of coating and substrate material as well as on the way the coating is produced . such details can influence droplets if their contact line is right at the edge or wedge but the effect is proportional to the square of the coating layer thickness @xmath67 . for simplicity we only consider systems with coating layers which are thin compared to the wetting film thickness ( see below ) , for which the contribution from the thin rod of coating material at the tip of the edge or in the corner of the wedge is irrelevant . according to eq . ( [ picoat ] ) the contribution to the disjoining pressure from the upper coating layer can be decomposed into @xmath102 . to first order in @xmath67 we obtain @xmath103\end{aligned}\ ] ] and @xmath104.\end{aligned}\ ] ] by symmetry one has @xmath105 for the contribution of the vertical part of the coating . the djp of a coated edge occupying @xmath106 is therefore given by @xmath107 the djp contribution from a coated edge occupying the right quarter space @xmath108 can be obtained analogously . however , since the integrals for the right part corresponding to eqs . ( [ eqpie ] ) and ( [ picoat ] ) are the mirror image ( with respect to the @xmath109-plane ) of their counterparts for the left hand side , the former ones can be expressed in terms of the latter ones . therefore the djp of the coated lower right quarter space @xmath110 is equal to @xmath111 . combining the contributions of the left and the right part leads to the following expression for the djp of a step of height @xmath5 : @xmath112 the last term on the right hand side of eq . ( [ dpstepeq ] ) removes the artificial extra coatings on the left and the right quarter spaces ( at @xmath48 , @xmath113 ) which get buried upon building the step out of the coated edges . figure [ dpstep ] shows typical examples for the djp . the djp is not only a function of the vertical distance from the substrate , but also of the lateral distance from the step . in this regard , the substrate in the vicinity of the step resembles a chemically structured substrate with laterally varying wettability @xcite . for positions far from the step the distribution of the djp resembles that of the @xmath73oated , laterally @xmath5omogeneous flat substrate obtained by setting @xmath114 in eq . ( [ dpstepeq ] ) . to linear order in @xmath67 one has @xmath115 since the repulsive contributions decay rapidly with distance from the substrate we neglect all those repulsive contributions which are shorter ranged than the corresponding term ( @xmath116 ) arising from @xmath117 @xcite , leading to @xmath118 with @xmath119 . the equilibrium thicknesses @xmath3 of the wetting film on such a substrate minimizes the effective interface potential @xcite @xmath120 with eq . ( [ piflat ] ) this leads to @xmath121 the second term is usually written as @xmath122 , where @xmath123 is the so - called hamaker constant . at this point we introduce dimensionless quantities ( marked by @xmath124 ) such that lengths are measured in units of @xmath125}^{1/6}$ ] which for @xmath126 and @xmath127 is the equilibrium wetting film thickness @xmath3 on the uncoated flat substrate . the djp is measured in units of the ratio @xmath13 where @xmath38 is the liquid - vapor surface tension . thus the dimensionless djp @xmath128 far from the edge has the form @xmath129 in the first and second term of eq . ( [ djpfarstar ] ) the upper ( lower ) sign corresponds to @xmath130 @xmath131 and @xmath132 @xmath133 , respectively . the dimensionless amplitude @xmath134 , with @xmath135 , compares the strength of the effective intermolecular forces in the uncoated case and of the surface tension forces . the amplitude @xmath136 measures the strength of the coating layer . since the molecular structure of the substrate and of the fluid yields a term of the same form @xcite we consider @xmath14 itself as a parameter independent of the actual properties of the coating layer . for the interactions considered here , @xmath137 is a necessary condition for the occurrence of an equilibrium wetting layer of nonzero thickness but @xmath138 can be positive or negative . therefore the first term in eq . ( [ djpfarstar ] ) can only be positive while the second term can be positive or negative . in the following we shall refer to these two cases simply as the minus ( ) and the plus ( ) case . in order to avoid a clumsy notation in the following we also drop the stars . with this , one has @xmath139 figure [ djpshapes ] shows the typical profile of @xmath140 for the minus and the plus case and also the corresponding equilibrium wetting layer thickness @xmath3 for which @xmath141 . while the parameter @xmath15 measures the strength of the djp , by changing @xmath14 one can modify the shape of the djp @xcite . in eq . ( [ djpfar ] ) the admissible value ranges of @xmath15 and @xmath14 which provide partial wetting can be inferred from considering the equilibrium contact angle @xmath142 @xcite : @xmath143 the admissible value ranges of @xmath14 and @xmath15 for which @xmath144 ( partial or incomplete wetting ) are given in fig . [ bc ] for both the minus and the plus case . in the minus case , for each value of @xmath14 one can find a value of @xmath15 such that the resulting substrate is partially wet . since the signs of the first two terms in eq . ( [ djpfar ] ) differ the disjoining pressure has a zero for any @xmath14 and the depth of the minimum of the corresponding effective interface potential can be tuned by choosing an appropriate value for @xmath15 . in the plus case , however , @xmath14 has to be negative in order to obtain a sign change of @xmath6 . the maximum admissible value of @xmath14 ( i.e. , @xmath145 ) can be obtained by simultaneously solving the following equations for @xmath3 and @xmath146 : @xmath147 @xmath148 from which one finds @xmath149 ( compare fig . [ djpshapes](d ) ) . in order to obtain dimensionless hydrodynamic equations ( see eqs . ( [ eq : stokes])([eq : surfacebc ] ) ) we choose @xmath150 as the velocity scale . with this , the dimensionless form of the stess tensor is given by @xmath151 and the surface tension coefficient drops out of eq . ( [ eq : surfacebc ] ) . the dimensionless time is given in units of @xmath152 . in order to study the dynamics of nanodroplets we solve the dimensionless hydrodynamic equations with a standard biharmonic boundary integral method described in more detail in [ numersec ] . ) djp ( in units of @xmath15 ) of a flat homogenous substrate for ( a ) the minus and ( b ) the plus case ( see eq . ( [ djpfar ] ) ) . the corresponding zeros @xmath3 of the djp for different values of @xmath14 are given in ( c ) and ( d ) for the minus and the plus case , respectively . in ( c ) and ( d ) full lines indicate stable wetting films and dashed lines unstable films . ] and @xmath15 for which the system exhibits a partial wetting ( pw ) situation , i.e. , @xmath153 for the minus ( a ) and the plus ( b ) case . ] on a flat homogeneous substrate for various values of @xmath14 and @xmath15 for the minus and the plus case . ( @xmath14 , @xmath15 ) for the minus case are i ( -1 , 7.7583 ) with @xmath154 , ii ( 0 , 2.6667 ) with @xmath155 , and iii ( 1 , 1.2703 ) with @xmath156 , and for the plus case i ( -2.5 , 4.2327 ) with @xmath157 and ii ( -4 , 0.9265 ) with @xmath158 . the values of @xmath14 and @xmath15 are chosen such that in all cases @xmath159 . ] in order to provide the information and terminology required for the subsequent considerations we first recall some basic results for the wetting of flat and homogeneous substrates . for this purpose a nanodroplet with @xmath160 and an initial configuration given by eq . ( [ inicond ] ) was positioned on the substrate . figure [ flat ] shows the equilibrium profile of the nanodroplet for various values of @xmath15 and @xmath14 resulting in an equilibrium contact angle @xmath159 for both the minus and the plus case , i.e. , the values @xmath17 lie on the dashed curves in figs . [ bc](a ) and [ bc](b ) . it is evident from the figure that the droplets have relaxed from the initial condition . the equilibrium profiles in all cases are roughly equal but the nanodroplets differ near their contact lines ( see the inset of fig . [ flat ] ) and with respect to their heights . the term proportional to @xmath14 in eq . ( [ djpfar ] ) is rather short - ranged and most important in the direct vicinity of the substrate . the top parts of the droplets are only influenced by the term @xmath161 such that the curvature at the peak changes with @xmath15 , independently of @xmath14 . this also changes the droplet height . however , also the wetting film thickness @xmath3 changes with @xmath14 , such that the differences in droplet height in fig . [ flat ] are a combined result of both effects . due to the translational symmetry of the substrate and due to the symmetry of the initial drop configuration the shape relaxation does not result in a lateral displacement of the droplets , in contrast droplets placed on heterogeneous substrates @xcite . previous studies of droplets near edges ( corresponding to steps of infinite height ) have shown that , in contrast to what is expected from a simple macroscopic model taking into account only interface energies , droplets are attracted towards the edge in the minus case and repelled from the edge in the plus case @xcite . in the minus case , the droplets move towards the edge with increasing velocity , but they stop rather abruptly before the leading contact line reaches the edge . the distance from the edge at which the droplets stop increases with decreasing @xmath14 , i.e. , with increasing strength of the coating layer . in the plus case , the droplets move away from the step with a velocity which decreases with the distance from the step . the strength of the attraction or repulsion is expected to be lower for steps of finite height . and @xmath162 , respectively . the droplets are initially positioned at a distance @xmath163 from the step . the droplets have an initial radius @xmath160 . @xmath164 and @xmath8 correspond to @xmath159 . the vertical scale is equal to the lateral scale . the corresponding lateral ( b ) and vertical ( c ) position of the center of mass ( @xmath20 , @xmath165 ) of the droplet relative to the step edge as a function of time . ] . initially it is positioned at @xmath163 ( @xmath166 , dash - dotted line ) . for the minus case it moves towards an isolated edge where it stops with the leading contact line pinned at the step edge ( full line ) . for this final configuration the arclength @xmath72 of the interface is measured as indicated from a certain position on the wetting layer on the vertical side of the step . ( b ) a comparison between djp induced ( eq . ( [ dpforce ] ) , full line ) and surface tension induced ( eq . ( [ sigmaforce ] ) , dashed line ) lateral force densities during the motion ( with the leading three - phase contact line still well separated from the edge ) expressed in terms of the position @xmath20 of the center of mass of the droplet . ( c ) laplace pressure @xmath167 ( dash - dotted line ) and djp @xmath6 ( full line ) on the surface of the droplet in the final equilibrium configuration as a function of the arclength @xmath72 . in the absence of other external forces ( e.g. , @xmath40 in eq . ( [ eq : surfacebc ] ) ) at each point on the droplet surface these add up to the constant pressure @xmath168 inside the droplet . @xmath8 and @xmath164 correspond to @xmath159 . force densities ( b ) and pressures ( c ) are measured in units of @xmath169 and @xmath13 , respectively . ] the effect of the contact angle on the dynamics of droplets on the top side of the step for a droplet with initial height @xmath160 and for the minus case . ( a ) the initial profile @xmath170 is shown in the top panel . the lower graphs show the configurations of the droplets after the initial relaxation ( @xmath171 , dashed lines ) and in the final stages ( @xmath1726200 , 7400 , and 7800 for i , ii , and iii , respectively , solid lines ) for @xmath173 ( i ) , @xmath174 ( ii ) , and @xmath175 ( iii ) from top to bottom . the corresponding substrate parameters are i @xmath176 , ii @xmath177 , and iii @xmath178 , respectively . in ( b ) and ( c ) as function of time the corresponding lateral and vertical positions @xmath20 and @xmath165 , respectively , of the center of mass of the droplets are shown relative to the step edge . the dips in ( c ) occur when the leading three - phase contact line reaches the edge ; then the droplet stops ( compare to ( b ) ) . ] ( while varying @xmath15 such that @xmath159 in all cases , see fig . [ bc](a ) ) on the dynamics of the droplets on the top side of an edge for an initial droplet with @xmath160 and for the minus case . the values of @xmath14 and @xmath15 are i ( @xmath179 , @xmath180 ) , ii ( @xmath8 , @xmath181 ) , and iii ( @xmath182 , @xmath183 ) . ( a ) the top panel depicts the initial profile ( @xmath163 ) . the dashed lines show the configuration of the droplet after the initial relaxation @xmath171 and the solid lines correspond to the final configuration of the droplets at @xmath184 , 7300 , and 2500 for i , ii , and iii , respectively . in ( b ) and ( c ) as a function of time the corresponding lateral and vertical positions @xmath20 and @xmath165 , respectively , of the center of mass of the droplets are shown relative to the step edge . ] in order to test the influence of the step height on the dynamics of nanodroplets identical droplets of half base width @xmath160 were placed at a distance @xmath163 from steps of height @xmath1852.5 , 5 , 10 , 15 , 20 , and @xmath186 . the results of the numerical solution of the mesoscopic hydrodynamic equations for the minus case are shown in fig . [ effstepp](a ) for @xmath164 and @xmath8 which corresponds to @xmath159 . in order to have a better view on the dynamics we monitor the time evolution of the position of the center of mass of the droplets ( @xmath20 , @xmath165 ) relative to the step edge in figs . [ effstepp](b ) and [ effstepp](c ) , where @xmath20 and @xmath165 are given by @xmath187 with @xmath60 denoting the droplet volume . since the droplets are smoothly connected to the wetting film , which on large substrates would influence the center of mass of the fluid , in calculating @xmath20 and @xmath165 we only consider the fluid above @xmath188 with @xmath189 , i.e. , only the fluid volume slightly above the wetting film . we selected @xmath190 ; but since we focus on substrates with equilibrium contact angles of about @xmath191 the results are only weakly affected by the precise choice of the value of @xmath192 . in all cases the dynamics of the droplets proceeds in three stages . the first stage is a fast initial shape relaxation , similar to the behavior on homogeneous substrates , which is accompanied by a lowering of the droplet center of mass @xmath165 without any considerable lateral motion . this is followed by a relatively slow lateral motion towards the edge , during which the changes in the droplet shape are almost unnoticeable . although the droplet shape is slightly asymmetric the lateral surface tension induced force density @xmath59 defined in eq . ( [ sigmaforce ] ) is much smaller than the force density @xmath58 induced by the djp ( see eq . ( [ dpforce ] ) ) as shown in figs . [ comppisigma](b ) . figure [ effstepp](b ) clearly shows that the lateral motion of the droplet slows down rapidly as soon as its leading three phase contact line reaches the edge . during this third and final stage a part of the droplet volume leaks into the wetting film on the vertical part of the step and as a result the droplet experiences a sudden drop in its height @xmath165 ( see fig . [ effstepp](c ) ) . the trailing three - phase contact line of the droplet still continues its motion towards the step and as a consequence the mean height of the droplet increases again and becomes even larger than during the migration stage . while the droplet contracts , its asymmetry gradually increases such that the surface tension induced force density @xmath59 grows and finally , as the equilibrium configuration is reached , cancels @xmath58 . ( this latter stage of cancelation is not visualized in fig . [ comppisigma](b ) due to numerical problems in evaluating the force densities on droplets once they have reached the step edge . ) in equilibrium , at each point on its surface the laplace pressure and the disjoining pressure ad up to the constant value of the hydrostatic pressure in the droplet ( see fig . [ comppisigma](c ) ) . increasing the step height from @xmath193 to 5 and 10 results in a significant increase in the droplet speed during the migration phase . the asymptotic speed for isolated edges ( corresponding to @xmath194 ) , i.e. , the maximum speed , is almost reached for @xmath195 . this height value is large compared to the thickness of the wetting layer but comparable with the droplet size ; here the base diameter is @xmath196 . however , in order to be able to conclude that the step height above which the droplet perceives the step as an isolated edge is comparable with the droplet size further calculations for droplets of different size are needed . changing the equilibrium contact angle @xmath16 by increasing @xmath15 while keeping @xmath8 does not qualitatively change the behavior of the droplets , as shown in fig . [ edgeeffc](a ) for droplets with @xmath160 near an isolated edge ( corresponding to @xmath194 ) , apart from the increase of @xmath165 during the initial relaxation process for large @xmath16 . the reason for this increase in droplet height is , that the initial shape of the droplet is not adapted to the substrate parameters . changing @xmath15 does not change the functional form of the djp , only its strength . consequently , droplets move faster for larger @xmath15 ( resulting in larger @xmath16 ) and their final shape is less symmetric . with the leading contact line pinned right at the step edge , large @xmath197 also result in an overhang over the step edge . since for fixed wetting film thickness @xmath12 on the uncoated flat substrate the time scale used to obtain dimensionless hydrodynamic equations depends on the substrate parameters in the same manner as the dimensionless parameter @xmath15 , we rescale time by @xmath15 in figs . [ edgeeffc](b ) and [ edgeeffc](c ) , as well as in all subsequent figures which compare @xmath198 and @xmath199 for different values of @xmath15 . this corresponds to changing the substrate material but keeping surface tension , viscosity , and wetting film thickness constant @xcite . figure [ edgediffb](a ) shows the effect of changing the value of @xmath14 ( while keeping the contact angle constant ) on the dynamics and on the final configuration of droplets which start with @xmath160 and for the minus case on the top side of an isolated edge ( @xmath194 ) . for each value of @xmath14 we choose @xmath15 such that @xmath159 , i.e. , corresponding to the dashed curve in fig . [ bc](a ) . for all values of @xmath14 the droplets move towards the edge . changing the values of @xmath14 and @xmath15 does not qualitatively change the behavior of the system . however , a closer examination of @xmath20 and @xmath165 ( see figs . [ edgediffb](b ) and [ edgediffb](c ) , respectively ) reveals quantitative differences in the dynamics and in the final configuration of the droplets despite the fact that the contact angle is the same for all these cases . for larger ( positive ) values of @xmath14 ( and thus @xmath15 , see fig . ( [ bc])(a ) ) droplets move faster in lateral direction although the contact angle equals @xmath159 for all of them . in addition , the final position of the droplets is closer to the step edge for larger values of @xmath14 , eventually leading to a slight overhang . the small differences in @xmath165 for different values of @xmath14 is related to the fact that the shape of nanodroplets is not only determined by @xmath16 , as shown in fig . [ flat ] , and that the wetting film thickness depends on @xmath14 . and @xmath200 , respectively . the droplets of size @xmath160 are initially positioned at a distance @xmath163 from the step . @xmath201 and @xmath10 result in an equilibrium contact angle @xmath159 . time evolution of the ( b ) horizontal position @xmath20 and ( c ) vertical position @xmath165 of the center of mass of the droplets relative to the step edge . since @xmath165 depends only weakly on @xmath5 only the case @xmath195 is shown . ] on the dynamics of the droplets near an edge for the plus case and for an initial droplet shape with @xmath160 . the values of @xmath14 and @xmath15 , i.e. , i ( @xmath10 , @xmath201 ) and ii ( @xmath202 , @xmath203 ) are selected such that the equilibrium contact angle is @xmath159 in both cases . ( a ) the initial distance from the step edge is @xmath163 . the dashed lines show the droplets after the initial relaxation at @xmath204 and the solid lines correspond to the droplets in the migration phase at @xmath205 . the horizontal position @xmath20 and the vertical position @xmath165 of the center of mass of the droplet are shown as a function of time in ( b ) and ( c ) , respectively . for less negative values of @xmath14 the velocity @xmath206 is larger ( i ) . [ edgeplus ] ] even if they exhibit the same equal equilibrium contact angles @xmath16 the behavior of droplets in the plus case differs substantially from that in the minus case : the direction of motion is reversed . however , apart from this sign change , the influences of the step height , the equilibrium contact angle , and @xmath14 are similar . the dependence of the droplet dynamics on the step height for the plus case is shown in fig . [ effstepplus](a ) . the initial size of the droplets is @xmath160 and the contact angle @xmath159 ( with @xmath201 , @xmath10 ) . the corresponding lateral and vertical positions of the center of mass of the droplets relative to the step edge are shown in figs . [ effstepplus](b ) and [ effstepplus](c ) , respectively . as in the minus case the migration phase is preceded by a fast initial relaxation process ( during which @xmath165 drops slightly ) . however , the droplets are repelled from the step . the lateral speed of the motion continuously decreases as the distance of the droplets from the step edge increases . for higher steps the droplets are faster . but as in the minus case , the maximum speed ( reached for @xmath194 , i.e. , in the case of an isolated edge ) is almost reached for @xmath207 ( see fig . [ effstepplus](b ) ) . the results for different values of @xmath14 , while keeping the contact angle @xmath159 fixed , are depicted in fig . [ edgeplus](a ) . the corresponding lateral @xmath20 and vertical @xmath165 positions of the center of mass of the droplet relative to the step edge are given in figs . [ edgeplus](b ) and [ edgeplus](c ) , respectively . for all the cases considered the droplets move away from the step . however , as in the minus case , the droplet speed increases with @xmath14 ( i.e. , for less negative values of @xmath14 ) , even though the contact angle is not changed . the reason for this is , that larger ( i.e. , less negative ) values of @xmath14 require larger values of @xmath15 in order to maintain the same @xmath16 . as in the minus case the droplet height , i.e. , @xmath165 , depends on @xmath14 as well . in ref . @xcite we have demonstrated that droplets near corners , i.e. , at the base of a step of infinite height , are attracted to the corner in the plus case and repelled from the corner in the minus case ( while in a macroscopic model taking into account only interface energies the free energy of the droplets is independent of their distance from the corner ) . in other words , the direction of motion is reversed as compared to the case of the edge . however , as we shall show in the following , at a step composed of an edge and a corner at its base , the direction of motion of nanodroplets is the same on both sides of the step . positioned at the base of topographic steps of different height for the minus case . the droplets start at a distance @xmath163 from the step . @xmath164 and @xmath8 correspond to @xmath159 . the dashed and the solid lines correspond to the configurations just after the initial relaxation at @xmath208 and to a later time @xmath209 , respectively . as function of time the horizontal position @xmath20 and the vertical position @xmath165 of the center of mass relative to the step base are shown in ( b ) and ( c ) , respectively . since @xmath165 depends only weakly on @xmath5 , in ( c ) only the trajectory for @xmath195 is shown . ] near a corner of substrates with different contact angles @xmath16 in the minus case . the top panel shows the initial droplet profile with @xmath210 . the other panels show the droplets after the initial relaxation ( @xmath211 ( i ) , @xmath212 ( ii ) , and @xmath213 ( iii ) , dashed lines ) and during the migration stage ( @xmath214 ( i ) , @xmath215 ( ii ) , and @xmath216 ( iii ) , solid lines ) for @xmath173 ( i , @xmath217 , @xmath8 ) , @xmath174 ( ii , @xmath218 , @xmath8 ) , and @xmath175 ( iii , @xmath219 , @xmath8 ) , respectively . the time evolution of the center of mass @xmath220 relative to the corner is shown in ( b ) and ( c ) for the lateral and vertical direction , respectively . [ wedgediffcminus ] ] but the same contact angle @xmath159 near a corner in the minus case : ( i : @xmath179 , @xmath180 ) , ( ii : @xmath8 , @xmath181 ) , and ( iii : @xmath182 , @xmath183 ) , top to bottom . the top panel depicts the initial droplet shape @xmath170 . the corresponding graphs show the droplets after the initial relaxation ( dashed lines , @xmath171 ) and in the migration process at @xmath221 ( solid lines , @xmath222 , 37500 , and 12700 for i , ii , and iii , respectively ) . the time evolution of the center of mass @xmath223 relative to the corner is shown in ( b ) and ( c ) for the lateral and vertical direction , respectively . [ wedgediffb ] ] of the center of mass as a function of time for droplets of size @xmath160 ( solid line ) and @xmath224 ( dashed line ) near the corner of a wedge in the minus case with @xmath8 , @xmath181 , and @xmath159 . ] as in the case of droplets on the top side of steps , the step height influences the droplet velocity but not the direction of motion and the transition from a planar substrate ( @xmath114 ) to an isolated wedge ( @xmath194 ) is continuous . this is demonstrated in fig . [ wedgediffhminus ] for droplets of size @xmath160 starting at a distance @xmath163 from the corner . the initial distance @xmath225 is chosen such that after the initial relaxation which precedes the migration phase the contact line facing the corner is well separated from the wetting layer on the vertical part of the step . for the minus case fig . [ wedgediffhminus](a ) presents the results of our stokes dynamics calculations for droplets at the step base for different step heights . the droplets are repelled from the step and move away with a speed which decreases continuously with the distance from the step . the differences in droplet speed are significant between step heights @xmath193 , 5 , and 10 ( see fig . [ wedgediffhminus](b ) ) . increasing the step height further influences the dynamics of the droplets only at large distances from the step . changing the equilibrium contact angle @xmath16 does not change the droplet dynamics qualitatively . this is demonstrated in fig . [ wedgediffcminus](a ) for droplets on substrates with different values of @xmath15 ( while keeping @xmath8 ) . the top panel shows the initial shape used in all cases considered here for the numerical solution of the stokes dynamics . the corresponding lateral position @xmath20 and vertical position @xmath165 of the center of mass of the droplets relative to the corner are depicted in figs . [ wedgediffcminus](b ) and [ wedgediffcminus](c ) , respectively . increasing @xmath16 ( by increasing @xmath15 ) results in faster droplet motion . since the initial droplet shape is not adapted to the modified contact angle @xmath226 , @xmath165 changes rapidly during the initial relaxation process for @xmath226 . the dynamics of droplets on substrates with the same contact angle @xmath159 but different values of @xmath14 ( with @xmath15 adapted accordingly ) is shown in fig . [ wedgediffb](a ) . the top panel shows the initial configuration . for all cases considered the droplets move away from the step . comparing @xmath198 for different values of @xmath14 ( see fig . [ wedgediffb](b ) ) shows , that the droplet velocity increases with @xmath14 ( which , for fixed @xmath159 , implies increasing @xmath15 ) . after the initial relaxation process the vertical coordinate @xmath165 of the center of mass does not vary in time ( see fig . [ wedgediffb](c ) ) . the droplet dynamics depends on the droplet size . this is demonstrated in fig . [ wedgedropsize ] for droplets of initial sizes @xmath224 and @xmath160 starting at @xmath227 near an isolated wedge for the minus case . the larger droplet moves faster because its two three - phase contact lines have a larger lateral distance from each other such that they experience a larger difference in the local disjoining pressure . on substrates with @xmath159 ( @xmath201 and @xmath10 ) starting at @xmath163 on the base of steps with heights varying between @xmath193 ( top ) and @xmath194 ( bottom , corresponding to an isolated wedge ) right after the initial relaxation process at @xmath228 ( dashed lines ) and during the migration process at @xmath229 ( solid lines ) . the horizontal position @xmath20 and the vertical position @xmath165 of the center of mass are shown in ( b ) and ( c ) , respectively , as a function of time . ] near isolated wedges ( @xmath194 ) in the plus case . @xmath159 on both substrates : ( i : @xmath10 , @xmath201 ) and ( ii : @xmath202 , @xmath203 ) . ( a ) the top panel shows the initial droplet shape and the lower panels show the droplets just after the initial relaxation at @xmath171 ( dashed lines ) and in their final configuration at @xmath230 and @xmath231 for substrate i and ii , respectively ( solid lines ) . the horizontal position @xmath20 and the vertical position @xmath165 of the center of the mass during the motion are shown in ( b ) and ( c ) , respectively . [ wedgediffbplus ] ] in the plus case the direction of motion of the droplets is reversed as compared to the minus case . as shown in fig . [ wedgediffhplus](a ) , the migration speed increases with the step height , but the droplets stop before the leading contact line reaches the wedge such that the droplets do not move into the corner . as in the other cases discussed so far , the droplet speed increases significantly as the step height is increased up to @xmath195 . in fig . [ wedgediffhplus](b ) the trajectories @xmath198 for @xmath195 and for @xmath194 almost coincide . the final distance of the droplets from the wedge decreases with the step height , but it remains finite in the limit @xmath232 . once the droplets reach the wedge there is a brief drop of @xmath165 due to fluid leaking out of the droplet into the corner area . after that their vertical position @xmath165 of the center of mass increases again ( see fig . [ wedgediffhplus](c ) ) . this increase is the result of a contraction of the droplets , which is also observed for droplets on the top side of steps in the minus case and which is more pronounced for higher steps . changing @xmath14 and @xmath15 such that @xmath159 remains the same does not change the dynamics of the droplets qualitatively . higher values of @xmath14 and @xmath15 result in larger droplet velocities ( see fig . [ wedgediffbplus ] concerning the example of droplets near an isolated wedge ) . beside this change of droplet speed we find that for larger values of @xmath15 the final position of the droplets is closer to the step and the final height @xmath165 of their center of mass is larger . the magnitude of the disjoining pressure and therefore the forces acting on the droplet increase with @xmath15 . in response to these forces the droplets deform more upon increasing @xmath15 . in the previous subsections we have discussed the behavior of nanodroplets originally positioned at a certain lateral distance from topographic features such as edges , wedges , and steps . their behavior suggests that these surface features provide migration barriers for droplets . even in those situations in which droplets migrate towards the edge or wedge , respectively , they stop just before reaching them . this result is also borne out in a macroscopic model which takes into account only interface energies : the free energy of a droplet positioned right on an edge is larger than that of a droplet of equal volume residing on a flat and homogeneous substrate , and the free energy of a drop in the corner of a wedge is even lower . as a consequence , we expect that droplets sitting on edges to be in an unstable and droplets sitting inside the corner of a wedge to be in a stable configuration . moving , by force , a droplet ( with the shape of the liquid - vapor interface remaining a part of a circle ) in the first case slightly to one side results in an increased contact angle on this side , while the contact angle on the other side decreases . however , with the leading contact angle being larger than the equilibrium one the corresponding contact line will move away from the edge , while the trailing contact line ( with the corresponding contact angle being smaller than the equilibrium one ) moves towards the edge . as a consequence , the droplet leaves its position at the edge . in the case of a droplet in a wedge , the situation is reversed : moving , by force , the droplet into one direction results in a decreased contact angle on this side and an increased contact angle at the trailing side , such that the droplet moves back into the corner of the wedge . accordingly one expects that a certain force has to be applied to push a droplet over an edge or to pull it out of the corner of a wedge . in the following our detailed numerical results indicate that this also holds for nanodroplets and they enable us to quantify those external forces . our analyses show that a nanodroplet positioned symmetrically on the tip of an edge is unstable on all types of substrates , regardless of whether the droplets migrate towards the edge or away from the edge ( see fig . [ overedge ] ; there a suitable , highly symmetric initial shape of the liquid - vapor interface has been chosen such that it indeed relaxes to the unstable state of a droplet sitting on the tip of the edge ) . due to the mirror symmetry with respect to the diagonal of the edge , a droplet right at the tip of an isolated edge is in mechanical equilibrium but in an unstable one . in the minus case , after a small perturbation the droplet flips either up or down but then rests next to the step , i.e. , in the position which it would assumes upon migrating towards the edge ( fig . [ overedge](a ) ) . in the plus case , as expected from the previous results the droplet migrates away from the edge after flipping to either side ( fig . [ overedge](b ) ) . at steps of finite height , this symmetry is broken by the presence of the wedge . in the minus case , the droplets are pushed away from the wedge , i.e. , upwards , which is consistent with the dynamics of droplets in the vicinity of isolated wedges of the same material . however , being attracted to the edge as shown in the previous subsections , they come to rest with the trailing contact line pinned to the step edge ( fig . [ overedge](c ) ) . in the plus case the droplets move in the opposite direction , i.e. , they are attracted by the wedge which they migrate to after leaving the edge area ( fig . [ overedge](d ) ) . in order to displace a droplet from one side of an edge to the other side ( as shown in fig . [ edgeforce](a ) ) one has to apply an external force , e.g. , a body force such as gravity or centrifugal forces , which we incorporate into the hydrodynamic equations via the boundary condition ( see eq . ( [ eq : surfacebc ] ) ) . if the applied force is small the droplets assume a new but distorted equilibrium position with the leading three - phase contact line still pinned at the edge . but there exists a threshold force density @xmath233 beyond which the configuration described above is unstable and the leading three - phase contact line depins from the step edge . as a consequence the droplet flips around the corner and ends up on the vertical side of the edge . since the applied body force has no component parallel to this vertical part of the substrate , the further fate of the droplet is determined by the action of the intermolecular forces . in the minus case considered in fig . [ edgeforce](a ) the droplet is attracted to the edge such that the new stable equilibrium configuration is that of a droplet residing on the vertical part of the step with the trailing three - phase contact line pinned at the step edge . in the plus case ( which we have not tested numerically ) the droplet is repelled from the edge and it is expected to move down the vertical part of the edge . as shown in fig . [ edgeforce](b ) we have determined the body force density @xmath40 needed to push the droplets over the edge for various types of substrates ( minus case with @xmath8 ) and for droplets of two different sizes . the threshold force density @xmath233 decreases both with @xmath15 ( i.e. , with @xmath16 ) and with the droplet size . both trends are also expected to occur for macroscopic droplets . in the limit @xmath234 the droplets loose contact with the substrate and the free energy of the droplet at the edge equals the free energy on a planar substrate . taking , however , the finite range of molecular interactions into account this no longer holds , but the barrier still decreases with increasing @xmath16 . since the force density @xmath40 is a body force , i.e. , a force density , the total force per unit ridge length @xmath235 ( with the ridge cross - sectional area @xmath236 ) acting on the droplet is proportional to the droplet cross - sectional area @xmath19 . in fig . [ edgeforce](b ) we observe that the total threshold force @xmath237 needed to push droplets over the edge increases with the droplet volume . apart from the effects of long - ranged intermolecular forces the main contribution to the barrier effect of the edge is the increase of the liquid - vapor surface area when the droplet is deformed as it passes over the edge . the square root of the ratio of the surface tension coefficient @xmath38 and the body force density @xmath40 defines a capillary length below which the surface tension dominates , while it is less important for larger drops . ( the surface area of three - dimensional droplets increases only quadratically with the droplet radius @xmath47 while the volume increases @xmath238 . ) from dimensional arguments @xmath239 one expects the threshold body force density @xmath233 needed to push droplets over an edge to decrease @xmath240 with the droplet radius , while for liquid ridges the total force per unit length @xmath237 should still increase linearly with the droplet radius @xmath47 . therefore the total threshold force @xmath237 needed for the larger droplet in fig . [ edgeforce](b ) should be about @xmath241 times the force needed for the smaller drop . the actual value is somewhat smaller and we attribute the difference to the effect of the long - ranged part of the intermolecular forces . and @xmath8 ) and in the ( b ) plus case ( @xmath201 and @xmath10 ) . a tiny perturbation at @xmath242 pushes the droplet either up or down the step . after that in the minus case the droplet stays next to the step ( a ) whereas in the plus case ( b ) it moves away from the edge ( solid lines , @xmath243 and @xmath244 in ( a ) and ( b ) , respectively ) . in the presence of a wedge , i.e. , for a finite step height @xmath5 , the droplet is ( c ) pushed onto the top side of the step in the minus case but ( d ) towards the corner of the step in the plus case ( solid lines , @xmath245 and @xmath246 in ( c ) and ( d ) , respectively ) . ] pushed over an edge ( minus case , @xmath8 and @xmath218 ) by an external , horizontal body force @xmath247 ( @xmath248 , direction indicated by the horizontal arrow ) . droplet shapes for @xmath242 ( indicated ) , 100 , 650 , 1050 , 1150 , and 6950 are shown ( from the upper left to the lower right ) . ( b ) the minimum total force per unit length @xmath249 ( @xmath19 is the droplet cross - sectional area ) to push droplets over the edge for two droplet cross - sectional areas @xmath250 ( dashed line ) and @xmath251 ( solid line ) corresponding to @xmath252 and 15 , respectively . the values of @xmath253 , @xmath254 , @xmath255 , and @xmath256 , correspond to @xmath257 , @xmath258 , @xmath174 , and @xmath175 , respectively . the force density ( force per unit volume ) @xmath40 is measured in units of @xmath259 . ] ( @xmath260 ) and @xmath193 ( @xmath261 ) ( ( a ) and ( d ) ) , @xmath262 ( @xmath263 ) and @xmath264 ( @xmath265 ) ( ( b ) and ( e ) ) , and @xmath266 ( @xmath267 , ( c ) and ( f ) ) in the minus case ( @xmath8 , ( a ) , ( b ) , and ( c ) ) and in the plus case ( @xmath10 , ( d ) , ( e ) , and ( f ) ) , for various values of @xmath15 . in the minus case @xmath268 , @xmath269 , and @xmath270 correspond to @xmath271 , @xmath272 , and @xmath175 , respectively , while for the plus case @xmath268 , @xmath269 , @xmath273 , and @xmath274 correspond to @xmath275 , @xmath276 , @xmath277 , and @xmath278 , respectively . ] pulled out of the corner of a wedge ( @xmath8 and @xmath218 ) by an external horizontal body force @xmath279 ( @xmath280 , direction indicated by the horizontal arrow ) . shown are droplet shapes for @xmath242 ( indicated ) , 400 , 1400 , 3700 , 4700 , and 4900 ( from left to right ) . ( b ) the minimum total force per unit ridge length @xmath281 required to extract the droplet from the corner as a function of @xmath15 for droplets of cross - sectional area @xmath265 ( corresponding to @xmath282 , dashed line ) and @xmath283 ( corresponding to @xmath284 , solid line ) in the minus case ( @xmath8 ) . the values @xmath9 , @xmath273 , @xmath285 , and @xmath270 correspond to @xmath257 , @xmath258 , @xmath174 , and @xmath175 , respectively . the force density @xmath40 is measured in units of @xmath259 . ] a macroscopic droplet spanning the whole topographic step ( i.e. , with one contact line on the top terrace and one on the base terrace ) moves downhill : since the surface of a macroscopic droplet is a part of a circle which is cut by the top side of the step at a higher level than by the base of the step the contact angle at the upper terrace is smaller than the contact angle at the lower terrace . this results in a net driving force in downhill direction . the final configuration is a droplet with the upper contact line pinned at the step edge . this is also true on the nano scale , as demonstrated in figs . [ overwedge](a ) and [ overwedge](b ) for the minus case and in figs . [ overwedge](d ) and [ overwedge](e ) for the plus case . the latter indicates that the difference in contact angle at the two contact lines due to the different height level at the top side and at the base side of the step provides a stronger driving force than the lateral action of the disjoining pressure which , in the plus case , moves droplets positioned next to the step in uphill direction . for all substrates the surface of droplets pinned at the edge becomes convex ( corresponding to a negative pressure in the droplet ) for small @xmath16 ( i.e. , small @xmath15 , see @xmath268 in figs . [ overwedge](c ) and [ overwedge](f ) ) . for very large @xmath16 the upper contact line depins from the edge and moves down towards the wedge ( see @xmath286 in figs . [ overwedge](c ) and [ overwedge](f ) ) . the result is a droplet sitting in the corner of the wedge area only . the critical value for @xmath16 between both types of configurations depends on the droplet volume and the step height : the smaller the droplet ( as compared to the step height ) the smaller is the value of @xmath16 at which the upper contact line depins and the larger the volume the smaller is the value of @xmath16 at which the droplet surface becomes convex . both phenomena are in qualitative agreement with macroscopic considerations which take into account interface energies only ( see refs . @xcite ) . droplets sitting in the corner of a wedge are in an energetically rather favorable situation as illustrated by the arguments given at the beginning of this subsection . however , even in the plus case , for which droplets are attracted by wedges , they stop before reaching the wedge and they do not move into the corner of the wedge . in any case , there is an energy barrier to overcome in order to move droplets out of wedges , as shown in fig . [ wedgeforce](a ) . if a small horizontal force is applied to a droplet sitting in the corner of a wedge it assumes a new , slightly distorted but stable shape . but there exists a threshold force density @xmath233 above which the distorted configuration becomes unstable and the droplet moves out of the corner . in the minus case considered in fig . [ wedgeforce ] , the droplet is repelled from the wedge such that the effect of the intermolecular forces adds to the external driving force and the droplet definitively moves out of the corner of the wedge . in contrast to the force required to push droplets over an edge , @xmath233 increases with @xmath15 ( i.e. , with @xmath16 ) . the total force per unit ridge length @xmath249 required to pull a droplet out of a wedge increases only slightly ( i.e. , less than linearly ) with the droplet volume ( see fig . [ wedgeforce](b ) ) , so that accordingly the required force density @xmath233 decreases significantly with volume . with the same dimensional arguments used above for droplets being pushed over edges one would expect the total force needed to pull two droplets of different volume out of the corner of a wedge to be proportional to the square root of the volume ratio . in particular for small @xmath287 according to fig . [ wedgeforce](b ) the total threshold force is almost independent of the droplet size , rather than to increase by a factor @xmath288 . we attribute this difference to the influence of the long - ranged part of the intermolecular forces . numerical solutions of the stokes dynamics of nanodroplets in the vicinity of edges , wedges , and steps are rather time consuming , even when using advanced numerical methods . as shown in fig . [ comppisigma](b ) , the main driving force for the migration of droplets is the disjoining pressure induced force density @xmath289 as defined in eq . ( [ dpforce ] ) . after the initial relaxation process , the shapes of the droplets hardly change during the migration process until the droplets either reach the edge ( minus case ) or the corner of the wedge area ( plus case ) . unfortunately the relaxed shape of the droplet is not available analytically , but for droplets on substrates with @xmath290 as mostly considered here the initial shape relaxations are rather mild . accordingly , as demonstrated in the following , the force on the droplets can be estimated rather accurately from calculating @xmath289 for droplets with a shape given by the initial profile in eq . ( [ inicond ] ) positioned at the distance @xmath49 from the edge or from the corner of the wedge . apparently this estimate becomes invalid for @xmath291 . figures [ dropsize](a ) and [ dropsize](b ) show the disjoining pressure induced force densities calculated along these lines for droplets of size @xmath160 and @xmath292 as a function of the distance @xmath225 of the right contact line to an edge for the minus and the plus case , respectively . since @xmath289 is proportional to @xmath15 only results for @xmath9 are shown . for the plus case the force is always negative for both droplet sizes and at all distances from the edge , with its strength increasing towards the edge . this means that droplets should move away from the edge with a speed which decreases continuously . this is in complete agreement with the numerical results presented in the previous section . for the minus case and for sufficiently large values of @xmath14 , the force is positive in accordance with the numerical results . however , as shown in fig . [ dropsize](a ) , for very small values of @xmath14 , i.e. , for @xmath293 the force in the direct vicinity of the edge becomes negative and droplets are expected to move away from the edge . indeed , as demonstrated in fig . [ edgediffb](b ) ( @xmath294 for i lies below the corresponding values for ii and iii ) , the final distance of the droplets from the step edge in the minus case increases with more negative values of @xmath14 . on the other hand , as shown in the following subsec . [ direction ] , for large distances from the edge , in the minus case even for arbitrarily small @xmath14 the force is positive so that droplets find an equilibrium position with vanishing force at a significant distance from the edge . the sign of the disjoining pressure induced force density does not depend on the droplet size . however , the equilibrium position changes as a function of droplet size . the force calculated for droplets of the same size but in the vicinity of a wedge for the minus and the plus case are shown in figs . [ dropsize](c ) and [ dropsize](d ) , respectively . for the minus case the force is positive for any droplet size and for any @xmath14 , which means that the droplets move away from the wedge . for the plus case the force is negative at large distances , but it changes sign close to the wedge at a distance which increases with decreasing the size of the droplets and with decreasing the value of @xmath14 . the latter relation is in agreement with the numerical results presented in fig . [ wedgediffbplus](b ) . the disjoining pressure induced force density @xmath289 presented in fig . [ dropsize ] has beeen calculated for droplets with a shape given by eq . ( [ inicond ] ) , i.e. , for droplets with equal height and half width . however , the substrate parameters used in fig . [ dropsize ] do not necessarily lead to @xmath159 , and droplets would adopt a very different shape even during the migration process . in order to check the influence of the droplet shape on the calculated disjoining pressure induced force density @xmath289 we also consider droplets which have a width @xmath295 different from their height @xmath47 ( compare with eq . ( [ inicond ] ) ) : @xmath296^{|x-\bar{x}|^m+1}.\ ] ] figure [ profsize ] compares the disjoining pressure induced force density @xmath289 on droplets in the vicinity of edges and wedges in both the minus and the plus case for different drop widths @xmath295 but for a fixed drop height @xmath160 . the results indicate that the form of the droplets does not change the sign of @xmath289 , and in particular in the vicinity of the wedge the droplet width has a rather small influence on the force . in the vicinity of topographic steps the dependence of @xmath289 on the step height @xmath5 is also in good agreement with the results of the full numerical solution of the stokes dynamics . figure [ stepsize ] shows @xmath289 above and below the step on substrates of the minus and plus type for step heights ranging from @xmath193 to @xmath186 ( i.e. , to isolated edges and wedges ) . the absolute value of the force increases with the step height with the force near isolated edges and wedges as the limiting values . this limiting value is almost reached for a step height @xmath195 ( not shown in fig . [ stepsize ] ) . ( in units of @xmath169 ) on droplets of size @xmath160 and @xmath297 in the vicinity of an edge ( ( a ) and ( b ) ) and a wedge ( ( c ) and ( d ) ) on substrates of the minus ( ( a ) and ( c ) ) and the plus ( ( b ) and ( d ) ) type with @xmath9 and various values of @xmath14 as indicated in the boxes as a function of the distance @xmath225 from the edge or the corner of the wedge . [ dropsize ] ] ( in units of @xmath169 ) on droplets of height @xmath160 and widths @xmath298 and @xmath299 in the vicinity of an edge ( ( a ) and ( b ) ) and a wedge ( ( c ) and ( d ) ) on substrates of the minus ( ( a ) and ( c ) ) and the plus ( ( b ) and ( d ) ) type with @xmath9 and various values of @xmath14 as indicated in the boxes as a function of the distance @xmath225 from the edge or wedge . [ profsize ] ] ( in units of @xmath169 ) on droplets of size @xmath300 on the top side ( ( a ) and ( b ) ) and on the bottom side ( ( c ) and ( d ) ) of steps of various heights @xmath5 with substrates of the minus ( @xmath179 , ( a ) and ( c ) ) and the plus ( @xmath10 , ( b ) and ( d ) ) type with @xmath9 as a function of the distance @xmath225 from the step . [ stepsize ] ] both the force calculations presented in the previous subsection as well as the results of the numerical solution of the mesoscopic hydrodynamic equations indicate , that the direction of motion of a nanodroplet far enough from the step does not depend on whether the droplet is positioned on the top side or on the bottom side of the step . in the minus case the droplets move in downhill direction ( i.e. , in the direction of positive @xmath11-values ) and in the plus case in the opposite direction , independent of the step height and of the values of @xmath14 and @xmath15 ( where the latter has to be positive ) . in order to understand this we further analyze the total force per unit ridge length @xmath18 on liquid ridges as defined in eq . ( [ dpforce ] ) for large droplets far from the step . asymptotically for large @xmath301 the djp reduces to its value on a flat substrate so that there the wetting film thickness assumes its value @xmath3 independent of @xmath11 up to @xmath302 . the leading order correction to the djp is @xmath303 for the plus and minus case , respectively . parameterizing the shape of the liquid - vapor interface to the left and to the right of the droplet apex by @xmath304 and @xmath305 , respectively , from eq . ( [ dpforce ] ) with @xmath306 we obtain : @xmath307\nonumber\\ & \approx & \pm { \mbox{sign}}(x)\,\int_{y_0}^{y_m } \frac{9\,h\,c}{16}\,\left [ \frac{1}{x_r(y)^4}-\frac{1}{x_\ell(y)^4 } \right]\,dy\nonumber\\ & \approx & \mp \frac{9\,h\,c}{16\,|\bar{x}|^5}\,\int_{y_0}^{y_m}\left [ x_r(y)-x_\ell(y)\right]\,dy=\mp \frac{9\,h\,c}{16\,|\bar{x}|^5}\,a_d\/ , \label{flimit0}\end{aligned}\ ] ] with the droplet apex height @xmath21 . in the last but one step we have approximated @xmath308 with @xmath309 . the force is proportional to the droplet cross - sectional area and its sign is determined by the sign of the hamaker constant ( i.e. , depending on the case ; plus or minus ) only : in the plus case the force is negative ( upper sign ) and in the minus case it is positive ( lower sign ) . this is in complete agreement with the numerical data . however , other than suggested by eq . ( [ flimit0 ] ) , the force on a droplet does not diverge in the limit @xmath232 as this limit has to be taken before taking the limit @xmath310 . at large distances from an isolated wedge as well as from an isolated edge , the disjoining pressure is to leading order given by the djp of the corresponding homogeneous substrate with @xmath311 as the leading order correction for the plus and the minus case , respectively . as in the case of the step , up to this order the thickness of the wetting film is independent of the ( large ) distances from the edge or wedge . using the same approximations as in the case of the step of finite height , the force on a droplet at a distance @xmath312 from an edge is given by @xmath313 with the upper sign corresponding to the plus case and the lower sign to the minus case . the sign of the force is the same as in the case of a step and it is also proportional to the droplet volume . however , it decreases less rapidly with the distance from the step . for very large , almost macroscopic droplets , the situation is again different from the previous two . in the following we follow the line of arguments developed in refs . @xcite for droplets in the vicinity of chemical steps . in this limit the droplets are approximately symmetric with respect to their apex and the main contribution to the force stems from the vicinity of the contact lines . for the wetting film as well as near the apex the @xmath11-component @xmath314 of the surface normal vector is zero and thus in the vicinity of the apex the djp is negligibly small . in most of the examples discussed here the equilibrium contact angle @xmath16 is about @xmath191 and , as a consequence , the lateral width of the contact lines ( i.e. , the range of @xmath11-values within which the drop profile crosses over to the flat one of the wetting film ) is small and the lateral variation of the djp within this region is negligible . therefore , after parameterizing the droplet surface in the vicinity of the left and right contact line ( at @xmath315 and @xmath316 , respectively ) by the corresponding function @xmath317 , the total force on a droplet can be approximated by @xmath318 \\ & = & -\phi(\bar{x}-a , y_0)+\phi(\bar{x}+a , y_0)\approx 2\,a\ , \partial_x \phi(\bar{x},y_0 ) , \label{forcemi}\end{aligned}\ ] ] where @xmath319 is the local effective interface potential at the level @xmath3 of the wetting film ( on the top side of the step one has to add @xmath5 to @xmath3 ) . extending eq . ( [ eqtheta ] ) to inhomogeneous substrates one can define a spatially varying `` equilibrium contact angle '' @xmath320 . in this sense , a droplet in the vicinity of a topographic step is exposed to an effective chemical wettability gradient which it follows . expanding @xmath321 for large @xmath301 yields @xmath322 with the upper sign corresponding to the plus case and the lower sign to the minus case . the force is equal on both sides of the step and it increases linearly with the step height but it decreases rather rapidly with the distance from the step , however , more slowly than in the case of nanodroplets . the force increases linearly with the base width @xmath323 rather than with the cross - sectional area @xmath19 . as in the case of nanodroplets the actual force on a droplet does not diverge in the limit @xmath232 . using the same approximations as in the case of the step of finite height , the force on a droplet at a distance @xmath301 from an isolated edge is given by @xmath324 with the upper sign corresponding to the plus case and the lower sign to the minus case . in the vicinity of a wedge the situation is more complicated . in order to obtain as in eq . ( [ forcemi ] ) the effective interface potential , the point @xmath325 corresponding to the upper limit of the integral there has to correspond to a point at infinite distance from the substrate . however , taking @xmath2 to @xmath186 for a fixed value @xmath20 does not change the distance from the vertical part of the wedge . at this point it is not clear whether the force integral in eq . ( [ dpforce ] ) can be approximated by the form given in eq . ( [ forcemi ] ) because the basic assumption , that the disjoining pressure is negligible at the apex , is probably not true . expanding the force as calculated from eq . ( [ forcemi ] ) for large distances from a step of very large height one obtains terms of the order @xmath326 competing with terms of order @xmath327 , which indicates that in the case of a wedge the approximations eq . ( [ forcemi ] ) is based on lead to a mathematically ill - posed problem . in all cases @xmath27 essentially depends on the ratio of the step height @xmath5 and the distance from the step @xmath20 as well as on the ratio of the apex height @xmath21 and @xmath20 . the asymptotic results are summarized in fig . [ asymptofig ] . @xmath27 varies according to a power law @xmath22 , @xmath328 , of the distance from the step . for finite sized droplet and steps of finite height ( @xmath329 and @xmath330 ) we obtain the fastest decay with @xmath331 . for almost macroscopic droplets ( @xmath332 ) in the vicinity of finite sized steps and for nanodroplets near isolated edges and wedges one has @xmath24 . for large drops ( @xmath25 ) next to an isolated edge we get the weakest decay with @xmath26 . in any case , the total force per unit length @xmath27 is proportional to the hamaker constant as observed in the numerical solution of the mesoscopic stokes dynamics as well as in the force analysis presented in subec . [ force ] . the total disjoining pressure induced force per unit ridge length @xmath18 at large distance from steps depends on two length ratios @xmath333 and @xmath334 ( the cross - sectional area @xmath19 is proportional to @xmath335 ) . this figure summarizes the analytical results obtained in subsec . [ direction ] . ] the driving force @xmath336 on the droplets is balanced by viscous forces . by applying a simple analysis within the lubrication approximation one can show that the rate of energy dissipation is proportional to the square of the velocity @xmath337 of the droplets @xcite . the form of this dependence can be expected to hold also for droplets with large contact angles on the basis of analyticity and symmetry arguments . by equating this dissipation with the work @xmath338 done by the driving force one finds @xmath339 for droplets far from the step with @xmath340 given by eqs . ( [ flimit1 ] ) and ( [ flimit2 ] ) as a power law one has @xmath341 and therefore @xmath342 with @xmath343 for large droplets in the vicinity of edges , @xmath344 for large droplets in the vicinity of steps of finite height as well as for nanodroplets near isolated edges and wedges , and @xmath345 for nanodroplets near steps of finite height . @xmath73 is a constant which also depends on whether there is an edge , wedge , or a step and whether the droplet is large or small . the functional form given by eq . ( [ xt ] ) with the corresponding value of @xmath346 can be fitted to the positions of nanodroplets as a function of time obtained by numerically solving the mesoscopic hydrodynamic equations , e.g. , to the data shown in fig . [ effstepplus ] ( droplet on the top side of steps , plus case ) and fig . [ wedgediffhminus ] ( droplet on the step base , minus case ) . however , the numerically available range of @xmath20 values is rather small so that the fits are consistent with the above values of @xmath346 but can not rule out different ones . in addition , it is not clear whether the distances considered in the numerical solutions of the mesoscopic hydrodynamic equations are large enough to reach the asymptotic regime considered here , and whether the droplets should be considered small or large in the above sense . a major and obvious driving force for studying the dynamics of nanodroplets on structured substrates is the rapid development and miniaturization of microfluidic devices , in particular of open microfluidic devices @xcite . but this is not the only research area for which a detailed understanding of the influence of the long - ranged part of the intermolecular interactions on fluids in the vicinity of lateral surface structures might be important . another example is the dynamics of nanodroplets at chemical surface structures as discussed in refs . moreover , dewetting processes are also strongly influenced by surface heterogeneities , both during the initial phase of film breakup @xcite as well as during hole growth @xcite . the latter example is particularly interesting in this respect because it reveals an intrinsic nanoscopic length scale which has to be understood : the receding contact line is pinned only by steps of a minimum height which increases with the size of the liquid molecules @xcite a clear indication that details of the intermolecular interactions in the vicinity of the step are relevant . all results presented in this article have been obtained for homogeneous straight liquid ridges . apart from the fact that such ridges are unstable with respect to breaking up into three - dimensional droplets @xcite , the question remains to check how relevant these results are for actual three - dimensional droplets . in this context we point out that the basic driving mechanism for droplets in the vicinity of steps is the difference of the disjoining pressure on that side of the droplet which is closer to the step and the side which is further away from the step . in such a situation also three - dimensional droplets move . however , the third dimension certainly changes the behavior of droplets spanning topographic steps @xcite : depending on the droplet volume , the droplet can spread along the step into a cigar shaped configuration . the influence of the long - ranged part of the intermolecular interactions on this phenomenon has not yet been studied . it is worthwhile to point out that , although the dynamics is different , there are strong similarities between nanodroplets and solid nanoclusters : their energetics on structured surfaces is determined by intermolecular forces as demonstrated in ref . @xcite by molecular dynamics simulations of gold clusters on graphite surfaces . unfortunately , in such simulations taking into account the long - ranged component of the intermolecular forces increases the numerical cost drastically , such that most of the effects discussed here are not accessible by molecular dynamics simulations @xcite . this leaves the question of experimental tests of our theoretical predictions presented here . as detailed in ref . @xcite the forces on the nanodroplets are of the order of @xmath347 n ( i.e. , about eight orders of magnitude stronger than the gravitational force on such a droplet ) and the resulting velocities range between @xmath348 and @xmath349 for viscosities between @xmath350 and @xmath351 . while topographic surface structures of almost any type can be produced with modern lithographic techniques , positioning nanodroplets with nanometer accuracy next to a step remains a tough challenge . most promising are techniques based on using atomic force microscopes as pens @xcite , but there are no experiments available yet . experimentally it is much easier to grow droplets from an aerosol or a vapor phase rather than to deposit them at a specific location . experimentally it has been shown , that water nanodroplets preferentially condense onto terrace steps of vicinal surfaces @xcite . however , these data do not allow one to determine whether the droplets reside on the top terrace , on the bottom terrace , or whether they span the step . this example shows , that condensation on ( and evaporation from ) nano - structured substrates is a challenging problem of its own . from the step . the boundary of the system @xmath352 decomposes into three different groups : liquid - liquid ( @xmath353 ) , liquid - solid ( @xmath354 ) , and liquid - vapor ( @xmath355 ) interfaces . the discretized node points considered in the numerical investigation are indicated ; @xmath36 and @xmath356 represents the normal and the tangential unit vectors on @xmath352 , respectively . ] in order to study the effect of the intermolecular forces on nanodroplets near a topographic step we solve eqs . ( [ eq : stokes])([eq : surfacebc ] ) numerically using a standard and accurate biharmonic boundary integral method ( bbim ) @xcite . to this end we introduce the stream function @xmath357 so that @xmath358 and @xmath359 as well as the vorticity @xmath360 which allows us to reformulate the dimensionless versions of eqs . ( [ eq : stokes ] ) and ( [ eq : incomp ] ) in terms the following harmonic and biharmonic equations @xcite : @xmath361 and @xmath362 the standard bbim relies on mapping the equations for @xmath363 and @xmath364 onto the boundary @xmath365 of the fluid , parameterized in terms of its contour length parameter @xmath72 . this results in an integral equation for @xmath363 , @xmath364 , and their derivatives @xmath366 and @xmath367 , with the surface normal vector pointing outwards of the liquid . by dividing the boundary of the system into a series of elements ( see fig . [ numfig ] ) one obtains a coupled system of algebraic equations which can be solved numerically . with the tangential velocity @xmath368 and the normal velocity @xmath369 ( with the index @xmath72 indicating the derivative in the direction tangential to the boundary ) the position of the liquid boundary after a time step can be calculated via the explicit euler scheme @xcite : @xmath370 in order to solve these equations the boundary conditions of the system must be expressed in terms of @xmath363 and @xmath364 . depending on the phases in contact with each other , three different types of boundary interfaces can be identified ( see fig . [ numfig ] ) : liquid - solid interfaces @xmath44 , liquid - liquid interfaces ( those boundaries @xmath371 and @xmath372 which are located at the end sides of the system ) , and liquid - vapor interfaces @xmath35 . for @xmath44 we impose the no - slip condition ( @xmath373 ) which corresponds to @xmath374 and @xmath375 . for @xmath371 and @xmath372 we apply a no - flux condition which corresponds to having a vertical symmetry plane there . such a system corresponds to a periodic repetition of the system attached to its mirror image . correspondingly the slope of the liquid - vapor interface at the side ends of the system is zero . these conditions can be implemented by setting @xmath374 and @xmath376 there . the tangential and the normal component of the boundary condition ( [ eq : surfacebc ] ) along the liquid - vapor interface @xmath35 in terms of the stream function and the vorticity read ( lower indices @xmath72 indicate derivatives with respect to the contour length parameter @xmath72 ) @xmath377 and @xmath378 respectively , with the local curvature @xmath379 in order to increase the efficiency of the numerical calculations we employ an adaptive time stepping : for any numerical step , the time step is selected such that the displacement of any node does not exceed @xmath380 percents of the length of the elements connected to that node ; @xmath380 can be changed during the numerical calculations . the starting value for @xmath380 and the rate of its increase depends on the actual situation but typically we have started with @xmath381 and then gradually increased it to 0.1 or even more . in order to avoid numerical instabilities , the position of the end points of the boundary elements are smoothed after a specified number of steps by fitting a spline through the points on the liquid - vapor interface , followed by selecting new and equally spaced points on the spline . dupuis a and yeomans j m in ed . bubak m , van albada g d , sloot p m a and dongarra j j , _ computational science - iccs 2004 _ vol . lecture notes in computer science _ pp . 556563 berlin 2004 .

mesoscopic hydrodynamic equations are solved to investigate the dynamics of nanodroplets positioned near a topographic step of the supporting substrate . our results show that the dynamics depends on the characteristic length scales of the system given by the height of the step and the size of the nanodroplets as well as on the constituting substances of both the nanodroplets and the substrate . the lateral motion of nanodroplets far from the step can be described well in terms of a power law of the distance from the step . in general the direction of the motion depends on the details of the effective laterally varying intermolecular forces . but for nanodroplets positioned far from the step it is solely given by the sign of the hamaker constant of the system . moreover , our study reveals that the steps always act as a barrier for transporting liquid droplets from one side of the step to the other .

multiple optimised parameter estimation and data compression ( moped ; @xcite ) is a patented algorithm for the compression of data and the speeding up of the evaluation of likelihood functions in astronomical data analysis and beyond . it becomes particularly useful when the noise covariance matrix is dependent upon the parameters of the model and so must be calculated and inverted at each likelihood evaluation . however , such benefits come with limitations . since moped only guarantees maintaining the fisher matrix of the likelihood at a chosen point , multimodal and some degenerate distributions will present a problem . in this paper we report on some of the limitations of the application of the moped algorithm . in the cases where moped does accurately represent the likelihood function , however , its compression of the data and consequent much faster likelihood evaluation does provide orders of magnitude improvement in runtime . in @xcite , the authors demonstrate the method by analysing the spectra of galaxies and in @xcite they illustrate the benefits of moped for estimation of the cmb power spectrum . the problem of `` badly '' behaved likelihoods was found by @xcite for the problem of light transit analysis ; nonetheless , the authors present a solution that still allows moped to provide a large speed increase . we begin by introducing moped in section 2 and define the original and moped likelihood functions , along with comments on the potential speed benefits of moped . in section 3 we introduce an astrophysical scenario where we found that moped did not accurately portray the true likelihood function . in section 4 we expand upon this scenario to another where moped is found to work and to two other scenarios where it does not . we present a discussion of the criteria under which we believe moped will accurately represent the likelihood in section 5 , as well as a discussion of an implementation of the solution provided by @xcite . full details of the moped method are given in @xcite , here we will only present a limited introduction . we begin by defining our data as a vector , @xmath0 . our model describes @xmath0 by a signal plus random noise , @xmath1 where the signal is given by a vector @xmath2 that is a function of the set of parameters @xmath3 defining our model , and the true parameters are given by @xmath4 . the noise is assumed to be gaussian with zero mean and noise covariance matrix @xmath5 , where the angle brackets indicate an ensemble average over noise realisations ( in general this matrix may also be a function of the parameters @xmath6 ) . the full likelihood for @xmath7 data points in @xmath0 is given by @xmath8^{\textrm{t } } \mathcal{n}(\btheta)^{-1 } [ { \bf x}-{\bf u}(\btheta)]\right\}}.\end{aligned}\ ] ] at each point , then , this requires the calculation of the determinant and inverse of an @xmath9 matrix . both scale as @xmath10 , so even for smaller datasets this can become cumbersome . moped allows one to eliminate the need for this matrix inversion by compressing the @xmath7 data points in @xmath0 into @xmath11 data values , one for each parameters of the model . additionally , moped creates the compressed data values such that they are independent and have unit variance , further simplifying the likelihood calculation on them to an @xmath12 operation . typically , @xmath13 so this gives us a significant increase in speed . a single compression is done on the data , @xmath0 , and then again for each point in parameter space where we wish to compute the likelihood . the compression is done by generating a set of weighting vectors , @xmath14 ( @xmath15 ) , from which we can generate a set of moped components from the theoretical model and data , @xmath16 note that the weighting vectors must be computed at some assumed fiducial set of parameter values , @xmath17 . the only choice that will truly maintain the likelihood peak is when the fiducial parameters are the true parameters , but obviously we will not know these in advance for real analysis situations . thus , we can choose our fiducial model to be anywhere and iterate the procedure , taking our likelihood peak in one iteration as the fiducial model for the next iteration . this process will converge very quickly , and may not even be necessary in some instances . for our later examples , since we do know the true parameters we will use these as the fiducial ( @xmath18 ) in order to remove this as a source of confusion ( all equations , however , are written for the more general case ) . note that the true parameters , @xmath4 , will not necessarily coincide with the peak @xmath19 of the original likelihood or the peak @xmath20 of the moped likelihood ( see below ) . the weighting vectors must be generated in some order so that each subsequent vector ( after the first ) can be made orthogonal to all previous ones . we begin by writing the derivative of the model with respect to the @xmath21th parameter as @xmath22 . this gives us a solution for the first weighting vector , properly normalised , of @xmath23 the first compressed value is @xmath24 and will weight up the data combination most sensitive to the first parameter . the subsequent weighting vectors are made orthogonal by subtracting out parts that are parallel to previous vectors , and are normalized . the resulting formula for the remaining weighting vectors is @xmath25 @xmath26 where @xmath27 . weighting vectors generated with equations and form an orthnomal set with respect to the noise covariance matrix so that @xmath28 this means that the noise covariance matrix of the compressed values @xmath29 is the identity , which significantly simplifies the likelihood calculation . the new likelihood function is given by @xmath30 where @xmath31 represents the compressed data and @xmath32 represents the compressed signal . this is a much easier likelihood to calculate and is time - limited by the generation of a new signal template instead of the inversion of the noise covariance matrix . the peak value of the moped likelihood function is not guaranteed to be unique as there may be other points in the original parameter space that map to the same point in the compressed parameter space ; this is a characteristic that we will investigate . moped implicity assumes that the covariance matrix , @xmath33 , is independent of the parameters . with this assumption , a full likelihood calculation with @xmath7 data points would require only an @xmath34 operation at each point in parameter space ( or @xmath35 if @xmath33 is diagonal ) . in moped , however , the compression of the theoretical data is an @xmath36 linear operation followed by an @xmath12 likelihood calculation . thus , one loses on speed if @xmath33 is diagonal but gains by a factor of @xmath37 otherwise . for the data sets we will analyze , @xmath38 . we begin , though , by assuming a diagonal @xmath33 for simplicity , recognizing that this will cause a speed reduction but that it is a necessary step before addressing a more complex noise model . one can iterate the parameter estimation procedure if necessary , taking the maximum likelihood point found as the new fiducial and re - analyzing ( perhaps with tighter prior constraints ) ; this procedure is recommended for moped in @xcite , but is not always found to be necessary . moped has the additional benefit that the weighting vectors , @xmath39 , need only to be computed once provided the fiducial model parameters are kept constant over the analysis of different data sets . computed compressed parameters , @xmath40 , can also be stored for re - use and require less memory than storing the entire theoretical data set . in order to demonstrate some of the limitations of the applicability of the moped algorithm , we will consider a few test cases . these originate in the context of gravitational wave data analysis for the _ laser interferometer space antenna _ ( _ lisa _ ) since it is in this scenario that we first discovered such cases of failure . the full problem is seven - dimensional parameter estimation , but we have fixed most of these variables to their known true values in the simulated data set in order to create a lower - dimensional problem that is simpler to analyse . we consider the case of a sine - gaussian burst signal present in the lisa detector . the short duration of the burst with respect to the motion of lisa allows us to use the static approximation to the response . in frequency space , the waveform is described by ( @xcite ) @xmath41 here @xmath42 is the dimensionless amplitude factor ; @xmath43 is the width of the gaussian envelope of the burst measured in cycles ; @xmath44 is the central frequency of the oscillation being modulated by the gaussian envelope ; and @xmath45 is the central time of arrival of the burst . this waveform is further modulated by the sky position of the burst source , @xmath46 and @xmath47 , and the burst polarisation , @xmath48 , as they project onto the detector . the one - sided noise power spectral density of the lisa detector is given by ( @xcite ) @xmath49 where @xmath50s is the light travel time along one arm of the lisa constellation , @xmath51hz@xmath52 is the proof mass acceleration noise and @xmath53hz@xmath52 is the shot noise . this is independent of the signal parameters and all frequencies are independent of each other , so the noise covariance matrix is constant and diagonal . this less computationally expensive example allows us to show some interesting examples . we begin by taking the one - dimensional case where the only unknown parameter of the model is the central frequency of the oscillation , @xmath44 . we set @xmath54 and @xmath55s ; we then analyze a @xmath56s segment of lisa data , beginning at @xmath57s , sampled at a @xmath58s cadence . for this example , the data was generated with random noise ( following the lisa noise power spectrum ) at an snr of @xmath59 with @xmath60hz ( thus @xmath61hz for moped ) . the prior range on the central frequency goes from @xmath62hz to @xmath63hz . @xmath64 samples uniformly spaced in @xmath44 were taken and their likelihoods calculated using both the original and moped likelihood functions . the log - likelihoods are shown in figure [ fig : likecomp ] . note that the absolute magnitudes are not important but the relative values within each plot are meaningful . both the original and moped likelihoods have a peak close to the input value @xmath65 . for the chosen template.,width=312 ] we see , however , that in going from the original to moped log - likelihoods , the latter also has a second peak of equal height at an incorrect @xmath44 . to see where this peak comes from , we look at the values of the compressed parameter @xmath66 as it varies with respect to @xmath44 as shown in figure [ fig : yf_vs_f ] . the true compressed value peak occurs at @xmath67hz , where @xmath68 . however , we see that there is another frequency that yields this exact same value of @xmath66 ; it is at this frequency that the second , incorrect peak occurs . by creating a mapping from @xmath44 to @xmath66 that is not one - to - one , moped has created the possibility for a second solution that is indistinguishable in likelihood from the correct one . this is a very serious problem for parameter estimation . interestingly , we find that even when moped fails in a one - parameter case , adding a second parameter may actually rectify the problem , although not necessarily . if we now allow the width of the burst , @xmath43 , to be a variable parameter , there are now two orthognal moped weighting vectors that need to be calculated . this gives us two compressed parameters for each pair of @xmath44 and @xmath43 . each of these may have its own unphysical degeneracies , but in order to give an unphysical mode of equal likelihood to the true peak , these degeneracies will need to coincide . in figure [ fig : ytruecontours ] , we plot the contours in @xmath69 space of where @xmath70 as @xmath6 ranges over @xmath44 and @xmath43 values . we can clearly see the degeneracies present in either variable , but since these only overlap at the one location , near to where the true peak is , there is no unphysical second mode in the moped likelihood . and @xmath71 as they vary over @xmath44 and @xmath43 . the one intersection is the true maximum likelihood peak.,width=312 ] hence , when we plot the original and moped log - likelihoods in figure [ fig : fqlikes ] , although the behaviour away from the peak has changed , the peak itself remains in the same location and there is no second mode . adding more parameters , however , does not always improve the situation . we now consider the case where @xmath43 is once again fixed to its true value and we instead make the polarisation of the burst , @xmath48 , a variable parameter . there are degeneracies in both of these parameters and in figure [ fig : ytruecontours3 ] we plot the contours in @xmath72-space where the compressed values are each equal to the value at the maximum moped likelihood point . these two will necessarily intersect at the maximum likelihood solution , near the true value ( @xmath73 hz and @xmath74 rad ) , but a second intersection is also apparent . this second intersection will have the same likelihood as the maximum and be another mode of the solution . however , as we can see in figure [ fig : fpslikes ] in the left plot , this is not a mode of the original likelihood function . moped has , in this case , created a second mode of equal likelihood to the true peak . and @xmath75 values as they vary as functions of @xmath44 and @xmath48.,width=312 ] for an even more extreme scenario , we now fix to the true @xmath48 and allow the time of arrival of the burst @xmath45 to vary ( we also define @xmath76 ) . in this scenario , the contours in @xmath77-space where @xmath70 are much more complicated . thus , we have many more intersections of the two contours than just at the likelihood peak near the true values and moped creates many alternative modes of likelihood equal to that of the original peak . this is very problematic for parameter estimation . in figure [ fig : ytruecontours2 ] we plot these contours so the multiple intersections are apparent . figure [ fig : ftlikes ] shows the original and moped log - likelihoods , where we can see the single peak becoming a set of peaks . and @xmath75 values as they vary as functions of @xmath44 and @xmath45 . we can see many intersections here other than the true peak.,width=312 ] what we can determine from the previous two sections is a general rule for when moped will generate additional peaks in the likelihood function equal in magnitude to the true one . for an @xmath11-dimensional model , if we consider the @xmath78-dimensional hyper - surfaces where @xmath70 , then any point where these @xmath11 hyper - surfaces intersect will yield a set of @xmath6-parameter values with likelihood equal to that at the peak near the true values . we expect that there will be at least one intersection at the likelihood peak corresponding to approximately the true solution . however , we have shown that other peaks can exist as well . the set of intersections of contour surfaces will determine where these additional peaks are located . this degeneracy will interact with the model s intrinsic degeneracy , as any degenerate parameters will yield the same compressed values for different original parameter values . unfortunately , there is no simple way to find these contours other than by mapping out the @xmath79 values , which is equivalent in procedure to mapping the moped likelihood surface . the benefit comes when this procedure is significantly faster than mapping the original likelihood surface . the mapping of @xmath79 can even be performed before data is obtained or used , if the fiducial model is chosen in advance ; this allows us to analyse properties of the moped compression before applying it to data analysis . if the moped likelihood is mapped and there is only one contour intersection , then we can rely on the moped algorithm and will have saved a considerable amount of time , since moped has been demonstrated to provide speed - ups of a factor of up to @xmath80 in @xcite . however , if there are multiple intersections then it is necessary to map the original likelihood to know if they are due to degeneracy in the model or were created erroneously by moped . in this latter case , the time spent finding the moped likelihood surface can be much less than that which will be needed to map the original likelihood , so relatively little time will have been wasted . if any model degeneracies are known in advance , then we can expect to see them in the moped likelihood and will not need to find the original likelihood on their account . one possible way of determining the validity of degenerate peaks in the moped likelihood function is to compare the original likelihoods of the peak parameter values with each other . by using the maximum moped likelihood point found in each mode and evaluating the original likelihood at this point , we can determine which one is correct . the correct peak and any degeneracy in the original likelihood function will yield similar values to one another , but a false peak in the moped likelihood will have a much lower value in the original likelihood and can be ruled out . this means that a bayesian evidence calculation can not be obtained from using the moped likelihood ; however , the algorithm was not designed to be able to provide this . the solution for this problem presented in @xcite is to use multiple fiducial models to create multiple sets of weighting vectors . the log - likelihood is then averaged across these choices . each different fiducial will create a set of likelihood peaks that include the true peaks and any extraneous ones . however , the only peaks that will be consistent between fiducials are the correct ones . therefore , the averaging maintains the true peaks but decreases the likelihood at incorrect values . this was tested with 20 random fiducials for the two - parameter models presented and was found to leave only the true peak at the maximum likelihood value . other , incorrect , peaks are still present , but at log - likelihood values five or more units below the true peak . when applied to the full seven parameter model , however , the snr threshold for signal recovery is increased significantly , from @xmath81 to @xmath82 . the moped algorithm for reducing the computational expense of likelihood functions can , in some examples , be extremely useful and provide orders of magnitude of improvement . however , as we have shown , this is not always the case and moped can produce erroneous peaks in the likelihood that impede parameter estimation . it is important to identify whether or not moped has accurately portrayed the likelihood function before using the results it provides . some solutions to this problem have been presented and tested , pg s phd is funded by the gates cambridge trust . feroz f , gair j , graff p , hobson m p , & lasenby a n , cqg , * 27 * 7 pp . 075010 ( 2010 ) , arxiv:0911.0288 [ gr - qc ] . gupta s & heavens a f , mnras * 334 * 167 - 172 ( 2002 ) , arxiv : astro - ph/0108315 . heavens a f , jimenez r , & lahav o , mnras * 317 * 965 - 972 ( 2000 ) , arxiv : astro - ph/9911102 . protopapas p , jimenez r , & alcock a , mnras * 362 * 460 - 468 ( 2005 ) , arxiv : astro - ph/0502301 .

we investigate the use of the multiple optimised parameter estimation and data compression algorithm ( moped ) for data compression and faster evaluation of likelihood functions . since moped only guarantees maintaining the fisher matrix of the likelihood at a chosen point , multimodal and some degenerate distributions will present a problem . we present examples of scenarios in which moped does faithfully represent the true likelihood but also cases in which it does not . through these examples , we aim to define a set of criteria for which moped will accurately represent the likelihood and hence may be used to obtain a significant reduction in the time needed to calculate it . these criteria may involve the evaluation of the full likelihood function for comparison . [ firstpage ] methods : data analysis methods : statistical

all massive galaxies appear to host a supermassive black hole ( with @xmath4 ) at their center @xcite . measuring the mass of central black holes in galaxies is of great importance , as the discovery of a relationship between mass and the velocity dispersion of the stars in the central bulge , the @xmath3 relation @xcite , reveals the possible co - evolution of black holes and their host galaxies @xcite . m31 , the andromeda galaxy , is an sb galaxy at a distance of 778 kpc and its nucleus can be observed with excellent spatial resolutions . @xcite , using data obtained with the stratoscope ii , revealed an asymmetry in the nuclear region of m31 , as the bright nucleus did not coincide with either the center of the bulge or the maximum of the stellar velocity dispersion . however , @xcite , using observations from the _ hubble space telescope _ ( _ hst _ ) , showed that the galaxy possesses a double nucleus , the two components being called p1 ( the brightest one ) and p2 ( located , approximately , at the center of the bulge ) . these two components are separated by about @xmath5 . a model to explain the morphology of the nucleus of m31 was proposed by @xcite and states that p1 and p2 are parts of an eccentric stellar disk around the black hole , with p1 coinciding with the apocenter and the black hole being located at p2 . several refinements to this model have been put forth @xcite ; @xcite , using _ hst _ data , revealed that the black hole is actually located in a structure embedded in p2 called p3 , which probably corresponds to a cluster of a - type stars . @xcite , using also _ hst _ data , confirmed that p3 corresponds to a cluster of blue stars around the central black hole . the mass of the central black hole of m31 has already been measured by , at least , six different techniques : ( 1 ) standard dynamical modeling ignoring asymmetries @xcite ; ( 2 ) the center of mass argument , which depends on the asymmetry of p1+p2 @xcite ; ( 3 ) dynamical modeling of the stellar nuclear disk taking into account the asymmetry of p1+p2 @xcite ; ( 4 ) complete dynamical modeling taking into account the asymmetries and the self - gravity of the nuclear stellar disk of p1+p2 @xcite ; ( 5 ) dynamical modeling of p3 , which is independent of p1+p2 @xcite ; ( 6 ) _ n_-body simulations @xcite . all of these methods involved stellar dynamics and resulted in values in the range @xmath6 for the mass of the central black hole in m31 . in this letter , we analyze a data cube of the nuclear region of m31 , obtained with the integral field unity ( ifu ) of the gemini multi - object spectrograph ( gmos ) of the gemini north telescope , and report the discovery of an eccentric h@xmath0 emitting disk around the central black hole . the observations of m31 were made on 2009 september 21 . we used the ifu of the gmos of the gemini north telescope , in the one - slit mode , in order to obtain data cubes , with two spatial dimensions and one spectral dimension . the science field of view ( fov ) has @xmath7 , while the sky fov ( observed simultaneously at a distance of @xmath8 from the science fov ) has @xmath9 . three 10 minute exposure of the nuclear region of m31 were made , with the grating b600-g5307 , in a central wavelength of @xmath10 . the final spectra had a coverage of @xmath11 and a resolution of @xmath12 . the estimated seeing for the night of observation was @xmath13 . standard calibration images were obtained during the observations . the data reduction was made in iraf environment . at the end of the process , three data cubes were obtained , with spaxels of @xmath14 . no sky subtraction was applied because the sky fov ( still inside the disk of m31 ) was contaminated with stellar emission from the galaxy . after the data reduction , we performed a procedure of data treatment . first , a correction of the differential atmospheric refraction was applied to all data cubes , using an algorithm developed by our group . in order to combine the three corrected data cubes into one , a median of these data cubes was calculated . after that , a butterworth spatial filtering @xcite , with order @xmath15 , was applied to all the images of the resulting data cube , in order to remove spatial high - frequency noise . finally , a richardson - lucy deconvolution @xcite was applied to all the images of the data cube , using a synthetic gaussian point - spread function ( psf ) . the psf of the final data cube has fwhm @xmath16 . figure [ fig1 ] shows an image of the final data cube of m31 ( obtained after the data treatment ) collapsed along the spectral axis and an average spectrum of this data cube . the brightest component of the nucleus , p1 , can be easily detected ; however , the fainter components , p2 and p3 , can not be seen , due to the spatial resolution and to the lack of spectral sensitivity in the blue ( below @xmath17 ) . a spectrum of p1 , extracted from a circular area with a radius of @xmath18 , is also shown in figure [ fig1 ] . the average signal - to - noise ratio ( s / n ) , between @xmath19 and @xmath20 , of the spectra of the data cube analyzed here is close to 50 . after the data treatment , a spectral synthesis was applied to the spectrum of each spaxel of the resulting data cube of m31 . this procedure was performed with the starlight software @xcite , which fits the stellar spectrum of a given object with a combination of template stellar spectra from a pre - established base . in this work , we used the base of stellar spectra miles ( medium resolution int library of empirical spectra ; snchez - blzquez et al . the spectral synthesis resulted in a synthetic stellar spectrum for each spaxel . these synthetic spectra were then subtracted from the observed ones , leaving a data cube with emission lines only . the non subtraction of the sky field during the data reduction had no observable effect in the results obtained with the spectral synthesis . in this residual data cube , we have detected a previously unreported weak extended h@xmath0 emission . the luminosity of this h@xmath0 emission line , within a radius of @xmath21 from the central black hole , is @xmath22 ; the equivalent width of this line is @xmath23 , while that of the absorption component is @xmath24 , indicating that the extended h@xmath0 emission is much weaker than the stellar component and , therefore , hard to detect . figure [ fig2 ] shows the spectral fit of the h@xmath0 absorption line in one spectrum of the data cube of m31 and also the fit residuals . it is possible to detect traces of an h@xmath0 emission line immersed in the absorption component , however , only the subtraction of the spectral fit allowed a clear visualization of this very weak emission line . figure [ fig2 ] also reveals the presence of h@xmath25 and [ o iii]@xmath26 emission lines . the spectral resolution of the base of stellar spectra miles ( fwhm = @xmath27 ) is very similar to our spectral resolution ( fwhm = @xmath28 ) and , because of that , the spectral fit is quite precise and allows an accurate subtraction of the stellar emission ( figure [ fig2 ] ) . the spectra of the data cube of m31 were resampled to @xmath29 ( the same spectral sampling of miles spectra ) , before the spectral synthesis was applied . in figure [ fig2 ] , we can see a map of the integrated flux of the h@xmath0 emission line . only the central region of the fov of the data cube was taken into account here , because the h@xmath0 emission was too faint farther away . we can see two bright areas in the vicinity of p1 and p2 , respectively . this reveals a certain similarity between the extended h@xmath0 emission and the double nucleus of m31 and suggests the existence of a possible h@xmath0 emitting disk . an elongated emitting region from se toward nw can also be seen ; however , we do not know the origin of this peculiarity in the morphology of the extended h@xmath0 emission . the area with masked values in the map corresponds to an emitting region , observed in previous studies @xcite , that is not associated to the extended h@xmath0 emission detected here . this emitting region can be identified by observing its intense , compact , and narrow [ o iii]@xmath30 emission . in order to obtain a velocity map for h@xmath0 , we fitted a gaussian function to the h@xmath0 emission line in each spectrum of the data cube . figure [ fig2 ] shows an example of the gaussian fit applied to the h@xmath0 emission line detected in one spectrum of the data cube , after the subtraction of the spectral fit . we can see that , despite the irregularities of the emission line , the gaussian fit provides ( with a considerable precision ) the wavelength corresponding to the peak of the emission line and , therefore , the radial velocity . on the other hand , the irregularities observed in the h@xmath0 emission lines , after the subtraction of the spectral fit , made impossible to determine reliable values for the velocity dispersion . therefore , such values were not taken into account in this work . the velocity map obtained for the h@xmath0 emission line is shown in figure [ fig3 ] and it clearly indicates the presence of an h@xmath0 emitting disk around the central black hole . the position angle of the line of nodes of this emitting disk is p.a.@xmath31 = @xmath32 , which is very close to the value measured for the stellar disk in previous studies ( p.a.@xmath33 = @xmath34 ) @xcite . the contours in figure [ fig3 ] show that the ascending node of the h@xmath0 emitting disk is very close to p1 , while p2 and p3 are located in an area with low velocities . this behavior is very similar to what is observed in the stellar disk . we extracted a velocity curve along the line of nodes ( figure [ fig3 ] ) and the modulus of the maximum and minimum velocities are considerably different , indicating that any kinematical model should be eccentric . we tried to reproduce the velocity curve and the velocity map of h@xmath0 using a model of a simple eccentric disk around the supermassive black hole . only a region within a radius of @xmath21 from the black hole was modeled because , as mentioned before , the disk was too faint farther away . we admitted that the stellar mass inside the radius of the modeled area was small compared to the mass of the black hole , so no stellar mass was taken into account in the model ( see more details below ) . the emitting disk was simulated by superposing 33 concentric keplerian orbits with the following free parameters : the argument of the pericenter @xmath35 , the inclination of the disk @xmath36 , the eccentricity of the disk @xmath37 and the mass of the central black hole @xmath38 . we measured the value of the longitude of the ascending node in the velocity map ( @xmath39 ) , so this parameter was fixed in our model . in each simulation , after all the orbits were superposed , the model was convolved with the estimated psf , in order to simulate the effect of the earth s atmosphere . the free parameters were varied and the simulations were made repeatedly , in order to minimize the value of the @xmath40 between the observed velocity map and the simulated one : + @xmath41 + where @xmath42 is the number of spaxels along the horizontal axis , @xmath43 is the number of spaxels along the vertical axis , @xmath44 is the uncertainty of the velocity of the spaxel @xmath45 , @xmath46 is the velocity of the spaxel @xmath45 of the observed velocity map , @xmath47 is the velocity of the spaxel @xmath45 of the simulated velocity map , @xmath48 is weight equal to 1 for areas near the line of nodes and equal to 0 for farther areas , @xmath49 is the value of the spaxel @xmath45 of the map of integrated fluxes of h@xmath0 ( figure [ fig2 ] ) , and @xmath50 is the sum of the values of all spaxels of the map of integrated fluxes of h@xmath0 ( figure [ fig2 ] ) . in the previous formula , the weight @xmath48 is represented by a step function , which is equal to 1 for spaxels closer than @xmath51 to the line of nodes and equal to 0 for farther spaxels . we used the values of the integrated flux of the h@xmath0 emission line ( @xmath49 ) in the calculation of the @xmath40 in order to give more weight to the spaxels with higher s / n . the best simulated velocity map and the corresponding best simulated velocity curve are also shown in figure [ fig3 ] . the parameters of the best model obtained , with @xmath52 , are shown in table [ tbl1 ] . the uncertainties ( 1@xmath2 ) of the parameters were estimated by plotting histograms ( probability distributions ) of each one of the parameters of the simulation , considering , only cases in which @xmath53 . after that , we fitted a gaussian function on each histogram and obtained the square deviation , which was taken as the uncertainty of each parameter @xcite . we can see that , despite some irregularities of the observed velocity map , our model of a simple eccentric disk reproduced the observed kinematical behavior considerably well . in figure [ fig3 ] , however , we can see a clear discrepancy between the observed velocity map and the simulated one in an area near the ascending node . this is the same emitting region that was masked in the map of the integrated flux of the h@xmath0 emission line ( figure [ fig2 ] ) , which is not associated to the eccentric disk detected here . a by - product of this modeling was an independent determination of the position of the black hole . we found that it is at a distance of @xmath54 from p2 , which is compatible ( at 1@xmath2 level ) with the position of p3 ( at a distance of @xmath55 from p2 ) from previous determinations @xcite . as mentioned before , in this model , we have not taken into account the effect of the mass of the stars . in order to evaluate this assumption , we used an _ hst _ image of the nucleus of m31 in _ v _ band , obtained with wfpc2 , to estimate the stellar mass within the radius of the simulated area . first , we decomposed this image into an asymmetric component ( containing the stellar disk around the black hole ) and a symmetric one ( containing the central part of the stellar bulge ) . after that , we de - projected the symmetric component using a multi - gaussian expansion approach @xcite . finally , the stellar masses of the two components were calculated using a mass - to - light ratio of @xmath56 @xcite . the results are @xmath57 for the bulge component and @xmath58 for the disk . together , these two components represent @xmath59 of the @xmath38 we obtained . that means that this value of @xmath38 resulting from our model can be , at most , @xmath59 too high , due to the neglect of the stellar mass . introducing this in the previous uncertainty for the black hole mass , we obtain a final value of @xmath60 . since the spectral synthesis performed on the data cube provided details about the stellar populations detected , we could , in principle , have used a mass - to - light ratio derived from these results to estimate the stellar mass in the nuclear region of m31 . however , we decided to use a mass - to - light ratio derived from _ observations @xcite because , due to our spatial resolution and lack of spectral sensitivity in the blue ( below @xmath17 ) , we did not observe certain young stellar populations in the nucleus of m31 that were clearly detected in _ hst _ observations . we performed an entirely analogous modeling on a stellar velocity map obtained from the data cube of m31 . only a region within a radius of @xmath61 from the black hole was modeled because the effect of the self - gravity of the stellar populations could be considerable at farther areas . the parameters of the best model obtained are shown in table [ tbl1 ] and the final mass obtained for the central black hole ( including the uncertainty introduced by the neglect of the stellar mass in the modeling ) is @xmath62 . we will show the details about this topic in r. b. menezes et al . ( 2013 , in preparation ) , since the stellar kinematics analysis is not the purpose of this letter . the mass of the central black hole in m31 we obtained by modeling the kinematics of the h@xmath0 emission as a simple eccentric disk ( @xmath60 ) is within the range of the values found in previous determinations using stellar kinematics ( @xmath63 in @xcite ; @xmath64 in @xcite ; @xmath65 in @xcite ; @xmath66 in @xcite ; @xmath67 in @xcite ; @xmath68 in @xcite ) and it is compatible ( at 1@xmath2 level ) with the measurement ( @xmath69 ) obtained with the most detailed models of the stellar disk elaborated so far @xcite . this result is also compatible ( at 1@xmath2 level ) with the value of the mass of the central black hole obtained by modeling the stellar kinematics as a simple eccentric disk ( @xmath62 ) . most of the recent models used to reproduce the stellar disk around the black hole in m31 did not use a single inclination and eccentricity for the entire disk . however , the values we obtained for @xmath36 and @xmath37 in our best models are in the range of the ones used in several studies to model the stellar disk ( @xmath70 for the eccentricity and @xmath71 for the inclination ) @xcite . the values of @xmath37 and @xmath36 obtained for the stellar and the h@xmath0 emitting disks with our models are compatible at 1@xmath2 level . this reveals a certain similarity between these two models . however , the values of @xmath35 obtained are not compatible at all . this suggests that the stellar and the h@xmath0 emitting disks are intrinsically different from each other . according to what was mentioned before , we have identified a weaker h@xmath25 emission line and also traces of even weaker nebular lines , like [ n ii ] @xmath72 , [ s ii ] @xmath73 , and [ o iii]@xmath30 ( figure [ fig2 ] ) , along the fov of the residual data cube of m31 . this , together with the intrinsic difference between the stellar and the h@xmath0 emitting disks , suggests that the h@xmath0 emission is associated with a gaseous disk . however , we can not exclude the possibility that this emission is , at least in part , originated by stars . the discovery of an h@xmath0 emitting disk in the nucleus of m31 , reported here , allows an independent measurement of the mass of the central black hole in m31 , and , therefore , has a considerable importance for the studies of this object . this work is based on observations obtained at the gemini observatory . we thank fapesp for support under grants 2008/11087 - 1 ( r.b.m . ) and 2008/06988 - 0 ( t.v.r ) and also an anonymous referee for valuable comments about this letter . bacon , r. , et al . 2001 , a&a , 371 , 409 bender , r. , et al . 2005 , , 631 , 280 cappellari , m. 2002 , , 333 , 400 cid fernandes , r. , mateus , a. , sodr , l. , stasinska , g. , & gomes , j. m. 2005 , , 358 , 363 de rosa , g. , et al . 2011 , , 739 , 56 del burgo , c. , mediavilla , e. , & arribas , s. 2000 , , 540 , 741 dressler , a. , & richstone , d. o. 1988 , , 324 , 701 ferrarese , l. , & merritt , d. 2000 , , 539 , l9 gebhardt , k. , et al . 2000 , , 539 , l13 gonzalez , r. c. , & woods , r. e. 2002 , digital image processing ( 2nd ed . ; upper saddle river , nj : prentice - hall ) granato , g. l. , de zotti , g. , silva , l. , bressan , a. , & danese , l. 2004 , , 600 , 580 gltekin , k. , et al . 2009 , , 698 , 198 kormendy , j. 1988 , , 325 , 128 kormendy , j. , & bender , r. 1999 , , 522 , 772 lauer , t. r. , et al . 1993 , , 106 , 1436 lauer , t. r. , bender , r. , kormendy , j. , rosenfield , p. , & green , r. f. 2012 , , 745 , 121 light , e. s. , danielson , r. e. , & schwarzschild , m. 1974 , , 194 , 257 lucy , l. b. 1974 , , 79 , 745 menezes , r. b. , steiner , j. e. , & ricci , t. v. 2012 , in preparation mcconnell , n. j. , et al . 2011 , nature , 480 , 215 peiris , h. , & tremaine , s. 2003 , , 599 , 237 richardson , w. h. 1972 , josa , 62 , 55 salow , r. m. , & statler , t. s. 2004 , , 611 , 245 snchez - blzquez , p. , et al . 2006 , , 371 , 703 tremaine , s. 1995 , , 110 , 628

due to its proximity , the mass of the supermassive black hole in the nucleus of andromeda galaxy ( m31 ) , the most massive black hole in the local group of galaxies , has been measured by several methods involving the kinematics of a stellar disk that surrounds it . we report here the discovery of an eccentric h@xmath0 emitting disk around the black hole at the center of m31 and show how modeling this disk can provide an independent determination of the mass of the black hole . our model implies a mass of @xmath1 for the central black hole , consistent with the average of determinations by methods involving stellar dynamics , and compatible ( at 1@xmath2 level ) with measurements obtained from the most detailed models of the stellar disk around the central black hole . this value is also consistent with the @xmath3 relation . in order to make a comparison , we applied our simulation on the stellar kinematics in the nucleus of m31 and concluded that the parameters obtained for the stellar disk are not formally compatible with the parameters obtained for the h@xmath0 emitting disk . this result suggests that the stellar and the h@xmath0 emitting disks are intrinsically different from each other . a plausible explanation is that the h@xmath0 emission is associated with a gaseous disk . this hypothesis is supported by the detection of traces of weaker nebular lines in the nuclear region of m31 . however , we can not exclude the possibility that the h@xmath0 emission is , at least partially , generated by stars .

recently conceptual issues of fundamental physical significance have been raised on the aharonov - bohm ( ab ) effect @xcite . nonlocality and/or independent physical reality of the electromagnetic potentials seemingly implied by the ab effect have been questioned and debated in the recent literature @xcite . the original paper on the ab effect @xcite is written with remarkable clarity ; to get proper perspective on the current controversy it would be appropriate to emphasize salient features contained in this article following recent historical account @xcite and theoretical study on gauge invariance @xcite . arbitrariness in choosing the potentials in classical electrodynamics embodied in the gauge transformation @xmath0 and the consequent invariance of the lorentz - maxwell theory unambiguously demonstrate the fact that the potentials are just auxiliary mathematical tools in the classical theory . the principle of gauge invariance in quantum mechanics acquires new significance first recognized by fock in 1926 @xcite since the schroedinger wave function of the charged particle , let us assume it to be electron , gets multiplied by a local phase factor @xmath1 though the potentials are needed in canonical formalism of the classical theory they do not appear in the equation of motion ; in contrast , the electromagnetic potentials can not be eliminated in quantum theory of electron interacting with the electromagnetic fields . however gauge invariance and the unobservability of phase factors would seem to deny the physical reality to the potentials even in quantum theory . the ingenuity in the aharonov - bohm argument lies in the consideration of loop integrals in the phase of the wavefunction . for example , the most celebrated one given schematically in figure 2 of @xcite for the vector potential defines the ab phase shift to be @xmath2 where the total magnetic flux confined to a small region inaccessible to the interfering electron beams is @xmath3 this phase shift would manifest as a shift in the whole interference pattern of the electron beams relative to that in the absence of the flux or to that obtained varying the magnetic flux . physical interpretation of the ab phase shift involves two important observations made in @xcite . the role of the pure gauge field in the field - free region amounts to a multiply connected space @xmath4 where @xmath5 is a multi - valued scalar field . now the electron wavefunction ( 2 ) is no longer a single - valued function demanded in quantum mechanics ; a novel suggestion is made by the authors to split the electron beam into two components encircling the flux region on opposite sides with the corresponding wavefunctions being single - valued . the phase shift in each beam is calculated in terms of a path - dependent phase factor @xmath6 . if the beams are recombined for quantum interference then the relative phase equals the ab phase shift for a closed path i. e. the expression ( 3 ) . the ideal ab scheme consists of double - slit interferometer , perfectly shielded magnetic flux confined in a small region behind the wall between two slits making it inaccessible to electrons on both sides , and static vector potential . the main conclusions drawn by the authors could be summarized as follows . shift in the whole interference pattern due to flux is an observable effect . the ab effect has no classical analog since the quantum wave mechanical nature of electrons is crucial for the interference phenomenon . * c2 . * topological nature is implied by multiply connected space . * gauge invariance is not violated . the absence of magnetic field on the path of electron beams implies either one postulates nonlocal interaction that conflicts with relativity principle or attributes physical reality to the potentials in the quantum domain . authors prefer second option @xcite stating that,we are led to regard @xmath7 as a physical variable. the most important physics issue is whether the ideal ab phenomenon could be realized in the laboratory experiments . numerous experiments performed over past more than five decades have very nearly implemented the ab scheme , and demonstrated the ab effect . nevertheless there remains the scope for a genuine doubt regarding the perfect shielding of the flux region @xcite . in view of the lack of the quantitative estimates on the empirical data seriously challenging the observed ab effect @xcite and the beauty of topological interpretation the alternatives advocating local interaction of fields receive very little attention . speculative arguments , however abound in the literature ; admittedly philosophical beliefs and thought experiments do have their own importance as exemplified in the current controversy @xcite . in such cases the pitfalls involving misinterpretations have to be carefully addressed . a recent experiment using time of flight measurement of electrons in the ab setup @xcite seems to rule out classical and semiclassical explanations of ab effect . in contrast , physical import of single slit diffraction experiment with quantum point contacts @xcite is misleading @xcite since the classical correspondence of the ab phenomenon based on the erroneous description given by feynman @xcite is employed . in the light of the categorical assertion c1 made in @xcite and the recent experimental result reported in @xcite it becomes crucial to examine the role of classical limit and classical interpretations of the ab effect . another equally important issue that has emerged concerns the meaning on nonlocality in relation to c4 . nonlocal exchange of modular momentum as a physical mechanism for the ab effect proposed in 1969 @xcite has been recently emphasized and two distinct aspects , namely the continuous and instantaneous ones are proposed in @xcite . note that there is no experimental evidence for such a nonlocal process . the aim of the present paper is two - fold : to offer a thorough reappraisal on the classical perspectives and nonlocal issues , and make new contributions . it is shown that feynman approach proving classical - quantum eqivalence of the ab effect is not just puzzling @xcite it is incorrect . the importance of a path dependent phase factor that we term as fock - london - weyl ( flw ) phase distinct from the topological ab phase is discussed in this context . recent insights gained on the double slit quantum interference @xcite lead us to understand the controversy on classical limit of the ab phase measurement . the ab phase evolution raised in @xcite is analyzed pointing out subtle difference between geometry and topology of the phenomenon . the effect of pure gauge field is discussed in analogy to angular momentum holonomy suggested for the geometric phases in optics @xcite . the paper is organized as follows . the basic concepts on double slit quantum interference of electrons based on the actual experiments @xcite , and on the role of gauge invariance in quantum mechanics following @xcite constitute next section . a detailed analysis of feynman approach to the classical limit of ab effect is presented in section iii . section iv deals with the classical perspective on the ab phenomenon . brief review on the nonlocal modular momentum is followed by a suggested new approach to explain ab effect in section v. concluding remarks constitute the last section . proposed experimental test of the ab phase shift ( 3 ) discussed in @xcite mentions the use of a solenoid or magnetized whiskers to create confined magnetic flux in the double slit experiment for the coherent electron beams . it is also pointed out that it would be convenient to observe the effect by varying the magnetic flux and that the induced electric field could be made negligible . authors explicitly state that vector potential is time - independent , therefore , at least in this kind of ab phenomenon their argument invoking relativity in the last section as regards to the objection to local interaction is unjustified . we re - emphasize that the ab effect discussed in @xcite is a nonrelativistic quantum phenomenon . to avoid confusions it is necessary to understand the delicate issues of double slit experiment in quantum mechanics and the salient features of gauge invariance . * a. double slit quantum interference * young s double slit interference experiment in optics , initially considered as a thought experiment for de broglie matter waves has been realized in the laboratory experiments for decades . tonomura et al @xcite reported an important experiment demonstrating wave - particle duality and quantum mechanical nature of the observed interference on the electron beams . let us briefly recall the elementary considerations of two beam interference based on the division of wave - front of a single beam of light in double slit experiment . the interference pattern consists of equidistant bright and dark bands on the plane of the screen . for small separation d between the slits @xmath8 and @xmath9 and the distance between the slits and the screen l the optical path difference from @xmath8 and @xmath9 to a point p(x , y ) on the screen is calculated to be @xmath10 in the approximation of short wavelength @xmath11 and @xmath12 . the corresponding phase difference is @xmath13 the light intensity maxima and minima for the superposition of the beams occur at @xmath14 , @xmath15 and @xmath16 respectively . fresnel biprism is another method in which refraction divides a beam of light into two coherent components and their superposition resulting into the interference phenomenon . it is known that one can treat electron optics in analogy to light , and define refractive index for an electron beam passing through a purely electrostatic field ; tonomura et al @xcite make use of this in the biprism experiment . assuming incident electron beam to be a plane wave @xmath17 propagating in the z - direction , after traversing a region with electrostatic potential v(x , z ) it is transformed to @xmath18\ ] ] the details of the numerical values and the approximate potential function can be found in @xcite , here we give a short account on their significant results . the actual interference is built from a succession of single electrons over a period of time . note that it is a single electron wave passing through both slits that forms quantum probability interference . on the screen a detector records an electron as a localized particle i. e. electron wavefunction collapses to a definite position . it is crucial that which - path information for electron trajectory in the classical sense does not exist ; thus quantum mechanical wave nature of an electron is essential for observing interference . position - sensitive electron counter on the 2-dimensional screen records particle nature of the electron . interference experiments in classical optics do not have the enigmatic role of wave - particle duality , of course , single photon quantum optics double slit experiments have similar interpretational issues as noted above for electrons . einstein - podolsky - rosen ( epr ) incompleteness argument on the foundations of quantum mechanics in 1935 @xcite are no longer philosophical ; a large volume of experimental work with the advances in technology throws light on them . the bohr - einstein debate could be addressed avoiding mysterious or counter - intuitive descriptions of the past @xcite . two representative experiments @xcite are discussed here which relate with the double slit interference . historically bohr in a detailed response to epr argument @xcite conceived single slit diffraction and double slit interference thought experiments to elucidate his complementarity principle . in a double slit experiment the knowledge of the path of a particle passing through one of the slits would wipe out the interference fringes since the position of the particle is ascertained from a measurement of the momentum transfer to the diaphragm and the heisenberg uncertainty principle comes into play . note that in tonomura et al experiment @xcite which - path knowledge is not known it is only on the screen that particle position is determined ; this experiment does not test bohr s prediction it just conforms to quantum mechanics . in eichmann et al experiment @xcite bohr s assertion is proved though position - momentum uncertainty is not invoked . two @xmath19 ions trapped in a linear paul trap act as slits for photons . internal electronic levels of the ions provide which - path information in a polarization - sensitive detection . a linearly polarized photon scattered from the ions is either @xmath20-polarized or @xmath21-polarized . in the case of @xmath20-polarized scattered photon the ions electronic levels remain unchanged and hence the knowledge as to which ion scattered the photon is unknown : one should observe interference pattern that is confirmed in the experiment . if the scattered photon is @xmath21-polarized one of the ions undergoes electronic level transition resulting into which - path information : observed disappearance of the fringes in this case validates bohr s prediction . authors are careful to mention that correlation between the object and the measuring instrument explains the destruction of the interference fringes since heisenberg uncertainty relation for position and momentum is not needed . schmidt et al experiment @xcite attempts to explore momentum transfer to the slits and study its effect on the interference . free floating diatomic ions @xmath22 act as slits for helium atoms . both quantum mechanical and semi - classical calculations are carried out by the authors and compared with the experimental observations : only former agrees with experiments . an interesting study is also reported : the semiclassical model is modified such that the momentum transfer is equally divided between both nuclei in each collision . this calculation gives good agreement with the observations . authors term this modification as classical analog of coherent momentum transfer . curiously though the force acts only on one of the scattering centers , the momentum is transferred to both . its similarity to the process in ab effect deserves attention . to summarize : classical correspondence to double slit quantum interference is intricate issue involving wave - particle duality . * b. gauge invariance * gauge invariance in electromagnetism is a standard textbook subject , however there exist subtle points that sometimes get overlooked or unrecognized as noted by wu and yang @xcite . motivated by the ab effect the authors review the role of the phase defined by a loop integral @xmath23 and the phase factor @xmath24 emphasizing the fact that though the phase ( 9 ) contains more information than the phase factor @xmath25 the additional information is not measurable . they further argue that one is naturally led to the basis for the description of electromagnetism to a path - dependent ( or nonintegrable ) phase factor @xmath26 this quantity as compared to @xmath25 is more generally applicable to the dynamics of a charged particle in an electromagnetic field . since this aspect is intrinsic to the ideas of fock , london , and in a more concrete form that of weyl @xcite we suggest that the path - dependent phase factor be termed as flw phase . this terminology would have the advantage that unnecessary confusion created in the literature by using the term ab phase for expression ( 11 ) would be avoided . moreover it becomes clear that flw phase factor describes charged particle interaction with the electromagnetic field in a transparent manner . note that the gauge transformation assumes the form @xmath27 the preceding discussion shows that flw phase factor can be used to obtain schroedinger wavefunction for an electron in the presence of the electromagnetic field in terms of the free electron wavefunction @xmath28 @xmath29 alternatively the schroedinger equation for an interacting electron can be derived from the free - particle schroedinger equation expressing @xmath28 in terms of @xmath30 from eq.(13 ) . for example , the free - electron wavefunction in the expression ( 8) expressed in terms of @xmath31 gives the interaction term @xmath32 in the schroedinger equation : the operator @xmath33 on @xmath34 $ ] yields @xmath32 noting that the integration variable @xmath35 can be changed to @xmath36 , where @xmath37 , and the partial time - derivative on the integral finally just gives @xmath38 . one can obtain the schroedinger equation for an electron in arbitrary electromagnetic field from eq.(13 ) using the free - particle wave equation @xmath39 note that the classical limit makes sense for eq.(14 ) as we discuss below in section iv . in one of the earliest textbook treatments feynman presents an expository discussion on the ab effect @xcite , and proceeds to raise the question of the classical correspondence of the quantum significance of the vector potential stating that, ... if we look at things on a large enough scale it will look as though the particles are acted on by a force equal to @xmath40 the curl of @xmath41. it turns out that this statement is incorrect and feynman s analysis has several flaws ; understanding them throws light on the claimed classical explanations of the ab effect as we argue in the following . a summary of feynman s analysis is given first . the ideal ab double slit electron interference setup proposed in @xcite is depicted in fig.1 ( feynman s fig.15 - 7 ) . the ab phase shift ( 3 ) is nicely elucidated by feynman . a modification is suggested by him , shown here in fig.2 that corresponds to fig.15 - 8 in @xcite such that a constant weak magnetic field extends over a long narrow strip of width w in the region behind the slits . there are two lines of thought in feynman s approach to analyze this scenario : i ) calculation of quantum phase shift using ab proposal and relate it with the deflection of the electron trajectories in the classical sense , and ii ) calculation of angular deflection of the electron motion in the impulse approximation , and relate it with the quantum phase shift using de broglie relation . ignoring for the moment the fundamental conceptual distinction between classical and quantum descriptions , let us reproduce the steps that go into the feynman s analysis . in the first step ab phase shift is obtained for the flux @xmath42 , and the shift in the interference pattern is easily calculated using eqs . ( 3 ) and ( 7 ) to be @xmath43 note that there is unstated assumption in this derivation that the magnetic field in the region adjacent to the slits does not affect electron motion though electrons actually traverse this region . this shift is rewritten as an angular deflection of electron trajectory @xmath44 in the second step treating electrons as newtonian particles the lorentz force due to the magnetic field adjacent to the slits that was neglected in the first step is considered . in the impulse approximation that this force lasts for a time @xmath45 the change in the transverse momentum is obtained to be @xmath46 and the corresponding angular deflection is given by @xmath47 it is straightforward to verify that de broglie relation @xmath48 and substitution of ( 17 ) in ( 18 ) immediately give @xmath49 remarkably the enclosed flux @xmath42 responsible for the ab phase shift in the first step has no role in the second step . the exact equivalence ( 19 ) seems impressive enough for feynman to conclude classical and quantum equivalence . the here highlighted crucial unjustified implicit assumptions in feynman s approach show that the equivalence ( 19 ) is physically meaningless : the physical situations analyzed in two steps are entirely different . even if we set the enclosed flux @xmath42 equal to zero , the value of @xmath50 does not change because of its different origin . to recognize the fallacy involved more clearly we split the feynman s modified setup ( fig.2 ) into two components : fig.3 represents the analysis given in the first step , and fig.4 that in the second step in feynman s approach . the incorrect implicit assumptions needed in fig.2 do not arise here . following the first step expression ( 15 ) is unambiguously obtained as the ab phase shift subject to the usual assumption of perfect shielding . derivation of the angular deflection ( 16 ) has only symbolic value as one can not use the classical notion of electron trajectories . recall the discussion in the preceding section that tonomura et al experiment specifically demands no which - path information otherwise the interference pattern gets destroyed . the recorded positions on the screen do represent localized particle nature of the electrons but these are not determined by the classical trajectories . thus the angular deflection @xmath51 has no physical significance . interestingly one can seek classical correspondence for the physical case shown in fig.4 , as well as one can treat this case in a purely classical manner . feynman s derivation of angular deflection using lorentz force would be correct but then it describes classical motion of electron in a magnetic field . mere use of de broglie relation to transform the angular deflection @xmath50 to the physical case of quantum interference phase shift is incorrect . quantum mechanically one can treat this problem as that of electrons having local interaction with the magnetic field : one can use flw phase factor for its description or equivalently use schroedinger equation ( 14 ) setting @xmath52 . in either case there exists a meaningful method to seek classical correspondence . for example , section 24 in schiff @xcite obtains the classical lorentz force and classical newtonian equation of motion from eq.(14 ) ; the classical correspondence is valid only if the electron wavepacket is well localized and ehrenfest theorem is applicable . naturally the question arises if the interference fringes could still be observed in the double slit experiment of fig.4 when this classical limit is taken . feynman does not address this important question . in a simple picture one would expect that in the physical situation that represents classical electron motion the quantum features would be lost . boyer @xcite in an early comment on @xcite questions the classical analog of the ab effect . we point out a simple but profound point in this connection . in the double slit experiment the copenhagen interpretation @xcite involves complementarity principle or heisenberg uncertainty relation to explain the quantum mechanical nature of the interference phenomenon . if there is a momentum transfer during the passage through a slit then it has to be less than @xmath53 for the fringes to exist . now the transverse momentum shift in eq.(17 ) must satisfy this inequality @xmath54 since the smallest aharonov - bohm flux unit is @xmath55 , the constraint ( 20 ) puts severe limitation on the physical validity of feynman s second step as the fringes would be wiped out in the presence of the magnetic field . the classical limit of a quantum system is said to be the limit @xmath56 . in some cases it is known that large quantum numbers have a classical correspondence ; the simple harmonic oscillator is one of the well known examples @xcite . in the ab effect there are intricacies due to two typical quantum characteristics : the quantum mechanical significance of the vector potential , and quantum interference . according to @xcite the vector potential can not be dispensed with in quantum theory , and it has observable significance in the ab phase . classical perspective could be put forward that the vector potential may have physical significance representing the interaction field momentum @xmath57 in view of its appearance in the canonical momentum @xcite . consider a solenoid and an electron system ; even for a force - free situation electron may impart rotation to the solenoid . however the classical significance of the vector potential can not be used to offer a classical explanation of the ab effect ; trammel is careful regarding this crucial point @xcite . the reason is that the observable significance of the ab effect depends on the quantum phase ( 9 ) or the ab phase factor ( 10 ) , on the other hand classically one has the following quantity different than the phase @xmath58 the classical expression of the energy interference term derived in @xcite and the time - lag effect @xcite suffer from the defect or inconsistency that the classical quantities are interpreted as quantum phases simply by putting @xmath59 . note that the deterministic trajectory does not make sense in quantum theory . boyer does discuss complementarity principle in the double slit experiment , however the technical limitations of the then reported experimental tests of the ab effect seem to have led him to argue that the complementarity is not relevant in the ab phase . in the light of tonomura et al experiment @xcite and modern developments on the double slit quantum interference @xcite briefly reviewed in section ii the role of wave - particle duality / complementarity in the ab phase shift can not be ruled out unless alternative interpretation other than the copenhagen interpretation , e. g. the statistical interpretation is adopted . local field interaction aims at the complete elimination of the vector potential the way one could do in classical theory . unfortunately quantum formalism purely in terms of the electromagnetic fields has not been successfully developed so far . there are nevertheless interesting ideas to understand ab effect as a local effect . vaidman s thought experiment @xcite and the example suggested in the prelude in the comment @xcite on closer examination reduce to the feynman s analysis . the postulated magnetic field shown in fig.4 would mimic the fields in @xcite . vaidman attributes the local field due to the change in the magnetic flux caused by the encircling electron , while aharonov et al suggest assuming a constant magnetic field . in either case the role of the mechanical aspect of the solenoid or that of vaidman s construction is not important in the ab phenomenon . in contrast , kang @xcite makes an important point that the relativistically moving electron in the real experiment @xcite makes the ideal shielding questionable ; however for a convincing argument there has to be a detailed empirical comparison with the experimental data of @xcite . it is also somewhat artifical to treat fluxon as a particle with mass in kang s formalism . the detailed discussion in the preceding section shows that the main problem with feynman s analysis is that the applicability of trajectory description and local field interaction amounts to the loss of the quantum phase informaion . the debate @xcite does not take notice of this most significant aspect . recent becker - batelaan experiment @xcite assumes great significance in the context of our arguments : the experiment demonstrates the absence of back - action and approximate dispersionless forces in the ab - like setup with macroscopic toroid , and the quantum interference signifying ab effect can not be observed in this setup . we suggest that the import of becker - batelaan experiment is that in the force - free region local interactions have no classical / semiclassical effect . the classical correspondence to the ab effect could be approached from a different point of view . recall that in the old quantum theory bohr - sommerfeld quantization played an important role . the most important distinguishing feature of this procedure is that the physical picture of the classical trajectories and orbits is still valid . applying bohr - sommerfeld quantum rule to the canonical momentum we are led to the aharonov - bohm flux quantization in the expression ( 4 ) @xmath60 for a large value of the quantum number ( integer n ) one would expect the classical limit corresponding to ( 21 ) . the flux unit is @xmath61 . a possible observable consequence could be the deflection of the electron trajectories in the force - free region which however is indicated to be absent in the experiment@xcite . we suggest the experiment with intense magnetic fields i. e. large magnetic flux , and also possibly the consideration of the transverse forces noted in @xcite . aharonov and his collaborators emphasizing nonlocality as a quantum feature with no classical analog have been trying to develop a modular variable theory since 1969 @xcite . in the recent paper the role of modular velocity is analyzed in the so called instantaneous aspect of the ab effect @xcite . a continuous physical quantity modulo its basic unit is defined to be a modular variable . modular momentum is defined as @xmath62 where @xmath63 , and l is a constant spatial length . transverse modular velocity is @xmath64 here m is the electron mass . this particular variable ( 24 ) is taken to explain the ab effect associated with the vector potential @xmath65 here @xmath66 is the heaviside step function . from a simple analysis of this example the argument is put forward that the relative ab phase shift occurs at the instant the two wavepackets cross the x - axis on which the vector potential ( 25 ) is nonzero . the above point with great clarity had been earlier made in @xcite where the question was raised whether the ab phase evolved continuously . we identify two important issues from this line of thinking . when does the ab effect occur ? what is the physical mechanism responsible for the ab effect ? we suggest that geometry and topology provide answer to the first question . to address the second question we propose that the angular momentum holonomy conjecture @xcite is applicable to the ab effect . topological aspect was already recognized in the original paper @xcite , however the rich and complex topological structure involved continues to offer new avenues . topology is a study in the global and the continuum though discrete invariants turn out to be more useful in physics . a point charge in electrostatics is the simplest example : the flux through any closed surface of arbitrary shape and area surrounding the charge is constant ; it represents the topology of a punctured 3-space . in the ab effect a circle in space encircling the magnetic flux relates with the u(1 ) phase of the quantum state space of the electron . in both examples , the local or the geometrical aspect too is of significance when the path dependent quantity is considered . a nontrivial geometry , for example , the surface of a 2-sphere gives rise to the change in the direction of a vector parallel transported from one point to another , and termed holonomy for a closed circuit . geometric phases in optics are the manifestations of this holonomy @xcite . altering the 2-sphere scenario with a magnetic monopole at its centre wu - yang analysis @xcite shows that the path dependent phase factor becomes undefined when the path crosses the singularity . in a construction having singularity - free potentials defined in two regions the dirac quantization condition emerges as a topological property . analogous to the monopole problem we associate geometrical aspect contained in the flw phase factor for a path on the circle such that the polar angle @xmath67 where the phase evolution is continuous . in the presence of the magnetic flux at the origin the topology is that of a 2-space with the origin removed . mathematically following @xcite single valued wavefunctions on the two paths encircling the singularity could be used to calculate the phase difference or one could treat the problem with a multivalued wavefunction with well defioned observables as shown by martin @xcite . thus the topological aspect is connected to the transition point in the path winding the circle . to see the topological aspect more clearly let us note that the flw phase factor is calculated by dividing the path into segments and integrating along them , however for the closed path the gauge invariance brings the gauge function @xmath68 into the picture at the crossing of the singularity or we say that a lift of the loop in the covering space where the real line is the covering space of the circle . martin shows the importance of the irreducible representations of the euclidean group e(2 ) and its covering groups parameterized by two real numbers for the ab effect ; a lucid and comprehensive account is given by kastrup @xcite . recognizing the necessity of a curved geometry ( i. e. circular arc ) and a topological defect causing a discontinuity as two elements in the ab effect we argue that the field angular momentum exchange similar to that proposed for the geometric phases in optics @xcite shown to have experimental support @xcite also provides a physical mechanism for this effect . though the idea of modular momentum exchange @xcite is interesting @xcite the role of angular momentum appears more natural . in fact , aharonov and cohen discuss rotating frame of reference and geometric effect in a recent paper @xcite . in this connection first we recall an elegant example @xcite . the hamiltonian of a two dimensional isotropic oscillator with unit mass @xmath69 where overdot denots the time derivative , is invariant under the rotation of the axes by an angle @xmath51 @xmath70 @xmath71 the constant of motion for this symmetry is obviously the angular momentum . if @xmath51 is made time dependent then @xmath72 is not invariant under the rotation , however a term @xmath73 added to @xmath72 makes the total hamiltonian invariant provided the frequency changes as @xmath74 here @xmath73 is given by @xmath75 the change in the angle variable after a period of time t from ( 29 ) gives @xmath76 the second term in ( 31 ) was interpreted to have topological origin in @xcite . this simple example illustrates the role of gauge invariance ( 29 ) and angular momentum exchange via ( 30 ) . martin s mathematical model @xcite would make the argument cogent . rotation ( 27)-(28 ) and translation for the group e(2 ) and its covering groups are defined by two real numbers @xmath77 . for e(2 ) itself @xmath78 , and for a rational number @xmath79 p and q being integers with no common divisors it is a q - fold covering group of e(2 ) . the self - adjoint generators of rotations and translations satisfy the lie algebra @xcite . of particular interest for the present discussion is the rotation generator @xmath80 in quantum mechanics for the state space we require a hilbert space on the circle . a hilbert space with square - integrable functions and a well defined scalar product can be constructed using the orthonormal basis vectors @xmath81 multiplying ( 32 ) by @xmath59 and interpreting it as angular momentum operator it is easily verified that @xmath82 the functions ( 33 ) are the eigenvectors of the angular momentum operator with noninteger eigenvalues @xmath83 . the nontrivial topology has the origin in the homotopy group of the nonsimply connected circle @xmath84 the number @xmath85 determines the different irreducible representations in the same hilbert space defined by the basis vectors ( 33 ) . using the unitary transformations on ( 33 ) separate hilbert spaces for each @xmath85 could be constructed , and the operator ( 32 ) becomes independent of @xmath85 @xmath86 the basis vectors now have a constant multiplicative phase factor @xmath87 . note that the eigenvalues do not change under the unitary transformation . it is remarkable that the eigenspectrum ( 34 ) of the angular momentum operator resembles the definition of a modular variable , e. g. the modular momentum defined by eq.(23 ) . a new interpretation of the ab effect emerges in terms of the modular angular momentum exchange . consider the flux line following @xcite in cylindrical coordinates and coulomb gauge the vector potential is @xmath88 the dynamics of an electron is described by @xmath89 @xmath90\ ] ] where @xmath91 . the casimir invariant is just @xmath92 , and the angular momentum operator becomes @xmath93 a simple calculation comparing ( 40 ) with ( 32 ) and noting the phase factor @xmath94 the expression for the phase shift @xmath95 is exactly the ab phase shift ( 3 ) . to gain further insight into the role of angular momentum in the ab effect we recall the arguments in @xcite the crucial difference between the kinetic and canonical momenta in the classical point dynamics manifests in a general form in the field theory @xcite . for free electromagnetic field the canonical angular momentum density tensor @xmath96 derived as a noether current for the invariance of the action under infinitesimal proper homogeneous lorentz transformation is not gauge invariant . a gauge invariant tensor @xmath97 can be constructed that differs from the canonical one by a pure divergence term @xmath98\ ] ] in the standard field theory @xcite it is assumed that the divergence term vanishes in the limit of rapidly falling fields at infinity . in a nontrivial geometry this assumption may no longer remain valid and the surface terms could result into what we termed as angular momentum holonomy @xcite . the 4-vector potential term in the expression ( 41 ) is interesting @xmath99 its 3-space analogue is just the term @xmath100 . this term also arises from the expression for the kinetic momentum @xmath101 considered by trammel @xcite . the angular momentum operator ( 40 ) is also related with the gauge covariant derivative . therefore the idea of modular angular momentum exchange seems quite logical . instead of a flux line having singularity , let us consider a solenoid with finite radius r and a constant magnetic field @xmath102 along z - axis . the physical situation for this case is represented by fig.1 . electrons move in the field - free region , i. e. outside the solenoid where the magnetic field is zero . the vector potential inside and outside regions is @xmath103 @xmath104 it can be seen that the vector potential in the field - free region @xmath105 is a pure gauge potential @xmath106 the quantity @xmath107 for the two regions is calculated to be @xmath108 @xmath109 where @xmath110 . interestingly using the flux quantum unit @xmath55 the expression ( 47 ) can be transformed to the modular form @xmath111 to picture the physical process of modular angular momentum exchange first it has to be realized that electrons in the field - free region do not undergo velocity change . however the motion on a curved trajectory with constant velocity brings the role of angular momentum and apparent forces . now the pure gauge potential , in general , could be viewed as a fictitious magnetic field @xcite using a vector identity . let @xmath112 then we have the following equality @xmath113 expression ( 49 ) provides alternative argument to explain the effect of the confined magnetic flux on the electron motion in the field - free region since the vector potential ( 45 ) is a pure gauge potential . recent experiment @xcite and the continued efforts seeking the classical limit of the ab effect by many authors motivated us to present a comprehensive discussion on the classical perspective . a critique on the feynman approach is offered and it is concluded that the ab effect has no classical analog consistent with the original claim @xcite . the idea of modular variables @xcite seems interesting , however based on topological arguments we suggest modular angular momentum exchange as a physical mechanism for the ab effect . the present work provides support to the speculation that potentials are fundamental in the microscopic domain . a typical gauge chosen in @xcite described by the vector potential ( 25 ) to explain the instantaneous aspect of the ab effect is not discussed here for the lack of the definitive result . the issue of canonical versus kinetic momentum and angular momentum is a delicate one having profound physical import as pointed out in the recent literature @xcite and the references cited therein . the present work may prove to be useful in this context . 99 w. eherenberg and r. e. siday , proc . london , section b 62 , 8 ( 1949 ) y. aharonov and d. bohm , phys . 115 , 485 ( 1959 ) l. vaidman , phys . a 86 , 040101(r ) , ( 2012 ) k. kang , phys . a 91 , 052116(2015 ) y. aharonov , e. cohen and d. rohrlich , phys . a 92 , 026101 ( 2015 ) l. vaidman , phys . a 92 , 026102 ( 2015 ) y. aharonov , e. cohen and d. rohrlich , phys . a 93 , 042110 ( 2016 ) j. d. jackson and l. b. okun , rev . 73 , 663 ( 2001 ) t. t. wu and c. n. yang , phys . d 12 , 3845 ( 1975 ) a. tonomura , n. osakabe , t. matsuda , t. kawasaki , j. endo , s. yano and h. yamada , phys . rev . 56 , 792 ( 1986 ) ; a. tonomura et al , phys . 48 , 1443 ( 1982 ) m. becker and h. batelaan , epl 115 , 10011 ( 2016 ) p. khatua , b. bansal and d. shahar , phys . 112 , 010403(2014 ) ; 113 , 158902(2014 ) s. c. tiwari , phys . 113 , 158901 ( 2014 ) r. p. feynman , r. b. leighton and h. sands , the feynman lectures on physics ( addison - 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recent literature on the aharonov - bohm effect has raised fundamental questions on the classical correspondence of this effect and the physical reality of the electromagnetic potentials in quantum mechanics . reappraisal on feynman s approach to the classical limit of ab effect is presented . the critique throws light on the significance of quantum interference and quantum phase shifts in any such classical correspondence . detailed analysis shows that feynman arguments are untenable on physical grounds and the claim made in the original ab paper that this effect had no classical analog seems valid . the importance of nonintegrable phase factor distinct from the ab phase factor , here termed as fock - london - weyl phase factor for the historical reasons , is underlined in connection with the classical aspects / limits . a topological approach incorporating the physical significance of the interaction field momentum is proposed . a new idea emerges from this approach that attributes the origin of the ab effect to the exchange of modular angular momentum .

over decades theoretical studies of the single - particle spectral properties of metallic one - dimensional ( 1d ) correlated electron systems so - called luttinger liquids ( lls)were ahead of the experimental attempts to find or synthesize appropriate quasi 1d materials and perform spectroscopy on them . in fact , while at the begining of the 1990 s a clear picture of the basic spectroscopic properties of translational invariant lls was established ( for reviews see e.g. refs . @xcite , @xcite and @xcite ) this period witnessed the first serious attempts to experimentally verify the specific spectroscopic signatures of lls @xcite . these are the ( i ) low - energy power - law suppression of the local spectral function @xmath0 for energies @xmath1 close to the chemical potential @xcite with @xmath2 depending on the two - particle interaction as well as ( ii ) the appearance of two dispersing features in the momentum resolved spectral function @xmath3 ( spin - charge separation ) @xcite instead of a single quasi - particle peak of a fermi liquid . for finite temperatures @xmath4 the suppression of the spectral weight as a function of @xmath1 is cut off by @xmath4 and one finds the scaling behavior @xmath5 with a @xmath2-dependent scaling function @xmath6 in which the two energy scales @xmath1 and @xmath4 only enter via their ratio @xcite . these results were exclusively obtained using bosonization within the tomonaga - luttinger ( tl ) model @xcite . using the modern language of renormalization group ( rg ) methods the ( translational invariant ) tl model is the exactly solvable effective low - energy fixed point model for a large class of metallic 1d correlated electron systems the lls @xcite . it thus plays the same role as the free fermi gas in fermi liquid theory . the model has two strictly linear branches of right- and left - moving fermions and two - particle scattering is restricted to processes with small momentum transfer @xmath7 , with the fermi momentum @xmath8 . these processes as well as the kinetic energy can be written as quadratic forms of the densities of right- and left - moving fermions which obey bosonic commutation relations . in most calculations in addition the momentum dependence of the low - momentum scattering processes @xmath9 and @xmath10 are neglected and momentum integrals are regularized in the ultraviolet introducing a cutoff by hand ( for an exception see ref . one can extend the resulting _ scale - free , _ field theoretical model by allowing for additional two - particle scattering processes . these turn out to be rg irrelevant in a wide parameter regime @xcite . the most important of these processes is the so - called @xmath11-process ( in the g - ology classification @xcite ) with momentum transfer @xmath12 between two scattering fermions of opposite spin . in some of the early experiments on 1d chains these were obviously interrupted by local impurities @xcite . a simple model of an _ inhomogeneous _ ll is the open boundary analog of the tl model . interestingly , a ll is very susceptible towards single - particle perturbations with momentum transfer @xmath12 @xcite and on asymptotically low energy scales even a single weak impurity has the same effects on the spectral properties as an open boundary @xcite . triggered by this theoretical insight and the early experiments , the spectral properties of the open boundary analog of the tl model were studied @xcite . the local spectral function close to the boundary shows power - law behavior as a function of @xmath1 but with an exponent @xmath13 different from the bulk one @xmath2 . as in the translational invariant case in this model only those low - energy scattering terms are kept which can be written as quadratic forms in bosonic densities . only recently it was shown that a large class of further two - particle processes appearing in a 1d system with an open boundary are indeed rg irrelevant @xcite . the latest scanning tunneling spectroscopy ( sts ) and photoemission spectroscopy ( pes ) measurements on different classes of 1d metallic systems @xcite impressively demonstrated that the experiments caught up and more refined questions must now be answered by theory . important ones are : how do effects which are not captured by the low - energy fixed point model , such as the momentum dependence of the two - particle interaction and the nonlinearity of the single - particle dispersion influence the spectral functions ? what is the energy scale of a given microscopic model on which the low - energy ll physics sets in ? how do scaling functions for lattice models look like in detail ? here we shed some light on the last two questions and briefly comment on the first one . it is widely believed that neglecting the momentum dependence of the interaction and regularizing momentum integrals in the ultraviolet by hand has no effect on the low - energy physics of ll s . this is indeed correct if all energy scales are sent to zero , that is for @xmath14 and @xmath15 : at small @xmath1 the spectral properties are unaffected by the details of the momentum dependence of the @xmath16 s . however , if @xmath3 as a function of @xmath1 is studied at fixed @xmath17 , as it is usually done in angular resolved pes , details of the momentum dependence of the interaction matter . this was investigated in ref . an overview on the effects of the nonlinearity of the single - particle dispersion can be found in the very recent review ref . @xcite . this paper is organized as follows . in sect . [ sec : scalfun ] we compute the local spectral function of the translationally invariant _ and _ the open boundary continuum tl model using bosonization . we show that both display scaling in @xmath18 and that the scaling functions have the same analytic form . we next compute the spectral function of the extended hubbard model on the lattice close to an open boundary as a function of energy , temperature and position in sect . [ sec : exthubbard ] . for this an approximate method is used which is based on the functional rg approach @xcite . it is devised for weak to intermediate two - particle interactions . in particular , we concentrate on inhomogeneous lls as the boundary exponent @xmath13 characterizing the spectral function close to an open boundary is _ linear _ in the two - particle interaction while the bulk exponent is _ quadratic _ ( see below ) . varying the microscopic parameters of the extended hubbard model we can tune the strength of the different scattering processes and thus study the crossover between nonuniversal behavior and the low - energy ll physics . we perform a scaling analysis of the spectral function as a function of @xmath1 and @xmath4 and show that the spectral weight close to the boundary follows the bosonization prediction within the universal low - energy regime . the position dependence of the spectral function is characterized by oscillatory behavior and a power - law envelope function in accordance with the result for the tl . interestingly , we additionally find a phase shift which is proportional to the two - particle interaction and not accounted for in the standard bosonization procedure . we summarize our results in sect . [ sec : summary ] and briefly comment on how spin - orbit interaction and several bands crossing the fermi surface both being potentially important effects in recent experiments influence the single - particle spectral functions . in this section we derive closed analytic expressions for the single - particle green function and the related local spectral function of the tl model with and without an open boundary at finite @xmath4 . we then closely inspect the scaling form of the spectral functions . in field theoretical notation ( see e.g. ref . @xcite ) the hamiltonian density of the tl model in spin - charge separated form reads @xmath19 , \label{eq : hamdens}\end{aligned}\ ] ] with the canonical bose fields @xmath20 and their dual fields @xmath21 . within the tl model the charge and spin velocities @xmath22 as well as the ll parameters @xmath23 are free parameters . if the model is used to describe the low - energy physics of an underlying microscopic model they become functions of the corresponding model parameters and the band filling @xcite . for spin - rotational invariant models on which we focus in the present and the next section @xmath24 . for repulsive interactions @xmath25 while @xmath26 in the attractive case . here we exclusively consider the former . the field operator @xmath27 annihilating an electron with spin direction @xmath28 at position @xmath29 is decomposed into a right- and a left - moving part @xmath30 the imaginary time fields @xmath31 and @xmath32 are bosonized according to @xmath33 where the klein factors @xmath34 satisfy anticommutation rules @xmath35 and @xmath36 . the fields @xmath37 and @xmath38 are the chiral components of @xmath20 and @xmath21 , @xmath39 for the translational invariant tl model the hamiltonian follows by integrating the density eq . ( [ eq : hamdens ] ) over @xmath40 . the tl model with an open boundary is obtained by integrating the density eq . ( [ eq : hamdens ] ) over @xmath41 and employing the boundary condition @xmath42 for the fermionic and @xmath43 for the bosonic fields , respectively . we are here interested in the imaginary time - ordered single - particle green function @xmath44 where @xmath45 denotes the expectation value in the canonical ensemble . from the decomposition eq . ( [ eq : lowenergy ] ) it follows that @xmath46 where , for example @xmath47 . as we are aiming at the local spectral function , we eventually set @xmath48 . for the translational invariant tl model the left- and right - moving fermion fields are independent and thus @xmath49 . in the presence of an open boundary the above mentioned boundary conditions imply @xmath50 . in this case and after setting @xmath48 the cross terms are characterized by a fast spatial oscillation with frequency @xmath12 . using standard methods ( see e.g. ref . @xcite ) one obtains for the green function of a translational invariant system ( @xmath51 , @xmath52 , @xmath53 , @xmath54 , @xmath55 ) @xmath56 } \\ & & \times \frac{1}{\sin^a\left [ \frac{\pi}{v_{\rm c } \beta } \left ( v_{\rm c } \tau - { \text{i}}r \right ) \right ] } \ , \frac{1}{\sin^b\left [ \frac{\pi}{v_{\rm c } \beta } \left ( v_{\rm c } \tau + { \text{i}}r \right ) \right ] } \ , \nonumber \\ \mbox { } \hspace{-1.5 cm } g^{ll}_{\sigma\sigma'}(\tau , x_1,x_2 ) & = & g^{rr}_{\sigma\sigma'}(\tau , x_2,x_1 ) \end{aligned}\ ] ] and for the case with boundary @xmath57_{\rm b}(\tau , x_1,x_2 ) & = & g^{rr}_{\sigma\sigma ' } ( \tau , x_1,x_2 ) \ , \left\ { \frac{\sinh\left ( \frac{2 \pi}{v_{\rm c } \beta } x_1 \right ) \sinh\left ( \frac{2 \pi}{v_{\rm c } \beta } x_2 \right ) } { \sin\left [ \frac{\pi}{v_{\rm c } \beta } \left ( v_{\rm c } \tau - 2 { \text{i}}r \right ) \right ] \sin\left [ \frac{\pi}{v_{\rm c } \beta } \left ( v_{\rm c } \tau + 2 { \text{i}}r \right ) \right ] } \right\}^c \\ \mbox { } \hspace{-2.5 cm } \left[g^{ll}_{\sigma\sigma'}\right]_{\rm b}(\tau , x_1,x_2 ) & = & \left [ g^{rr}_{\sigma\sigma'}\right]_{\rm b}(\tau , x_2,x_1 ) \end{aligned}\ ] ] the cross terms @xmath58_{\rm b}$ ] and @xmath59_{\rm b}$ ] are equal to @xmath60_{\rm b}$ ] and @xmath61_{\rm b}$ ] after interchanging @xmath62 . the appearing exponents are given by @xmath63 the main steps to obtain the local spectral function from the imaginary time green function are the analytic continuation @xmath64 followed by fourier transformation with respect to @xmath65 . mathematically the real part @xmath66 corresponds to the ultraviolet cutoff introduced to regularize momentum intergrals . from an experimental perspective it can be considered as the resolution of the setup ( at @xmath67 ) . for the translational invariant model one obtains ( @xmath48 ) @xmath68 \right|_{\tau \rightarrow { \text{i}}t+ \delta } \nonumber \\ \mbox { } \hspace{-2.cm } & = & \frac{1}{\pi^2 } \left ( 1 + e^{-\beta \omega } \right ) \left ( \frac{\pi}{v_{\rm c } \beta } \right)^{a+b } \left ( \frac{\pi}{v_{\rm s } \beta } \right)^{1/2 } \left . \int_{-\infty}^\infty dt \ , \frac { e^{{\text{i}}\omega t}}{\sin^{a+b+1/2 } \left ( \frac{\pi \tau}{\beta } \right ) } \right|_{\tau \rightarrow { \text{i}}t+ \delta } \nonumber \\ & = & \frac{4 \pi^{a+b+1/2}}{\pi^2 v_{\rm c}^{a+b } v_{\rm s}^{1/2 } } \ , t^\alpha \ , s_\alpha(\omega / t ) , \label{eq : rhotransinv}\end{aligned}\ ] ] with @xmath69 } \cosh{\left ( \frac{u}{2 } \right ) } \ , \left| \gamma \left ( \frac{1+\gamma}{2}+{\text{i}}\frac{u}{2 \pi}\right ) \right|^2 , \label{eq : sdef } \\ & & \alpha = a+b-\frac{1}{2 } = \frac{1}{4 } \left ( k_{\rm c } + \frac{1}{k_{\rm c } } -2 \right ) . \label{eq : diveresedeffs } \end{aligned}\ ] ] the _ position independent _ local spectral weight of the translational invariant tl model thus shows _ scaling behavior _ : @xmath70 is a function of @xmath18 only . the amplitude of the scaling function depends on @xmath71 and @xmath72 , which in turn are functions of the interaction strength , while its shape is given by @xmath72 only . this result was first derived in ref . a similar expression was later used to describe transport properties of lls @xcite . taking the @xmath73 limit of eq . ( [ eq : rhotransinv ] ) one obtains the well known power - law suppression of the spectral weight @xcite @xmath74 for @xmath75 . for fixed small @xmath76 this is cut off by temperature and @xmath77 saturates for @xmath78 . for the energy set to the chemical potential , that is @xmath79 , one finds a power - law suppression of @xmath77 for @xmath73 ( see eq . ( [ eq : rhotransinv ] ) ) : @xmath80 . from studies of microscopic models it is known that ( see e.g. ref . @xcite ) @xmath81 ^ 2\right ) , \label{eq : kexp}\end{aligned}\ ] ] where @xmath82 is a measure of the two - particle interaction and @xmath83 is a scale which depends on the other model parameters . using eq . ( [ eq : diveresedeffs ] ) one thus obtains @xmath84 for the exponent characterizing the low - energy behavior of the local spectral function of the translational invariant tl model . verifying the power - law suppression of the spectral weight as a function of @xmath4 and @xmath1 with the same exponent @xmath2 as well as the scaling property of measured sts and/or pes data provide strong indications that the system under investigation is indeed a ll @xcite . it was very recently argued that these characteristics are still not unique to lls as other mechanisms than 1d electronic correlations might lead to similar behavior @xcite . we therefore suggest further consistency checks by in addition measuring spectra close to the end points of cut 1d chains . the local spectral function becomes position @xmath29 dependent when considering a chain with an open boundary . in this case @xmath85 has three contributions @xmath86 where the first follows from fourier transforming @xmath87 and the last two from transforming @xmath88 and @xmath89 , respectively ( @xmath48 ) . following the same steps as above we obtain @xmath90 with @xmath91 and @xmath92 where @xmath93 in the limit @xmath73 these expressions simplify to the ones given in refs . @xcite and @xcite . for distances from the boundary beyond the thermal length @xmath94 , that is @xmath95 we expect @xmath96 to become equal to @xmath85 of eq . ( [ eq : rhotransinv ] ) ( exponentially fast ) . that this is indeed the case follows from @xmath97 eqs . ( [ eq : summe ] ) , ( [ eq : term1 ] ) , ( [ eq : term2 ] ) and ( [ eq : sdef ] ) . we thus end up with @xmath98 we next consider the limit @xmath99 , that is the local spectral function close to the open boundary . then @xmath100 interestingly the remaining integral has the same form as the one appearing in the second line of eq . ( [ eq : rhotransinv ] ) but with @xmath101 replaced by @xmath102 . for fixed @xmath29 close to the boundary @xmath103 thus displays scaling with the _ same _ scaling function as the one found in the bulk but @xmath2 replaced by @xmath104 explicitely one obtains @xmath105 with eq . ( [ eq : kexp ] ) the boundary exponent @xmath13 eq . ( [ eq : alphabdef ] ) has , in contrast to the bulk one @xmath2 , a contribution _ linear _ in the interaction @xmath106 and one finds @xmath107 . to show that for fixed @xmath29 close to the boundary @xmath108 indeed follows the same scaling function as in the bulk ( with @xmath109 ) we still have to analyze @xmath110 for @xmath99 . in the simplest approximation we neglect the @xmath111-dependence in the integral eq . ( [ eq : gdef ] ) and obtain @xmath112 using eq . ( [ eq : summe ] ) this completes our proof that for fixed @xmath113 with @xmath114 given in eq . ( [ eq : sdef ] ) . showing the consistency of the scaling of spectra measured in the two spatial regimes is within reach of the latest sts experiments @xcite . combined with a consistency check of the two exponents @xmath2 and @xmath13 , which both depend on @xmath72 only and which was already achieved in ref . @xcite ( see also sect . [ sec : summary ] ) , this would provide a stringent experimental verification of ll physics . one can _ improve _ the analysis of @xmath110 close to the boundary by keeping the phase factor @xmath115 of the integral eq . ( [ eq : gdef ] ) . ) on @xmath116 , @xmath111 , and the model parameters are irrelevant for the regimes studied here . ] a numerical evaluation of the integral shows that @xmath117 is a function of @xmath72 and @xmath118 with @xmath119 . taking all terms together we find @xmath120 \right\ } , \label{eq : rhocompl}\end{aligned}\ ] ] where the overall amplitude @xmath121 depends on @xmath72 , @xmath122 and @xmath123 but _ not _ on the variables @xmath1 , @xmath29 , and @xmath4 . in the @xmath73 limit eqs . ( [ eq : summe ] ) to ( [ eq : gdef ] ) give close to the boundary @xmath124 \right\ } . \label{eq : rhocomplt0}\end{aligned}\ ] ] at fixed @xmath29 , @xmath125 thus vanishes @xmath126 . as in the translational invariant case this power law is cut off by a finite temperature and @xmath125 saturates for @xmath127 . for fixed @xmath29 close to the boundary and @xmath79 eq . ( [ eq : tgives ] ) gives @xmath128 for @xmath73 . at @xmath67 and deep in the bulk , that is for @xmath129 , @xmath130 can be written as a sum of terms which vanish algebraically in @xmath29 @xcite . they show a power - law dependence on @xmath131 in general each with a different exponent . the contribution @xmath132 to @xmath133 becomes position independent and goes as @xmath134 ( instead of @xmath13 close to the boundary ) . for sufficiently large @xmath135 ( such that algebraically decaying terms can be neglected ) one thus finds ( @xmath136 ) @xmath137 for @xmath67 and in the _ noninteracting _ limit @xmath138 does not decay and one obtains @xmath139 \right\ } , \label{eq : rhocompl0}\end{aligned}\ ] ] for all @xmath29 and @xmath1 . it is often argued , that the contribution @xmath110 to @xmath133 can be neglected when comparing to experiments . the electrons in pes and sts do not come from a specific location @xmath29 but rather from an extended spatial range . if this is large enough compared to the characteristic length @xmath140 , @xmath110 averages out due to the fast spatial oscillations with frequency @xmath12 . it is not obvious that the criterion for neglecting @xmath110 is fulfilled in the latest sts experiments @xcite ( see sect . [ sec : summary ] ) . equation ( [ eq : rhocompl ] ) ( or eq . ( [ eq : rhocomplt0 ] ) for @xmath67 ) allows for another consistency check of ll behavior . it predicts a spatial power - law dependence of @xmath133 close to the boundary with exponent @xmath141 superimposed by oscillations . if it is possible to measure the envelope function of the spatial dependence of the spectral weight close to a boundary and extract the power - law exponent it would allow to relate the resulting @xmath72 to the ones obtained from @xmath2 and/or @xmath13 . we note in passing that performing a spatial fourier transform of @xmath96 reveals characteristic informations of the bulk state of a ll including its elementary excitations ( see ref . @xcite and references therein ) . in the next section we show that the spectral function of microscopic lattice models of interacting 1d electrons , in our case the extended hubbard model , indeed shows scaling behavior as a function of @xmath18 . up to a subtlety in the spatial dependence , namely an interaction dependent phase shift in the oscillatory factor , the lattice spectral function falls on top of the above computed scaling function of the tl model . this holds in the low - energy regime . we discuss the crossover between this universal behavior and the nonuniversal regime at higher energies . the crossover scale @xmath142 depends on the parameters of the microscopic model . partly aiming at the low - energy ll physics of microscopic lattice models of interacting electrons different groups computed the local spectral function @xcite as well as the momentum resolved one @xcite for such models using very accurate numerical methods ( often denoted as numerically exact ) . quantum monte - carlo ( qmc ) @xcite and ( dynamical ) density - matrix renormalization group ( dmrg ) @xcite were used . the results obtained by these methods turned out to be very useful for understanding the spectral features of the studied models over the entire band width . unfortunately , due to system size restrictions ( dmrg and qmc ) , artificial broadenings of the spectra ( dmrg ) , as well as the problem of analytic continuation of numerical data ( qmc ) , it was not possible to reach the low - energy regime ; in none of the calculations it was possible to convincingly demonstrate power - law behavior of the spectra . rephrasing this in experimental terms one can say that the energy resolution of these methods is not high enough . for ( partly ) technical reasons the focus of the numerical approaches lies on the hubbard model with a local two - particle interaction . as the crossover scale @xmath142 between ll behavior and nonuniversal physics in this model is very small ( see below ) reaching the ll regime is particularly challenging . we here use a method which allows to obtain _ approximate _ results for the spectral function of the extended hubbard model with a local @xmath82 and nearest - neighbor @xmath143 interaction . our approximation is based on the functional rg approach to quantum many - body physics @xcite . functional rg allows to set up a hierarchy of approximation schemes with the two - particle interaction being the small parameter . the one we are using here is controlled to leading order and can thus only be used for small to intermediate @xmath82 and @xmath143 ( compared to the band width ) . the method was mainly applied in the context of transport through inhomogeneous lls and there it was shown to reproduce typical impurity strength independent @xcite ll exponents to _ leading order _ in the interaction @xcite . due to appropriate resummations of classes of diagrams the rg procedure thus goes way beyond standard perturbation theory . as the exponent of the bulk local spectral function is of _ second order _ in the interaction ( see eq . ( [ eq : u_2 ] ) ) the ll physics of translationally invariant systems can not be assessed in our approximation . in the following we therefore restrict ourselves to the model with open boundaries characterized by the exponent @xmath13 which has a _ linear _ contribution ( see eq . ( [ eq : u_1 ] ) ) and study @xmath133 _ close _ to one of the boundaries . we here refrain from giving any further technical details on how the spectral function can be computed within our functional rg approach . those can be found in ref . @xcite . the hamiltonian of the extended hubbard model with two open boundaries is given by @xmath144 where @xmath145 and @xmath146 are creation and annihilation operators for fermions with spin @xmath147 on lattice site @xmath148 , while @xmath149 , and @xmath150 is the local density operator on site @xmath148 . for the ( nonextended ) hubbard model the nearest neighbor interaction @xmath143 vanishes . the number of lattice sites is denoted by @xmath151 . the noninteracting tight - binding part gives the standard dispersion @xmath152 with the hopping matrix element @xmath153 ( the lattice constant is chosen to be unity ) . under the _ assumption _ that a given microscopic model is a ll ( at low energy scales ) one can use general relations between the exact ground state energy @xmath154 and @xmath72 @xcite to extract the dependence of the ll parameter @xmath72 on the parameters of the model considered . in general however @xmath154 of a many - body system is not known analytically . the translational invariant hubbard model constitutes one of the rare exceptions and closed expressions for @xmath154 in form of integral equations can be determined using bethe ansatz @xcite . the integral equations can easily be solved numerically which gives access to the dependence of @xmath72 on the parameters @xmath155 and the band filling @xmath156 ( @xmath156 can vary between 0 and 2 ) @xcite . we emphasize that this _ only _ implies that on _ asymptotically small _ scales one can expect power - law behavior with @xmath2 ( bulk ) or @xmath157 ( close to the boundary ) while no information on the crossover scale @xmath142 from nonuniversal to universal ll behavior can be extracted this way . furthermore , this expectation holds under the assumption that the hubbard model is a ll , which away from half - filling @xmath158for which it is a mott insulator @xcite is not doubted seriously , but also not proven rigorously . for the ( translational invariant ) extended hubbard model @xmath159 can only be computed numerically along similar lines @xcite . the exponents @xmath2 and @xmath13 for the hubbard @xcite can not become large enough to match the exponents inferred from experiments on different systems @xcite , or , putting it differently , @xmath72 can not become small enough . for the extended hubbard model @xmath72 s of roughly the correct order can be achieved for @xmath82 and @xmath143 of the order of the band width or larger . this part of the parameter space lies very close to the mott transition of the model @xcite . one can expect that this effects the spectral properties . when aiming at a typical ll spectral function with @xmath2 and @xmath13 of experimental size it is thus advisable to study models with interaction of longer spatial range . we note that within our approximate approach @xmath82 and @xmath143 are bound to be sufficiently smaller than the band width . the extended hubbard model is spin - rotational invariant which implies @xmath51 . for the hubbard model with an open boundary the scale @xmath142 was earlier computed in the small @xmath82 limit and it was estimated to be exponentially small @xcite @xmath160 ^ 2 } { [ u/(8 v_{\rm f})]^2 } } \right\ } } . \label{eq : crossoverscale}\end{aligned}\ ] ] in fact , approaching the chemical potential @xmath136 the local spectral weight first increases before the ll power - law suppression sets in for @xmath161 ( see the solid line in fig . [ fig1 ] which does not look ll - like around @xmath79 ; the power - law suppression is beyond the energy resolution ) . this is consistent with the observation of a small crossover scale and a peak close to @xmath79 in the local spectra of the _ translational invariant _ hubbard model obtained numerically by qmc and dmrg @xcite . to compute the finite temperature @xmath162 ( here the continuous position @xmath29 is replaced by the discrete lattice site index @xmath148 ) of the extended hubbard model we consider a chain of @xmath151 lattice sites described by eq . ( [ eq : model ] ) . for this the spectrum is discrete and the spectral function consists of @xmath66-peaks . due to even - odd effects the spectral weight might vary quickly from one eigenvalue to the next one . a smooth function of @xmath1 is obtained by averaging the weight over neighboring eigenvalues . to obtain the local spectral function as defined in the continuum one furthermore has to devide the weights by the level spacing between eigenvalues . the energy scale @xmath163 associated to the chain length becomes irrelevant as we always consider sufficiently large systems with @xmath164 for fixed @xmath4 . our results are thus not influenced by finite size effects ( for an exception , see the discussion of fig . [ fig4 ] ) . typical experimental temperatures are in the few to few ten kelvin range which corresponds to @xmath165 to @xmath166 for our model . , @xmath167 , @xmath168 , @xmath169 , @xmath170 and different @xmath171 . for filling @xmath172 the optimal nearest - neighbor interaction is given by @xmath173 . only for sizable nearest - neighbour interaction @xmath143 we observe the ll suppression of the weight at @xmath79 . the suppression close to @xmath174 is a lattice effect @xcite . ] in fig . [ fig1 ] @xmath162 is shown for filling @xmath172 , lattice site @xmath167 next to the boundary , @xmath168 , @xmath169 , @xmath170 and different @xmath171 . for the hubbard model with @xmath175 no suppression of the spectral weight is observable as @xmath142 is much smaller than temperature . obviously , @xmath142 increases with increasing @xmath171 and the ll suppression at @xmath79 becomes apparent . this can be understood as follows . the crossover scale @xmath142 is strongly affected by the size of the open boundary analog of a @xmath11 two - particle scattering process @xcite which can not be written quadratically in the bosonic densities . its initial value ( with respect to an rg flow ) in the extended hubbard model is given by @xmath176 with @xmath177 . it is large for @xmath178 . at @xmath179 $ ] it vanishes ( see the dashed - dotted line in fig . [ fig1 ] ) . under an rg procedure this non - ll term flows to zero and is thus rg irrelevant . however , the flow is only logarithmically . this implies that for sizable initial @xmath11 ll physics sets in on exponentially small scales consistent with eq . ( [ eq : crossoverscale ] ) for the hubbard model @xcite . for small initial @xmath11 , @xmath133 appears ll - like with the characteristic power - law behavior of the spectral weight close to @xmath79 . in fig . [ fig1 ] the spectral weight at @xmath79 remains finite due to the finite temperature . we conclude that to observe ll physics on moderate scales in the present model the interaction should not be too local . in particular , to demonstrate power - law behavior and obtain an estimate of the exponent by fitting @xmath133 for fixed @xmath148 ( close to the boundary ) and @xmath180 as a function of @xmath1 ( in the range @xmath181 ) one should consider fine - tuned parameters with @xmath182 . the spectral functions of fig . [ fig1 ] show another high - energy nonanalyticity . a similar feature was observed for a model of spinless fermions in ref . @xcite and was explained there as a lattice effect . , @xmath168 , @xmath183 ( for @xmath172 ) , @xmath169 , @xmath170 and different @xmath148 close to the open boundary at @xmath167 . ] in fig . [ fig2 ] the low - energy regime of @xmath133 is shown for the same parameters as in fig . [ fig1 ] but optimal @xmath184 ( for @xmath172 ) , which allows for the largest low - energy regime , and varying position @xmath148 close to the boundary site @xmath167 . for fixed @xmath1 we observe strong variations of the weight with @xmath148 and pronounced @xmath185 asymmetries which is consistent with the result from the tl model eq . ( [ eq : rhocompl ] ) . below we return to the spatial dependence of @xmath133 . to confirm scaling in @xmath18 at fixed @xmath148 as predicted in eq . ( [ eq : rhocompl ] ) we computed @xmath133 for the parameters of fig . [ fig2 ] but with @xmath167 and for different @xmath4 . by fitting @xmath186 as a function of @xmath1 for the smallest @xmath4 in the range @xmath181 we can extract a functional rg estimate of the boundary exponent @xmath187 . for the given parameters we obtain @xmath188 in good agreement with the dmrg result @xmath189 obtained from eq . ( [ eq : alphabdef ] ) and @xmath190 derived as explained above @xcite . the scaling obtained with this @xmath187 is shown in fig . [ fig3 ] . in the inset the unscaled data are displayed . the thick solid line is the prediction eq . ( [ eq : rhocompl ] ) of the tl model with an open boundary , where we replaced @xmath191 . the data nicely collapse on the tl model curve in the low - energy regime . , @xmath168 , @xmath183 ( for @xmath172 ) , @xmath169 , @xmath167 and different @xmath4 . the inset shows the unscaled spectral function as a function of @xmath1 for different @xmath4 . ] we next take a closer look at the @xmath148 dependence of @xmath133 at fixed @xmath1 . as emphasized in the last section measuring the local spectral weight as a function of @xmath148 offers another possibility for a consistency check that the system under consideration is a ll : eq . ( [ eq : rhocompl ] ) predicts power - law behavior of the envelope with exponent @xmath192 . figure [ fig4 ] shows @xmath133 for @xmath172 , @xmath168 , @xmath169 , @xmath170 and @xmath193 as a function of @xmath148 ( filled circles ) . here @xmath193 refers to taking the eigenvalue of the finite system closest to @xmath79 , which might be of order @xmath194 away from zero . we again tune @xmath143 to the optimal value @xmath195 ( for @xmath172 ) providing the largest @xmath142 . the spatial oscillations with frequency @xmath196 are apparent . we fitted the envelope to a power law @xmath197 and obtained @xmath198 in excellent agreement with @xmath199 . the power - law fit is shown as the thick solid line in fig . as mentioned above we control the different exponents only to leading order in the interaction . to this order the analytic expressions for @xmath13 and @xmath200 agree , as is apparent from eqs . ( [ eq : expcdef ] ) , ( [ eq : alphabdef ] ) and ( [ eq : kexp ] ) . the numerical values for @xmath201 and @xmath187 still differ by roughly 2% as the rg produces higher than linear order terms in the different exponents as well . we emphasize that for @xmath178 , that is for the hubbard model , in a similar plot no spatial suppression of the envelope of the spectral weight at small @xmath148 is visible . in fact , the envelope of the spectral weight at @xmath193 even _ increases _ for @xmath148 approaching the boundary site @xmath167 . for @xmath172 , two different @xmath155 , @xmath184 ( for @xmath172 ) , @xmath169 , @xmath170 and @xmath193 . the solid line is a power - law fit to the envelope . ] a comparison of the data for @xmath168 ( filled circles ) and for @xmath202 ( crosses ) additionally presented in fig . [ fig4 ] shows that a _ phase shift _ @xmath203 of the spatial @xmath12 oscillations appears which is _ not _ captured by the result for the tl model eq . ( [ eq : rhocompl ] ) . the latter was derived using standard bosonization with the boundary conditions on the continuum fields given after eq . ( [ eq : gaussfields ] ) . the deviation of the @xmath202 curve from eq . ( [ eq : rhocompl0 ] ) for @xmath204 ( splitting of degenerate values of the spectral weight ) is a finite size effect ; @xmath1 is not exactly zero but of order @xmath194 ( eigenvalue closest to zero ) . the phase shift can be most easily identified from the observation that @xmath133 vanishes on every eighth lattice site for @xmath202 ( crosses ) , as it is supposed to according to eq . ( [ eq : rhocompl0 ] ) with @xmath79 and @xmath205 , but not so for @xmath168 ( filled circles ) , where the same should hold according to eq . ( [ eq : rhocompl ] ) . the phase @xmath203 turns out to be _ linearly _ dependent on @xmath82 ( for small @xmath155 and @xmath206 ) . as our functional rg approximation scheme is controlled to this order , the appearance of @xmath203 is a reliable finding . considering @xmath67 at different system sizes @xmath151 we furthermore verified that @xmath203 does not vanish for decreasing @xmath194 . the phase shift is thus not a finite size effect . a phase shift as observed in the extended hubbard model ( with @xmath206 ) can be accounted for in bosonization by adding a _ local single - particle forward scattering term _ @xmath207 to the hamiltonian density eq . ( [ eq : hamdens ] ) . the phase shift is then given by @xmath208 . to match the result of the extended hubbard model ( with @xmath206 ) @xmath209 has to be chosen @xmath82-dependent , in particular @xmath210 for small @xmath155 . the sts and pes experiments always imply a spatial averaging . as the phase shift becomes illusive even after averaging over only a few lattice sites and as the main focus of the present paper is on relating theoretical spectral functions to experimental ones we here do not further investigate this issue . for @xmath211 , two different @xmath1 , @xmath168 , @xmath212 ( for @xmath211 ) , @xmath169 , @xmath170 . ] finally we compare the spatial dependence of @xmath133 for two different @xmath1 in fig . apparently the frequency of the spatial oscillations depends on @xmath1 which is consistent with eq . ( [ eq : rhocompl ] ) . as this is more transparent for a more commensurable filling , the parameters of this figure are @xmath211 , @xmath168 , @xmath212 ( for @xmath211 ) , @xmath169 and @xmath170 . in the first part of this paper we have derived analytic expressions for the power - law behavior of the local spectral weight @xmath77 of the translational invariant tl model as well as the one with an open boundary @xmath133 as a function of @xmath1 , @xmath4 and @xmath29 . the results provide a variety of possibilities for consistency checks of experimental sts and pes data on 1d electron systems . the first is to show scaling of data for different @xmath1 and @xmath4 taken in the bulk or at fixed position close to the boundary onto the bosonization predictions eqs . ( [ eq : rhotransinv ] ) and ( [ eq : diveresedeffs ] ) ( bulk ) or eq . ( [ eq : rhocompl ] ) ( boundary ) . scaling of bulk spectra was e.g. demonstrated in ref . if the same could be achieved for boundary spectra the _ same _ scaling function with @xmath2 replaced by @xmath13 should appear if the system is a ll . experimentally showing this together with the required consistency of @xmath2 and @xmath13 ( for a spin - rotational invariant model with @xmath51 both given by a single number @xmath72 ; see eqs . ( [ eq : diveresedeffs ] ) and ( [ eq : alphabdef ] ) ) would constitute a second highly nontrivial check that the studied system indeed is a ll . a consistency of bulk and boundary exponents within the experimental error bars ( but not the entire scaling function ) was achieved in refs . @xcite and @xcite ( also see below ) . a third consistency check is provided by the predicted spatial power - law behavior with the exponent @xmath213 which again can be expressed solely in terms of @xmath72 ( see eq . ( [ eq : expcdef ] ) ) . it is often argued that the spatially oscillating contributions @xmath110 and @xmath214 to @xmath133 with frequency @xmath12 can be neglected when comparing to experimental spectra due to spatial averaging effects . taking the numbers from ref . @xcite this is not apparent . in this experiment @xmath215 m while the range of spatial averaging is estimated as @xmath216 m , leading to @xmath217 this is not a very large number and one would thus conclude that @xmath110 and @xmath214 can not be dropped . a quantitative picture of the averaging effects can easily be obtained by integrating eq . ( [ eq : summe ] ) over an appropriate spatial range . we have explicitely verified that averaging over @xmath216 m does not significantly smear out the boundary ( @xmath218 ) and bulk ( @xmath219 ) exponents in @xmath1 at low temperatures ( here @xmath67 ) . in particular , this shows that the spatial resolution of the experiment is high enough to detect @xmath13 ( as it is implicit to the analysis presented in ref . @xcite ) . in connection with the comparison of the experimental results on gold chains on a germanium surface of ref . @xcite to the ll predictions one might be worried about two effects which are not included in the tl model of sect . [ sec : scalfun ] . one is rashba spin - orbit interaction ( soi ) , which in a surface setup can become sizable . along the lines of refs . @xcite and @xcite one can bosonize the 1d electron gas with rashba soi . an important effect is the appearance of _ two _ different fermi velocities @xmath220 due to subband mixing and the soi splitting @xcite . here @xmath221 is a measure of the strength of the soi . the local spectral function ( with and without an open boundary ) shows the same characteristics as a function of @xmath1 as discussed in sect . [ sec : scalfun ] , but with modified exponents @xmath222 taking realistic numbers for the velocities it turns out that @xmath223 and the effects of rashba soi on the exponents are negligible . we note in passing that also the momentum resolved spectral function of a translational invariant ll is barely modified by soi of realistic size @xcite . the other issue is the observation of _ four _ electron branches ( instead of two in the tl model ) crossing the fermi surface in pes measurements on the gold chains @xcite . the four branch situation can also be accounted for in bosonization @xcite . one then has to introduce even and odd pairs of charge and spin density bosons as well as the related @xmath224 s and velocities . under the plausible _ assumption _ that only the even charge channel ll parameter is different from the noninteracting value @xmath225 , that is @xmath226 , one finds the same power - law behavior for the local spectral functions as a function of @xmath1 as the ones given in sect . [ sec : scalfun ] but with @xmath227 interestingly taking these expressions would significantly improve the consistency of the experimentally determined @xmath2 and @xmath13 @xcite . from the measured value @xmath228 of the bulk spectra one finds @xmath229 which nicely agrees to the measured value @xmath230 . in particular , the agreement is improved compared to taking the two - branch expressions eqs . ( [ eq : diveresedeffs ] ) and ( [ eq : alphabdef ] ) which gives @xmath231 @xcite . in the second part of our paper we have shown that the behavior of @xmath133 as a function of @xmath1 , @xmath4 and @xmath148 close to an open boundary predicted by the bosonization solution of the tl model can indeed be found in an example of a microscopic lattice model , namely the extended hubbard model . interestingly , a linear - in-@xmath82 phase shift @xmath203 not captured by standard open - boundary bosonization appears in the spatial oscillations of the spectral weight of the extended hubbard model . the energy resolution required to access the low - energy ll regime strongly depends on the model parameters . even for the fairly high energy resolution we can achieve within our approximate method , convincingly demonstrating ll power - law behavior and reliably extracting exponents requires fine - tuning of the parameters . roughly speaking the crossover scale becomes small if the two - particle interaction becomes too local . this suggests that to access the low - energy ll regime at a given experimental energy resolution one should consider systems with poor screening properties . * acknowledgements : * we are grateful to jrg schfer , ralph claessen , kurt schnhammer and christian ast for very fruitful discussions . the numerical calculations were performed on the machines of the max - planck institute for solid state research , stuttgart . this work was supported by the dfg via the emmy - noether program ( d.s . ) and for 723 ( s.a . and v.m .

motivated by recent scanning tunneling and photoemission spectroscopy measurements on self - organized gold chains on a germanium surface we reinvestigate the local single - particle spectral properties of luttinger liquids . in the first part we use the bosonization approach to exactly compute the local spectral function of a simplified field theoretical low - energy model and take a closer look at scaling properties as a function of the ratio of energy and temperature . translational invariant luttinger liquids as well as those with an open boundary ( cut chain geometry ) are considered . we explicitly show that the scaling functions of both setups have the same analytic form . the scaling behavior suggests a variety of consistency checks which can be performed on measured data to experimentally verify luttinger liquid behavior . in a second part we approximately compute the local spectral function of a microscopic lattice model the extended hubbard model close to an open boundary using the functional renormalization group . we show that as a function of energy and temperature it follows the field theoretical prediction in the low - energy regime and point out the importance of nonuniversal energy scales inherent to any microscopic model . the spatial dependence of this spectral function is characterized by oscillatory behavior and an envelope function which follows a power law both in accordance with the field theoretical continuum model . interestingly , for the lattice model we find a phase shift which is proportional to the two - particle interaction and not accounted for in the standard bosonization approach to luttinger liquids with an open boundary . we briefly comment on the effects of several one - dimensional branches cutting the fermi energy and rashba spin - orbit interaction .

for ultra - cold atoms in an optical lattice @xcite dynamical aspects include transverse resonances @xcite density waves @xcite , the evolution of quantum fluctuations @xcite , the speed of sound @xcite and time - resolved observation and control of superexchange interactions @xcite . the aim of the present manuscript is to perform exact two - particle dynamics in an optical lattice similar to what has been suggested in ref . @xcite , a bright soliton in a one - dimensional waveguide . as the dispersion relation for the bound two - particle states in the lattice approach case without lattice for suitable parameters , this can be used to quantitatively test the @xmath0-particle predictions of ref . @xcite via exact numerics on the two - particle level for which a soliton is simply a dimer . besides the analytical @xmath0-particle quantum mechanical calculations @xcite , the scattering of the soliton has also been investigated via numerical methods on the @xmath0-particle level @xcite . different approaches to obtain such schrdinger cat states or related fragmentations have been investigated in refs . contrary to schrdinger cat states of a single atom @xcite , cat - like states of radiation @xcite or mesoscopic spin - squeezed states ( which have already been realised experimentally @xcite ) , the experimental realisation of schrdinger cat states of say , 100 atoms , is still a challenge of fundamental research . suggestions how interesting quantum superpositions might be obtained can be found , e.g. , in refs . @xcite and references therein . for bright quantum matter wave solitons @xcite , the mean - field ( gross - pitaevskii ) limit has been shown to be achieved already for particle numbers as low as @xmath1 @xcite . many of the papers published after the ground - breaking experiments @xcite solve the gross - pitaevskii equation for solitons . however , any mesoscopic entangled state which involves superpositions of wavefunctions can not be described by a non - linear equation and therefore the reasoning of ref . @xcite is not valid in the situation considered here . thus , instead of applying the gross - pitaevskii equation , the @xmath0-particle schrdinger equation has to be used to reveal true quantum behaviour of a soliton created from a bose - einstein condensate . under experimentally realistic conditions , the schrdinger equation is given by the analytically solvable lieb - liniger(-mcguire ) model . the challenge of the generation of mesoscopic superpositions via scattering of solitons is that to add a scattering potential removes the separability of the centre - of - mass motion and the relative motion ; in order to avoid that the scattering potential acts like a beam splitter on each single atom ( rather than the entire soliton ) , the initial state has to be prepared carefully . mesoscopic entangled states with the soliton being in a quantum superposition with @xmath2 probability of moving to the right / left should thus be obtainable . the probability to find in a _ single measurement _ ( at least ) one particle moving to the right and at ( at least ) one particle moving in the other direction will be negligible . however , this will not be enough to prove that the two parts of the wavefunction really are in a quantum superposition if someone claims that a coin is in a quantum superposition of heads and tails , an experiment showing only the classical outcomes would hardly convince anyone . the experimental verification could be delivered via interference experiments @xcite . rather than dealing with bright @xmath0-particle quantum solitons , this paper treats a simpler but nevertheless instructive case : dimers in an optical lattice . the paper is organised as follows : after a short summary of how to describe the scattering of bright solitons analytically @xcite ( sec . [ sec : liebliniger ] ) , the two - particle bound states used to describe the scattering of the dimer are introduced in sec . [ sec : two ] . section [ sec : results ] shows the numeric results in the limit where the motion in the optical lattice mimics the motion without lattice . the hamiltonian of the lieb - liniger - mcguire @xcite model with attractive interaction and an additional scattering - potential @xmath3 is given by @xmath4 bright solitons @xcite are well described by this model . for @xmath5 , exact eigenfunctions of this hamiltonian are known . solutions corresponding to @xmath0-particle solitons with momentum @xmath6 read : @xmath7 where @xmath8 the corresponding energies are given by @xmath9 where @xmath10 is the ground state energy of the system @xcite . as long as the kinetic energy is not too large , these states are separated from the first excited internal state ( which corresponds to one particle having left the soliton ) by a finite energy barrier @xmath11 ( see , e.g. , ref . @xcite ) . had the scattering potential been a function of the centre of mass of all @xmath0 particles ( @xmath12 ) , the situation would have been easy as the centre of mass and relative coordinates then still separate . however , the potential in the hamiltonian ( [ eq : h ] ) is given by @xmath13 it would nevertheless be tempting to argue that , given the fact that the particles are tightly bound , they behave essentially as a single particle and one could thus approximate @xmath14 by @xmath15 and thus @xmath16 where @xmath17 is the centre - of - mass coordinate . however , this approximation can give wrong results ( as will be shown towards the end of this paper ) and the mathematically justified @xcite effective potential approach : @xmath18 has to be used . the effective potential is given by the convolution @xcite @xmath19 this approach is valid for sufficiently well behaved potentials ( like a laser focus ) and for solitons which can not break apart for energetic reasons ( see the paragraph below eq . ( [ eq : e0 ] ) ) . two - particle bound states in optical lattices are interesting both experimentally @xcite and theoretically @xcite ; recently even three - particle bound states @xcite have been investigated . within a bose - hubbard hamiltonian , @xmath20 one can use _ exact _ eigen - states to do the numerics ( the restriction to negative pair interactions @xmath21 is not necessary , however the idea is to discuss a case close to the bright solitons created from attractive bose - einstein condensates ; the creation / annihilation operators of particles at lattice cite @xmath22 are denoted by @xmath23/@xmath24 ) . rather than using the approach via green s functions of ref . @xcite , we proceeded in ref . @xcite along the lines of ref . @xcite to show in a straight - forward but somewhat lengthy calculation that the dimer wavefunctions in a tight - binding lattice with band - width @xmath25 and lattice spacing @xmath26 are given by @xmath27&:\;\mu\ne\nu\\ \exp\left[ikb(\nu+\mu)\right]/\sqrt{2}&:\ ; \mu = \nu \end{array}\right.,\ ] ] where @xmath28 the coordinates @xmath29 and @xmath30 label the lattice points on which the particles `` sit '' and as a basis the fock basis is used ( _ i.e. _ @xmath31 refers to the fock state with two particles at the lattice - site @xmath29 ) . the interaction energy of two particles sitting at the same lattice site is @xmath32 ; for the wavefunction to be normalisable for fixed centre of mass , @xmath33 ( which implies @xmath34 ) . to avoid a breaking of the dimer , the energy @xmath35 should be lower than two times the minimal possible energy of two particles : @xmath36 as for the single particle energy band , @xmath37 the dispersion relation for two particles approaches the free - particle behaviour in the limit @xmath38 which can thus be used to model particles not restricted by optical potentials : @xmath39 by choosing the parameters to approximate the dispersion relation for a two - particle `` soliton '' ( cf . ( [ eq : psink ] ) ) @xmath40 the effective potential approach of ref . @xcite ( cf . ( [ eq : veffend ] ) ) will thus be valid for small enough @xmath26 . ] the initial centre - of - mass wavefunction in ref . @xcite is suggested to be a gaussian ( obtained by preparing the system in a swallow harmonic oscillator potential ) . if the system can be described within a centre - of - mass approximation , the expected result will always look similar to fig . [ fig : refl ] . for the initial gaussian to be realisable experimentally , the length scale on which the wavefunctions of the relative coordinates decays has to be smaller than the width of the initial wavefunction . given the fact that the dimer can not break into two free particles , this implies that in the final wavefunction measuring one particle on one side of the barrier would also lead to measuring the other particle on the same side - thus the final state indeed is a schrdinger cat state . figure [ fig : three ] shows the result for a two - particle soliton scattered off a step potential , @xmath41 the initial wavefunction is constructed as a superposition of the eigen - solutions ( [ eq : eigenfunkt ] ) with @xmath42 which @xmath43 ensures the validity of eq . ( [ eq : bedb ] ) ( cf . ( [ eq : zweisoliton ] ) ) : @xmath44 if @xmath45 is not too small this leads to a gaussian centre - of - mass wavefunction initially centred at @xmath46 . surprisingly , for the parameters chosen in fig . [ fig : three ] the wavefunction splits into three rather than the expected two parts . if one of the particles was measured , the second would be much closer to the location of the first than the width of the three moving wave - packets . the strange behaviour of fig . [ fig : three ] clearly demonstrates that the centre - of - mass approximation is not valid here : the effective potential ( [ eq : veffend ] ) as displayed in the inset of fig . [ fig : three ] at least qualitatively explains why resonances can occur which make parts the dimer wave - function stay at the potential . however , rather than being able to qualitatively understand the scattering behaviour , the effective potential approach can also be tested quantitatively for realistic scattering potentials . if the scattering potential is realised via the focus of a laser beam , the potential can be approximated by a gaussian or even the potential @xmath47 which is analytically solvable on the single particle level . figure [ fig : miaou ] a compares the wave - function after scattering for the exact potential @xmath48 the center of mass approximation @xmath49 and the effective potential @xcite @xmath50 where @xmath43 is given by eq . ( [ eq : betatilde ] ) . the effective potential clearly is a huge improvement over the center - of - mass approximation . it still remains to be shown that the final state indeed is entangled in the sense that whenever one measures one particle on the right ( left ) side , the other particle will be on the same side.to do this , a measure if an outgoing wave is unentangled , @xmath51 where the wavefunction is taken to be normalised to one . figure [ fig : miaou ] b demonstrates that for the wave - function to be entangled , the initial velocity can not be too large . experimental verifications that such states are indeed quantum superpositions could be done via interference experiments @xcite . to summarise , scattering of bright quantum matter wave solitons @xcite has been investigated for two attractive atoms in an optical lattice via exact numerics . it has been demonstrated that the beyond centre - of - mass approximation approach of ref . @xcite is indeed necessary to describe the physical situation at least qualitatively : the effective potential derived in ref . @xcite can explain the break - down of the centre - of - mass approximation observed for specific parameters in the numerics ( fig . [ fig : three ] ) . for a scattering potential given by a laser focus ( the potential would then be approximately gaussian ) such an effect does not occur . the wavefunction then splits into two parts as shown for a single particle in fig . [ fig : refl ] and the effective potential even gives excellent quantitative agreement ( fig . [ fig : miaou ] ) with exact numerics . i acknowledge insightful discussions with y. castin , a. sinatra and c. salomon at the laboratoire kastler brossel at the ens in paris where this research was started ( funded by the european union via contract meif - ct-2006 - 038407 ) . furthermore , i would like to thank m. holthaus for his continuous support .

the motion of two attractively interacting atoms in an optical lattice is investigated in the presence of a scattering potential . the initial wavefunction can be prepared by using tightly bound exact two - particle eigenfunction for vanishing scattering potential . this allows to numerically simulate the dynamics in the generation of two - particle schrdinger cat states using a scheme recently proposed for scattering of quantum matter wave solitons .

the color fields of hadrons boosted to the light cone are thought to grow very strong , parametrically of order @xmath2 where @xmath3 is the coupling @xcite . the fields of nuclei are enhanced further by the high density of valence charges per unit transverse area , which is proportional to the thickness @xmath4 of a nucleus @xcite . in collisions of such strong color fields a large number of soft gluons is released . due to the genuinely non - perturbative dynamics of the strong color fields a semi - hard `` saturation scale '' @xmath5 emerges ; it corresponds to the transverse momentum where the phase space density of produced gluons is of order @xmath6 . the mean multiplicity per unit rapidity in high - energy collisions is then @xmath7 . below we argue that a semi - classical effective theory of valence color charge fluctuations predicts that the variance of the multiplicity distribution is of order @xmath8 so that the perturbative expansion of @xmath9 begins at order @xmath10 . we show that in the strong field limit then a gaussian effective theory leads to koba - nielsen - olesen ( kno ) scaling @xcite . this relates the emergence of kno scaling in @xmath11-integrated multiplicity distributions from high - energy collisions to properties of soft gluons around the saturation scale . collisions at various energies as measured by the ua5 @xcite , alice @xcite and cms @xcite collaborations , respectively . note that we restrict to the bulk of the distributions up to 3.5 times the mean multiplicity.,scaledwidth=50.0% ] the kno scaling conjecture refers to the fact that the particle multiplicity distribution in high - energy hadronic collisions is _ universal _ ( i.e. , energy independent ) if expressed in terms of the fractional multiplicity @xmath12 . this is satisfied to a good approximation in the central ( pseudo- ) rapidity region at center of mass energies of 900 gev and above @xcite as shown in fig . [ fig : kno_lhcdata ] . on the other hand , ua5 data @xcite taken at @xmath13 gev appears to show a slightly distorted multiplicity distribution . this is in line with the observation that at lower energies higher - order factorial moments @xmath14 of the distribution are energy dependent and significantly _ different _ from the reduced moments @xmath15 @xcite : g_q , c_q . in fact , since the difference of @xmath14 and @xmath15 is subleading in the density of valence charges one may interpret this finding to indicate that the high density approximation is less accurate for @xmath16 gev @xmath0 collisions . approximate kno scaling has been predicted to persist also for min - bias @xmath17 collisions ( at lhc energies ) in spite of additional glauber fluctuations of the number of participants and binary collisions @xcite . a more detailed discussion of multiplicity distributions at tev energies is given in refs . @xcite , and references therein . transverse momentum integrated multiplicities in inelastic hadronic collisions are not governed by an external hard scale , unlike say multiplicity distributions in @xmath18 annihilation or in jets @xcite . hence , the explanation for the experimental observation should relate to properties of the distribution of produced gluons around the saturation scale @xmath5 . we shall first discuss the multiplicity distribution of small-@xmath19 gluons obtained from a gaussian effective action for the color charge fluctuations of the valence charge densities @xmath20 @xcite , z & = & e^-s_mv [ ] , + s_mv [ ] & = & d^2x_^_- dx^- . [ eq : s_mv ] in the strong field limit a semi - classical approximation is appropriate and the soft gluon field ( in covariant gauge ) can be obtained in the weizscker - williams approximation as a^+(z^-,x _ ) = - g ^a(z^-,x _ ) = g d^2z _ ^a(z^-,z _ ) . parametrically , the mean multiplicity obtained from the action ( [ eq : s_mv ] ) is then [ eq : nbar ] |n ~ q_s^2 s _ , where @xmath21 denotes a transverse area and @xmath22 . the prefactor in ( [ eq : nbar ] ) can be determined numerically @xcite but is not required for our present considerations . one can similarly calculate the probability to produce @xmath23 particles by considering fully connected diagrams with @xmath23 valence sources @xmath20 in the amplitude and @xmath23 sources @xmath24 in the conjugate amplitude ( for both projectile and target , respectively ) . these can be expressed as @xcite of the @xmath23 particles should be similar . here we assume that all particles are in the same rapidity bin . ] _ conn . = c_q , where the reduced moments [ eq : c_q_mv ] c_q = . this expression is valid with logarithmic accuracy and was derived under the assumption that all transverse momentum integrals over @xmath25 are effectively cut off in the infrared at a scale @xmath26 due to non - linear effects . the fluctuation parameter @xmath27 in eq . ( [ eq : c_q_mv ] ) is of order k ~(n_c^2 - 1 ) q_s^2 s _ . once again , the precise numerical prefactor ( in the classical approximation ) has been determined by a numerical computation to all orders in the valence charge density @xmath20 @xcite . the multiplicity distribution is therefore a negative binomial distribution ( nbd ) @xcite , [ eq : nbd ] p(n ) = . indeed , multiplicity distributions observed in high - energy @xmath0 collisions ( in the central region ) can be described quite well by a nbd , see for example refs . the parameter @xmath28 determines the variance of the distribution ; the latter approximation applies in the limit @xmath29 , see below . ] and can be obtained from the ( inclusive ) double - gluon multiplicity : [ eq : dn2_k ] _ conn . = . from this expression it is straightforward to see that the perturbative expansion of @xmath28 starts at @xmath30 since the connected diagrams on the lhs of eq . ( [ eq : dn2_k ] ) involve the same number of sources and vertices as the disconnected diagrams on the rhs of that equation ( also see appendix ) . this observation is important since _ in general _ the nbd ( [ eq : nbd ] ) exhibits kno scaling only when @xmath29 , and if @xmath27 is not strongly energy dependent . a numerical analysis of the multiplicity distribution at 2360 gev , for example , achieves a good fit to the data for @xmath31 @xcite , which we confirm below . such values for @xmath9 have also been found for peripheral collisions of heavy ions from ab initio solutions of the classical yang - mills equations @xcite ; furthermore those solutions predict that @xmath32 for central collisions of @xmath33 nuclei . to illustrate how deviations from kno scaling arise it is instructive to consider a `` deformed '' theory with an additional contribution to the quadratic action . we shall add a quartic operator @xcite , s_q [ ] = d^2v_^_- dv_1 ^ - \ { + ^_-dv_2 ^ - } . [ eq : squartic ] we assume that the contribution from the quartic operator is a small perturbation since @xmath34 while @xmath35 . in the classical approximation the mean multiplicity is unaffected by the correction as it involves only two - point functions in the theories ( [ eq : s_mv ] ) and ( [ eq : squartic ] ) need to be matched . thus , the `` bare '' parameters @xmath36 in ( [ eq : s_mv ] ) and @xmath37 in ( [ eq : squartic ] ) are _ different _ as the latter absorbs some self - energy corrections . we refer to ref . @xcite for details . ] . on the other hand , @xmath28 as defined in ( [ eq : dn2_k ] ) now becomes [ eq : k_quartic ] q_s^2 s _ & = & 1 - 3 ( n_c^2 + 1 ) ( ^2 ) . in nsd collisions at various energies and nbd fits ; @xmath38 and @xmath39 . note that the mean multiplicity quoted for the fits has been rescaled by 1.5 to include neutral particles ; also , that here @xmath27 is integrated over the transverse plane of the collision.,title="fig:",scaledwidth=47.0% ] in nsd collisions at various energies and nbd fits ; @xmath38 and @xmath39 . note that the mean multiplicity quoted for the fits has been rescaled by 1.5 to include neutral particles ; also , that here @xmath27 is integrated over the transverse plane of the collision.,title="fig:",scaledwidth=47.0% ] therefore , in the classical approximation [ eq : quartic_nbar_k ] ~ ( 1 - 3 ( n_c^2 + 1 ) ) . this result illustrates that @xmath9 decreases as the contribution of the @xmath40 operator increases . we repeat that the derivation assumed that the correction is small so that ( [ eq : quartic_nbar_k ] ) does not apply for large values of @xmath41 . ref . @xcite estimated by entirely different considerations that for protons @xmath42 at @xmath43 . that would correspond to a smaller value of @xmath9 by a factor of 1.43 than for the gaussian theory . assuming that rg flow with energy approaches a gaussian action @xcite , @xmath9 should increase by about this factor . nbd fits to the data shown in fig . [ fig : knofits ] confirm that @xmath9 indeed increases with energy , which might indicate flow towards a gaussian action ; however , the observed increase from @xmath44 gev to 7 tev is much stronger : a factor of about 3 . this apparent discrepancy could be resolved at least partially by running of the coupling in eqs . ( [ eq : nbar],[eq : quartic_nbar_k ] ) with @xmath5 but this requires more careful analysis at the effective scale @xmath5 is taken into account if the mean multiplicity is computed with energy evolved unintegrated gluon distributions like e.g. in refs . @xcite . ] . in the previous section we considered the multiplicity distribution of `` produced gluons '' in a collision of classical ym fields sourced by classical color charges @xmath20 moving on the light - cone . at high energies though ( i.e. , when @xmath45 ) the classical fields are modified by quantum fluctuations @xcite . resummation of boost - invariant quantum fluctuations leads to an energy dependent saturation scale , for example , as required in order to reproduce the growth of the multiplicity @xmath46 with energy . in particular , the energy dependence of the _ mean _ saturation scale , averaged over all `` evolution ladders '' ( distribution of quantum emissions ) , can be obtained by solving the running - coupling bk equation @xcite . instead , in this section we shall solve a _ evolution equation which accounts both for saturation ( non - linear ) effects as well as for the fluctuations of the rapidities and transverse momenta of the virtual gluons in the wave function of a hadron before the collision . we do this in order to determine the multiplicity distribution ( rather than just the mean number ) of dipoles in a hadronic wave function boosted to rapidity @xmath47 . we shall do so by solving via monte - carlo techniques the following evolution equation for @xmath48 $ ] , which is the probability for the dipole size distribution @xmath49 to occur : [ eq : plevol ] = _ z f_z[n(x)-_xz]p[n(x)-_xz , y ] -_z f_z[n(x)]p[n(x),y ] . note that in this section @xmath50 denotes the logarithmic dipole size ( conjugate to its transverse momentum ) rather than to a light - cone momentum fraction . this equation has been studied before in ref . @xcite for fixed @xmath51 and in ref . @xcite for running @xmath52 . those papers also provide references to related earlier work . the first term in ( [ eq : plevol ] ) is a gain term due to dipole splitting while the second term corresponds to loss due to `` recombination '' . f_z[n(x ) ] = is the splitting rate and t_z[n(x ) ] = 1 - _ x n(x ) ( 1-(z|x ) ) is the dipole scattering amplitude for a dipole projectile of size @xmath53 to scatter off the target with the dipole distribution @xmath49 . note that @xmath54 $ ] is non - linear in the dipole density as it involves also the _ pair _ ( and higher ) densities . finally , ( x|y ) = ( x ) ( y ) ( -|x - y| ) ( r^2 _ < ) ( r^2 _ > ) is the elementary dipole - dipole scattering amplitude at lo in perturbative qcd . for more details we refer to ref . @xcite . here , we recall only that it was found there that evolution with a running coupling suppresses fluctuations in the tails of the travelling waves and so restores approximate geometric scaling @xcite . we have determined the multiplicity distribution of dipoles with size @xmath55 , n_i(y ) = _ 1/q_s^2(y ) dx n_i(x , y ) ( i=110 ^ 5 ) by evolving a given initial configuration @xmath56 @xmath57 times . despite starting with a fixed initial condition , evolution introduces fluctuations in the rapidities where splittings occur , and in the sizes of the emerging dipoles . in the wave function . left : evolution with trivial @xmath1-function , @xmath58const . right : qcd @xmath1-function.,title="fig:",scaledwidth=47.0% ] in the wave function . left : evolution with trivial @xmath1-function , @xmath58const . right : qcd @xmath1-function.,title="fig:",scaledwidth=47.0% ] in fig . [ fig : pl_kno ] we show that fixed - coupling evolution does not obey kno scaling of the distribution of virtual quanta while running - coupling evolution does . the shape of the distribution however looks different than the measured distribution of produced particles from fig . [ fig : kno_lhcdata ] . this could be due to the fact that our evolution model does not treat diffusion in impact parameter space . hence , @xmath59 shown in fig . [ fig : pl_kno ] should be interpreted as the multiplicity distribution at the center of the hadron . our work is supported by the doe office of nuclear physics through grant no . de - fg02 - 09er41620 and by the city university of new york through the psc - cuny research award program , grant 65041 - 0043 . we can obtain the fluctuation parameter @xmath27 by calculating the inclusive double gluon multiplicity and expressing it in terms of the single inclusive or mean multiplicity . the connected two particle production cross section for gluons with rapidity @xmath61 and @xmath62 has the form : n_2(p , q ) - _ conn . [ eq : c_2 ] @xmath63 is the mean multiplicity and the brackets denote an average over events . @xmath64 is given by : n_2(p , q)=f_gaaf_gbbf_gccf_gdd _ i=1 ^ 4 + _ 1^*a(k_2 ) _ 1^*b(k_4 ) _ 1^c(k_1 ) _ 1^d(k_3 ) _ 2^*a(p - k_2 ) _ 2^*b(q - k_4 ) _ 2^c(p - k_1 ) _ 2^d(q - k_3 ) . @xmath65 denotes the lipatov vertex , for which : l_(p , k)l^(p , k)=-(p - k)^2 .[eq : lipatovvertex ] for the four - point function in the target and projectile fields we use @xcite ^*a_2(p - k_2 ) ^*b_2(q - k_4 ) ^c_2(p - k_1 ) ^d_2(q - k_3 ) = + ( 2)^4 ^2 + -(2)^4 ^2 ( ^ab^cd + ^ac^bd + ^ad^bc ) ( k_1+k_3-k_2-k_4 ) . [ eq:4ptcorr ] the first two lines on the rhs of the above equation originate from the quadratic part of the action while the third line is due to the quartic operator . the product of the gaussian parts of the two four - point functions gives nine terms , one of which @xmath66 corresponds to a disconnected contribution . it exactly cancels the second term in eq . ( [ eq : c_2 ] ) . four of the other eight terms @xmath67 or @xmath68 give identical leading contributions to double gluon production . they correspond to a `` rainbow '' diagram like the one shown in fig . [ fig : gaussianconn ] . in the `` rainbow '' diagram , on one side ( target or projectile ) , the @xmath20 s corresponding to the same gluon momentum are contracted with each other . the remaining four `` non - rainbow '' diagrams are suppressed relative to the terms we keep at large @xmath69 and @xmath23 @xcite . hence , the leading gaussian contribution is : ~^4 . [ eq : gaussianconn ] the same reasoning applies also for the additional quartic contribution and only `` rainbow '' diagrams are considered , like the one in fig . [ fig : quarticconn ] . there are two of them ( one for the projectile and one for the target ) to first order in @xmath70 , and their contribution is : ~-f_gaaf_gbbf_gccf_gdd _ i=1 ^ 4 + ^2 ^2 ^ac^bd(k_1-k_2)(k_3-k_4 ) + ( ^ab^cd + ^ac^bd + ^ad^bc)(k_1+k_3-k_2-k_4 ) . the color factor evaluates to f_gaaf_gbbf_gccf_gdd ^ac^bd ( ^ab^cd+^ac^bd+ ^ad^bc)=2 n_c^2(n_c^2 - 1)+n_c^2(n_c^2 - 1)^2 .using eq . ( [ eq : lipatovvertex ] ) we get : -^2 ^2 + . the integral over the ladder momentum is again cut off at the saturation scale @xmath5 : . then , the quartic contribution to connected two gluon production becomes - ^2 ^2 . [ eq : quarticconn ] the last step is to express the fully connected diagrams in terms of the single inclusive cross section : = ^2 n_c(n_c^2 - 1 ) . [ eq : single ] summing eq . ( [ eq : gaussianconn ] ) and eq . ( [ eq : quarticconn ] ) and using eq . ( [ eq : single ] ) we get : _ the fluctuation parameter @xmath28 is now identified with the expression in the square brackets . we rewrite it in terms of [ eq : beta ] ^2 , and use [ eq : qs ] q_s^2=dz^- ^2(z^- ) , to arrive at the final expression [ eq:1/k ] q_s^2s _ = 1 - 3(n_c^2 + 1 ) . in this section we are going to calculate the connected diagrams for three - gluon production to obtain the correction to the reduced moment @xmath71 at order @xmath72 $ ] , assuming as before that @xmath73 . at the end of this section we also outline corrections suppressed by higher powers of @xmath74 . we are looking for the contribution of the connected diagrams to the following expression @xcite : = f_gaaf_gbbf_gccf_gfff_geef_gdd + _ i=1 ^ 6 + _ 1^*f(p - k_2 ) _ 1^*e(q - k_4 ) _ 1^*d(l - k_6 ) _ 1^a(p - k_1)_1^b(q - k_3 ) _ 1^c(l - k_5 ) + _ 2^*f(k_2 ) _ 2^*e(k_4 ) _ 2^*d(k_6 ) _ 2^a(k_1)_2^b(k_3 ) _ 2^c(k_5 ) . [ eq : c3 ] as before , the @xmath20 correlators of the target and the projectile consist of two parts , one from the quadratic operator in the action and another from the additional @xmath75 operator : ^*f ^*e ^*d ^a^b ^c = ^*f ^*e ^*d ^a^b ^c _ + ^*f ^*e ^*d ^a ^b ^c _ . the product of the two gaussian contributions from the target and the projectile , to leading order in the gluon momenta , gives rise to 16 `` rainbow '' diagrams . the result has been obtained previously @xcite and reads ( expressed in terms of the mean multiplicity ) : _ [ eq : gaussian3gluon ] the correction , to first order in @xmath70 is ~2 ^*f ^*e ^*d ^a^b ^c _ ^*f ^*e ^*d ^a ^b ^c _ . [ eq : simcorr ] again , we are considering only rainbow diagrams , so for the gaussian six - point function in the above expression , from all possible contractions , we keep only the term ( 2)^6 ^3 ^af^be^cd(k_1-k_2)(k_3-k_4)(k_5-k_6 ) . to calculate the correction to the six - point function to first order in @xmath70 we factorize it into a product of two- and four - point functions . there are fifteen possible factorizations of that kind . three of them are disconnected diagrams and the remaining twelve give identical contributions . we consider , for example , the following combination : _ 1^*f(p - k_2 ) _ 1^*e(q - k_4 ) _ 1^*d(l - k_6 ) _ 1^a(p - k_1)_1^b(q - k_3 ) _ 1^c(l - k_5 ) + = _ 1^a(p - k_1)_1^b(q - k_3 ) _ 1^*f(p - k_2 ) _ 1^*e(q - k_4 ) _ 1^*d(l - k_6 ) _ 1^c(l - k_5 ) . the two point function is _ 1^a(p - k_1)_1^b(q - k_3 ) = ( 2)^2 ^ab ( p+q - k_1-k_3 ) , and for the correction to the four - point function we use the last line from eq . ( [ eq:4ptcorr ] ) . the color factor is f_gaaf_gbbf_gccf_gfff_geef_gdd^ab(^cd^ef + ^ce^df + ^cf^de ) ^af^be^cd + = 2n_c^3(n_c^2 - 1)+n_c^3(n_c^2 - 1)^2 . putting everything together into eq . ( [ eq : c3 ] ) and multiplying by two [ because of ( [ eq : simcorr ] ) ] and by twelve ( which is the number of possible diagrams ) we get : _ = ^4 ^ 2 + _ i=1 ^ 6 + ( k_1-k_2)(k_3-k_4)(k_5-k_6)(p+q - k_1-k_3)(p+q - k_2-k_4-k_6+k_5 ) + + = -^4 ^ 2 + . again , we regularize the ladder integrals at the saturation scale , . finally , using expression ( [ eq : single ] ) for the mean multiplicity the @xmath75 contribution to three - gluon production becomes _ = - . [ eq : correction3gluon ] summing ( [ eq : gaussian3gluon ] ) and ( [ eq : correction3gluon ] ) , _ = . from the above equation the third reduced moment is : c_3 = - , or [ eq : c3final ] q_s^4s_^2= 1 - 9(n_c^2 + 1 ) , where we have used expressions ( [ eq : beta ] ) and ( [ eq : qs ] ) . for a nbd we have that @xmath76 but if we compare ( [ eq : c3final ] ) to the square of eq . ( [ eq:1/k ] ) , which is q_s^4s_^2=1 - 6(n_c^2 + 1 ) , we see that the coefficients of the corrections at order @xmath77 differ . that means that the @xmath75 operator in the action provides a correction to the negative binomial distribution . in fact , such deviation from a nbd is more obvious if even higher order operators are added to the action . dropping the longitudinal dependence of the operators for simplicity , such an action would have the form sd^2v _ . the additional terms are suppressed by powers of @xmath74 @xcite : ^2~g(ga^1/3 ) , _ 3~g(ga^1/3)^2 , _ 4~g(ga^1/3)^3 , _ 5~g(ga^1/3)^4 , _ 6~g(ga^1/3)^5 . the cubic operator gives a correction to the six - point function , i.e. to @xmath71 at order @xmath78 but does not correct the four - point function , i.e. @xmath79 ( it only renormalizes @xmath36 ) . the same applies to the @xmath80 operator : @xmath71 will contain a term @xmath81 but @xmath82 does not . hence , beyond a quadratic action the relation @xmath83 is not exact . z. koba , h. b. nielsen and p. olesen , nucl . b * 40 * , 317 ( 1972 ) . r. e. ansorge _ et al . _ [ ua5 collaboration ] , z. phys . c * 43 * , 357 ( 1989 ) . k. aamodt _ et al . _ [ alice collaboration ] , eur . j. c * 68 * , 89 ( 2010 ) . v. khachatryan _ et al . _ [ cms collaboration ] , jhep * 1101 * , 079 ( 2011 ) . w. a. zajc , phys . b * 175 * , 219 ( 1986 ) . a. dumitru and y. nara , phys . c * 85 * , 034907 ( 2012 ) . r. ugoccioni and a. giovannini , j. phys . * 5 * , 199 ( 2005 ) [ hep - ph/0410186 ] ; d. prorok , int . j. mod . a * 26 * , 3171 ( 2011 ) [ arxiv:1101.0787 [ hep - ph ] ] ; t. mizoguchi and m. biyajim , arxiv:1207.0916 [ hep - ph ] . a. bassetto , m. ciafaloni and g. marchesini , nucl . b * 163 * , 477 ( 1980 ) ; y. l. dokshitzer , v. s. fadin and v. a. khoze , z. phys . c * 18 * , 37 ( 1983 ) ; y. l. dokshitzer , phys . b * 305 * , 295 ( 1993 ) ; g. p. salam , nucl . b * 449 * , 589 ( 1995 ) . a. krasnitz , y. nara and r. venugopalan , phys . * 87 * , 192302 ( 2001 ) ; nucl . a * 727 * , 427 ( 2003 ) ; t. lappi , phys . c * 67 * , 054903 ( 2003 ) . d. kharzeev , e. levin and m. nardi , nucl . a * 730 * , 448 ( 2004 ) [ erratum - ibid . a * 743 * , 329 ( 2004 ) ] ; nucl . a * 747 * , 609 ( 2005 ) ; a. dumitru , d. e. kharzeev , e. m. levin and y. nara , phys . rev . c * 85 * , 044920 ( 2012 ) ; j. l. albacete , a. dumitru , h. fujii and y. nara , arxiv:1209.2001 [ hep - ph ] . f. gelis , t. lappi and l. mclerran , nucl . a * 828 * , 149 ( 2009 ) . p. tribedy and r. venugopalan , nucl . a * 850 * , 136 ( 2011 ) [ erratum - ibid . a * 859 * , 185 ( 2011 ) ] ; arxiv:1112.2445 [ hep - ph ] . t. lappi , s. srednyak and r. venugopalan , jhep * 1001 * , 066 ( 2010 ) . b. schenke , p. tribedy and r. venugopalan , arxiv:1206.6805 [ hep - ph ] . a. dumitru and e. petreska , nucl . a * 879 * , 59 ( 2012 ) . e. iancu , k. itakura and l. mclerran , nucl . a * 724 * , 181 ( 2003 ) ; a. dumitru , j. jalilian - marian , t. lappi , b. schenke and r. venugopalan , phys . b * 706 * , 219 ( 2011 ) ; e. iancu and d. n. triantafyllopoulos , jhep * 1111 * , 105 ( 2011 ) ; jhep * 1204 * , 025 ( 2012 ) . i. balitsky , phys . d * 75 * , 014001 ( 2007 ) ; y. v. kovchegov and h. weigert , nucl . a * 784 * , 188 ( 2007 ) ; nucl . a * 789 * , 260 ( 2007 ) ; j. l. albacete and y. v. kovchegov , phys . rev . d * 75 * , 125021 ( 2007 ) . e. iancu , j. t. de santana amaral , g. soyez and d. n. triantafyllopoulos , nucl . a * 786 * , 131 ( 2007 ) . a. dumitru , e. iancu , l. portugal , g. soyez and d. n. triantafyllopoulos , jhep * 0708 * , 062 ( 2007 ) . a. m. stasto , k. j. golec - biernat and j. kwiecinski , phys . lett . * 86 * , 596 ( 2001 ) . k. dusling , d. fernandez - fraile and r. venugopalan , nucl . * a828 * , 161 ( 2009 ) [ arxiv:0902.4435 [ nucl - th ] ] .

transverse momentum integrated multiplicities in the central region of @xmath0 collisions at lhc energies satisfy koba - nielsen - olesen scaling . we attempt to relate this finding to multiplicity distributions of soft gluons . kno scaling emerges if the effective theory describing color charge fluctuations at a scale on the order of the saturation momentum is approximately gaussian . from an evolution equation for quantum corrections which includes both saturation as well as fluctuations we find that evolution with the qcd @xmath1-function satisfies kno scaling while fixed - coupling evolution does not . thus , non - linear saturation effects and running - coupling evolution are both required in order to reproduce geometric scaling of the dis cross section and kno scaling of virtual dipoles in a hadron wave function .

despite its still limited extent , the sloan digital sky survey ( sdss ) continues to be a remarkable resource for studies of galactic structure . in addition to the large scale features attributed to past galaxy accretion events @xcite , sdss data were used to detect the remarkably strong tidal tails of palomar 5 @xcite . tidal tails of globular clusters are particularly interesting from a galactic structure standpoint as they are expected to be very numerous and to sample the galactic potential much more uniformly than satellite galaxies . moreover , such tidal tails are dynamically very cold @xcite , making them useful for constraining not only the global galactic potential , but also its lumpiness @xcite . substantial tidal streams have now been found associated with at least two of the eight globular clusters in the sdss area ; pal 5 @xcite and ngc 5466 @xcite . in this letter we examine a much larger region of the sdss to search for more extended structures . we briefly describe our analysis in section [ analysis ] . we discuss the detection of a new stellar stream in section [ discussion ] , make some initial distance estimates in section [ distance ] , attempt to identify a progenitor in section [ progenitor ] , and put initial constraints on the orbit in section [ orbit ] . we make concluding remarks section [ conclusions ] . data comprising @xmath1 , and @xmath2 photometry for @xmath3 stars in the region @xmath4 and @xmath5 were extracted from the sdss database using the sdss casjobs query system . the data were analyzed using the matched filter technique employed by @xcite and @xcite , and described in detail by @xcite . this technique is made necessary by the fact that , over the magnitude range and over the region of sky we are considering , the foreground disk stars outnumber the more distant stars in the galactic halo by three orders of magnitude . applied to the color - magnitude domain , the matched filter is a means by which we can optimally differentiate between two populations , provided we know the color - magnitude distribution of each . in practise , we use the sdss photometry to create a color - magnitude density or hess diagram for both the stars of interest ( e.g. those in a globular cluster ) and the foreground population . dividing the former by the latter , we generate an array of relative weights that constitute an optimal color - magnitude filter . using this filter , every star in the survey can be assigned a weight or probability of being a member of our chosen globular cluster based on its measured magnitude and color . having used observed data to generate it , the filter implicitly includes the effects of photometric uncertainties , and even though a particular star may lie 3@xmath6 away from the main sequence of a globular cluster of interest , its relative weight ( and thus probability of being a cluster star ) may still be high if the number of foreground stars in this region of the hess diagram is relatively low ( e.g. near the horizontal branch , or blueward of the main sequence turnoff ) . as we were initially interested in searching for tidal streams associated with globular clusters , we constructed hess diagrams for each of the eight globular clusters in the sdss dr4 area ( ngc 2419 , pal 3 , ngc 5272 , ngc 5466 , pal 5 , ngc 6205 , ngc 7078 , and ngc 7089 ) . a single hess diagram for field stars was generated using @xmath7 stars spread over @xmath8 deg@xmath9 of dr4 . we then applied each of the eight resulting optimal filters in turn to the entire survey area . the resulting weighted star counts were summed by location on the sky to produce eight different , two dimensional weight images or probability maps . we used all stars with @xmath10 . we dereddended the sdss photometry as a function of position on the sky using the dirbe / iras dust maps of @xcite . we optimally filtered the @xmath11 , @xmath12 , @xmath13 , and @xmath14 star counts independently and then co - added the resulting weight images . in figure 1 we show the final , combined , filtered star count distribution , using a filter matched to the color magnitude distribution of stars in ngc 6205 ( m 13 ) . the image has been smoothed with a gaussian kernel with @xmath15 . a low - order , polynomial surface has been subtracted from the image to approximately remove large scale gradients due to the galactic disk and bulge . quite obvious in figure 1 is a long , remarkably smooth , curving stream of stars , extending over @xmath16 . on the sky the stream runs in an almost straight line through the whole of ursa major and leo minor , ending in cancer and spanning a total of @xmath17 . the stream is most evident when we use a filter that is matched to the color - magnitude distribution and luminosity function of stars in m 13 , although shifted faintwards by 0.2 mag . optimal filters based on the other seven globular clusters in dr4 did not yield the level of contrast that we see in figure 1 . the stream is easily visible in each individual color pair , including @xmath18 . there may be a second , more diffuse feature with @xmath19 about @xmath20 to the north of the stream , but we defer analysis of this feature to a future paper . the stream is not a product of our dereddening procedure ; careful examination of the reddening map of @xcite shows no correlation between this feature and the applied reddening corrections . the maximum values of @xmath21 are @xmath22 , with typical values in the range 0.01 - 0.02 over the length of the stream . rerunning the matched filter analysis without reddening corrections yielded little more than a slight reduction in the apparent strength of the stream . we also ran our optimum filter against the sdss dr4 galaxy catalog to investigate whether the stream could be due to confusion with faint galaxies . ( such a structure in the distribution of galaxies would be no less interesting than a stellar stream of these dimensions ! ) . however , we found no feature in the filtered galaxy counts that could mimic the stream apparent in figure 1 . at its southwestern end the stream is truncated by the limits of the available data . we attempted to trace the stream in the portion of dr4 with @xmath23 but could find no convincing continuation . plausible orbits for the stream ( see below ) predict a fairly narrow range of possible paths across this region , and generally a rather sharp increase in sun - stream distance . we attempted to recover the stream by shifting our filter from -1.0 to + 3.0 mags , but to no avail . a continuation of the stream may well be there , but the power of the matched filter is significantly reduced as the bulk of the main sequence drops below the survey data s 50% completeness threshold . combined with the rapid rise in the number of contaminating galactic disk stars in this region , there appears to be little chance of recovering the stream until much deeper surveys become available . on the northeastern end , the stream becomes indiscernible beyond ra = @xmath24 . we attempted to enhance the northeastern end of the stream by shifting the filter to both brighter and fainter magnitudes , but again to no avail . experiments in which we inserted artificial stellar streams with surface densities similar to those observed in figure 1 revealed that they too largely vanished beyond r.a . = @xmath24 . hence , while it is conceivable that we are seeing the physical end of the stream , it is equally possible that our failure to trace the stream any further reflects once again the rapidly increasing contamination by field stars at lower galactic latitudes . sampling at several representative points , the stream appears to be @xmath25 wide ( fwhm ) on average . this width is similar to those observed in the tidal tails of the globular clusters pal 5 and ngc 5466 @xcite . on the other hand , the width is much narrower than the tidal arms of the sagittarius dwarf @xcite ( one of which runs along the southern edge of the field shown in figure 1 ) . this suggests that the stars making up the stream have a low random velocities , and that they were probably weakly stripped from a relatively small potential . combining this with a location high above the galactic plane ( see below ) suggests that the parent body is or was a globular cluster . integrating the background subtracted , weighted star counts over a width of @xmath26 we find the total number of stars in the discernible stream to be @xmath27 . as is evident in figure 1 , the surface density of stars fluctuates considerably along the stream . for stars with @xmath28 the average surface density is @xmath29 stars deg@xmath30 , with occasional peaks of over 70 stars deg@xmath30 . the power of the matched filter resides primarily at the main sequence turnoff and below , where the luminosity function increases rapidly and the stars lie blueward of the bulk of foreground population . the blue horizontal branch can generate much higher weights per star , but the typical numbers of horizontal branch stars in any likely progenitor are too low to account for such a continuous and well - populated stream ( e.g. @xcite ) . assuming that our filter is indeed beating against the main sequence of the stream population , we can use the filter response to estimate distances . we have attempted to extract the color magnitude distribution for the stream stars directly , but contamination by foreground stars is so high as to make differentiation between stream and field distributions highly uncertain . varying the shift applied to the m 13 matched filter from -0.3 to + 0.7 mags , we measured the background - subtracted , mean weighted star counts along the stream in the regions @xmath31 , @xmath32 , and @xmath33 . we also measured a @xmath34 segment centered on the strongest concentration of stream stars at ( r.a , decl . ) = ( 144.1 , 30.3 ) . to avoid potential problems related to a difference in age between m 13 and the stream stars , we used only the portion of the filter with @xmath35 , where the bright cutoff is 0.8 mags below m 13 s main sequence turn off . this reduces the contrast between the stream and the background by about 40% , but still provides sufficient signal strength to enable a reasonably precise measurement of peak response . if our assumptions above are valid , we are effectively measuring relative distances by main sequence fitting . the mean stream surface densities as a function of filter magnitude shift are shown in figure 2 . using a distance to m 13 of 7.7 kpc and fitting gaussians to the individual profiles , we find sun - stream distances of 7.3 , 8.5 , and 9.1 kpc , respectively , for the three stream segments identified above . the high density clump at r.a . @xmath36 yields a distance of 7.7 kpc . this puts the mean distance of the stream high above the galactic disk at @xmath37 kpc , with the stream oriented almost perpendicularly to our line of sight . the corresponding galactocentric distances range from 13.5 to 15 kpc . the measured distances rise more or less monotonically from southwest to northeast , as might be expected for a feature that traces a small part of an extended galactic orbit . based on the widths of the peaks in figure 2 we estimate our random measurement uncertainties to be @xmath38 pc . while the match between the color - magnitude distributions of stars in m 13 and those in the stream is uncertain , the relative line - of - sight distances along the stream should be fairly robust . the measurement of relative distances rests only on the assumption that the color - magnitude distribution of stream stars is uniform , and that different parts of the stream will respond to filter shifts in the same way . of course , this may not be valid if the luminosity function of stars escaping from the parent cluster changed as a function of time due to the dynamical evolution of the cluster , and a forthcoming paper will deal with possible observational support for this . our relative distance estimates may also be subject to variations in sdss sensitivity and completeness at faint magnitudes , although it seems reasonable to suppose that such variations will have largely averaged out on the scales with which we are dealing . the location and the narrowness of the stream lead us to believe that the progenitor likely is or was a globular cluster . on the other hand , the stream does not pass near any of the eight globular clusters in dr4 . in particular , despite the apparent similarity in color magnitude distribution , there appears to be no way to dynamically associate the stream with m 13 . the distance , radial velocity , and proper motion measurements for m 13 are among the best for any globular cluster @xcite . using these measurements , the projected orbital path of m 13 is neither near nor aligned with the stream for either the last two or the next two orbits of the cluster . we conclude that there is almost no possibility that m 13 could be the progenitor of the current stream . could it be that the stream might be an apogalactic concentration of tidally stripped stars from a globular cluster that is currently elsewhere in its orbit ? we have integrated orbits for 39 globular clusters with measured proper motions @xcite to see whether their orbits are aligned with the stream at its current location . this is fraught with considerable uncertainty as even small errors in the proper motions measurements can lead to rather large departures between the predicted and true orbits of clusters . nonetheless , as a first attempt to associate this stream with a progenitor , we integrated orbits for the 39 clusters , projected them onto the sky , and compared them with the location and orientation of the stream . we used the galactic model of @xcite , which includes a disk and bulge and assumes a spherical halo . only two of the 39 clusters , ngc 1904 and ngc 4590 , have predicted orbits whose projections lie near the stream on the sky and are roughly aligned with it . both of these clusters show evidence of tidal tails @xcite . however , for ngc 1904 the orbital path in the region of the stream passes within 5 kpc of the sun . this would require a brightward shift of the m 13 filter of over a magnitude , a shift that we have tested and for which the stream all but disappears . for ngc 4590 , the apparent alignment with the stream is much closer , but the predicted distances at the eastern and western ends of the stream are 8.2 and 19 kpc , respectively . thus , not only does the mean distance disagree with our estimate above , but the spatial orientation of the stream is at odds with the predicted orbit by more than @xmath39 . where then is the progenitor of the stream ? it is certainly possible that the stream represents the leavings of one of the @xmath40 clusters for which we can not yet estimate orbits . alternatively , the source of the stream may be embedded in the stream itself . the densest portions of the stream occur at ( r.a . , decl . ) = ( @xmath41 ) and ( @xmath42 ) , each with surface densities of over 70 stars deg@xmath30 . however , 70 stars deg@xmath30 to almost 4 mags below the turn off would barely qualify as an open cluster . examining the distribution of sdss catalog stars directly , there appears to be no tendency for the stars to cluster in these regions . the highest density peaks in figure 1 are probably not bound clusters , and may rather be entirely analogous to similar peaks seen in the tidal stream of pal 5 , which are interpreted as a natural consequence of the episodic nature of tidal stripping . alternatively , one of the peaks in figure 1 could be the _ remnant _ of the parent cluster . the existence of such disrupted clusters would not be unexpected @xcite , and pal 5 itself is believed to have lost at least 90% of its mass and to be on its last orbit around the galaxy @xcite . lacking velocity measurements , we can not `` solve '' for the orbit of the stream . however , the apparent orientation of the stream , along with our estimates of its distance and curvature , can yield some constraints . again using the galactic model of @xcite ( which @xcite and @xcite found to work reasonably well for ngc 5466 and pal 5 ) , we use a least squares method to fit both the orientation on the sky and the distance measurements in section [ distance ] . the tangential velocity at each point is primarily constrained by the projected path of the stream on the sky , while our distance estimates help to limit the range of possible radial velocities . we fit to a number of normal points lying along the centerline of the stream and adopt a radial velocity fiducial point at the midpoint of the northeastern segment above , with r.a . , dec = ( 202.0 , 58.4 ) . if we give no weight to our relative distance estimates but use only the @xmath43 kpc estimate for the fiducial point , then for a reasonable range of radial velocities ( @xmath44 km s@xmath45 km s@xmath46 ) , the perigalacticon of the stream s orbit must lie in the range 6.5 kpc @xmath47 kpc . for all lsr radial velocities in the range -270 km s@xmath48 km s@xmath46 , the apogalactic radius is constrained to fall within 17 kpc @xmath49 kpc . on the other hand , positive radial velocities lead to orbits with apogalactica rising from 100 to 500 kpc . all of these orbits give excellent fits to the observed path of the stream on the sky . if we constrain the model using our relative distance estimates ( allowing the proper motions to become free ranging and uninteresting parameters ) we find a best fit value for the radial velocity at the fiducial point of @xmath50 km s@xmath46 , where the uncertainty corresponds to the 95% confidence interval . the orbit corresponding to this radial velocity has @xmath51 kpc and @xmath52 kpc , where the uncertainties reflect only our estimated uncertainty in the radial velocity . for this orbit , the physical length of the visible stream would be @xmath53 kpc . applying optimal contrast filtering techniques to sdss data , we have detected a stream of stars some @xmath0 long on the sky . we are unable to identify a progenitor for this stream , although from its appearance and location on the sky , we believe it to be a either an extant or disrupted globular cluster . based on a good match to the color - magnitude distribution of stars in m 13 , we conclude that the stars making up the stream are primarily old and metal poor , and that the stream as a whole is about 8.5 kpc distant and roughly perpendicular to our line of sight . refinement of the stream s orbit will require radial velocity measurements of individual stars along its length . ultimately , the vetted stream stars will become prime targets for the _ space interferometry mission _ , whose proper motion measurements will enable very much stronger constraints to be placed on both the orbit of the progenitor and on the potential field of the galaxy . we are grateful to helio rocha - pinto for comments which significantly improved the manuscript . 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 .

we report on the detection in sloan digital sky survey data of a @xmath0-long tidal stream of stars , extending from ursa major to cancer . the stream has no obvious association with the orbit of any known cluster or galaxy . the contrast of the detected stream is greatest when using a star count filter that is matched to the color - magnitude distribution of stars in m 13 , which suggests that the stars making up the stream are old and metal poor . the visible portion of the stream is very narrow and about 8.5 kpc above the galactic disk , suggesting that the progenitor is or was a globular cluster . while the surface density of the stream varies considerably along its length , its path on the sky is very smooth and uniform , showing no evidence of perturbations by large mass concentrations in the nearby halo . while definitive constraints can not be established without radial velocity information , the stream s projected path and estimates of its distance suggest that we are observing the stream near the perigalacticon of its orbit .

entanglement renormalization@xcite is a renormalization group ( rg ) approach to quantum many - body systems on a lattice . as with most rg methods @xcite , it proceeds by coarse - graining the microscopic degrees of freedom of a many - body system , and thus also their hamiltonian @xmath0 , to produce a sequence of effective systems , with hamiltonians @xmath1 that define a flow towards larger length scale / lower energies . entanglement renormalization operates in _ real space _ ( it does not rely on fourier space analysis ) and it is a _ non - perturbative _ approach ( that is , it can handle interactions of any strength ) . as a result , it has a wide range of applicability , from quantum criticality @xcite to emergent topological order @xcite , from frustrated antiferromagnets @xcite to interacting fermions @xcite and even to interacting anyons @xcite . entanglement renormalization produces an efficient ( approximate ) representation of the ground state of the system in terms of a variational tensor network , the multi - scale entanglement renormalization ansatz ( mera ) @xcite , from which one can extract expectation values of arbitrary local observables . most applications of the mera have so far focused on systems that are translation invariant . here we will consider instead systems where translation invariance is explicitly broken by the presence of a defect . for simplicity , we assume that the defect is placed on an infinite quantum critical system that , in the absence of the defect , would be both homogeneous ( that is , translation invariant ) and a fixed point of the rg ( that is , scale invariant ) . under that assumption , the mera offers a shockingly simple description : in the absence of the defect it is completely characterized by a single pair of tensors @xmath2 and , in the presence of the defect , by just one additional tensor @xmath3 if the defect is also itself at a ( scale invariant ) fixed point of the rg flow ; or by a sequence of a few additional tensors @xmath4 that describe its flow towards an rg fixed point . in this paper we propose and benchmark algorithms for quantum critical systems in the presence of defects that exploit the simple description afforded by the mera . we start by briefly reviewing the required background material on entanglement renormalization , including a recently proposed theory of minimal updates @xcite that is at the core of the surprisingly compact mera description of defects in quantum critical systems . two distinctive aspects of entanglement renormalization are the _ tensor network _ structure of the coarse - graining transformation and the _ variational _ nature of the approach . the coarse - graining transformation is implemented by a linear ( isometric ) map @xmath5 , relating the hilbert spaces of the lattice system before and after coarse - graining . as illustrated in fig . [ fig : er](a ) , the linear map @xmath5 decomposes as a network of tensors , called disentanglers @xmath6 and isometries @xmath7 . the structure of the network has been designed with the important property that @xmath5 preserves locality : local operators are mapped into local operators . thus , if @xmath0 is a short - ranged hamiltonian , then the effective hamiltonians @xmath8,@xmath9 , etc , are also short - ranged . on the other hand , the approach is variational . the disentanglers @xmath6 and isometries @xmath7 are loaded with variational parameters , which are determined through energy minimization . this ensures that the coarse - graining transformation @xmath5 is properly adapted to the system under consideration . that is , instead of deciding a priori which degrees of freedom should be kept and which should be thrown away , the method proceeds by asking the hamiltonian @xmath0 which part of many - body hilbert space corresponds to low energies and proceeds to safely remove the rest . for a lattice in @xmath10 dimensions that decomposes as a tensor network made of disentanglers @xmath6 , depicted as squares , and isometries @xmath7 , depicted as triangles . ( b ) the mera on a @xmath10 dimensional lattice made of @xmath11 sites , obtained by collecting together a sequence of coarse - graining transformations @xmath12.,width=321 ] however , the most prominent feature of entanglement renormalization , setting it apart from other real space rg approaches , is its handling of short - range entanglement . while isometries @xmath7 map a block of sites into an effective site , and thus play a rather standard role in a coarse - graining transformation , disentanglers @xmath6 perform a more singular task : the removal of short - range entanglement from the system . thanks to this removal , the coarse - graining transformation @xmath5 constitutes a proper implementation of the rg @xcite , in that the sequence of effective systems , with hamiltonians @xmath13 , only retain degrees of freedom corresponding to increasing length scales . in particular , at fixed - points of the rg flow , entanglement renormalization explicitly realizes scale invariance : the system before coarse - graining and the system after coarse - graining are seen to be locally identical . the mera @xcite is the class of tensor network state that results from joining the sequence of coarse - graining transformations @xmath14 , see fig . [ fig : er](b ) . it is a variational ansatz for ground states ( or , more generally , low energy states ) of many - body systems on a lattice in @xmath15 spatial dimensions . by construction , the mera extends in @xmath16 dimensions , where the additional dimension corresponds to length scale or rg flow . as a result , it is distinctly well suited to study systems where several length scales are relevant , because the information related to each length scale is stored in a different part of the network . in particular , the mera offers an extremely compact description of ground states of homogeneous systems at fixed points of the rg flow , that is , in systems with both translation invariance and scale invariance . these encompass both stable ( gapped ) rg fixed points , which include topologically ordered systems @xcite , and unstable ( gapless ) rg fixed points , corresponding to quantum critical systems @xcite . indeed , translation invariance leads to a position - independent coarse - graining transformation @xmath5 , made of copies of a single pair of tensors @xmath17 , whereas scale invariance implies that the same @xmath5 can be used at all scales . as a result , the single pair @xmath18 completely characterizes the state of an infinite system . the study of quantum critical systems is therefore among the natural targets of the mera . until now , most applications of the mera to quantum criticality have focused on systems that are invariant under translations ( see , however , refs . ) . in translation invariant systems , the mera provides direct access to the universal information of the quantum phase transition , as often encoded in the conformal data of an underlying conformal field theory@xcite ( cft ) ( see appx . [ sect : mera ] for a review ) . in particular , in one spatial dimension one can extract the central charge and identify the set of primary scaling operators @xmath19 ( both local @xcite and non - local @xcite ) together with their scaling dimensions @xmath20 ( from which most critical exponents of the theory follow ) as well as the corresponding operator product expansion coefficients . this data completely characterizes the underlying cft . the goal of this manuscript is to address quantum critical systems where the translation invariance of a system is explicitly broken by the presence of a _ boundary _ , an _ impurity _ , an _ interface _ , etc . we refer to any such obstruction to translation invariance generically as a _ defect _ , and to the system in the absence of the defects as the _ host _ system . methods for simulating quantum critical systems with such defects are important in order to understand and model their effects in realistic settings . a major difficulty in addressing such systems is that , since the presence of a defect manifestly breaks the translation invariance of the host hamiltonian , the ground state is no longer homogeneous . instead , expectation values of local observables differ from the homogeneous case throughout the whole system by an amount that only decays as a power law with the distance to the defect . in this scenario a natural option ( which we will not follow here ) would be to choose a coarse - graining map @xmath5 with position - dependent disentanglers and isometries that adjust to the power law profile of ground state expectation values . notice that the resulting mera would be made of a large number ( proportional to the system size ) of inequivalent disentanglers and isometries , and would therefore incur much larger computational costs ( again , proportional to the system size ) than in a homogeneous system . importantly , we would not be able to study infinite systems directly , and when extracting the low energy properties of the defect , these would be significantly contaminated by ubiquitous finite size effects , which vanish as a power law with the system size . what one would like , then , is a mera description of many - body systems with defects that is nearly as compact as in the homogeneous case . fortunately , a recent theory of _ minimal updates _ in holography @xcite provides us with a recipe to obtain such a description . let @xmath0 denote a local hamiltonian for an extended many - body system on a @xmath15-dimensional lattice , and let @xmath21 @xmath22 denote the hamiltonian for the same system after we added a new term @xmath23 localized in region @xmath24 . in addition , let @xmath25 and @xmath26 denote the ground states of the hamiltonian @xmath0 and of hamiltonian @xmath21 ( the modified hamiltonian ) , respectively . then , the theory of minimal updates in holography @xcite argues in favor of the following conjecture . _ * conjecture ( minimal update ) : * a mera for @xmath26 can be obtained from a mera for @xmath25 by modifying the latter only in the causal cone @xmath27 of region @xmath24 . _ here , the causal cone @xmath27 of region @xmath24 is the part of the mera that describes the successive coarse - graining of region @xmath28 . for instance , for a region @xmath24 consisting of two contiguous sites , fig . [ fig : directedmera ] illustrates the causal cone @xmath27 . the figure also shows how a mera for @xmath25 should be modified to obtain a mera for @xmath26 . of a lattice hamiltonian @xmath0 in @xmath10 space dimensions . scale and translation invariance result in a compact description : two tensors @xmath29 are repeated throughout the infinite tensor network . ( b ) the theory of minimal updates dictates that the ground state @xmath26 of the hamiltonian @xmath30 is represented by a mera with the same tensors @xmath29 outside the causal cone @xmath31 ( shaded ) , whereas inside @xmath31 two new tensors @xmath32 are repeated throughout the semi - infinite causal cone . ( c - d ) the same illustrations , without drawing the tensors of the network.,width=321 ] in this paper we propose and benchmark mera algorithms for quantum critical system with one or several defects . the theoretical foundation of the algorithms is the above conjecture on minimal updates , specialized to a hamiltonian of the form @xmath33 where @xmath0 is the hamiltonian for the host system and @xmath34 is the hamiltonian describing the localized defect . more specifically , we will assume that the host hamiltonian @xmath0 , which describes an infinite system on a lattice , is a homogeneous , critical , fixed - point hamiltonian , so that its ground state @xmath25 can be succinctly described by a mera that is characterized in terms of just a single pair of tensors @xmath17 . region @xmath24 will typically consists of one or two sites . then , following the above conjecture , a mera for the ground state @xmath35 of the hamiltonian @xmath36 , which we call _ modular _ mera and will be further described in sect . [ sect : modularity ] , is completely characterized in terms of two sets of tensors , see fig . [ fig : directedmera ] . first , the pair of tensors @xmath17 corresponding to the ( scale and translation invariant ) host system , is repeated throughout the outside of the causal cone of the defect . second , ( for a defect that is scale invariant , that is , a fixed point of the rg flow ) another pair of tensors @xmath37 is repeated throughout the inside of the causal cone of the defect . after some rewiring of the modular mera , this second pair @xmath38 will be replaced by a single tensor @xmath3 . [ some settings will require slight modifications of this simple description . for instance , in the case of interfaces involving several types of system , each system will contribute a different pair of tensors for the outside of the causal cone . on the other hand , if the defect is not yet at a fixed - point of the rg flow , then instead of a single tensor @xmath3 , a sequence of scale - dependent tensors @xmath4 will be used to account for the flow of the defect into the rg fixed - point . ] the modular mera leads to simple numerical algorithms for quantum critical systems in the presence of one of several defects , which complement and generalize those discussed in ref . for homogeneous systems . as in the homogeneous case , the computational cost of the new algorithms is independent of the system size , allowing us to address infinite systems . in this way , we can extract the universal , low energy properties associated to a defect directly in the thermodynamic limit , where they are free of finite - size effects . although in this paper we restrict our attention to systems in @xmath10 dimensions for simplicity , the key idea of the algorithms can also be applied to systems in @xmath39 dimensions . in the discussion in sect . [ sect : conclusion ] we will also address how to lift the assumption , present throughout this work , that the host system is both translation and scale invariant . the algorithms proposed in this paper are thus based on assuming the validity of the conjectured theory of minimal updates in holography of ref . . we contribute to that theory in two ways . first , by applying the above conjecture recursively , we will investigate applications that go well beyond the simple scenario described in ref . , namely that of a single impurity . specifically , the _ modular _ mera describes the ground state of a complex system , such as an interface between two systems @xmath40 and @xmath41 , by combining ` modules ' obtained by studying simpler systems , such as homogeneous versions of system @xmath40 and system @xmath41 , separately . modularity is central to the algorithms proposed in this work and key to their computational efficiency . second , the benchmark results presented here constitute solid evidence that the conjectured minimal updates are indeed sufficient to accurately represent a large variety of defects . this contributes significantly to establishing the theory of minimal updates , which so far was supported mostly by the theoretical arguments provided in ref . . in this paper we assume that the reader is already familiar with the scale invariant mera for translation invariant systems ( a detailed introduction to which can be found in ref . ) . however , for completeness , we have also included a brief review to the mera in the presence of scale and translation invariance in appx . [ sect : mera ] . [ sect : modularity ] introduces the modular mera and describes how they can be applied to quantum critical systems with an impurity , boundary , interface , and more complex settings , such as several defects or y - interfaces involving three systems ( also called y - junctions ) . it also explains how to extract the low energy , universal properties of the defect . [ sect : optmod ] discusses how to optimize the modular mera . this is illustrated with the paradigmatic case of a single impurity . the first step involves optimizing a mera for the homogeneous system ( refs . ) so as to obtain the pair of tensors @xmath17 . then an effective hamiltonian for the causal cone of the impurity , or _ wilson chain _ , is produced by properly coarse - graining the host hamiltonian @xmath0 and adding the impurity term @xmath23 . finally , a simplified tensor network ansatz for the ground state of the wilson chain is optimized by energy minimization , from which one would be able to extract tensor @xmath3 ( or tensors @xmath42 . [ sect : bench ] benchmarks the modular mera algorithm for a number of quantum critical systems in @xmath10 spatial dimension . these include systems with one and several impurities , systems with one or two boundaries , interfaces between two systems , and y - interfaces between three systems . for each type of defect , we outline how the basic algorithm of sect . [ sect : optmod ] needs to be modified . the approach is seen to provide accurate numerical results for ground state properties , both for expectation values of local observables and for low energy , universal properties ( e.g. in the form of conformal data describing an underlying cft , including the critical exponents associated to the defect ) . finally , sect . [ sect : conclusion ] concludes the paper with a discussion and a summary of results . we have also included three appendices . [ sect : mera ] provides a basic introduction to key aspects of er and mera used throughout the manuscript , and reviews how to extract universal properties ( conformal data ) from a translation and scale invariant mera . b and c provide technical details on certain aspects of the modular mera . in this section we introduce the modular mera for homogeneous systems with one or several defects . we also explain how to extract the universal properties of a defect , including its set of scaling dimensions , from which one can derive all critical exponents associated to the defect . for simplicity , we only consider lattice systems in one spatial dimension . the modular mera is built upon the conjecture that the presence of a defect can be accurately accounted for by only updating the interior of the causal cone @xmath27 of the region @xmath24 on which the defect is supported . below we will argue that , when applied recursively , this minimal update implies that we can describe e.g. an interface between two semi - infinite quantum critical spin chains by combining ` modules ' that describe the two systems individually , that is , in the absence of an interface . we refer to this property as _ modularity _ in the holographic description of quantum states . next we describe the modular mera for systems with a single impurity , an open boundary , or an interface of two different quantum systems ( notice that the impurity system can be considered as an interface of two identical systems , while the open boundary can be considered as an interface with a trivial system ) , before discussing more general applications of modularity , such as systems with multiple impurities or y - interfaces of three quantum chains . a note on terminology. we call _ modular _ mera any mera for a system with one or several defects that , following the theory of minimal updates of ref . , has been obtained from a mera for the host system ( that is , without the defects ) by modifying only the tensors in the causal cone of the defects . on the other hand , for specific types of defects , such as an impurity , a boundary , etc , we also occasionally use the more specific terms _ impurity _ mera , _ boundary _ mera , etc , to denote the corresponding specific type of modular meras . throughout this section , the quantum critical , homogeneous host system is described by an infinite lattice @xmath43 in one dimension , with a fixed - point hamiltonian @xmath44 made of constant nearest neighbor couplings @xmath45 , such that its the ground state @xmath25 of @xmath0 can be represented by a ( scale invariant and translation invariant ) mera with a single pair of tensors @xmath17 . let us first consider an impurity problem in one spatial dimension , with hamiltonian @xmath46 where @xmath47 accounts for an impurity that is supported on a small region @xmath24 , which in the following is supposed to be made of two contiguous sites . let @xmath48 denote the ground state of hamiltonian @xmath49 . then , the theory of minimal updates in holography @xcite asserts that a mera for the ground state @xmath48 can be obtained by modifying the mera for @xmath50 only in the causal cone @xmath27 of region @xmath24 , which we assume to also be scale invariant . accordingly , the impurity mera is fully described by two pairs of tensors @xmath51 and @xmath52 . [ if the impurity is not scale invariant , then additional pairs of scale - dependent tensors @xmath53 inside the causal cone will be required in order to describe the non - trivial rg flow of the impurity to a scale invariant , rg fixed point . [ fig : defectmera](a ) depicts the impurity mera . in practical computations , we find it more convenient to apply cosmetic changes inside the causal cone of the tensor network , as described in fig . [ fig : defectmera](b - c ) , and work instead with the impurity mera depicted in fig . [ fig : defectmera](c ) . this requires first splitting the isometries @xmath7 within the causal cone @xmath27 into pairs of binary isometries @xmath54 and @xmath55 , as described in appendix [ sect : isodecomp ] , and then further simplifying the tensor network inside the causal cone replacing the pair of tensors @xmath37 by a single tensor @xmath3 . [ if the impurity is not scale invariant , then additional scale - dependent tensors @xmath4 will be required ] . notice that figs . [ fig : defectmera](a ) and [ fig : defectmera](c ) represent two essentially equivalent forms of the modular mera . however , the latter form is slightly simpler and , accordingly , we will use it in the theoretical discussion of sect . [ sect : critmod ] and in the benchmark results of sect . [ sect : benchimpurity ] . of hamiltonian @xmath49 , eq . ( a ) regular form of an impurity mera for @xmath56 , originating in the mera for a scale - invariant , translation invariant state @xmath25 described by a pair of tensors @xmath17 , and that has a different pair of tensors @xmath37 inside the causal cone @xmath27 ( shaded ) of the local region @xmath24 associated to the impurity . ( b ) prior to modifying the homogeneous mera , we can decompose some of its isometries @xmath7 into upper @xmath54 and lower @xmath55 isometries , as described in appendix [ sect : isodecomp ] . ( c ) a slightly different impurity mera for the same ground state @xmath56 is obtained by replacing the tensors within the causal cone @xmath27 of the tensor network in ( b ) with a new set of isometric tensors @xmath3.,width=321 ] let us now consider a modular mera for a semi - infinite chain with a boundary . notice that a special case of the impurity hamiltonian of eq . [ s3e2 ] corresponds to an impurity that cancels out the interaction between the two sites in region @xmath28 , @xmath57 where @xmath58 denotes the part of the homogeneous hamiltonian @xmath59 that is supported on @xmath24 . [ more generally , @xmath60 could also contain additional single - site terms , such as a single - site magnetic field , etc . ] notice that , since we are dealing with a special case of the impurity hamiltonian of eq . [ s3e2 ] , the impurity mera of fig . [ fig : boundarymera](a ) could be used as an ansatz for its ground state . however , since there is no interaction ( and therefore no entanglement ) between the left and right semi - infinite halves of the system , we can simplify the impurity mera by setting the disentanglers @xmath61 within the causal cone to identity , resulting in the ( doubled ) boundary mera depicted in fig . [ fig : boundarymera](b ) . in other words , the theory of minimal updates @xcite asserts that a modular mera , consisting of ` half ' a homogeneous mera and a single column of boundary tensors @xmath3 , can be used to represent the ground state @xmath62 of a homogeneous hamiltonian with an open boundary , @xmath63 where the additional ( and completely unconstrained ) one - site term @xmath64 is included to set the boundary condition . this form of modular mera for boundary problems , boundary mera , was first proposed and tested in ref . . there , however , no theoretical justification of its remarkable success was provided . in sect . [ sect : benchbound ] we expand upon these previous results for boundary mera , by benchmarking the ansatz both for semi - infinite chains and for finite systems with two open boundaries . note that a related form of boundary mera was also proposed in ref . . of hamiltonian @xmath65 ( a ) an impurity mera can be used as an ansatz for the ground state @xmath56 of a homogeneous hamiltonian @xmath0 that has an impurity @xmath66 added on region @xmath24 , see also fig . [ fig : defectmera ] . ( b ) as a special case of the impurity mera , if the impurity @xmath66 is chosen such as to remove all interaction between the left and right halves of the chain , as described in eq . [ s3e3 ] , then the disentanglers @xmath61 from ( a ) can be set to identity . in this way we obtain ( two copies of ) the boundary mera , an ansatz for the ground state @xmath62 of a semi - infinite system with a single open boundary.,width=321 ] next we describe a modular mera for an interface between two semi - infinite , homogeneous systems @xmath40 and @xmath41 . consider an infinite chain with hamiltonian @xmath67 where @xmath68 ( @xmath69 ) is the restriction to the left ( right ) semi - infinite half of the chain of a hamiltonian for a scale and translation invariant system @xmath40 ( @xmath41 ) , and where @xmath70 describes a coupling between @xmath40 and @xmath41 across the interface @xmath24 . if the strength @xmath71 of the interface coupling is set at @xmath72 , then hamiltonian @xmath73 reduces to a pair of non - interacting open boundary hamiltonians of the form described in eq . [ s3e3b ] . in this case , the ground state could be represented with two ( different ) boundary meras , as depicted in fig . [ fig : interfacemera](a ) . if we now consider switching on the interface coupling , i.e. @xmath74 , then the theory of minimal updates asserts that only the inside of the causal cone of @xmath24 in fig . [ fig : interfacemera](a ) needs be modified . similar to the approach with the impurity mera in fig . [ fig : defectmera](c ) , we replace the structure within the causal cone by a new set of isometric tensors @xmath3 , which leads to the interface mera as shown in fig . [ fig : interfacemera](b ) . the performance of the interface mera is benchmarked in sect . [ sect : benchtwo ] . in eq . ( b ) if a non - zero interface coupling @xmath75 is introduced , then the mera from ( a ) is modified within the causal cone @xmath27 of region @xmath24 with the introduction of a new set of isometric tensors @xmath3 . the resulting ansatz is an interface mera.,width=321 ] the theory of minimal updates produces a modular mera also for more complex problems , such as systems involving multiple impurities , or for systems with several types of defects , such a system with both a boundary and an impurity . in the benchmark results of sect . [ sect : bench ] we describe a modular mera for a system with two impurities , for a finite system with two open boundaries , and for a y - interface of three semi - infinite quantum spin chains . a summary of several types of modular mera , together with the corresponding hamiltonians , is depicted in fig . [ fig : meratypes ] . notice that in all instances , the modular mera is characterized by a small number of tensors that does not scale with the system size . thus it can be used to address thermodynamically large systems directly , as shall be demonstrated in the benchmark results . from a homogeneous system , and dark shading indicates regions occupied by tensors associated to a defect . ( a ) mera for the scale and translation invariant ground state @xmath76 of a homogeneous hamiltonian @xmath77 . ( b ) impurity mera for the ground state @xmath56 of an impurity hamiltonian @xmath49 , eq . [ s3e2 ] , see also fig . [ fig : defectmera ] . ( c ) modular mera for the ground state @xmath78 of a hamiltonian @xmath79 with two impurities localized on disjoint regions @xmath80 and @xmath81 . ( d ) tensor product of two boundary meras for the ground state @xmath82 of an impurity hamiltonian @xmath49 in which the impurity is used to remove any interaction between the left and right halves of the chain . ( e ) modular mera for the ground state @xmath83 of the hamiltonian @xmath84 for a finite chain with two open boundaries at @xmath85 and @xmath86 . ( f ) interface mera for the ground state @xmath87 of an interface hamiltonian @xmath73 , eq . [ s3e4 ] , describing the interface between two two homogeneous systems @xmath40 and @xmath41.,width=321 ] located at the impurity site of an impurity mera , is coarse - grained into one - site operators @xmath88 , then @xmath89 , and so forth . ( b ) the scaling superoperator @xmath90 associated to the impurity . ( c ) an operator at the site of the impurity @xmath91 and an operator @xmath92 some distance @xmath93 from the impurity become nearest neighbors after @xmath94 coarse - graining steps.,width=321 ] next we explain how to extract the large length scale , universal properties of a defect from the modular mera . we will see that the structure of the ansatz automatically implies ( i ) the existence of a new set of scaling operators and scaling dimensions associated to the defect [ that is , in addition to the ( so - called _ bulk _ ) scaling operators and scaling dimensions associated to the host system , see appx . [ sect : scalemera ] ] ; ( ii ) that the expectation values of local observables differ from those in the absence of the defect by an amount that decays as a power - law with the distance to the defect . these properties , which match those obtained in the context of boundary conformal field theory ( bcft ) @xcite , indicate that the modular mera is a very natural ansatz to describe ground states of quantum critical systems in the presence of a defect , and further justifies the validity of the theory of minimal updates of ref . . for concreteness , let us consider the impurity mera in fig . [ fig : defectmera](c ) , which is fully characterized by the ( homogeneous ) tensors @xmath95 and the impurity tensor @xmath3 . let @xmath96 be a local operator that is measured on the region @xmath24 where the impurity is located ( which we effectively collapse into a single site ) . each layer @xmath5 of the impurity mera can be interpreted as a coarse - graining transformation that will map @xmath96 into a new local operator , @xmath97 as also illustrated in fig . [ fig : twocorrloc](a ) . the coarse - graining of one - site operators located at the impurity is achieved by means of a scaling superoperator @xmath98 associated to the impurity , @xmath99 where the form of @xmath98 is depicted in fig . [ fig : twocorrloc](b ) . notice that @xmath98 depends only on the impurity tensor @xmath3 ( i.e. it does not depend on tensors @xmath95 ) . one can diagonalize the impurity superoperator @xmath98 ( as was done with the scaling superoperator @xmath100 in appx . [ sect : scalemera ] ) to obtain its scaling operators @xmath101 and scaling dimensions @xmath102 , which are defined as @xmath103 let us now evaluate the ground state correlator between an impurity scaling operator @xmath101 located at the site of the impurity ( @xmath104 ) , and a bulk scaling operator @xmath105 located at site @xmath93 , @xmath106 , as illustrated in fig . [ fig : twocorrloc](c ) . for convenience we choose @xmath107 for a integer @xmath108 . after applying one layer of coarse - graining the distance between the scaling operators is reduced to @xmath109 , @xmath110 , which leads to the equality , @xmath111 where @xmath112 and @xmath113 are eigenvalues of the scaling superoperators @xmath98 and @xmath114 , respectively . after @xmath115 coarse - graining transformations , the two scaling operators become nearest neighbors in the ( effective ) lattice . iterating eq . [ s3e7 ] that many times , we obtain @xmath116 in the last step we have ignored a subdominant term that becomes negligible in the large @xmath93 limit , and have introduced the constant @xmath117 . the constant @xmath118 is defined as the correlator for the scaling operators on adjacent sites , @xmath119 here @xmath120 is the two - site reduced density matrix on the site of the impurity and the adjacent site . [ s3e8 ] reproduces a well - established result from bcft @xcite : the correlator between a scaling operator at the impurity and a scaling operator outside the impurity decays polynomially with the distance @xmath93 , with an exponent that is the sum of the corresponding impurity scaling dimension @xmath121 and bulk scaling dimension @xmath122 . let us now specialize eq . [ s3e8 ] by setting the impurity scaling operator to the identity , @xmath123 . this leads to @xmath124 i.e. , the expectation value of a bulk scaling operator @xmath105 tends to zero polynomially in distance @xmath125 from the impurity with an exponent equal to its scaling dimension @xmath122 . recall that in a bulk critical system all bulk scaling operators ( with the exception of the identity ) have vanishing expectation value , @xmath126 . thus , in the large @xmath93 limit , the expectation value of arbitrary local operator @xmath127 located at site @xmath93 of the impurity mera differs from its bulk expectation value @xmath128 as , @xmath129 where the exponent @xmath130 of the decay represents the dominant ( smallest , non - zero ) scaling dimension of the operator @xmath96 when decomposed in a basis of bulk scaling operators . [ s3e11 ] shows that in the modular mera the expectation values of local observables deviate from bulk expectation values everywhere , with a magnitude that decays polynomially with respect to the distance @xmath93 from the defect . in this section we describe how the modular mera can be optimized . for concreteness , we focus on the optimization of the impurity mera depicted in fig . [ fig : logscale](a ) , noting that other modular meras , such as those introduced in sect . [ sect : modularity ] , can be optimized using a similar approach . in the following , the impurity mera will be optimized so as to approximate the ground state of an impurity hamiltonian @xmath0 of the form , @xmath131 where @xmath132 is the hamiltonian of a translation invariant , quantum critical host system and the term @xmath133 represents a local impurity localized on a region @xmath24 of the lattice . the proposed optimization algorithm is a direct implementation of the theory of minimal updates . first , a scale - invariant mera for the ground state @xmath76 of the host hamiltonian @xmath77 is obtained , which is then modified within the causal cone @xmath27 of region @xmath24 in order to account for the impurity @xmath133 and obtain the ground state @xmath56 of @xmath49 . the three steps for optimizing the impurity mera are thus as follows : 1 . the tensors @xmath17 describing the host system are obtained through optimization of a scale - invariant mera for the ground state @xmath76 of the host hamiltonian @xmath77 . [ step : s1e1 ] 2 . the original impurity hamiltonian @xmath134 , defined on the infinite lattice @xmath135 , is mapped to an effective hamiltonian @xmath136 on a semi - infinite wilson chain @xmath137 ( to be introduced below ) , @xmath138 through an inhomogeneous coarse - graining @xmath139 defined in terms of tensors @xmath17 . [ step : s1e2 ] 3 . the impurity tensors @xmath3 are obtained through optimization of a tensor network approximation to the ground state @xmath140 of the effective problem @xmath136 on the wilson chain . [ step : s1e3 ] the optimization of the mera for the host hamiltonian , step [ step : s1e1 ] above , has been covered extensively in e.g. refs . @xcite to which we refer the reader . we now describe in sect . [ sect : logscale ] the details of step [ step : s1e2 ] , and in sect . [ sect : optlog ] the optimization algorithm for step [ step : s1e3 ] . and impurity tensors @xmath3 for a @xmath141 lattice @xmath135 . the causal cone @xmath27 of the impurity region @xmath24 is shaded ; the wilson chain @xmath142 is the @xmath141 lattice formed along the boundary of this causal cone . ( b ) the inhomogeneous coarse - graining @xmath139 maps the initial hamiltonian @xmath0 , here partitioned into shells @xmath143 of varying size ( see eq . [ eq : shells ] ) , to the effective hamiltonian @xmath136 defined on the wilson chain @xmath144 . ( c ) a schematic depiction of the coarse - graining of a term from the local hamiltonian @xmath145 , assuming scale invariance of the hamiltonian @xmath0 , to a local coupling on the wilson chain , see eq . [ eq : ad6 ] . ( d ) diagrammatic representation of the coarse - graining described in eq . [ eq : ad7 ] for @xmath146 . ( e ) diagrammatic representation of the coarse - graining described in eq . [ eq : ad7 ] for @xmath147 . ( g ) a diagrammatic representation of @xmath148.,width=321 ] consider a mera on lattice @xmath135 , and a region @xmath24 with corresponding causal cone @xmath27 . we call the _ wilson chain _ of region @xmath28 , denoted @xmath149 , the one - dimensional lattice obtained by following the surface of the causal cone @xmath150 , see fig . [ fig : logscale](a ) . that is , the hilbert space for the wilson chain is built by coarse - graining the hilbert space of the initial lattice @xmath43 with an _ inhomogeneous _ ( logarithmic scale ) coarse - graining transformation @xmath139 , which is comprised of all the tensors in the mera that lay outside the causal cone @xmath150 , see fig . [ fig : logscale](b ) . in the following we describe how the hamiltonian @xmath49 defined on lattice @xmath135 is coarse - grained to an effective hamiltonian @xmath136 on this wilson chain , which , by construction , can be seen to be only made of nearest neighbor terms , @xmath151 here the nearest neighbor coupling @xmath152 depends on @xmath153 . however , below we will see that scale invariance of the host hamiltonian @xmath0 implies that for all values of @xmath153 , @xmath152 is proportional to a constant coupling @xmath154 . obtaining the effective hamiltonian @xmath136 for the wilson chain is a preliminary step to optimizing the impurity tensors @xmath3 . it is convenient to split the hamiltonian @xmath49 intro three pieces , @xmath155 where @xmath156 collects the impurity hamiltonian @xmath157 and the restriction of the host hamiltonian @xmath0 on region @xmath24 , and @xmath158 and @xmath159 contain the rest of hamiltonian terms to the left and two the right of region @xmath24 , respectively . for simplicity , we shall only consider explicitly the contribution to the effective hamiltonian @xmath136 that comes from @xmath159 , @xmath160 where @xmath161 measures the distance from the impurity region @xmath24 . we note that @xmath158 in eq . [ eq : split ] yields an identical contribution , whereas @xmath156 is not touched by the coarse - graining transformation @xmath139 . let us rewrite @xmath159 as @xmath162 here @xmath163 denotes the sum of all terms in @xmath159 supported on the sites of lattice @xmath135 that are in the interval @xmath164 $ ] to the right of @xmath165 , where @xmath166 is @xmath167 for instance , @xmath168 is the sum of hamiltonian terms in the interval @xmath169=[1,2]$ ] , which is actually just a single term , @xmath170 while @xmath171 is the sum of terms in the interval @xmath172=[2,5]$ ] , @xmath173 and so forth . let @xmath174 denote the ascending superoperator that implements one step of coarse - graining of @xmath163 ( the explicit forms of @xmath175 , @xmath176 and @xmath177 are depicted in fig . [ fig : logscale](d - f ) , respectively ) . then the term @xmath178 of the effective hamiltonian @xmath136 is obtained by coarse - graining @xmath179 a total of @xmath153 times , @xmath180 as an example , fig . [ fig : logscale](c ) depicts the coarse - graining of the term @xmath145 , @xmath181 through use of eq . [ eq : ad5 ] one can evaluate all the terms @xmath182 for @xmath183 that define the effective hamiltonian @xmath136 on the wilson chain @xmath142 . let us now specialize the analysis to the case where the original hamiltonian on @xmath43 is scale invariant ( see appx . [ sect : scalemera ] ) . in this case , @xmath163 transforms in a precise way under coarse - graining , namely @xmath184 for all @xmath185 . let us define @xmath186 . then all the terms @xmath187 of the effective hamiltonian @xmath136 are seen to be proportional to this same term @xmath188 , @xmath189 and the effective hamiltonian @xmath136 for the wilson chain is , @xmath190 that is , in the scale invariant case , we have obtained a nearest neighbor hamiltonian where each nearest neighbor term is proportional to @xmath188 , with a proportionality constant that decays exponentially with @xmath153 . [ if the scale invariant mera contained @xmath191 transitional layers before reaching scale invariance ( see appx . [ sect : scalemera ] ) then the form of the terms in @xmath136 would be position dependent for @xmath192 , and only become proportional to a fixed @xmath193 for @xmath194 . ] the hamiltonian @xmath136 is analogous to the effective hamiltonian wilson obtained , and subsequently solved , in his celebrated solution to the kondo impurity problem @xcite . this observation was central to the proposal and justification of minimal updates in mera in ref . . , defined on lattice @xmath135 , is mapped to an effective hamiltonian @xmath136 defined on the wilson chain @xmath142 via the inhomogeneous coarse - graining @xmath139 . ( b ) the set of impurity tensors @xmath195 form a tree tensor network state @xmath140 on @xmath142 . we denote by @xmath196 the block of radius @xmath153 about @xmath24 . ( c - d ) the block hamiltonian @xmath197 , defined as the part of @xmath136 supported on block @xmath196 , is coarse - grained to the one - site block hamiltonian @xmath198 using the impurity tensors @xmath199 . ( e - f ) the reduced density matrix @xmath200 on block @xmath196 is coarse - grained to the one - site reduced density matrix @xmath201 using the impurity tensors @xmath199.,width=321 ] once we have constructed the effective hamiltonian @xmath136 for the ( logarithmic scale ) wilson chain @xmath142 , as represented schematically in fig . [ fig : defectalga](a ) , we can proceed to optimize for the impurity tensors @xmath3 . the impurity tensors @xmath3 form a tensor network known as tree tensor network @xcite ( ttn ) , which we use as a variational ansatz for the ground state @xmath140 on the wilson hamiltonian @xmath136 , see fig . [ fig : defectalga](b ) . specifically the impurity tensors @xmath3 will be obtained through the energy minimization @xmath202 notice that , if folded through the middle , this ttn is equivalent to a matrix product state ( mps ) @xcite . therefore , its optimization can be accomplished using standard variational mps methods @xcite , once they have been properly adapted to a semi - infinite chain . here , for concreteness , we describe in detail an optimization algorithm that is similar to the techniques employed in the optimization algorithm for scale invariant mera @xcite . we assume that the state @xmath140 can be described by the above ttn made of tensors @xmath203 , where all the tensors for @xmath204 are given by a fixed tensor @xmath205 . the number of required transitional tensors @xmath206 will in general depend on both the details of the mera for the state @xmath76 of the lattice @xmath43 ( more specifically , on the number @xmath191 of transitional layers required before reaching scale invariance , see appx . [ sect : scalemera ] ) , as well as the details of the specific impurity under consideration . in practice the appropriate @xmath206 is found heuristically : one starts with a small @xmath206 , minimizes the energy ( using e.g. the algorithm provided below ) and then iteratively increases @xmath206 until the corresponding optimized energy does no longer depend on @xmath206 . in total , @xmath207 distinct tensors @xmath208 need be optimized . this is achieved by iteratively optimizing one tensor at a time , so as to minimize the energy , @xmath209 . if @xmath195 is the tensor to be optimized , then we proceed by computing its linearized environment @xmath210 , which is the tensor obtained by removing tensor @xmath195 ( but not its conjugate @xmath211 ) from the tensor network describing the energy @xmath209 , and that therefore fulfills @xmath212 , where ttr denotes a tensor trace . an updated @xmath195 that minimizes the energy is then obtained through the singular value decomposition ( svd ) of @xmath210 . let us define the nested set of blocks @xmath213 as block of radius @xmath153 around @xmath24 with @xmath214 , see fig . [ fig : defectalga](b ) . then the process of computing linearized environments @xmath210 is simplified by first computing the coarse - grained block hamiltonians @xmath198 and reduced density matrices @xmath201 supported on @xmath196 , as described in sects.[sect : optham ] and [ sect : optden ] respectively . [ sect : optlin ] discusses details of the construction of linearized environments and the svd update , while sect . [ sect : optalg ] describes how these steps can be composed into the full optimization algorithm . let us denote by @xmath197 the part of the hamiltonian @xmath136 that is supported on block @xmath196 , and by @xmath198 its effective , one - site version that results from coarse - graining @xmath197 by the first @xmath153 impurity tensors @xmath199 , see fig . [ fig : defectalga](c - d ) for examples . the block hamiltonian @xmath216 for a larger block @xmath217 can be computed from the smaller block hamiltonian @xmath218 by @xmath219 where @xmath220 is the one - site impurity ascending superoperator associated to @xmath195 , and @xmath221 and @xmath222 are left and right ascending superoperators that add the contributions from the local couplings @xmath223 to the block hamiltonian . the forms of these ascending superoperators are depicted as tensor network diagrams in fig . [ fig : defectalgb](a ) . we us denote by @xmath200 the reduced density matrix that is obtained from @xmath140 by tracing out the sites outside the block @xmath196 , and by @xmath201 as its effective , one - site version that results from coarse - graining @xmath200 with the first @xmath153 impurity tensors @xmath199 , see fig . [ fig : defectalga](e - f ) for examples . the one - site density matrix @xmath225 for a smaller block @xmath226 can be obtained from the density matrix @xmath227 for the larger region @xmath228 by fine - graining it with isometry @xmath195 , then tracing out the boundary sites . this can be achieved by applying the one - site descending superoperator @xmath229 associated to the impurity tensor @xmath195 , @xmath230 see fig . [ fig : defectalgb](b ) . notice that scale invariance , such that @xmath231 for scales @xmath232 , implies that @xmath233 for all @xmath234 , where the fixed - point density matrix @xmath235 satisfies @xmath236 here @xmath237 is the one - site scaling superoperator ( as introduced in sect . [ sect : critmod ] when studying scale invariant properties of modular mera ) , which is just the impurity ascending superoperator @xmath238 constructed from @xmath205 . we can thus obtain @xmath239 as the dominant eigenvector of @xmath240 ( e.g. by diagonalizing @xmath240 ) . from @xmath239 , one can then sequentially compute the density matrices @xmath241 by using eq . [ eq : densitya ] . ( a ) the tensor contractions required for evaluating the block hamiltonian @xmath242 , see also eq . [ eq : blockham ] . ( b ) the tensor contraction required for evaluating the reduced density matrix @xmath243 from @xmath244 , see also eq . [ eq : densitya ] . ( c ) the five contributions to the linearized environment @xmath210 of the impurity tensor @xmath195.,width=321 ] fig . [ fig : defectalgb](c ) shows the linearized environment @xmath210 for the impurity tensor @xmath195 . @xmath210 decomposes into a sum of five terms , each of which corresponds to a small tensor network , and it depends on the effective hamiltonian @xmath245 , the reduced density matrices @xmath201 and @xmath246 , the hamiltonian terms @xmath223 and @xmath247 , and the impurity tensors @xmath248 , @xmath249 , and @xmath250 , @xmath251 let us consider first the optimization of @xmath195 for @xmath252 . in this case , the updated impurity tensor is chosen as @xmath253 , where @xmath254 and @xmath255 are isometric tensors obtained from the svd of the linearized environment @xmath210 , namely @xmath256 , see ref . for further details . for @xmath204 , the impurity tensor @xmath195 is a copy of the impurity tensor @xmath205 . in order to update @xmath205 we should construct the environment as the sum of environments for each @xmath257 , @xmath258 obtaining the environment @xmath259 directly through this infinite summation may only be possible at a very large computational cost . however , since the system is assumed to be scale invariant , the environments @xmath260 in eq . [ s4e11 ] should quickly converge to a fixed environment as we increase @xmath153 . thus one can obtain an approximate environment @xmath261 of the scaling impurity tensor @xmath205 through a partial summation of eq . [ s4e11 ] , @xmath262 the number @xmath263 of terms in this partial summation , required in order to obtain a sufficiently accurate environment , will in general depend on the problem under consideration . however , for the numerical results of sect . [ sect : bench ] we find that keeping @xmath264 is sufficient in most cases . once the linearized environment @xmath261 has been computed , the tensor @xmath265 is updated by taking the svd of the environment as in the case @xmath252 . let us then review the algorithm to optimize the tensors @xmath208 of the ttn of fig . [ fig : defectalga](b ) for the ground state @xmath140 of the effective hamiltonian @xmath136 . the optimization is organized in sweeps through the ttn , where each sweep consists of a sequence of single tensor updates for each @xmath195 , from @xmath266 to @xmath267 . we iterate these optimization sweeps until the state @xmath140 has converged sufficiently . recall that the effective hamiltonian @xmath136 generically takes the form of eq . [ eq : ad9 ] , with nearest neighbor coupling strengths that decay geometrically with the distance to the origin . thus , a very good approximation to the ground state of @xmath136 can be obtained using wilson s numerical renormalization group@xcite ( nrg ) . here we use the nrg to initialize the impurity tensors @xmath195 , and then apply the variational sweeping to further improve the approximation to the ground state . each iteration of the variational sweep is comprised of the following steps : 1 . compute the fixed - point density matrix @xmath239 through diagonalization of the ( adjoint ) impurity scaling superoperator @xmath268 . 2 . compute the block density matrices @xmath269 for all @xmath270 using eq . [ eq : densitya ] . sequentially update @xmath195 , starting from @xmath266 and proceeding to @xmath271 . for each such values of @xmath153 , first compute the linearized environment @xmath210 and then update the impurity tensor @xmath195 via the svd of this environment . then compute the effective hamiltonian @xmath198 from @xmath245 using the updated isometry @xmath195 , as described in eq . [ eq : blockham ] . 4 . update the fixed - point tensor @xmath205 : compute an approximate environment @xmath259 as described in eq . [ s4e12 ] , and then update the fixed - point tensor @xmath205 via the svd of this environment . notice that this algorithm is analogous to the one introduced to optimize the scale - invariant mera as described in ref . . in this section we benchmark the use of the modular mera for several types of defect in quantum critical systems ; specifically we consider impurities , boundaries , and interfaces . in the case of a single impurity , a single boundary , and a simple interface , we use the corresponding modular meras introduced in sects . [ sect : impuritymera ] , [ sect : boundmera ] , and [ sect : interfacemera ] . for multiple impurities , two boundaries , and y - interfaces , we use more complicated modular meras that result from a recursive use of the theory of minimal updates , as outlined in sect . [ sect : furgen ] . in several cases , we also specify how to modify the basic optimization algorithm of sect . [ sect : optmod ] . we start by benchmarking the use of the modular mera to describe a quantum critical system in the presence of a single impurity first , and then in the presence of multiple impurities . let us first consider a quantum critical system with a hamiltonian of the form @xmath272 where @xmath0 is a fixed - point hamiltonian that describes the host system ( which is invariant both under translations and changes of scale ) , and @xmath273 accounts for an impurity localized on region @xmath24 of the lattice . specifically , we test the impurity mera in the case where @xmath0 corresponds to the critical ising hamiltonian , @xmath274 where @xmath275 and @xmath276 are pauli matrices , and the impurity hamiltonian @xmath277 acts on two adjacent lattice sites @xmath278 , where it weakens or strengthens the nearest neighbor term , @xmath279 for some real number @xmath71 . the quantum critical ising model with an impurity of this form , which is in direct correspondence with the @xmath280 classical ising model with a defect line , has been studied extensively in the literature @xcite . we refer the reader to ref . for a review of the problem . we optimize the impurity mera for the ground state @xmath56 of this impurity problem using the strategy outlined in sect . [ sect : optmod ] . we fist find tensors @xmath17 for the ground state of the homogeneous critical ising model using a scale invariant mera with bond dimension @xmath281 . this mera incorporated both the @xmath282 ( spin flip ) global on - site symmetry and the reflection symmetry ( see appendix [ sect : refsym ] ) of @xmath283 . this optimization required approximately 1 hour of computation time on a 3.2 ghz desktop pc with 12 gb of ram . the mapping of the initial impurity hamiltonian @xmath49 to the effective problem @xmath136 on the wilson chain @xmath144 , as described in sect . [ sect : logscale ] , was accomplished in negligible computation time ; it is less expensive than a single iteration of the optimization of the scale invariant mera . optimization of the impurity tensors @xmath195 , as discussed in sect . [ sect : optlog ] , was performed for a range of impurity strengths , namely the two series @xmath284 and @xmath285 , which required approximately 20 minutes of computation time for each value of @xmath71 . evaluated from an impurity mera ( @xmath286 s ) versus the exact solutions ( solid lines ) for the critical ising model with an impurity @xmath287 , as described eq . [ s5e4 ] , located on lattice sites @xmath278 . the magnetization approaches the bulk value @xmath288 polynomially as @xmath289 , for all values of @xmath71 considered.,width=321 ] for the critical ising model with a conformal defect @xmath287 , comparing results from the impurity mera ( @xmath286 s ) with the exact results of eq . [ s5e5 ] ( solid lines ) . note that only scaling dimensions in the @xmath290 parity sector of the @xmath282 global symmetry of the ising model are plotted , as those in the @xmath291 parity sector are invariant under addition of the conformal defect . ( b ) the complete spectrum of scaling dimensions obtained from the mera , organized according to parity sector @xmath292 , for values of @xmath293.,width=321 ] from the optimized impurity mera we compute the magnetization profiles @xmath294 , as shown fig . [ fig : defectzmag ] , which match the exact profiles ( obtained by solving the free fermion problem , see ref . ) with high precision . for all defect strengths @xmath71 considered , the magnetization approaches the constant bulk value @xmath295 as @xmath289 , _ i.e. _ with scaling dimension @xmath296 . this result , consistent with the behavior of modular mera predicted in sect . [ sect : critmod ] , is in agreement with the scaling of the magnetization @xmath297 predicted from study of the ising cft ( where the @xmath276 operator is related to the energy density operator @xmath298 of the ising cft with scaling dimension @xmath296 ) . for each value of the impurity coupling @xmath71 , we also compute the scaling dimensions @xmath299 associated to the impurity by diagonalizing the impurity scaling superoperator @xmath90 , as described sect . [ sect : critmod ] . in refs . the spectrum of scaling dimensions for the critical ising model associated to the impurity @xmath287 have been derived analytically , @xmath300 where @xmath301 is a positive integer and @xmath302 is a phase associated to the strength of the impurity @xmath71 , @xmath303 a comparison of the scaling dimensions obtained from mera and the exact scaling dimensions is presented in fig . [ fig : defectcritexp ] . remarkably , the impurity mera accurately reproduces the smallest scaling dimensions ( all scaling dimensions @xmath304 ) for the full range of @xmath71 considered , which include the special cases of ( i ) an impurity that removes any interaction between the left and right halves of the chain @xmath305 , ( ii ) the case with no impurity @xmath306 , and ( iii ) an impurity which sets an infinitely strong ising interaction over two spins @xmath307 . these results confirm that the impurity mera accurately approximates the ground state of the impurity system , both in terms of its local expectation values ( e.g. magnetization profile @xmath294 ) and its long distance , universal properties ( e.g. scaling dimensions @xmath308 ) . and @xmath81 , here separated by @xmath309 lattice sites . the causal cones of the individual impurities fuse at a depth @xmath310 . at small depth , @xmath311 , the mera has two types of impurity tensor , @xmath312 and @xmath313 , one associated to each of the impurities . at greater depth , @xmath314 , the mera has one type of impurity tensor , @xmath315 , associated to a fusion of the two impurities . ( b ) an inhomogeneous coarse - graining @xmath139 , defined from the bulk tensors , maps the original two impurity hamiltonian @xmath0 to an effective two impurity hamiltonian @xmath136 . a subsequent coarse - graining @xmath316 , defined from the impurity tensors @xmath312 and @xmath313 , maps @xmath136 into an effective _ single _ impurity hamiltonian @xmath317.,width=321 ] next we consider a system with two impurities , with hamiltonian @xmath318 where @xmath319 and @xmath320 represent the distinct impurities located on separate local regions @xmath80 and @xmath81 of the lattice . the two - impurity mera for the ground state @xmath321 of hamiltonian @xmath322 is depicted in fig . [ fig : twodefectcg](a ) . in this more complex modular mera the tensors have been modified within the causal cone @xmath323 of the union of regions @xmath80 and @xmath81 . for length scales @xmath324 , where @xmath93 is the distance separating the two regions @xmath325 and @xmath326 , the causal cones @xmath327 and @xmath328 are distinct , while for length scales @xmath329 the causal cone have fused into a single cone . thus for short length scales , @xmath324 , there are two distinct types of impurity tensor : tensors @xmath312 associated to the impurity @xmath40 and tensors @xmath313 associated to the impurity @xmath41 . for longer length scales , @xmath329 , there is a single type of impurity tensor @xmath315 which is associated to the fusion of the two impurities @xmath40 and @xmath41 into a new impurity @xmath330 . the steps for optimizing the two - impurity mera are as follows : 1 . optimize a scale - invariant mera for the ground state @xmath25 of the homogeneous host hamiltonian @xmath0 to obtain tensors @xmath331 . [ step : s3e1 ] 2 . optimize a ( single ) impurity mera for the single impurity hamiltonian @xmath332 to obtain the impurity tensors @xmath312 . [ step : s3e2 ] 3 . optimize a ( single ) impurity mera for the single impurity hamiltonian @xmath333 to obtain the impurity tensors @xmath313 . [ step : s3e3 ] 4 . map the original two - impurity hamiltonian @xmath322 of eq . [ s5e7 ] to an effective single impurity hamiltonian @xmath317 , @xmath334 as depicted in fig . [ fig : twodefectcg](b ) , where @xmath139 is an inhomogeneous coarse - graining defined in terms of the bulk tensors , and @xmath316 is a coarse - graining defined in terms of the impurity tensors @xmath312 and @xmath313 . [ step : s3e4 ] 5 . optimize a ttn for the effective single impurity problem @xmath317 to obtain the impurity tensors @xmath315 . [ step : s3e5 ] thus , by exploiting minimal updates and the modular character of mera , the two - impurity problem is addressed by solving a sequence of three single impurity problems : two single impurity problems for impurities @xmath40 and @xmath41 separately , and a third single impurity problem for the effective impurity @xmath330 that results from coarse - graining together impurities @xmath40 and @xmath41 . to test the validity of this approach , we investigate the case where @xmath77 in eq . [ s5e7 ] is the critical ising model @xmath335 of eq . [ s5e3 ] , and @xmath336 and @xmath337 are each defects of the form described in eq . conformal field theory predicts@xcite that , when viewed at distances much larger than the separation @xmath93 between the two impurities , the two - impurity ising model is equivalent to an ising model with a single impurity @xmath330 with effective hamiltonian @xmath338 . the strength @xmath339 of the fused impurity @xmath330 relates to the strength @xmath340 and @xmath341 of the original impurities @xmath40 and @xmath41 according to @xcite @xmath342 where @xmath343 is the phase associated to the defect as described by eq . we employ the mera to test a special case of eq . [ s5e12 ] in which we choose the weight of the second impurity as the inverse of the first , @xmath344 , such that @xmath345 is the unique solution to eq . [ s5e12 ] . in other words , we test the case where the two impurities are predicted to fuse to identity ( i.e. no impurity ) at large distances . obtained from the mera for the ground state of the critical ising model with two conformal impurities @xmath346 and @xmath347 ( see eq . [ s5e4 ] ) with strengths @xmath348 and @xmath349 respectively , which are located @xmath350 lattice sites apart . the magnetization profile when both impurities are present is represented with @xmath351 s , while the two solid lines each represent magnetization profiles when only one of the impurities is present . ( b ) the spectra of scaling dimensions associated to the impurities : @xmath352 and @xmath353 are the single impurity spectra for impurities of strength @xmath348 and @xmath349 respectively , while @xmath354 is the spectrum arising from the fusion of these conformal impurities . it is seen that @xmath354 matches the scaling dimensions of the bulk ( i.e. impurity free ) critical ising model.,width=321 ] we optimize the two - impurity mera for the case @xmath355 and @xmath356 , where the impurities are set a distance of @xmath350 sites apart . tensors @xmath357 , and the single impurity tensors @xmath312 and @xmath313 are recycled from the single impurity calculations of sect . [ sect : benchsingle ] . thus the only additional work to address the two - impurity problem , provided the individual impurities have been previously addressed , is to perform steps [ step : s3e4 ] and [ step : s3e5 ] above , namely producing an effective , single impurity hamiltonian @xmath317 , and then optimizing the impurity tensors @xmath315 for the ` fused ' impurity @xmath330 . the scaling superoperator @xmath358 associated to the fused impurity was diagonalized to obtain the scaling dimensions @xmath354 associated to the fused impurity @xmath330 . these scaling dimensions , together with the magnetization profile @xmath359 of the two impurity system , are plotted in fig . [ fig : twodefectcritexp ] . it can be seen that the scaling dimensions @xmath354 reproduce the spectrum of scaling dimensions for the homogeneous ising model @xcite , as predicted by eq . [ s5e12 ] , thus indicating that the two - impurity mera accurately captures the universal properties of the ground state . the method outlined to address a two - impurity problem can be easily generalized to the case of a system with any finite number of impurities . the many - impurity problem can likewise be reduced to a first sequence of single impurity problems that , under fusion , give rise to a second sequence of single impurity problems , and so on . next we benchmark the use of the modular mera to describe a quantum critical system in the presence of one boundary ( semi - infinite chain ) and in the presence of two boundaries ( finite chain ) . let us first consider a semi - infinite lattice @xmath360 with hamiltonian @xmath0 , @xmath361 where the hamiltonian term @xmath64 at site @xmath362 describes the boundary ( and can be chosen so as to describe certain types of open boundary conditions , such as ` fixed ' or ` free ' open boundary conditions ) , and @xmath363 is a nearest neighbor hamiltonian term such that the hamiltonian @xmath364 represents the host system , which is invariant under translations and under changes of scale . the boundary mera for the ground state @xmath62 of hamiltonian @xmath65 , as described sect . [ sect : boundmera ] , was initially introduced and tested in ref . . here we shall both reproduce and expand upon the results in that paper . a similar construction was proposed also in ref . . in order to optimize the boundary mera depicted in fig . [ fig : boundeffective](a ) , which is fully characterized in terms of the tensors @xmath365 for the homogeneous system and the tensors @xmath3 for the boundary , we follow the following steps : 1 . optimize tensors @xmath17 by energy minimization of a mera for the homogeneous host system with hamiltonian @xmath59 . [ step : s4e1 ] 2 . map the original boundary hamiltonian @xmath65 to the effective boundary hamiltonian @xmath136 on the wilson chain @xmath142 , @xmath366 through the inhomogeneous coarse - graining @xmath139 , as depicted in fig . [ fig : boundeffective](b ) . [ step : s4e2 ] 3 . optimize the tensors @xmath3 by energy minimization on the effective hamiltonian @xmath136 . [ step : s4e3 ] these steps can be accomplished with only minor changes to the method presented in sect . [ sect : optmod ] . of which is described by a pair of bulk tensors @xmath367 and a boundary tensor @xmath195 . the causal cone @xmath368 of the boundary @xmath369 , which only contains boundary tensors @xmath195 , is shaded , and the associated wilson chain @xmath142 is indicated . ( b ) an inhomogeneous coarse - graining @xmath139 , defined in terms of the bulk tensors , is used to map the original boundary hamiltonian @xmath0 to an effective boundary hamiltonian @xmath136 on the wilson chain @xmath142.,width=321 ] for the critical ising model with free and fixed bc obtained with a boundary mera . the exact solution approaches the bulk value @xmath370 as @xmath371 . right : error in @xmath372 for free bc ( similar to that for fixed bc ) . the non - vanishing expectation value of bulk scaling operators is accurately reproduced even thousands of sites away from the boundary.,width=321 ] we consider two quantum critical models for the host hamiltonian @xmath59 : the critical ising model @xmath373 of eq . [ s5e3 ] and the quantum xx model , @xmath374 where @xmath275 and @xmath375 are pauli matrices . the boundary condition at site @xmath362 are set either as free boundary , in which case @xmath376 in eq . [ s6e1 ] , or fixed boundary , @xmath377 . tensors @xmath17 for the ising model can be recycled from the calculations of sect . [ sect : benchimpurity ] , while for the quantum xx model they are obtained from a mera with @xmath378 that exploits both reflection symmetry and a global @xmath379 spin symmetry and required approximately 2 hours of optimization time on a 3.2 ghz desktop pc with 12 gb of ram . optimization of the effective boundary problem @xmath136 for the boundary tensors @xmath3 required less than 10 minutes of computation time for each of the critical models , under each of the boundary conditions tested . [ fig : zmag ] displays the magnetization profile @xmath380 for the ising model with both free and fixed bc , which are compared against the exact magnetization profiles ( obtained using the free fermion formalism ) , @xmath381 the optimized boundary mera accurately reproduces the effect of the boundary on the local magnetization even up to very large distances . specifically , the exact magnetization profile is reproduced within @xmath382 accuracy up to distances of @xmath383 sites from the boundary . [ fig : scaledim ] shows the boundary scaling dimensions @xmath130 for critical ising and quantum xx models , obtained by diagonalizing the scaling superoperator @xmath98 associated to the boundary . the boundary scaling dimensions obtained from the boundary mera also reproduce the known results from cft @xcite with remarkable accuracy . for the ising model the smallest scaling dimensions @xmath384 are reproduced with less than @xmath385 error while for the quantum xx model @xmath386 the error is less than @xmath387 . finally , we analyze the boundary contribution @xmath388 to the ground state energy , @xmath389 defined as the difference between the energy @xmath390 of the semi - infinite chain with the boundary term @xmath64 , eq . [ s6e1 ] , and one half of the ground state energy for the host hamiltonian on the infinite chain @xmath391 , eq . [ s6e1b ] . since both @xmath390 and @xmath391 are infinite quantities , we can not compute @xmath388 through the evaluation of the individual terms in eq . [ s6e6 ] . instead , we estimate @xmath388 by comparing the energy of the first @xmath93 sites of the semi - infinite chain to the energy of @xmath93 sites of the infinite homogeneous system , and increase the value of @xmath93 until the energy difference is converged within some accuracy . for the quantum ising model on a semi - infinite lattice we obtain the following results : for free bc , a value @xmath392 , which is remarkably close to the exact solution@xcite , @xmath393 , and for fixed boundary conditions , a value @xmath394 which , based upon the exact solution for finite chains of over a thousand sites , we estimate to carry an error of less than @xmath395 . let us now consider a finite lattice @xmath43 made of @xmath396 sites and with two boundaries , with hamiltonian @xmath397 where @xmath398 and @xmath399 at sites @xmath362 and @xmath400 describe the left and right boundaries , respectively , and the @xmath363 is a nearest neighbor hamiltonian term as in eq . [ s6e1b ] . a two - boundary mera for the ground state @xmath401 of a finite chain with hamiltonian @xmath402 is depicted in fig . [ fig : boundfinite](a ) . each layer of tensors consists of tensors @xmath403 in the bulk and tensors @xmath404 and @xmath405 at the left and right boundaries , respectively . the two - boundary mera is organized into a finite number , @xmath406 , of layers , and has an additional tensor @xmath407 at the top . the steps for optimizing this particular form of modular mera are as follows : 1 . optimize tensors @xmath17 by energy minimization of a mera for the homogeneous infinite host system with hamiltonian @xmath0 . [ step : s5e1 ] 2 . optimize the left boundary tensors @xmath404 by energy minimization on an effective semi - infinite , single boundary problem with boundary term @xmath408 , as described in sect . [ sect : benchsemi ] . 3 . optimize the right boundary tensors @xmath405 by energy minimization on an effective semi - infinite , single boundary problem with boundary term @xmath409 , as described in sect . [ sect : benchsemi ] . coarse - grain the original boundary problem @xmath410 , defined on the @xmath396-site lattice @xmath411 , into an effective boundary problem @xmath412 defined on the coarse - grained lattice @xmath413 , @xmath414 where each @xmath415 is a layer of the two - boundary mera , as depicted in fig . [ fig : boundfinite](b ) . [ step : s5e3 ] 5 . compute the top tensor @xmath407 through diagonalization of the effective hamiltonian @xmath416 for its ground state or excited states . [ step : s5e4 ] in summary , to treat a finite chain with open boundaries with the mera , one should first address an infinite system , then two semi - infinite systems , and finally a coarse - grained version of the original hamiltonian , which is reduced to a small number of sites . the mera is organized into @xmath417 layers , where each layer @xmath5 is described by a pair of bulk tensors @xmath403 and ( left , right ) boundary tensors @xmath404 and @xmath405 . the boundary mera also has a top - tensor @xmath407 at the final level . ( b ) the original boundary problem @xmath418 defined on an @xmath396-site lattice @xmath419 can be mapped into an effective open boundary problem @xmath420 defined on a @xmath421-site lattice @xmath422 through coarse - graining with mera layers @xmath423 and @xmath424 , see also eq . [ s6e9].,width=321 ] of @xmath425 sites with different combinations of open boundary conditions . the energy is expressed in units such that the gap between descendants is a multiple of unity . all non - equivalent combinations of open bc are considered . the different open bc are @xmath426.,width=321 ] to test the validity of the two - boundary mera to finite systems with open boundary conditions , we investigate the low energy spectrum of the critical ising model under different fixed and free boundary conditions , as defined sect . [ sect : benchsemi ] . we are able to recycle the tensors @xmath17 for the homogeneous host system , as well as the boundary tensors @xmath404 and @xmath405 obtained from the previous investigation of semi - infinite ising chains in sect . [ sect : benchsemi ] . thus , we only need to perform steps [ step : s5e3 ] and [ step : s5e4 ] above . we proceed by constructing the effective hamiltonians @xmath416 for a two - boundary mera with @xmath427 total layers , which equates to a total system size of @xmath428 sites , for all non - equivalent combinations of boundary conditions . there are four such non - equivalent combinations : free - free , fixed(up)-fixed(down ) , fixed(up)-fixed(up ) and free - fixed . the low - energy spectra of the effective hamiltonians @xmath416 are then computed with exact diagonalization based on the lanczos method . these low - energy spectra , displayed in fig . [ fig : finitespect ] , match the predictions from cft @xcite to high precision . these results indicate that the two - boundary mera is not only a good ansatz for the ground states of finite systems with open boundary conditions , but also for their low - energy excited states . furthermore , only the top tensor @xmath407 of the mera needs to be altered in order to describe different excited states . next we benchmark the use of the modular mera to describe the interface between two or more quantum critical systems . let us first consider the interface between two systems @xmath40 and @xmath41 , described by an infinite lattice @xmath360 with a hamiltonian of the form @xmath429 where the hamiltonian term @xmath430 couples two ( left and right ) semi - infinite chains @xmath431 and @xmath432 , @xmath433 , and the nearest neighbor terms @xmath434 and @xmath435 are such that on an infinite lattice , the hamiltonians @xmath436 describe homogeneous , quantum critical host systems that are invariant under translations and changes of scale . the interface mera for the ground state @xmath87 of hamiltonian @xmath73 , depicted in fig . [ fig : interfaceeffective](a ) , is made of the following tensors : two sets of tensors @xmath437 and @xmath438 corresponding to the mera for the ground state of the host hamiltonians @xmath439 and @xmath440 , respectively , and the interface tensors @xmath3 . optimization of the interface mera can be accomplished through a straightforward generalization of the approach described in sect . [ sect : optmod ] for an impurity . the only differences here are that one needs to address first two different homogeneous systems , and that the coarse - graining of @xmath73 into the effective hamiltonian @xmath136 on the wilson chain @xmath142 , see fig . [ fig : interfaceeffective](b ) , uses one set of host tensors @xmath437 on the left and the other @xmath438 on the right . , supported on semi - infinite chain @xmath431 , with a different critical system @xmath69 , supported on semi - infinite chain @xmath432 . each layer @xmath5 of the interface mera is described by a pair of tensors @xmath437 associated to host system ` @xmath40 ' , a pair of tensors @xmath438 associated to host system ` @xmath41 ' , and an interface tensor @xmath3 , which resides in the causal cone @xmath27 of the interface region @xmath24 . the wilson chain @xmath142 associated to the interface @xmath24 is indicated . ( b ) the inhomogeneous coarse - graining @xmath139 , defined in terms of the host tensors @xmath441 and @xmath442 , maps original interface hamiltonian @xmath0 to an effective interface hamiltonian @xmath136 defined on the wilson chain @xmath142.,width=321 ] , as defined eq . [ s7e3 ] , of the interface between a quantum xx model chain ( on sites @xmath443 ) and the critical ising chain ( on sites @xmath444 ) , coupled across the interface @xmath278 . the parameter @xmath71 relates to the strength of the interface coupling . in all cases the magnetization decays to the bulk value , @xmath445 for quantum xx and @xmath446 for ising , as @xmath447.,width=321 ] we test the validity of the interface mera by choosing as quantum critical systems @xmath40 and @xmath41 the quantum xx model in eq . [ s6e4 ] and the critical ising model in eq . [ s5e3 ] , respectively , and as the coupling at the interface the two - site term @xmath448 for several values of @xmath449 . the tensors @xmath450 for the quantum xx model and @xmath451 for the ising model are recycled from previous computations in sect . [ sect : benchbound ] . thus the only additional work required is to produce the effective interface hamiltonian @xmath136 , and then to optimize the interface tensors @xmath3 by energy minimization over @xmath136 . the later , undertaken on a 3.2 ghz desktop pc with 12 gb of ram , required only approximately 20 minutes of computation time for every value of @xmath71 . fig . [ fig : interfacemag ] plots the magnetization profile @xmath452 , @xmath453 obtained from the optimized interface mera . associated to the interface of the quantum xx and ising model , as a function of the coupling strength @xmath71 across the interface . ( b ) the scaling dimensions for the interface with no coupling ( @xmath72 ) , which take on integer and half - integer values , are seen to be the product of the boundary scaling dimensions for quantum xx and ising models with free bc . ( c ) under interaction strength @xmath454 much of the degeneracy of the @xmath72 case is lifted , yet the scaling dimensions remain organized in conformal towers.,width=321 ] for @xmath72 in eq . [ s7e2 ] ( that is , two decoupled semi - infinite chains ) , we recover indeed the magnetization profiles for the semi - infinite quantum xx chain and semi - infinite ising chain with a free boundary , as expected . for @xmath75 , the quantum xx chain acquires a non - zero magnetization near the interface , and the magnetization of the ising chain near the interface is reduced with respect to the case @xmath72 . however , away from the interface , the magnetizations still decay polynomially to their values for a homogeneous system : @xmath445 for the quantum xx model and @xmath446 for the critical ising model . we also computed the scaling dimensions @xmath130 associated to the interface , as plotted in fig . [ fig : interfacecritexp ] , through diagonalization of the scaling superoperator @xmath90 associated to the interface . the exact scaling dimensions are only known to us for the case of interface strength @xmath72 ( decoupled case ) , where one would expect the spectrum of scaling dimensions to be the product of spectra for the open boundary ising and open boundary quantum xx models on a semi - infinite chains , see fig . [ fig : scaledim ] . the numerical results of fig . [ fig : interfacecritexp ] match this prediction . for @xmath75 , we no longer have exact scaling dimensions to compare with . however , we see that these are still organized in conformal towers , where the scaling dimensions for descendant fields differ by an integer from the scaling dimensions of the corresponding primary fields @xcite , and where the scaling dimensions of the primary fields depend on @xmath71 . this is a strong indication that the results from the interface mera are correct . interestingly , those scaling dimensions that correspond to an integer value for @xmath72 , remain unchanged for @xmath75 , up to small numerical errors . these are likely to be protected by a symmetry ( the interface hamiltonian has a global @xmath282 , spin flip symmetry ) similar to the case of the critical ising impurity model described in sect . [ sect : benchimpurity ] . , see eq . [ s7e1b ] . ( b ) under action of the inhomogeneous coarse - graining @xmath139 the hamiltonian @xmath0 is mapped to an effective y - interface hamiltonian @xmath136 on the wilson chain . ( c ) the y - interface tensors @xmath195 , which form a peculiar tree tensor network on the wilson chain , are obtained through optimization of the effective hamiltonian @xmath136.,width=321 ] let us now consider a y - interface ( also called y - junction ) between three systems , as described by a lattice @xmath360 made of the union of three semi - infinite lattices @xmath431 , @xmath432 , and @xmath455 , @xmath456 , see fig . [ fig : ymera](a ) , with hamiltonian @xmath457^a)\label{s7e1b}\\ & + & \sum_{r = 1}^{\infty } h_b^{{}}(r^b,[r + 1]^b ) + \sum_{r = 1}^{\infty } h_c^{{}}(r^c,[r + 1]^c)\nonumber\end{aligned}\ ] ] here we use @xmath458 ( and @xmath459 , @xmath460 ) to denote site @xmath161 of lattice @xmath431 ( respectively , @xmath432 , @xmath455 ) . the term @xmath461 describes the coupling between the three semi - infinite chains @xmath431 , @xmath432 , and @xmath455 , whereas the nearest neighbor terms @xmath434 , @xmath435 , and @xmath462 are such that on an infinite lattice , the hamiltonians @xmath463 describe homogeneous , quantum critical host systems that are invariant under translations and changes of scale . the y - interface mera for the ground state @xmath464 of hamiltonian @xmath465 is a straightforward generalization of the interface mera considered in sect . [ sect : benchtwo ] . it is characterized by three sets of tensors @xmath450 , @xmath451 , and @xmath466 that describe the mera for the ground states of the host hamiltonians @xmath467 , @xmath468 , and @xmath469 , and a set of tensors @xmath3 at the y - interface . upon optimizing tensors @xmath450 , @xmath451 , and @xmath466 in three independent optimizations , they are used to map the initial y - interface hamiltonian @xmath465 to an effective hamiltonian @xmath136 , see fig . [ fig : ymera](b ) , now by employing three copies of the mapping depicted in fig . [ fig : logscale](b ) . the y - interface tensors @xmath3 , which are arranged in the ttn structure depicted in fig . [ fig : ymera](c ) , are then optimized to minimize the energy according to the effective hamiltonian @xmath136 using the approach described in sect . [ sect : optlog ] . obtained for the y - interface of three ising chains , with @xmath71 the strength of the coupling at the y - interface . the scaling dimensions are organized according to parity sectors @xmath292 of the global @xmath282 symmetry of the ising model . ( left ) for the case of @xmath72 , i.e. no coupling between different chains , the spectrum is seen to be a product of three times the spectrum of the free boundary ising chain , see fig . [ fig : scaledim](a ) , where some numeric error is evident for the larger @xmath470 scaling dimensions . ( right ) the cases of coupling strength @xmath471 all converge to the same spectrum , which symmetric between the @xmath292 parity sectors.,width=321 ] we benchmark the y - interface mera for an interface of three identical semi - infinite chains , where the each of the chains is a critical ising model as defined in eq . [ s5e3 ] and the interface coupling is given by @xmath472 , \label{s7e10}\end{aligned}\ ] ] where the pauli operators @xmath473 , @xmath474 , and @xmath475 act on the first site of the semi - infinite lattices @xmath431 , @xmath432 , and @xmath455 respectively . once again , tensors @xmath450 , @xmath451 , and @xmath466 for the critical ising model are recycled from previous calculations . we optimize the y - interface tensors @xmath3 by minimizing the energy of the effective hamiltonian @xmath136 for interface coupling strengths @xmath476 . for each value of @xmath71 we compute the spectrum of scaling dimensions @xmath130 associated to the interface by the usual diagonalization of the corresponding scaling superoperator . the results for are plotted in fig . [ fig : ychaincritexp ] . for @xmath72 , which corresponds to three uncoupled semi - infinite ising chains with free boundary conditions , the spectrum of scaling dimensions obtained from the y - interface mera is seen to be indeed the product of three copies of the spectrum of scaling dimensions for free bc ising model , see fig . [ fig : scaledim ] , as expected . for all non - zero interface couplings @xmath75 , the scaling dimensions converged to an identical spectrum ( independent of @xmath71 ) , with smaller values of @xmath71 however requiring more transitional layers @xmath206 to reach the fixed point , indicating an rg flow to the strong coupling ( or large @xmath71 ) limit . indeed , choosing a very large coupling strength , @xmath477 , reproduces the same spectrum of scaling dimensions with only @xmath478 transitional layers required . notice that the spectrum obtained for @xmath75 , which is identical between @xmath292 parity sectors of the @xmath282 symmetry of the ising model , is somewhat similar to that in fig . [ fig : defectcritexp](b ) for the ising chain with an infinitely strong bond impurity , @xmath479 , between two sites . in this manuscript we have built on the theory of minimal updates in holography proposed in ref . , and have argued that a recursive use of the conjectured minimal updates leads to the modular mera , a surprisingly simple ansatz to describe the ground state of a quantum critical system with defects such as impurities , boundaries , and interfaces . we then have provided compelling numerical evidence that the modular mera is capable of accurately describing these ground states , by considering a large list of examples . notice that the modular mera is , at its core , a concatenation of two conjectures regarding the structure of the ground state wave - function of quantum critical systems . the first conjecture , embodied in the specific of tensors of the mera , is that the ground state of a quantum critical system contains entanglement that can be removed by means of unitary transformations ( disentanglers ) acting locally on each length scale @xcite . the second conjecture , the theory of minimal updates @xcite , is that in order to account for a change of the hamiltonian in region @xmath24 , only the tensors inside the causal cone @xmath27 of region @xmath24 need to be modified . the results in this paper provide evidence that these two conjectures are correct , and thus teach us about the structure of the ground state wave - function . the modular mera is characterized by a small number of unique tensors that is _ independent _ of the system size @xmath396 . similarly , the computational cost of the optimization algorithms is also independent of the system size . as a result , the effects of local defects in an otherwise homogeneous system can be studied directly in the thermodynamic limit , avoiding finite size effects when extracting the universal properties of defects . furthermore , modularity has the useful implication that tensors can be recycled from one problem to another . for instance , the same tensors @xmath17 for the homogeneous critical ising model were used in sect . [ sect : benchimpurity ] for impurity problems , in sect . [ sect : benchbound ] for boundary problems , and in sect . [ sect : benchinterface ] for interface problems . similarly , the impurity tensors @xmath3 obtained from a single impurity problem in sect . [ sect : benchsingle ] were later reused in a multiple impurity problem in sect . [ sect : benchmultiple ] . in this manuscript we have assumed for simplicity that the quantum critical host system was described by a homogeneous hamiltonian @xmath0 that was a fixed point of the rg flow , and exploited translation and scale invariance to obtain a mera for its ground state @xmath25 that was fully characterized in terms of just one single pair of tensors @xmath17 . this had the advantage that a finite number of variational parameters ( encoded in the pair @xmath17 ) was sufficient to completely describe an infinite system . however , the theory of minimal updates does not require scale or translation invariance . let us first remove the assumption that the host system is a fixed point of the rg flow . in this case , each layer of tensors of the mera , corresponding to a different length scale @xmath153 , will be described by a different pair @xmath480 . assuming that after some finite scale @xmath191 the system can effectively be considered to have reached an rg fixed point , characterized by fixed - point tensors @xmath17 , we still obtain a finite description of the ground state of an infinite system in terms of the tensors @xmath481 and @xmath17 . the effect of a defect on a region @xmath24 can still be accounted for by a modular mera where the tensors in the causal cone @xmath27 are modified , again by energy minimization over the wilson hamiltonian @xmath136 described in sect . [ sect : logscale ] . however , in this case @xmath136 will not have the simple form of eq . [ eq : ad9 ] , but instead will consist of @xmath153-dependent terms @xmath482 for @xmath483 , after which all its terms will be proportional to some coupling @xmath154 . this case was briefly mentioned in sect . [ sect : optmod ] . let us now also remove the assumption of translation invariance in the host system . then the mera for the ground state @xmath25 of the host hamiltonian @xmath0 requires tensors @xmath484 that depend both on the scale @xmath153 and position @xmath161 . in this case the mera for @xmath25 depends on a number of tensors that grows linearly in the system size . in the presence of a defect added to the host hamiltonian @xmath0 , we can still obtain a modular mera for the system with the defect by applying a minimal update to the mera for @xmath25 . however , in this case we can not take the thermodynamic limit . although in this manuscript we focused in exploring modularity in @xmath10 spatial dimension , the theory of minimal updates , as proposed in ref . , applies to any spatial dimension @xmath15 , and thus the modular mera can be also used in systems in @xmath39 dimensions . the algorithms we presented here can be easily generalized to study e.g. a system in @xmath485 dimensions with an impurity ( in @xmath486 dimensions ) . following the outline described in sect . [ sect : optmod ] , here one would first optimize the mera for the ( impurity free ) homogeneous system , and then re - optimize the tensors within the causal cone of the impurity . notice that , since the causal cone of the impurity is a one - dimensional structure , one would build an effective system ( wilson chain ) which is again a semi - infinite chain , as in the @xmath10 case . instead , the study of a boundary or of an interface in @xmath485 dimensions requires the study of a more complex , @xmath485 effective hamiltonian , where one dimension corresponds to the extension of the boundary and the other corresponds to scale . the authors acknowledge kouichi okunishi for helpful discussions regarding wilson s solution to the kondo problem , and helpful input from masaki oshikawa regarding the two - impurity ising model . support from the australian research council ( apa , ff0668731 , dp0878830 ) is acknowledged . is supported by the sherman fairchild foundation . this research was supported in part by perimeter institute for theoretical physics . research at perimeter institute is supported by the government of canada through industry canada and by the province of ontario through the ministry of research and innovation . 99 g. evenbly and g. vidal , arxiv:1307.0831 ( 2013 ) . g. vidal , phys . lett . , 99 , 220405 ( 2007 ) . for a review of the renormalization group see : m.e . fisher , rev . mod . phys . 70 , 653 ( 1998 ) . g. evenbly and g. vidal , phys . rev . b , 81 , 235102 ( 2010 ) . l. cincio , j. dziarmaga , and m. m. rams phys . , 100 , 240603 ( 2008 ) . g. evenbly and g. vidal , new j. phys . , 12 , 025007 ( 2010 ) . g. evenbly and g. vidal , phys . rev . b , 79 , 144108 ( 2009 ) . r.n.c . pfeifer , g. evenbly , and g. vidal , phys . a , 79 , 040301(r ) ( 2009 ) . s. montangero , m. rizzi , v. giovannetti , and r. fazio , phys . b , 80 , 113103 ( 2009 ) . g. evenbly and g. vidal , phys . lett . , 102 , 180406 ( 2009 ) . g. evenbly , r. n. c. pfeifer , v. pico , s. iblisdir , l. tagliacozzo , i. p. mcculloch , and g. vidal , phys . b 82 , 161107(r ) ( 2010 ) . p. silvi , v. giovannetti , p. calabrese , g. e. santoro , and r. fazio , j. stat . l03001 ( 2010 ) . g. vidal , in _ understanding quantum phase transitions _ , edited by l. d. carr ( taylor @xmath487 francis , boca raton , 2010 ) . g. evenbly , p. corboz , and g. vidal , phys . rev . b 82 , 132411 ( 2010 ) . g. evenbly and g. vidal , chapter 4 in _ strongly correlated systems : numerical methods _ , edited by a. avella and f. mancini ( springer series in solid - state sciences , vol . 176 2013 ) , arxiv:1109.5334 . m. aguado and g. vidal , phys . rev . 100 , 070404 ( 2008 ) . r. koenig , b. w. reichardt , and guifre vidal , phys . b 79 , 195123 ( 2009 ) . m. aguado , annals of physics , volume 326 , issue 9 , pages 2444 - 2473 ( 2011 ) . l. tagliacozzo and g. vidal , phys.rev.b 83 , 115127 ( 2011 ) . o. buerschaper , j. m. mombelli , m. christandl , and miguel aguado , j. math . 54 , 012201 ( 2013 ) . j. haah , arxiv:1310.4507 ( 2013 ) . h. chang , y. hsieh , and y. kao , arxiv:1305.2663 ( 2013 ) . g. evenbly and g. vidal , phys . lett . , 104 , 187203 ( 2010 ) . k. harada , phys . b 86 , 184421 ( 2012 ) . j. lou , t. suzuki , k. harada , and n. kawashima , arxiv:1212.1999 ( 2012 ) . p. corboz , g. evenbly , f. verstraete , and g. vidal , phys . a 81 , 010303(r ) ( 2010 ) . c. pineda , t. barthel , and j. eisert , phys . a 81 , 050303(r ) ( 2010 ) . p. corboz and g. vidal , phys . b 80 , 165129 ( 2009 ) . r. n. c. pfeifer , p. corboz , o. buerschaper , m. aguado , m. troyer , and g. vidal , physical review b 82 , 115126 ( 2010 ) . r. koenig and e. bilgin , phys . b 82 , 125118 ( 2010 ) . g. vidal , phys . 101 , 110501 ( 2008 ) . without the action of disentanglers @xmath6 , short - range entanglement is preserved under coarse - graining , leading to an effective description that still contains some of the original , small scale degrees of freedom . as a result , the coarse - graining transformation is not a proper realization of the rg . indeed , two many - body systems that differ in irrelevant short - range details but behave identically at low energies [ that is , two many - body systems that flow to the same fixed - point of the rg ] will flow to different fixed - points of the coarse - graining transformation , because after being coarse - grained they still retain small scale details that reveal their origin . j. i. cirac and f. verstraete , j. phys . a : math . theor . 42 , 504004 ( 2009 ) . g. evenbly and g. vidal , j. stat . 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[ fig : hamintro](a ) . specifically , we consider a transformation @xmath5 that decomposes into the product of local transformations , known as disentanglers @xmath6 and isometries @xmath7 . disentangles @xmath6 are unitary transformations that act across the boundaries between blocks in @xmath360 , @xmath495 where @xmath496 is identity on @xmath488 , while isometries @xmath7 implement an isometric mapping of a block of three sites in @xmath360 to a single site in @xmath490 , @xmath497 where @xmath498 is the identity operator on @xmath492 . the isometric constraints on disentanglers @xmath6 and isometries @xmath7 are expressed pictorially in fig . [ fig : hamintro](b ) . , based on entanglement renormalization , maps a lattice @xmath360 made of @xmath396 sites into a coarse - grained lattice @xmath490 made of @xmath491 sites . ( b ) the isometries @xmath7 and disentanglers @xmath6 that constitute the coarse - graining transformation @xmath5 are constrained to be isometric , see also eqs . [ eq : b1 ] and [ eq : b2 ] . ( c ) an operator @xmath499 , supported on a local region @xmath500 made of two contiguous sites , is coarse - grained to a new local operator @xmath501 , supported on a local region @xmath502 made also of two contiguous sites . ( d ) a nearest neighbor hamiltonian @xmath503 is coarse - grained to a nearest neighbor hamiltonian @xmath504 . ( e ) the left , center and right ascending superoperators @xmath505 , @xmath506 and @xmath507 can be used to compute the new coupling @xmath508 from the initial coupling @xmath45 , see also eq . [ eq : b5].,width=321 ] an important property of the coarse - graining transformation @xmath5 is that , by construction , it preserves _ locality_. let @xmath499 be a local operator defined on a region @xmath24 of two contiguous sites of lattice @xmath43 . this operator transforms under coarse - graining as , @xmath509 where the new operator @xmath501 is supported on a region @xmath510 of two contiguous sites in lattice @xmath511 , see fig . [ fig : hamintro](c ) . the coarse - grained operator @xmath501 remains local due to the specific way in which transformation @xmath5 decomposes into local isometric tensors @xmath6 and @xmath7 . indeed , in @xmath512 , most tensors in @xmath5 annihilate to identity with their conjugates in @xmath513 . the causal cone @xmath27 of a region @xmath24 is defined as to include precisely those tensors that do not annihilate to identity when coarse - graining an operator supported on @xmath24 , and it thus tracks how region @xmath24 itself evolves under coarse - graining . in particular , a local hamiltonian @xmath0 on @xmath360 will be coarse - grained into a local hamiltonian @xmath8 on @xmath490 , @xmath514 see fig . [ fig : hamintro](d ) . the local coupling @xmath508 of the coarse - grained hamiltonian @xmath8 can be computed by applying the ( left , center , right ) ascending superoperators @xmath515 , @xmath516 and @xmath517 to the coupling @xmath45 of the initial hamiltonian , @xmath518 see fig . [ fig : hamintro](e ) . the coarse - graining transformation @xmath5 can be repeated @xmath519 times to obtain a _ sequence _ of local hamiltonians , @xmath520 where each of the local hamiltonian @xmath521 is defined on a coarse - grained lattice @xmath522 of @xmath523 sites . notice the use of subscripts to denote the level of coarse - graining , with the initial lattice @xmath524 and hamiltonian @xmath525 . the final coarse - grained hamiltonian @xmath416 in this sequence , which is defined on a lattice @xmath526 of @xmath527 sites , can be exactly diagonalized so as to determine its ground state @xmath528 . as a linear ( isometric ) map , each transformation @xmath415 can also be used to fine - grain a quantum state @xmath529 defined on @xmath530 into a new quantum state @xmath531 defined on @xmath532 , @xmath533 thus a quantum state @xmath534 defined on the initial lattice @xmath419 can be obtained by fine graining state @xmath528 with the transformations @xmath415 as , @xmath535 if each of the transformations @xmath415 has been chosen as to properly preserve the low energy subspace of the hamiltonian @xmath536 , such that @xmath521 is a low - energy effective hamiltonian for @xmath536 , then @xmath537 is a representation of the ground state of the initial hamiltonian @xmath538 . more generally , the multi - scale entanglement renormalization ansatz ( mera ) is the class of states that can be represented as eq . [ eq : b7 ] for some choice of @xmath539 and @xmath528 . for a generic choice of local hilbert space dimensions @xmath540 ( where @xmath541 ) , only a subset of all states of lattice @xmath43 can be represented in eq . [ eq : b7 ] , whereas the choice @xmath542 allows for a ( computationally inefficient ) representation of any state of the lattice . we now move to discussing the mera for a quantum critical system that is both scale invariant and translation invariant . we describe how universal information of the quantum critical point can be evaluated , by characterizing the scaling operators and their scaling dimensions . we also review the power - law scaling of two - point correlators . in this appendix , fixed - point objects ( e.g. @xmath5 , @xmath0 , @xmath17 , etc ) are denoted with a star superscript ( as @xmath543 , @xmath544 , @xmath545 , etc ) , whereas in the main text of this manuscript we did not use a star superscript to ease the notation . let @xmath546 be an infinite lattice and let @xmath538 denote a translation invariant , quantum critical hamiltonian . we assume that this hamiltonian tends to a fixed point of the rg flow of eq . [ eq : b6 ] , such that all coarse - grained hamiltonians @xmath521 are proportionate to a fixed - point hamiltonian @xmath544 for some sufficiently large @xmath153 . specifically , the coarse - grained hamiltonians in the scale invariant regime are related as @xmath547 , where @xmath548 with @xmath549 is the dynamic critical exponent of the hamiltonian ( i.e. @xmath550 for a lorentz invariant quantum critical point ) . equivalently , the local couplings that define that hamiltonians are related as @xmath551 . for concreteness , let us assume that the initial hamiltonian @xmath538 reaches the scale invariant ( lorentz invariant ) fixed point after @xmath147 coarse - grainings , such that its rg flow can be written , @xmath552 where @xmath543 represents the scale invariant coarse - graining transformation for @xmath544 . in this case , the ground state @xmath553 of the hamiltonian @xmath538 can be represented by the infinite sequence of coarse - graining transformations , @xmath554 see fig . [ fig : scaleintro ] . the class of states that can be represented as eq . [ eq : b9 ] are called scale invariant mera . the scale - dependent transformations before scale invariance , here @xmath423 and @xmath424 , correspond to transitional layers of the mera . these are important to diminish the effect of any rg irrelevant terms potentially present in the initial hamiltonian , which break scale invariance at short distances . in general , the number @xmath191 of transitional layers required will depend on the specific critical hamiltonian under consideration . [ strictly speaking , scale invariance is generically only attained after infinitely many transitional layers , but in practice a finite number @xmath191 of them often offers already a very good approximation ] . we call the fixed - point coarse - graining transformation @xmath543 scale invariant . notice that the scale invariant mera , which describes a quantum state on an infinite lattice , is defined in terms of a small number of unique tensors . each transitional map @xmath415 is described by a pair of tensors @xmath555 and the scale invariant map @xmath556 is described by the pair @xmath557 . of transitional layers with coarse - graining maps @xmath558 , here @xmath559 , followed by an infinite sequence of scaling layers , with a scale invariant map @xmath543 . ( b ) each @xmath415 of the scale invariant mera is a coarse - graining transformation composed of local tensors @xmath555.,width=321 ] we now discuss how scaling operators and their scaling dimensions can be evaluated from the scale - invariant mera . this is covered in more detail in e.g. refs . . for simplicity , let us consider a scale invariant mera with no transitional layers , that is composed of an infinite sequence of a scale invariant map @xmath543 , described by a single pair @xmath545 . as shown in fig . [ fig : twocorr](a ) , a one - site operator @xmath96 , placed on certain lattice sites , is coarse - grained under the action of layer @xmath543 into new one - site operator @xmath88 . this coarse - graining is implemented with the one - site scaling superoperator @xmath100 , @xmath560 where @xmath100 is defined in terms of the isometry @xmath561 and its conjugate , see also fig . [ fig : twocorr](b ) . the ( one - site ) scaling operators @xmath562 are defined as those operators that transform covariantly under action of @xmath100 , @xmath563 where @xmath564 is the scaling dimension of scaling operator @xmath562 . as is customary in rg analysis , the scaling operators @xmath562 and their scaling dimensions @xmath564 can be obtained through diagonalization of the scaling superoperator @xmath565 . one can obtain explicit expressions for two - point correlation functions of the scale invariant mera based upon their scaling operators , as we now describe . let us suppose that two scaling operators @xmath562 and @xmath566 are placed on special sites @xmath161 and @xmath567 that are at a distance of @xmath568 sites apart for positive integer @xmath569 , as shown in fig . [ fig : twocorr](c ) . the correlator @xmath570 can be evaluated by coarse - graining the scaling operators until they occupy adjacent sites , where the expectation value @xmath571 can then be evaluated with the local two - site density matrix @xmath572 ( which is the same at every level of the mera due to scale invariance ) . for each level of coarse - graining applied to the scaling operators @xmath562 and @xmath566 , we pick up a factor of the eigenvalues of the scaling operators , as described eq . [ eq : b10b ] , and the distance @xmath93 between the scaling operators shrinks by a factor of 3 , see fig . [ fig : twocorr](c ) , which leads to the relation @xmath573 notice that the scaling operators are coarse - grained onto adjacent sites after @xmath574 levels , thus through iteration of eq . [ eq : b11 ] we have @xmath575 where constant @xmath576 is the expectation value of the correlators evaluated on adjacent sites , @xmath577 thus it is seen that the correlator of two scaling operators @xmath562 and @xmath566 scales polynomially in the distance between the operators , with an exponent that is the sum of their corresponding scaling dimensions @xmath20 and @xmath122 , in agreement with predictions from cft @xcite . notice that eq . [ eq : b12 ] was derived from structural considerations of the mera alone and , as such , holds regardless of how the tensors in the scale invariant mera have been optimized . this argument is only valid for the chosen special locations @xmath161 and @xmath567 . for a generic pair of locations , the polynomial decay of correlations may only be obtained after proper optimization ( for instance , via energy minimization ) of the mera so as to approximate the ground state of a translation invariant , quantum critical hamiltonian @xmath0 . , which are defined in terms of a single pair of tensors @xmath578 . a one - site operator @xmath96 is coarse - grained into new one - site operators @xmath88 and @xmath89 . ( b ) the scaling superoperator @xmath100 acts covariantly upon scaling operators @xmath19 , see also eq . [ eq : b10b ] . ( c ) two scaling operators @xmath19 and @xmath579 that are separated by @xmath93 lattice sites are coarse - grained onto neighboring sites after @xmath580 maps @xmath556.,width=321 ] , which involves spatial permutation of indices as well as enacting a unitary matrix @xmath581 on each index . ( b ) the definition of reflection symmetry for a disentangler @xmath6.,width=188 ] in this appendix we describe how symmetry under spatial reflection can be exactly enforced into the mera . this is done by directly incorporating reflection symmetry in each of the tensors of the mera ( note that an equivalent approach , dubbed inversion symmetric mera , was recently proposed in ref . ) . such a step was found to be key in applications of the modular mera to quantum critical systems with a defect , as considered in sect . [ sect : bench ] . indeed , we found that in order for the modular mera to be an accurate representation of the ground state of a quantum critical system with a defect , the homogeneous system ( that is , the system in the absence of the defect ) had to be addressed with a reflection invariant mera . let us describe how the individual tensors of the mera , namely the isometries @xmath7 and disentanglers @xmath6 , can be chosen to be reflection symmetric , i.e. @xmath582 see fig . [ fig : refsym ] . here @xmath583 is a superoperator that denotes spatial reflection , which squares to the identity . the spatial reflection on a tensor involves permutation of its indices , as well as a ` reflection ' within each index , as enacted by a unitary matrix @xmath581 such that @xmath584 . the latter is needed because each index of the tensor effectively represents several sites of the original system , which also need to be reflected ( permuted ) . matrix @xmath581 has eigenvalues @xmath292 corresponding to reflection symmetric and reflection antisymmetric states , respectively . it is convenient , though not always necessary , to work within a basis such that each @xmath489-dimensional index @xmath585 decomposes as @xmath586 , where @xmath587 labels the parity ( @xmath588 for even parity and @xmath290 for odd parity ) and @xmath589 labels the distinct values of @xmath585 with parity @xmath587 . in such a basis , @xmath581 is diagonal , with the diagonal entries corresponding to the eigenvalues @xmath292 . let us turn our attention to the question of how reflection symmetry , as described in eq . [ s8e1 ] , can be imposed on the mera tensors . for concreteness , we consider an isometry @xmath7 ( analogous considerations apply to a disentangler ) . notice that we can not just symmetrize @xmath7 under reflections directly , @xmath590 because the new , reflection symmetric tensor @xmath591 will no longer be isometric . instead , we can include an additional step in the optimization algorithm that symmetrizes the _ environment _ of the tensors before each tensor is updated . in the optimization of the mera @xcite , in order to update an isometry @xmath7 one first computes its linearized environment @xmath592 . now , to obtain an updated isometry that is reflection symmetric , we first symmetrize its environment , @xmath593 in this way we ensure that the updated isometry @xmath591 ( which is obtained through a svd of @xmath594 , see ref . ) , is reflection symmetric , yet also retains its isometric character . likewise the environments @xmath595 of disentanglers @xmath6 should also be symmetrized . from the ternary mera , which coarse - grains three @xmath489-dimensional lattice sites into a single @xmath489 dimensional lattice site , is decomposed into upper and lower binary isometries , @xmath596 and @xmath597 . the index connecting the upper and lower binary isometries is chosen at an independent dimension @xmath598 . ( b ) the upper and lower binary isometries @xmath596 and @xmath597 should be chosen to maximize their overlap with the ternary isometry @xmath7 against the one - site density matrix @xmath572 , see eq . [ s9e1].,width=321 ] in the formulation of modular mera described in sect . [ sect : modularity ] it was convenient to decompose some of the isometries @xmath7 of the mera used to describe the homogeneous system into pairs of upper and lower isometries @xmath596 and @xmath597 , as depicted in fig . [ fig : isosplit](a ) . in this section we discuss how this can be accomplished . let @xmath489 denote the bond dimension of the indices of the isometry @xmath7 , and let @xmath493 denote the index connecting the upper and lower isometries @xmath596 and @xmath597 . since @xmath493 effectively represents two sites with bond dimension @xmath489 , we have that the isometric character of @xmath54 requires @xmath599 . we should perform this decomposition such that it does not change the quantum state described by the mera ( perhaps to within some very small error ) . therefore the best choice of upper @xmath596 and lower @xmath597 isometries follows from maximizing their overlap with the isometry @xmath7 against the one - site density matrix @xmath572 . that is , we choose them such that they maximize @xmath600 see fig . [ fig : isosplit](b ) . given the density matrix @xmath572 and isometry @xmath7 , one can obtain @xmath54 and @xmath55 by iteratively maximizing the above trace over each of the two tensors , one at a time . ideally , we would like the decomposition of @xmath7 into the product of @xmath54 and @xmath55 to be exact , that is , such that such that @xmath601 . this is typically only possible for @xmath602 . however , in practice we find that for choice of bond dimension @xmath598 between one or two times the dimension @xmath489 , i.e. @xmath603 , the above trace is already @xmath604 with @xmath605 negligibly small . the use of a @xmath493 smaller than @xmath606 results in a reduction of computational costs .

we propose algorithms , based on the multi - scale entanglement renormalization ansatz , to obtain the ground state of quantum critical systems in the presence of boundaries , impurities , or interfaces . by exploiting the theory of minimal updates [ ref . : g. evenbly and g. vidal , arxiv:1307.0831 ] , the ground state is completely characterized in terms of a number of variational parameters that is independent of the system size , even though the presence of a boundary , an impurity , or an interface explicitly breaks the translation invariance of the host system . similarly , computational costs do not scale with the system size , allowing the thermodynamic limit to be studied directly and thus avoiding finite size effects e.g. when extracting the universal properties of the critical system .

we are living in the golden age of cosmology . various data sets from precision measurements of temperature and polarization anisotropy in the cosmic microwave background ( cmb ) radiation as well as those of matter density fluctuations in the large - scale structure of the universe mapped by galaxy redshift surveys , lyman-@xmath11 forests and weak gravitational lensing observations are in a spectacular agreement with the concordance @xmath12cdm model @xcite . these results assure that theory of cosmological linear perturbations is basically correct , and can accurately describe the evolution of photons , neutrinos , baryons , and collisionless dark matter particles @xcite , for given initial perturbations generated during inflation @xcite . the predictions from linear perturbation theory can be compared with the precision cosmological measurements , in order to derive stringent constraints on the various basic cosmological parameters . future observations with better sensitivity and higher precision will continue to further improve our understanding of the universe . fluctuations in different cosmic fluids ( dark matter , photons , baryons , and neutrinos ) imprint characteristic features in their power spectra , owing to their interaction properties , thermal history , equation of state , and speed of sound . a remarkable example is the acoustic oscillation in the photon - baryon fluid that was generated before the decoupling epoch of photons , @xmath13 , which has been observed in the power spectrum of cmb temperature anisotropy @xcite , temperature polarization cross correlation @xcite , and distribution of galaxies @xcite . yet , the latest observations have shown convincingly that we still do not understand much of the universe . the standard model of cosmology tells us that the universe has been dominated by four components . in chronological order the four components are : early dark energy ( also known as `` inflaton '' fields ) , radiation , dark matter , and late - time dark energy . the striking fact is that we do not understand the precise nature of three ( dark matter , and early and late - time dark energy ) out of the four components ; thus , understanding the nature of these three dark components has been and will continue to be one of the most important topics in cosmology in next decades . of which , one might be hopeful that the next generation particle accelerators such as the large hadron collider ( coming on - line in 2007 ) would find some hints for the nature of dark matter particles . on the other hand , the nature of late - time dark energy , which was discovered by measurements of luminosity distance out to distant type ia supernovae @xcite , is a complete mystery , and many people have been trying to find a way to constrain properties of dark energy ( see , e.g. , @xcite for a review ) . how about the early dark energy , inflaton fields , which caused the expansion of the universe to accelerate in the very early universe ? we know little about the nature of inflaton , just like we know little about the nature of late - time dark energy . the required property of inflaton fields is basically the same as that of the late - time dark energy component : both must have a large negative pressure which is less than @xmath14 of their energy density . to proceed further , however , one needs more information from observations . different inflation models make specific predictions for the shape of the power spectrum @xcite ( see also appendix b ) as well as for other statistical properties @xcite of primordial perturbations . therefore , one of the most promising ways to constrain the physics of inflation , hence the nature of early dark energy in the universe , is to determine the shape of the primordial power spectrum accurately from observations . for example , the cmb data from the wilkinson microwave anisotropy probe @xcite , combined with the large - scale structure data from the two - degree field galaxy redshift survey @xcite , have already ruled out one of the popular inflationary models driven by a self - interacting massless scalar field @xcite . understanding the physics of inflation better will likely provide an important implication for late - time dark energy . `` radiation '' in the universe at around the matter - radiation equality mainly consists of photons and neutrinos ; however , neutrinos actually stop being radiation when their mean energy per particle roughly equals the temperature of the universe . the physics of neutrinos has been revolutionized over the last decade by solar , atmospheric , reactor , and accelerator neutrino experiments having provided strong evidence for finite neutrino masses via mixing between different neutrino flavors , the so - called neutrino oscillations @xcite . these experiments are , however , only sensitive to mass square differences between neutrino mass eigenstates , implying @xmath15 ev@xmath16 and @xmath17 ev@xmath16 ; thus , the most fundamental quantity of neutrinos , the absolute mass , has not been determined yet . cosmological neutrinos that are the relic of the cosmic thermal history have distinct influences on the structure formation . their large energy density , comparable to the energy density of photons before the matter - radiation equality , determines the expansion history of the universe . even after the matter - radiation equality , neutrinos having become non - relativistic affect the structure formation by suppressing the growth of matter density fluctuations at small spatial scales owing to their large velocity dispersion @xcite ( see sec . ii and appendix a for more details ) . therefore , the galaxy redshift surveys , combined with the cmb data , provide a powerful , albeit indirect , means to constraining the neutrino properties @xcite . this approach also complements the theoretical and direct experimental efforts for understanding the neutrino physics . in fact , the cosmological constraints have placed the most stringent upper bound on the total neutrino mass , @xmath18 ( @xmath19 ) @xcite , stronger than the direct experiment limit @xmath20 @xcite . in addition , the result obtained from the liquid scintillator neutrino detector ( lsnd ) experiment , which implies @xmath21 to @xmath22 oscillations with @xmath23ev@xmath16 @xcite in an apparent contradiction with the other neutrino oscillation experiments mentioned above , potentially suggests the need for new physics : the cosmological observations will provide independent tests of this hypothesis . in this paper we shall study the capability of future galaxy surveys at high redshifts , combined with the cmb data , for constraining ( 1 ) the neutrino properties , more specifically the total neutrino mass , @xmath24 , and the number of non - relativistic neutrino species , @xmath25 , and ( 2 ) the shape of the primordial power spectrum that is parameterized in terms of the spectral tilt , @xmath26 , and the running index , @xmath9 , motivated by inflationary predictions ( see appendix b ) . for the former , we shall pay particular attention to our ability to simultaneously constrain @xmath24 and @xmath25 , as they will provide important clues to resolving the absolute mass scale as well as the neutrino mass hierarchy . the accuracy of determining the neutrino parameters and the power spectrum shape parameters will be derived using the fisher information matrix formalism , including marginalization over the other cosmological parameters as well as the galaxy bias . our analysis differs from the previous work on the neutrino parameters in that we fully take into account the two - dimensional nature of the galaxy power spectrum in the line - of - sight and transverse directions , while the previous work used only spherically averaged , one - dimensional power spectra . the geometrical distortion due to cosmology and the redshift space distortion due to the peculiar velocity field will cause anisotropic features in the galaxy power spectrum . these features help to lift degeneracies between cosmological parameters , substantially reducing the uncertainties in the parameter determinations . this is especially true when variations in parameters of interest cause modifications in the power spectrum shape , which is indeed the case for the neutrino parameters , tilt and running index . the usefulness of the two - dimensional power spectrum , especially for high - redshift galaxy surveys , has been carefully investigated in the context of the prospected constraints on late - time dark energy properties @xcite . we shall show the parameter forecasts for future wide - field galaxy surveys that are already being planned or seriously under consideration : the fiber multiple object spectrograph ( fmos ) on subaru telescope @xcite , its significantly expanded version , wfmos @xcite , the hobby ebery telescope dark energy experiment ( hetdex ) @xcite , and the cosmic inflation probe ( cip ) mission @xcite . to model these surveys , we consider three hypothetical galaxy surveys which probe the universe over different ranges of redshift , ( 1 ) @xmath27 , ( 2 ) @xmath28 and ( 3 ) @xmath29 . we fix the sky coverage of each survey at @xmath30 deg@xmath16 in order to make a fair comparison between different survey designs . as we shall show below , high - redshift surveys are extremely powerful for precision cosmology because they allow us to probe the linear power spectrum down to smaller length scales than surveys at low redshifts , protecting the cosmological information against systematics due to non - linear perturbations . we shall also study how the parameter uncertainties are affected by changes in the number density of sampled galaxies and the survey volume . the results would give us a good guidance to defining the optimal survey design to achieve the desired accuracies in parameter determinations . the structure of this paper is as follows . in sec . [ nu ] , we review the physical pictures as to how the non - relativistic ( massive ) neutrinos lead to scale - dependent modifications in the growth of mass clustering relative to the pure cdm model . sec . [ pps ] defines the parameterization of the primordial power spectrum motivated by inflationary predictions . in sec . [ formalism ] we describe a methodology to model the galaxy power spectrum observable from a redshift survey that includes the two - dimensional nature in the line - of - sight and transverse directions . we then present the fisher information matrix formalism that is used to estimate the projected uncertainties in the cosmological parameter determination from statistical errors on the galaxy power spectrum measurement for a given survey . after survey parameters are defined in sec . [ survey ] , we show the parameter forecasts in sec . [ results ] . finally , we present conclusions and some discussions in sec . we review the basic properties of cosmological neutrinos in appendix a , the basic predictions from inflationary models for the shape of the primordial power spectrum in appendix b , and the relation between the primordial power spectrum and the observed power spectrum of matter density fluctuations in appendix c. in the following , we assume an adiabatic , cold dark matter ( cdm ) dominated cosmological model with flat geometry , which is supported by the wmap results @xcite , and employ the the notation used in @xcite : the present - day density of cdm , baryons , and non - relativistic neutrinos , in units of the critical density , are denoted as @xmath31 , @xmath32 , and @xmath33 , respectively . the total matter density is then @xmath34 , and @xmath35 is the ratio of the massive neutrino density contribution to @xmath36 : @xmath37 . throughout this paper we assume the standard thermal history in the early universe : there are three neutrino species with temperature equal to @xmath38 of the photon temperature . we then assume that @xmath39 species are massive and could become non - relativistic by the present epoch , and those non - relativistic neutrinos have equal masses , @xmath40 . as we show in appendix [ app : neutrino ] , the density parameter of the non - relativistic neutrinos is given by @xmath41 , where we have assumed 2.725k for the cmb temperature today @xcite , and @xmath42 is the hubble parameter defined as @xmath43kms@xmath44mpc@xmath44 . the neutrino mass fraction is thus given by @xmath45 structure formation is modified by non - relativistic neutrinos on scales below the hubble horizon size when the neutrinos became non - relativistic , @xmath46mpc@xmath44 ( see eq . [ [ eq : knr ] ] ) . in particular , the characteristic scale imprinted onto the galaxy power spectrum at a given redshift @xmath0 is the neutrino free - streaming scale , which is defined by eq . ( [ eq : kfs ] ) : @xmath47 therefore , non - relativistic neutrinos with lighter masses suppress the growth of structure formation on larger spatial scales at a given redshift , and the free - streaming length becomes shorter at a lower redshift as neutrino velocity decreases with redshift . the most important property of the free - streaming scale is that it depends on the mass of each species , @xmath40 , rather than the total mass , @xmath48 ; thus , measurements of @xmath49 allow us to distinguish different neutrino mass hierarchy models . fortunately , @xmath49 appears on the scales that are accessible by galaxy surveys : @xmath50 at @xmath51 for @xmath52 . on the spatial scales larger than the free - streaming length , @xmath53 , neutrinos can cluster and fall into gravitational potential well together with cdm and baryonic matter . in this case , perturbations in all matter components ( cdm , baryon and neutrinos , denoted as ` cb@xmath54 ' hereafter ) grow at the same rate given by @xmath55 where @xmath56 is the usual linear growth factor ( see , e.g. , eq . ( 4 ) in @xcite ) . on the other hand , on the scales smaller than the free - streaming length , @xmath57 , perturbations in non - relativistic neutrinos are absent due to the large velocity dispersion . in this case , the gravitational potential well is supported only by cdm and baryonic matter , and the growth of matter perturbations is slowed down relative to that on the larger scales . as a result , the matter power spectrum for @xmath58 is suppressed relative to that for @xmath59 . in this limit the total matter perturbations grow at the slower rate given by @xmath60^{1-p}\qquad k\gg k_{\rm fs}(z ) , \label{eqn : d_cbnu}\ ] ] where @xmath61 @xcite . in @xcite an accurate fitting function for the scale - dependent growth rate was derived by matching these two asymptotic solutions . we shall use the fitting function throughout this paper . = 8.5 cm figure [ fig : growth ] shows suppression in the growth rate of total matter perturbations at @xmath62 , 0.1 , and 1 @xmath42mpc@xmath44 due to the neutrino free - streaming . the suppression becomes more significant at lower redshifts for a given wavenumber , or for higher frequency perturbations at a given redshift , because neutrino can grow together with cdm and baryonic matter after the spatial scale of a given perturbation has become larger than the neutrino free - streaming scale that varies with redshift as given by eq . ( [ eq : kfs * ] ) . it is thus expected that a galaxy survey with different redshift slices can be used to efficiently extract the neutrino parameters , @xmath63 and @xmath40 . the upper and middle panels of figure [ fig : pk ] illustrate how free - streaming of non - relativistic neutrinos suppresses the amplitude of linear matter power spectrum , @xmath64 , at @xmath65 . note that we have normalized the primordial power spectrum such that all the power spectra match at @xmath66 ( see [ pps ] ) . to illuminate the dependence of @xmath64 on @xmath40 , we fix the total mass of non - relativistic neutrinos , @xmath48 , by @xmath67 and @xmath68 in the upper and middle panels , respectively , and vary the number of non - relativistic neutrino species as @xmath69 , @xmath70 and @xmath71 . the suppression of power is clearly seen as one goes from @xmath72 to @xmath73 ( see eq . [ [ eq : kfs * ] ] for the value of @xmath49 ) . the way the power is suppressed may be easily understood by the dependence of @xmath74 on @xmath40 ; for example , @xmath64 at smaller @xmath75 is more suppressed for a smaller @xmath40 , as lighter neutrinos have longer free - streaming lengths . on very small scales , @xmath76 ( @xmath77 and 0.1mpc@xmath44 for @xmath67 and @xmath68 , respectively ) , however , the amount of suppression becomes nearly independent of @xmath75 , and depends only on @xmath35 ( or the total neutrino mass , @xmath78 ) as @xmath79 \approx 8f_\nu . \label{eq : overallsuppression}\ ] ] we therefore conclude that one can extract @xmath35 and @xmath63 separately from the shape of @xmath64 , if the suppression `` pattern '' in different regimes of @xmath75 is accurately measured from observations . are observations good enough ? the shaded boxes in the upper and middle panels in figure [ fig : pk ] represent the 1-@xmath80 measurement errors on @xmath64 expected from one of the fiducial galaxy surveys outlined in sec . [ survey ] . we find that @xmath64 will be measured with @xmath81 accuracy in each @xmath75 bin . if other cosmological parameters were perfectly known , the total mass of non - relativistic neutrinos as small as @xmath82 would be detected at more than 2-@xmath80 . this limit is much smaller than the lower mass limit implied from the neutrino oscillation experiments , 0.06ev . this estimate is , of course , unrealistic because a combination of other cosmological parameters could mimic the @xmath63 or @xmath35 dependence of @xmath64 . the lower panel in figure [ fig : pk ] illustrates how other cosmological parameters change the shape of @xmath64 . in the following , we shall extensively study how well future high - redshift galaxy surveys , combined with the cosmic microwave background data , can determine the mass of non - relativistic neutrinos and discriminate between different @xmath25 , fully taking into account degeneracies between cosmological parameters . inflation generally predicts that the primordial power spectrum of curvature perturbations is nearly scale - invariant . different inflationary models make specific predictions for _ deviations _ of the primordial spectrum from a scale - invariant spectrum , and the deviation is often parameterized by the `` tilt '' , @xmath26 , and the `` running index '' , @xmath9 , of the primordial power spectrum . as the primordial power spectrum is nearly scale - invariant , @xmath83 and @xmath84 are predicted to be much less than unity . this , however , does not mean that the observed matter power spectrum is also nearly scale - invariant . in appendix [ app : norm ] , we derive the power spectrum of total matter perturbations that is normalized by the primordial curvature perturbation ( see eq . [ [ eq : pknorm ] ] ) @xmath85 where @xmath86 mpc@xmath44 , @xmath87 , and @xmath88 is the normalization parameter given by the wmap collaboration @xcite . we adopt @xmath89 , which gives @xmath90 . ( in the notation of @xcite @xmath91 . ) the linear transfer function , @xmath92 , describes the evolution of the matter power spectrum during radiation era and the interaction between photons and baryons before the decoupling of photons . note that @xmath92 depends only on non - inflationary parameters such as @xmath93 and @xmath94 , and is independent of @xmath26 and @xmath9 . also , the effects of non - relativistic neutrinos are captured in @xmath95 ; thus , @xmath92 is independent of time after the decoupling epoch . we use the fitting function found in @xcite for @xmath92 . note that the transfer function and the growth rate are normalized such that @xmath96 and @xmath97 as @xmath66 during the matter era . in appendix [ app : inflation ] we describe generic predictions on @xmath26 and @xmath9 from inflationary models . for example , inflation driven by a massive , self - interacting scalar field predicts @xmath98 and @xmath99 for the number of @xmath100-foldings of expansion factor before the end of inflation of 50 . this example shows that precision determination of @xmath26 and @xmath9 allows us to discriminate between candidate inflationary models ( see @xcite for more details ) . suppose now that we have a redshift survey of galaxies at some redshift . galaxies are biased tracers of the underlying gravitational field , and the galaxy power spectrum measures how clustering strength of galaxies varies as a function of 3-dimensional wavenumbers , @xmath75 ( or the inverse of 3-dimensional length scales ) . we do not measure the length scale directly in real space ; rather , we measure ( 1 ) angular positions of galaxies on the sky , and ( 2 ) radial positions of galaxies in redshift space . to convert ( 1 ) and ( 2 ) to positions in 3-dimensional space , however , one needs to assume a reference cosmological model , which might be different from the true cosmology . an incorrect mapping of observed angular and redshift positions to 3-dimensional positions produces a distortion in the measured power spectrum , known as the `` geometrical distortion '' @xcite . the geometrical distortion can be described as follows . the comoving size of an object at redshift @xmath0 in radial , @xmath101 , and transverse , @xmath102 , directions are computed from the extension in redshift , @xmath103 , and the angular size , @xmath104 , respectively , as @xmath105 where @xmath106 is the comoving angular diameter distance given in the spatial sector of the friedmann - robertson - walker line element , @xmath107 ( @xmath108 is the comoving radial distance ) . we assume a flat universe throughout this paper , in which case @xmath109 . the comoving angular distance out to a galaxy at redshift @xmath0 is @xmath110 where @xmath111 is the hubble parameter given by @xmath112.\ ] ] here @xmath113 , and @xmath114 is the present - day density parameter of a cosmological constant , @xmath12 . a tricky part is that @xmath111 and @xmath115 in eq . ( [ eq : perp ] ) depend on cosmological models . it is therefore necessary to assume some fiducial cosmological model to compute the conversion factors . in the following , quantities in the fiducial cosmological model are distinguished by the subscript ` fid ' . then , the length scales in fourier space in radial , @xmath116 , and transverse , @xmath117 , directions are estimated from the inverse of @xmath118 and @xmath119 . these fiducial wavenumbers are related to the true wavenumbers by @xmath120 therefore , any difference between the fiducial cosmological model and the true model would cause anisotropic distortions in the estimated power spectrum in ( @xmath121 , @xmath122 ) space . in addition , shifts in @xmath0 due to peculiar velocities of galaxies distort the shape of the power spectrum along the line - of - sight direction , which is known as the `` redshift space distortion '' @xcite . from azimuthal symmetry around the line - of - sight direction , which is valid when a distant - observer approximation holds , the linear power spectrum estimated in redshift space , @xmath123 , is modeled in @xcite as @xmath124 ^ 2\nonumber\\ & & \times b_1 ^ 2 p(k , z ) , \label{eqn : ps}\end{aligned}\ ] ] where @xmath125 and @xmath126 is a function characterizing the linear redshift space distortion , and @xmath127 is a scale - independent , linear bias parameter . note that @xmath128 depends on both redshift and wavenumber via the linear growth rate . in the infall regime , @xmath129 , we have @xmath130 , while in the free - streaming regime , @xmath76 , we have @xmath131 , where @xmath132 is defined below eq . ( [ eqn : d_cbnu ] ) . one might think that the geometrical and redshift - space distortion effects are somewhat degenerate in the measured power spectrum . this would be true only if the power spectrum was a simple power law . fortunately , characteristic , non - power - law features in @xmath64 such as the broad peak from the matter - radiation equality , scale - dependent suppression of power due to baryons and non - relativistic neutrinos , the tilt and running of the primordial power spectrum , the baryonic acoustic oscillations , etc . , help break degeneracies quite efficiently @xcite . in this paper , we employ the linear transfer function with baryonic oscillations _ smoothed out _ ( but includes non - relativistic neutrinos ) @xcite . as extensively investigated in @xcite , the baryonic oscillations can be used as a standard ruler , thereby allowing one to precisely constrain @xmath111 and @xmath115 separately through the geometrical distortion effects ( especially for a high - redshift survey ) . therefore , our ignoring the baryonic oscillations might underestimate the true capability of redshift surveys for constraining cosmological parameters . we have found that the constraints on @xmath26 and @xmath9 from galaxy surveys improve by a factor of 23 when baryonic oscillations are included . this is because the baryonic oscillations basically fix the values of @xmath36 , @xmath133 and @xmath134 , lifting parameter degeneracies between @xmath133 , @xmath134 , @xmath26 , and @xmath9 . however , we suspect that this is a rather optimistic forecast , as we are assuming a flat universe dominated by a cosmological constant . this might be a too strong prior , and relaxing our assumptions about geometry of the universe or the properties of dark energy will likely result in different forecasts for @xmath26 and @xmath9 . in this paper we try to separate the issues of non - flat universe and/or equation of state of dark energy from the physics of neutrinos and inflation . we do not include the baryonic oscillations in our analysis , in order to avoid too optimistic conclusions about the constraints on the neutrino parameters , @xmath26 , and @xmath9 . eventually , the full analysis including non - flat universe , arbitrary dark energy equation of state and its time dependence , non - relativistic neutrinos , @xmath26 , and @xmath9 , using all the information we have at hand including the baryonic oscillations , will be necessary . we leave it for a future publication ( takada and komatsu , in preparation ) . in order to investigate how well one can constrain the cosmological parameters for a given redshift survey design , one needs to specify measurement uncertainties of the galaxy power spectrum . when non - linearity is weak , it is reasonable to assume that observed density perturbations obey gaussian statistics . in this case , there are two sources of statistical errors on a power spectrum measurement : the sampling variance ( due to the limited number of independent wavenumbers sampled from a finite survey volume ) and the shot noise ( due to the imperfect sampling of fluctuations by the finite number of galaxies ) . to be more specific , the statistical error is given in @xcite by @xmath135 ^ 2= \frac{2}{n_k}\left[1+\frac1{\bar{n}_gp_s(k_i)}\right]^2 , \label{eqn : pkerror}\ ] ] where @xmath136 is the mean number density of galaxies and @xmath137 is the number of independent @xmath138 modes within a given bin at @xmath139 : @xmath140 here @xmath141 is the size of the fundamental cell in @xmath75 space , @xmath142 is the comoving survey volume , and @xmath143 is the cosine of the angle between @xmath144 and the line - of - sight . note that we have assumed that the galaxy selection function is uniform over the redshift slice we consider and ignored any boundary effects of survey geometry for simplicity . the first term in eq . ( [ eqn : pkerror ] ) represents sampling variance . errors become independent of the number density of galaxies when sampling variance dominates ( i.e. , @xmath145 over the range of @xmath75 considered ) , and thus the only way to reduce the errors is to survey a larger volume . on the other hand , the second term represents shot noise , which comes from discreteness of galaxy samples . when shot noise dominates ( @xmath146 ) , the most effective way to reduce noise is to increase the number density of galaxies by increasing exposure time per field . note that for a fixed @xmath147 the relative importance of shot noise contribution can be suppressed by using galaxies with larger bias parameters , @xmath127 , as @xmath148 . in sec . [ survey ] we shall discuss more about the survey design that is required to attain the desired parameter accuracy . we use the fisher information matrix formalism to convert the errors on @xmath149 into error estimates of model parameters @xcite . the fisher matrix is computed from @xmath150 ^ 2 , \label{eqn : fisher}\end{aligned}\ ] ] where @xmath151 expresses a set of parameters . one may evaluate some derivative terms analytically : @xmath152 the @xmath153 error on @xmath151 marginalized over the other parameters is given by @xmath154 , where @xmath155 is the inverse of the fisher matrix . it is sometimes useful to consider projected constraints in a two - parameter subspace to see how two parameters are correlated . we follow the method described around eq . ( 37 ) in @xcite for doing this . another quantity to describe degeneracies between given two parameters , @xmath156 and @xmath157 , is the correlation coefficient defined as @xmath158 if @xmath159 , the parameters are totally degenerate , while @xmath160 means they are uncorrelated . to calculate @xmath161 using eq . ( [ eqn : fisher ] ) , we need to specify @xmath162 and @xmath163 for a given galaxy survey . we use the upper limit , @xmath163 , to exclude information in the non - linear regime , where the linear theory prediction of density fluctuations , eq . ( [ eqn : ps ] ) , becomes invalid . following @xcite , we adopt a conservative estimate for @xmath163 by imposing the condition @xmath164 , where @xmath165 is the r.m.s . mass fluctuation in a sphere of radius @xmath166 at a given redshift @xmath0 . all the fourier modes below @xmath163 are considered as in the linear regime . this idea is partly supported by the simulation - based work in the literature @xcite , while a more careful and quantitative study is needed to understand the impact of non - linearities on cosmological parameter estimates as well as to study how to protect the cosmological information against the systematics . table [ tab : survey ] lists @xmath163 for each redshift slice of galaxy surveys we shall consider . in addition , we shall show how the results will change with varying @xmath163 . as for the minimum wavenumber , we use @xmath167 mpc@xmath44 , which gives well - converged results for all the cases we consider . the parameter forecasts derived from the fisher information formalism depend on the fiducial model and are also sensitive to the choice of free parameters . we include a fairly broad range of the cdm dominated cosmology : the density parameters are @xmath168 , @xmath169 , and @xmath170 ( note that we assume a flat universe ) ; the primordial power spectrum shape parameters are the spectral tilt , @xmath171 , the running index , @xmath172 , and the normalization of primordial curvature perturbation , @xmath173 ( the numbers in the parentheses denote the values of the fiducial model ) . the linear bias parameters , @xmath127 , are calculated for each redshift slice as given in table [ tab : survey ] ; the fiducial values of the neutrino parameters , @xmath35 and @xmath63 , are allowed to vary in order to study how the constraints on @xmath35 and @xmath63 change with the assumed fiducial values . for a survey which consists of @xmath174 redshift slices , we have @xmath175 parameters in total . as we shall show later , a galaxy survey alone can not determine all the cosmological parameters simultaneously , but would leave some parameter combinations degenerated . this is especially true when non - relativistic neutrinos are added . therefore , it is desirable to combine the galaxy survey constraints with the constraints from cmb temperature and polarization anisotropy , in order to lift parameter degeneracies . when computing the fisher matrix of cmb , we employ 7 parameters : 6 parameters ( the parameters above minus the neutrino parameters and the bias parameters ) plus the thomson scattering optical depth to the last scattering surface , @xmath176 . note that we ignore the effects of non - relativistic neutrinos on the cmb power spectra : their effects are small and do not add very much to the constraints from the high-@xmath0 galaxy survey . we then add the cmb fisher matrix to the galaxy fisher matrix as @xmath177 . we entirely ignore the contribution to the cmb from the primordial gravitational waves . we use the publicly - available cmbfast code @xcite to compute the angular power spectrum of temperature anisotropy , @xmath178 , @xmath179-mode polarization , @xmath180 , and their cross correlation , @xmath181 . specifically we consider the noise per pixel and the angular resolution of the planck experiment that were assumed in @xcite . note that we use the cmb information in the range of multipole @xmath182 . 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high-@xmath0 galaxy redshift surveys open up exciting possibilities for precision determinations of neutrino masses and inflationary models . the high-@xmath0 surveys are more useful for cosmology than low-@xmath0 ones owing to much weaker non - linearities in matter clustering , redshift - space distortion and galaxy bias , which allows us to use the galaxy power spectrum down to the smaller spatial scales that are inaccessible by low-@xmath0 surveys . we can then utilize the two - dimensional information of the linear power spectrum in angular and redshift space to measure the scale - dependent suppression of matter clustering due to neutrino free - streaming as well as the shape of the primordial power spectrum . to illustrate capabilities of high-@xmath0 surveys for constraining neutrino masses and the primordial power spectrum , we compare three future redshift surveys covering 300 square degrees at @xmath1 , @xmath2 , and @xmath3 . we find that , combined with the cosmic microwave background data expected from the planck satellite , these surveys allow precision determination of the total neutrino mass with the projected errors of @xmath4 , 0.043 , and 0.025 ev , respectively , thus yielding a positive _ detection _ of the neutrino mass rather than an upper limit , as @xmath5 is smaller than the lower limits to the neutrino masses implied from the neutrino oscillation experiments , by up to a factor of 4 for the highest redshift survey . the accuracies of constraining the tilt and running index of the primordial power spectrum , @xmath6 and @xmath7 at @xmath8 , respectively , are smaller than the current uncertainties by more than an order of magnitude , which will allow us to discriminate between candidate inflationary models . in particular , the error on @xmath9 from the future highest redshift survey is not very far away from the prediction of a class of simple inflationary models driven by a massive scalar field with self - coupling , @xmath10 .

there has been an intense interest to understand the superconductivity of the recently discovered lafeaso.@xcite experiments have found values of the curie temperature ( t@xmath5 ) as large as 26 k for electron doping of lafeaso@xmath0f@xmath1 , 0.04 @xmath6 0.12@xcite . similar values of t@xmath5 are found for hole doping of la with sr but not with ca@xcite . neutron scattering@xcite and optical measurements@xcite find an antiferromagnetic ( afm ) ground state which has been confirmed by previous electronic structure calculations.@xcite the nature of the superconductivity has not been understood , though evidence suggests its unconventional character.@xcite the understanding of the normal - state electronic structure is important and serves as the foundation for understanding the superconductivity . one important question is what happens to the electronic structure when the extra electrons are added to the system via the fluorine dopants . a number of band structure studies have been performed to date to address these questions ; however , most of them use either the simple rigid - band picture of shifting the fermi energy in the band structure of the undoped system or the virtual crystal approximation.@xcite while these methods are expected to describe the rough picture , the actual positions of the dopants could make significant differences to the band structure as compared to the rigid - band shift or to the vca band structure , which is well known from the work on other systems.@xcite in this work , we investigate the band structure using full supercell calculations and study the changes in the fermi surface and the energetics with electron doping , with the fluorine substitution of the oxygen sites . lafeaso forms in the @xmath7 structure@xcite with ( fe@xmath8as@xmath9)@xmath10 layers lying between ( la@xmath11o@xmath12)@xmath10 layers , each of the atoms forming a square sublattice . half of the as atoms belonging to the feas layer occur above the center of the fe squares and the other half below it in an alternating pattern . they belong to a class of materials@xcite formed by one layer of a rare - earth atom with oxygen and another layer with late transition metal with a pnictogen atom . each fe atom , lying at the middle of a layer as seen in fig . [ figcrystal ] , is coordinated with four as atoms in distorted tetrahedral bonds above and below ; o also lies in a distorted tetrahedron of la atoms . the doping of la ( with sr ) or o ( with f ) is not in the magnetic feas layer but changes the magnetic properties nonetheless . experimental lattice parameters of @xmath13 = 4.035 and @xmath14 = 8.739 were used . the internal parameters were relaxed by total energy minimization , the results of which agreed with the values reported in the literature@xcite , viz . , @xmath15 = 0.142 and @xmath16 = 0.633 . electronic structure calculations were performed using the linearized augmented plane wave ( lapw ) method as implemented in the wien2k@xcite program . the unit cell contains two formula units and for studying the effects of the dopants we used two supercells , a 16-atom supercell ( four formula units ) formed by doubling the cell in the @xmath17 or @xmath18 direction and a 32-atom supercell ( eight formula unit ) formed by doubling the cell in the @xmath19 plane in each direction . these two supercells correspond , respectively , to 25% and 12.5% f doping when one o atom is replaced by f. calculations were also performed with the virtual crystal approximation ( vca)@xcite with the standard unit cell . these two methods were used to understand the effects of f doping on the o sites . in the vca the nuclear and the electron charge of the o atoms are increased continuously to approximate the additional electrons introduced by the f dopants . for example , a 5% concentration of f would change the nuclear and electronic charge of the o atoms from 8.0 to 8.05 . since superconductivity is expected to arise in the nonmagnetic ( nm ) state , we have focused on the electronic structure in the nm state . . , title="fig:",width=317 ] . , title="fig:",width=317 ] in order to understand the effect of electron doping , we first discuss the results for the density of states obtained from the supercell calculation of f - doped lafeaso . the density of states ( dos ) for lafeaso given in fig . [ figdos1]a shows la @xmath20 and @xmath21 states lying above the fermi level , while the o @xmath22 and as @xmath22 states occur below it . the o @xmath23 and as @xmath23 states lie well below , outside the range of the figure . the fe @xmath21 states hybridize with the as @xmath22 states , though the size of the as sphere in the lapw method leaves much of the as @xmath22 character outside the spheres , reducing its weight in the plot . this leaves the primary character of the bands observed in the calculated dos near @xmath24 as fe @xmath21 . strong fe - fe interactions cause the fe @xmath21 states not to split apart into @xmath25 and @xmath26 states . the positions of these states agree very well with those reported for the undoped lafeaso@xcite and lafeasp.@xcite a full supercell calculation with 25% f replacing o , shown in fig . [ figdos1]b , finds that the f @xmath22 levels lie far below @xmath24 and act only to add electrons to the system , appearing to cause a rigid shift of the bands . as mentioned by previous authors@xcite , although the total number of carriers increases , the electron doping shifts @xmath24 to a lower dos , making it hard to understand how the superconducting state can arise . however , while the dos has a minimum at @xmath24 , there is no evidence that the system is close to a metal - insulator transition.@xcite fe@xmath27as@xmath27o@xmath28f ) in violet and for the undoped material ( la@xmath27fe@xmath27as@xmath27o@xmath27 ) with rigid shift in black and ( b ) the corresponding fermi surfaces given on the @xmath29 plane . the symmetry points are for the supercell brillouin zone , which has the same symmetry points as in the original unit cell but with half the magnitudes for the @xmath30 and @xmath31 components.,title="fig:",width=226 ] fe@xmath27as@xmath27o@xmath28f ) in violet and for the undoped material ( la@xmath27fe@xmath27as@xmath27o@xmath27 ) with rigid shift in black and ( b ) the corresponding fermi surfaces given on the @xmath29 plane . the symmetry points are for the supercell brillouin zone , which has the same symmetry points as in the original unit cell but with half the magnitudes for the @xmath30 and @xmath31 components.,title="fig:",width=226 ] from the calculated dos ( fig . [ figdos1 ] ) , it might appear that the band structure for lafeaso is relatively unaffected by f doping , so that a rigid band shift of @xmath24 to accommodate the added electrons might be good enough to describe the states at the fermi energy . we find that while the overall shapes of the bands are about the same , there are enough differences in the states near @xmath24 to produce significant differences in the fermi surface for the doped case . the band structure has been plotted in fig . [ figbands2]a for the 32-atom supercell with one f atom on an o site and a calculation without f doping but with the bands rigidly shifted . in comparing the two cases , we have aligned the bands so that the energies of the deep oxygen core levels ( o 1@xmath23 and 2@xmath23 ) remain the same , in view of the fact that the deep core levels are very narrow in energy and they are not affected by the f substitution . comparing the two sets of bands , the bands with f doping are sometimes above the shifted bands and sometimes below , so a better agreement is not possible simply by shifting the bands further . an important difference is the increased splitting of bands halfway between @xmath32 and @xmath33 at @xmath24 . previous calculations@xcite have predicted that a rigid shift would lead to no separation between these two bands at @xmath24 , but the supercell calculations show that these two bands remain apart . turning now to the fermi surface , in the original brillouin zone of the standard unit cell , the fermi surface consists of two hole sheets around @xmath32 and two electron sheets around @xmath33 . all sheets now occur around the @xmath32 point of the supercell brillouin zone , since the original @xmath33 point gets folded to @xmath32 . most of the fermi sheets in the full calculation have larger radii than that predicted from a rigid shift as the bands move further away from @xmath32 as seen from fig . [ figbands2]b . thus the rigid band shift does not describe very well the changes in the fermi surface due to the doping . fe@xmath27as@xmath27o@xmath28f ) ( violet lines ) compared with the equivalent vca calculation ( black lines ) with changed o nuclear charge . ( b ) same as ( a ) except that the vca calculation was done with changed la nuclear charge ( black lines ) . no difference is seen between the two sets of band structures near @xmath24.,title="fig:",width=226 ] fe@xmath27as@xmath27o@xmath28f ) ( violet lines ) compared with the equivalent vca calculation ( black lines ) with changed o nuclear charge . ( b ) same as ( a ) except that the vca calculation was done with changed la nuclear charge ( black lines ) . no difference is seen between the two sets of band structures near @xmath24.,title="fig:",width=226 ] our calculations of a rigid shift of the bands show significant changes in the fermi surface compared to the full supercell calculation with the dopants included . in view of the fact that the states at @xmath24 are predominantly fe @xmath21 and the f dopants are far from the feas layers , one might expect that the dopants could affect the band structure near @xmath24 in two ways : ( a ) by changing the coulomb potential on different fe sites by different amounts depending on their locations or ( b ) by introducing the extra electrons in the fe layers which can then modify the on - site energies of different fe orbitals differently because of their selective occupation of the various fe(@xmath21 ) orbitals . quite interestingly , we find that there is a remarkable agreement between the vca and the supercell results for states close to @xmath24 ( fig . [ figbands1 ] ) . in both cases , we have the same number of electrons in the feas layer and this agreement does not change even if we introduce the extra carriers in the vca by changing the la nuclear charge instead of the o nuclear charge . this shows that the band structure is sensitive only to the electron concentration in the feas layer , so the coulomb shift due to the relative position of the f dopants is lost by the dielectric screening due to the intermediate la and as layers . by the same token , the rigid band shift does not describe the band structure accurately because of the different concentration of the electrons implicit in the rigid band shift vs. the full calculation . with and without the spin - orbit interaction.,title="fig:",width=207 ] with and without the spin - orbit interaction.,title="fig:",width=207 ] while comparisons of vca and rigid shifts of the bands are important to the fermi surface , spin - orbit effects can also change the details of the fermi surface . since small changes to the fermi surface can play an important role in superconductivity , spin - orbit effects can not be ignored . spin - orbit has not been investigated in lafeaso in any detail . band structure calculations for the 8-atom unit cell given in fig . [ figbands2]a , b show that spin - orbit lifts degeneracies for bands lying near @xmath24 . one can see that the splitting is larger at @xmath24 along @xmath34 and @xmath35 . significant changes occur along @xmath2 where the more dispersive fe @xmath36 band hybridizes with much less dispersive fe @xmath37 and @xmath38 bands , separating the third hole pocket ( with strong @xmath36 character ) from the rest of the fermi surface . the other bands at @xmath24 are relatively unchanged . the calculated fermi surface for the standard 8-atom unit cell corresponding to the undoped ( lafeaso)@xmath10 agrees well with previous calculations@xcite . we here show the fermi surface calculated using the vca for 12.5% f concentration in fig . [ figfermi1]a . the fermi surface consists of two cylindrical hole sheets lying along @xmath2 and two cylindrical electron sheets lying along @xmath4 . by doubling the unit cell in the both directions of the @xmath19 plane to form the supercell , the fermi surface undergoes band folding , as seen in fig . [ figfermi1]b . this causes the elliptical electron pockets around @xmath33 to now surround the two cylindrical hole sheets at @xmath32 . we note that the crystal symmetry of the supercell is the same as that as the smaller unit cell . therefore , a point in the brillouin zone , such as @xmath33 ( @xmath39,@xmath39,0 ) is used in both figures , but corresponds to the fraction of the reciprocal lattice vectors in each case . therefore the @xmath33 point in fig . [ figfermi1]a is not the same @xmath33 point in fig . [ figfermi1]b . at lower concentrations , there exists a hole cap around the @xmath3 point , as has been mentioned in previous calculations@xcite . plane of the supercell brillouin zone for the 32-atom supercell of lafeaso with ( a ) no f doping and ( b ) with f replacing one of the 8 o sites.,title="fig:",width=226 ] plane of the supercell brillouin zone for the 32-atom supercell of lafeaso with ( a ) no f doping and ( b ) with f replacing one of the 8 o sites.,title="fig:",width=226 ] adding electrons to the feas plane via f doping increases the size of the elliptical electron pockets and reduces the size of the hole pockets . full calculations performed in the 32-atom supercell with no f doping ( fig . [ figfermi3]a ) shows the two elliptical electron pockets surrounding the two nearly circular hole pockets . the smaller electron pocket and the larger hole pocket nearly overlap . when we replace one f for an o atom ( la@xmath27fe@xmath27as@xmath27o@xmath28f ) in fig . [ figfermi3]b , the overall shape of the fermi surface remains unchanged , but the electron pockets become larger while the hole pockets shrink . as we can see in the plot of the fermi surface ( fig . [ figfermi1 ] ) , the hole pockets and electron pockets are narrower in the @xmath29 plane and become larger in the @xmath40 plane . addition of electrons reduces the differences between the sizes of the pockets in these two planes , so the fermi surface looks more column - like in the @xmath2 or @xmath4 direction , consistent with previous calculations.@xcite .radius @xmath41 of the hole pockets in in the @xmath42 and @xmath43 planes at different doping levels @xmath17 . the full supercell calculations with f dopants agree with the vca results for the cases where we have compared them ( @xmath17=0 , 0.125 , and 0.25 ) . the third hole pocket on the second plane disappears beyond @xmath44 . [ cols="^,^,^,^,^,^",options="header " , ] [ tabl1 ] c|ccc|ccc x&&@xmath45 plane&&&@xmath46 plane & + & a ( )&b ( )&@xmath47&a ( )&b ( )&@xmath47 + 0.00&0.102&0.128&0.601&0.102&0.152&0.744 + 0.125&0.104&0.127&0.571&0.104&0.147&0.707 + 0.25&0.122&0.148&0.571&0.122&0.172&0.707 + [ tabl2 ] the size of the hole and electron pockets were calculated using the vca and the supercell calculation with f substitution on o sites and have been shown in tables i and ii . since the fermi surfaces obtained from the vca and the full supercell calculations are substantially the same , only one number is given for each concentration . the size of the electron and hole pockets were calculated along the @xmath35 direction . the hole pockets as shown in fig . [ figfermi1 ] are circular ( or nearly so ) lying along @xmath2 and consist of hybridized fe @xmath37 and @xmath38 states . the @xmath37 and @xmath38 orbitals are degenerate due to the point group symmetry of fe . this is consistent with previous calculations@xcite . there exists a third band which forms the cap around the @xmath3 point , mostly of @xmath36 character . this third fermi surface sheet disappears below @xmath24 with 7 - 8% electron doping . the electron pockets are elliptical with significant nesting characteristics . several proposed superconducting theories require understanding of the eccentricity of electron pockets which affects the fermi surface nesting and in addition may be important for magnetic instabilities@xcite . we list in table ii the calculated eccentricity as a function of electron doping using the standard definition @xmath47 = @xmath48 , where @xmath13 and @xmath49 are , respectively , the major and the minor axes . the two electron fermi surface sheets surround the @xmath4 points in the standard unit cell ( fig . [ figfermi1]a ) . the electron cylinders arise out of two bands , one of which is primarily of fe @xmath50 character and the other , of mixed fe @xmath37 and @xmath38 character . these two bands can be identified as those lying along m@xmath32 in fig . [ figbands2]a , b crossing about 0.25 ev above @xmath24 . the eccentricity of the ellipse arises due to different dispersion along different directions in the @xmath19 plane.@xcite with electron doping , the separation between these two bands decreases along @xmath35 , reducing the eccentricity . however , unlike what was seen in a rigid shift of the bands@xcite , the eccentricity never disappears or begins to increase with electron doping . , width=317 ] conventional theories of the superconductivity describe the superconducting state to arise from the fermi surface instability of the paramagnetic normal state , while density functional calculations show the ground state of the undoped material to be an antiferromagnetic metal . therefore the question arises as to whether the electron doping destabilizes the afm state in favor of a paramagnetic state thereby facilitating the formation of the superconducting state . to address this question , we have performed calculations of the total energy with and without electron doping in the supercell geometry and have shown these results in fig . [ figene ] along with the vca results . the results of the full supercell calculation and the vca energies agree quite well , which is consistent with the excellent agreement between their two band structures ( fig . [ figbands1 ] ) . we find that even though the afm state is stable for all dopant concentrations , the energy of the nm state is significantly reduced as compared to that of the afm state . these results suggest that the the electron doping might serve to destabilize the afm state in favor of the nonmagnetic state thereby facilitating superconductivity . in summary , from density - functional supercell calculations we have studied the changes in the fermi surface of lafeaso as a function of electron doping . important differences in the fermi surface were found from results obtained with the simple rigid - band shift , while the virtual crystal approximation yielded reasonable results . finally , our total energy results suggest that electron doping might provide an extra degree of stability to the superconducting state by making the afm normal state less favorable . p. blaha , k. schwarz , g.k.h . madsen , d. kvasnicka , and j. luitz in _ wien2k , an augmented plane wave plus local orbitals program for calculating crystal properties _ , edited by k. schwarz ( technische universitt wien , austria , 2001 ) .

we study the changes in the fermi surface with electron doping in the lafeaso@xmath0f@xmath1 superconductors with density - functional supercell calculations using the linearized augmented planewave ( lapw ) method . the supercell calculations with explicit f substitution are compared with those obtained from the virtual crystal approximation ( vca ) and from a simple rigid band shift . we find significant differences between the supercell results and those obtained from the rigid - band shift with electron doping , although quite remarkably the supercell results are in good agreement with the virtual crystal approximation ( vca ) where the nuclear charges of the o atoms are slightly increased to mimic the addition of the extra electrons . with electron doping , the two cylindrical hole pockets along @xmath2 shrink in size , and the third hole pocket around @xmath3 disappears for an electron doping concentration in excess of about 7 - 8% , while the two elliptical electron cylinders along @xmath4 expand in size . the spin - orbit coupling does not affect the fermi surface much except to somewhat reduce the size of the third hole pocket in the undoped case . we find that with the addition of the electrons the antiferromagnetic state becomes energetically less stable as compared to the nonmagnetic state , indicating that the electron doping may provide an extra degree of stability to the formation of the superconducting ground state .

a comprehensive understanding of hamiltonian dynamics is a long outstanding problem in nonlinear and statistical physics , which has important applications in various other areas of physics . typical hamiltonian systems are nonhyperbolic as they exhibit mixed phase space with coexisting regular and chaotic regions . over the past years , a number of ground - breaking works @xcite have increasingly elucidated the asymptotic behavior of such systems and it is now well understood that , because of the stickiness due to kolmogorov - arnold - moser ( kam ) tori , the chaotic dynamics of typical hamiltonian systems is fundamentally different from that of hyperbolic , fully chaotic systems . here `` asymptotic '' means in the limit of large time scales and small length scales . but in realistic situations , the time and length scales are limited . in the case of hyperbolic systems , this is not a constraint because the ( statistical ) self - similarity of the underlying invariant sets guarantees the fast convergence of the dynamical invariants ( entropies , lyapunov exponents , fractal dimensions , escape rates , etc ) and the asymptotic dynamics turns out to be a very good approximation of the dynamics at finite scales . in nonhyperbolic systems , however , the self - similarity is usually lost because the invariant sets are not statistically invariant under magnifications . as a result , the finite - scale behavior of a hamiltonian system may be fundamentally different from the asymptotic behavior considered previously , which is in turn hard to come by either numerically @xcite or experimentally @xcite . the aim of this paper is to study the dynamics of hamiltonian systems at finite , physically relevant scales . to the best of our knowledge , this problem has not been considered before . herewith we focus on hamiltonian chaotic scattering , which is one of the most prevalent manifestations of chaos in open systems , with examples ranging from fluid dynamics @xcite to solid - state physics @xcite to general relativity @xcite . we show that the finite - scale dynamics of a hamiltonian system is characterized by _ effective _ dynamical invariants ( e.g. , effective fractal dimension ) , which : ( 1 ) may be significantly different from the corresponding invariants of the asymptotic dynamics ; ( 2 ) depend on the resolution but can be regarded as constants over many decades in a given region of the phase space ; and ( 3 ) may change drastically from one region to another of the _ same _ dynamically connected ( ergodic ) component . these features are associated with the slow and nonuniform convergence of the invariant measure due to the breakdown of self - similarity in nonhyperbolic systems . to illustrate the mechanism behind the properties of the effective invariants , we introduce a simple deterministic model which we build on the observation that a hamiltonian system can be represented as a chain of hyperbolic systems . the paper is organized as follows . we start , in sec . [ s2 ] , with the analysis of the invariant measure and the outline of the transport structures underlying its convergence . our chain model is introduced and analyzed in sec . [ 3 ] . the effective fractal dimension is defined in sec . [ 4 ] and its properties are verified for a specific system in sec . [ 5 ] . conclusions are presented in the last section . for concreteness , consider a two - dimensional area preserving map with a major kam island surrounded by a chaotic region . one such map captures all the main properties of a wide class of hamiltonian systems with mixed phase space . when the system is open ( scattering ) , almost all particles initialized in the chaotic region eventually escape to infinity . we first study this case with a diffusive model for the transversal motion close to the main kam island , obtaining an analytical expression for the probability density @xmath0 of particles remaining in the scattering region at time @xmath1 and distance @xmath2 from the island [ see appendix ] . we find that , in the case of chaotic scattering , a singularity develops and the invariant measure , given by @xmath3 , accumulates on the outermost kam torus of the kam island [ appendix ] . physically , this corresponds to the tendency of nonescaping particles to concentrate around the regular regions . dynamically , the stickiness due to kam tori underlies two major features of hamiltonian chaotic scattering , namely the algebraic decay of the survival probability of particles in the scattering region @xcite and the integer dimension of the chaotic saddle @xcite , and distinguishes this phenomenon from the hyperbolic chaotic scattering characterized by exponential decay and noninteger fractal dimension . however , the convergence of the measure is rather slow and highly nonuniform , as shown in fig . [ fig1 ] for typical parameters , which is in sharp contrast with the fast , uniform convergence observed in hyperbolic systems . our main results are ultimately related to this slow and nonuniform convergence of the invariant measure . previous works on transport in hamiltonian systems have used stochastic models , where invariant structures around kam islands are smoothened out and the dynamics is given entirely in terms of a diffusion equation @xcite or a set of transition probabilities ( markov chains or trees ) @xcite . the stochastic approach is suitable to describe transport properties ( as above ) , but can not be used to predict the behavior of dynamical invariants such as lyapunov exponents and fractal dimensions . here we adopt a deterministic approach where we use the cantori surrounding the kam islands to split the nonhyperbolic dynamics of the hamiltonian system into a chain of hyperbolic dynamical systems . cantori are invariant structures that determine the transversal transport close to the kam islands @xcite . there is a hierarchy of infinitely many cantori around each island . let @xmath4 denote the area of the scattering region outside the outermost cantorus , @xmath5 denote the annular area in between the first and second cantorus , and so on . as @xmath6 is increased , @xmath7 becomes thinner and approaches the corresponding island . for simplicity , we consider that there is a single island @xcite and that , in each iteration , a particle in @xmath7 may either move to the outer level @xmath8 or the inner level @xmath9 or stay in the same level @xcite . let @xmath10 and @xmath11 denote the transition probabilities from level @xmath6 to @xmath12 and @xmath13 , respectively . a particle in @xmath4 may also leave the scattering region , and in this case we consider that the particle has escaped . the escaping region is denoted by @xmath14 . the chaotic saddle is expected to have points in @xmath7 for all @xmath15 . it is natural to assume that the transition probabilities @xmath10 and @xmath11 are constant in time . this means that each individual level can be regarded as a hyperbolic scattering system , with its characteristic exponential decay and noninteger chaotic saddle dimension . therefore , a nonhyperbolic scattering is in many respects similar to a sequence of hyperbolic scatterings . we now introduce a simple deterministic model that incorporates the above elements and reproduces essential features of the hamiltonian dynamics . our model is depicted in fig . [ fig2 ] and consists of a semi - infinite chain of 1-dimensional `` @xmath16-shaped '' maps , defined as follows : @xmath17 where @xmath18 and @xmath19 ( @xmath20 ) . if @xmath2 falls in the interval @xmath21 , where @xmath22 is not defined , the `` particle '' is considered to have crossed a cantorus to the `` outer level '' @xmath12 . this interval is mapped uniformly to @xmath23 $ ] , and the iteration proceeds through @xmath24 . symbolically , this is indicated by @xmath25 . similarly , if @xmath2 falls into @xmath26 , where @xmath27 , the particle goes to the `` inner level '' , and @xmath28 . particles that reach @xmath29 are considered to have escaped . the domain of @xmath22 is denoted by @xmath30 and is analogous to @xmath7 in a hamiltonian system , where @xmath10 and @xmath11 represent the transition probabilities . the transition rate ratios @xmath31 and @xmath32 are taken in the interval @xmath33 and are set to be independent of @xmath6 , where @xmath34 . the parameter @xmath35 is a measure of the fraction of particles in a level @xmath6 that will move to the inner level @xmath13 when leaving level @xmath6 , while @xmath36 is a measure of how much longer it takes for the particles in the inner level to escape . the nondependence on @xmath6 corresponds to the approximate scaling of the cantori suggested by the renormalization theory @xcite . despite the hyperbolicity of each map , the entire chain behaves as a nonhyperbolic system . for a uniform initial distribution in @xmath37 , it is not difficult to show @xcite that the number of particles remaining in the chain after a long time @xmath1 decays algebraically as @xmath38 , and that the initial conditions of never escaping particles form a zero lebesgue measure fractal set with box - counting dimension 1 . however , the finite - scale behavior may deviate considerably from these asymptotics , as shown in fig . [ fig3 ] . in fig . [ fig3](a ) we show the survival probability @xmath39 as a function of time . for small @xmath35 and @xmath36 , the curve is composed of a discrete sequence of exponentials with scaling exponents @xmath40 , which decrease ( in absolute value ) as we go forward in the sequence . the length of each exponential segment is of the order of @xmath35 in the decay of @xmath39 and @xmath41 in the variation of @xmath42 . this striking behavior is related to the time evolution of the density of particles inside the chain . this is shown in fig . [ fig3](b ) , where we plot the average position @xmath43 of an ensemble of particles initialized in @xmath37 ( i.e. , @xmath44 ) . the transitions between successive exponentials in the decay of @xmath39 [ fig . [ fig3](a ) ] match the transitions from a level @xmath6 to the next in the average position of the remaining particles [ fig . [ fig3](b ) ] . in a hamiltonian system , the increase of @xmath43 in time is related to the development of the singular invariant measure anticipated in our diffusion analysis [ see fig . [ fig1 ] ] . the piecewise exponential behavior of @xmath39 is smoothened out for large @xmath35 and @xmath45 [ figs . [ fig3](a ) and [ fig3](b ) ] . in fig . [ fig3](c ) we show the fractal dimension of the set of initial conditions of never escaping particles as computed from the uncertainty algorithm @xcite , which consists in measuring the scaling of the fraction @xmath46 of _ @xmath47-uncertain _ points ( initial points whose escaping time is different from the escaping time of points taken @xmath47 apart ) . the scaling is statistically well defined over decades and the exponent @xmath48 can be computed accurately . however , the resulting dimension @xmath49 is not only significantly smaller than 1 but also depends critically on the region @xmath50 of the phase space where it is computed . the convergence of the dimension is indeed so slow that it can only be noticed when observed over very many decades of resolution , as shown in fig . [ fig3](d ) where data of fig . [ fig3](c ) is plotted over 35 decades ! initially smaller , the dimension measured for @xmath51 approaches the dimension measured for @xmath52 as the scale @xmath47 is reduced beyond @xmath53 ( i.e. the corresponding curves in fig . [ fig3](d ) become parallel ) . as shown in fig . [ fig3](d ) , this behavior is related to a transition in the average innermost level @xmath54 reached by the particles launched from @xmath47-uncertain points . as @xmath47 is further reduced , new transitions are expected . the dimension measured in between transitions is mainly determined by the dimension @xmath55 , @xmath56 , of the corresponding element of the chain . for given @xmath6 and @xmath47 , the measured dimension is larger when @xmath50 is taken in a denser part of the invariant set , such as in the subinterval of @xmath37 first mapped into @xmath57 [ fig . [ fig3](c ) ; diamonds ] , because @xmath58 is larger in these regions . in some regions , however , the measured dimension is quite different from the asymptotic value even at scales as small as @xmath59 . this slow convergence of the dimension is due to the slow increase of @xmath58 , which in a hamiltonian system is related to the slow convergence of the invariant measure [ fig . [ fig1 ] ] . the convergence is even slower for smaller @xmath35 and larger @xmath36 . incidentally , the experimental measurements of the fractal dimension are usually based on scalings over less than two decades @xcite . therefore , at realistic scales the dynamics is clearly not governed by the asymptotic dynamical invariants . our results on the chain model motivate us to introduce the concept of effective dynamical invariants . as a specific example , we consider the _ effective _ fractal dimension , which , for the intersection of a fractal set @xmath60 with a @xmath61-dimensional region @xmath50 , we define as @xmath62 where @xmath63 , and @xmath64 and @xmath65 are the number of cubes of edge length @xmath66 needed to cover @xmath67 and @xmath50 , respectively @xcite . we take @xmath50 to be a generic segment of line [ i.e. , @xmath68 in eq . ( [ 2 ] ) ] intersected by @xmath60 on a fractal set . in the limit @xmath69 , we recover the usual box - counting dimension @xmath70 of the fractal set @xmath67 , which is known to be 1 for all our choices of @xmath50 . however , for any practical purpose , the parameter @xmath47 is limited and can not be made arbitrarily small ( e.g. , it can not be smaller than the size of the particles , the resolution of the experiment , and the length scales neglected in modeling the system ) . at scale @xmath47 the system behaves as if the fractal dimension were @xmath71 ( therefore `` effective '' dimension ) . in particular , the final state sensitivity of particles launched from @xmath50 , with the initial conditions known within accuracy @xmath72 , is determined by @xmath73 rather than @xmath74 : as @xmath47 is variated around @xmath72 , the fraction of particles whose final state is uncertain scales as @xmath75 , which is different from the prediction @xmath76 . this is important in this context because , as shown in fig . [ fig3 ] ( where the effective dimension is given by @xmath49 ) , the value of @xmath71 may be significantly different from the asymptotic value @xmath77 even for unrealistically small @xmath47 and may also depend on the region of the phase space . similar considerations apply to many other invariants as well . we now return to the hamiltonian case . consider a scattering process in which particles are launched from a line @xmath50 transversal to the stable manifold @xmath78 of the chaotic saddle . based on the construction suggested by the chain model , it is not difficult to see that @xmath79 exhibits a hierarchical structure which is not self - similar and is composed of infinitely many nested cantor sets , each of which is associated with the dynamics inside one of the regions @xmath7 . as a consequence , the effective dimension @xmath71 in hamiltonian systems is expected to behave similarly to the effective dimension in the chain model [ figs . [ fig3](c ) and [ fig3](d ) ] . in particular , @xmath71 is expected to display a strong dependence on @xmath50 and a weak dependence on @xmath47 . we test our predictions on the area preserving hnon map : @xmath80 , where @xmath81 is the bifurcation parameter . in this system , typical points outside kam islands are eventually mapped to infinity . because of the symmetry @xmath82 , where @xmath83 , the stable and unstable manifolds of the chaotic saddle are obtained from each other by exchanging @xmath2 and @xmath84 . for @xmath85 , the system displays a period - one and a period - four major island , as shown in fig . [ fig4](a ) . in the same figure we also show the complex invariant structure around the islands , the stable manifold of the chaotic saddle , and three different choices for the line of starting points : a large interval away from the islands ( @xmath86 ) , a small subinterval of this interval where the stable manifold appears to be denser ( @xmath87 ) , and an interval closer to the islands ( @xmath88 ) . the corresponding effective dimensions are computed for a wide interval of @xmath47 . the results are shown in fig . [ fig4](b ) : @xmath89 , @xmath90 , and @xmath91 for @xmath92 . these results agree with our predictions that the effective fractal dimension has the following properties : @xmath93 may be significantly different from the asymptotic value @xmath94 of the fractal dimension ; @xmath93 depends on the resolution @xmath47 but is nearly constant over decades ; @xmath93 depends on the region of the phase space under consideration and , in particular , is larger in regions closer to the islands and in regions where the stable manifold is denser . similar results are expected for any typical hamiltonian system with mixed phase space . we have shown that the finite - scale dynamics of hamiltonian systems , relevant for realistic situations , is governed by effective dynamical invariants . the effective invariants are not only different from the asymptotic invariants but also from the usual hyperbolic invariants because they strongly depend on the region of the phase space . our results are generic and expected to meet many practical applications . in particular , our results are expected to be relevant for fluid flows , where the advection dynamics of tracer particles is often hamiltonian @xcite . in this context , a slow nonuniform convergence of effective invariants is expected not only for time - periodic flows , capable of holding kam tori , but also for a wide class of time - irregular incompressible flows with nonslip obstacles or aperiodically moving vortices . the diffusion model is : @xmath95 $ ] , where @xmath96 is the probability density of all particles , @xmath97 , and @xmath98 @xcite . the outermost torus of the kam island is at @xmath99 , where the diffusion rate ( proportional to @xmath100 ) vanishes . in a chaotic scattering process the initial distribution of particles is localized apart from the confining islands . we take @xmath101 , @xmath102 , and consider a particle to escape when it reaches @xmath103 . under the approximation that for large @xmath104 the return of particles can be neglected , we disregard the boundary condition @xmath105 and we take the solution to be the corresponding green function : @xmath106 , where @xmath107 , @xmath108 is @xmath84 at @xmath109 , @xmath110 , and @xmath111 is the modified bessel function , which scales as @xmath112 for small @xmath113 @xcite . for any fixed @xmath114 , we can show that the distribution for large @xmath1 decreases as @xmath115 , where @xmath110 . on the other hand , as shown in ref . @xcite , the fraction of particles in the interval @xmath116 decays algebraically as @xmath117d@xmath118 . combining these two results , it follows that the normalized probability density @xmath119 decreases as @xmath120 at each fixed @xmath121 for large enough @xmath1 and diverges arbitrarily close to @xmath99 . chain models have also been used to describe power laws generated by a hierarchy of repellers in the motion towards the feigenbaum attractor [ p. grassberger and m. scheunert , j. stat . * 26 * , 697 ( 1981 ) ] and to study deterministic diffusion [ r. klages and j. r. dorfman , phys lett . * 74 * , 387 ( 1995 ) ] . the escape rate follows from the same argument used in [ j. d. meiss and e. ott , physica d * 20 * , 387 ( 1986 ) ] . the fractal dimension follows from the observation that the dimension of the invariant set of map @xmath22 goes to @xmath94 as @xmath122 . lai , m. ding , c. grebogi , and r. blmel , phys . a * 46 * , 4661 ( 1992 ) ; w. breymann , z. kovcs , and t. tl , phys . e * 50 * , 1994 ( 1994 ) ; p. gaspard and j. r. dorfman , phys . e * 52 * , 3525 ( 1995 ) ; v. constantoudis and c. a. nicolaides , phys . e * 64 * 056211 ( 2001 ) .

an adequate characterization of the dynamics of hamiltonian systems at physically relevant scales has been largely lacking . here we investigate this fundamental problem and we show that the finite - scale hamiltonian dynamics is governed by effective dynamical invariants , which are significantly different from the dynamical invariants that describe the asymptotic hamiltonian dynamics . the effective invariants depend both on the scale of resolution and the region of the phase space under consideration , and they are naturally interpreted within a framework in which the nonhyperbolic dynamics of the hamiltonian system is modeled as a chain of hyperbolic systems . [ phys . rev . e * 71 * , 036215 ( 2005 ) ]

grb light curves measured with swift consist of a bat light curve in the 15 150 kev range followed , after slewing within @xmath2 s , by a detailed 0.3 10 kev xrt x - ray light curve @xcite . this information supplements our knowledge of the highly variable hard x - ray and @xmath0-ray light curves measured from many grbs with batse and other grb detectors . about one - half of swift grbs show x - ray flares or short timescale structure , sometimes hours or later after the onset of the grb . approximately @xmath3% of the swift grbs display rapid x - ray declines , and an additional @xmath4% display features unlike simple blast wave model predictions @xcite . we make three points in this paper : 1 . highly variable light curves can be produced by an external shock under the assumption that the grb blast wave does not spread , or spreads much more slowly than assumed from gas - dynamic or relativistic hydrodynamic models that do not take into account magnetic effects in grb blast waves . if this assumption is valid , then it is wrong to conclude that highly variable @xmath0-ray emissions , x - ray flares with @xmath5 , or late time x - ray flares require delayed central engine activity or colliding shells . 2 . external shocks in grb blast waves can accelerate cosmic ray protons and ions to @xmath1 ev , making grbs a logical candidate to accelerate the highest energy cosmic rays . 3 . escape of ultra - high energy cosmic rays ( uhecrs ) takes place from an external shock formed by an expanding grb blast wave on time scales of a few hundred seconds for the observer . blast - wave deceleration due to the loss of the internal hadronic energy is proposed @xcite to be the cause of x - ray declines in grb light curves observed with swift . we have performed a detailed analysis of the interaction between a grb blast - wave shell and an external stationary cloud @xcite . the analysis is performed under the assumption that the cloud width @xmath6 , where @xmath7 is the distance of the cloud from the grb explosion . the interaction is divided into three phases : ( 1 ) a collision phase with both a forward and reverse shock ; ( 2 ) a penetration phase where either the reverse shock has crossed the shell while the forward shock continues to cross the cloud , or vice versa ; and ( 3 ) an expansion phase , where both shocks have crossed the cloud and shell , and the shocked fluid expands . the shell width is written as @xmath8 and the proper number density of the relativistic shell is given by @xmath9 where @xmath10 is the coasting lorentz factor of the grb blast wave , and @xmath11 is the apparent isotropic energy release . short timescale flaring requires ( a ) a strong forward shock , which from the relativistic shock jump conditions @xcite imply a maximum cloud density given by @xmath12 and ( b ) significant blast - wave deceleration to provide efficient energy extraction , which occurs in clouds with thick columns @xcite , that is , with densities @xmath13 these two conditions translate into the requirement that @xmath14 in order to produce short timescale variability . the short timescale variabilty condition @xcite for quasi - spherical clouds is @xmath15 using eq . ( [ delta(x ) ] ) for the shell width , eqs . ( [ deltacl ] ) and ( [ deltacl ] ) imply the requirement that @xmath16 in order to produce rapid variability from an external shock . hence the production of @xmath0-ray pulses and x - ray flares from external shocks depends on whether the grb blast - wave width spreads in the coasting phase according to eq . ( [ delta(x ) ] ) , with @xmath17 , as is generally argued . in the gas - dynamical study of @xcite , inhomogeneities in the grb fireball produce a spread in particle velocities of order @xmath18 , so that @xmath19 when @xmath20 . this dependence is also obtained in a hydrodynamical analysis @xcite . two points can be made about these relations . first , the spread in @xmath21 considered for a spherical fireball is averaged over all directions . as the fireball expands and becomes transparent , the variation in fluid motions or gas particle directions over a small solid angle @xmath22 of the full sky becomes substantially less . second , the particles within a magnetized blast - wave shell will expand and adiabatically cool so that the fluid will spread with thermal speed @xmath23 . the comoving width of the blast wave is @xmath24 , so that the spreading radius @xmath25 . adiabatic expansion of nonrelativistic particles can produce a very cold shell with @xmath26 , leading to very small shell widths . the requirement on the thinness of @xmath27 does not apply to the adiabatic self - similar phase , where the width is necessarily @xmath28 , as implied by the relativistic shock hydrodynamic equations @xcite . even in this case , however , @xmath29 if the blast wave is highly radiative @xcite . under the assumption of a strong forward shock and small clouds in the vicinity of a grb , highly variable grb light curves are formed with reasonable efficiency ( @xmath30% ) to transform blast wave energy into @xmath0 rays @xcite . the maximum particle energy for a cosmic ray proton accelerated by an external shock in a grb blast wave is derived . consider a grb blast wave with apparent isotropic energy release @xmath31 ergs , ( initial ) coasting lorentz factor @xmath32 , and external medium density @xmath33 @xmath34 . the comoving blast wave volume for the assumed spherically symmertric explosion , after reaching distance @xmath7 from the center of the explosion , is @xmath35 where the shell width @xmath36 ( the factor @xmath37 is the product of the geometrical factor @xmath38 and the factor @xmath39 from the continuity equations of relativistic hydrodynamics ; @xmath40 is the evolving grb blast wave lorentz factor ) . the hillas condition @xcite for maximum particle energy @xmath41 is that the particle larmor radius is less than the size scale of the system ; @xmath42 in the stationary frame ( primes refer to the comoving frame ) is given by @xmath43 the largest particle energy is reached at the deceleration radius @xmath44 when @xmath45 , where the deceleration radius @xmath46 hence @xmath47 the mean magnetic field @xmath48 in the grb blast wave is assigned in terms of a magnetic field parameter @xmath49 that gives the magnetic field energy density in terms of the energy density of the downstream shocked fluid , so @xmath50 thus @xmath51 @xcite , so that external shocks of grbs can accelerate particles to ultra - high and , indeed , super - gzk energies . implicit in this result is that acceleration occurs within the grb blast wave through , for example , second - order fermi acceleration @xcite . acceleration to ultra - high energy through first - order relativistic shock acceleration requires a highly magnetized surrounding medium @xcite . if uhecrs are accelerated by grb blast waves , then blast - wave dynamics will be affected by the loss of internal energy when the uhecrs escape . this effect is proposed to explain the rapid x - ray declines in the swift grb light curves @xcite . photohadronic processes become important when the threshold condition @xmath52 , where @xmath53 is the dimensionless photon energy , @xmath54 is the proton energy , and @xmath55 is the proton lorentz factor . for protons interacting with photons at the peak photon energy @xmath56 of the @xmath57 spectrum , @xmath58 the comoving timescale for a proton to lose a significant fraction of its energy through photohadronic processes is given by @xmath59 , where @xmath60 c$ ] , @xmath61 @xmath62b is the product of the photohadronic cross section and inelasticity , and the comoving energy density of photons with energy @xmath63 is @xmath64 . the relation between the measured @xmath65 flux @xmath66 and internal energy density is @xmath67 , where @xmath68 cm is the luminosity distance of the grb . for protons interacting with photons with energy @xmath69 , we therefore find that the comoving time required for a proton with energy @xmath70 ( as measured by an observer outside the blast wave ) to lose a significant fraction of its energy through photohadronic processes is @xmath71 where @xmath72 cm and @xmath73 ergs @xmath74 s@xmath75 is the @xmath57 flux measured at @xmath76 ; the relation between @xmath70 and @xmath76 is given by eq . ( [ epk ] ) . the dependence of the terms @xmath77 , @xmath78 , @xmath79 , and @xmath80 on observer time in eq . ( [ tprimephipi ] ) can be analytically expressed for the external shock model in terms of the grb blast wave properties @xmath11 , @xmath10 , environmental parameters , e.g. , @xmath81 , and microphysical blast wave parameters @xmath49 and @xmath82 @xcite . this can also be done for other important timescales , for example , the ( available ) comoving time @xmath83 since the start of the grb explosion , the comoving acceleration time @xmath84 , written as a factor @xmath85 times the larmor timescale @xcite , the escape timescale @xmath86 in the bohm diffusion approximation , and the proton synchrotron energy loss timescale @xmath87 . 1 shows the rates ( or the inverse of the timescales ) for @xmath88 ev protons in the case of an adiabatic blast wave that decelerates in a uniform surrounding medium . the left - hand panel of fig . 1 uses the parameter set @xmath89 and the right - hand panel uses the parameter set @xmath90 the characteristic deceleration timescale in the left and right cases , given by @xmath91 s , is @xmath92 s and @xmath93 s , respectively . for these parameters , it takes a few hundred seconds to accelerate protons to energies @xmath94 ev , at which time photohadronic losses and escape start to be important . photohadronic losses inject electrons and photons into the grb blast wave . the electromagnetic cascade emission , in addition to hyperrelativistic electron synchrotron radiation from neutron escape followed by subsequent photohadronic interactions @xcite , makes a delayed anomalous @xmath0-ray emission component as observed in some grbs @xcite . ultra - high energy neutrino secondaries are produced by the photohadronic processes . detection of high - energy neutrinos from grbs would confirm the importance of hadronic processes in grb blast waves . the ultra - high energy neutrons and escaping protons form the uhecrs with energies @xmath1 ev . the grb blast wave rapidly loses internal energy due to the photohadronic processes and particle escape . the blast wave will then rapidly decelerate , producing a rapidly decaying x - ray flux . as argued in more detail elsewhere @xcite , the rapidly decaying fluxes in swift grbs are signatures of uhecr acceleration by grbs . if this scenario is correct , glast will detect anomalous @xmath0-ray components , particularly in those grbs that undergo rapid x - ray declines in their x - ray light curves . this work is supported by the office of naval research , by nasa _ glast _ science investigation no . dpr - s-1563-y , and nasa swift guest investigator grant no . dpr - nng05ed41i . thanks also to guido chincarini for the kind invitation .

highly variable @xmath0-ray pulses and x - ray flares in grb light curves can result from external shocks rather than central engine activity under the assumption that the grb blast - wave shell does not spread . acceleration of cosmic rays to @xmath1 ev energies can take place in the external shocks of grbs . escape of hadronic energy in the form of uhecrs leads to a rapidly decelerating grb blast wave , which may account for the rapid x - ray declines observed in swift grbs . [ 1999/12/01 v1.4c il nuovo cimento ]

the mathematical theory of compressive sensing ( cs ) @xcite asserts that one can acquire signals from measurements whose rate is much lower than the total bandwidth . whereas the cs theory is now well developed , challenges concerning hardware implementations @xcite of cs - based acquisition devices , especially in optics , have only started being addressed . this paper will introduce a color video cs camera capable of capturing low - frame - rate measurements at acquisition , with high - frame - rate video recovered subsequently via computation ( decompression of the measured data ) . the coded aperture compressive temporal imaging ( cacti ) @xcite system uses a moving binary mask pattern to modulate a video sequence within the integration time @xmath0 many times prior to integration by the detector . the number of high - speed frames recovered from a coded - exposure measurement depends on the speed of video modulation . within the cacti framework , modulating the video @xmath1 times per second corresponds to moving the mask @xmath1 pixels within the integration time @xmath0 . if @xmath2 frames are to be recovered per compressive measurement by a camera collecting data at @xmath3 frames - per - second ( fps ) , the time variation of the code is required to be @xmath4 fps . the liquid - crystal - on - silicon ( lcos ) modulator used in @xcite can modulate as fast as @xmath5 fps by pre - storing the exposure codes , but , because the coding pattern is continuously changed at each pixel throughout the exposure , it requires considerable energy consumption ( @xmath6 ) . the mechanical modulator in @xcite , by contrast , modulates the exposure through periodic mechanical translation of a single mask ( coded aperture ) , using a pizeoelectronic translator that consumes minimal energy ( @xmath7 ) . the coded aperture compressive temporal imaging ( cacti ) @xcite now has been extended to the color video @xcite , which can capture r " , g " and b " channels of the context . by appropriate reconstruction algorithms @xcite , we can get @xmath1 frames color video from a single gray - scale measurement . while numerous algorithms have been used for cs inversion , the bayesian cs algorithm @xcite has been shown with significant advantages of providing a full posterior distribution . this paper develops a new bayesian inversion algorithm to reconstruct videos based on raw measurements acquired by the color - cacti camera . by exploiting the hybrid three dimensional ( 3d ) tree - structure of the wavelet and dct ( discrete cosine transform ) coefficients , we have developed a hidden markov tree ( hmt ) @xcite model in the context of a bayesian framework . research in @xcite has shown that by employing the hmt structure of an image , the cs measurements can be reduced . this paper extends this hmt to 3d and a sophisticated 3d tree - structure is developed for video cs , with color - cacti shown as an example . experimental results with both simulated and real datasets verify the performance of the proposed algorithm . the basic model and inversion method may be applied to any of the compressive video cameras discussed above . let @xmath8 be the continuous / analog spatiotemporal volume of the video being measured ; @xmath9 represents a moving mask ( code ) with @xmath10 denoting its spatial translation at time @xmath11 ; and @xmath12 denotes the camera spatial sampling function , with spatial resolution @xmath13 . the coded aperture compressive camera system modulates each temporal segment of duration @xmath14 with the moving mask ( the motion is periodic with the period equal to @xmath14 ) , and collapses ( sums ) the coded video into a single photograph ( @xmath15 @xmath16 ) : @xmath17 @xmath18 and @xmath19 , with the detector size @xmath20 pixels . the set of data @xmath21 , which below we represent as @xmath22 , corresponds to the @xmath23th compressive measurement . the code / mask @xmath24 is here binary , corresponding to photon transmission and blocking ( see figure [ fig : dec ] ) . denote @xmath25 , defining the original continuous video @xmath8 sampled in space @xmath26 and in time ( @xmath1 discrete temporal frames , @xmath27 , within the time window of the @xmath23th compressive measurement ) . we also define @xmath28 we can rewrite ( [ eq : cacti - measurement ] ) as @xmath29 where @xmath30 is an added noise term , @xmath31 , and @xmath32 denotes element - wise multiplication ( hadamard product ) . in ( [ eq : cacti - measurement - discrete ] ) , @xmath33 denotes the mask / code at the @xmath34th shift position ( approximately discretized in time ) , and @xmath35 is the underlying video , for video frame @xmath34 within cs measurement @xmath23 . dropping subscript @xmath23 for simplicity , ( [ eq : cacti - measurement - discrete ] ) can be written as @xmath36\\ \mathbf{x}&=&\mathrm{vec}([\mathbf{z}_{1},\cdots,\mathbf{z}_{n_{t } } ] ) , \vspace{-3mm}\end{aligned}\ ] ] where @xmath37 and @xmath38 is standard vectorization . we record temporally compressed measurements for rgb colors on a bayer - filter mosaic , where the three colors are arranged in the pattern shown in the right bottom of figure [ fig : dec ] . the single coded image is partitioned into four components , one for r and b and two for g ( each is @xmath39 the size of the original spatial image ) . the cs recovery ( video from a single measurement ) is performed separately on these four mosaiced components , prior to demosaicing as shown in figure [ fig : dec](b ) . one may also jointly perform cs inversion on all 4 components , with the hope of sharing information on the importance of ( here wavelet and dct ) components ; this was also done , and the results were very similar to processing r , b , g1 and g2 separately . note that this is the key difference between color - cacti and the previous work of cacti in @xcite . an image s zero - tree structure @xcite has been investigated thoroughly since the advent of wavelets @xcite . the 3d wavelet tree structure of video , an extension of the 2d image , has also attracted extensive attention in the literature @xcite . introduced a tree - based representation to characterize the block - dct transform associated with jpeg @xcite . for the video representation , we here use the wavelet in space and dct in time . considering the video sequence has @xmath1 frames with spatial @xmath40 pixels , and let @xmath41 denote the indices of the dct / wavelet coefficients . assume there are @xmath42 levels ( scales ) of the coefficients ( @xmath43 in figure [ fig:3d_tree ] ) . the parent - children linkage of the coefficients are as follows : a ) a root - node @xmath44 has 7 children , @xmath45 , where @xmath46 denotes the size of scaling ( ll ) coefficients ; b ) an internal node @xmath44 has 8 children @xmath47 ; and c ) a leaf - node has no children . when the tree structure is used in 3d dct , we consider the block size of the 3d dct is @xmath48 , and @xmath49 . the parent - children linkage is the same as with the wavelet coefficients @xcite . the properties of wavelet coefficients that lead to the bayesian model derived in the following section are @xcite : + 1 ) large / small values of wavelet coefficients generally persist across the scales of the wavelet tree ( the two states of the binary part of the model developed in the following section ) . + 2 ) persistence becomes stronger at finer scales ( the confidence of the probability of the binary part is proportional to the number of coefficients at that scale ) . + 3 ) the magnitude of the wavelet coefficients _ decreases exponentially _ as we move to the finer scales . in this paper , we use a multiplicative gamma prior @xcite , a typical shrinkage prior , for the non - zero wavelet coefficients at different scale to embed this decay . let @xmath50 , @xmath51 , @xmath52 be orthonormal matrices defining bases such as wavelets or the dct @xcite . define @xmath53 where @xmath54 symbolizes the 3d wavelet / dct coefficients corresponding to @xmath55 and @xmath56 and @xmath57 denotes the kronecker product . it is worth noting here the @xmath58 is the 3d transform of the projection matrix @xmath59 . unlike the model used in @xcite , where the projection matrix is put directly on the wavelet / dct coefficients , in the coding strategy of color - cacti , we get the projection matrix @xmath60 from the hardware by capturing the response of the mask at different positions . following this , we transform @xmath59 row - by - row to the wavelet / dct domain , to obtain @xmath58 . the measurement noise is modeled as zero mean gaussian with precision matrix ( inverse of the covariance matrix ) @xmath61 , where @xmath62 is the identity matrix . we have : @xmath63 to model the sparsity of the 3d coefficients of wavelet / dct , the _ spike - and - slab _ prior is imposed on @xmath64 as : @xmath65 where @xmath66 is a vector of non - sparse coefficients and @xmath67 is a binary vector ( zero / one indicators ) denoting the two state of the hmt @xcite , with zero " signifying the low - state " in the hmt and one " symbolizing the high - state " . note when the coefficients lie in the low - state " , they are explicitly set to zero , which leads to the sparsity . to model the linkage of the tree structure across the scales of the wavelet / dct , we use the the binary vector , @xmath68 , which is drawn from a bernoulli distribution . the parent - children linkage is manifested by the probability of this vector . we model @xmath69 is drawn from a gaussian distribution with the precision modeled as a multiplicative gamma prior . the full bayesian model is : @xmath70 where @xmath71 denotes the @xmath72th component at level @xmath73 , and @xmath74 denotes the scaling coefficients of wavelet ( or dc level of a dct ) . @xmath75 in the experiments , we use the following settings : @xmath76 where @xmath77 is the number of coefficients at @xmath73th level , and @xmath78 is the length of @xmath54 . we developed the variational bayesian methods to infer the parameters in the model as in @xcite . the posterior inference of @xmath79 , thus @xmath80 is different from the model in @xcite , and we show it below : @xmath81 where @xmath82 denotes the expectation in @xmath83 . both simulated and real datasets are adopted to verify the performance of the proposed model for video reconstruction . the hyperparameters are setting as @xmath84 ; the same used in @xcite . best results are found when @xmath85 and @xmath86 are wavelets ( here the daubechies-8 @xcite ) and @xmath56 corresponds to a dct . the proposed tree - structure bayesian cs inversion algorithm is compared with the following algorithms : @xmath72 ) generalized alternating projection ( gap ) algorithm @xcite ; @xmath87 ) two - step iterative shrinkage / thresholding ( twist ) @xcite ( with total variation norm ) ; @xmath88 ) k - svd @xcite with orthogonal matching pursuit ( omp ) @xcite used for inversion ; @xmath89 ) a gaussian mixture model ( gmm ) based inversion algorithm @xcite ; and @xmath90 ) the linearized bregman algorithm @xcite . the @xmath91-norm of dct or wavelet coefficients is adopted in linearized bregman and gap with the same transformation as the proposed model . gmm and k - svd are patch - based algorithms and we used a separate dataset for training purpose . a batch of training videos were used to pre - train k - svd and gmm , and we selected the best reconstruction results for presentation here . we consider a scene in which a basketball player performs a dunk ; this video is challenging due to the complicated motion of the basketball players and the varying lighting conditions ; see the example video frames in figure [ fig : dec](a ) . we consider a binary mask , with 1/0 coding drawn at random bernoulli(0.5 ) ; the code is shifted spatially via the coding mechanism in figure [ fig : dec](a ) ) , as in our physical camera . the video frames are @xmath92 spatially , and we choose @xmath93 . it can be seen clearly that the proposed tree - structure bayesian cs algorithm demonstrates improved psnr performance for the inversion . we test our algorithm using real datasets captured by our color - cacti camera , with selected results shown in figures [ fig:3balls]-[fig : hammer ] . figure [ fig:3balls ] shows low - framerate ( captured at 30fps ) compressive measurements of fruit falling / rebounding and corresponding high - framerate reconstructed video sequences . in the left are shown four contiguous measurements , and in the right are shown 22 frames reconstructed per measurement . note the spin of the red apple and the rebound of the orange in the reconstructed frames . figure [ fig : hammer ] shows a process of a purple hammer hitting a red apple with 3 contiguous measurements . we can see the clear hitting process from the reconstructed frames . we have implemented a color video cs camera , color - cacti , capable of compressively capturing and reconstructing videos at low - and high - framerates , respectively . a tree - structure bayesian compressive sensing framework is developed for the video cs inversion by exploiting the 3d tree structure of the wavelet / dct coefficients . both simulated and real datasets demonstrate the efficacy of the proposed model . x. yuan , p. llull , x. liao , j. yang , g. sapiro , d. j. brady , and l. carin , `` low - cost compressive sensing for color video and depth , '' in _ ieee conference on computer vision and pattern recognition ( cvpr ) _ , 2014 .

a bayesian compressive sensing framework is developed for video reconstruction based on the color coded aperture compressive temporal imaging ( cacti ) system . by exploiting the three dimension ( 3d ) tree structure of the wavelet and discrete cosine transformation ( dct ) coefficients , a bayesian compressive sensing inversion algorithm is derived to reconstruct ( up to 22 ) color video frames from a _ single _ monochromatic compressive measurement . both simulated and real datasets are adopted to verify the performance of the proposed algorithm . compressive sensing , video , bayesian , tree structure , wavelet

dpvs were discovered in the small magellanic cloud after a search for be stars in the ogle - ii database ( mennickent et al . they were clearly distinguished from other variables by showing 2 linked photometric cycles ( @xmath1 and @xmath2 ) . a spectroscopic monitoring of some of them allowed to associate the short periodicity to the orbital period of a binary ( mennickent et al . 2005 ) whereas the long term variations were found to be reddish and non strictly constant ( mennickent , assman , & sabogal 2006 , michalska et al.2009 ) . the current census of dpvs amounts to 114 in the magellanic clouds and 11 in our galaxy ( see magnitud - color and period - period diagrams in mennickent & koaczkowski 2009a ) . after the discovery of additional variability in v393sco ( pilecki & szczygiel 2007 ) , we recognized it as the first dpv in our galaxy . later we found that in the past an additional long cycle was also reported for the galactic dpv aumon ( lorenzi 1985 ) . both stars were also found during our independent search for galactic dpvs in the asas database . few dpvs have been studied in detail , but we can get insights on dpvs as a class based on well studied and representative cases . cumulative evidence indicates that dpvs are interacting binaries with a component ( the donor ) filling their roche lobe and transferring mass to the gainer ( the primary ) . broad and variable hei lines probably probe an accretion disc that sometimes hides the primary . hi ( sometimes hei ) line emission is the rule ( although usually not quite prominent ) . it is probable that the deeper dpv eclipse corresponds mostly to the occultation of the circumprimary disc . we observed a loop in the color - magnitude diagram of lmc - dpv1 during the long cycle that interpreted in terms of mass loss ( mennickent et al . 2008 , hereafter m08 ) . the same star shows discrete pa@xmath3 and pa@xmath4 absorption components following a saw - teeth pattern with the orbital period indicating outflows through the outer lagrangian points ( m08 ) . the same phenomenon could explain the depressed blue wings observed in the hei 10833 infrared spectra of v393sco near secondary eclipse ( fig.1 ) . we observe ( minor ) variability in the shape of the light curve for v393sco during main minima that could indicate changes in the properties of the circumprimary disc . in addition , the h@xmath5 emission line strength increases during supercycle maximum ( mennickent & koaczkowski 2009b ) . most dpvs with 2mass data seems to show infrared excess . in the studied cases mass ratios ( donor / gainer ) are always less than one . all these singular characteristics , plus the presence of two distinct periodicities , suggest that dpvs can be observed as a new class of interacting binaries , at least from the observational point of view . we propose that dpvs are case - a / b mass transfer binaries after mass ratio reversal in algol - like configurations . they are more massive than ordinary algols ( mennickent & koaczkowski 2009a ) , so it is possible that the mass transfer rate is larger , and the primary is rotating at critical velocity . under these circumstances , accretion is stopped and the disc starts cumulating mass that is periodically ejected from the system . our observations indicate that mass loss occurs permanently in dpvs , mainly through the outer lagrangian points . however , the long term periodicity implies that there is another clock governing mass loss in the long term . we believe that during the supercycle the disc cumulates extra matter that is is expelled from the binary during supermaximum . the remarkable behavior of hei in v393sco ( fig.2 ) suggests that the rotational velocity of the circumprimary disc is larger during supermaximum and modulated with supercycle phase . the mechanism for this supercycle is unknown . we analyzed the possibility that the disc outer radius grows until the 3:1 resonance radius and disc starts to precess , as happens in low mass ratio suuma stars ( mennickent & koaczkowski 2009a ) . in this view precession enhances mass loss into the interstellar medium . however , the fact that @xmath0 maintains the same orbital behavior during supercycle ( fig.2 ) suggests that there is no disc precession . other more speculative hypothesis is that the primary experiences instabilities around critical velocity , gaining extra momentum during accretion until attains a velocity just above the critical one , then relaxes below critical velocity giving the extra momentum to the disc that partly escapes from the binary . additional studies are needed to confirm this view . we have initiated a program to study dpvs with high resolution optical / infrared spectrographs and robotic telescopes with the aim of shedding light on this phenomenon . in table 1 we summarize our view for dpvs in the context of algols and w serpentid stars . it is possible that the critical velocity of the gainer can be maintained only until the mass of the donor ( @xmath6 ) drops to certain value . during this period of high mass transfer rate the star behaves as a w serpentid ( with a thick circumprimary disc and chaotic mass loss ) or as a dpv ( with slightly lower @xmath7 allowing the dpv instability to operate ) . when @xmath7 drops even more , tidal forces spin down the gainer , the orbital separation ( @xmath8 ) increases , and the system becomes a typical algol star . in algols mass transfer rate if present is comparatively small , partly due to the less massive donor and also to the larger @xmath8 . @xmath3 lyr probably still is not a dpv , as suggested by their very small long period amplitude and position in the @xmath9 diagram ( fig.1 ) . ccccc systems & key facts & @xmath6 , @xmath7 & mass loss & age , @xmath10 + w ser & polar jets , variable eclipse , large @xmath11 & large & large & young , yes + dpv & 2-periods , small ecl . variability , @xmath12 & medium & cyclic & middle , yes + algol & small / no additional variability , @xmath12 & small & small & old , no + _ s.m . : what about blends ? they are always of concern in other galaxies . + _ mennickent _ : dpvs are also observed in the galaxy . we have discarted the possibility of blends in the case of dpvs .

we introduce the class of intermediate mass binaries named double periodic variables ( dpvs ) , characterized by orbital photometric variability ( ellipsoidal or eclipsing ) in time scales of few days and a long photometric cycle lasting roughly 33 times the orbital period . after a search conducted in the ogle and asas catalogues , we identified 114 of these systems in the magellanic clouds and 11 in the galaxy . we present results of our photometric and spectroscopic campaigns on dpvs conducted during the last years , outlining their main observational characteristics . we present convincing evidence supporting the view that dpvs are semidetached interacting binaries with optically thick discs around the gainer , that experience regular cycles of mass loss into the interstellar medium . the mechanism regulating this long - term process still is unknown but probably is related to relaxation cycles of the circumprimary disc . a key observational fact is the modulation of the @xmath0 of hei 5875 with the long cycle in v393sco . the dpv evolution stage is investigated along with their relationship to algols and w serpentid stars . we conclude that dpvs can be used to test models of non - conservative binary evolution including the formation of circumbinary discs .

main sequence stars with mass in the range 0.9 - 9 m@xmath2 evolve through a double shell burning phase , refered to as the asymptotic giant branch ( agb ) phase of evolution . this phase is characterized by carbon dredge up of the core to the surface after each thermal pulse - helium shell flash - ( iben & renzini 1983 ) . the temperatures of these objects are very badly known . although they are highly variable , their determination from static models such as assumed in the basel library can be justified as a first approximation . in order to explore the capabilities of the basel library ( lejeune , cuisinier & buser 1997 , 1998 and references therein , see also lastennet , lejeune & cuisinier , these proceedings ) to predict correct temperatures for such cool agb stars , we compare our results from synthetic infrared photometry of the stellar photosphere with the detailed study of lorenz - martins & lefvre ( 1994 ) of the agb carbon star r fornacis . their work is based on a modelling of the spectral energy distribution of the dust envelope , where they put tight constraints on the temperature of the heating source . table 1 gives the jhklm photometry of r for ( hip 11582 ) that we used ( le bertre , 1992 ) . the photometric errors in the individual jhklm magnitudes are not provided so we assume an error of 0.2 on each magnitude , according to the maximum uncertainty estimated from fig . 1 of le bertre ( 1988 ) . ccccccc j & h & k & l & m & t@xmath0@xmath3 & t@xmath0@xmath4 + & & & & & ( k ) & ( k ) + 5.76 & 3.97 & 2.32 & 0.21 & @xmath50.28 & 2650 & 2440 - 2520 + @xmath3 lorenz - martins & lefvre ( 1994 ) ; + @xmath4 basel jhkm synthetic photometry ( this work , see text for details ) . although the dust may have a significant contribution in the ir _ bands _ of this star , especially l and m , it should only have a secondary influence on the photospheric _ colours_. we intend of course to correct for the predicted differences by a dust model ( lorenz - martins & lefvre , 1993 ) due to the envelope . however in a first step we merely compare the observed colours of r fornacis with the photospheric predictions of the basel library ( basel-2.2 version , with spectral corrections ) by minimizing their @xmath6 differences . + this @xmath6-minimization method is similar to the one applied in lastennet et al . ( 2001 ) : we derived the t@xmath0 and log g values matching simultaneously the observed jhklm photometry listed in tab . 1 , assuming a solar metallicity ( [ fe / h]@xmath70 ) . we have tested various colour combinations of the j ( 1.25 @xmath8 ) , h ( 1.65 @xmath8 ) , k ( 2.2 @xmath8 ) , l ( 3.4 @xmath8 ) , and m ( 5.0 @xmath8 ) magnitudes : ( j@xmath5h ) , ( h@xmath5k ) , ( k@xmath5l ) , ( j@xmath5k ) and ( k@xmath5 m ) . they all give t@xmath0 estimates in agreement with the work of lorenz - martins & lefvre ( 1994 ) . + since better constraints should be obtained by matching more than 1 colour , we chose the ( j@xmath5h ) and ( k@xmath5 m ) colours which give the best @xmath6-scores . the solutions we get to match simultaneously the observed ( j@xmath5h ) and ( k@xmath5 m ) are presented in fig . our best basel - infrared solution is t@xmath0@xmath72440k , but all the solutions inside the 1-@xmath9 contour are good fits to the observed photometric data . the effective temperature of the central star of r for found by lorenz - martins & lefvre is t@xmath0@xmath72650 k ( shown as a vertical line on fig . 1 ) . this is larger by @xmath1100k than the 1-@xmath9 basel contour but still inside the 2-@xmath9 contour . additionally the basel models show that this star has a surface gravity log g @xmath1@xmath50.5@xmath100.4 , which is what one expects for carbon stars . we reported a preliminary study to determine the t@xmath0 and surface gravity of the central star of r fornacis by exploring the best @xmath6-fits to the infrared photometric data . these results are in a surprising good agreement - given the approximation we made ( no envelope absorption / emission correction ) - with the detailed study of lorenz - martins & lefvre ( 1994 ) . therefore , while detailed spectra studies are obviously highly preferred ( see e.g. loidl , lanon & jrgensen , 2001 ) , our method may provide a good starting point . if our r fornacis result is confirmed with other agb stars , this would mean that the basel jhklm synthetic photometry is suited to derive ( teff - log g ) estimates for cool agb stars . iben i. , renzini a. , 1983 , ara&a , 21 , 271 lastennet e. , lignires f. , buser r. , lejeune th . , lftinger th . , cuisinier f. , vant veer - menneret c. , 2001 , , 365 , 535 le bertre t. , 1988 , , 190 , 79 le bertre t. , 1992 , , 94 , 377 lejeune th . , cuisinier f. , buser r. , 1997 , , 125 , 229 lejeune th . , cuisinier f. , buser r. , 1998 , , 130 , 65 loidl r. , lanon a. , jrgensen u.g . , 2001 , , 371 , 1065 lorenz - martins s. , lefvre j. , 1993 , , 280 , 567 lorenz - martins s. , lefvre j. , 1994 , , 291 , 831

we discuss the possibilities of the basel models in its lowest temperature boundary ( t@xmath0@xmath12500 k for cool giants ) to provide the t@xmath0 of agb stars . we present the first step of our work , by comparing our predictions for the agb star r fornacis with the results of lorenz - martins & lefvre ( 1994 ) based on the dust spectral energy distribution .

percolation@xcite describes the passage of an influence through a medium which is irregularly structured in the sense that the influence can propagate through some regions whereas it can not pass other areas . prominent examples for such media are computer networks like the internet where information propagates and irregularity can be caused by random switch failures or other technical problems . a particularly simple percolation model is the random resistor network ( rrn ) . in this model the irregular medium is given by a , say hypercubic , lattice in which bonds between nearest neighboring sites are randomly occupied with a probability @xmath4 . the influence corresponds to an external current @xmath5 , which is injected at a terminal site @xmath6 and withdrawn at another terminal site @xmath7 . depending on the occupation probability @xmath4 the resistors ( bonds ) are likely to either be isolated or to form clusters . two sites belong to the same cluster if they are connected by a path of bonds and hence current can flow between them . at low @xmath4 two infinitely separated terminal sites @xmath6 and @xmath7 are not connected by such a path and the network behaves as an insulator . for large @xmath4 , on the other hand , many paths between @xmath6 and @xmath7 may exist and the network is a conductor . therefore , at some probability in between , a threshold @xmath8 must exist where for the first time current can percolate from @xmath6 to @xmath7 . the threshold probability is called the percolation threshold . since it separates the conducting and the insulating phase , it is also referred to as the critical probability . in rrns the influence can percolate through occupied bonds in all directions . the resulting clusters are typically isotropic in space . this kind of percolation is referred to as isotropic percolation ( ip ) . the linear extend of the isotropic clusters can be characterized by the correlation length @xmath9 , where @xmath10 is the correlation length exponent of the ip universality class . directed percolation ( dp)@xcite is an anisotropic variant of percolation . the bonds function as diodes so that the current can percolate only along a given distinguished direction . the critical properties of isotropic and directed percolation are very different . typical dp clusters are anisotropic and they are characterized by two different correlation lengths : @xmath11 ( parallel to the distinguished direction ) and @xmath12 ( perpendicular to it ) . as one approaches the critical probability , the two correlation lengths diverge with the exponents @xmath13 and @xmath14 of the dp universality class . the apparent success of dp might be attributed to the fact that it is perhaps the simplest model resulting in branching self - affine objects . it has many potential applications , including fluid flow through porous media under gravity , hopping conductivity in a strong electric field@xcite , crack propagation@xcite , and the propagation of surfaces at depinning transitions@xcite . dp has a dynamic interpretation in which the distinguished direction is viewed as time . a dp cluster then represents the history of a stochastic process . in this dynamic interpretation the dp universality class is the generic universality class for phase transitions from an active to an absorbing inactive state . for example the epidemic spreading of an infectious desease without immunization@xcite may be described by dp@xcite . moreover , dp is related to self - organized critical models @xcite . in the early 1980 s redner introduced the random resistor diode network ( rdn ) which comprises both , ip and dp . a rdn is a bond percolation model where nearest - neighbor sites are connected by a resistor , a positive diode ( conducting only in the distinguished direction ) , a negative diode ( conducting only opposite to the distinguished direction ) , or an insulator with respective probabilities @xmath4 , @xmath15 , @xmath16 , and @xmath17 . in the three dimensional phase diagram ( pictured as a tetrahedron spanned by the four probabilities ) one finds a nonpercolating and three percolating phases . the percolating phases are isotropic , positively directed , or negatively directed . between the phases there are surfaces of continuous transitions . all four phases meet along a multicritical line , where @xmath18 and @xmath19 . on the entire multicritical line , i.e. , independently of @xmath2 , one finds the scaling properties of usual isotropic percolation ( @xmath20 ) . for the crossover from ip to dp see , e.g. , ref.@xcite . in this paper we focus exclusively on the vicinity of the critical surface separating the non - percolating and the positively directed phase . an important notion in the theory of rdn is the average resistance @xmath21 between two connected terminal sites @xmath22 and @xmath23 of the network . the critical behavior of this average resistance is well known@xcite . if @xmath21 is measured for example in the distinguished , timelike direction then it scales like @xmath24 with @xmath25 being the dp resistance exponent . here in this paper we consider a generalized rdn in which the resistor - like bonds and the diode - like bonds under forward bias voltage obey a generalized ohm s law , @xmath0 . our motivation to assume this non linear ohm s law is twofold . first , real circuit elements have non linear current - voltage characteristics . this is obviously the case for diodes but it also applies to a certain extend to resistors , in particular for large currents . our main motivation is , however , that the generalized average resistance @xmath26 is related for certain values of @xmath2 to important fractal substructures of dp clusters . this relation provides us with an elegant way to determine the fractal dimensions of the red bonds , the chemical path , and the dp backbone . parts of this work have been presented briefly in ref . @xcite . the plan of presentation is the following . in sec . [ themodel ] we give background on non linear rdn and set up our field theoretic model . section [ rga ] sketches our renormalization group improved perturbation calculation . we derive the scaling behavior of @xmath26 which is governed by a generalized resistance exponent @xmath1 . we present our one - loop result for @xmath1 . in sec . [ fracdim ] we calculate the fractal dimensions of the red bonds , the chemical path , and the dp backbone by considering the limits @xmath27 , @xmath28 , and @xmath29 , respectively . section [ concls ] contains our conclusions . consider a @xmath30-dimensional hypercubic lattice in which the direction @xmath31 is distinguished . assume that the bonds @xmath32 between two nearest neighboring sites @xmath33 and @xmath34 are directed so that @xmath35 . suppose that the directed bonds obey the non - linear ohm s law@xcite [ ohmr ] @xmath36 or , equivalently , @xmath37 @xmath38 , where @xmath39 denotes the potential at site @xmath33 , is the voltage drop over the bond between sites @xmath33 and @xmath34 . @xmath40 denotes the current flowing from @xmath34 to @xmath33 . in the following we drop the subscript @xmath41 whenever it is save . the bond conductances @xmath42 are random variables taking on the values @xmath43 , @xmath44 , @xmath45 , and @xmath46 with respective probabilities @xmath4 , @xmath47 , @xmath48 , and @xmath49 . @xmath43 is a positive constant and @xmath50 denotes the heaviside function . the exponents @xmath2 and @xmath51 are describing the non linearity with @xmath52 . the bond resistances @xmath53 are related to the conductances via @xmath54 . note that the diodes are idealized : under forward - bias voltage they behave for @xmath55 as `` ohmic '' resistors whereas they are insulating under backward - bias voltage . we point out that the round brackets in eq . ( [ ohmr ] ) contain the argument of the bond conductance or resistance , respectively . it is important to realize that these quantities depend on the voltages or currents by means of the step function and that @xmath56 . hence we may write @xmath57 and @xmath58 . assume that an external current @xmath5 is injected at @xmath6 and withdrawn at @xmath7 . it is understood that @xmath6 and @xmath7 are connected . the power dissipated on the network is by definition @xmath59 using ohm s law it may be expressed entirely in terms of the voltages as @xmath60 the sum is taken over all bonds on the cluster connecting @xmath6 and @xmath7 and @xmath61 denotes the corresponding set of voltages . @xmath62 stands for the macroscopic resistance when @xmath5 is inserted at @xmath6 and withdrawn at @xmath7 . similarly one may define @xmath63 as the macroscopic resistance when @xmath5 is inserted at @xmath7 and withdrawn at @xmath6 . the two quantities are related by @xmath64 . from the power one obtains kirchhoff s first law @xmath65 as a consequence of the variation principle @xmath66 = 0 \ .\end{aligned}\ ] ] the summation in eq . ( [ kirchhoff ] ) extends over the nearest neighbors of @xmath33 and @xmath67 is given by @xmath68 . alternatively to eq . ( [ power1 ] ) , the power can be expressed in terms of the currents as @xmath69 with @xmath70 denoting the set of currents flowing through the individual bonds . it is understood that @xmath71 whenever @xmath72 . kirchhoff s second law , saying that the voltage drops along closed loops vanish , can be stated in terms of the variation principle@xmath73 i.e. , there are no independent loop currents @xmath74 circulating around a complete set of independent closed loops . now we take a short detour and show that the average resistance @xmath26 is related for specific values of @xmath2 to the mass , i.e. , the average number of bonds , of important substructures of dp clusters . the arguments below are well established for rrn and we simply adapt them here to rdn . we start with the backbone . the ( forward ) backbone between two sites @xmath6 and @xmath7 is defined , in principle , as the union of all current carrying bonds when @xmath5 is injected at @xmath6 and withdrawn at @xmath7 . consider @xmath29 . one obtains immediately as a consequence of eq . ( [ power2 ] ) , that @xmath75 with only those bonds carrying non zero current contributing to the sum on the right hand side . hence @xmath76 where @xmath77 stands for the mass of the backbone . now we turn to @xmath27 and @xmath28 following the lines of blumenfeld and aharony@xcite . on the backbone between two sites @xmath6 and @xmath7 one may distinguish between two different substructures : blobs formed by multi - connected bonds and singly connected bonds which are referred to as red bonds . both substructures are contributing to the resistance of the backbone @xmath78 where the sums are taken over all bonds belonging to blobs and over all red bonds respectively . since sites on a blob are multi - connected by definition , @xmath79 , and thus @xmath80 in conclusion , @xmath81 is related to the mass of the red bonds @xmath82 via @xmath83 consider now the first site @xmath6 at some end of a blob . an entering current @xmath5 splits into currents @xmath84 flowing to nearest neighbors @xmath33 with @xmath85 in the limit @xmath86 the ratios @xmath87 vanish whenever @xmath88 . thus , current flows only through the resistor with the largest @xmath89 . this argument may be iterated through the entire blob . one identifies either a single self avoiding chain through which @xmath5 flows , with @xmath90 being the power dissipated on the chain , or several of such chains with identical power . the expression in eq . ( [ powerofchain ] ) is minimal for minimal @xmath91 , i.e. , the current chooses the shortest path through the blob and one is led to @xmath92 where @xmath93 stands for the mass of the chemical path . our aim now is to calculate the average resistance between two ports @xmath6 and @xmath7 which is precisely defined by @xmath94 @xmath95 denotes the average over all configurations of the diluted lattice . @xmath96 is an indicator function that takes the value one if @xmath6 and @xmath7 are positively connected , i.e. , if @xmath5 can percolate from @xmath6 to @xmath7 , and zero otherwise . note that @xmath97 is nothing more than the usual dp correlation function . now we follow an idea by stephen@xcite and its generalization to networks of nonlinear resistors by harris@xcite and exploit correlation functions @xmath98 of @xmath99 as generating functions of @xmath100 . in writing eqs . ( [ defcorr ] ) and ( [ defpsi ] ) we switched to @xmath101-fold replicated voltages , @xmath102 , and imaginary currents , @xmath103 . the correlation functions are given by @xmath104 \bigg\rangle_c \ , \end{aligned}\ ] ] where @xmath105 and @xmath106 is the normalization @xmath107 \ .\end{aligned}\ ] ] the additional power term @xmath108 which we have introduced in eqs . ( [ erzeugendefunktion ] ) and ( [ norm ] ) is necessary to give the voltage integrals a well defined meaning . without this term the integrands depend only on voltage differences and the integrals are divergent . physically the new term corresponds to grounding each lattice site by a capacitor of unit capacity . the original situation can be retrieved by taking the limit of vanishing frequency , @xmath109 . because the integrations in eqs . ( [ erzeugendefunktion ] ) and ( [ norm ] ) are not gaussian , we employ the saddle point method . the saddle point equation is nothing more than the variation principle stated in eq . ( [ variationprinciple1 ] ) . thus , the maximum of the integrand is determined by the solution of the circuit equations ( [ kirchhoff ] ) . provided that the condition @xmath110 holds , we obtain , up to an unimportant multiplicative constant which goes to one in the limit @xmath111 , @xmath112 \right\rangle_c \ , \end{aligned}\ ] ] where @xmath113 now we may expand @xmath114 about @xmath115 , @xmath116 this shows us that @xmath114 is indeed the desired generating function from which the average resistance may be calculated via @xmath117 here we would like to comment on the nature of @xmath118 . we work near the limit when all the components of @xmath118 are equal and continue to large imaginary values . accordingly we set@xcite @xmath119 with real @xmath120 and @xmath121 , and impose the condition @xmath122 . the saddle point approximation may be justified by demanding @xmath123 substitution of eq . ( [ crazylambda ] ) into the definition of @xmath124 leads to @xmath125 thus , one can justify the expansion in eq . ( [ expansionofg ] ) by invoking the conditions @xmath126 note that the replica limit @xmath127 allows for a simultaneous fulfillment of the conditions ( [ cond1 ] ) and ( [ cond2 ] ) . however , we will not only rely exclusively on these conditions on @xmath118 . we will provide several consistency checks for the validity of harris saddle point approach as we go along and reproduce known results . since infinite voltage drops between different clusters may occur , it is not guaranteed that @xmath106 stays finite , i.e. , the limit @xmath128 is not well defined . this problem can be regularized by switching to voltage variables @xmath129 taking discrete values on a @xmath101-dimensional torus @xcite . the voltages are discretized by setting @xmath130 , where @xmath131 is the gap between successive voltages , @xmath132 is a voltage cutoff , @xmath133 is a @xmath101-dimensional integer , and @xmath134 a positive integer . the components of @xmath133 are restricted to @xmath135 and periodic boundary conditions are realized by equating @xmath136 . the continuum may be restored by taking @xmath137 and @xmath138 . by setting @xmath139 , @xmath140 , and , respectively , @xmath141 , the two limits can be taken simultaneously via @xmath142 . note that the limit @xmath111 has to be taken before any other limit . since the voltages and @xmath118 are conjugated variables , @xmath118 is affected by the discretization as well : @xmath143 where @xmath144 is a @xmath101-dimensional integer taking the same values as @xmath133 . this choice guarantees that the completeness and orthogonality relations [ complete ] @xmath145 and @xmath146 do hold . equation ( [ complete ] ) provides us with a fourier transform between the @xmath129- and @xmath118-tori . after taking care of these regularization issues , we now carry out the average over the diluted lattice configurations in eq . ( [ erzeugendefunktion ] ) . this provides us with the effective hamiltonian @xmath147 \right\rangle_c \nonumber \\ & = & - \sum_{\underline{b } } k \left ( \vec{\vartheta}_{\underline{b } } \right ) - \frac{i\omega}{2 } \sum_i \vec{\vartheta}_{i}^2 \ , \end{aligned}\ ] ] where @xmath148 + p_+ \prod_{\alpha = 1}^d \exp \left [ - \frac{\sigma}{s+1 } \theta \left ( \vartheta^{(\alpha ) } \right ) \left| \vartheta^{(\alpha ) } \right|^{s+1 } \right ] \nonumber \\ & + & p_- \prod_{\alpha = 1}^d \exp \left [ - \frac{\sigma}{s+1 } \theta \left ( - \vartheta^{(\alpha ) } \right ) \left| \vartheta^{(\alpha ) } \right|^{s+1 } \right ] \bigg\ } \ .\end{aligned}\ ] ] in order to proceed further we recall the choice for @xmath118 made in eq . ( [ crazylambda ] ) . because @xmath118 and @xmath129 are related by ohm s law , we have to make a consistent choice for @xmath129 : @xmath149 with real @xmath150 and @xmath151 , and where @xmath152 . upon imposing the condition @xmath153 for all @xmath154 we may write @xmath155 \nonumber \\ & + & p_- \left [ \theta \left ( \vartheta_0 \right ) + \theta \left ( - \vartheta_0 \right ) \exp \left ( - \frac{\sigma}{s+1 } \left| \vartheta \right|^{s+1 } \right ) \right ] \bigg\ } \ , \end{aligned}\ ] ] where we have introduced the abbreviation @xmath156 . after doing a little straightforward algebra and by dropping a term @xmath157 + \theta \left ( - \vartheta_0 \right ) \ln \left [ 1 - p - p_- \right ] \end{aligned}\ ] ] which does not depend on the bond conductances we obtain @xmath158 the @xmath159 in eq . ( [ kern3 ] ) are given by @xmath160 \ .\end{aligned}\ ] ] note that these are exponentially decreasing functions in replica space with a decay rate proportional to @xmath161 . in order to refine @xmath162 towards a field theoretic hamiltonian we now expand @xmath163 in terms of @xmath164 : @xmath165 k \left ( \vec{\vartheta } \right ) \ .\end{aligned}\ ] ] upon exploiting that @xmath166 we obtain @xmath167 \nonumber \\ & & + \frac{1}{2 } \left [ \theta \left ( \lambda_0 \right ) - \theta \left ( -\lambda_0 \right ) \right ] \left [ \widetilde{k}_+ \left ( \vec{\lambda } \right ) - \widetilde{k}_- \left ( \vec{\lambda } \right ) \right ] \bigg\ } \ .\end{aligned}\ ] ] where @xmath168 stands for the fourier transform of @xmath169 , @xmath170 k \left ( \vec{\vartheta } \right ) \ .\end{aligned}\ ] ] the fourier transform can be carried out by switching back to continuous currents and expanding the logarithm in eq . ( [ defkpm ] ) . the so obtained result has the taylor expansion @xmath171 where @xmath172 and @xmath173 are expansion coefficients depending on @xmath4 and @xmath174 with @xmath175 and @xmath176 . now we insert eq . ( [ kern5 ] ) into eq . ( [ effhamil ] ) . we also carry out a gradient expansion in position space . this is justified because only nearest neighbor pairs enter in the power @xmath177 , i.e. , the interaction is short ranged not only in replica but also in position space . we find @xmath178 \psi_{-\vec{\lambda } } \left ( i \right ) \left [ 1 + \frac{1}{2 } \left ( \underline{b}_i \cdot \nabla \right)^2 + \cdots \right ] \psi_{\vec{\lambda } } \left ( i \right ) \nonumber \\ & & + \frac{1}{2 } \left [ \theta \left ( \lambda_0 \right ) - \theta \left ( -\lambda_0 \right ) \right ] \left [ \widetilde{k}_+ \left ( \vec{\lambda } \right ) - \widetilde{k}_- \left ( \vec{\lambda } \right ) \right ] \psi_{-\vec{\lambda } } \left ( i \right ) \left [ \underline{b}_i \cdot \nabla + \cdots \right ] \psi_{\vec{\lambda } } \left ( i \right ) \ , \end{aligned}\ ] ] with @xmath168 given by eq . ( [ kernelexpansion ] ) . we proceed with the usual coarse graining step and replace the @xmath179 by order parameter fields @xmath180 which inherit the constraint @xmath181 . we model the corresponding field theoretic hamiltonian @xmath182 in the spirit of landau as a mesoscopic free energy and introduce the landau - ginzburg - wilson type functional @xmath183 \psi_{\vec{\lambda } } \left ( { \rm{\bf x } } \right ) \nonumber \\ & & + \ , \frac{g}{6 } \sum_{\vec{\lambda } , \vec{\lambda}^\prime , \vec{\lambda } + \vec{\lambda}^\prime \neq \vec{0 } } \psi_{-\vec{\lambda } } \left ( { \rm{\bf x } } \right ) \psi_{-\vec{\lambda}^\prime } \left ( { \rm{\bf x } } \right ) \psi_{\vec{\lambda } + \vec{\lambda}^\prime } \left ( { \rm{\bf x } } \right ) + \frac{i \omega}{2 } \nabla^2_{\vec{\lambda } } \psi_{\vec{\lambda } } \left ( { \rm{\bf x } } \right ) \bigg\ } \ .\end{aligned}\ ] ] as usual we have neglected all terms which are irrelevant in the sense of the renormalization group . the parameter @xmath184 is the coarse grained ancestor of @xmath185 . it specifies the `` distance '' from the critical surface under consideration . @xmath186 is the coarse grained analog of @xmath187 . the vector @xmath188 lies in the distinguished direction , @xmath189 . @xmath190 depends as @xmath184 and @xmath191 on the three probabilities @xmath4 , @xmath47 , and @xmath48 . for @xmath192 it vanishes . in the limit @xmath193 our hamiltonian @xmath194 describes the usual purely geometric dp . indeed @xmath194 leads for @xmath193 to exactly the same perturbation series as obtained in@xcite . in the limit @xmath195 @xmath194 describes resistor diode percolation as studied in refs . @xcite . in the remainder of this paper we drop the regularization term proportional to @xmath196 for simplicity . now we are in the position to set up a perturbation calculation . this perturbation calculation can be simplified from the onset by manipulating @xmath194 in such a way that it takes the form of a dynamic functional . we assume that @xmath197 and introduce new variables by setting @xmath198 by substituting eq . ( [ subst ] ) into eq . ( [ hamiltonian ] ) we obtain @xmath199 s_{\vec{\lambda } } \left ( { \rm{\bf x}}_\perp , t \right ) \nonumber \\ & & + \ , \frac{\rho \overline{g}}{6 } \sum_{\vec{\lambda } , \vec{\lambda}^\prime , \vec{\lambda } + \vec{\lambda}^\prime \neq \vec{0 } } s_{-\vec{\lambda } } \left ( { \rm{\bf x}}_\perp , t \right ) s_{-\vec{\lambda}^\prime } \left ( { \rm{\bf x}}_\perp , t \right ) s_{\vec{\lambda } + \vec{\lambda}^\prime } \left ( { \rm{\bf x}}_\perp , t \right ) \bigg\ } \ , \end{aligned}\ ] ] where @xmath200 . note that we have neglected a term containing a second derivative with respect to the `` time '' @xmath201 . this is justified because this term is less relevant than the one with the first `` time '' derivative which we kept . we proceed with standard methods of field theory@xcite . from eq . ( [ dynfktnal ] ) we gather the diagrammatic elements contributing to our perturbation series . the first element is the vertex @xmath202 . dimensional analysis shows that the vertex @xmath203 is marginal in four transverse dimensions . hence @xmath204 is the upper critical dimension as it is well known for dp . the second diagrammatic element is the gaussian propagator @xmath205 which is determined by the equation of motion @xmath206 + \left [ \theta \left ( \lambda_0 \right ) - \theta \left ( -\lambda_0 \right ) \right ] \frac{\partial}{\partial t } \right\ } g \left ( { \rm{\bf x}}_\perp , t , \vec{\lambda } \right ) = \delta \left ( { \rm{\bf x}}_\perp \right ) \delta \left ( t \right ) .\end{aligned}\ ] ] for the fourier transformed @xmath207 of @xmath205 , where @xmath208 is the momentum conjugate to @xmath209 , one obtains readily @xmath210 the quantities on the right hand side are given by @xmath211 \left ( 1 - \delta_{\vec{\lambda } , \vec{0 } } \right ) \ .\end{aligned}\ ] ] for the diagrammatic expansion it is sufficient to keep either @xmath212 or @xmath213 . we choose to keep @xmath212 . from the vertex @xmath202 and the propagator @xmath212 we now assemble the feynman graphs constituting our diagrammatic expansion . as in our previous work on transport in ip@xcite these feynman diagrams have a real - world interpretation : they may be viewed as being directed resistor networks themselves . this real - world interpretation has basically two roots . the first one is that the principal propagator @xmath212 decomposes into two parts : @xmath214 \nonumber \\ & & - \theta \left ( t \right ) \exp \left [ - t \rho \left ( \tau + { \rm{\bf p}}^2 \right ) \right ] \delta_{\vec{\lambda } , \vec{0 } } \ .\end{aligned}\ ] ] one of these parts is carrying @xmath118 s and hence we call it conducting . the other one is not carrying @xmath118 s and accordingly we call it insulating . equation ( [ decogp ] ) allows for a schematic decomposition of the principal diagrams into sums of conducting diagrams consisting of conducting and insulating propagators . in fig . 1 we list the conducting diagrams resulting from the decomposition procedure up to two - loop order . the second root of the real - world interpretation is that the replica currents @xmath118 are conserved in each vertex just as currents are conserved in nodes of real networks . hence we may write for each edge @xmath33 of a diagram , @xmath215 , where @xmath118 is an external current and @xmath216 denotes a complete set of independent loop currents . the @xmath118-dependent part of each conducting diagram then takes the form @xmath217 \ .\end{aligned}\ ] ] now it is important to realize that @xmath218 resembles the structure of a power ( cf . eq . ( [ power2 ] ) ) . thus , we interpret the `` time '' associated with a conducting propagator as its resistance and write @xmath219 \ .\end{aligned}\ ] ] the real - world interpretation provides for an alternative way of computing the conducting feynman diagrams . to evaluate the sums over independent loop currents , @xmath220 \ , \end{aligned}\ ] ] we employ the saddle point method under the conditions discussed at the end of sec . [ replicaformalism ] . note that the saddle point equation is nothing more than the variation principle stated in eq . ( [ variationprinciple2 ] ) . thus , solving the saddle point equations is equivalent to determining the total resistance @xmath221 of a diagram , and the saddle point evaluation of ( [ toevaluate ] ) yields @xmath222 \ , \end{aligned}\ ] ] where we have omitted once more multiplicative factors which go to one for @xmath223 . a completion of squares in the momenta renders the momentum integrations , which remain to be done to compute the diagrams , straightforward . equally well we can use the saddle point method which is exact here since the momentum dependence is purely quadratic . after an expansion for small @xmath224 all diagrammatic contributions are of the form @xmath225 d \left ( { \rm{\bf p}}^2,t ; \left\ { t_i \right\ } \right ) \ .\end{aligned}\ ] ] @xmath226 is a typical integrand as known from the field theory of dp@xcite . we proceed with standard techniques of renormalized field theory@xcite . the ultraviolet divergences occurring in the diagrams can be absorbed by dimensional regularization . we employ the renormalization scheme [ renorscheme ] @xmath227 where @xmath228 and @xmath229 is the usual inverse length scale . the factor @xmath230 , with @xmath231 denoting the gamma function , is introduced for convenience . @xmath106 , @xmath232 , @xmath233 , and @xmath234 are the usual dp @xmath106 factors known to second order in @xmath235@xcite . in ref . @xcite we determined @xmath236 to second order in @xmath235 . here , we calculate @xmath237 for arbitrary @xmath2 to order @xmath235 . this calculation is straightforward because we can determine the total resistance of the one - loop diagrams by using simple rules . for example , two nonlinear resistors with resistances @xmath238 and @xmath239 added in series have a total resistance @xmath240 given by @xmath241 whereas two such resistors in parallel give @xmath242 by exploiting eq . ( [ addrule2 ] ) find @xmath243 + { \sl o } \left ( u^2 \right ) \ .\end{aligned}\ ] ] calculating @xmath237 for general @xmath2 to higher loop orders appears to be beyond possibility . the reason is , that conducting diagrams like c appear . the total resistance of these diagrams can not be determined by using simple rules like eqs . ( [ addrule1 ] ) and ( [ addrule2 ] ) . instead , one has to solve the set of nonlinear circuit equations which is hardly feasible in closed form . now we set up in a standard fashion the renormalization group equation for our problem . the unrenormalized theory has to be independent of the length scale @xmath244 introduced by renormalization . in particular , the unrenormalized connected @xmath245 point correlation functions must be independent of @xmath229 , i.e. , @xmath246 for all @xmath245 . ( [ independence ] ) translates via the wilson functions @xmath247 where the bare quantities are kept fix while taking the derivatives , into the gell - mann - low renormalization group equation @xmath248 } \nonumber \\ & & \times \ , g_n \left ( \left\ { { \rm{\bf x}}_\perp , \rho t , w_r \lambda_r \left ( \vec{\lambda } \right ) \right\ } ; \tau , u , \mu \right ) = 0 \ .\end{aligned}\ ] ] the particular form of the wilson functions can be extracted from the renormalization scheme and the @xmath106 factors . the critical behavior of the correlation functions is determined by the infrared stable fixed point solutions of eq . ( [ rge ] ) . this fixed point @xmath249 is readily extracted from the condition @xmath250 . then eq . ( [ rge ] ) is solved at @xmath249 by the method of characteristics which gives @xmath251 where @xmath252 , @xmath253 , @xmath254 , and @xmath255 . to analyze the scaling behavior of the correlation functions completely , the solution ( [ solofrgg ] ) has to be supplemented by a dimensional analysis : @xmath256 equation ( [ solofrgg ] ) in conjunction with eq . ( [ dimana ] ) now gives @xmath257 @xmath258 , @xmath259 , and @xmath260 are the well known the critical exponents for dp which have been calculated previously to second order in @xmath235@xcite . these dp exponents , however , are not sufficient to specify the critical behavior of the rdn correlation functions completely . in eq . ( [ scaling ] ) we introduced the additional nonlinear resistance exponent @xmath261 note that @xmath262 is in conformity to order @xmath235 with our result for the resistance exponent for the usual `` ohmic '' rdn , i.e. , eq . ( [ resphi ] ) satisfies an important consistency check . since we are primarily interested in the critical behavior of the average two - port resistance , we now take a closer look at the two point correlation function @xmath263 . equation ( [ scaling ] ) implies for @xmath114 at @xmath264 that @xmath265 where we dropped several arguments for notational simplicity . in the following we set @xmath266 and @xmath267 , once more for the sake of simplicity . the choice @xmath268 and a taylor expansion of the right hand side of eq . ( [ scalerel ] ) lead to @xmath269 where the @xmath270 s are scaling functions . equally well we can choose @xmath271 which then leads to @xmath272 with other scaling functions @xmath273 and @xmath274 and where @xmath275 . now we can extract the critical behavior of @xmath100 . for measurements in the distinguished direction we straightforwardly exploit eq . ( [ scalet ] ) via eq . ( [ shit ] ) and find that @xmath276 for all other directions we determine @xmath100 from eq . ( [ scalex ] ) . with help of eq . ( [ shit ] ) we find that @xmath277 here it is convenient to choose a common length scale @xmath278 and to express both , @xmath279 and @xmath201 , in terms of it : @xmath280 and @xmath281 . this choice guarantees that the scaling functions @xmath282 is constant and eq . ( [ brauchnoch ] ) simplifies to @xmath283 in this section we calculate @xmath1 for @xmath28 , @xmath27 , and @xmath29 to two - loop order . this provides us with the fractal dimension of the backbone , the red bonds , and the chemical length respectively . for self - affine objects the notion of fractal dimension is less straightforward than for self - similar objects . to determine , for example , the fractal dimension of the red bonds in dp one considers a @xmath284-dimensional hyper - plane with an orientation perpendicular to @xmath285 . the cut through the red bonds is a self similar object with the fractal dimension @xmath286 where @xmath287 is the local fractal dimension@xcite of the red bonds . by virtue of eqs . ( [ massred ] ) and ( [ brauchnoch ] ) the mass of the red bonds scales as @xmath288 accordingly the mass of the cut scales like @xmath289 by choosing once more @xmath280 and @xmath290 we find that @xmath291 this leads via eq . ( [ cutdim ] ) to @xmath292 it remains to compute @xmath293 . to do so we take direct advantage of our view of the feynman diagrams as being resistor networks themselves . as argued in sec . [ clusterproperties ] , blobs do not contribute to the total resistance for @xmath27 . in analogy only singly connected conducting propagators contribute to the total resistance of a diagram , i.e , @xmath294 with the sum being taken exclusively over singly connected conducting propagators . the contribution of a diagram to the renormalization factor @xmath295 takes the form @xmath296 note that a factor @xmath297 in eq . ( [ expansionofdiagrams_r_infty ] ) corresponds to the insertion ( cf . ref.@xcite ) of @xmath298 into the @xmath33th edge of the diagram . thus , we can generate @xmath299 for a given conducting diagram by inserting @xmath298 into its singly connected conducting propagators . this procedure is carried out up to two - loop order , i.e. , every conducting propagator in fig . 1 that does not belong to a closed loop gets an insertion . the resulting diagrams are displayed in fig . 2 . at this point it is instructive to consider the contributions of the diagrams listed in fig . 1 to @xmath232 . these can be generated by inserting @xmath298 in conducting as well as in insulating propagators . again , one obtains the diagrams depicted in fig . 2 with the same fore - factors . consequently , @xmath295 and @xmath232 are identical at least up to two - loop order . the same goes for the corresponding wilson functions @xmath300 and @xmath301 . from the definition of @xmath1 it follows that @xmath302 upon inserting eq . ( [ resultforphiinfty ] ) into eq . ( [ dredfoermelchen ] ) we obtain the prime result of sec . [ redbonds ] , @xmath303 holding at least to second order in @xmath235 . by substituting the results of ref . @xcite in eq . ( [ dredresult ] ) we obtain the following @xmath235-expansion for @xmath304 : @xmath305 \ , \epsilon \right\ } + { \sl o } \left ( \epsilon^3 \right ) \ .\end{aligned}\ ] ] note that the scaling relation ( [ dredresult ] ) holds rigorously and its validity is not restricted to second order in the @xmath235-expansion . coniglio proved for ip that the mass of the red bonds scales as @xmath306 for ip this is equivalent to saying that the fractal dimension of the red bonds is @xmath307 . since coniglio s arguments do not rely on the isotropy of the system , they can be adapted to apply to dp . for the dp problem eq . ( [ iso ] ) has to be modified to @xmath308 this in turn leads again to eq . ( [ dredresult ] ) . next we address the fractal dimension of the chemical length . equation ( [ dminrel ] ) in conjunction with eq . ( [ brauchnoch ] ) provides us with @xmath309 by applying the same reasoning as in sec . [ backbone ] we learn that the local fractal dimension of the chemical length is given by @xmath310 in order to calculate @xmath311 we determine the shortest self - avoiding path of conducting propagators connecting the external legs of a diagram . due to the dynamic structure all of these paths for a given diagram have the same total resistance that is nothing more than the total `` time '' between the external legs . hence , we can choose any self - avoiding path connecting the external legs . we work with the diagrammatic expansion depicted in fig . 3 . minimal subtraction provides us with the renormalization factor @xmath312 + { \sl o } \left ( u^3 \right ) \ .\end{aligned}\ ] ] note that @xmath313 as stated in eq . ( [ renfuck ] ) is identical to the field renormalization @xmath106 given to two - loop order in refs . @xcite . by virtue of the renormalization scheme ( [ renorscheme ] ) we deduce that @xmath314 at least to second order in @xmath315 . equation ( [ trivialren ] ) leads via @xmath316 to @xmath317 from the definitions of @xmath1 and @xmath318 it follows immediately that @xmath319 in conjunction with eq ( [ dminrel ] ) this leads finally to @xmath320 this result is intuitively plausible because the chemical distance in dp is basically equivalent to the `` time '' @xmath201 . we conclude sec . [ fracdim ] by studying the backbone dimension . by virtue of eqs . ( [ massbb ] ) and ( [ brauchnoch ] ) the mass of the backbone scales as @xmath321 accordingly the local fractal dimension of the backbone is given by @xmath322 it remains to compute @xmath323 . once more we exploit our real - world interpretation . as argued in sec . [ clusterproperties ] , the resistance of the backbone between two sites @xmath6 and @xmath7 is given by @xmath324 with the sum running over all current carrying bonds of the underlying cluster . in analogy , the resistance of a conducting feynman diagram is given by @xmath325 where the sum is extending over all conducting propagators of the diagram . this tells us that the contribution of a diagram to @xmath326 takes the form @xmath327 hence we can generate @xmath299 of any conducting diagram by inserting @xmath298 into its conducting propagators . all conducting propagators in fig . 1 get such an insertion . then it is a matter of simple counting to see that the individual contributions cancel each other . thus , we find @xmath328 as a consequence we obtain @xmath329 at least to second order in @xmath235 . equation ( [ skalenrel ] ) leads by virtue of eq . ( [ dbfoermelchen ] ) to @xmath330 where @xmath331 is the dp order parameter exponent known to second order in @xmath235@xcite . from the scaling relation eq . ( [ ergebnisdb ] ) , which is the main result of this section , the @xmath235-expansion of @xmath332 is readily obtained by inserting the @xmath235-expansion for @xmath333@xcite : @xmath334 \epsilon \right\ } + { \sl o } \left ( \epsilon^3 \right ) \ .\end{aligned}\ ] ] equation ( [ ergebnisdb ] ) is in agreement with scaling arguments yielding that the fractal dimension of the transverse cut through a dp cluster with local dimension @xmath335 is @xmath336 . the analogous cut through the backbone can be viewed as the intersection of the cut through the cluster and the clusters backward oriented pendant . hence , the codimension of the backbone cut is twice the codimension @xmath337 of the cluster cut , which leads again to eq . ( [ ergebnisdb ] ) . it is interesting to compare the @xmath235-expansion result to numerical estimates . we are not aware , however , of any simulations in which @xmath332 itself was determined . et al_. presented numerical results for the scaling exponent of the backbone mass when measured in the longitudinal direction . in the following we call this exponent @xmath338 . formally one can define @xmath338 via @xmath339 . from eqs . ( [ massbb ] ) , ( [ keinelustmehr ] ) , and ( [ skalenrel ] ) it follows that @xmath340 at least to second order in @xmath235 . crudely evaluating the corresponding @xmath235-expansion of @xmath338 for small spatial dimensions leads to poor quantitative predictions . therefore it is appropriate to improve the @xmath235-expansion by incorporating rigorously known features . we carry out a rational approximation which takes into account that obviously @xmath341 . practically this is done by adding an appropriate third order term . by this procedure we obtain the interpolation formula @xmath342 evaluation in two dimensions leads to @xmath343 , where the error is based on a subjective estimate . this result is , within the errors , in agreement with the numerical result @xmath344 . our result though appears to be somewhat small . in this paper we studied a nonlinear version of resistor diode percolation where ohm s law is generalized to @xmath0 . we investigated the critical behavior of the average two - port resistance @xmath100 at the transition from the non percolating to the directed percolating phase . by employing our real - world interpretation of feynman diagrams we calculated the resistance exponent @xmath1 for arbitrary @xmath2 to one - loop order . to our knowledge this is the first time that @xmath1 has been determined for the rdn while @xmath1 is known for rrn , also to one - loop order , since the 1980s@xcite . extending either of these results to higher loop orders seems to be beyond possibility because not all conducting diagrams appearing at higher loop orders can be assembled by simply adding resistors in parallel and in series . for these diagrams one has to solve the set of nonlinear kirchhoff s equations to obtain their total resistance . in closed form , however , this is hardly feasible . the relation of @xmath100 to the mass of the red bonds , the chemical length , and the backbone , respectively , provided us with alternative means to extract the fractal dimensions of these substructures of dp clusters . by computing @xmath1 for @xmath27 , @xmath28 , and @xmath29 we determined @xmath304 , @xmath345 , and @xmath332 to two - loop order . the fractality in dp and ip is qualitatively different . dp clusters are self affine rather than self similar objects . hence , the notion of fractal dimensions in more subtle for dp than for ip . moreover , dp has a markovian character which is evident in the dynamic interpretation . this markovian character provides for scaling relations which do not have an analog in ip . the dp backbone dimension for example can be expressed entirely in terms of the usual ( purely geometric ) critical exponents of the dp universality class , @xmath346 . within the renormalization group framework such a scaling relation is typically associated with a ward identity . the fact that @xmath347 renormalizes trivially to two - loop order is reminiscent of this ward identity . it is an interesting issue for future work to identify the ward identity and its underlying symmetry . another consequence of the markovian character of dp is that the fractal dimension of the chemical length is identical to one . this is intuitively plausible since the shortest longitudinal path through a dp cluster corresponds to the time in the dynamical interpretation . the fractal dimension of the red bonds in dp obeys the scaling relation @xmath348 similar to the @xmath349 for ip . the ward identities corresponding to either of these scaling relations are not known to date . again , this leaves interesting and challenging opportunities for future studies . for a review on percolation see , e.g. , a. bunde and s. havlin , _ fractals and disordered systems _ ( springer , berlin , 1991 ) ; d. stauffer and a. aharony , _ introduction to percolation theory _ ( taylor&francis , london , 1992 ) ; b. d. hughes , _ random walks and random environments _ ( clarendon , oxford , 1995 ) . for a review on dp see , e.g. , h. hinrichsen , adv . in phys . * 49 * , 815 ( 2000 ) . n. van lien and b. i. shklovskii , solid state commun . * 38 * , 99 ( 1981 ) . j. kertsz and t. vicsek , j. phys . c * 13 * , l343 ( 1980 ) . see , e.g. , a .- l . barabasi _ et al_. , in _ surface disordering : growth , roughening , and phase transitions _ , edited by r. jullien _ et al_. ( nova science , new york , 1992 ) ; s. havlin _ et al_. , in _ growth patterns _ , edited by j. m. garcia - ruiz _ et al_. ( plenum , new york , 1993 ) . see , e.g. , j. d. murray , _ mathematical biology _ ( springer , berlin , 1988 ) . p. grassberger , j. phys . a * 18 * , l215 ( 1985 ) . k. sneppen and m. h. jensen , phys . lett . * 70 * , 3833 ( 1993 ) ; * 71 * , 101 ( 1993 ) ; phys . rev . e * 49 * , 919 ( 1994 ) ; p. bak and k. sneppen , phys . lett . * 71 * , 4083 ( 1993 ) ; m. paczuski , s. maslov , and p. bak , europhys . lett . * 27 * , 97 ( 1994 ) . s. redner , j. phys . a : math . * 14 * , l349 ( 1981 ) ; phys . b * 25 * , 3242 ( 1982 ) . s. redner , in _ percolation structures and processes _ , edited by g. deutscher _ et al_. ( adam hilger , bristol , 1983 ) . rdns were already contained implicitly in the pioneering work of s. r. broadbent and j. m. hammersley on percolation , proc . * 53 * , 629 ( 1957 ) . h. k. janssen and o. stenull , phys . rev . e * 62 * , 3173 ( 2000 ) . h. k. janssen and o. stenull , phys . e * 63 * , 025103(r ) ( 2001 ) . o. stenull and h. k. janssen , to appear in j. stat . see also s. w. kenkel and j. p. straley , phys . * 49 * , 767 ( 1982 ) . r. blumenfeld and a. aharony , j. phys . a * 18 * , l443 ( 1985 ) . m. j. stephen , phys . b * 17 * , 4444 ( 1978 ) . a. b. harris , phys . b * 35 * , 5056 ( 1987 ) . see , e.g. , a. b. harris and t. c. lubensky , phys . b * 35 * , 6964 ( 1987 ) . j. l. cardy and r. l. sugar , j. phys . a * 13 * , l423 ( 1980 ) . h. k. janssen , z. phys . b * 42 * , 151 ( 1981 ) . h. k. janssen , j. stat . phys . * 103 * , 801 ( 2001 ) . h. k. janssen , z. phys . b * 23 * , 377 ( 1976 ) ; r. bausch , h. k. janssen , and h. wagner , _ ibid_. * 24 * , 113 ( 1976 ) ; h. k. janssen , in _ dynamical critical phenomena and related topics _ , edited by c. p. enz , lecture notes in physics , vol . 104 ( springer , heidelberg , 1979 ) . c. de dominicis , j. phys . ( paris ) colloq . * 37 * , c-247 ( 1976 ) ; c. de dominicis and l. peliti , phys . b * 18 * , 353 ( 1978 ) . h. k. janssen , in _ from phase transitions to chaos _ , edited by g. gyrgyi , i. kondor , l. sasvri , and t. tl ( world scientific , singapore , 1992 ) . see , e.g. , d. j. amit , _ field theory , the renormalization group , and critical phenomena _ ( world scientific , singapore , 1984 ) ; j. zinn - justin , _ quantum field theory and critical phenomena _ ( clarendon , oxford , 1989 ) . o. stenull , h. k. janssen , and k. oerding , phys . e * 59 * , 4919 ( 1999 ) . h. k. janssen , o. stenull , and k. oerding , phys . e * 59 * , r6239 ( 1999 ) . h. k. janssen and o. stenull , phys . e * 61 * , 4821 ( 2000 ) . o. stenull , _ renormalized field theory of random resistor networks _ , ph.d . thesis , universitt dsseldorf , ( shaker , aachen , 2000 ) . o. stenull and h. k. janssen , europhys . * 51 * , 539 ( 2000 ) . see , e.g. , j. kertsz , and t. vicsek , in _ fractals in science _ , edited by a. bunde and s. havlin ( springer , berlin , 1994 ) . a. coniglio , phys . * 46 * , 250 ( 1981 ) ; j. phys . a * 15 * , 3829 ( 1982 ) . see , e.g. , b. hede , j. kertsz , and t. vicsek , j. stat . * 64 * , 829 ( 1991 ) . b. m. arora , m. barma , d. dhar , and m. k. phani , j. phys . c * 16 * , 2913 ( 1983 ) . g. huber , m. h. jensen , and k. sneppen , phys . e * 52 * , r2133 ( 1995 ) .

we study nonlinear random resistor diode networks at the transition from the non percolating to the directed percolating phase . the resistor - like bonds and the diode - like bonds under forward bias voltage obey a generalized ohm s law , @xmath0 . based on general grounds as symmetries and relevance we develop a field theoretic model . we focus on the average two - port resistance , which is governed at the transition by the resistance exponent @xmath1 . by employing renormalization group methods we calculate @xmath1 for arbitrary @xmath2 to one - loop order . then we address the fractal dimensions characterizing directed percolation clusters . via considering distinct values of the nonlinearity @xmath2 , we determine the dimension of the red bonds , the chemical path and the backbone to two - loop order . # 1@xmath3#1

the environments of quasars provide important clues to the physical processes of their formation and also yield important information about the relations between the distribution of quasars and the large - scale structure of the universe . for more than three decades , we have known that quasars are associated with enhancements in the spatial distributions of galaxies ( @xcite ) . studies of the environments of quasars in the nearby universe ( @xmath2 ) have shown that quasars reside in environments ranging from small to moderate groups of galaxies rather than in rich clusters ( e.g. @xcite ; @xcite ; @xcite ) . in order to interpret the observational results of the environments of quasars at low redshifts and predict the environments of quasars at high redshifts , a physical model of quasar formation based on cosmological context is required . it has become widely accepted that quasars are fueled by accretion of gas onto supermassive black holes ( smbhs ) in the nuclei of host galaxies since @xcite proposed this idea on quasars . recent observations of galactic centers suggest that a lot of nearby galaxies have central black holes and their estimated masses correlate with the luminosities of spheroids of their host galaxies ( e.g. @xcite ; @xcite ; @xcite ) . the connection between smbhs and their host spheroids suggests that the formation of smbhs physically links the formation of the spheroids which harbor the smbhs . thus , this implies that the formation of quasars is closely related to the formation of galaxies , especially of spheroids . therefore , in order to study the formation and evolution of quasars , it is necessary to construct a unified model which includes both galaxy formation and quasar formation . recently , some authors have tried to construct galaxy formation models on the basis of the theory of hierarchical structure formation in cold dark matter ( cdm ) universe . these efforts are referred to as semi - analytic models ( sams ) of galaxy formation . in the cdm universe , dark matter halos cluster gravitationally and merge together in a manner that depends on the adopted power spectrum of initial density fluctuations . in each of the merged dark halos , radiative gas cooling , star formation , and supernova feedback occur . the cooled dense gas and stars constitute _ galaxies_. these galaxies sometimes merge together in a common dark halo and more massive galaxies form . in sams , the merger trees of dark matter halos are constructed using a monte - carlo algorithm and simple models are adopted to describe the above gas processes . stellar population synthesis models are used to calculate the luminosities and colors of model galaxies . it is therefore straightforward to understand how galaxies form and evolve within the context of this model . sams successfully have reproduced a variety of observed features of local galaxies such as their luminosity functions , color distribution , and so on ( e.g. @xcite ; @xcite , ; @xcite ; @xcite , ) . in these models , it is assumed that disk stars are formed by cooling of gas in the halo . if two galaxies of comparable mass merge , it is assumed that starbursts occur and form the spheroidal component in the center of the galaxy . @xmath3-body simulations have shown that a merger hypothesis for the origin of spheroids can explain their detailed internal structure ( e.g. @xcite ; @xcite , ; @xcite ) . kauffmann and charlot ( ) have demonstrated that the merger scenario for the formation of elliptical galaxies is consistent with the color - magnitude relation and its redshift evolution ( see also @xcite ) . on the other hand , hydrodynamical simulations have shown that a merger of galaxies drives gas to fall rapidly to the center of a merged system and to fuel nuclear starburst ( @xcite ; @xcite , ; @xcite ) . moreover , observed images of quasar hosts show that many quasars reside in interacting systems or elliptical galaxies ( @xcite ) . therefore , it has often been thought that the major merger of galaxies would be a possible mechanism for quasar and spheroid formation . so far , a lot of studies on quasar evolution based on the hierarchical clustering scenario have been carried out with the assumption that the formation of quasars is linked to the first collapse of dark matter halos with galactic mass and that these models can explain the decline of quasar number density at @xmath4 ( e.g. @xcite ; @xcite ) and properties of luminosity functions of quasars ( e.g. @xcite ; @xcite ; @xcite ) . however , if quasars are directly linked to spheroids of host galaxies rather than to dark matter halos , the approximation of a one - to - one relation between quasar hosts and dark matter halos would be very crude , especially at low redshift . therefore , it is necessary to construct a model related to spheroid formation and smbh formation directly . kauffmann and haehnelt ( ) introduced a unified model of the evolution of galaxies and quasars within the framework of sam ( see also @xcite ) . they assumed that smbhs are formed and fueled during major galaxy mergers and their model reproduces quantitatively the observed relation between spheroid luminosity and black hole mass in nearby galaxies , the strong evolution of the quasar population with redshift , and the relation between the luminosities of nearby quasars and those of their host galaxies . in this paper , we investigate properties of quasar environments , using a sam incorporated simple quasar evolution model . we assume that smbhs are formed and fueled during major galaxy mergers and the fueling process leads quasar activity . while this assumption is similar to the model of kauffmann and haehnelt ( ) , our galaxy formation model and the adopted model of fueling process are different from their model . here we focus on optical properties of quasars and attempt to consider the number of quasars per halo , effective bias parameter of quasars and the number of galaxies around quasars as characterizations of environments of quasars , because a ) these quantities provide a direct measure of bias in their distribution with respect to galaxies and b ) comparing results of the model with observations will enable us to constrain our quasar formation model . the paper is organized as follows : in [ model ] we briefly review our sam for galaxy formation ; in [ qsomodel ] we introduce the quasar formation model ; in [ env ] we calculate the galaxy number distribution function around quasars ; in [ disc ] we provide a summary and discussion . in this study , we use a low - density , spatially flat cold dark matter ( @xmath5cdm ) universe with the present density parameter @xmath6 , the cosmological constant @xmath7 , the hubble constant in units of @xmath8 @xmath9 and the present rms density fluctuation in spheres of @xmath10 radius @xmath11 . in this section we briefly describe our sam for the galaxy formation model , details of which are shown in @xcite . our present sam analysis obtains essentially the same results as the previous sam analyses , with minor differences in a number of details . first , we construct monte carlo realizations of merging histories of dark matter halos using the method of @xcite , which is based on the extended press - schechter formalism ( @xcite ; @xcite ; @xcite ; @xcite ) . we adopt the power spectrum for the specific cosmological model from @xcite . halos with circular velocity @xmath1240 km s@xmath13 are treated as diffuse accretion matter . the evolution of the baryonic component is followed until the output redshift coincides with the redshift interval of @xmath14 , corresponding to the dynamical time scale of halos which collapse at the redshift @xmath15 . note that @xcite recently pointed out that a much shorter timestep is required to correctly reproduce the mass function given by the press - schechter formalism . however , a serious problem exists only at small mass scales ( @xmath16 ) . thus we use the above prescription of timestep . if a dark matter halo has no progenitor halos , the mass fraction of the gas in the halo is given by @xmath17 , where @xmath18 is the baryonic density parameter constrained by primordial nucleosynthesis calculations ( e.g. @xcite ) . note that a recent measurement of the anisotropy of the cosmic microwave background by the boomerang project suggests a slight higher value , @xmath19 ( @xcite ) . @xcite have already investigated the effect of changing @xmath20 and showed that this mainly affects the value of the invisible stellar mass fraction such as brown dwarfs parameterized by @xmath21 ( see below ) . when a dark matter halo collapses , the gas in the halo is shock - heated to the virial temperature of the halo . we refer to this heated gas as the _ hot gas_. at the same time , the gas in dense regions of the halo cools due to efficient radiative cooling . we call this cooled gas the _ cold gas_. assuming a singular isothermal density distribution of the hot gas and using the metallicity - dependent cooling function by @xcite , we calculate the amount of cold gas which eventually falls onto a central galaxy in the halo . in order to avoid the formation of unphysically large galaxies , the above cooling process is applied only to halos with @xmath12400 km s@xmath13 . this handling would be needed because the simple isothermal distribution forms so - called `` monster galaxies '' due to the too efficient cooling at the center of halos . while this problem will probably solved by adopting another isothermal distribution with central core ( @xcite ) , we take the above approach for simplicity . stars are formed from the cold gas at a rate of @xmath22 , where @xmath23 is the mass of cold gas and @xmath24 is the time scale of star formation . we assume that @xmath24 is independent of @xmath15 , but dependent on @xmath25 as follows : @xmath26 the free parameters of @xmath27 and @xmath28 are fixed by matching the observed mass fraction of cold gas in neutral form in the disks of spiral galaxies . in our sam , stars with masses larger than @xmath29 explode as type ii supernovae ( sne ) and heat up the surrounding cold gas . this sn feedback reheats the cold gas to hot gas at a rate of @xmath30 , where @xmath31 is the efficiency of reheating . we assume that @xmath31 depends on @xmath25 as follows : @xmath32 the free parameters of @xmath33 and @xmath34 are determined by matching the local luminosity function of galaxies . with these @xmath35 and @xmath36 thus determined , we obtain the masses of hot gas , cold gas , and disk stars as a function of time during the evolution of galaxies . given the star formation rate as a function of time , the absolute luminosity and colors of individual galaxies are calculated using a population synthesis code by @xcite . the initial stellar mass function ( imf ) that we adopt is the power - law imf of salpeter form with lower and upper mass limits of @xmath37m@xmath38 and @xmath39m@xmath38 , respectively . since our knowledge of the lower mass limit is incomplete , there is the possibility that many brown dwarf - like objects are formed . therefore , following @xcite , we introduce a parameter defined as @xmath40 , where @xmath41 is the total mass of luminous stars with @xmath42 and @xmath43 is that of invisible brown dwarfs . to account for extinction by internal dust we adopt a simple model by @xcite in which the optical depth in @xmath44-band is related to the luminosity as @xmath45 . optical depths in other bands are calculated by using the galactic extinction curve , and the dust distribution in disks is assumed to be the slab model considered by @xcite . when several progenitor halos have merged , the newly formed larger halo should contain at least two or more galaxies which had originally resided in the individual progenitor halos . we identify the central galaxy in the new common halo with the central galaxy contained in the most massive of the progenitor halos . other galaxies are regarded as satellite galaxies . these satellites merge by either dynamical friction or random collision . the time scale of merging by dynamical friction is given by @xmath46 where @xmath47 and @xmath25 are the radius and the circular velocity of the new common halo , respectively , @xmath48 is the coulomb logarithm , and @xmath49 is the mass of the satellite galaxy including its dark matter halo @xcite . when the time passed after a galaxy becomes a satellite exceeds @xmath50 , a satellite galaxy infalls onto the central galaxy . on the other hand , the mean free time scale of random collision is given by @xmath51 where @xmath3 is the number of satellite galaxies , @xmath52 is their radius , and @xmath53 and @xmath54 are the 1d velocity dispersions of the common halo and satellite galaxies , respectively @xcite . with a probability of @xmath55 , where @xmath56 is the timestep corresponding to the redshift interval @xmath57 , a satellite galaxy merges with another randomly picked satellite . consider the case that two galaxies of masses @xmath58 and @xmath59 merge together . if the mass ratio @xmath60 is larger than a certain critical value of @xmath61 , we assume that a starburst occurs and all the cold gas turns into stars and hot gas , which fills the halo , and all of the stars populate the bulge of a new galaxy . on the other hand , if @xmath62 , no starburst occurs and a smaller galaxy is simply absorbed into the disk of a larger galaxy . these processes are repeated until the output redshift . we classify galaxies into different morphological types according to the @xmath44-band bulge - to - disk luminosity ratio @xmath63 . in this paper , galaxies with @xmath64 , and @xmath65 are classified as ellipticals / s0s and spirals , respectively . this method of type classification well reproduces the observed type mix . the above procedure is a standard one in the sam for galaxy formation . model parameters are determined by comparison with observations of the local universe . in this study , we use the astrophysical parameters determined by @xcite from local observations such as luminosity functions , and galaxy number counts in the hubble deep field . the adopted parameters of this model are tabulated in table [ tab : astro ] . in figure [ fig : gal - lum ] we plot the results of local luminosity functions of galaxies represented by solid lines . note that the resultant luminosity functions hardly change if the smbh formation model is included ( dashed lines ; see the next section ) . symbols with errorbars indicate observational results from the @xmath44-band redshift surveys ( apm , @xcite ; 2df , @xcite ) and from the @xmath66-band redshift surveys ( @xcite ; 2mass , @xcite ) . as can be seen , the results of our model using these parameters are generally consistent with the observations , at least with the apm result . [ tab : astro ] .model parameters [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^ " , ] ( 120mm,80mm)fig1.eps in this section , we introduce a quasar formation and evolution model into our sam . as mentioned earlier , the masses of smbhs have tight correlation with the spheroid masses of their host galaxies ( e.g. @xcite ; @xcite ; @xcite ) and the hosts of quasars found in the local universe are giant elliptical galaxies or galaxies displaying evidence of major mergers of galaxies ( @xcite ) . moreover , in sams for galaxy formation , it is assumed that a galaxy - galaxy major merger leads to the formation of a spheroid . therefore , we assume that smbhs grow by merging and are fueled by accreted cold gas during major mergers of galaxies . when host galaxies merge , pre - existing smbhs sink to the center of the new merged galaxy owing to dynamical friction and finally coalesce . the timescale for this process is unknown , but for the sake of simplicity we assume that smbhs merge instantaneously . gas - dynamical simulations have demonstrated that the major merger of galaxies can drive substantial gaseous inflows and trigger starburst activity ( @xcite ; @xcite , ; @xcite ) . thus , we assume that during major merger , some fraction of the cold gas that is proportional to the total mass of stars newly formed at starburst is accreted onto the newly formed smbh . under this assumption , the mass of cold gas accreted on a smbh is given by @xmath67 where @xmath68 is a constant and @xmath69 is the total mass of stars formed at starburst . @xmath69 is derived in the appendix . the free parameter of @xmath68 is fixed by matching the observed relation between a spheroid luminosity and a black hole mass found by @xcite and we find that the favorable value of @xmath68 is nearly @xmath70 . in figure [ fig : bulge - bh ] we show scatterplots ( open circles ) of the absolute @xmath71-band magnitudes of spheroids versus masses of smbhs of model for @xmath72 . the thick solid line is the observational relation and the dashed lines are the @xmath73 scatter in the observations obtained by @xcite . for @xmath74 , changing @xmath68 shifts the black hole mass almost linearly . the obtained gas fraction ( @xmath75 ) is so small that the inclusion of smbh formation does not change the properties of galaxies in the local universe . in figure [ fig : gal - lum ] , the dashed lines show the results of the model with the smbh formation . this result differs negligibly from the result of the model without smbh formation . therefore , we use the same astrophysical parameters tabulated in table [ tab : astro ] regardless of inclusion of the smbh formation model . figure [ fig : bh - mass ] ( a ) shows black hole mass functions in our model at a series of redshifts . this indicates that the number density of the most massive black holes increases monotonically with time in the scenario where smbhs grow by accretion of gas and by merging . in figure [ fig : bh - mass ] ( a ) , we superpose the present black hole mass function obtained by @xcite . they derived this black hole mass function from the observed radio luminosity function of nearby radio - quiet galaxies and the empirical correlation between radio luminosities from the nuclei of radio - quiet galaxies and the mass of their black holes . our model result is consistent with their mass function . for comparison , we also plot the mass functions of bulge and disk for all galaxies in figure [ fig : bh - mass ] ( b ) and ( c ) , respectively . the steep slopes at low masses of mass functions of black hole and bulge are mainly due to random collisions between satellite galaxies in this model . to obtain the observed linear relation between a spheroid luminosity and a black hole mass , kauffmann and haehnelt ( ) adopted model of fueling process in which the ratio of accreted mass to total available cold gas mass scales with halo circular velocity in the same way as the mass of stars formed per unit mass of cooling gas . while their approach is similar to ours , their star formation and feedback models are different from ours and they do not consider random collisions . therefore , their resultant model description is slightly differ from ours in equation ( [ eq : bhaccret ] ) . ( 70mm,70mm)fig2.eps ( 120mm,80mm)fig3.eps next , we consider the light curve of quasars . we assume that a fixed fraction of the rest mass energy of the accreted gas is radiated in the @xmath44-band and the quasar life timescale @xmath76 scales with the dynamical time scale @xmath77 of the host galaxy where @xmath78 . here we adopt the @xmath44-band luminosity of a quasar at time @xmath79 after the major merger as follows ; @xmath80 the peak luminosity @xmath81 is given by @xmath82 where @xmath83 is the radiative efficiency in @xmath44-band , @xmath84 is the quasar life timescale and @xmath85 is the speed of light . in order to determine the parameter @xmath83 and the present quasar life timescale @xmath86 , we have chosen them to match our model luminosity function with the observed abundance of bright quasars at @xmath87 . we obtain @xmath88 and @xmath89 . the resulting luminosity functions at four different redshifts are shown in figure [ fig : qso - lum ] . we superpose the luminosity functions derived from the 2df 10k catalogue ( @xcite ) for a cosmology with @xmath90 and @xmath9 , which is analyzed and kindly provided by t. t. takeuchi . he used the method of @xcite for the estimation of the luminosity functions . in order to reanalyze the error with greater accuracy , they applied bootstrap resampling according to the method of @xcite . absolute @xmath44-band magnitudes were derived for the quasars using the @xmath91-corrections derived by @xcite . our model reproduces reasonably well the evolution of observed luminosity functions . thus , in the next section , we use these model parameters in order to investigate the environments of quasars . for comparison , we also plot the result of model with @xmath88 and @xmath92 in figure [ fig : qso - lum ] ( dot - dashed lines ) . in this case , the abundance of luminous quasars decreases . to prolong a quasar life timescale affects the quasar luminosity function due to the following two factors : a decrease in the peak luminosity @xmath93 ( eq.[[eq : qso - peak ] ] ) and an increase in the exponential factor @xmath94 in equation ( [ eq : qso - lc ] ) . for the majority of bright quasars , the elapsed time @xmath79 since the major merger is much smaller than the quasar life timescale @xmath95 , @xmath96 . therefore , the former factor dominates the latter and the number of luminous quasars decreases . thus , a long quasar life timescale results in a very steep quasar luminosity function . note that if we change the radiative efficiency @xmath83 , the quasar luminosities simply scale by a constant factor in our model . thus , changing @xmath83 shifts the luminosity function horizontally . in this section , we investigate the environments of quasars using our model . we consider the halo mass dependence of the mean number of quasars per halo and the probability distribution of the number of galaxies around quasars as characterizations of the environments of quasars . this is because the former is one of measures of the relation between quasars and dark matter distributions and the latter reflects the relationship between galaxies and quasars . in figure [ fig : number - gal ] , we plot @xmath97 and @xmath98 that denote the mean number of galaxies and quasars per halo with mass @xmath99 , respectively , at ( a ) @xmath100 and ( b ) @xmath101 . we select galaxies with @xmath102 and quasars with @xmath103 , where @xmath104 is absolute @xmath44-band magnitude . it should be noted that changing the magnitude of selection criteria for galaxies and quasars would alter these results , but qualitative features are not altered . as is seen in figure [ fig : number - gal ] , there are more galaxies and quasars at high @xmath15 . at higher redshift , halos have more cold gas available to form stars and to fuel smbhs because there has been relatively little time for star formation to deplete the cold gas at these redshifts . thus , the number of luminous galaxies grows . furthermore , at higher redshift , both timescales of the dynamical friction and the random collisions are shorter because the mass density of a halo is higher . therefore , the galaxy merging rate increases . consequently , the number of quasars also grows . moreover , the decrease in the quasar life timescale @xmath105 with redshift also contributes to the increase in the number of quasars because quasars become brighter as a result of decrease in @xmath105 ( eq . [ [ eq : qso - peak ] ] ) . from figure [ fig : number - gal ] , we find that the dependence of @xmath106 on halo mass @xmath99 is different from the dependence of @xmath97 . furthermore , figure [ fig : qg - ratio ] shows that the ratio of @xmath107 to @xmath108 varies with redshift and halo mass . @xcite used a combination of cosmological @xmath3-body simulation and semi - analytic modeling of galaxy formation and showed that the galaxy spatial distribution is sensitive to the efficiency with which galaxies form in halos with different mass . @xcite also obtained the same conclusion using an analytic model of galaxy clustering . these results are applicable to the quasar spatial distribution . therefore , our result indicates that the clustering properties of galaxies are not the same as those of quasars and that the bias in the spatial distribution of galaxies relative to that of dark matter is not the same as the bias in the spatial distribution of quasars . assumed that biases are independent of scale , we can calculate effective biases using the method of @xcite as follows ; @xmath109 where @xmath110 is the bias parameter for dark matter halos of mass @xmath99 at @xmath15 , @xmath111 denotes the mean number of objects ( galaxies or quasars ) in a halo of mass @xmath99 at @xmath15 that satisfy the selection criteria and @xmath112 is the dark halo mass function at @xmath15 . our sam adopts the press - schechter mass function which is given by @xmath113 dm , \label{eq : psmass}\ ] ] where @xmath114 is the present mean density of the universe , @xmath115 is the rms linear density fluctuation on the scale @xmath99 at @xmath116 and @xmath117 . @xmath118 is the linear growth factor , normalized to unity at the present day and @xmath119 is the linear critical density contrast at the collapse epoch . here , we use an approximate formula of @xmath120 for spatially flat cosmological model ( @xcite ) . the bias parameter for dark matter halos is given by @xcite ; @xmath121 \right\ } \left [ \frac{\sigma^{4}(m)}{2 \delta^{4}_{c}(z ) } + 1 \right]^{(0.06 - 0.02n_{\rm eff } ) } , \label{eq : bias}\ ] ] where @xmath122 is the effective spectral index of the power spectrum , @xmath123 , at the wavenumber defined by the lagrangian radius of the dark matter halo , @xmath124 and @xmath125 . figure [ fig : bias ] shows the evolution of effective bias for galaxies with @xmath102 and quasars with @xmath103 . as is seen in figure [ fig : bias ] , quasars are higher biased tracer than galaxies . furthermore , the evolution of quasar bias is different from that of galaxy bias . this reflects the difference in th dependence on halo mass @xmath99 and redshift of @xmath126 and @xmath127 . note that these effective biases are valid for large scale where objects ( galaxies or quasars ) which contribute two - point correlation function populate different halos . ( 120mm,80mm)fig5.eps ( 70mm,70mm)fig6.eps ( 70mm,70mm)fig7.eps next , we formulate the conditional probability that a halo with @xmath128 quasars has @xmath129 galaxies . the number density of the halos which contains @xmath129 galaxies and @xmath130 quasars at @xmath15 is obtained from the following expression : @xmath131 where @xmath132 denotes the number of the halos with mass @xmath99 which contains @xmath133 galaxies and @xmath134 quasars at @xmath15 and @xmath112 is the dark halo mass function at @xmath15 . the number density of the halos which contain @xmath130 quasars at @xmath15 is obtained from the following expression : @xmath135 where @xmath136 denotes the number of the halos with mass @xmath99 which contain @xmath137 quasars at @xmath15 . from equation ( [ eq : ng - q ] ) and ( [ eq : nq ] ) , the conditional probability that the halo with @xmath130 quasars has @xmath138 galaxies at @xmath15 is given by @xmath139 as is seen in the above formulation , given @xmath140 and @xmath141 from the quasar formation model , one can calculate the probability distribution for the number of galaxies around quasars . figure [ fig : gnd ] shows these galaxy number distribution functions around quasars estimated by our model . the results are shown for quasars brighter than @xmath142 and for galaxies brighter than @xmath143 . note that at @xmath144 and @xmath100 @xmath145 and @xmath146 for all @xmath129 ( fig . [ fig : gnd](a ) and ( b ) ) and that at @xmath147 @xmath146 for all @xmath129 ( fig . [ fig : gnd](c ) ) . at lower redshift , a halo has at most one quasar . fig [ fig : gnd](a ) and ( b ) show that the halo which has one quasar contains several galaxies by high probability . these results indicate that most quasars tend to reside in groups of galaxies at @xmath0 and is consistent with the observation at @xmath148 ( e.g. @xcite ; @xcite ; @xcite ) . on the other hand , at higher redshift , the numbers of galaxies in the halo with one or two quasars is from several to dozens ( fig [ fig : gnd](c ) and ( d ) ) . these results indicate that quasars locate in ranging from small groups of galaxies to clusters of galaxies . thus at @xmath1 quasars seem to reside in more varied environments than at lower redshift . kauffmann and haehnelt ( ) used a combination of cosmological @xmath3-body simulation and semi - analytic modeling of galaxy and quasar formation , and showed that the ratio of the amplitude of the quasar - galaxy cross correlation function to that of the galaxy autocorrelation function decrease with redshift . this indicates that the difference between galaxy and quasar distribution becomes smaller at higher redshift . thus , our results obtained by @xmath149 is not in conflict with their results . we have constructed a unified semi - analytic model for galaxy and quasar formation and have predicted the mean number of quasars per halo with mass @xmath99 , @xmath107 , the effective bias parameter of quasars @xmath150 and probability distribution of the number of galaxies around quasars , @xmath151 , as characterizations of the environments of quasars . these quantities reflect the processes of quasar formation such as the amount of cold gas available for fueling , the galaxy merger rate and the quasar life timescale . therefore , by comparing these predictions with observations , one will be able to constrain quasar formation models . our model can reproduce not only general form of the galaxy luminosity functions in the local universe but also the observed relation of the smbh mass to spheroid luminosity , and the quasar luminosity functions at different redshifts ( fig.[fig : bulge - bh ] and fig.[fig : qso - lum ] ) . using this model , we have shown @xmath107 and @xmath151 . the ratio of @xmath107 to @xmath97 varies with halo mass in our model ( fig[fig : number - gal ] ) . these results of our model suggest that the clustering of galaxies is not the same as the clustering of quasars and the effective bias parameter of quasars and its evolution are different from these of galaxies ( fig.[fig : bias ] ) . furthermore , we predict the galaxy number distribution function around quasars , @xmath151 ( fig[fig : gnd ] ) . at lower redshifts ( @xmath0 ) , most halos which have quasars have at most several galaxies . this indicates that most quasars reside in groups of galaxies . on the other hand , at higher redshift ( @xmath152 ) , the number of galaxies in the halo with quasars is from several to dozens ; quasars reside in ranging from small groups of galaxies to clusters of galaxies . these results show that most quasars at higher redshift reside in more varied environments than at lower redshift . this model prediction is checkable by statistics of galaxies around quasars which will be obtained in future . it is still controversial whether the environments of quasars depend on their optical and radio luminosities . some authors have claimed that radio - loud quasars were located in richer environments than radio - quiet quasars at at @xmath153 ( e.g. @xcite ; @xcite ; @xcite ; @xcite ) . however , other people obtained a different result . for example , @xcite observed the galaxy environment of radio - loud quasars and radio - quiet quasars and concluded that there is no significant difference in the richness . recent studies support this conclusion ( e.g. @xcite ) . the discrepancies between different studies may be caused partly by too small quasar samples and by differences in sample selection of quasars . this situation will soon improve with the availability of a new generation of very large quasar surveys such as the 2df quasar redshift survey ( @xcite ) and the sloan digital sky survey ( @xcite ) . although we do not deal with radio properties of quasars in this paper , our investigation of quasar environments will also provide a clue for understanding the radio character of quasar environments . the mean number of quasars per halo , @xmath107 , and probability distribution of the number of galaxies around quasars , @xmath151 , used in this study can provide some useful features of the quasar environments . furthermore , the spatial galaxy - quasar correlation function is used in order to quantify the galaxy environments around a quasar . therefore , for the further investigation of environments and clustering of quasars and in order to constrain the quasar formation model , it is also necessary to predict spatial distribution of galaxies and quasars . we will show the results using the combination of cosmological @xmath3-body simulation and sam for formation of galaxy and quasar in the near future . we would like to thank t. t. takeuchi for providing us with the reanalyzed data of the quasar luminosity functions derived from the 2df 10k catalogue . we are also grateful to k. okoshi , h. yahagi and s. yoshioka for useful comments and discussions . we also thank to the anonymous referee for a thorough reading of the manuscript and for his valuable suggestions and comments , which improved our paper very much . numerical computations in this work were partly carried out at the astronomical data analysis center of the national astronomical observatory , japan . this work has been supported in part by the grant - in - aid for the scientific research funds ( 13640249 ) of the ministry of education , culture , sports , science and technology of japan . in this appendix , we summarize our model of star formation and gas evolution . we use a simple instantaneous recycling approximation of model star formation , feedback and chemical enrichment . the following difference equations describe the evolution of the mass of cold gas @xmath23 , hot gas @xmath154 , and long lived stars @xmath155 at each time step . @xmath156 where @xmath157 is star formation rate , @xmath158 is the gas fraction returned by evolved stars , and @xmath31 is the efficiency of reheating . in this paper , @xmath159 . the solutions of these equations are the following : @xmath160 , \label{eq : coldsol } \\ m_{\rm hot } & = & m_{\rm hot}^{0 } + \beta \delta m _ { * } , \label{eq : hotsol } \\ m_{\rm star } & = & m_{\rm star}^{0 } + ( 1-r)\delta m _ { * } , \label{eq : starsol } \end{aligned}\ ] ] where @xmath161 and @xmath162 are the masses of cold gas , hot gas and long - lived stars from the previous time step , @xmath79 is the time sine the start of the time step , and @xmath163 is the mass of total formed stars . when a starburst occurs , stars are formed in a very short timescale . thus , the starburst corresponds to @xmath164 in the above solutions . in this case , the changes of masses are given by @xmath165 and the total star mass formed at starburst becomes @xmath166 from equation ( [ eq : totstar ] ) , we can obtain the mass of accreted cold gas onto a black hole ( eq.[[eq : bhaccret ] ] ) .

we investigate the environments of quasars such as number distribution of galaxies using a semi - analytic model which includes both galaxy and quasar formations based on the hierarchical clustering scenario . we assume that a supermassive black hole is fueled by accretion of cold gas and that it is a source of quasar activity during a major merger of the quasar host galaxy with another galaxy . this major merger causes spheroid formation of the host galaxy . our model can reproduce not only general form of the galaxy luminosity functions in the local universe but also the observed relation between a supermassive black hole mass and a spheroid luminosity , the present black hole mass function and the quasar luminosity functions at different redshifts . using this model , we predict the mean number of quasars per halo , bias parameter of quasars and the probability distribution of the number of galaxies around quasars . in our model , analysis of the mean number of quasars per halo shows that the spatial distribution of galaxies is different from that of quasars . furthermore , we found from calculation of the probability distribution of galaxy numbers that at @xmath0 , most quasars are likely to reside in galaxy groups . on the other hand , at @xmath1 most quasars seem to reside in more varied environments than at a lower redshift ; quasars reside in environments ranging from small groups of galaxies to clusters of galaxies . comparing these predictions with observations in future will enable us to constrain our quasar formation model .

kullback - leibler ( kl ) divergence ( relative entropy ) can be considered as a measure of the difference / dissimilarity between sources . estimating kl divergence from finite realizations of a stochastic process with unknown memory is a long - standing problem , with interesting mathematical aspects and useful applications to automatic categorization of symbolic sequences . namely , an empirical estimation of the divergence can be used to classify sequences ( for approaches to this problem using other methods , in particular true metric distances , see @xcite , @xcite ; see also @xcite ) . in @xcite ziv and merhav showed how to estimate the kl divergence between two sources , using the parsing scheme of lz77 algorithm @xcite on two finite length realizations . they proved the consistence of the method by showing that the estimate of the divergence for two markovian sources converges to their relative entropy when the length of the sequences diverges . furthermore they proposed this estimator as a tool for an `` universal classification '' of sequences . a procedure based on the implementations of lz77 algorithm ( gzip , winzip ) is proposed in @xcite . the estimate obtained of the relative entropy is then used to construct phylogenetic trees for languages and is proposed as a tool to solve authorship attribution problems . moreover , the relation between the relative entropy and the estimate given by this procedure is analyzed in @xcite . two different algorithms are proposed and analyzed in @xcite , see also @xcite . the first one is based on the burrows - wheeler block sorting transform @xcite , while the other uses the context tree weighting method . the authors proved the consistence of these approximation methods and show that these methods outperform the others in experiments . in @xcite it is shown how to construct an entropy estimator for stationary ergodic stochastic sources using non - sequential recursive pairs substitutions method , introduced in @xcite ( see also @xcite and references therein for similar approaches ) . in this paper we want to discuss the use of similar techniques to construct an estimator of relative ( and cross ) entropy between a pair of stochastic sources . in particular we investigate how the asymptotic properties of concurrent pair substitutions might be used to construct an optimal ( in the sense of convergence ) relative entropy estimator . a second relevant question arises about the computational efficiency of the derived indicator . while here we address the first , mostly mathematical , question , we leave the computational and applicative aspects for forthcoming research . the paper is structured as follows : in section [ sec : notations ] we state the notations , in section [ sec : nsrps ] we describe the details of the non - sequential recursive pair substitutions ( nsrps ) method , in section [ sec : scaling ] we prove that nsrps preserve the cross and the relative entropy , in section [ sec : convergence ] we prove the main result : we can obtain an estimate of the relative entropy by calculating the 1-block relative entropy of the sequences we obtain using the nsrps method . we introduce here the main definitions and notations , often following the formalism used in @xcite . given a finite alphabet @xmath0 , we denote with @xmath1 the set of finite words . given a word @xmath2 , we denote by @xmath3 its length and if @xmath4 and @xmath5 , we use @xmath6 to indicate the subword @xmath7 . we use similar notations for one - sided infinite ( elements of @xmath8 ) or double infinite words ( elements of @xmath9 ) . often sequences will be seen as finite or infinite realizations of discrete - time stochastic stationary , ergodic processes of a random variable @xmath10 with values in @xmath0 . the @xmath11-th order joint distributions @xmath12 identify the process and its elements follow the consistency conditions : @xmath13 when no confusion will arise , the subscript @xmath11 will be omitted , and we will just use @xmath14 to denote both the measure of the cylinder and the probability of the finite word . equivalently , a distribution of a process can also be defined by specifying the initial one - character distribution @xmath15 and the successive conditional distributions : @xmath16 given an ergodic , stationary stochastic source we define as usual : @xmath17 @xmath18 where @xmath19 denotes the concatenated word @xmath20 and @xmath21 is just the process average . @xmath22 the following properties and results are very well known @xcite , but at the same time quite important for the proofs and the techniques developed here ( and also in @xcite ) : * @xmath23 * a process @xmath24 is @xmath25-markov if and only if @xmath26 . * _ entropy theorem _ : for almost all realizations of the process , we have @xmath27 in this paper we focus on properties involving pairs of stochastic sources on the same alphabet with distributions @xmath24 and @xmath28 , namely _ cross entropy _ and the related _ relative entropy _ ( or _ kullback leibler divergence _ ) : _ n - conditional cross entropy _ @xmath29 _ cross entropy _ @xmath30 _ relative entropy ( kullback - leibler divergence ) _ @xmath31 note that @xmath32 moreover we stress that , if @xmath28 is k - markov then , for any @xmath24 @xmath33 namely @xmath34 for any @xmath35 : @xmath36 & = - \sum_{\omega \in a^{l - k},\,b\in a^k,\,a\in a } \mu ( \omega ba ) \log \nu(a\vert b ) \\ & = - \sum_{b\in a^k,\,a\in a } \mu(ba ) \log \nu(a\vert b)= h_k(\mu\|\nu ) \end{array}\ ] ] note that @xmath37 depends only on the two - symbol distribution of @xmath24 . entropy and cross entropy can be related to the asymptotic behavior of properly defined _ returning times _ and _ waiting times _ , respectively . more precisely , given an ergodic , stationary process @xmath24 , a sample sequence @xmath38 and @xmath39 , we define the returning time of the first @xmath11 characters as : @xmath40 similarly , given two realizations @xmath41 and @xmath42 of @xmath24 and @xmath28 respectively , we define the @xmath43 obviously @xmath44 . we now have the following two important results : [ returning ] if @xmath24 is a stationary , ergodic process , then @xmath45 [ waiting ] if @xmath24 is stationary and ergodic , @xmath28 is k - markov and the marginals @xmath12 of @xmath24 are dominated by the corresponding marginals @xmath46 of @xmath28 , i.e. @xmath47 , then @xmath48 we now introduce a family of transformations on sequences and the corresponding operators on distributions : given @xmath49 ( including @xmath50 ) , @xmath51 and @xmath52 , a _ pair substitution _ is a map @xmath53 which substitutes sequentially , from left to right , the occurrences of @xmath54 with @xmath55 . for example @xmath56 or : @xmath57 @xmath58 is always an injective but not surjective map that can be immediately extended also to infinite sequences @xmath59 . the action of @xmath60 shorten the original sequence : we denote by @xmath61 the inverse of the contraction rate : @xmath62 for @xmath24-_typical _ sequences we can pass to the limit and define : @xmath63 an important remark is that if we start from a source where admissible words are described by constraints on consecutive symbols , this property will remain true even after an arbitrary pair substitution . in other words ( see theorem 2.1 in @xcite ) : a pair substitution maps pair constraints in pair constraints . a pair substitution @xmath64 naturally induces a map on the set of ergodic stationary measures on @xmath65 by mapping typical sequences w.r.t . the original measure @xmath24 in typical sequences w.r.t . the transformed measure @xmath66 : given @xmath67 then ( theorem 2.2 in @xcite ) @xmath68 exists and is constant @xmath24 almost everywhere in @xmath69 , moreover @xmath70 are the marginals of an ergodic measure on @xmath71 . again in @xcite , the following results are proved showing how entropies transform under the action of @xmath72 , with expanding factor @xmath73 : _ invariance of entropy _ @xmath74 _ decreasing of the 1-conditional entropy _ @xmath75 moreover , @xmath76 maps 1-markov measures in 1-markov measures . in fact : @xmath77 _ decreasing of the k - conditional entropy _ @xmath78 moreover @xmath76 maps @xmath25-markov measures in @xmath25-markov measures . while later on we will give another proof of the first fact , we remark that this property , together with the decrease of the 1-conditional entropy , reflect , roughly speaking , the fact that the amount of information of @xmath79 , which is equal to that of @xmath80 , is more concentrated on the pairs of consecutive symbols . as we are interested in sequences of recursive pair substitutions , we assume to start with an initial alphabet @xmath0 and define an increasing alphabet sequence @xmath81 , @xmath82 , @xmath83 , . given @xmath84 and chosen @xmath85 ( not necessarily different ) : * we indicate with @xmath86 a new symbol and define the new alphabet as @xmath87 ; * we denote with @xmath88 the substitution map @xmath89 which substitutes whit @xmath90 the occurrences of the pair @xmath91 in the strings on the alphabet @xmath92 ; * we denote with @xmath93 the corresponding map from the measures on @xmath94 to the measures on @xmath95 ; * we define by @xmath96 the corresponding normalization factor @xmath97 . we use the over - line to denote iterated quantities : @xmath98 and also @xmath99 the asymptotic properties of @xmath100 clearly depend on the pairs chosen in the substitutions . in particular , if at any step @xmath84 the chosen pair @xmath91 is the pair of maximum of frequency of @xmath101 then ( theorem 4.1 in @xcite ) : @xmath102 regarding the asymptotic properties of the entropy we have the following theorem that rigorously show that @xmath103 becomes asymptotically 1-markov : if @xmath102 then @xmath104 the main results of this paper is the generalization of this theorem to the cross and relative entropy . before entering in the details of our construction let us sketch here the main steps . in particular let us consider the cross entropy ( the same argument will apply to the relative entropy ) of the measure @xmath24 with respect to the measure @xmath28 : i.e. @xmath105 . as we will show , but for the normalization factor @xmath106 , this is equal to the cross entropy of the measure @xmath107 w.r.t the measure @xmath108 : @xmath109 moreover , as we have seen above , if we choose the substitution in a suitable way ( for instance if at any step we substitute the pair with maximum frequency ) then @xmath110 and the measure @xmath108 becomes asymptotically 1-markov as @xmath111 . interestingly , we do not know if @xmath112 also diverges ( we will discuss this point in the sequel ) . nevertheless , noticing that the cross entropy of a 1-markov source w.r.t a generic ergodic source is equal to the 1-markov cross entropy between the two sources , it is reasonable to expect that the cross entropy @xmath105 can be obtained as the following limit : @xmath113 this is exactly what we will prove in the two next sections . we first show how the relative entropy between two stochastic process @xmath24 and @xmath28 scales after acting with the _ same _ pair substitution on both sources to have @xmath66 and @xmath114 . more precisely we make use of theorem [ waiting ] and have the following : [ main1 ] if @xmath24 is ergodic , @xmath28 is a markov chain and @xmath47 , then if @xmath60 is a pair substitution @xmath115 _ proof . _ to fix the notations , let us denote by @xmath116 and @xmath117 the infinite realizations of the process of measure @xmath24 and @xmath28 respectively , and by @xmath118 and @xmath119 the corresponding finite substrings . let us denote by @xmath49 the characters involved in the pair substitution @xmath58 . moreover let us denote the waiting time with the shorter notation : @xmath120 we now explore how the waiting time rescale with respect to the transformation @xmath60 : we consider the first time we see the sequence @xmath121 inside the sequence @xmath122 . to start with , we assume that @xmath123 as we can always consider th . [ waiting ] for realizations with a fixed prefix of positive probability . moreover we choose a subsequence @xmath124 such that @xmath125 is the smallest @xmath126 such that @xmath127 . of course @xmath128 as @xmath129 . in this case , it is easy to observe that @xmath130 then , using theorem [ waiting ] @xmath131 = \nonumber\\ & = & \lim_{i\to + \infty } \frac{n_i}{|g(w_1^{n_i})|}\frac{\log|g(w_1^{t_{n_i}})|}{n_i}= \nonumber\\ & = & \lim_{i\to + \infty } \frac{n_i}{|g(w_1^{n_i})|}\left[\frac{1}{n_i}\log ( t_{n_i } ) + \frac{1}{n_i}\log\left(\frac{|g(w_1^{t_{n_i}})|}{t_{n_i}}\right)\right]=\nonumber\\ & = & z^{\mu } h(\mu\|\nu ) \label{kl1}\end{aligned}\ ] ] where in the last step we used the fact that @xmath132 as @xmath129 , the definition of @xmath133 and theorem [ waiting ] for @xmath24 and @xmath28 . note that for @xmath134 , equation ( [ kl1 ] ) reproduces the content of theorem 3.1 of @xcite : @xmath135 that thus implies @xmath136 note that the limit in th . [ waiting ] is almost surely unique and then the initial restrictive assumption @xmath137 and the use of the subsequence @xmath125 have no consequences on the thesis ; this concludes the proof . @xmath138 before discussing the convergence of relative entropy under successive substitutions we go thorough a simple explicit example of the theorem [ main1 ] , in order to show the difficulties we deal with , when we try to use the explicit expressions of the transformed measures we find in @xcite . _ example . _ we treat here the most simple case : @xmath24 and @xmath28 are bernoulli binary processes with parameters @xmath139 and @xmath140 respectively . we consider the substitution @xmath141 given by @xmath142 . it is long but easy to verify that @xmath66 is a stationary , ergodic , 1-markov with equilibrium state @xmath143 where @xmath144 . for example , given a @xmath66-generic sequence @xmath145 , corresponding to a @xmath24-generic sequence @xmath146 ( @xmath147 ) : @xmath148 clearly : @xmath149 using the same argument as before , it is now possible to write down the probability distribution of pair of characters for @xmath66 . again the following holds for a generic process : @xmath150 \frac{{\mathcal g}\mu(10)}z= \mu(10)-\mu(010 ) -\mu(101)+\mu(0101 ) & \frac{{\mathcal g}\mu(11)}z= \mu(11)-\mu(011 ) & \frac{{\mathcal g}\mu(12)}z= \mu(101)-\mu(0101)\\[4pt ] \frac{{\mathcal g}\mu(20)}z= \mu(010)-\mu(0101 ) & \frac{{\mathcal g}\mu(21)}z= \mu(011 ) & \frac{{\mathcal g}\mu(22)}z= \mu(0101 ) \end{array}\ ] ] it is easy to see that @xmath151 . now we can write the transition matrix @xmath152 for the process @xmath66 as @xmath153 : @xmath154 for bernoulli processes : @xmath155 we now denote with @xmath156 the transition matrix for @xmath157 . for the two 1-markov processes , we have @xmath158 via straightforward calculations , using the product structure of the measure @xmath24 : @xmath159\\ + z\mu(11)\left[\mu(00)\log\frac{\mu(00)}{\nu(00)}+\mu(1)\log\frac{\mu(1)}{\nu(1)}+\mu(01)\log\frac{\mu(01)}{\nu(01)}\right]\\ + z\mu(01)\left[\mu(00)\log\frac{\mu(00)}{\nu(00)}+\mu(1)\log\frac{\mu(1)}{\nu(1)}+\mu(01)\log\frac{\mu(01)}{\nu(01)}\right]\\ = z\mu(00 ) d(\mu\vert\vert\nu)+ z\mu(1)\left[\mu(00)\log\frac{\mu(00)}{\nu(00)}+\mu(1)\log\frac{\mu(1)}{\nu(1)}+\mu(01)\log\frac{\mu(01)}{\nu(01)}\right]\\ = z\mu(00 ) d(\mu\vert\vert\nu)+z\mu(1)\left[\mu(0)d(\mu\vert\vert\nu)+ d(\mu\vert\vert\nu)\right]\\ = z d(\mu\vert\vert\nu ) ( \mu(00)+\mu(10)+\mu(1))\\ = z d(\mu\vert\vert\nu)\end{aligned}\ ] ] we now prove that the renormalized 1-markov cross entropy between @xmath12 and @xmath46 converges to the cross - entropy between @xmath160 and @xmath161 as the number of pair substitution @xmath11 goes to @xmath162 . more precisely : [ main2 ] if @xmath163 as @xmath164 , @xmath113 _ proof . _ let us define , as in @xcite the following operators on the ergodic measures : @xmath165 is the projection operator that maps a measure to its 1-markov approximation , whereas @xmath166 is the operator such that for any arbitrary @xmath28 @xmath167 we notice ( see @xcite for the details ) that the normalization constant for @xmath168 is the same of that for @xmath28 : @xmath169 the measure @xmath168 is not @xmath170-markov , but we know that it becomes 1-markov after @xmath84 steps of substitutions , in fact it becomes @xmath171 . moreover , as discussed in @xcite , it is an approximation of @xmath28 if @xmath172 diverges : for any @xmath80 of length @xmath25 , @xmath173 now it is easy to establish the following chain of equalities : @xmath174 where we have used the conservation of the cross entropy @xmath175 and the fact that @xmath176 if @xmath177 are 1-markov , as shown in eq . [ h - k - markov ] . to conclude the proof we have to show that @xmath178 this is an easy consequence of eq . [ convergenza ] the definition [ hk ] and eq . [ hktoh ] . it is important to remark that we are assuming the divergence of @xmath179 too , as not being necessary for the convergence to the ( rescaled ) two - characters relative entropy . nevertheless , it would be interesting to understand both the topological and statistical constraints that prevent or permit the divergence of the expanding factor @xmath179 . experimentally , it seems that if we start with two measures with finite relative entropy ( i.e. with absolutely continuous marginals ) , then if we choose the standard strategy ( most frequent pair substitution ) for the sequence of pair substitutions that yields the divergence of @xmath180 , we also simultaneously obtain the divergence of @xmath181 ( see for instance fig . [ fig : z ] ) . on the other hand , it seems possible to consider particular sources and particular strategies of pairs substitutions withdiverging @xmath180 , that prevent the divergence of @xmath181 . at this moment we do not have conclusive rigorous mathematical results on this subject . finally , let us note that th . [ main2 ] do not give directly an algorithm to estimate the relative entropy : in any implementation we would have to specify the `` optimal '' number of pairs substitutions , with respect to the length of initial sequences and also with respect to the dimension of the initial alphabet . namely , in the estimate we have to take into account at least two correction terms , which diverges with @xmath84 : the entropy cost of writing the substitutions and the entropy cost of writing the frequencies of the pairs of characters in the alphabet we obtain after the substitutions ( or equivalent quantities if we use , for instance , arithmetic codings modeling the two character frequencies ) . for what concerns possible implementations of the method it is important to notice that the nsrps procedure can be implemented in linear time @xcite . therefore it seems reasonable that reasonably fast algorithms to compute relative entropy via nsrps can be designed . anyway , preliminary numerical experiments show that for sources of finite memory this method seems to have the same limitations of that based on parsing procedures , with respect to the methods based on the analysis of context introduced in @xcite . in fig . [ fig : h ] we show the convergence of the estimates of the entropies of the two sources and of the cross entropy , given th . [ main2 ] , for two markov process of memory 5 . in this case , the numbers of substitutions @xmath182 is small with respect to the length of the sequences @xmath183 , then the correction terms are negligible . let us finally note that the cross entropy estimate might show large variations for particular values of @xmath84 . this could be interpreted by the fact that for these values of @xmath84 pairs with particular relevance for one source with respect to the other have been substituted . this example suggest that the nsrps method for the estimation of the cross entropy should be useful in sequences analysis , for example in order to detect strings with a peculiar statistical role . 99 d. benedetto , e. caglioti , d. gabrielli : non - sequential recursive pair substitution : some rigorous results . _ issn : 1742 - 5468 ( on line ) * 09 * pp . 121 doi:10.1088/1742.-5468/2006/09/p09011 ( 2006 )

the entropy of an ergodic source is the limit of properly rescaled 1-block entropies of sources obtained applying successive non - sequential recursive pairs substitutions @xcite,@xcite . in this paper we prove that the cross entropy and the kullback - leibler divergence can be obtained in a similar way . _ keywords _ : information theory , source and channel coding , relative entropy .

in the last years , metamaterials have generated a huge interest among communities of physicists and mathematicians , owing to their extraordinary properties such as negative refraction @xcite , allowing the design of spectacular devices like the perfect flat lens @xcite or the cylindrical cloak in @xcite . such properties result from the possibility of creating artificially microscopic structures whose macroscopic electromagnetic behavior amounts to negative electric permittivity @xmath0 and/or negative magnetic permeability @xmath1 within some frequency range . such a phenomenon can also be observed in metals in the optical frequency range @xcite : in this case one says that this material is _ a negative material _ @xcite . thanks to these negative electromagnetic coefficients , waves can propagate at the interface between such a negative material and a usual dielectric material @xcite . these waves , often called _ surface plasmon polaritons _ , are localized near the interface and allow then to propagate signals in the same way as in an optical fiber , which may lead to numerous physical applications . mathematicians have so far little explored these negative materials and most studies in this context are devoted to the frequency domain , that is , propagation of time - harmonic waves @xcite . in particular , it is now well understood that in the case of a smooth interface between a dielectric and a negative material ( both assumed non - dissipative ) , the time - harmonic maxwell s equations become ill - posed if both ratios of @xmath0 and @xmath1 across the interface are equal to @xmath2 which is precisely the conditions required for the perfect lens in @xcite . this result raises a fundamental issue which can be seen as the starting point of the present paper . indeed , for numerous scattering problems , a time - harmonic wave represents the large time asymptotic behavior of a time - dependent wave resulting from a time - harmonic excitation which is switched on at an initial time . such a property is referred to as the _ limiting amplitude principle _ in the context of scattering theory . it has been proved for a large class of physical problems in acoustics , electromagnetism or elastodynamics @xcite . but what can be said about the large time behavior of the time - dependent wave if the frequency of the excitation is such that the time - harmonic problem becomes ill - posed , that is , precisely in the situation described above ? what is the effect of the surface plasmons on the large time behavior ? our aim is to give a precise answer to these questions in an elementary situation . to reach this goal , several approaches are possible . the one we adopt here , which is based on spectral theory , has its own interest because it provides us a very powerful tool to represent time - dependent waves and study their behavior , not only for large time asymptotics . our aim is to make the spectral analysis of a simple model of interface between a negative material and the vacuum , more precisely to construct a _ generalized fourier transform _ for this model , which is the keystone for a time frequency analysis . indeed this transform amounts to a _ generalized eigenfunction expansion _ of any possible state of the system , which yields a representation of time - dependent waves as superpositions of time - harmonic waves . from a mathematical point of view , this transform offers a _ diagonal _ form of the operator that describes the dynamics of the system . the existence of such a transform is ensured in a very general context @xcite , but its practical construction highly depends of the considered model . the situation studied here consists in the basic case of a plane interface between the vacuum and a negative material filling respectively two half - spaces . our negative material is described by a non - dissipative drude model , which is the simplest model of negative material . the technique we use to construct the generalized fourier transform is inspired by previous studies in the context of stratified media @xcite . compared to these studies , the difficulty of the present work relies in the fact that in the drude material , @xmath0 and @xmath1 depend on the frequency and become negative for low frequencies . for the sake of simplicity , instead of considering the complete three - dimensional physical problem , we restrict ourselves to the so - called transverse electric ( te ) two - dimensional problem , _ i.e. _ , when the electric field is orthogonal to the plane of propagation . the transverse magnetic ( tm ) case can be studied similarly . as shown in @xcite , in stratified media , the spectral theory of the three - dimensional problem follows from both te and tm cases , but is not dealt with here . the present paper is devoted to the construction of the generalized fourier transform of the te maxwell s equations . it will be used in a forthcoming paper @xcite to study the validity of a limiting amplitude principle in our medium , but the results we obtain in the present paper are not limited to this purpose . the generalized fourier transform is also the main tool to study scattering problems as in @xcite , numerical methods in stratified media as in @xcite and has many other applications . in let us mention that both present and forthcoming papers are an advanced version of the preliminary study presented in @xcite . the paper is organized as follows . in [ s - mod - meth ] , we introduce the above mentioned plane interface problem between a drude material and the vacuum , more precisely the te maxwell s equations . these equations are formulated as a conservative schrdinger equation in a hilbert space , which involves a self - adjoint hamiltonian . we briefly recall some basic notions of spectral theory which are used throughout the paper . section [ s - spec - th - ak ] is the core of our study : we take advantage of the invariance of our medium in the direction of the interface to reduce the spectral analysis of our hamiltonian to the analysis of a family of one - dimensional _ reduced hamiltonians_. we diagonalize each of them by constructing a adapted generalized fourier transform . we finally bring together in [ s - spec - th - a ] this family of results to construct a generalized fourier transform for our initial hamiltonian and conclude by a spectral representation of the solution to our schrdinger equation . we consider a metamaterial filling the half - space @xmath3 and whose behavior is described by a drude model ( see , _ e.g. _ , @xcite ) recalled below . the complementary half - space @xmath4 is composed of vacuum ( see figure [ fig.med ] ) . the triplet @xmath5 stands for the canonical basis of @xmath6 . we denote respectively by @xmath7 and @xmath8 the electric and magnetic inductions , by @xmath9 and @xmath10 the electric and magnetic fields . we assume that in the presence of a source current density @xmath11 , the evolution of @xmath12 in the whole space is governed by the macroscopic maxwell s equations ( in the following , the notation @xmath13 refers to the usual @xmath14d curl operator)@xmath15 which must be supplemented by the constitutive laws of the material @xmath16 involving two additional unknowns , the electric and magnetic polarizations @xmath17 and @xmath18 . the positive constants @xmath19 and @xmath20 stand respectively for the permittivity and the permeability of the vacuum . in the vacuum , @xmath21 so that maxwell s equations become @xmath22 on the other hand , for a homogeneous non - dissipative drude material , the fields @xmath17 and @xmath18 are related to @xmath9 and @xmath10 through @xmath23 where the two unknowns @xmath24 and @xmath25 are called usually the induced electric and magnetic currents . both parameters @xmath26 and @xmath27 are positive constants which characterize the behavior of a drude material . we can eliminate @xmath7 , @xmath8 , @xmath17 and @xmath28 which yields the time - dependent maxwell equations in a drude material : @xmath29 \mu_0 \:\partial_t { \mathbb{h}}+{\operatorname{\bf curl}}{\mathbb{e}}+{\mathbb{k}}=0 & \quad \partial_t { \mathbb{k}}= \mu_0 \ , { \omega_{\rm m}}^2\,{\mathbb{h}}\end{array } \right . \quad\mbox { in } { \mathbb{r}}^3_+.\ ] ] the above equations in @xmath30 and @xmath31 must be supplemented by the usual transmission conditions @xmath32_{x=0}=0 \quad\mbox{and}\quad [ \boldsymbol{e_x } \times { \mathbb{h}}]_{x=0}=0,\ ] ] which express the continuity of the tangential electric and magnetic fields through the interface @xmath33 ( the notation @xmath34_{x=0}$ ] designates the gap of a quantity @xmath35 across @xmath36 _ i.e. _ , @xmath37 when looking for time - harmonic solutions to these equations for a given ( circular ) frequency @xmath38 _ i.e. _ , @xmath39 for a periodic current density @xmath40 , we can eliminate @xmath41 and @xmath42 and obtain the following time - harmonic maxwell equations : @xmath43 where @xmath44 and @xmath45 if @xmath46 whereas @xmath47 functions @xmath48 and @xmath49 define the frequency - dependent electric permittivity and magnetic permeability of a drude material ( see figure [ fig.drude ] ) . several observations can be made . first notice that one recovers the permittivity and the permeability of the vacuum if @xmath50 then , a drude material behaves like the vacuum for high frequencies ( since @xmath51 and @xmath52 whereas for low frequencies , it becomes a _ negative material _ in the sense that @xmath53 note that if @xmath54 , there is a frequency gap @xmath55 of width @xmath56 where @xmath48 and @xmath49 have opposite signs . at these frequencies , waves can not propagate through the material : by this we means that corresponding plane waves are necessarily evanescent , in other words associated to non real wave vectors . it is precisely what happens in metals at optical frequencies @xcite . finally there exists a unique frequency for which the relative permittivity @xmath57 ( respectively the relative permeability @xmath58 ) is equal to @xmath59 : @xmath60 note that both ratios can be simultaneously equal to @xmath59 at the same frequency if and only if @xmath61 as a function of the frequency @xmath62.,scaledwidth=35.0% ] in the physical literature , the drude model ( [ eq.drude ] ) consists in a simple but useful approximation of a metamaterial s behavior @xcite . but one can find more intricate models to express the frequency dependency of @xmath48 and @xmath49 in the time - harmonic maxwell s equations , for instance , the lorentz model @xcite : @xmath63 where @xmath64 and @xmath65 are non negative parameters . for generalized lorentz materials @xcite , functions @xmath48 and @xmath49 are defined by finite sums of similar terms for various poles @xmath64 and @xmath65 . as mentioned in [ s - intro ] , in this paper , we restrict ourselves to the study of the so - called transverse electric ( te ) equations which result from equations ( [ eq.maxwelldiel ] ) , ( [ eq.maxwelldr ] ) and ( [ eq.transmission ] ) by assuming that @xmath66 and searching for solutions independent of @xmath67 in the form @xmath68 setting @xmath69 and @xmath70 we obtain a two - dimensional problem for the unknowns @xmath71 which will be written in the following concise form : @xmath72 \mu_0\ : \partial_t { \boldsymbol{h}}+ { \operatorname{\bf curl}}e + { \boldsymbol{\pi}}\ , { \boldsymbol{k}}= 0 & \mbox{in } { \mathbb{r}}^2,\\[5pt ] \partial_t j = { \varepsilon}_0 { \omega_{\rm e}}^2 \ , \rop \ , e & \mbox{in } { \mathbb{r}}_+^2 , \\[5pt ] \partial_t { \boldsymbol{k}}= \mu_0 { \omega_{\rm m}}^2 \ , \br \ , { \boldsymbol{h } } & \mbox{in } { \mathbb{r}}_+^2 , \end{array } \right.\ ] ] where we have used the 2d curl operators of scalar and vector fields respectively : @xmath73 moreover , @xmath74 ( respectively , @xmath75 ) denotes the extension by @xmath76 of a scalar function ( respectively , a 2d vectorial field ) defined on @xmath77 to the whole space @xmath78 , whereas @xmath79 ( respectively , @xmath80 ) stands for the restriction to @xmath77 of a function defined on the whole plane @xmath81 note that in ( [ te ] ) where equations are understood in the sense of distributions , we assume implicitly that the two - dimensional version of the transmission conditions ( [ eq.transmission ] ) are satisfied , namely @xmath82_{x=0}=0 \quad\mbox{and}\quad [ h_{y}]_{x=0}=0.\ ] ] the theoretical study of ( [ te ] ) is based on a reformulation of this system as a schrdinger equation @xmath83 where the _ hamiltonian _ @xmath84 is an unbounded operator on the hilbert space @xmath85 we assume that this space is equipped with the inner product defined for all @xmath86 and @xmath87 by @xmath88 where @xmath89 denotes the usual @xmath90 inner product , with @xmath91 or @xmath92 we easily verify that ( [ te ] ) writes as the schrdinger equation ( [ eq.schro ] ) with @xmath93 if we choose for @xmath84 the operator defined by @xmath94 where @xmath95 and @xmath96 is the following matrix differential operator ( all derivatives are understood in the distributional sense ) : @xmath97 note that the transmission conditions ( [ eq.transte ] ) are satisfied as soon as @xmath98 [ prop.autoadjoint ] the operator @xmath99 is self - adjoint . the symmetry property @xmath100 for all @xmath101 follows from our choice ( [ eq.inn-prod-2d ] ) of an inner product and the fact that the operators of each of the pairs @xmath102 @xmath103 and @xmath104 are adjoint to each other . besides , it is readily seen that the domain of the adjoint of @xmath84 coincide with @xmath105 by virtue of the hille yosida theorem @xcite , proposition [ prop.autoadjoint ] implies that the schrdinger equation ( [ eq.schro ] ) is well - posed , hence also the evolution system ( [ te ] ) . more precisely , we have the following result . [ cor.hil ] if @xmath106 , then the schrdinger equation ( [ eq.schro ] ) with zero initial condition @xmath107 admits a unique solution @xmath108 given by the duhamel integral formula : @xmath109 where @xmath110 is the group of unitary operators generated by the self - adjoint operator @xmath84 . as a consequence of the duhamel formula ( [ eq.duhamel ] ) , we see that if @xmath111 is bounded on @xmath112 ( for instance a time - harmonic source ) , then @xmath113 increases at most linearly in time . more precisely , as @xmath114 is unitary , we have @xmath115 by spectral decomposition of the operator @xmath84 , we mean its _ diagonalization _ with generalized eigenfunctions , which extends the usual diagonalization of matrices in the sense that @xmath116 where @xmath117 is a unitary transformation from the physical space @xmath118 to a spectral space @xmath119 and @xmath120 is a multiplication operator in this spectral space ( more precisely , the multiplication by the spectral variable ) . the operator @xmath117 is often called a _ generalized fourier transform _ for @xmath121 the above decomposition of @xmath84 will lead to a modal representation of the solution @xmath122 to ( [ eq.duhamel ] ) . this spectral decomposition of @xmath84 relies on general results on spectral theory of self - adjoint operators @xcite , mainly the so - called _ spectral theorem _ which roughly says that any self - adjoint operator is _ diagonalizable_. for non - expert readers , we collect below some basic materials about elementary spectral theory which allow to understand its statement , using elementary measure theory . the starting point is the notion of _ spectral measure _ ( also called _ projection valued measure _ or _ resolution of the identity _ ) . [ def.spec ] a spectral measure on a hilbert space @xmath123 is a mapping @xmath9 from all borel subsets of @xmath124 into the set of orthogonal projections on @xmath123 which satisfies the following properties : 1 . @xmath125 , 2 . @xmath126 for any @xmath127 and any sequence @xmath128 of disjoint borel sets , where the convergence of the series holds in the space @xmath123 . property 2 is known as @xmath129-additivity property . note that 1 and 2 imply @xmath130 and @xmath131 for any borel sets @xmath132 and @xmath133 . suppose that we know some such @xmath134 choose some @xmath135 and define @xmath136 for @xmath137 ( where @xmath138 and @xmath139 are the inner product and associated norm in @xmath123 ) , which maps all the borel sets of @xmath124 into positive real numbers . translated into @xmath140 the above properties for @xmath9 mean exactly that @xmath141 satisfies the @xmath129-additivity property required to become a ( positive ) _ measure _ , which allows us to define integrals of the form @xmath142 for any measurable function @xmath143 and any @xmath144 . measure theory provides the limiting process which yields such integrals starting from the case of simple functions : @xmath145 where @xmath146 denotes the indicator function of @xmath147 ( the @xmath147 s are assumed disjoint to each other ) . going further , choose now two elements @xmath148 and @xmath149 in @xmath123 and define @xmath150 which is no longer positive . integrals of the form @xmath151 can nevertheless be defined for @xmath152 they are simply deduced from the previous ones thanks to the the polarization identity @xmath153 consider then the subspace @xmath154 of @xmath155 defined by : @xmath156 by the cauchy schwarz inequality , this subspace is included in @xmath157 and one can prove that it is dense in @xmath123 . the key point is that for @xmath158 the linear form @xmath159 is continuous in @xmath123 . thus , by riesz theorem , we can define an operator denoted @xmath160 with domain @xmath154 by @xmath161 the operator usually associated to the spectral measure @xmath9 corresponds to the function @xmath162 . this operator is shown to be self - adjoint . if we choose to denote it @xmath84 , one has @xmath163 the above construction provides us a _ functional calculus _ , _ i.e. _ , a way to construct _ functions _ of @xmath84 defined by @xmath164 these operators satisfy elementary rules of composition , adjoint and normalization : @xmath165 the first rule confirms in particular that this functional calculus is consistent with composition and inversion , that is , the case of rational functions of @xmath121 the second one shows that @xmath166 is self - adjoint as soon as @xmath35 is real - valued . the third one tells us that @xmath166 is bounded if @xmath35 is bounded on the support of @xmath134 whereas it becomes unbounded if @xmath35 is unbounded . the functions which play an essential role in this paper are the functions @xmath167 associated with the _ resolvent _ of @xmath168 that is , @xmath169 for @xmath170 exponential functions @xmath171 which appears in the solution to schrdinger equations and the indicator function @xmath172 of an interval @xmath173 for which we have by construction @xmath174 we have shown above that every spectral measure give rise to a self - adjoint operator . the _ spectral theorem _ tells us that the converse statement holds true . [ th.spec ] for any self - adjoint operator @xmath84 on a hilbert space @xmath175 there exists a spectral measure @xmath9 which diagonalizes @xmath84 in the sense of ( [ eq.defaauto ] ) and ( [ eq.def-functiona ] ) . [ rem.spec ] the support the spectral measure @xmath9 is defined as the smallest closed borel set @xmath173 of @xmath124 such that @xmath176 . one can show that the spectrum @xmath177 of @xmath84 coincide with the support of @xmath9 . moreover the point spectrum @xmath178 is the set @xmath179 . theorem [ th.spec ] does not answer the crucial issue : how can we find @xmath9 if we know @xmath180 a common way to answer is to use the following stone s formulas . let @xmath84 be a self - adjoint operator on a hilbert space @xmath123 . its associated spectral measure @xmath9 is constructed as follows , for all @xmath181 @xmath182)\big ) \,u \big \|^2 = \lim_{\eta \searrow 0}\ \frac{1}{\pi}\int_{a}^{b}{\operatorname{im}}\big(r(\lambda+{{\rm i}}\eta ) \ , u , u\big)\ , { { \mathrm{d}}}\lambda , \label{eq.stone - ab}\\ \mbox { if } a\in { \mathbb{r } } : & \displaystyle \big \|{\mathbb{e}}(\left\{a\right\ } ) u \big \|^2 = \lim_{\eta \searrow 0}\ \eta \ , { \operatorname{im}}\big(r(a+{{\rm i}}\eta)u , u\big ) . \label{eq.stone - a}\end{aligned}\ ] ] note that formulas @xmath183 and @xmath184 are sufficient by @xmath129-additivity to know the spectral measure @xmath9 on all borel sets . according to remark [ rem.spec ] , formula @xmath184 permits to characterize the point spectrum @xmath185 whereas @xmath183 enables us to determine the whole spectrum @xmath177 and thus its continuous spectrum . the invariance of our medium in the @xmath186-direction allows us to reduce the spectral theory of our operator @xmath84 defined in ( [ eq.defa ] ) to the spectral theory of a family of self - adjoint operators @xmath187 defined on functions which depend only on the variable @xmath188 in the present section , we introduce this family and perform the spectral analysis of each operator @xmath189 in [ s - spec - th - a ] , we collect all these results to obtain the spectral decomposition of @xmath121 let @xmath191 be the fourier transform in the @xmath186-direction defined by @xmath192 which extends to a unitary transformation from @xmath193 to @xmath194 for functions of both variables @xmath195 and @xmath196 we still denote by @xmath191 be the partial fourier transform in the @xmath186-direction . in particular , the partial fourier transform of an element @xmath197 is such that @xmath198 where the hilbert space @xmath199 is endowed with the inner product @xmath200 defined by the same expression ( [ eq.inn-prod-2d ] ) as @xmath201 except that @xmath90 inner products are now defined on one - dimensional domains . applying @xmath191 to our transmission problem ( [ te ] ) leads us to introduce a family of operators @xmath187 in @xmath199 related to @xmath84 ( defined in ( [ eq.defa ] ) ) by the relation @xmath202 therefore @xmath190 is deduced from the definition of @xmath84 by replacing the @xmath186-derivative by the product by @xmath203 _ i.e. _ , @xmath204 where @xmath205 -\mu_0^{-1}\,{\operatorname{{\bf curl}_{\mathit k } } } & 0 & 0 & - \mu_0^{-1 } \,{\boldsymbol{\pi}}\\[4pt ] { \varepsilon}_0 { \omega_{\rm e}}^2 \,\rop & 0 & 0 & 0 \\[4pt ] 0 & \mu_0 { \omega_{\rm m}}^2\,\br & 0 & 0 \end{pmatrix},\ ] ] @xmath206 and the operators @xmath74 , @xmath75 , @xmath79 and @xmath80 are defined as in ( [ eq.opa ] ) but for functions of the variable @xmath195 only . finally , @xmath207 note again that the transmission conditions ( [ eq.transte ] ) are satisfied as soon as @xmath208 as in proposition [ prop.autoadjoint ] , it is readily seen that @xmath209 is self - adjoint for all @xmath210 the following proposition shows the particular role of the values @xmath76 and @xmath211 in the spectrum of @xmath212 [ prop.eigvalak ] for all @xmath213 , the values @xmath76 and @xmath211 are eigenvalues of infinite multiplicity of @xmath214 whose respective associated eigenspaces @xmath215 and @xmath216 are given by @xmath217 where @xmath218 , @xmath219 is the extension by @xmath76 of a 2d vector field defined on @xmath220 to the whole line @xmath124 and @xmath221 [ p.vp-de-ak ] moreover the orthogonal complement of the direct sum of these three eigenspaces , _ i.e. _ , @xmath222 is @xmath223 where @xmath224 . we detail the proof only for @xmath225 the case of the eigenvalue @xmath76 can be dealt with in the same way . suppose that @xmath226 satisfies @xmath227 which is equivalent to i _ 0 ^ -1 ( - j ) = _ m e , [ eq.syst-eigen-1 ] + -i _ 0 ^ -1 ( e + ) = _ m , [ eq.syst-eigen-2 ] + i _ 0 _ e^2 e = _ m j , [ eq.syst-eigen-3 ] + i _ 0 _ m^2 = _ m , [ eq.syst-eigen-4 ] thanks to the above definition of @xmath212 using ( [ eq.syst-eigen-3 ] ) and ( [ eq.syst-eigen-4 ] ) , we can eliminate the unknowns @xmath228 and @xmath229 in ( [ eq.syst-eigen-1 ] ) and ( [ eq.syst-eigen-2 ] ) which become @xmath230 where @xmath231 denotes the indicator function of @xmath232 in particular , we have @xmath233 in @xmath234 thus @xmath235 ( by defintion of @xmath236 ) , so @xmath237 by ( [ eq.syst-eigen-3 ] ) . in @xmath238 we can eliminate @xmath239 between the two equations of ( [ eq.vpeh ] ) , which yields @xmath240 where the last condition follows from ( [ eq.transte ] ) and @xmath235 . obviously the only solution in @xmath241 is @xmath242 hence @xmath243 vanishes on both sides @xmath244 and @xmath245 the second equation of ( [ eq.vpeh ] ) then tells us that @xmath246 whereas the first one ( together with ( [ eq.transte ] ) ) shows that @xmath247 which implies that @xmath248 where @xmath249 hence @xmath250 by ( [ eq.syst-eigen-4 ] ) . conversely , for all @xmath251 the vector @xmath252 belongs to @xmath253 and satisfies ( [ eq.syst-eigen-1])([eq.syst-eigen-4 ] ) . using these characterizations of @xmath254 and @xmath255 we finally identify the orthogonal complement of their direct sum , or equivalently , the intersection of their respective orthogonal complements . we have @xmath256 in the same way , @xmath257 this yields the definition ( [ eq.vuk ] ) of @xmath258 . in order to apply stone s formulas @xmath183 and @xmath184 to @xmath259 we first derive an integral representation of its resolvent @xmath260 we begin by showing how to reduce the computation of @xmath261 to a scalar sturm liouville equation , then we give an integral representation of the solution of the latter and we finally conclude . let @xmath262 . suppose that @xmath263 for some @xmath264 or equivalently that @xmath265 setting @xmath266 @xmath267 and using definition ( [ eq.opak ] ) , the latter equation can be rewritten as i _ 0 ^ -1 ( - j ) - e = f_e , + -i _ 0 ^ -1 ( e + ) - = , + i _ 0 _ e^2 e- j = f_j , + i _ 0 _ m^2 - = . the last two equations provide us expressions of @xmath228 and @xmath229 that can be substituted in the first two which become a system for both unknowns @xmath239 and @xmath268 we can then eliminate @xmath239 and obtain a sturm liouville equation for @xmath269 @xmath270 where @xmath271 and the following notations are used hereafter : @xmath272 \displaystyle { \varepsilon}_{\zeta}^{+}:={\varepsilon}_0 \left ( 1-\frac { { \omega_{\rm e}}^2}{\zeta^2}\right ) & \mbox { if } x>0,\\ \end{array } \right . \label{eq.def - eps } \\ \mu_{\zeta}(x ) & : = & \left\lbrace\begin{array}{ll } \mu_{\zeta}^{-}:=\mu_{0 } & \mbox { if } x<0,\\[2pt ] \displaystyle \mu_{\zeta}^{+}:=\mu_0 \left ( 1-\frac { { \omega_{\rm m}}^2}{\zeta^2}\right ) & \mbox { if } x>0 , \end{array } \right.\label{eq.def - mu } \\ { \theta_{k,\zeta}}(x ) & : = & k^2-{\varepsilon}_{\zeta}(x)\,\mu_{\zeta}(x)\,\zeta^2 = \left\lbrace\begin{array}{ll } { \theta_{k,\zeta}}^{- } : = k^2-{\varepsilon}_0\,\mu_0 \,\zeta^2 & \mbox{if } x < 0,\\[4pt ] { \theta_{k,\zeta}}^{+ } : = k^2-{\varepsilon}_{\zeta}^{+}\,\mu_{\zeta}^{+ } \,\zeta^2 & \mbox{if } x > 0 . \end{array } \right.\label{eq.deftheta}\end{aligned}\ ] ] the eliminated unknowns @xmath273 @xmath228 and @xmath229 can finally be deduced from @xmath243 by the relations @xmath274 j & = & \displaystyle \frac{{{\rm i}}\ , { \varepsilon}_0 \,{\omega_{\rm e}}^2}{\zeta } \,\rop \,e&\displaystyle - & \displaystyle\frac{1}{\zeta}\,{f_j } , \\ [ 10pt ] { \boldsymbol{k } } & = & \displaystyle \frac{\mu_0\ , { \omega_{\rm m}}^2 } { \mu_{\zeta}^+\,\zeta^2}\,\br \,{\operatorname{{\bf curl}_{\mathit k}}}e & -&\displaystyle \frac{{{\rm i}}\mu_0 ^ 2 \ , { \omega_{\rm m}}^2}{\mu_{\zeta}^+\,\zeta^2}\,\br{\boldsymbol{f_h}}- \frac{\mu_0}{\mu_{\zeta}^+\,\zeta}\,{\boldsymbol{f_k}}. \end{array}\ ] ] we can write these results in a condensed form by introducing several operators . first , we denote by @xmath275 the operator which maps the right - hand side @xmath35 of the sturm liouville equation ( [ eq.e ] ) to its solution : @xmath276 by the lax milgram theorem , it is easily seen that @xmath275 is continuous from @xmath277 to @xmath278 ( where @xmath279 denotes the dual space of @xmath280 next , associated to the expression ( [ eq.righthandsidesturm ] ) for the right - hand side of the sturm liouville equation ( [ eq.e ] ) , we define @xmath281 the operator @xmath282 is a `` scalarizator '' since it maps the vector datum @xmath283 to a scalar quantity . it is clearly continuous from @xmath199 to @xmath284 finally , associated to the two columns of the right - hand side of ( [ eq.vectorialization ] ) which distinguish the role of the electrical field @xmath285 from the one of the vector datum @xmath283 , we define @xmath286 the operator @xmath287 is a `` vectorizator '' since it maps the scalar field @xmath243 to a vector field of @xmath199 . it is continuous from @xmath288 to @xmath289 finally @xmath290 maps the vector datum @xmath283 to a vector field of @xmath199 and is continuous from @xmath199 to @xmath289 the solution of our sturm liouville equation ( [ eq.e ] ) can now be expressed as @xmath291 so that @xmath292 to sum up , we have the following proposition . [ prop.res ] let @xmath210 the resolvent of the self - adjoint operator @xmath190 can be expressed as @xmath293 where @xmath294 is the solution to the sturm liouville equation ( [ eq.e ] ) and the operators @xmath282 , @xmath290 and @xmath287 are respectively defined in ( [ eq.ops ] ) , ( [ eq.opv ] ) and ( [ eq.opt ] ) . it is readily seen that the respective adjoints of the above operators satisfy the following relations : @xmath295 from which we retrieve the usual formula @xmath296 which is valid for any self - adjoint operator . notice that in comparison with previous studies on stratified media which inspire our approach @xcite , the essential difference lies in the fact that our sturm liouville equation ( [ eq.e ] ) depends nonlinearly on the spectral variable @xmath297 which is a consequence of the frequency dispersion in a drude material . this dependence considerably complicates the spectral analysis of @xmath189 in order to use the expression of @xmath261 given by proposition [ prop.res ] in stone s formulas , we need an explicit expression of @xmath298 we recall here some classical results about the solution of a sturm liouville equation , which provide us an integral representation of @xmath299 @xmath300 where the kernel @xmath301 is the green function of the sturm liouville equation ( [ eq.e ] ) . for all @xmath302 function @xmath303 is defined as the unique solution in @xmath288 to @xmath304 where @xmath305 is the dirac measure at @xmath306 . note that formula ( [ eq.kernelop ] ) is only valid for @xmath307 if @xmath308 , we just have to replace the integral by a duality product between @xmath277 and @xmath309 in order to express @xmath301 , we first introduce the following basis of the solutions to the homogeneous sturm liouville equation associated to ( [ eq.e ] ) , _ i.e. _ , for @xmath310 @xmath311 where @xmath312 { \theta_{k,\zeta}}^{+ } : = \sqrt{k^2-{\varepsilon}_{\zeta}^{+}\,\mu_{\zeta}^{+ } \,\zeta^2 } & \mbox{if } x > 0 , \end{array } \right.\ ] ] and @xmath313 denotes the principal determination of the complex square root , _ i.e. _ , @xmath314 the special feature of the above basis is that both @xmath315 and @xmath316 are analytic functions of @xmath317 for all @xmath195 ( since they can be expanded as power series of @xmath318 in particular , they do not depend on the choice of the determination of @xmath319 whereas @xmath320 depends on it . note that this definition of @xmath320 makes sense since @xmath321 for all @xmath262 and @xmath322 this is obvious for @xmath323 since @xmath324 implies @xmath325 hence @xmath326.$ ] on the other hand , by ( [ eq.def-eps ] ) , ( [ eq.def-mu ] ) and ( [ eq.deftheta ] ) , we have @xmath327 which can not belong to @xmath328 $ ] for the same reasons , since the imaginary parts of both quantities @xmath329 and @xmath330 have the same sign as @xmath331 [ prop.green ] let @xmath332 and @xmath333 . for all @xmath334 the function @xmath335 satisfies the integral representation ( [ eq.kernelop ] ) where the green function @xmath301 of the sturm liouville equation ( [ eq.e ] ) is given by @xmath336 where @xmath337 is the ( constant ) wronskian of @xmath338 and @xmath339 _ i.e. _ , @xmath340 . we omit the proof of this classical result ( see , e.g. , @xcite ) . the expression ( [ eq.defgreen ] ) of @xmath301 involves another basis @xmath341 of the solutions to the homogeneous sturm liouville equation associated to ( [ eq.e ] ) which has the special feature to be evanescent as @xmath195 tends either to @xmath342 or @xmath343 more precisely , @xmath344 a_{k,\zeta,-1}\ { { \rm e}}^{+{\theta_{k,\zeta}}^{+}\ , x } + b_{k,\zeta,-1}\ { { \rm e}}^{-{\theta_{k,\zeta}}^{+}\ , x } & \mbox { if } x>0 , \end{array } \right . \label{eq.psim } \\ \psip(x ) & = & \left\lbrace\begin{array}{ll } a_{k,\zeta,+1}\ { { \rm e}}^{-{\theta_{k,\zeta}}^{-}\ , x } + b_{k,\zeta,+1}\ { { \rm e}}^{+{\theta_{k,\zeta}}^{-}\ , x } & \mbox { if } x<0,\\[2pt ] { { \rm e}}^{-{\theta_{k,\zeta}}^{+}\,x } & \mbox { if } x>0 , \end{array } \right . \label{eq.psip } \end{aligned}\ ] ] where @xmath345 formulas ( [ eq.psim ] ) and ( [ eq.psip ] ) show that @xmath346 decreases exponentially when @xmath347 since @xmath348 note that the wronskian @xmath349 can not vanish for @xmath350 otherwise the resolvent @xmath261 would be singular , which is impossible , because it is analytic in @xmath351 since @xmath190 is self - adjoint . we are now able to give an explicit expression of @xmath352 more precisely of the quantity @xmath353 involved in stone s formulas @xmath183 and @xmath184 . thanks to proposition [ prop.res ] and ( [ eq.adjoints ] ) , we can rewrite this quantity as @xmath354 where @xmath355 denotes the duality product between @xmath278 and @xmath284 in order to express this duality product as an integral ( and to avoid other difficulties which occur when applying stone s formulas ) , we restrict ourselves to particular @xmath122 chosen in the _ dense _ subspace of @xmath199 defined by @xmath356 where @xmath357 for @xmath358 or @xmath234 denotes the space of bump functions in @xmath359 ( compactly supported in @xmath359 and smooth ) and the condition @xmath360 ensures that @xmath361 using the integral representation ( [ eq.kernelop ] ) , the above formula becomes @xmath362 before applying stone s formulas @xmath183 and @xmath184 , we have to make clear the behavior of the resolvent @xmath261 as @xmath363 tends to the real axis , hence the behavior of all quantities involved in the integral representation ( [ eq.resint2 ] ) , in particular the green function @xmath301 . this is the subject of this paragraph . in the following , for any quantity @xmath364 depending on the parameter @xmath365 , we choose to denote @xmath366 the one - sided limit ( if it exists ) of @xmath364 when @xmath363 tends to @xmath367 from the _ upper half - plane _ , _ i.e. _ , @xmath368 notice that @xmath282 @xmath287 and @xmath290 defined in ( [ eq.ops ] ) , ( [ eq.opv ] ) and ( [ eq.opt ] ) have obviously one - sided limits @xmath369 @xmath370 and @xmath371 if @xmath372 differs from 0 and @xmath373 ( they actually depend analytically on @xmath363 outside these three values ) . the determination of the one - sided limit of @xmath375 defined in ( [ eq.deftheta ] ) requires us to identify the zones , in the @xmath376 plane , where @xmath377 or @xmath378 ( defined in ( [ eq.deftheta ] ) ) is located on the branch cut @xmath379 of the complex square root ( [ eq.defrac ] ) . using ( [ eq.def-eps ] ) and ( [ eq.def-mu ] ) , one computes that @xmath380 { \theta_{k,\lambda}}^{+ } = \displaystyle \frac{-{\varepsilon}_0\mu_0\ , \lambda^4 + \left(k^2+{\varepsilon}_0\mu_0({\omega_{\rm e}}^2+{\omega_{\rm m}}^2)\right ) \lambda^2 - { \varepsilon}_0\mu_0 \ , { \omega_{\rm e}}^2{\omega_{\rm m}}^2}{\lambda^2 } & \mbox{if } x > 0 . \end{array } \right.\ ] ] we denote by @xmath381 ( respectively , @xmath382 and @xmath383 , with @xmath384 ) the non - negative values of @xmath372 for which @xmath385 ( respectively , @xmath386 vanishes , _ i.e. _ , @xmath387 \lambda_\scd(k ) & : = & \sqrt{\frac{k^2}{2{\varepsilon}_0\mu_0 } + \frac{{\omega_{\rm e}}^2+{\omega_{\rm m}}^2}{2 } + \sqrt{\left ( \frac{k^2}{2{\varepsilon}_0\mu_0 } + \frac{{\omega_{\rm e}}^2-{\omega_{\rm m}}^2}{2 } \right)^2 + \frac{k^2\,{\omega_{\rm m}}^2}{{\varepsilon}_0\mu_0 } } } , \\ \lambda_\sci(k ) & : = & \sqrt{\frac{k^2}{2{\varepsilon}_0\mu_0 } + \frac{{\omega_{\rm e}}^2+{\omega_{\rm m}}^2}{2 } -\sqrt{\left ( \frac{k^2}{2{\varepsilon}_0\mu_0 } + \frac{{\omega_{\rm e}}^2-{\omega_{\rm m}}^2}{2 } \right)^2 + \frac{k^2\,{\omega_{\rm m}}^2}{{\varepsilon}_0\mu_0}}},\end{aligned}\ ] ] where @xmath383 and @xmath382 are related by @xmath388 . in the @xmath376-plane , the graphs of these functions are curves through which the sign of @xmath385 or @xmath389 changes . the main properties of these functions are summarized in the following lemma , whose proof is obvious . [ lem.disp ] for all @xmath390 we have @xmath391 \big).\ ] ] the function @xmath392 is a @xmath393 strictly decreasing function on @xmath394 whose range is @xmath395 $ ] , whereas @xmath396 is a @xmath393 strictly increasing function on @xmath394 whose range is @xmath397 and @xmath398 as @xmath399 moreover , denoting @xmath400 the unique non negative value of @xmath401 for which @xmath402 , that is to say @xmath403 one has @xmath404 finally , @xmath405 if and only if @xmath406 and @xmath407 grey color , vertical and horizontal hatched lines mean respectively propagation in the vacuum , direct propagation and inverse propagation in the drude material . ] grey color , vertical and horizontal hatched lines mean respectively propagation in the vacuum , direct propagation and inverse propagation in the drude material . ] ( left ) and @xmath408 ( left ) and @xmath408 ( right ) . see caption of figure [ fig.dieldru ] . [ fig.speczones ] figure [ fig.dieldru ] illustrates this lemma in the quarter @xmath376-plane @xmath409 ( all zones are symmetric with respect to both @xmath401 and @xmath372 axes , since @xmath410 are even functions of @xmath401 and @xmath411 the signs of @xmath412 and @xmath413 indicate the regime of vibration in the vacuum and in the drude material : _ propagative _ or _ evanescent_. as it will be made clear in [ s.phys-spect-zones ] , two kinds of propagative waves occur in the drude material , which will be called _ direct _ and _ inverse _ , whereas propagative waves can only be _ direct _ in the vacuum . this explains the choice of the indices @xmath414 @xmath415 and @xmath416 used hereafter , which mean respectively _ direct _ , _ inverse _ and _ evanescent_. figure [ fig.speczones ] , constructed as the superposition of the left and right graphics of figure [ fig.dieldru ] , then shows the coupling of both half - spaces . this leads us divide the @xmath376-plane in several _ spectral zones _ defined as follows ( see [ s.phys-spect-zones ] for the justification of the notations ) : @xmath417 \zde & : = & \left\{(k,\lambda ) \in { \mathbb{r}}^2 \mid |\lambda| \neq { \omega_{\rm m}}\mbox { and } \max\big(\lambda_0(k),\lambda_\sci(k)\big ) < |\lambda| < \lambda_\scd(k)\right\},\\[4pt ] \zdi & : = & \left\{(k,\lambda ) \in { \mathbb{r}}^2 \mid |k| < k_{\rm c } \mbox { and } \lambda_0(k ) < |\lambda| < \lambda_\sci(k)\right\ } , \\[4pt ] \zei & : = & \left\{(k,\lambda ) \in { \mathbb{r}}^2 \mid 0 < |\lambda| < \min\big(\lambda_0(k),\lambda_\sci(k)\big)\right\}. \end{array } \label{eq.defspeczones}\ ] ] note that the eigenvalues @xmath418 and @xmath419 of @xmath190 ( see proposition [ p.vp-de-ak ] ) are excluded from these zones . with our choice ( [ eq.defrac ] ) for the complex square root , we notice that if @xmath420 is positive , then @xmath421 is a positive real number ( more precisely , @xmath422 on the other hand , if @xmath423 is negative , then @xmath421 is purely imaginary and its sign coincides with the sign of @xmath424 for small positive @xmath425 from ( [ eq.def-eps ] ) , ( [ eq.def-mu ] ) and ( [ eq.deftheta ] ) , one computes that @xmath426 note that for @xmath427 , @xmath428 has the same sign as @xmath372 , as a consequence @xmath429 has the opposite sign of @xmath372 . moreover , the sign of @xmath430 , the limit as @xmath431 of @xmath432 when @xmath427 , depends on the position of @xmath433 with respect @xmath434 . from lemma [ lem.disp ] , we deduce that @xmath435 thus the sign of @xmath430 is positive in the spectral zone @xmath436 ( where @xmath437 ) and negative ( where @xmath438 ) in the spectral zones @xmath439 and @xmath440 . finally , we obtain @xmath441 \end{array}\right . \label{eq.expr - thetam } \\ { \theta_{k,\lambda}}^+ & = & \left\lbrace \begin{array}{ll } + { { \rm i}}\,{\operatorname{sgn}}(\lambda)\,|{\theta_{k,\lambda}}^+|^{1/2 } & \mbox{if } ( k,\lambda ) \in \zei \cup \zdi,\\[4pt ] - { { \rm i}}\,{\operatorname{sgn}}(\lambda)\,|{\theta_{k,\lambda}}^+|^{1/2 } & \mbox{if } ( k,\lambda ) \in \zdd,\\[4pt ] \end{array}\right . \label{eq.expr - thetap}\end{aligned}\ ] ] in addition to the spectral zones defined above , we have to investigate the possible singularities of the green function in the @xmath376-plane , that is , the pairs @xmath376 for which the one - sided limit @xmath442 of @xmath443 ( see ( [ eq.coeff-psi ] ) ) vanishes . hence we have to solve the following _ dispersion equation _ , seen as an equation in the @xmath376-plane : @xmath444 [ lem.sing ] define @xmath445 the solutions @xmath376 to ( [ eq.disp ] ) are given by @xmath446 where ( see figure [ fig.speczones ] for an illustration ) @xmath447 the critical value @xmath400 has been defined in lemma [ lem.disp ] and @xmath448 the function @xmath449 is strictly decreasing on @xmath450 if @xmath451 _ i.e. _ , @xmath452 ( respectively , strictly increasing if @xmath453 _ i.e. _ , @xmath454 moreover @xmath455 and @xmath456 as @xmath399 first notice that if @xmath376 is a solution to ( [ eq.disp ] ) , then @xmath457 the proof is made of two steps . + _ step 1 . _ we first show that @xmath458 . + indeed , ( [ eq.disp-carre ] ) implies that @xmath385 and @xmath389 either have the same sign or vanish simultaneously . hence * @xmath459 and @xmath440 contain no solution since @xmath385 and @xmath389 have opposite signs in these zones ( first statement of lemma [ lem.disp ] ) . * if @xmath460 @xmath461 ( from ( [ eq.expr-thetam ] ) and ( [ eq.expr-thetap ] ) ) and @xmath462 ( since @xmath463 . this shows that @xmath376 can not satisfy ( [ eq.disp ] ) . a similar argument works for @xmath439 ( @xmath464 and @xmath465 have opposite signs in this zone , but @xmath466 and @xmath467 also have opposite signs ) . on the other hand , the curves @xmath468 @xmath469 and @xmath470 contain no solution to ( [ eq.disp ] ) , except the _ critical points _ @xmath471 @xmath472 @xmath473 and @xmath474 where both @xmath410 vanish . to sum up , apart from the critical points , the possible solutions to ( [ eq.disp ] ) are located outside the closure of all previously defined zones , that is , in the `` white area '' of figure [ fig.speczones ] . + _ step 2 . _ in this `` white area '' , @xmath475 so solutions may occur only if @xmath464 and @xmath465 have opposite signs , that is , in the sub - area located in @xmath476 in this sub - area , ( [ eq.disp ] ) and ( [ eq.disp-carre ] ) are equivalent since @xmath477 . using ( [ eq.def-eps ] ) , ( [ eq.def-mu ] ) and ( [ eq.deftheta ] ) , ( [ eq.disp-carre ] ) can be written as @xmath478 from which we infer that @xmath479 if @xmath480 the only positive root is @xmath481 which is strictly decreasing on @xmath450 with range @xmath482.$ ] moreover , the curve @xmath483 crosses @xmath470 only at @xmath471 which shows that the former is located in the above mentioned sub - area . on the other hand , if @xmath484 both roots @xmath485 and @xmath481 are positive , but @xmath486 so the curve @xmath483 is now outside the sub - area . the other root yields the only solution in the admissible sub - area and it is strictly increasing on @xmath450 with range @xmath487 . finally , if k=0 , the only root of @xmath488 is @xmath489 the various zones introduced above are related to various types of waves in both media , which can be either propagative or evanescent . as already mentioned , the indices @xmath414 @xmath415 and @xmath416 we chose to qualify these zones stand respectively for _ direct _ , _ inverse _ and _ evanescent_. the first two , @xmath490 and @xmath491 are related to propagative waves which can be either direct or inverse waves ( in the drude medium ) , whereas @xmath416 means evanescent , that is , absence of propagation . as shown below , for each pair of indices characterizing the various zones @xmath493 @xmath494 @xmath440 and @xmath495 the former indicates the behavior of the vacuum @xmath496 and the latter , that of the drude material @xmath497 to see this , substitute formulas ( [ eq.expr-thetam ] ) and ( [ eq.expr-thetap ] ) in ( [ eq.psim ] ) and ( [ eq.psip ] ) , which yields the one - sided limits @xmath498 of @xmath346 in the sense of ( [ eq.convention ] ) . the aim of this section is to show the physical interpretation of these functions as superpositions of elementary waves . for simplicity , we restrict ourselves to @xmath499 when interpreting their direction of propagation . first notice that in each half line @xmath500 and @xmath501 , function @xmath502 is a linear combination of terms of the form @xmath503 or @xmath504 , where for a fixed @xmath401 , @xmath505 is a certain function of @xmath372 , denoted @xmath506 . * when @xmath505 is purely imaginary , @xmath507 is an _ evanescent _ wave in the direction @xmath501 or @xmath500 . * when @xmath505 is real , @xmath507 is a _ propagative _ wave whose phase velocity is given by @xmath508 locally ( if @xmath509 ) , @xmath506 defines @xmath372 as a function of @xmath505 and the group velocity of @xmath507 is defined by @xmath510 if the product @xmath511 is positive , ( _ i.e. _ , when the group and phase velocities point in the same direction ) , one says that @xmath507 is a _ direct propagative _ wave . if the product @xmath511 is negative , ( _ i.e. _ , when the group and phase velocities point in opposite directions ) , one says that @xmath507 is an _ inverse propagative _ wave . + one uses also the notion of group velocity , which characterizes _ the direction of propagation _ ( the direction of the energy transport ) , to distinguish between incoming and outgoing propagative waves . one says that a propagative wave @xmath507 is _ incoming _ ( respectively , _ outgoing _ ) in the region @xmath512 if its group velocity points towards the interface ( respectively , towards @xmath513 ) . in our case , for a real @xmath505 , one denotes by @xmath514 and @xmath515 , the phase and group velocities of a propagative wave @xmath507 in the region @xmath512 . let us derive a general formula for the product @xmath516 . in our case , the dispersion relation ( [ eq.deftheta ] ) can be rewritten as @xmath517 by differentiating this latter expression with respect to @xmath518 one gets @xmath519 from ( [ eq.defphasevel ] ) and ( [ eq.defphasegroup ] ) , this yields @xmath520 in the vacuum , one easily check that formula ( [ eq.vphvg ] ) leads to the classical relation @xmath521 thus all the propagative waves are direct propagative . in the drude material , one has @xmath522 and by expressing the derivatives in the latter expression , one can rewrite ( [ eq.vphvg ] ) as @xmath523^{-1}.\ ] ] one sees with this last expression that the sign of @xmath524 depends on the sign of @xmath525 and @xmath526 . let us look at what happens in the different spectral zones . this study is summarized in the tables of figures [ fig.modes ] and [ fig.plasmons ] . @xmath527 suppose first that @xmath528 in this spectral zone , @xmath529 thus the corresponding waves @xmath507 are propagative on both side of the interface . moreover , as @xmath530 and @xmath531 , the product ( [ eq.sign ] ) is positive : the propagative waves are direct propagative waves in both media and thus their direction of propagation is the sign of their wave number @xmath505 . ( [ eq.psim ] ) shows that for @xmath532 function @xmath533 is an oscillating wave of amplitude 1 and wave number @xmath534 which propagates towards @xmath342 , whereas for @xmath535 it is a superposition of a wave of amplitude @xmath536 and wave number @xmath537 which propagates towards the origin and a wave of amplitude @xmath538 and wave number @xmath539 which propagates towards @xmath343 in other words , @xmath533 can be interpreted as an _ incoming incident _ wave of amplitude @xmath536 propagating from @xmath540 whose diffraction on the interface @xmath541 generates two _ outgoing _ waves : a _ reflected _ wave of amplitude @xmath538 and a _ transmitted _ wave of amplitude 1 . similarly , @xmath542 can be interpreted as an _ incoming incident _ wave of amplitude @xmath543 and wave number @xmath544 propagating from @xmath342 whose diffraction on the interface @xmath541 generates two _ outgoing _ waves : a _ reflected _ wave of amplitude @xmath545 and wave number @xmath534 and a _ transmitted _ wave of amplitude 1 and wave number @xmath539 . @xmath527 if @xmath546 we still have @xmath547 but now @xmath548 which means that propagative waves are no longer allowed in the drude material : the only possible waves are exponentially decreasing or increasing . proceeding as above , we see that on the one hand , @xmath533 can be interpreted as an _ incident _ wave of amplitude @xmath536 which is exponentially increasing as @xmath549 whose diffraction on the interface @xmath541 generates two waves : a _ reflected evanescent _ wave of amplitude @xmath538 and a _ transmitted outgoing _ wave of amplitude 1 and wave number @xmath534 . on the other hand , @xmath542 can be interpreted as an _ incoming incident _ wave of amplitude @xmath543 and wave number @xmath544 propagating from @xmath342 whose diffraction on the interface @xmath541 generates two waves : a _ reflected outgoing _ wave of amplitude @xmath545 and wave number @xmath534 and a _ transmitted evanescent _ wave of amplitude 1 . @xmath527 suppose now that @xmath550 we still have @xmath547 but propagative waves are again allowed in the drude material since @xmath551 compared with the case where @xmath460 we simply have a change of sign in the expression of @xmath552 which amounts to reversing all sign of the wave numbers @xmath505 associated to the propagative waves in the drude material . hence for @xmath553 one might be tempted to interchange the words _ incoming _ and _ outgoing _ in the above interpretation of @xmath498 for @xmath528 it would be wrong ! indeed , the change of sign of the imaginary part of @xmath554 yields an opposite sign of the _ phase velocity _ in the drude material , but not of the _ group velocity _ which characterizes the direction of the energy transport . as we see in ( [ eq.sign ] ) , since @xmath555 and @xmath465 are both negative in this spectral zone , the phase and group velocity are pointing in opposite directions . hence , the waves @xmath507 are _ inverse propagative _ in the drude material . @xmath527 assuming now that @xmath556 we have @xmath557 and @xmath551 the drude material behaves behaves as a negative material as in the previous case ( since @xmath555 and @xmath465 are also both negative in this spectral zone ) , but propagative waves are no longer allowed in the vacuum . in the different spectral zones @xmath558 , @xmath559 , @xmath560 and @xmath561 . propagative waves are represented by an oscillating function and the larger and smaller arrow indicate respectively the direction of the group velocity and the phase velocity . evanescent waves are represented with decreasing exponential . the `` unphysical '' functions @xmath498 which contain increasing exponential behavior are colored in gray.,scaledwidth=80.0% ] in the spectral zone @xmath562.,scaledwidth=80.0% ] @xmath527 finally , if @xmath563 both @xmath564 are real and positive , so that waves are evanescent on both sides of the interface @xmath565 in this case , functions @xmath566 and @xmath567 are equal and real . such a wave is referred as plasmonic wave in the physical literature ( see @xcite ) . note that in the particular case @xmath568 , we have @xmath569 and @xmath570 so that @xmath571 . this shows that @xmath572 is an even function of @xmath188 + among the various categories of waves described above , some of them may be called `` unphysical '' since they involve an exponentially increasing behavior at infinity , which occurs for @xmath533 if @xmath546 as well as @xmath542 if @xmath573 fortunately these waves will disappear by limiting absoption in the upcoming proposition [ prop.vecgen1 ] : only the `` physical '' ones are needed to describe the spectral behavior of @xmath189 that is why in figure [ fig.speczones ] , all the curves represent _ spectral cuts _ since they are the boundaries where some @xmath498 appear or disappear . we are now able to express the one - sided limit @xmath574 of the green function defined in ( [ eq.defgreen ] ) . the following two propositions , which distinguish the case of @xmath575 from the other zones , provide us convenient expressions of the quantities related to @xmath301 which are needed in stone s formulas . [ prop.vecgen1 ] for all @xmath576 with @xmath577 , we have @xmath578 @xmath579 suppose first that @xmath528 if this case , ( [ eq.expr-thetam ] ) and ( [ eq.expr-thetap ] ) tell us that @xmath580 in order to express the imaginary part of the green function , we rewrite the expression of proposition [ prop.green ] in terms of functions @xmath581 and @xmath582 which are real - valued . we obtain that @xmath583 is equal to @xmath584 this expression shows that we can replace @xmath585 and @xmath586 by @xmath195 and @xmath306 respectively . it can be written in matrix form as @xmath587 where the symbol @xmath588 denotes the conjugate transpose of a matrix . note that the conjugation could be omitted since @xmath581 and @xmath582 are real - valued . however it is useful for the next step which consists in rewriting this expression in terms of @xmath498 by noticing that @xmath589which follows from the definition ( [ eq.coeff-psi ] ) of @xmath590 therefore , after some calculations exploiting that @xmath591 is purely imaginary , one obtains @xmath592 which yields the announced result , for @xmath593 ( since @xmath594 + suppose now that @xmath595 . compared with the previous case , we have now @xmath596 and @xmath597 becomes negative ( since @xmath598 it is readily seen that in this case , the calculations above hold true if we replace @xmath597 by @xmath599 + when @xmath600 , we still have @xmath601 but @xmath602 is now a positive number , which shows that @xmath542 is a real - valued function . following the same steps as above , we obtain @xmath603 finally , if @xmath604 , we have @xmath605 and @xmath606 as in @xmath494 the drude material behaves as a negative material @xmath607 in this case , we obtain @xmath608 which completes the proof . [ prop.vecgen2 ] for all @xmath609 , we have @xmath610 where the real - valued function @xmath611 function is given by @xmath612 and the remainder @xmath613 is uniform in @xmath614 on any compact set of @xmath78 . finally , if @xmath615 , then @xmath616 as @xmath617 from the upper - half plane uniformly with respect to @xmath614 on any compact set of @xmath81 as a consequence , @xmath618 let @xmath619 it is readily seen that the definition ( [ eq.coeff-psi ] ) of @xmath349 can be written equivalently ( see also the proof of lemma [ lem.sing ] ) @xmath620 where @xmath621 is the polynomial defined in ( [ eq.def-qk ] ) . as @xmath622 and one can compute that @xmath623 we deduce after some manipulations that , for @xmath624 @xmath625 ( where we used the dispersion relation ( [ eq.disp ] ) and the fact that @xmath626 the announced result follows from the expression ( [ eq.coeff-psi ] ) of @xmath627 since @xmath346 are analytic functions of @xmath363 near @xmath372 which both tend to @xmath628 + finally , if @xmath629 , @xmath630 and @xmath631 are simple zeros of @xmath632 it is clear ( see formula ( [ eq.coeff-psi ] ) ) that @xmath633 as @xmath617 from the upper - half plane . thus , the conclusion follows again from the expression ( [ eq.coeff-psi ] ) of @xmath627 since @xmath346 are here uniformly bounded in @xmath363 , @xmath195 and @xmath306 when @xmath363 belongs to a vicinity of @xmath634 and @xmath614 to any compact set of @xmath81 as we shall see , the spectral zones introduced in [ s.spectr-zones ] actually show us the location of the spectrum of @xmath190 for each @xmath635 we simply have to extract the associated sections of these zones , that is , the sets @xmath636 which all are unions of symmetric intervals with respect to @xmath637 this is a by - product of the following proposition which tells us that apart from the three eigenvalues @xmath638 and @xmath27 ( see proposition [ prop.eigvalak ] ) , the spectrum of @xmath214 is composed of two parts : an _ absolutely continuous spectrum _ defined by @xmath639 if @xmath640 and by @xmath641 if @xmath642 and a _ pure point spectrum _ given by @xmath643 if @xmath642 ( we point out that there is no _ singularly continuous spectrum _ ) . this proposition yields a convenient expression of the spectral projection @xmath644 of @xmath214 for @xmath645 as @xmath644 is a projection on an invariant subspace by @xmath646 the canonical way to express such a projection is to use a _ spectral basis_. propositions [ prop.vecgen1 ] and [ prop.vecgen2 ] provide us such a basis : these are the vector fields deduced from the @xmath647 s ( see ( [ defwkl ] ) and ( [ defwklzero ] ) ) by the `` vectorizator '' defined in ( [ eq.opv ] ) , _ i.e. _ , @xmath648 for each zone @xmath649 as we shall see afterwards , the knowledge of these vector fields leads to a diagonal form of @xmath212 these are _ generalized eigenfunctions _ of @xmath212 [ prop.specmeasureak ] let @xmath650 @xmath651 the spectral measure of @xmath214 and @xmath652 ( see ( [ eq.defdx ] ) ) . for all interval @xmath653 with @xmath654 we have @xmath655 0 & \mbox{if } |k| = k_{\rm c}. \end{array}\right . \label{eq.specmesak - ponct}\end{aligned}\ ] ] moreover , @xmath656 for all interval @xmath657 , which does not contain @xmath76 or @xmath658 [ p.specmesak ] [ rem.wl21d ] note the symbol @xmath659 in the index `` @xmath660 '' in ( [ eq.specmesak-cont ] ) . it indicates that @xmath661 is a slight adaptation of the inner product @xmath662 which is necessary because @xmath663 since these are oscillating ( bounded ) functions at infinity ( that is why they are called _ generalized _ ) . a simple way to overcome this difficulty is to introduce _ weighted _ @xmath90 spaces @xmath664 we can then define @xmath665 it is readily seen that for positive @xmath666 the spaces @xmath667 and @xmath668 are dual to each other if @xmath199 is identified with its own dual space , which yields the continuous embeddings @xmath669 the notation @xmath661 represents the duality product between them , which extends the inner product of @xmath199 in the sense that @xmath670 the above proposition holds true as soon as we choose @xmath659 so that the @xmath671 s belong to @xmath672 that is if @xmath673 let @xmath674 @xmath675 \subset \lambda_\scz(k)$ ] with @xmath676 and @xmath652 . we show in the second part of the proof that @xmath677 which implies that @xmath678)\,{\boldsymbol{u}}.$ ] hence stone s formula @xmath183 together with the integral representation ( [ eq.resint2 ] ) show that the quantity @xmath679 is given by the following limit , where @xmath680 @xmath681 the first step is to permute the limit and the integrals in this formula thanks to the lebesgue s dominated convergence theorem . according to the foregoing , as @xmath682 and @xmath683 the integrand is a continuous function of @xmath684 for all @xmath685,$ ] @xmath195 and @xmath306 in the compact support of @xmath122 ( recall that @xmath686 moreover , the integrand is dominated by a constant ( provided @xmath684 remains bounded ) . therefore the permutation is justified : in the above formula , we can simply replace @xmath363 and @xmath687 by @xmath688 formula ( [ eq.adjoints ] ) tells us that @xmath371 is self - adjoint , hence @xmath689 besides , notice that @xmath690 where @xmath691 ( this is easily deduced from the symmetry of @xmath692 _ i.e. _ , @xmath693 see ( [ eq.defgreen ] ) ) . this allows us to use proposition [ prop.vecgen1 ] so that @xmath694 by fubini s theorem , this expression becomes @xmath695 which is nothing but ( [ eq.specmesak-cont ] ) , since an integration by parts shows that @xmath696 by virtue of the @xmath129-additivity of the spectral measure ( see definition [ def.spec ] ) , formula ( [ eq.specmesak-cont ] ) holds true for any interval @xmath697 even if the resolvent is singular . indeed , this singular behavior occurs only if @xmath615 at @xmath698 where proposition [ prop.vecgen2 ] tells us that in this case , @xmath699 , which is an integrable singularity . suppose now that for @xmath676 , @xmath700 $ ] is located outside @xmath701 and does not contain @xmath76 or @xmath658 in this case , the one - sided limit @xmath702 of the green function is real - valued for all @xmath703 ( since @xmath704 see ( [ eq.expr-thetam])-([eq.expr-thetap ] ) ) and the same steps as above yield @xmath705 consider finally singletons @xmath706 for @xmath707 stone s formula @xmath184 together with ( [ eq.resint2 ] ) show that for all @xmath708 the quantity @xmath709 is given by the following limit , where @xmath710 @xmath711 this shows that @xmath712 can be nonzero only if @xmath713 is a singularity of @xmath714 @xmath282 or @xmath715 from ( [ eq.ops ] ) and ( [ eq.opt ] ) , the singularities of @xmath290 and @xmath282 are @xmath716 and @xmath717 but these points are excluded from our study since proposition [ p.vp-de-ak ] . hence , we are only interested in the singularities of @xmath627 that is , the zeros @xmath718 of @xmath349 defined in lemma [ lem.sing ] . suppose then that @xmath719 and @xmath720 as above , using the lebesgue s dominated convergence theorem and proposition [ prop.vecgen2 ] , we obtain @xmath721 where the second inequality follows from fubini s theorem . integrating by parts , formula ( [ eq.specmesak-cont ] ) follows . on the other hand , if @xmath722 and @xmath723 the green function is singular near @xmath724 but proposition [ prop.vecgen2 ] tells us that @xmath725 which shows that @xmath726 the aim of this subsection is to deduce from the knowledge of the spectral measure @xmath727 a _ generalized fourier transform _ @xmath728 for @xmath646 that is , an operator which provides us a _ diagonal _ form of the reduced hamiltonian @xmath214 as @xmath729 this transformation @xmath728 maps @xmath199 to a _ spectral space _ @xmath730 which contains fields that depend on the spectral variable @xmath688 in the above diagonal form of @xmath646 `` @xmath372 '' denotes the operator of multiplication by @xmath372 in @xmath731 in short , @xmath728 transforms the action of @xmath214 in the physical space @xmath199 into a linear spectral amplification in the spectral space @xmath731 we shall see that @xmath728 is a partial isometry which becomes unitary if we restrict it to the orthogonal complement of the eigenspaces associated with the three eigenvalues 0 and @xmath732 that is , the space @xmath258 defined in proposition [ p.vp-de-ak ] . the definition of @xmath728 comes from formulas ( [ eq.specmesak-cont ] ) and ( [ eq.specmesak-ponct ] ) : for a fixed @xmath733 and all @xmath734 , we denote @xmath735 which represents the `` decomposition '' of @xmath122 on the family of _ generalized eigenfunctions _ @xmath736 of @xmath214 defined in ( [ eq : defwijk ] ) . we show below that the codomain of @xmath728 is given by @xmath737 l^{2}(\lambda_{\dd}(k))^{2 } \oplus l^{2}(\lambda_{\de}(k ) ) \oplus l^{2}(\lambda_{\ei}(k ) ) \oplus l^{2}(\lambda_{\ee } ( k ) ) & \mbox { if } |k|>k_{\rm c } . \end{array } \right.\ ] ] we point out that the space @xmath738 is here isomorphic to @xmath739 since @xmath740 . we denote by @xmath741 the fields of @xmath730 , where it is understood that @xmath742 and @xmath743 for @xmath744 if @xmath745 and for @xmath746 if @xmath747 . the hilbert space @xmath730 is endowed with the norm @xmath748 defined by @xmath749 \displaystyle \sum_{\scz \in \ { \dd,\de,\ei\ } } \int_{\lambda_\scz(k)}\sum \limits_{j\in j_{\scz } } | { \boldsymbol{\hat{u}}}(\cdot , j ) |^2\ , { { \mathrm{d}}}\lambda + \sum_{\lambda \in \lambda_{\ee}(k)}|{\boldsymbol{\hat{u}}}(\lambda,0)|^2 & \mbox{if } |k| > k_{\rm c}. \end{array}\right . \label{eq.defnormhathk}\ ] ] the following theorem expresses the diagonalization of the reduced hamiltonian . its proof is classical ( see , e.g. , @xcite ) and consists of two steps . we first deduce from theorem [ th.spec ] that @xmath728 is an isometry from @xmath258 to @xmath730 which diagonalizes @xmath212 we then prove that @xmath728 is surjective . [ th.diagak ] for @xmath213 , let @xmath750 denote the orthogonal projection on the subspace @xmath258 of @xmath751 _ i.e. _ , @xmath752 where @xmath753 is the indicator function of @xmath754 . the operator @xmath728 defined in ( [ eq.vecgen1d ] ) extends by density to a partial isometry from @xmath199 to @xmath730 whose restriction to the range of @xmath750 ( that is , @xmath258 ) is unitary . moreover , @xmath728 diagonalizes the reduced hamiltonian @xmath214 in the sense that for any measurable function @xmath143 , we have @xmath755 where @xmath756 stands for the operator of multiplication by the function @xmath35 in the spectral space @xmath730 . first , notice that the orthogonal projection @xmath750 onto @xmath258 is indeed the spectral projection @xmath757 ( by ( [ eq.indicator ] ) and proposition [ prop.eigvalak ] ) . from proposition [ p.specmesak ] and the definition ( [ eq.vecgen1d ] ) of @xmath758 one can rewrite the spectral measure of @xmath214 for any interval @xmath759 and for all @xmath652 as @xmath760 \displaystyle \sum_{\scz \in \ { \dd,\de,\ei\ } } \int_{\lambda \cap \lambda_\scz(k ) } \sum \limits_{j\in j_{\scz } } |{\mathbb{f}}_k{\boldsymbol{u}}(\lambda , j ) |^2 \,{{\mathrm{d}}}\lambda+ \sum_{\lambda \cap \lambda_{k}(\ee ) } |{\mathbb{f}}_k{\boldsymbol{u}}(\lambda,0 ) |^2 & \mbox{if } |k| > k_{\rm c}. \end{array}\right.\ ] ] in the particular case @xmath761 , we have @xmath762 for all @xmath763 , thus using the definition ( [ eq.defnormhathk ] ) of the norm @xmath764 leads to the following identity @xmath765 hence , as @xmath766 is dense in @xmath199 , @xmath728 extends to a bounded operator on @xmath199 and the latter formula holds for all @xmath767 . thus , @xmath728 is a partial isometry which satisfies @xmath768 and its restriction to the range of @xmath750 is an isometry . in the sequel , as the above expression of the spectral measure depends on @xmath401 , we only detail the case @xmath769 . the case @xmath770 can be dealt with in the same way . using the polarization identity , the above expression of @xmath771 yields that of @xmath772 and the spectral theorem [ th.spec ] then shows that @xmath773 which holds for all @xmath774 ( note that @xmath775 and @xmath750 commute , thus @xmath776 for @xmath774 ) and @xmath777 . using the definition of the inner product in @xmath730 ( see ( [ eq.defnormhathk ] ) ) , this latter formula can be rewritten as @xmath778 which yields ( [ eq.projdiagak1 ] ) . let us prove now that the isometry @xmath779 is unitary , _ i.e. _ , that @xmath728 is surjective or equivalently that @xmath780 is injective . let @xmath781 such that @xmath782 , then @xmath783 we now choose a spectral zone @xmath784 . for any interval @xmath653 , one denotes @xmath785 , the orthogonal projection in @xmath730 corresponding to the multiplication by the indicator function of @xmath173 . we shall show at the end of the proof the commutation property @xmath786 using this relation and ( [ eq.scalarzero ] ) for @xmath787 instead of @xmath788 we get @xmath789 where we have used the definition of inner product in @xmath730 ( cf ( [ eq.defnormhathk ] ) ) . hence , as the last formula holds for any interval @xmath790 , we get @xmath791 @xmath792 in @xmath668 for a.e . @xmath793 the family @xmath794 is clearly linearly independent ( see ( [ eq.psim ] ) and ( [ eq.psip ] ) ) , so is also the family @xmath795 ( see ( [ defwkl ] ) ) . then it follows from ( [ eq : defwijk ] ) that the @xmath796 are linearly independent too . therefore @xmath797 as it holds for any @xmath784 , @xmath798 . hence , @xmath799 is injective and @xmath117 is surjective . it remains to prove ( [ eq.comut ] ) . using ( [ eq.projdiagak1 ] ) with @xmath800 leads to @xmath801 and therefore to @xmath802 where @xmath803 is the orthogonal projection onto the ( closed ) range of @xmath804 to remove @xmath805 , we point out that @xmath806 where the first equality is an immediate consequence of the relations @xmath807 and @xmath808 , whereas the second one is readily deduced from ( [ eq.thspecak ] ) by taking @xmath809 and @xmath810 . in the following proposition , we give an explicit expression of the adjoint @xmath780 of the generalized fourier transform @xmath728 which is a `` recomposition operator '' in the sense that its `` recomposes '' a function @xmath767 from its spectral components @xmath811 which appear as `` coordinates '' on the spectral basis @xmath812 . as @xmath799 is bounded in @xmath730 , if suffices to know it on a dense subspace of @xmath730 . we first introduce @xmath813 the subspace of @xmath730 made of compactly supported functions . then we consider the subspace of functions whose support `` avoids '' values of @xmath372 , namely @xmath814 { \boldsymbol{\hat{{\mathcal{h}}}}}_{k,{\rm d } } : = \big \ { { \boldsymbol{\hat{u}}}\in { \boldsymbol{\hat{{\mathcal{h}}}}}_{k,{\rm c } } \ ; | \ ; \mbox{supp } { \boldsymbol{\hat{u}}}\cap \big ( \ { -{\omega_{\rm m } } , 0 , { \omega_{\rm m}}\ } \cup \ { -\lambda_{\rm c } , \lambda_{\rm c } \ } \big ) \big \ } & \mbox{if } |k| = k_{\rm c}\ , . \end{array}\ ] ] note that @xmath815 is clearly dense in @xmath731 [ prop.adjointtransformfk ] for all @xmath816 , we have @xmath817 \displaystyle \sum_{\scz \in \ { \dd,\de,\ei\ } } \sum \limits_{j\in j_{\scz } } \int _ { \lambda_\scz(k ) } { \boldsymbol{\hat{u}}}(\lambda , j ) \ , \wlkj \,{{\mathrm{d}}}\lambda + \sum_{\lambda \in \lambda_\ee(k ) } { \boldsymbol{\hat{u}}}(\lambda,0)\ , \wlkz & \mbox{if } |k| > k_{\rm c}. \end{array}\right . \label{eq.adjointtransformfk}\ ] ] where the integrals in the right - hand side of ( [ eq.adjointtransformfk ] ) are vector - valued integrals ( bochner integrals @xcite ) with values in @xmath818 [ rem.fk ] ( i ) the reason why we have to restrict ourselves to functions of @xmath815 in ( [ eq.adjointtransformfk ] ) is that the @xmath668-norm of @xmath819 remains uniformly bounded if @xmath376 is restricted to vary in a compact set of @xmath78 that does not intersect the points @xmath820 , neither the points @xmath821 when @xmath615 . on the other hand , these norms blow up as soon as @xmath372 approaches any of these points . for @xmath821 this results from the presence of the wronskian @xmath442 in the denominator of the expression ( [ defwkl ] ) of @xmath822 . for @xmath211 , this is due to the term @xmath823 ( which vanishes for @xmath824 in the same denominator . for @xmath76 , this follows from the fact that @xmath554 blows up when @xmath372 tends to @xmath76 ( cf . ( [ eq.deftheta ] ) and ( [ eq.drude ] ) ) . \(ii ) hence , when @xmath825 , the integrals considered in ( [ eq.adjointtransformfk ] ) , whose integrands are valued in @xmath668 , are bochner integrals @xcite in @xmath668 . however , as @xmath799 is bounded from @xmath730 to @xmath751 the values of these integrals belongs to @xmath199 . by virtue of the density of @xmath815 in @xmath826 the expression of @xmath827 for any @xmath781 follows by approximating @xmath828 by its restrictions to an increasing sequence of compact subsets of @xmath124 as in the definition of @xmath829 of course , the limit we obtain belongs to @xmath199 and does not depend on the sequence ( note that this is similar to the limiting process used to express the usual fourier transform of a @xmath90 function ) . this limit process will be implicitly understood in the sequel . we prove this result in the case @xmath769 , the case @xmath770 can be dealt with in the same way . let @xmath816 . by definition of the adjoint , for all @xmath830 one has @xmath831 and the expression of @xmath832 yields @xmath833 one can permute the duality product in @xmath195 and the bochner integral in @xmath372 to obtain @xmath834 as it holds for any @xmath835 , this yields ( [ eq.adjointtransformfk ] ) . the permutation is here justified by the following arguments : for any @xmath836 and @xmath743 , @xmath837 is a @xmath838 compactly supported function in @xmath372 and the generalized eigenfunctions @xmath671 are uniformly bounded for @xmath839 and @xmath372 on the compact support of @xmath837 . thus , the left - hand side of the duality product is a finite sum of bochner integrals , since the considered integrands ( which are vector - valued in @xmath668 ) are integrable . hence , the permutation follows from a standard property ( fubini s like ) of bochner integrals @xcite . the hard part of the work is now done : for each fixed non zero @xmath401 , we have obtained a diagonal form of the reduced hamiltonian @xmath214 . it remains to gather this collection of results for @xmath213 , which yields a diagonal form of the full hamiltonian @xmath84 . the proper tools to do so are the notions of direct integrals of hilbert spaces and operators ( see , e.g. @xcite ) that we implicitly assume to be known by the reader ( at least their definition and elementary properties ) . the first step consists in rewriting the link ( [ eq.atoak ] ) between @xmath84 and @xmath214 in an abstract form using direct integrals . the partial fourier transform in the @xmath186-direction @xmath191 led us to define @xmath214 for each fixed @xmath401 as an operator in @xmath199 ( see ( [ eq.defh1d ] ) ) . actually , the initial space @xmath118 introduced in ( [ eq.defhxy ] ) is nothing but the tensor product of hilbert spaces @xmath840 or equivalently the ( constant fiber ) integral @xmath841 hence the partial fourier transform @xmath191 appears as a unitary operator from @xmath118 to @xmath842 a vector @xmath843 is simply a @xmath844-uplet analogous to a @xmath197 but depending on the pair of variables @xmath845 instead of @xmath846 . for a.e . @xmath733 , we denote @xmath847 so that @xmath848 in this functional framework , we can gather the family of reduced hamiltonian @xmath214 for @xmath733 as a direct integral of operators @xmath849 defined by @xmath850 which means that for a.e . @xmath733 , @xmath851 where @xmath852 relation ( [ eq.atoak ] ) can then be rewritten in the concise form @xmath853 , or equivalently @xmath854 general properties can now be applied to obtain a diagonal expression of @xmath84 summarized in the following theorem and illustrated by the commutative diagram of figure [ fig.diag2 ] . [ th.diaga ] let @xmath17 denote the orthogonal projection defined in @xmath118 by @xmath855 where @xmath753 is the indicator function of @xmath856 . consider the direct integral @xmath857 of the family of all generalized fourier transforms @xmath728 for @xmath858 ( see theorem [ th.diagak ] ) , that is , @xmath859 then , for any measurable function @xmath143 , we have @xmath860 where @xmath861 thus , the restriction of @xmath117 on the range of @xmath17 is a unitary operator . from the definition of @xmath862 the diagonalization formula ( [ eq.diaga ] ) amounts to @xmath863 to prove this , we start from formula ( [ eq.rel-a-aoplus ] ) which shows that @xmath84 and @xmath849 are unitarily equivalent . so the same holds for @xmath166 and @xmath864 for any measurable function @xmath143 ( see @xcite ) . more precisely , using @xmath865 instead of @xmath866 we have @xmath867 it remains to diagonalize @xmath868 we first use the essential property ( see @xcite ) @xmath869 ( where the domain @xmath870 is defined as @xmath871 in ( [ eq.domaoplus ] ) by replacing @xmath849 and @xmath214 by @xmath864 and @xmath775 ) . roughly speaking , this latter relation means that the functional calculus `` commutes '' with direct integrals of operators . we deduce from this relation that for @xmath872 and @xmath873 , one has @xmath874 hence , using the family of diagonalization formulas ( [ eq.projdiagak1 ] ) , _ i.e. _ , @xmath875 we obtain @xmath876 which shows that @xmath877 where @xmath857 is defined in ( [ eq.defspectralspaceh ] ) . note that this operator is bounded since @xmath878 ( where @xmath879 denotes the essential sup ) because @xmath880 for all @xmath881 . combining the latter relation with ( [ eq.calfuncaaplus ] ) yields ( [ eq.diaga ] ) . theorem [ th.diaga ] is the main result of the present paper . it is formulated here in an abstract form which will become clearer if we make more explicit the various objects involved in this theorem . this is the subject of this section . by ( [ eq.calfuncaaplus ] ) , the orthogonal projection @xmath887 can be equivalently written as @xmath888 where the @xmath750 s are defined in theorem [ th.diagak ] . this shows in particular that the range of @xmath17 is given by @xmath889 where @xmath890 is defined in ( [ eq.vuk ] ) . as @xmath891 , we deduce that @xmath892 in the same way , @xmath893 is described by @xmath894 where @xmath215 and @xmath895 are characterized in proposition [ prop.eigvalak ] . as @xmath896 , we have @xmath897 where @xmath219 is the extension by @xmath76 of a 2d vector field defined on @xmath898 to the whole plane @xmath78 and @xmath899 consider now the spectral space @xmath119 defined in ( [ eq.defspectralspaceh ] ) where each fiber @xmath730 is given in ( [ eq.specspacehk ] ) . the elements of @xmath119 are then vector fields @xmath900 such that @xmath901 each space @xmath730 is composed of @xmath90-spaces defined on the various zones @xmath902 which are vertical sections of the spectral zones @xmath903 represented in figure [ fig.speczones ] ( and defined in ( [ eq.defspeczones ] ) and ( [ eq.defspeczoneszee ] ) ) . the above formula gathers the spaces associated with all sections to create a space of fields defined on the zones @xmath904 @xmath649 indeed , by fubini s theorem , we see that @xmath119 can be identified with the following direct sum : @xmath905 as we did for the generalized eigenfunctions @xmath906 , we denote somewhat abusively by @xmath907 the fields of @xmath119 , where it is understood that @xmath743 while @xmath908 for the various zones @xmath903 , @xmath763 . the norm in @xmath119 can then be rewritten as @xmath909 we show here that , as for the reduced hamiltonian , the generalized fourier transform @xmath117 appears as a `` decomposition '' operator on a family of generalized eigenfunctions of @xmath168 denoted by @xmath910 and constructed from the generalized eigenfunctions @xmath812 of the reduced hamiltonian @xmath214 ( see ( [ eq : defwijk ] ) ) via the following formula : @xmath911 similarly the adjoint @xmath799 is a `` recomposition '' operator in the sense that its `` recomposes '' a function @xmath197 from its spectral components @xmath912 which appears as `` coordinates '' on the spectral basis @xmath910 of @xmath84 . as @xmath117 ( respectively , @xmath799 ) is bounded in @xmath913 ( respectively , @xmath119 ) , if suffices to define it on a dense subspace of @xmath913 ( respectively , @xmath119 ) . consider first the case of the physical space @xmath914 in the same way as the @xmath915-case ( see remark [ rem.wl21d ] ) , we define @xmath916 where @xmath917 for @xmath91 or @xmath918 note that @xmath919 and @xmath920 are dual spaces , the corresponding duality bracket being denoted @xmath921 it is clear that @xmath922 is dense in @xmath118 for all @xmath923 but we shall actually choose @xmath924 the key point is that each function @xmath925 , being bounded , belongs to @xmath919 . for the space @xmath119 , this is a little more tricky . we first define @xmath926 the subspace of @xmath119 made of compactly supported functions and we introduce the lines @xmath927 as well as the finite set @xmath928 . then we define the space @xmath929 since @xmath930 , @xmath931 and @xmath932 have lebesgue measure 0 in @xmath78 , @xmath933 is clearly dense in @xmath934 let @xmath673 the generalized fourier transform @xmath935 of all @xmath936 is explicitly given in each zone @xmath937 @xmath938 by @xmath939 where the @xmath925 s are defined in ( [ eq.defvecgen2d ] ) . furthermore , for all @xmath940 we have @xmath941 where the integrals are bochner integrals with values in @xmath919 . [ p.fandadjoint ] [ propbbw ] the content of remark [ rem.fk ] could be transposed here with obvious changes . in particular , for general @xmath197 or @xmath942 the expressions of @xmath935 or @xmath827 are deduced from the above ones by a limit process on the domain of integration ( exactly as for the usual fourier transform of a square integrable function ) . in the sequel , this process will be implicitly understood when applying formulas ( [ eq.vecgen2d ] ) and ( [ eq.adjointtransform ] ) for general @xmath122 and @xmath943 let @xmath936 with @xmath944 by definition , @xmath945 where @xmath857 is defined in ( [ eq.defspectralspaceh ] ) . hence , in each zone @xmath937 @xmath946 we have @xmath947 note that @xmath948 and @xmath949 where @xmath950 stands for the sobolev space of index @xmath659 , thus @xmath951 . as @xmath952 , @xmath950 is included in @xmath953 the space of continuous function on @xmath954 , so @xmath955 and one can define @xmath956 for all real @xmath401 . hence , using the respective definitions ( [ eq.deffour ] ) and ( [ eq.vecgen1d ] ) of @xmath191 and @xmath728 leads to @xmath957 which yields ( [ eq.vecgen2d ] ) thanks to a fubini s like theorem for bochner integrals ( which applies here since @xmath936 and @xmath958 ) . we now prove ( [ eq.adjointtransform ] ) . recall that the adjoint of a direct integral of operators is the direct integral of their adjoints @xcite . hence , @xmath959 let @xmath960 for a.e . @xmath733 , @xmath961 and its support in @xmath372 is compact . moreover @xmath962 vanishes for @xmath401 large enough . so @xmath963 note that the support of @xmath964 does not contain @xmath76 and @xmath211 , neither @xmath821 when @xmath615 . thus , by proposition [ prop.adjointtransformfk ] , @xmath965 is given by formula ( [ eq.adjointtransformfk ] ) with @xmath964 instead of @xmath828 . then it suffices to apply fubini s theorem again to obtain formula ( [ eq.adjointtransform ] ) , using the fact that the function @xmath966 is bounded in @xmath919 when @xmath376 varies in the support of @xmath828 . the preceding results actually show that the spectrum @xmath177 of @xmath84 is obtained by superposing the spectra @xmath967 of @xmath214 for all @xmath210 more precisely , we have the following property . the spectrum of @xmath84 is the whole real line : @xmath968 the point spectrum @xmath185 is composed of eigenvalues of infinite multiplicity : @xmath969 if @xmath970 and @xmath971 if @xmath61 [ c.spectruma ] it is based on remark [ rem.spec ] . first consider an interval @xmath972 with @xmath973 the diagonalization formula ( [ eq.diaga ] ) applied to the indicator function @xmath974 shows that the spectral projection @xmath975 is given by @xmath976 ( since @xmath977 moreover , from the identification ( [ eq.ident-hath ] ) of the spectral space @xmath978 we see that the operator of multiplication by @xmath979 in @xmath119 can not vanish . hence @xmath980 for all non empty @xmath981 as @xmath177 is closed , we conclude that @xmath968 suppose now that @xmath982 we still have @xmath976 but here , the operator of multiplication by @xmath983 in @xmath119 always vanishes except if @xmath984 and @xmath985 to understand this , first consider a two - dimensional zone @xmath986 for @xmath987 as @xmath988 is one - dimensional , its intersection with @xmath986 has measure zero in @xmath937 so the operator of multiplication by @xmath983 in @xmath989 vanishes . on the other hand , for the one - dimensional zone @xmath990 several situations may occur . if @xmath991 the intersection of @xmath988 and @xmath992 is either empty or consists of two points ( which are symmetric with respect to the @xmath372-axis ) , hence this intersection still have measure zero in @xmath993 if @xmath994 this intersection is empty when @xmath995 whereas it is one half of @xmath992 when @xmath996 ( the half located in the half - plane @xmath997 in the latter case , we see that the range of the projection @xmath998 is isomorphic , via the generalized fourier transform @xmath117 , to the infinite dimensional space @xmath999 . to sum up , if @xmath991 there is no eigenvalue of @xmath84 in @xmath1000 whereas if @xmath994 the only eigenvalues of @xmath84 located in @xmath1001 are @xmath1002 and they both have infinite multiplicity . consider first the free evolution of the system , that is , when @xmath1004 in this case , @xmath1005 for all @xmath1006 where @xmath1007 is the initial state . theorem [ th.diaga ] provides us a diagonal expression of @xmath1008 more precisely , @xmath1009 hence , if we restrict ourselves to initial conditions @xmath1010 ( see ( [ eq.hxydiv ] ) ) , we simply have @xmath1011 thanks to ( [ eq.vecgen2d ] ) and ( [ eq.adjointtransform ] ) , this expression becomes @xmath1012 for simplicity , we have condensed here the various sums in the right - hand side of ( [ eq.adjointtransform ] ) in a single sum which includes the last term corresponding to @xmath1013 ( which is of course abusive for this term , since it is a single integral represented here by a double integral ) . the above expression is a _ generalized eigenfunction expansion _ of @xmath1014 it tells us that @xmath1003 can be represented as a superposition of the time - harmonic waves @xmath1015 modulated by the spectral components @xmath1016 of the initial state . strictly speaking , the above expression is valid if we are in the context of proposition [ p.fandadjoint ] , _ i.e. _ , if @xmath1017 with @xmath1018 and @xmath1019 for general @xmath1020 the limit process mentioned in remark [ propbbw ] is implicitly understood . consider now equation ( [ eq.schro ] ) with a non - zero excitation @xmath1021 for simplicity , we assume zero initial conditions . in this case , the duhamel integral formula ( [ eq.duhamel ] ) writes as @xmath1022 which yields the following generalized eigenfunction expansion : @xmath1023 as mentioned in the introduction , we are especially interested in the case of a time - harmonic excitation which is switched on at an initial time , that is , @xmath1024 for given @xmath1025 and @xmath1026 where @xmath1027 denotes the heaviside step function ( _ i.e. _ , @xmath1028 in this particular situation , duhamel s formula simplifies as @xmath1029 where @xmath1030 is the bounded continuous function defined for non negative @xmath1031 and real @xmath372 by @xmath1032 t\ , { { \rm e}}^{-{{\rm i}}\omega \,t } & \mbox { if } \lambda = \omega . \end{array } \right.\ ] ] the generalized eigenfunction expansion of @xmath1003 then takes the form @xmath1033 in the second part @xcite of the present paper , we will study the asymptotic behavior of this quantity for large time , in particular the validity of the _ limiting amplitude principle _ , that is , the fact that @xmath1003 becomes asymptotically time - harmonic . we can already predict that this principle fails in the particular case @xmath984 if we choose @xmath1034 since @xmath1002 are eigenvalues of infinite multiplicity of @xmath84 ( see corollary [ c.spectruma ] ) . indeed , if @xmath1035 is an eigenfunction associated to @xmath1036 it is readily seen that the above expression becomes @xmath1037 using the expression of @xmath1038 we obtain @xmath1039 which shows that the amplitude of @xmath1003 increases linearly in time . this _ resonance _ phenomenon is compatible with formula ( [ eq.incrlint ] ) which tells us that @xmath113 increases at most linearly in time . it is similar to the resonances which can be observed in a bounded electromagnetic cavity filled with a non - dissipative dielectric medium , when the frequency of the excitation coincides with one of the eigenfrequencies of the cavity . but it is related here to _ surface plasmon polaritons _ which are the waves that propagate at the interface of our drude material and that are described by the eigenfunctions associated with the eigenvalues @xmath1002 of @xmath84 . moreover , unlike the eigenfrequencies of the cavity , these eigenvalues are of infinite multiplicity and embedded in the continuous spectrum . in the case of an unbounded stratified medium composed of standard non - dissipative dielectric materials such nonzero eigenvalues do not exist ( see @xcite ) . in @xcite , we will explore more deeply this phenomenon and we will investigate all the possible behaviors of our system submitted to a periodic excitation . this forthcoming paper is devoted to both _ limiting absorption _ and _ limiting amplitude _ principles , which yield two different but concurring processes that characterize time - harmonic waves . in the limiting absorption principle , the time - harmonic regime is associated to the existence of one - sided limits of the resolvent of the hamiltonian on its continuous spectrum , whereas in the limiting amplitude principle , it appears as an asymptotic behavior for large time of the time - dependent regime . m. cassier , tude de deux problmes de propagation dondes transitoires . 1 : focalisation spatio - temporelle en acoustique . 2 : transmission entre un dilectrique et un mtamatriau ( in french ) . ph.d . thesis , cole polytechnique , 2014 , available online at https://pastel.archives-ouvertes.fr/pastel-01023289 . m. cassier , c. hazard , p. joly and v. vinoles , spectral theory for maxwell s equations at the interface of a metamaterial . part ii : principles of limiting absorption and limiting amplitude . in preparation . y. dermenjian and j. c. guillot , scattering of elastic waves in a perturbed isotropic half space with a free boundary . the limiting absorption principle . math . methods appl . 10 ( 2 ) ( 1988 ) , 87124 . j. dixmier , les algbres doprateurs dans lespace hilbertien , gauthier villars , paris , 1969 . n. dunford and j. t. schwartz , linear operators . part 2 : spectral theory . self adjoint operators in hilbert space , interscience publishers , 1963 . d. m. eidus , the principle of limiting absorption , amer . , 47(2 ) ( 1965 ) , 157191 . d. m. eidus , the principle of limit amplitude , russ . surv . , 24(3 ) ( 1969 ) , 97167 . j. d. jackson , classical electrodynamics , 3rd ed . , wiley , new york , 1998 . b. gralak and a. tip , macroscopic maxwell s equations and negative index materials , j. math . phys . , 51 ( 2010 ) , 052902 . b. gralak and d. maystre , negative index materials and time - harmonic electromagnetic field , c.r . physique , 13(8 ) ( 2012 ) , 786799 . j. c. guillot and p. joly , approximation by finite differences of the propagation of acoustic waves in stratified media , numer . math , 54 ( 1989 ) , 655702 . c. hazard and f. loret , generalized eigenfunction expansions for conservative scattering problems with an application to water waves . section a mathematics , 137(5 ) ( 2007 ) , 9951035 . j. li et y. huang , time - domain finite element methods for maxwell s equations in metamaterials , springer series in computational mathematics , 2013 . s. a. maier , plasmonics : fundamentals and applications , springer , new york , 2007 . k. morgenrther and p. werner , on the principles of limiting absorption and limit amplitude for a class of locally perturbed waveguides . part 2 : time - dependent theory , math . method appl . , 11(1 ) ( 1989 ) , 125 . j. b. pendry , negative refraction makes a perfect lens , phys . lett . , 85(18 ) ( 2000 ) , 3966 . m. radosz , new limiting absorption and limit amplitude principles for periodic operators , z. angew . , 66(2 ) ( 2015 ) , 253275 . j. sanchez hubert and e. sanchez palencia , vibration and coupling of continuous systems , asymptotic methods , springer - verlag , berlin , 1989 . k. schmdgen , unbounded self - adjoint operators on hilbert space , springer , dordrecht , 2012 . v. g. veselago , the electromagnetics of substance with simultaneously negative values of @xmath0 and @xmath1 , soviet physics uspekhi 10(4 ) ( 1968 ) , 509514 . k. k. wan , from micro to macro quantum systems : a unified formalism with superselection rules and its applications , world scientific imperial college press , 2006 . r. weder , spectral and scattering theory for wave propagation in perturbed stratified media , applied mathematical sciences 87 , springer - verlag , new york , 1991 . c. h. wilcox , scattering theory for diffraction gratings , applied mathematical sciences 46 , springer - verlag , new york , 1984 . c. h. wilcox , sound propagation in stratified fluids , applied mathematical sciences 50 , springer - verlag , new york , 1984 . k. yosida , functional analysis , springer - verlag , berlin , 1974 .

we explore the spectral properties of the time - dependent maxwell s equations for a plane interface between a metamaterial represented by the drude model and the vacuum , which fill respectively complementary half - spaces . we construct explicitly a generalized fourier transform which diagonalizes the hamiltonian that describes the propagation of transverse electric waves . this transform appears as an operator of decomposition on a family of generalized eigenfunctions of the problem . it will be used in a forthcoming paper to prove both limiting absorption and limiting amplitude principles . * keywords : * negative index materials ( nims ) , drude model , maxwell equations , generalized eigenfunctions .

bars within bars appear to be a common phenomenon in galaxies . recent surveys indicate that up to 30% of early - type barred galaxies contain such double bars ( erwin & sparke 2002 ; laine et al . inner bars remain distinct in near infrared ( wozniak et al . 1995 ) , therefore fairly old stars must contribute to their light . the relative orientation of the two bars in doubly barred galaxies is random , therefore it is likely that the bars rotate with different pattern speeds . the origin of multiply barred systems remains unclear . a bar takes away angular momentum from gas very efficiently , and in the young universe bars might have been responsible for the early rapid growth of the massive black holes ( begelman , volenteri & rees 2006 ) . if the innermost parts of galactic discs formed first , then early instabilities there might have lead to the formation of small - scale bars , which may be surviving in the present - day universe as nuclear bars , often nested inside larger , outer bars , that might have formed later . studying the dynamics of nested bars can therefore help us to understand the formation of galaxies and of their central massive black holes . our understanding of how double bars are sustained has significantly improved in the recent years . maciejewski & sparke ( 1997 , 2000 ) have developed a formalism that enables finding families of stable regular orbits in such systems . stable regular orbits are robust structures , which define the shape of the galaxy , and therefore they can serve as a backbone for double bars . so far they have been analyzed for the case when the pattern speed of the inner bar is higher than that of the outer bar ( maciejewski & sparke 2000 ) . using n - body simulations , rautiainen et al . ( 2002 ) confirmed that stars get trapped around these orbits , and form long - lasting doubly barred systems . however , in this scenario the kinematics of the inner bar is not a scaled - down copy of that of the main bar , since the inner bar can not extend to its corotation . other numerical simulations have shown that systems of two counter - rotating bars are also possible ( sellwood & merritt 1994 ; friedli 1996 ) , and systems with secondary bars rotating slower than the outer , main bars , have never been excluded on theoretical grounds . various dynamical scenarios for doubly barred galaxies may have different implications for the evolution of the galactic centres . in order to discriminate between them , one should measure the pattern speed of the inner bar . if only one pattern speed is present in the system , the tremaine - weinberg ( 1984 ) method allows to derive it using a set of simple kinematical measurements . recent kinematical observations of the doubly barred galaxy ngc 2950 ( corsini , debattista & aguerri 2003 ; hereafter cda03 ) are inconsistent with one pattern speed there . the observations are suggestive of another pattern speed in the area of the inner bar . cda03 attempted to estimate it , but they concluded a wide range of values , consistent with a fast - rotating prograde secondary bar as well as with a retrograde one . in this paper , i show that a simple extension of the tremaine - weinberg method to multiple pattern speeds , based on the separation of tracer s density , is sufficient to discriminate between prograde and retrograde inner bars . similar extension has been already considered by cda03 , but they did not explore its implications . in section 2 , i outline the extended method , and i show that it immediately gives a qualitative information about the sign of rotation of the inner bar . in section 3 , i calculate the integrals in the original tremaine - weinberg method for the case of a realistic doubly barred galaxy , and i show that their deviation from values for a single rotating pattern is the same as predicted by the extension to the tremaine - weinberg method proposed in section 2 . in section 4 , i examine one method to measure the pattern speed of the inner bar , and i show that it recovers the pattern speed in the model with an acceptable accuracy . limitations of the extended method , other attempts to estimate multiple pattern speeds using the tremaine - weinberg formalism , and consequences of counter - rotating inner bars for the evolution of galactic centres are discussed in section 5 . when the extended tremaine - weinberg method proposed here is applied to ngc 2950 , it indicates that the pattern speed of the inner bar there is smaller than the positive pattern speed of the outer bar , and that it is most likely negative , i.e. the inner bar is counter - rotating with respect to the outer bar and to the disc ( see also maciejewski 2004 ) . the tremaine - weinberg method is designed for one pattern speed , which it derives from the luminosity centroid and the luminosity - weighted line - of - sight velocity of a chosen tracer moving in the galaxy s potential . the method rests on three assumptions : the disc of the galaxy is flat , it has a well - defined pattern speed , and the tracer obeys the continuity equation . under these assumptions , the surface density of the tracer , @xmath2 , can be written as @xmath3 where @xmath4 is the pattern speed , @xmath5 are cartesian coordinates in the disc plane , @xmath6 are polar coordinates there , centred on the galactic centre , and @xmath7 is time . when another pattern speed is introduced , the two patterns can not rotate rigidly one through another ( louis & gerhardt 1988 ; sridhar 1989 ; maciejewski & sparke 2000 ) , and in principle one can not split the right - hand side of ( 1 ) into components with different pattern speeds . however , if a rough separation of patterns is possible , and if one neglects secular evolution , this system is periodic with period @xmath8 , where @xmath9 and @xmath10 are pattern speeds of the two bars . consequently , the surface density of the tracer can generally be written as @xmath11 where @xmath12 denotes dependence on time with periodicity @xmath13 . in appendix a , i evaluate correction to the tremaine - weinberg integrals that arises because of this periodic oscillation of realistic double bars . this correction turns out to be small , and i will neglect it in the following argument . once periodic oscillations of nested bars are neglected , ( 2 ) gets reduced to @xmath14 twofold interpretation of such a separation of tracer s density is possible . either the tracer in the disc ( e.g. stars ) can be divided into subgroups belonging to two unchanging patterns , each rotating with a constant pattern speed , or one can separate radial zones in the galactic disc that rotate with constant pattern speeds , with no net exchange of the tracer between the zones . neither of these interpretations is consistent with the dynamics of galaxies , because patterns rotating one through another change with time , and tracers of each pattern are often present at the same radius . however , as i show in appendix a , these inconsistencies are likely to be small , and the separation ( 3 ) can still be approximately valid . below i will show that although the extension of the tremaine - weinberg method based on the separation of tracer s density ( 3 ) can not recover the two pattern speeds , it can yield a qualitative prediction of how the integrals in the standard tremaine - weinberg method change when a second rotating pattern is introduced . in section 3 , i will show that this change , calculated properly for a realistic model of doubly barred galaxy , with bars oscillating in time , and with tracers of each bar overlapping , is the same as predicted by the extended method proposed here . thus the separation of tracer s density may help in interpreting the tremaine - weinberg integrals in the models and in the observed galaxies . once tracer s density is separated according to ( 3 ) , one can write two separate continuity equations . further derivation is identical to the one performed by tremaine & weinberg ( 1984 ) , and it can be conducted for each tracer s density separately , leading to two equations : @xmath15 @xmath16 where @xmath17 are coordinates on the sky ( @xmath18 running parallel to the line of nodes and @xmath19 being the offset , perpendicular to the line of nodes , of the slit along which the integration is performed ) , @xmath20 is the observed line - of - sight velocity , and @xmath21 is the inclination of the disc . let the tracer be made of stellar - light emission ; then one can define for each bar component , as well as for the whole galaxy , the luminosity - density @xmath22 the luminosity centroid @xmath23 and the luminosity - weighted line - of - sight velocity @xmath24 where the index @xmath25 is @xmath26 for the outer bar , @xmath27 for the inner bar , and @xmath28 for the whole galaxy . obviously , only values with index @xmath28 can be observed , for example by placing a slit parallel to the line of nodes ( fig.1 ) , and deriving along it the values of luminosity , centroid and line - of - sight velocity , where all integrals include integration over the width of the slit . with the definitions above , the sum of ( 4 ) and ( 5 ) takes a form @xmath29 where @xmath30 . cda03 gave a similar equation , but in their convention there are no fractions @xmath31 . note that equation ( 6 ) is not sufficient to derive two pattern speeds @xmath32 by measuring the luminosity centroid @xmath33 and the luminosity - weighted line - of - sight velocity @xmath34 along the slits . although @xmath35 , and it can be measured directly , the pattern speeds are combined with the unknown @xmath36 and @xmath37 . however , the form of equation ( [ twsum ] ) , together with the morphology of the doubly barred galaxy , whose pattern speeds we want to measure , already reveals some information about the relation between the two pattern speeds . the line of nodes passing through the centre of the galaxy sets a division of this galaxy into four quadrants . consider for example the case when the two bars lie in opposite quadrants of the galaxy ( fig.1 ) . this is the case of ngc 2950 . even if we can not measure @xmath38 or @xmath39 , we know that they are always of opposite signs , no matter how the slit is placed ( parallel to the line of nodes ) . then if @xmath10 and @xmath9 are of the same sign , adding the inner bar should bring the sum on the left of ( [ twsum ] ) closer to zero ( or even through zero for strong fast inner bars ) when compared to the contribution of the outer bar alone . this implies that the observed @xmath40 should be smaller when entering the region of the inner bar than that interpolated from the outer bar . to the contrary , cda03 observe @xmath40 _ increasing _ around this transition region in ngc 2950 , consistent with @xmath9 having the sign opposite to @xmath10 , i.e. with the counter - rotating secondary bar . in the argument above , no reference point has been fixed , since the sums on each side of ( [ twsum ] ) get modified by the introduction of the secondary bar . we do not know the contribution from the outer bar in the region where the inner bar is present , and interpolation can be misleading . here i show that ( [ twsum ] ) allows to tell whether @xmath42 or @xmath43 . equation ( [ twsum ] ) can be rewritten as @xmath44 this form shows how the relation between the integrals @xmath34 and @xmath33 gets modified by the presence of the secondary bar : the second term in the sum is a correction term . for slits outside the inner bar this correction is zero ( because @xmath45 is zero there ) , and for those slits the observed values of @xmath33 and @xmath34 should lie on a straight line of inclination @xmath46 . when the slit passes through the inner bar , @xmath47 . if for such slits @xmath40 is larger than the values given by the linear relation ( 7 ) without the correction factor , then the correction has to have the same sign as @xmath48 . for the bars that lie in opposite quadrants , like in ngc 2950 ( fig.1 ) , @xmath38 and @xmath39 are of opposite signs , and @xmath33 has the same sign as @xmath38 throughout the galaxy ( fig.3 in cda03 ) . therefore the two components of the sum in ( 7 ) can only be of the same sign when @xmath10 has opposite sign to @xmath41 . if we take a convention that @xmath49 then @xmath43 . this argument can be extended to the case when @xmath33 changes sign for slits passing through the secondary bar . in ngc 2950 , the values of @xmath33 for these slits are very small , which is consistent with the observed geometry of bars . a similar argument can be applied to galaxies with bars in the same quadrants , where @xmath33 for innermost slits is not that small . in this case , @xmath38 , @xmath39 and @xmath33 are all of the same sign . if an increase of @xmath40 is observed in the region of the inner bar , @xmath10 must now have the same sign as @xmath41 , which means @xmath42 for the assumed @xmath49 . thus we see that the same increase of @xmath40 in the region of the inner bar can either indicate the inner bar rotating slower or faster than the outer bar , depending on the relative orientation of the bars . in some cases , an argument can be made about corotation or counter - rotation of the inner bar with respect to the inertial frame . it relies on the variation of the integrals in the tremaine - weinberg method with the distance of the slit from the centre of the galaxy , @xmath19 . this argument can be made under an assumption that @xmath50 does not increase when we march with the slit through the galaxy toward the line of nodes ( in the direction marked by an arrow on the left of fig.1 ) . this is a justified assumption , since @xmath51 decreases inward for the very reason of the introduction of the secondary bar , and @xmath52 should not go up inward , since early - type galaxies like ngc 2950 have flat bars with a nearly constant surface brightness as a function of radius ( elmegreen et al . 1996 ) . consider again equation ( [ twsum ] ) . normally , for slits that avoid the secondary bar , @xmath40 decreases inward ( i.e. when shifting the slit in the direction given by the arrow in fig.1 ) . this is also the case in ngc 2950 ( fig.3 in cda03 ) . if then @xmath40 _ increases _ inward when the slit reaches the secondary bar ( like in ngc 2950 ) , this can be caused by either of the two components of the sum on the left of ( [ twsum ] ) . i already argued that the first component can not be the cause , since @xmath51 decreases , @xmath52 is unlikely to increase , and @xmath53 . the second component can only cause the increase of the sum when it is of the same sign as the first one . but again , @xmath38 and @xmath39 are of opposite sign . therefore @xmath9 and @xmath10 have to be of opposite sign , too . one can repeat this reasoning for a galaxy with bars in the same quadrants , to show that in that case @xmath40 increasing in the region of the inner bar indicates that the inner bar is corotating in the inertial frame . similar to section 2.1 , the conclusion about the sense of rotation of the inner bar depends on the relative orientation of the bars . regular motion of a particle in a potential of a doubly barred galaxy has two frequencies associated with it , each related to one of the bars , in addition to the frequency of its free oscillations ( maciejewski & sparke 1997 ) . in the linear approximation this motion corresponds to epicyclic oscillations with these frequencies around the guiding radius . particles with the same guiding radii are bound to closed curves ( loops ) that oscillate in the pulsating potential of double bars . loops are also observed in nonlinear analysis ( maciejewski & sparke 2000 ; maciejewski & athanassoula 2006 ) . the orbital approach directly indicates that the motion of a particle associated with one bar has always a component coming from the other bar . amplitudes of the oscillations can be easily evaluated in the linear approximation ( maciejewski 2003 ) , giving particle s position and velocity at each relative position of the bars . in fig.2 , i plot the loops populated by particles moving in the potential of a doubly barred galaxy constructed by maciejewski & sparke ( 2000 ; model 2 ) . this is a realistic potential , since it admits orbits that support the outer bar , as well as orbits supporting the inner bar , throughout the extent of each bar . the two bars in this model rotate in the same direction with pattern speeds @xmath54 km s@xmath1 kpc@xmath1 and @xmath55 km s@xmath1 kpc@xmath1 . i plot the loops for two relative orientations of the bars : the bars parallel ( fig.2 , left ) , and the bars at the angle of @xmath56 rad , the value similar to that observed in ngc 2950 after deprojection ( fig.2 , right ) . the loops change shapes as the bars rotate one with respect to another , and loops associated with one bar intersect the loops associated with the other bar . thus formally one can not perform here the separation of tracer s density postulated in ( 3 ) . however , changes in the shapes of the bars , and zones where both tracers coexist , may be small , and then , what formally prohibits the separation , may turn into a higher - order correction to it ( see also appendix a ) . in order to check whether it is the case for the realistic model 2 ( maciejewski & sparke 2000 ) , i calculated the positions and velocities of some @xmath57 points on 200 loops in that model , from which i obtained the centroids @xmath33 and the line - of - sight velocities @xmath34 along slits placed at the same relative angle to the outer bar as in the cda03 observations of ngc 2950 ( dotted straight lines in fig.2 ) . for clarity , only 28 loops are displayed in fig.2 , out of 200 used in the calculations . the 200 loops were populated with particles , so that a smooth density distribution in the bars is recovered . several recipes have been adopted to populate loops with particles , in order to make sure that the results below do not depend on the way in which the model is constructed . in the model , i use deprojected velocities , which replaces @xmath58 by @xmath34 , and i measure @xmath19 in the plane of the galaxy . in fig.3b , i plot the value of the centroid , @xmath33 , as a function of the offset of the slit from the galaxy centre , @xmath19 . when the inner bar is placed at the position similar to the one observed in ngc 2950 , the value of @xmath33 ( solid line ) is closer to zero in the region of the inner bar ( @xmath59 ) , than when the inner bar is absent ( dot - dashed line ) . however , @xmath33 does not change the sign in the region of the inner bar , which is also observed in ngc 2950 ( fig.3 in cda03 ) . this indicates that the light integrated along a slit passing through the inner bar is still dominated by the outer bar . in the case of the two bars parallel ( fig.2 , left panel ) there is no such decrease of @xmath33 in the region of the inner bar ( dashed line in fig.3b ) . in fig.3a , i plot the luminosity - weighted line - of - sight velocity , @xmath34 , as a function of the same offset of the slit from the galaxy centre , @xmath19 , as in fig.3b . when the inner bar , rotating in the same direction as the outer bar , but faster , is placed at the position similar to the one observed in ngc 2950 , the value of @xmath34 ( solid line ) rapidly approaches zero in the region of the inner bar ( @xmath59 ) , and in most of this region it has the sign opposite to the value of @xmath34 for slits not passing through the inner bar ( @xmath60 ) . this behaviour of @xmath34 is opposite to that observed in ngc 2950 by cda03 . for the model of galaxy with @xmath42 , changes of @xmath34 with @xmath19 observed by cda03 can only be reproduced when both bars lie in the same quadrants of the galaxy . for the particular case of bars parallel , presented here , @xmath40 increases in the region of the inner bar ( dashed line in fig.3a ) , relative to the case when there is no inner bar ( dot - dashed line ) . this dependence of deviations in @xmath34 on the relative position of the two bars is exactly as expected from the simple extension of the tremaine - weinberg method derived section 2 , based on the separation of tracer s density ( 3 ) . namely , if the inner bar rotates faster than the outer bar then , when both bars are in the same quadrants ( defined by the line of nodes and the centre of the galaxy ) , @xmath40 increases in the slits passing through the inner bar , but it _ decreases _ , when the bars are in the opposite quadrants . the plot of @xmath34 as a function of @xmath33 is presented in fig.3c . one may attempt to fit a second straight line in the region of the inner bar , but this fit will not yield the actual pattern speed of the inner bar . in fig.3d , i plot the ratios of @xmath61 as estimators of the derived @xmath62 . for a single bar ( dot - dashed line ) , there is one pattern speed independent of the offset of the slit , and equal to the one assumed in the model . if there are two pattern speeds in the system , the inner bar does not alter significantly the pattern speed of the outer bar derived from the slits that do not pass through this inner bar . however , the induced deviations are systematic : pattern speed of the outer bar is spuriously increased when the bars lie in the same quadrants , and decreased when they lie in quadrants opposite . the effect is small though , below 5% . for the slits passing through the inner bar , the derived @xmath62 rapidly decreases if the bars lie in opposite quadrants ( solid line ) , and it becomes negative , not giving any information about the pattern speed of the inner bar . if the bars lie in the same quadrants ( dashed line ) , the derived @xmath62 is larger than @xmath10 for the slits passing through the inner bar , but it varies with @xmath19 , and never reaches the value of @xmath63 km s@xmath1 kpc@xmath1 assumed in the model , remaining at slightly over half of that value . . values of @xmath64 are plotted against @xmath65 . ] the main goal of this paper is to show that with a simple extension of the tremaine - weinberg method to multiple pattern speeds one can derive at least rough qualitative information about the secondary pattern rotation . for the data on ngc 2950 presented by cda03 this information is that the inner bar rotates in the opposite direction than the outer bar . on the other hand , getting the numerical value of the pattern speed of the secondary bar , @xmath9 , may not be possible without making additional assumptions . here i analyse one possible method to calculate @xmath9 . if the tracer of the secondary bar does not extend beyond some radius @xmath66 in the galaxy plane ( outlined in projection by the dotted ellipse in fig.1 ) , then ( 5 ) can be rewritten as @xmath67 where @xmath68 . summing ( 4 ) and ( 8) leads to @xmath69 however , aside for @xmath9 , we still do not know the integral in ( 9 ) . it can be approximated when one assumes that only tracers of the inner bar and of the axisymmetric component are present inside the dotted ellipse in fig.1 , i.e. that the outer bar is almost axisymmetric in this region . note that this is a strong and poorly founded assumption . however , if we take it , and since the contribution of the axisymmetric component to the integral in ( 9 ) cancels out , one can substitute there the observable @xmath70 for the unknown @xmath45 , and rewrite ( 9 ) as a linear regression of @xmath41 : @xmath71 thus @xmath41 can be obtained as a slope of a straight line fitted to the data from slits passing through the inner bar . a similar equation has been used by cda03 to estimate @xmath9 , but here the coefficient at @xmath41 is defined differently . i tested equation ( 10 ) on the model examined in section 3 , for the position angles of the bars like the ones observed in ngc 2950 . as in section 3 , in the model i use deprojected velocities , which replaces @xmath58 by @xmath34 , and i measure @xmath19 in the plane of the galaxy . on the basis of the shapes of the loops , i chose @xmath72 kpc for this model , and i fixed @xmath73 km s@xmath1 kpc@xmath1 . in fig.4 , i plot the data points for the regression ( 10 ) . they follow a straight line well , except for the points around zero values . however , these points come from @xmath74 , where the integration is over a small number of particles , hence random error is large there . if we exclude these points , the derived @xmath41 oscillates between 60 and 70 km s@xmath1 kpc@xmath1 for @xmath75 kpc . this gives @xmath9 between 95 and 105 km s@xmath1 kpc@xmath1 , consistent with the input value of 110 km s@xmath1 kpc@xmath1 , with an error of about 10% . in order to apply the same method to the observed data on ngc 2950 from cda03 , the value of @xmath66 can be estimated by checking how the integral in ( 10 ) changes with varying @xmath66 it should have a plateau in the transition area between the bars . i used the r - band image of ngc 2950 ( erwin & sparke 2003 ) to calculate this integral and @xmath76 for each of the slits passing through the secondary bar , placed at positions reported by cda03 . the values of @xmath77 as a function of @xmath66 are plotted in fig.5 . all curves indeed have a plateau around the same @xmath66 of 56 arcsec . it is located just outside a nuclear stellar ring at @xmath78 arcsec , reported by erwin & sparke ( 2003 ) , which is likely circular in the plane of the galaxy . thus the method proposed here may be particularly well suited for ngc 2950 . as a function of @xmath79 for ngc 2950 . the curves are derived from the r - band image of ngc 2950 ( erwin & sparke 2003 ) , for slits placed parallel to the line of nodes , and offset from it by values given on the right of the plot in arcsec . two dotted vertical lines indicate a plateau range , common for all the curves . ] each curve in fig.5 has a characteristic u - shape , whose depth depends on the relative contribution of the inner bar to the integral in ( 10 ) , and therefore it vanishes outside the inner bar , and also very close to the line of nodes , where the integral tends to be zero . from the r - band image , the depth is largest for slits offset by @xmath80 arcsec . the linear fit to ( 10 ) , with the data for @xmath34 and @xmath33 from each slit passing through the inner bar provided by cda03 , reveals a large negative value of @xmath41 , about @xmath81 km s@xmath1 arcsec@xmath1 . for @xmath82 km s@xmath1 arcsec@xmath1 , measured by cda03 , this gives @xmath83 km s@xmath1 arcsec@xmath1 . this result , although with much poorer grounds than the qualitative arguments from section 2 , reaffirms counter - rotation of the inner bar . note that the unrealistically large value of @xmath9 derived with this method is likely an overestimate : in order to get it , the integral @xmath84 has been substituted for @xmath85 . the substituted integral is most likely smaller than the original one , because the contributions of the two bars are of opposite signs . the original tremaine - weinberg method , and its extension proposed here , are based on the continuity equation applied to a tracer moving in the gravitational field of the galaxy . if old stars are taken as the tracer , the continuity equation is well satisfied globally . if tracers associated with each bar are separated as proposed in ( 2 ) , continuity of each tracer can only be violated when there is a net flux of stars from one bar to the other , consistent over many rotations of the bars . this corresponds to a secular strengthening of one bar at the cost of the other , hence such a system is no longer periodic . however , if the mass transfer is slow , the analysis presented in this paper is still applicable . this can be supported by an argument similar to that presented in appendix a , but for densities of the bars monotonically changing . moreover , studies of orbits in self - consistent models of double bars show that mass transfer between the bars is likely to be small , because orbits that are trapped and oscillate around one or the other bar , populate large fraction of phase space ( maciejewski & sparke 2000 ; maciejewski & athanassoula 2006 ) . note that the separation of tracers in ( 2 ) is only formal , and ( 2 ) applies to any system that is periodic with period @xmath13 . however , in order to follow the tremaine - weinberg formalism for each tracer , one has to approximate each tracer as a solid - body rotator , which leads to ( 3 ) . the analysis of a realistic doubly barred galactic potential , presented in section 3 , indicates that although this approximation is incorrect in a rigorous sense , deviations from solid - body rotation are small , and they only contribute to higher - order terms in the tremaine - weinberg integrals . namely , even if each particle in the system oscillates with frequencies related to both bars , for most of the particles one frequency dominates , and the separation can be performed . although the presence of two forcing frequencies leads to noticeable pulsation of the inner bar ( e.g. rautiainen et al . 2002 ) , the magnitude of additional terms in the tremaine - weinberg integrals , that this pulsation gives rise to , is much smaller than the magnitude of the leading terms , as argued in appendix a. the radial separation of the tracers , necessary for the estimate of the pattern speed of the inner bar in section 4 , can also be possible , if the zones , where two tracers coexist , occupy small fraction of the galaxy . the larger this fraction , the poorer is the estimate of the inner bar s pattern speed . further testing of this method will require a fully self - consistent model with an inner bar . this paper shows that the change in value of the integrals @xmath33 and @xmath34 in the original tremaine - weinberg formalism , caused by the second pattern speed , depends on the relative position of the bars . in particular , i showed that fitting another straight line in the @xmath86 diagram to the data from slits passing through the inner bar _ does not _ yield the pattern speed of this bar . therefore recent derivations of multiple pattern speeds based on such analysis ( rand & wallin 2004 ; hernandez et al . 2005 ) have to be treated with caution . the retrograde rotation of the inner bar in ngc 2950 proposed here is inconsistent with the first estimate of @xmath9 by cda03 . however , in that estimate it is assumed that the inner bar dominates the light in the slits passing through it . this can not be correct , since @xmath33 and @xmath39 are of opposite sign there , which indicates that the light from the outer bar still dominates in the slits passing through the inner bar . the second estimate of cda03 , like the estimate given in this paper , indicates that the inner bar is counter - rotating . the obvious next step in verifying counter - rotation of the inner bar in ngc 2950 would be an examination of orbital structure of a galaxy with a retrograde inner bar , following the method presented in section 3 . however , a bar counter - rotating with respect to its stellar disc has to be built out of so called @xmath87 orbits , which are normally very close to circular ( see e.g. sellwood & wilkinson 1993 ) . in terms of the linear approximation of section 3 , the amplitude of oscillation around the guiding radius generated by a retrograde bar is very small ( see equations ( 12 ) and ( 13 ) in maciejewski 2003 ) . thus a construction of a retrograde bar in a prograde stellar disc is unlikely . this is consistent with n - body simulations , in which multiple pattern speeds form naturally in stellar discs ( rautiainen et al . 2002 ) these simulations do not recover retrograde pattern speeds . this leads to another possibility , namely that the inner bar is formed out of an inner disc that counter - rotates with respect to the outer disc . numerical results indicate that two counter - rotating bars , formed in two counter - rotating stellar discs that overlap each other , can survive in galaxies for many rotation periods ( sellwood & merritt 1994 ; friedli 1996 ) . if this is the case , one should find out whether a retrograde inner disc is consistent with the rotation curve and velocity dispersion in ngc 2950 . the data presented by cda03 do not indicate counter - rotation in the innermost few arc - seconds of ngc 2950 . if further kinematical studies of ngc 2950 confirm counter - rotation in its innermost parts , then extending the tremaine - weinberg method along the lines proposed here may yield an efficient way of detecting counter - rotation in galaxies with nested bars . thus far all detections of counter - rotation in galaxies come directly from the observed velocity fields obtained with the long - slit ( e.g. kuijken , fisher & merrifield 1996 ) or integral - field ( e.g. emsellem et al . 2004 ) spectroscopy . however , if the counter - rotating population is small , it may not be recognized with those methods . on the other hand , the tremaine - weinberg method goes beyond the raw observed velocity field by finding integrals that are useful in detecting a rotating pattern . in the presence of rotating patterns counter - rotation should be spotted more easily with this method . the use of this method , like of the original tremaine - weinberg method , is limited to galaxies with bars considerably inclined to both the major and minor axes of the disc . ngc 2950 fulfils this requirement particularly well , but there is a number of other galaxies to which this method can be readily applied . for example ngc 3368 , ngc 3941 , ngc 5365 , ngc 5850 , and ngc 6684 all have the position angles of both bars separated by 20 to 70 degrees from the position angle of the line of nodes ( see the catalogue by erwin 2004 for the parameters ) . the bar inside the oval in ngc 3081 has also parameters favourable for this method . in addition , a modification of the tremaine - weinberg method along the lines proposed in section 4 can be applied to the interiors of nuclear rings that often host nuclear bars ( e.g. ngc 1097 , ngc 6782 ) . the fraction of counter - rotating inner bars can constrain theories of galaxy formation and evolution . currently the most accepted view on the origin of the inner bar is that it forms through instabilities in gas inflowing along the outer bar ( shlosman , frank & begelman 1989 ) . however , if inner bars form on early stages of galaxy assembling , and outer bars form from material that settled on the galaxy later , the spins of the two bars may be unrelated , leading to a large fraction of counter - rotating inner bars . determining this fraction may help to distinguish between these two evolutionary scenarios . in this paper i presented an argument that a simple extension of the tremaine - weinberg method to multiple pattern speeds can provide an information about the sense of rotation of the inner bar in doubly barred galaxies . the extended formula advocated here links the relative position of the bars to the sign of rotation of the inner bar . this extension can not be as rigorous as the original method , because it assumes that the patterns do not change as they rotate one through another , which is not true in general . however , it predicts the same deviations of the integrals in the tremaine - weinberg method from their values for the single rotating pattern , as in the orbital model of a realistic doubly barred galaxy , which does not involve this assumption . this indicates that the degree of change that the patterns rotating one through another undergo is small , and that it does not affect significantly the results of the extended method proposed here . application of the extended method to ngc 2950 implies that the inner bar there counter - rotates with respect to the outer bar and to the large - scale disc . since a retrograde bar is unlikely to be supported in a prograde disc , a retrograde inner disc may be hiding in the central kiloparsecs of ngc 2950 . * acknowledgements . * i wish to thank peter erwin for letting me use the r - band image of ngc 2950 , which he obtained with the wyin telescope , and for the list of other galaxies , to which this method can be applied . i am grateful to the authors of the cda03 paper for sharing the details of their observations with me . discussions with linda sparke improved presentation of this argument . this work was partially supported by the polish committee for scientific research as a research project 1 p03d 007 26 in the years 20042007 . begelman m.c . , volonteri m. , rees m.j . , 2006 , mnras , 10.1111/j.1365 - 2966.2006.10467.x corsini e.m . , debattista v.p . , aguerri j.a.l . , 2003 , apj , 599 , l29 elmegreen d.m . , elmegreen b.g , chromey f.r . , hasselbacher d.a . , bisssel b.a . , 1996 , aj , 111 , 2233 emsellem e. , et al . , 2004 , mnras , 352 , 721 erwin p. , 2004 , a&a , 415 , 941 erwin p. , sparke , l.s . , 2002 , aj , 124 , 65 erwin p. , sparke , l.s . , 2003 , apjs , 146 , 299 friedli d. , 1996 , a&a , 312 , 761 hernandez o. , wozniak h. , carignan c. , amram p. , chemin l. , daigle o. , 2005 , apj , 632 , 253 kuijken k. , fisher d. , merrifield m.r . , 1996 , mnras , 283 , 543 laine s. , shlosman i. , knapen j.h . , peletier , r.f . , 2002 , apj , 567 , 97 louis p.d . , gerhard o.e . , 1988 , mnras , 233 , 337 maciejewski w. , 2003 , in : contopoulos g. , voglis n. ( eds . ) lecture notes in physics vol . 626 , springer verlag , berlin , 91 maciejewski w. , 2004 , in : block d.l . , puerari i. , freeman k.c . , groess r. , block e.k . astrophys . and space sci . lib . vol.319 , springer verlag , berlin , 175 maciejewski w. , athanassoula e. , 2006 , in preparation maciejewski w. , sparke l.s . , 1997 , apj , 484 , l117 maciejewski w. , sparke l.s . , 2000 , mnras , 313 , 745 rautiainen p. , salo h. , laurikainen e. , 2002 , mnras , 337 , 1233 rand r.j . , wallin j.f . , 2004 , apj , 614 , 142 sellwood j. a. , merritt , d. , 1994 , apj , 425 , 530 sellwood j. a. , wilkinson a. , 1993 , rep . , 56 , 173 shlosman i. , frank j. , begelman m.c . , 1989 , nature , 338 , 45 sridhar s. , 1989 , mnras , 238 , 1159 tremaine s. , weinberg m.d . , 1984 , apj , 282 , l5 wozniak h. , friedli d. , martinet l. , martin p. , bratschi p. , 1995 , a&as , 111 , 115 consider a ferrers bar with major and minor axes @xmath88 and @xmath89 . its surface density can be written in the cartesian coordinates @xmath5 in the plane of the galaxy as @xmath90 where @xmath91=const is the central density of the bar . pulsation of the bar can be described by periodic variation of the length of its axes @xmath92 where @xmath93 and @xmath94 are periodic functions of time with period @xmath13 . thus non - zero density , corresponding to the top line in ( a1 ) , of a pulsating bar that rotates with pattern speed @xmath10 , can be written in polar coordinates @xmath95 as @xmath96 , \nonumber\end{aligned}\ ] ] where @xmath97 note that ( a2 ) is one of the components of ( 2 ) for the particular case of a ferrers bar . let ( a2 ) , wherever larger than zero , describe non - zero surface density of a tracer in a rotating and pulsating bar . in the tremaine - weinberg method , the time derivative of this surface density enters the continuity equation . simple partial derivation of ( a2 ) gives @xmath98 further on in the tremaine - weinberg method , the continuity equation is integrated over @xmath99 and @xmath100 , which in the case considered here leads to @xmath101 where @xmath102 is the velocity of the tracer along the @xmath100 axis . ( a4 ) is the counterpart of ( 4 ) and ( 5 ) , and since bar s pulsation is explicitly included here , it contains two additional correction terms ( second and third term ) , involving integration over @xmath103 and @xmath104 . this integration , although formally extending to infinity , in this case is limited to the regions where bar s surface density is non - zero . thus since @xmath103 and @xmath104 contain only rotated cartesian coordinates , the integrals that involve them are finite , and will be denoted as @xmath105 and @xmath106 . moreover , let us represent the pulsation of the bar by a simple oscillatory form of @xmath93 and @xmath94 : @xmath107 where @xmath108 , and @xmath109 controls the amplitude of the oscillation . then , with the use of notation from section 2 , ( a4 ) takes the form @xmath110 similar exercise can be done for the second bar , indexed by @xmath27 , leading to an equation being a counterpart to ( a6 ) @xmath111 the sum of these two equations gives ( 6 ) , the equation of the extended tremaine - weinberg method , but here with correction terms , one for each bar . in both ( a6 ) and ( a7 ) , the correction term is the second term . for each bar , the integrals @xmath112 and @xmath113 can be evaluated explicitly , indicating that @xmath114 is about twice smaller than @xmath115 throughout each bar . numerical simulations indicate that the overall pulsation of the inner bar is more noticeable than that of the outer bar , hence the magnitude of the coefficient @xmath116 should be larger for the inner bar . in the numerical n - body simulations by rautiainen et al.(2002 ) , the axial ratio of the inner bar , @xmath117 , varies roughly between 0.52 and 0.72 . in the notation adopted here , this corresponds to @xmath118 . finally , @xmath119 is always smaller than @xmath120 . thus the magnitude of the second , correction term in ( a7 ) should not exceed about 15% of the magnitude of the first , leading term in the case of the inner bar . for the outer bar , the ratio of @xmath121 is larger , but since the pulsation of that bar has smaller amplitude , @xmath109 is significantly smaller . this example shows that periodic changes in time of surface density of the tracer of each bar in this bar s reference frame can be accommodated as correction terms in the extended tremaine - weinberg formalism proposed in this paper , given that the amplitude of these changes is sufficiently small . this amplitude , as observed in numerical simulations , is indeed small enough , and it leads to correction terms that are one order of magnitude smaller than the leading terms in the extended tremaine - weinberg equation ( 6 ) .

when integrals in the standard tremaine - weinberg method are evaluated for the case of a realistic model of a doubly barred galaxy , their modifications introduced by the second rotating pattern are in accord with what can be derived from a simple extension of that method , based on separation of tracer s density . this extension yields a qualitative argument that discriminates between prograde and retrograde inner bars . however , the estimate of the value of inner bar s pattern speed requires further assumptions . when this extension of the tremaine - weinberg method is applied to the recent observation of the doubly barred galaxy ngc 2950 , it indicates that the inner bar there is counter - rotating , possibly with the pattern speed of @xmath0 km s@xmath1 arcsec@xmath1 . the occurrence of counter - rotating inner bars can constrain theories of galaxy formation . galaxies : individual ( ngc 2950 ) galaxies : kinematics and dynamics galaxies : structure

one of the most important goals in the study of star formation is to understand the state and physical conditions of the molecular cloud cores from which the stars form . the prevailing view concerning low - mass - star - forming cores is that they are quasi - static equilibrium configurations supported against gravitational collapse by a combination of magnetic , thermal and turbulent pressures ( e.g. , mouschovias 1976a , b ; shu , adams & lizano 1987 ) . when considering only thermal pressure , two variants of the equilibrium structures are usually discussed : either singular isothermal structures , with diverging central densities and smooth @xmath1 density dependence extending to infinity ( e.g. , shu et al . 1987 ) , or finite - central density structures , truncated at some finite radius and confined by the pressure of some external medium , generally assumed to be at higher temperatures and lower densities than the isothermal core ( ebert 1955 ; bonnor 1956 ) . more recently , the equilibria of non - axisymmetric configurations have also been studied ( e.g. , fiege & pudritz 2000 ; curry 2000 ; galli et al . 2001 ; shadmehri & ghanbari 2001 ; lombardi & bertin 2001 ; curry & stahler 2001 ) . the support from magnetic fields is generally included through the consideration of the mass - to - magnetic flux ratio of the core , since , assuming that the latter has a fixed mass , the flux freezing condition implies that its mass - to - flux ratio is constant ( chandrasekhar & fermi 1953 ; mestel & spitzer 1956 ) . under isothermal conditions , the magnetic pressure and the gravitational energy scale as the same power of the core s volume ; thus , self - gravity can not overcome the magnetic support if the mass - to - flux ratio is smaller than some critical value , and collapse can only occur as the magnetic flux diffuses out of the cloud by ambipolar diffusion ( see , e.g. , mestel & spitzer 1956 ; mouschovias & spitzer 1976 ; shu , adams & lizano 1987 ) . on the other hand , it is well established that the molecular clouds within which the cores form are turbulent , with linewidths that are supersonic for scales @xmath2 pc ( e.g. , larson 1981 ) , and with ( magnetohydrodynamic ) turbulent motions providing most of the support against gravity , with only a minor role of thermal pressure at all but the smallest ( @xmath3 pc ) scales . thus , there appears to be a conceptual gap between the turbulent nature of the clouds and the quasi - hydrostatic assumed nature of the cores . the cores in molecular clouds must be subject to global motions and distortions , as well as mass exchange with its surroundings ( in general , to continuous `` morphing '' ) , and , in fact , are likely to be themselves the turbulent density fluctuations within the clouds ( von weizscker 1951 ; bania & lyon 1980 ; scalo 1987 ; elmegreen 1993 ; , & scalo 1999 , hereafter bvs99 ; padoan et al.2001 ) . at present , one interpretation is that the cores are the dissipative end of the turbulent cascade , because the velocity dispersion within them becomes sonic or subsonic ( e.g. , goodman et al.1998 ) . however , in actuality , substructure is seen down to the smallest resolved scales ( e.g. , falgarone , puget & prault 1992 ) , and appears even within what were previously considered to be `` smooth '' cores , as the resolution is improved ( wilner et al . also , inflow motions , themselves with substructure , are generally seen around these cores ( e.g. myers , evans & ohashi 2000 ) . moreover , if the transonic cores are part of a compressible cascade , they do not need to be the dissipative end of it , but may simply mark the transition to a regime of nearly incompressible turbulence ( , & klessen 2002 , 2003 ) . this issue also poses a problem for the idea of confining clumps by turbulent pressure , since the latter is in general anisotropic and transient at large scales . in this regard , it is worth remarking that a frequent interpretation of the role of turbulent pressure in `` confining '' cores is that the total thermal - plus - turbulent pressure is larger outside a core than inside it , because the turbulent velocity dispersion increases with size . this is , however , an incorrect interpretation , as the dependence of turbulent pressure with size scale is a non - local property referring to statistical averages over domains of a given size , not to a gradient of the local value of the velocity dispersion as larger distances from the core s center are considered . if the density peaks ( clumps and cores ) within molecular clouds have a dynamic origin , then an immediate question is whether they can ever reach hydrostatic equilibrium . several pieces of evidence suggest that this is not possible . first , tohline et al . ( 1987 ) considered the potential energy curve of an initially gravitationally - stable fluid parcel in a radiative medium characterized by an effective adiabatic ( or `` polytropic '' ) exponent , showing that it has a `` thermal energy barrier '' that must be overcome , say by an increase in the external turbulent ram pressure , in order to push the parcel into gravitational collapse . in particular , these authors estimated the mach numbers required for this to occur . although those authors did not discuss it , the production of a hydrostatic configuration within this framework would require hitting precisely the tip of such `` barrier '' , the probability of which is vanishingly small , because the tips of potential barriers constitute unstable equilibria . second , although shu ( 1977 ) has argued that the singular isothermal sphere is the state asymptotically approached by the flow as it seeks to establish detailed mechanical balance when its parts can communicate subsonically with one another , the maintenance of this configuration for long times seems highly unlikely , as this configuration constitutes an _ unstable _ equilibrium , being the precursor of gravitational collapse . if the formation of the core is a dynamical process , no reason exists for the flow to relax onto an unstable equilibrium . such a state can be used as an initial condition in simulations of gravitational collapse , but does not represent itself a realistic state that can be reached by a gas parcel in a turbulent medium . third , clarke & pringle ( 1997 ) have pointed out that cores cool mainly through optically thick lines , but are heated by cosmic rays , and therefore may be dynamically unstable , as velocity gradients may enhance local cooling . fourth , numerical simulations of self - gravitating , turbulent clouds ( e.g. , et al . 1996 ; klessen , heitsch & mac low 2000 ; heitsch , mac low & klessen 2001 ; bate et al . 2002 ) never show the production of hydrostatic objects . instead , once a fluid parcel is compressed strongly enough to become gravitationally bound , it proceeds to collapse right away . specifically , bvs99 suggested that hydrostatic structures can not be formed by turbulent compressions in polytropic flows , in which the pressure is given by @xmath4 , where @xmath5 is the mass density and @xmath6 is the effective polytropic exponent . this is because the collapse of an initially_stable _ gas parcel can only be induced ( i.e , the parcel made unstable ) by a ( strong enough ) mechanical compression if @xmath7 , where the value of @xmath8 depends on the dimensionality of the compression and the specific heat ratio of the gas ( see , e.g. , , passot & pouquet 1996 ) . however , once collapse has been initiated , it can not be halted unless @xmath6 changes in the process , to become larger than @xmath8 again . in other words , for non - isothermal situations , with @xmath9 , equilibria can be found even if the external pressure is time variable . this is why stars can be formed as stable entities from highly anisotropic , dynamic , time - dependent accretion ( hartmann , & bergin 2001 ) . for systems that are much closer to isothermal , such as molecular cloud cores , the boundary pressures are indispensible in establishing stable equilibria , which are therefore not expected to exist in an isothermal turbulent medium with a fluctuating ram pressure . in fact , an analysis of the energy contents of the clouds in numerical simulations shows that they are in near energy equipartition but nowhere near virial _ equilibirum _ ( ve ) ( see ballesteros - paredes & vzquez - semadeni 1995 ; 1997 ; shadmheri , vzquez - semadeni & ballesteros - paredes 2003 ) . this suggests that observations of rough energy equipartition ( e.g. , myers & goodman 1988 ) does not necessarily imply that clouds are in such detailed mechanical balance . the only case when numerical studies show the formation of ( magneto)static structures occurs in simulations of super - jeans , yet subcritical clouds ( e.g. , ostriker , gammie & stone 1999 ) , in which the whole box is subcritical . however , as we discuss in [ sec : magn ] , we believe that this is an artifact of the simulations being performed in closed boxes that do not allow further mass accretion until the system becomes supercritical . in this paper , we provide further arguments against the possibility of molecular cloud cores being hydrostatic entities , and argue in favor of them being instead transients , although with low ( subsonic ) internal velocity dispersion . the plan of the paper is as follows : in [ sec : trunc_ext ] we argue against the possibility of truncated bonnor - ebert - type configurations arising in nearly single - temperature molecular clouds , suggesting instead that cores must either be shock - confined or else have smooth(extended ) density profiles , and then discuss the stability of extended structures , noting that unstable equilibria are not expected to arise in turbulent media . in [ sec : re - exp ] we give a crude estimate of the re - expansion time of density peaks ( cores ) that are not sufficiently compressed to undergo gravitational collapse . in [ sec : magn ] we then discuss the magnetic case , arguing that the subcritical case is also just a transient , on the basis of previous results existing in the literature . then , in [ sec : dyn_scen ] we discuss how the proposed dynamical nature of the cores is not inconsistent with observations , and finally , in [ sec : conclusions ] , we summarize our results and give some conclusions . the notion that molecular cloud cores are nearly - hydrostatic structures can probably be traced back to the classical work of ebert ( 1955 ) and bonnor ( 1956 ) . these authors independently studied the stability _ against gravitational collapse _ of truncated isothermal configurations ( bonnor - ebert , or be , spheres ) bounded by a wall or by a tenuous hot medium capable of maintaining pressure balance at the sphere s boundary but without contributing appreciably to the self - gravity of the system . the be analysis shows that such structures are stable for @xmath10 , where @xmath11 $ ] is a nondimensional radial variable , normalized by the jeans length at the central density @xmath12 . a less remembered fact is that the presence of the wall or hot confining medium is indispensible in order to prevent instability towards _ reexpansion_. indeed , a simple application of the virial theorem to the case of a hydrostatic self - gravitating sphere in the _ absence _ of a confining medium shows that this case is always unstable for media obeying a polytropic equation ( @xmath4 ) with @xmath13 ( which includes the isothermal case , @xmath14 ) . upon a perturbation in its volume , such a hydrostatic sphere will engage in either runaway expansion or runaway contraction ( collapse ) . now , since molecular clouds are quite close to being isothermal ( e.g. , myers 1978 ; pratap et al . 1997 ; scalo et al . 1998 ) , the cores within them are essentially at the same temperature as their `` confining '' medium . thus , no thermal discontinuity can occur analogous to that required by the be configuration . the only possibility for a discontinuity within a single - temperature medium is for there to exist a shock at the boundary , across which both the density and the pressure change discontinuously , with the pressure jump being supplemented by ram pressure . in what follows , we do not discuss this possibility any further , as it already amounts to a dynamic , rather than hydrostatic , situation . we should note that another possibility exists for the realization of a be - type configuration , namely that the core is really `` trapped '' within a hotter region , as is the case of the well known globule barnard 68 ( alves , lada & lada 2001 ) . in this case , the surrounding medium can indeed confine the globule without contributing significantly to the gravitational potential . however , in the general molecular - cloud case of a confining medium at the same temperature as the core , this possibility is excluded . thus , we conclude that the cores within molecular clouds must either be shock - confined ( and thus transient ) , or have smooth density profiles , extending in principle to infinity , asymptotically approaching a zero background density . we refer to these as `` extended '' profiles . in practice , molecular cloud cores have densities at least 100 times larger than the average molecular cloud density , so the assumtion of vanishing background density appears reasonable . we now wish to consider the stability of extended density configurations . the best - known of these is the singular isothermal sphere , which has an @xmath1 density profile , and is known to be gravitationally unstable ( see , e.g. , shu 1977 ) . since we are interested in density fluctuations produced by turbulent compressions , we require them to have a finite central density . these configurations can be obtained by numerically integrating the lane - emden equation to arbitrarily large radii . however , this implies that they are equivalent to a be sphere of arbitrarily large radius , and are thus gravitationally unstable ( bonnor 1956 ) is well into the unstable regime for be - spheres . ] ( we thank d. galli for suggesting this argument ) . we conclude that _ all _ extended hydrostatic configurations in a single - temperature medium are gravitationally unstable . it is worth comparing this situation with that of the be configurations , which have a well - defined range of ratios of central - to - boundary densities for which the configuration is stable . this occurs because the confinement of the sphere by a hot tenuous medium circumvents the need to satisfy hydrostatic equilibrium at all distances from the center ; i.e. , the hydrostatic condition is only imposed out to the `` boundary '' radius . we see that the existence of a stable range of be spheres is precisely allowed by the truncation . in the standard picture of low - mass star formation , some equilibrium state of the kind described above is usually taken as the initial condition for subsequent collapse . however , we see that , since extended configurations are unstable , no reason exists for them to ever be reached if they are originated by dynamic turbulent compressions . moreover , as we have discussed , truncated configurations are inapplicable within single - temperature media . thus , the above arguments are suggestive that _ the initial conditions for star formation should be dynamical in general , rather than quasi - static . _ the arguments above would seem to suggest that all density fluctuations within a turbulent molecular cloud should collapse . however , it is easy to see that this need not be the case , if one is willing to relax the requirement of hydrostatic equilibrium . for example , consider any stable be sphere , i.e. , of radius smaller than the critical one , and surround it with a medium with a steeper density profile than that of equilibrium , say a power law @xmath15 , with @xmath16 , but maintaining pressure continuity . in this case , the configuration has less than one jeans mass at every radius , and will proceed to re - expansion . thus , if such a configuration is produced by a turbulent compression , its fate is to re - expand , after the compression subsides . it is important to note , however , that the re - expansion process must occur on a longer time scale than the compression because of the retarding action of self - gravity . this implies that cores that are compressed to conditions close to those of instability are expected to have somewhat longer lifetimes than those that proceed to collapse , and thus suggests that possibly the majority of the cores observed in molecular clouds are _ not _ on their way to collapse , but rather towards re - expansion and merging into their ambient medium . it is thus of interest to estimate their extended lifetimes . a crude estimate of the re - expansion time can be given in terms of the virial theorem ( vt ) , because in this case we are interested in the characteristic growth time of an unstable equilibrium configuration . the vt allows the description of this situation , in particular of the case in which the evolution is towards re - expansion , by consideration of a gas sphere in equilibrium between its self - gravity and internal pressure exclusively . as mentioned above , this configuration is unstable , and can evolve either towards collapse or re - expansion upon a perturbation of the sphere s volume . thus , the vt in this case provides a reasonable approximation to the case of an unstable extended configuration . we can then proceed as follows . the vt for an isothermal spherical gas mass ( `` cloud '' ) of volume @xmath17 and mean density @xmath18 in empty space is @xmath19 where the overhead dots indicate time derivatives , @xmath20 is the cloud s mass , @xmath21 is its radius , @xmath22 is its moment of inertia , @xmath23 is the sound speed , and @xmath24 is a factor of order unity . we can obtain an evolution equation for the cloud s radius by replacing the radius - dependent density by its mean value in the expression for @xmath25 , to find @xmath26 . thus , @xmath27 . \label{eq : i_r}\ ] ] equating equations ( [ eq : vt ] ) and ( [ eq : i_r ] ) , we obtain @xmath28 = 3c^2 -\alpha gm / r . \label{eq : r_evol}\ ] ] this equation can be integrated analytically , with solution @xmath29 \label{eq : sol_r_evol}\end{aligned}\ ] ] where @xmath30 and @xmath31 are the initial and final radii of expansion , normalized to the equilibrium radius @xmath32 , and @xmath33 is the time , non - dimensionalized to the free - fall time @xmath34 . the characteristic re - expansion time can be defined as the time required to double the initial radius ( i.e. , @xmath35 ) , starting from an initial condition @xmath36 . figure [ fig : re - exp ] shows this characteristic time as a function of @xmath30 . we see that when @xmath30 is very close to unity ( i.e. , linear perturbations from the equilibrium radius ) , the re - expansion time can be up to a few times the free - fall time . moderately nonlinear perturbations have the shortest re - expansion times , because the initial force imbalance is greater , yet the final size is still not much larger than twice the equilibrium radius . finally , for larger initial radii ( far from the equilibrium value ) , the re - expansion time approaches that of free expansion at the sound speed . we conclude that the re - expansion time is at least larger than twice the free - fall time , making the probability of observing a core in this process larger by this factor than that of observing a free - falling core , in agreement with the fact that molecular clouds are generally observed to contain more starless than star - forming cores ( e.g. , taylor , morata & williams 1996 ; lee & myers 1999 ; see also evans 1999 and references therein ) . in the magnetic case , the classical virial - theorem ( vt ) analysis ( chandrasekhar & fermi 1953 ; spitzer 1968 ; mouschovias 1976a , b ; mouschovias & spitzer 1976 ; zweibel 1990 ) predicts the existence of sub- and super - critical configurations ( shu et al . 1987 ; lizano & shu 1989 ) depending on whether the mass - to - magnetic flux ratio is below or above a critical value @xmath37 . subcritical configurations are known not to be able to collapse gravitationally unless the magnetic flux is lost by some process such as ambipolar diffusion . supercritical configurations , on the other hand , are analogous to the non - magnetic case , except for the fact that the cloud behaves as if having an `` effective '' mass , reduced by an amount equal to the critical mass ( which depends on the magnetic field strength ) . the vt analysis , however , has the same problem as the be treatment in that it neglects to satisfy the hydrostatic condition beyond the cloud radius , and so it is not applicable for cores within clouds at their same temperature . we are thus faced again with the need to consider extended configurations . the equilibria of magnetically supported cores is significantly more complicated than that of non - magnetic ones because of the anisotropy introduced by the field , and the many possible field geometries . instability analyses of extended magneto - static layers and cylinders with a variety of geometrical field configurations have been performed by many workers ( e.g. , chandrasekhar & fermi 1953 ; nakamura , hanawa & nakano 1991 , 1993 ; nakajima & hanawa 1996 ; gehman , adams & watkins 1996 ; nagai , inutsuka & miyama 1998 ) . the layers are in general unstable , although long - lived intermediate filamentary states in route to collapse have been reported ( nakajima & hanawa 1996 ) . nevertheless , it is clear that in a turbulent medium , there is no reason for the unstable equilibrium configurations to arise . the structures of greatest interest here may be the intermediate , long - lived structures mentioned above , arising not from the collapse of magneto - static initial states , but of dynamically - produced structures . moreover , additional considerations suggest that the very concept of subcritical cores may not be realized in practice within molecular clouds if the latter are supercritical as a whole , as it appears to be both from obervations ( crutcher 1999 ; bourke et al . 2001 ; crutcher , heiles & troland 2002 ) and from theoretical considerations ( nakano 1998 ) . indeed , the vt treatment giving stability below the critical mass - flux ratio assumes a fixed mass for the cloud or core under consideration . instead , a core that forms part of a larger cloud , has a mass that is not fixed , and continued accretion along magnetic field lines can occur until the core becomes supercritical ( hartmann et al . 2001 ) . this possibility appears to be supported by recent numerical simulations of mhd turbulent flows ( padoan & nordlund 1999 ; ostriker , stone & gammie 2001 ; passot & 2002 ) , which have shown that the magnetic field is essentially decorrelated from the density . passot & ( 2002 ) have explained this phenomenon in terms of the different scalings of the magnetic pressure with density for the different mhd wave modes , and shown that for the slow mode the magnetic pressure can actually be _ anti-_correlated with the density . additionally , numerical simulations in which the entire computational domain is supercritical systematically show the collapse of the _ local _ density peaks ( heitsch , mac low & klessen 2001 ) , while magnetostatic cores are not observed ( r. klessen , 2002 , private communication ) . only when the entire computational box is artificially constrained to be subcritical by the ( usually periodic ) boundary conditions ( which do not allow further accretion along field lines ) the collapse of both the large and the small scales is prevented ( e.g. , ostriker et al . 1999 ) , giving rise to flattened structures , and having led some groups to consider two - dimensional models of molecular clouds ( e.g. , shu et al.1999 ) . however , if accretion were allowed from the surrounding medium , a supercritical configuration can eventually be reached , provided the entire cloud is supercritical , in order for there to be enough mass available . these considerations suggest that the subcritical state is in fact a transient stage prior to the formation of supercritical structures that can subsequently collapse . the suggestion that molecular cloud cores can not be in hydrostatic equilibrium immediately raises two questions : one , how should we then interpret the low ( subsonic ) velocity dipersions observed within the cores ? two , if the time scale for external pressure variations were sufficiently large , should we not then expect quasi - hydrostatic cores that are hydrostatic for all practical purposes ? the answer to the first question can be found in the scenario outlined in bvs99 and et al . ( 2002 , 2003 ) . this is based on the suggestion that turbulence plays a dual role in structures from giant molecular clouds to cloud clumps ( sasao 1973 ; falgarone et al . 1992 ; & passot 1999 ; klessen et al . 2000 ) , in such a way that it contributes to the global support while promoting fragmentation into smaller - scale structures . the process is hierarchical , repeating itself towards smaller scales ( kornreich & scalo 2000 ) until the turbulent velocity dispersion within the next level of structures becomes subsonic , as dictated by a turbulent cascade in which smaller have smaller velocity dispersions . at this point , no further sub - fragmentation can occur , because subsonic isothermal turbulence can not produce large density fluctuations , and morever it becomes sub - dominant in the support of the structure ( padoan 1995 ) . in this scenario , a core is made by an initially supersonic velocity fluctuation _ at larger scale _ , but during the process the compression slows down , because of generation of smaller - scale motions , dissipation , and transfer to internal energy , which is however readily radiated away . thus , in this scenario , subsonic cores ( some of them collapsing , some re - expanding ) are a natural outcome and the ending point of the compressible , lossy turbulent cascade . work is in progress for the formulation of a quantitative model . concerning the second question , it is important to remark that , in order to make a significant density fluctuation from a turbulent compression , the latter must have an appreciable ( @xmath38 ) _ external _ mach number . therefore , the formation of the cores is expected to occur on short time scales , of the order of the crossing time of the next large scales in the hierarchy ( elmegreen 1993 ) , and the possibility of long time scales for the pressure variations is excluded . in this paper we have argued that the final state of isothermal fluid parcels compressed into `` cores '' by turbulent velocity fluctuations can not remain in equilibrium . in the non - magnetic case , this is due to the isothermality of the flow , which implies that the a continuous pressure profile requires a continuous density profile , except if it is supplemented by ram pressure in a shock . since shocks are already non - hydrostatic features , thus agreeing with our claim , we focus on continuous - profile ( `` extended '' ) structures . for these , we argued that all equilibrium configurations are unstable , contrary to the stability range found for truncated be - type structures , and thus are not expected to arise in a dynamic , fluctuating medium . thus , cores must in general collapse or re - expand , but can not remain in equilibrium , unless they happen to enter ( or be `` captured '' in ) a hotter region , as is the case of the much - discussed b68 globule . in the magnetic case , we have recalled several recent results suggesting that all cores are critical or supercritical , thus being qualitatively equivalent to the non - magnetic case regarding their possibility of collapse . although our arguments are conceptually very simple , we believe they have been overlooked in the literature because the hydrostatic state is normally considered as an _ initial _ condition , accepted without questioning how such state can be arrived at , and because the turbulent pressure is implicitly assumed to be `` microscopic '' ( i.e. , of characteristic scales much smaller than the core ) , neglecting the fact that molecular clouds are globally turbulent and that the bulk of the turbulent energy is at the largest scales , as clearly suggested by the observed velocity dispersion - size scaling relation ( larson 1981 ) , implying that the cores themselves _ are _ the turbulent density fluctuations . our results have the implication that many observed cores are not on route to forming stars , but instead `` fail '' , and must re - expand and merge back into the general molecular cloud medium . for these , the re - expansion time is expected to be larger than the compression time due to the retarding action of self - gravity . a simple estimate based on virial balance suggests that the re - expansion time is of the order of a few free - fall times . this is consistent with the facts that molecular clouds typically contain more starless than star - forming cores ( e.g. , taylor , morata & williams 1996 ; lee & myers 1999 ; see also evans 1999 and references therein ) , and that most of the cores do not appear to be gravitationally bound ( e.g. , blitz & williams 1999 ) . it is worthwhile to note that these time scales are over one order of magnitude shorter than estimates based on ambipolar diffusion ( see , e.g. , mckee et al . indeed , the long ambipolar diffusion time scales were necessary to explain the low efficiency of star formation in the old hydrostatic paradigm , but in the dynamic scenario of star formation , the low efficiency is a natural consequence that only a small fraction of the mass in a molecular cloud is deposited by the turbulence in collapsing cores ( padoan 1995 ; et al . 2002 , 2003 ) , and does not need to rely on magnetic support of the cores . we thus suggest that hydrostatic configurations have no room in the process of star formation in turbulent , isothermal molecular clouds . theories of core structure and star formation should consider the fact that core formation is a dynamical process . this probably implies that the density profile in cores is a function of time , and therefore _ not unique_. this may be in agreement with the fact that recent surveys find _ distributions _ of the scaling exponent , rather than clearly defined unique values ( e.g. , shirley et al . 2002 ) . another implication is that fundamental properties like the star formation efficiency may be _ statistical _ consequences of the turbulence in molecular clouds ( elmegreen 1993 ; padoan 1995 ; vzquez - semadeni et al . 2002 ) , rather than depending on ambipolar diffusion to break the equilibrium state . we have benefitted from comments and criticisms by shantanu basu , daniele galli , susana lizano and lee mundy . we especially thank lee hartmann , for valuable suggestions for the contents and presentation . we also thank the anonymous referee for an exceptionally deep report ( including plots and calculations ! ) which showed holes in the arguments presented in the initial version of the paper , prompting us to find a more direct argumentation . we acknowledge partial financial support from conacyt grants 27752-e to e.v .- s and i 39318-e to j. b .- p . , and from ferdowsi university to m.s . this work has made extensive use of nasa s astrophysics abstract data service . ballesteros - paredes , j. & vzquez - semadeni , e. 1995 , in `` fifth mex - tex meeting in astrophysics . gaseous nebulae and star formation '' , ed . m. pea y s. kurtz , rev . ser . , 3 , 105 shirley , y. l. , mueller , k. e. , young , c. h. & evans , n. j. ii , 2002 , in galactic star formation across the stellar mass spectrum , asp conf . series , ed . j. m. de buizer ( san francisco : asp ) , in press ( astro - ph/0205519 ) vzquez - semadeni , e. , ballesteros - paredes , j. & klessen , r. 2002 , in galactic star formation across the stellar mass spectrum , asp conf . series , ed . j. m. de buizer ( san francisco : asp ) , in press ( astro - ph/0206038 )

under the assumptions that molecular clouds are nearly spatially and temporally isothermal and that the density peaks ( `` cores '' ) within them are formed by turbulent fluctuations , we argue that cores can not reach a hydrostatic ( or magneto - static ) state as a consequence of their formation process . in the non - magnetic case , this is a consequence of the fact that , for cores at the same temperature of the clouds , the necessary bonnor - ebert truncation at a finite radius is not feasible , unless it amounts to a shock , which is clearly a dynamical feature , or the core is really embedded in hotter gas . otherwise , quiescent cores must have non - discontinuous density profiles until they merge with their parent cloud , constituting extended structures . for these , we argue that any equilibrium configuration with non - vanishing central density is unstable . since the cores environment ( the molecular cloud ) is turbulent , no reason exists for them to settle into an unstable equilibrium . instead , in this case , cores must be dynamical entities that can either be pushed into collapse , or else `` rebound '' towards the mean pressure and density as the parent cloud . nevertheless , rebounding cores are delayed in their re - expansion by their own self - gravity . we give a crude estimate for the re - expansion time as a function of the closeness of the final compression state to the threshold of instability , finding typical values @xmath0 myr , i.e. , of the order of a few free - fall times . our results support the notion that not all cores observed in molecular clouds need to be on route to forming stars , but that instead a class of `` failed cores '' should exist , which must eventually re - expand and disperse , and which can be identified with observed starless cores . in the magnetic case , recent observational and theoretical work suggests that all cores are critical or supercritical , and are thus qualitatively equivalent to the non - magnetic case . this is , however , not a problem for the efficiency of star formation : within the turbulent scenario the low efficiency of star formation does not need to rely on magnetic support of the cores , but instead is a consequence of the low probability of forming collapsing cores in a medium that is globally supported by turbulence . our results support the notion that the entire star formation process is dynamical , with no intermediate hydrostatic stages . ' # 1#1i"13i

smoothed particle hydrodynamics ( sph ) is a particle - based numerical method , pioneered by @xcite and @xcite , for solving the equations of hydrodynamics ( recent reviews include @xcite ; @xcite ; @xcite ; @xcite ) . in sph , the particles trace the flow and serve as interpolation points for their neighbours . this lagrangian nature of sph makes the method particularly useful for astrophysics , where typically open boundaries apply , though it becomes increasingly popular also in engineering ( e.g. * ? ? ? the core of sph is the density estimator : the fluid density is _ estimated _ from the masses @xmath3 and positions @xmath4 of the particles via ( the symbol @xmath5 denotes an sph _ estimate _ ) @xmath6 where @xmath7 is the _ smoothing kernel _ and @xmath8 the _ smoothing scale _ , which is adapted for each particle such that @xmath9constant ( with @xmath10 the number of spatial dimensions ) . similar estimates for the value of any field can be obtained , enabling discretisation of the fluid equations . instead , in _ conservative _ sph , the equations of motion for the particles are derived , following @xcite , via a variational principle from the discretised lagrangian @xmath11\ ] ] @xcite . here , @xmath12 ) is the internal energy as function of density and entropy @xmath13 ( and possibly other gas properties ) , the precise functional form of which depends on the assumed equation of state . the euler - lagrange equations then yield @xmath14,\ ] ] where @xmath15 and @xmath16 , while the factors @xmath17 ( @xcite ; @xcite ) arise from the adaption of @xmath8 ( @xcite ) such that @xmath18constant . equation ( [ eq : hydro ] ) is a discretisation of @xmath19 , and , because of its derivation from a variational principle , conserves mass , linear and angular momentum , energy , entropy , and ( approximately ) circularity . however , its derivation from the lagrangian is only valid if all fluid variables are smoothly variable . to ensure this , in particular for velocity and entropy , artificial dissipation terms have to be added to @xmath20 and @xmath21 . recent progress in restricting such dissipation to regions of compressive flow @xcite have greatly improved the ability to model contact discontinuities and their instabilities as well as near - inviscid flows . sph is _ not _ a monte - carlo method , since the particles are not randomly distributed , but typically follow a semi - regular glass - like distribution . therefore , the density ( and pressure ) error is much smaller than the @xmath22 expected from poisson noise for @xmath23 neighbours and sph obtains @xmath24 convergence . however , some level of particle disorder can not be prevented , in particular in shearing flows ( as in turbulence ) , where the particles are constantly re - arranged ( even in the absence of any forces ) , but also after a shock , where an initially isotropic particle distribution is squashed along one direction to become anisotropic . in such situations , the sph force ( [ eq : hydro ] ) in addition to the pressure gradient contains a random ` e@xmath25 error ' @xcite error ' term of @xcite is only the dominant contribution to the force errors induced by particle discreteness . ] , and sph converges more slowly than @xmath24 . since shocks and shear flows are common in star- and galaxy - formation , the ` e@xmath25 errors ' may easily dominate the overall performance of astrophysical simulations . one can dodge the ` e@xmath25 error ' by using other discretisations of @xmath26 @xcite . however , such approaches unavoidably abandon momentum conservation and hence fail in practice , in particular , for strong shocks @xcite . furthermore , with such modifications sph no longer maintains particle order , which it otherwise automatically achieves . thus , the ` e@xmath25 error ' is sph s attempt to resurrect particle order @xcite and prevent shot noise from affecting the density and pressure estimates . another possibility to reduce the ` e@xmath25 error ' is to subtract an average pressure from each particle s @xmath27 in equation ( [ eq : hydro ] ) . effectively , this amounts to adding a negative pressure term , which can cause the tensile instability ( see [ sec : stable : cont ] ) . moreover , this trick is only useful in situations with little pressure variations , perhaps in simulations of near - incompressible flows ( e.g. * ? ? ? the only remaining option for reducing the ` e@xmath25 error ' appears an increase of the number @xmath0 of particles contributing to the density and force estimates ( contrary to naive expectation , the computational costs grow sub - linear with @xmath0 ) . the traditional way to try to do this is by switching to a smoother and more extended kernel , enabling larger @xmath0 at the same smoothing scale @xmath28 ( e.g. * ? ? ? * ) . however , the degree to which this approach can reduce the ` e@xmath25 errors ' is limited and often insufficient , even with an infinitely extended kernel , such as the gaussian . therefore , one must also consider ` stretching ' the smoothing kernel by increasing @xmath28 . this inevitably reduces the resolution , but that is still much better than obtaining erroneous results . of course , the best balance between reducing the ` e@xmath25 error ' and resolution should be guided by results for relevant test problems and by convergence studies . unfortunately , at large @xmath0 the standard sph smoothing kernels become unstable to the pairing ( or clumping ) instability ( a cousin of the tensile instability ) , when particles form close pairs reducing the effective neighbour number . the pairing instability ( first mentioned by @xcite ) has traditionally been attributed to the diminution of the repulsive force between close neighbours approaching each other ( @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ) . such a diminishing near - neighbour force occurs for all kernels with an inflection point , a necessary property of continuously differentiable kernels . kernels without that property have been proposed and shown to be more stable ( e.g. @xcite ) . however , we provide demonstrably stable kernels with inflection point , disproving these ideas . instead , our linear stability analysis in section [ sec : linear ] shows that non - negativity of the kernel fourier transform is a necessary condition for stability against pairing . based on this insight we propose in section [ sec : smooth ] kernel functions , which we demonstrate in section [ sec : test ] to be indeed stable against pairing for all neighbour numbers @xmath0 , and which possess all other desirable properties . we also present some further test simulations in section [ sec : test ] , before we discuss and summarise our findings in sections [ sec : disc ] and [ sec : conc ] , respectively . ll@@xmath29lc@c@cc@c@cc@c@c & & & & + & @xmath30 & @xmath31 & @xmath32 & @xmath30 & @xmath31 & @xmath32 & @xmath30 & @xmath31 & @xmath32 + cubic spline & @xmath33 & @xmath34 & @xmath35 & @xmath36 & @xmath37 & @xmath38 & @xmath39 & @xmath40 & 1.732051 & 1.778002 & 1.825742 + quartic spline & @xmath41 & @xmath42 & @xmath43 & @xmath44 & @xmath45 & @xmath46 & @xmath47 & @xmath48 & 1.936492 & 1.977173 & 2.018932 + quintic spline & @xmath49 & @xmath50 & @xmath51 & @xmath52 & @xmath53 & @xmath54 & @xmath55 & @xmath56 & 2.121321 & 2.158131 & 2.195775 + + wendland @xmath57 , @xmath30 & @xmath58 & @xmath59 & @xmath60 & & & @xmath61 & & & 1.620185 & & + wendland @xmath62 , @xmath30 & @xmath63 & @xmath64 & @xmath65 & & & @xmath46 & & & 1.936492 & & + wendland @xmath66 , @xmath30 & @xmath67 & @xmath68 & @xmath69 & & & @xmath70 & & & 2.207940 & & + + wendland @xmath57 , @xmath71 & @xmath72 & @xmath73 & & @xmath74 & @xmath75 & & @xmath76 & @xmath46 & & 1.897367 & 1.936492 + wendland @xmath62 , @xmath71 & @xmath77 & @xmath78 & & @xmath79 & @xmath80 & & @xmath81 & @xmath70 & & 2.171239 & 2.207940 + wendland @xmath66 , @xmath71 & @xmath82 & @xmath83 & & @xmath84 & @xmath85 & & @xmath86 & @xmath87 & & 2.415230 & 2.449490 sph smoothing kernels are usually isotropic and can be written as @xmath88 with a dimensionless function @xmath89 , which specifies the functional form and satisfies the normalisation @xmath90 . the re - scaling @xmath91 and @xmath92 with @xmath93 leaves the functional form of @xmath94 unchanged but alters the meaning of @xmath28 . in order to avoid this ambiguity , a definition of the smoothing scale in terms of the kernel , i.e. via a functional @xmath95 $ ] , must be specified . in this study we use two scales , the smoothing scale @xmath28 , defined below , and the _ kernel - support radius _ @xmath96 , the largest @xmath97 for which @xmath98 . for computational efficiency , smoothing kernels used in practice have compact support and hence finite @xmath96 . for such kernels @xmath99 where @xmath100 for @xmath101 and @xmath102 for @xmath103 . @xmath96 is related to the average number @xmath0 of neighbours within the smoothing sphere by @xmath104 with @xmath105 the volume of the unit sphere . @xmath96 and @xmath0 are useful quantities in terms of kernel computation and neighbour search , but not good measures for the smoothing scale @xmath28 . unfortunately , there is some confusion in the sph literature between @xmath96 and @xmath28 , either being denoted by ` @xmath28 ' and referred to as ` smoothing length ' . moreover , an appropriate definition of @xmath28 in terms of the smoothing kernel is lacking . possible definitions include the kernel standard deviation @xmath106 the radius of the inflection point ( maximum of @xmath107 ) , or the ratio @xmath108 at the inflection point . for the gaussian kernel @xmath109 all these give the same result independent of dimensionality , but not for other kernels ( ` triangular ' kernels have no inflection point ) . because the standard deviation ( [ eq : sigma ] ) is directly related to the numerical resolution of sound waves ( [ sec : stable : long ] ) , we set @xmath110 in practice ( and in the remainder of our paper ) , the neighbour number @xmath0 is often used as a convenient parameter , even though it holds little meaning by itself . a more meaningful quantity in terms of resolution is the average number @xmath111 of particles within distance @xmath28 , given by @xmath112 for kernels with compact support , or the ratio @xmath113 between @xmath28 and the average particle separation . after these definitions , let us list the desirable properties of the smoothing kernel ( cf . * ? ? ? * ; * ? ? ? a. [ enum:1 ] equation ( [ eq : rho ] ) obtains an accurate density estimate ; b. [ enum:2 ] @xmath7 is twice continuously differentiable ; c. [ enum:4 ] sph is stable against pairing at the desired @xmath0 ; d. [ enum:5 ] @xmath7 and @xmath114 are computationally inexpensive . here , condition ( [ enum:1 ] ) implies that @xmath115 as @xmath116 but also that @xmath117 is monotonically declining with @xmath97 ; condition ( [ enum:2 ] ) guarantees smooth forces , but also implies @xmath118 . the most used sph kernel functions are the @xcite b - spline functions , generated as 1d fourier transforms - fold convolution ( in one dimension ) of @xmath119 with itself ( modulo a scaling ) , and hence are identical to the @xcite-@xcite probability density for the sum @xmath120 of @xmath121 independent random variables , each uniformly distributed between @xmath122 and @xmath123.[foot : irwin : hall ] ] @xcite @xmath124 with normalisation constant @xmath125 . these kernels consist of @xmath126 piece - wise polynomials of degree @xmath127 ( see table [ tab : kernel ] ) and are @xmath128 times continuously differentiable . thus , the cubic spline ( @xmath129 ) is the first useful , but the quartic and quintic have also been used . for large @xmath121 , the b - splines approach the gaussian : @xmath130 ( this follows from footnote [ foot : irwin : hall ] and the central limit theorem ) . following @xcite , @xmath131 is conventionally used as smoothing scale for the b - splines independent of @xmath10 . this is motivated by their original purpose to interpolate equidistant one - dimensional data with spacing @xmath132 , but can not be expressed via a functional @xmath133 $ ] . moreover , the resulting ratios between @xmath132 for the @xmath134 do not match any of the definitions discussed above . however , this is just a coincidence for @xmath135 ( quintic spline ) since @xmath136 for the b - splines in 1d . ] . instead , we use the more appropriate @xmath137 also for the b - spline kernels , giving @xmath138 for the cubic spline in 3d , close to the conventional @xmath139 ( see table [ tab : kernel ] ) . at low order @xmath121 the b - splines are only stable against pairing for modest values of @xmath0 ( we will be more precise in section [ sec : linear ] ) , while at higher @xmath121 they are computationally increasingly complex . therefore , alternative kernel functions which are stable for large @xmath0 are desirable . as the pairing instability has traditionally been associated with the presence of an inflection point ( minimum of @xmath140 ) , functions @xmath141 without inflection point have been proposed . these have a triangular shape at @xmath142 and necessarily violate point ( ii ) of our list , but avoid the pairing instability ) , but to keep a smooth kernel for the density estimate . however , such an approach can not be derived from a lagrangian and hence necessarily violates energy and/or entropy conservation @xcite . ] . for comparison we consider one of them , the ` hoct4 ' kernel of @xcite . the linear stability analysis of the sph algorithm , presented in the next section , shows that a necessary condition for stability against pairing is the non - negativity of the multi - dimensional fourier transform of the kernel . the gaussian has non - negative fourier transform for any dimensionality and hence would give an ideal kernel were it not for its infinite support and computational costs . therefore , we look for kernel functions of compact support which have non - negative fourier transform in @xmath10 dimensions and are low - order polynomials would avoid the computation of a square root . however , it appears that such functions can not possibly have non - negative fourier transform ( h. wendland , private communication ) . ] in @xmath120 . this is precisely the defining property of the @xcite functions , which are given by @xmath143 with @xmath144 and the linear operator @xmath145(r ) \equiv \int_r^\infty sf(s)\,\mathrm{d}s.\ ] ] in @xmath10 spatial dimensions , the functions @xmath146 with @xmath147 have positive fourier transform and are @xmath148 times continuously differentiable . in fact , they are the unique polynomials in @xmath97 of minimal degree with these properties @xcite . for large @xmath149 , they approach the gaussian , which is the only non - trivial eigenfunction of the operator @xmath150 . we list the first few wendland functions for one , two , and three dimensions in table [ tab : kernel ] , and plot them for @xmath32 in fig . [ fig : kernel ] . fig . [ fig : kernel ] plots the kernel functions @xmath141 of table [ tab : kernel ] , the gaussian , and the hoct4 kernel , all scaled to the same @xmath137 for @xmath32 . amongst the various scalings ( ratios for @xmath151 ) discussed in [ sec : smooth : scale ] above , this gives by far the best match between the kernels . the b - splines and wendland functions approach the gaussian with increasing order . the most obvious difference between them in this scaling is their central value . the b - splines , in particular of lower order , put less emphasis on small @xmath120 than the wendland functions or the gaussian . obviously , the hoct4 kernel , which has no inflection point , differs significantly from all the others and puts even more emphasis on the centre than the gaussian ( for this kernel @xmath152 ) . for spherical kernels of the form ( [ eq : w ] ) , their fourier transform only depends on the product @xmath153 , i.e. @xmath154 . in 3d ( @xmath155 denotes the fourier transform in @xmath10 dimensions ) @xmath156(\kappa ) = 4\pi \kappa^{-1 } \int_0^\infty \sin(\kappa r)\,w(r)\,r\,\mathrm{d}r\ ] ] which is an even function and ( up to a normalisation constant ) equals @xmath157/\mathrm{d}\kappa$ ] . for the b - splines , which are defined via their 1d fourier transform in equation ( [ eq : b - spline ] ) , this gives immediately @xmath158(\kappa ) = 3 \left(\frac{\textstyle n}{\textstyle\kappa}\right)^{n+2 } \sin^n\!\frac{\textstyle\kappa}{\textstyle n } \left(1- \frac{\textstyle\kappa}{\textstyle n}\cot\frac{\textstyle\kappa}{\textstyle n}\right)\ ] ] ( which includes the normalisation constant ) , while for the 3d wendland kernels @xmath159(\kappa ) = \left(-\frac{1}{\kappa}\frac{\mathrm{d}}{\mathrm{d}\kappa}\right)^{k+1 } \mathcal{f}_1\left[(1-r)^\ell_+\right](\kappa)\ ] ] ( we abstain from giving individual functional forms ) . all these are plotted in fig . [ fig : fk ] after scaling them to a common @xmath137 . notably , all the b - spline kernels obtain @xmath160 and oscillate about zero for large @xmath161 ( which can also be verified directly from equation [ eq : w : kappa : b ] ) , whereas the wendland kernels have @xmath162 at all @xmath161 , as does the hoct4 kernel . as non - negativity of the fourier transform is necessary ( but not sufficient ) for stability against pairing at large @xmath0 ( see [ sec : stable : cont ] ) , in 3d the b - splines ( of any order ) fall prey to this instability for sufficiently large @xmath0 , while , based solely on their fourier transforms , the wendland and hoct4 kernels may well be stable for all neighbour numbers . at large @xmath163 ( small scales ) , the hoct kernel has most power , caused by its central spike , while the other kernels have ever less small - scale power with increasing order , becoming ever smoother and approaching the gaussian , which has least small - scale power . the scaling to a common @xmath137 in fig . [ fig : fk ] has the effect that the @xmath164 all overlap at small wave numbers , since their taylor series @xmath165 the sph force ( [ eq : hydro ] ) is inseparably related , owing to its derivation via a variational principle , to the _ derivative _ of the density estimate . another important role of the sph density estimator is to obtain accurate values for @xmath27 in equation ( [ eq : hydro ] ) , and we will now assess the performance of the various kernels in this latter respect . in fig . [ fig : rho ] , we plot the estimated density ( [ eq : rho ] ) vs. neighbour number @xmath0 for the kernels of table [ tab : kernel ] and particles distributed in three - dimensional densest - sphere packing ( solid curves ) or a glass ( squares ) . while the standard cubic spline kernel under - estimates the density ( only values @xmath166 are accessible for this kernel owing to the pairing instability ) , the wendland kernels ( and gaussian , not shown ) tend to over - estimate it . it is worthwhile to ponder about the origin of this density over - estimation . if the particles were randomly rather than semi - regularly distributed , @xmath167 obtained for an unoccupied position would be unbiased ( e.g. * ? ? ? * ) , while at a particle position the self contribution @xmath168 to @xmath169 results in an over - estimate . of course , in sph and in fig . [ fig : rho ] particles are not randomly distributed , but at small @xmath0 the self - contribution still induces some bias , as evident from the over - estimation for _ all _ kernels at very small @xmath0 . the hoct4 kernel of read et al . ( 2010 , _ orange _ ) with its central spike ( cf . [ fig : kernel ] ) shows by far the worst performance . however , this is not a peculiarity of the hoct4 kernel , but a generic property of all kernels without inflection point . these considerations suggest the _ corrected _ density estimate @xmath170 which is simply the original estimate ( [ eq : rho ] ) with a fraction @xmath171 of the self - contribution subtracted . the equations of motion obtained by replacing @xmath169 in the lagrangian ( [ eq : l ] ) with @xmath172 are otherwise identical to equations ( [ eq : hydro ] ) and ( [ eq : omega ] ) ( note that @xmath173 , since @xmath174 and @xmath175 differ only by a constant ) , in particular the conservation properties are unaffected . from the data of fig . [ fig : rho ] , we find that good results are obtained by a simple power - law @xmath176 with constants @xmath177 and @xmath178 depending on the kernel . we use @xmath179 = ( 0.0294,0.977 ) , ( 0.01342,1.579 ) , and ( 0.0116,2.236 ) , respectively , for the wendland @xmath57 , @xmath62 , and @xmath66 kernels in @xmath32 dimensions . the dashed curves and triangles in fig . [ fig : rho ] demonstrate that this approach obtains accurate density and hence pressure estimates . the sph linear stability analysis considers a plane - wave perturbation to an equilibrium configuration , i.e. the positions are perturbed according to @xmath180\big)\ ] ] with displacement amplitude @xmath181 , wave vector @xmath161 , and angular frequency @xmath182 . equating the forces generated by the perturbation to linear order in @xmath181 to the acceleration of the perturbation yields a dispersion relation of the form @xmath183 this is an eigenvalue problem for the matrix @xmath184 with eigenvector @xmath181 and eigenvalue @xmath185 . the exact ( non - sph ) dispersion relation ( with @xmath186 , @xmath187 at constant entropy ) @xmath188 has only one non - zero eigenvalue @xmath189 with eigenvector @xmath190 , corresponding to longitudinal sound waves propagating at speed @xmath191 . the actual matrix @xmath184 in equation ( [ eq : dispersion ] ) depends on the details of the sph algorithm . for conservative sph with equation of motion ( [ eq : hydro ] ) , @xcite gives it for @xmath192 in one spatial dimension . we derive it in appendix [ app : linear ] for a general equation of state and any number @xmath10 of spatial dimensions : @xmath193 where @xmath194 is the outer product of a vector with itself , bars denote sph estimates for the unperturbed equilibrium , @xmath195 , and [ eq : uupi ] @xmath196 here and in the remainder of this section , curly brackets indicate terms not present in the case of a constant @xmath197 , when our results reduce to relations given by @xcite and @xcite . since @xmath184 is real and symmetric , its eigenvalues are real and its eigenvectors mutually orthogonal ) one omits the factors @xmath198 but still adapts @xmath8 to obtain @xmath18constant , as some practitioners do , then the resulting dispersion relation has an asymmetric matrix @xmath184 with potentially complex eigenvalues . ] . the sph dispersion relation ( [ eq : dispersion ] ) can deviate from the true relation ( [ eq : p : exact ] ) in mainly two ways . first , the longitudinal eigenvalue @xmath199 ( with eigenvector @xmath200 ) may deviate from @xmath201 ( wrong sound speed ) or even be negative ( pairing instability ; @xcite ) . second , the other two eigenvalues @xmath202 may be significantly non - zero ( transverse instability for @xmath203 or transverse sound waves for @xmath204 ) . the matrix @xmath184 in equation ( [ eq : p ] ) is not accessible to simple interpretation . we will compute its eigenvalues for the various sph kernels in [ sec : stable : kern]-3 and figs . [ fig : stable : s3]-[fig : disprel ] , but first consider the limiting cases of the dispersion relation , allowing some analytic insight . + + there are three spatial scales : the wavelength @xmath205 , the smoothing scale @xmath28 , and the nearest neighbour distance @xmath206 . we will separately consider the limit @xmath207 of well resolved waves , the continuum limit @xmath208 of large neighbour numbers , and finally the combined limit @xmath209 . if @xmath210 , the argument of the trigonometric functions in equations ( [ eq : uupi]a , b ) is always small and we can taylor expand them regardless of @xmath28 . ] . if we also assume a locally isotropic particle distribution , this gives to lowest order in @xmath163 ( @xmath211 is the unit matrix ; see also [ app : limit ] ) @xmath212\ ] ] with the eigenvalues @xmath213 the error of these relations is mostly dictated by the quality of the density estimate , either directly via @xmath214 , @xmath215 , and @xmath216 , or indirectly via @xmath217 . the density correction method of equation ( [ eq : rho : corr ] ) can only help with the former , but not the latter . the difference between constant and adapted @xmath28 is a factor 4/9 ( for 3d ) in favour of the latter . for large neighbour numbers @xmath0 , @xmath218 , @xmath219 , @xmath220 and the sums in equations ( [ eq : uupi]a , b ) can be approximated by integrals , is to assume some _ radial distribution function _ @xmath221 ( as in statistical mechanics of glasses ) for the probability of any two particles having distance @xmath120 . such a treatment may well be useful in the context of sph , but it is beyond the scope of our study . ] @xmath222 with @xmath164 the fourier transform of @xmath7 . since @xmath223 , we have @xmath224 and thus from equation ( [ eq : p ] ) @xmath225.\ ] ] @xmath226 , but towards larger @xmath163 the fourier transform decays , @xmath227 , and in the limit @xmath228 or @xmath229 , @xmath230 : short sound waves are not resolved . negative eigenvalues of @xmath184 in equation ( [ eq : p : fourier ] ) , and hence linear instability , occur only if @xmath231 itself or the expression within square brackets are negative . since @xmath232 , the latter can only happen if @xmath233 , which does usually not arise in fluid simulations ( unless , possibly , one subtracts an average pressure ) , but possibly in elasticity simulations of solids @xcite , when it causes the _ tensile _ instability ( an equivalent effect is present in smoothed - particle mhd , see @xcite ) . @xcite proposed an artificial repulsive short - range force , effectuating an additional pressure , to suppress the tensile instability . the pairing instability , on the other hand , is caused by @xmath160 for some @xmath234 . this instability can be avoided by choosing the neighbour number @xmath0 small enough for the critical wave number @xmath235 to remain unsampled , i.e. @xmath236 or @xmath237 ( though such small @xmath96 is no longer consistent with the continuum limit ) . however , if the fourier transform of the kernel is non - negative everywhere , the pairing instability can not occur for large @xmath0 . as pairing is typically a problem for large @xmath0 , this suggests that kernels with @xmath162 for every @xmath161 are stable against pairing for _ all _ values of @xmath0 , which is indeed supported by our results in [ sec : test : noise ] . the combined limit of @xmath209 is obtained by inserting the taylor expansion ( [ eq : w : taylor ] ) of @xmath231 into equation ( [ eq : p : fourier ] ) , giving @xmath238 + \mathcal{o}(h^4|{\boldsymbol{k}}|^4 ) \right).\ ] ] @xcite gave an equivalent relation for @xmath30 when the expression in square brackets becomes @xmath239 or @xmath240 ( for adapted or constant @xmath28 , respectively ) , which , he argues , bracket all physically reasonable values . however , in 3d the value for adaptive sph becomes @xmath241 , i.e. _ vanishes _ for the most commonly used adiabatic index . in general , however , the relative error in the frequency is @xmath242 . this shows that @xmath137 is indeed directly proportional to the resolution scale , at least concerning sound waves . we have evaluated the eigenvalues @xmath199 and @xmath243 of the matrix @xmath184 in equation ( [ eq : p ] ) for all kernels of table [ tab : kernel ] , as well as the hoct4 and gaussian kernels , for unperturbed positions from densest - sphere packing ( face - centred cubic grid ) simply because the configuration itself was unstable , not the numerical scheme . ] . in figs . [ fig : stable : s3]&[fig : stable : kernel ] , we plot the resulting contours of @xmath244 over wave number @xmath163 and smoothing scale @xmath28 ( both normalised by the nearest - neighbour distance @xmath206 ) or @xmath0 on the right axes ( except for the gaussian kernel when @xmath0 is ill - defined and we give @xmath111 instead ) for two wave directions , one being a nearest - neighbour direction . the top sub - panels of figs . [ fig : stable : s3]&[fig : stable : kernel ] refer to the longitudinal eigenvalue @xmath199 , when green and red contours are for , respectively , @xmath245 and @xmath246 , the latter indicative of the pairing instability . for the gaussian kernel ( truncated at @xmath247 ; fig . [ fig : stable : s3 ] ) @xmath248 everywhere , proving its stability at values for @xmath28 larger than plotted . in agreement with our analysis in [ sec : stable : cont ] , this is caused by truncating the gaussian , which ( like any other modification to avoid infinite neighbour numbers ) invalidates the non - negativity of its fourier transform . these theoretical results are confirmed by numerical findings of d. price ( referee report ) , who reports pairing at large @xmath249 for the truncated gaussian . ] , similar to the hoct4 and , in particular the higher - degree , wendland kernels . in contrast , all the b - spline kernels obtain @xmath246 at sufficiently large @xmath0 . the quintic spline , wendland @xmath57 , and hoct4 kernel each have a region of @xmath246 for @xmath163 close to the nyquist frequency and @xmath250 , @xmath251 , and @xmath252 , respectively . in numerical experiments similar to those described in [ sec : test : noise ] , the corresponding instability for the quintic spline and wendland @xmath57 kernels can be triggered by very small random perturbations to the grid equilibrium . however , such modes are absent in glass - like configurations , which naturally emerge by ` cooling ' initially random distributions . this strongly suggests , that these kernel-@xmath0 combinations can be safely used in practice . whether this also applies to the hoct4 kernel at @xmath252 we can not say , as we have not run test simulations for this kernel . note , that these islands of linear instability at small @xmath0 are not in contradiction to the relation between kernel fourier transform and stability and are quite different from the situation for the b - splines , which are only stable for sufficiently small @xmath0 . the bottom sub - panels of figs . [ fig : stable : s3]&[fig : stable : kernel ] show @xmath253 , when both families of kernels have @xmath254 with either sign occurring . @xmath255 implies growing transverse modes with a ` banding instability ' which appeared near a contact discontinuity in some of their simulations . however , they fail to provide convincing arguments for this connection , as their stability analysis is compromised by the use of the unstable cubic lattice . ] , which we indeed found in simulations starting from a slightly perturbed densest - sphere packing . however , such modes are not present in glass - like configurations , which strongly suggests , that transverse modes are not a problem in practice . the dashed lines in figs . [ fig : stable : s3]&[fig : stable : kernel ] indicate sound with wavelength @xmath256 . for @xmath257 , such sound waves are well resolved in the sense that the sound speed is accurate to @xmath258 . this is similar to grid methods , which typically require about eight cells to resolve a wavelength . the effective sph sound speed can be defined as @xmath259 . in fig . [ fig : disprel ] we plot the ratio between @xmath260 and the correct sound speed as function of wave number for three different wave directions and the ten kernel-@xmath0 combinations of table [ tab : nh : kern ] ( which also gives their formal resolutions ) . the transition from @xmath261 for long waves to @xmath262 for short waves occurs at @xmath263 , but towards longer waves for larger @xmath249 , as expected . for resolved waves ( @xmath264 : left of the thin vertical lines in fig . [ fig : disprel ] ) , @xmath260 obtains a value close to @xmath191 , but with clear differences between the various kernel-@xmath0 combinations . surprisingly , the standard cubic spline kernel , which is used almost exclusively in astrophysics , performs very poorly with errors of few percent , for both @xmath265 and 55 . this is in stark contrast to the quartic spline with similar @xmath266 but @xmath260 accurate to @xmath267 . moreover , the quartic spline with @xmath266 resolves shorter waves better than the cubic spline with a smaller @xmath268 , in agreement with table [ tab : nh : kern ] . we should note that these results for the numerical sound speed assume a perfectly smooth simulated flow . in practice , particle disorder degrades the performance , in particular for smaller @xmath0 , and the resolution of sph is limited by the need to suppress this degradation via increasing @xmath28 ( and @xmath0 ) . .some quantities ( defined in [ sec : smooth : scale ] ) for kernel-@xmath0 combinations used in fig . [ fig : disprel ] and the test simulations of [ sec : test ] . @xmath206 is the nearest - neighbour distance for densest - sphere packing , which has number density @xmath269 . the cubic spline with @xmath270 is the most common choice in astrophysical simulations , the other @xmath0 values for the b - spline are near the pairing - stability boundary , hence obtaining close to the greatest possible reduction of the ` e@xmath25 errors ' . for @xmath271 , 200 , 400 , we picked the wendland kernel which gave best results for the vortex test of [ sec : test : vortex ] . [ tab : nh : kern ] [ cols= " < , > , > , > , < " , ] in order to assess the wendland kernels and compare them to the standard b - spline kernels in practice , we present some test simulations which emphasise the pairing , strong shear , and shocks . all these simulations are done in 3d using periodic boundary conditions , @xmath272 , conservative sph ( equation [ eq : hydro ] ) , and the @xcite artificial viscosity treatment , which invokes dissipation only for compressive flows , and an artificial conductivity similar to that of @xcite . for some tests we used various values of @xmath0 per kernel , but mostly those listed in table [ tab : nh : kern ] . in order to test our theoretical predictions regarding the pairing instability , we evolve noisy initial conditions with 32000 particles until equilibrium is reached . initially , @xmath273 , while the initial @xmath4 are generated from densest - sphere packing by adding normally distributed offsets with ( 1d ) standard deviation of one unperturbed nearest - neighbour distance @xmath206 . to enable a uniform - density equilibrium ( a glass ) , we suppress viscous heating . the typical outcome of these simulations is either a glass - like configuration ( right panel of fig . [ fig : noise : xy ] ) or a distribution with particle pairs ( left panel of fig . [ fig : noise : xy ] ) . in order to quantify these outcomes , we compute for each particle the ratio @xmath274 between its actual nearest - neighbour distance and kernel - support radius . the maximum possible value for @xmath275 occurs for densest - sphere packing , when @xmath276 with @xmath121 the number density . replacing @xmath169 in equation ( [ eq : nh ] ) with @xmath277 , we obtain @xmath278 thus , the ratio @xmath279 is an indicator for the regularity of the particle distribution around particle @xmath280 . it obtains a value very close to one for perfect densest - sphere packing and near zero for pairing , while a glass typically gives @xmath281 . [ fig : noise : qmin ] plots the final value for the overall minimum of @xmath282 for each of a set of simulations . for all values tested for @xmath0 ( up to 700 ) , the wendland kernels show no indication of a single particle pair . this is in stark contrast to the b - spline kernels , all of which suffer from particle pairing . the pairing occurs at @xmath283 and 190 for the quartic , and quintic spline , respectively , whereas for the cubic spline @xmath284 approaches zero more gradually , with @xmath285 at @xmath286 . these thresholds match quite well the suggestions of the linear stability analysis in figs . [ fig : stable : s3]&[fig : stable : kernel ] ( except that the indications of instability of the quintic spline at @xmath250 and the wendland @xmath57 kernel at @xmath251 are not reflected in our tests here ) . the quintic ( and higher - order ) splines are the only option amongst the b - spline kernels for @xmath0 appreciably larger than @xmath287 . we also note that @xmath284 grows substantially faster , in particularly early on , for the wendland kernels than for the b - splines , especially when operating close to the stability boundary . as discussed in the introduction , particle disorder is unavoidably generated in shearing flows , inducing ` e@xmath25 errors ' in the forces and causing modelling errors . a critical test of this situation consists of a differentially rotating fluid of uniform density in centrifugal balance ( @xcite , see also @xcite , @xcite , and @xcite ) . the pressure and azimuthal velocity are [ eqs : gresho ] @xmath288 with @xmath289 and @xmath290 the cylindrical radius . we start our simulations from densest - sphere packing with effective one - dimensional particle numbers @xmath291 , 102 , 203 , or 406 . the initial velocities and pressure are set as in equations ( [ eqs : gresho ] ) . there are three different causes for errors in this test . first , an overly viscous method reduces the differential rotation , as shown by @xcite ; this effect is absent from our simulations owing to the usage of the @xcite dissipation switch . second , the ` e@xmath25 error ' generates noise in the velocities which in turn triggers some viscosity . finally , finite resolution implies that the sharp velocity kinks at @xmath292 and 0.4 can not be fully resolved ( in fact , the initial conditions are not in sph equilibrium because the pressure gradient at these points is smoothed such that the sph acceleration is not exactly balanced with the centrifugal force ) . in fig . [ fig : gresho : dv ] we plot the azimuthal velocity at time @xmath293 for a subset of all particles at our lowest resolution of @xmath291 for four different kernel-@xmath0 combinations . the leftmost is the standard cubic spline with @xmath268 , which considerably suffers from particle disorder and hence e@xmath25 errors ( but also obtains too low @xmath294 at @xmath295 ) . the second is the wendland @xmath57 kernel with @xmath271 , which still suffers from the ` e@xmath25 error ' . the last two are for the wendland @xmath66 kernel with @xmath296 and the wendland @xmath57 kernel with @xmath271 but with @xmath297 in equation ( [ eq : gresho : p ] ) . in both cases , the ` e@xmath25 error ' is much reduced ( and the accuracy limited by resolution ) either because of large neighbour number or because of a reduced pressure . in fig . [ fig : gresho ] , we plot the convergence of the @xmath298 velocity error with increasing numerical resolution for all the kernels of table [ tab : kernel ] , but with another @xmath0 for each , see also table [ tab : nh : kern ] . for the b - splines , we pick a large @xmath0 which still gives sufficient stability against pairing , while for @xmath271 , 200 , and 400 we show the wendland kernel that gave best results . for the cubic spline , the results agree with the no - viscosity case in fig . 6 of @xcite , demonstrating that our dissipation switch effectively yields inviscid sph . we also see that the rate of convergence ( the slope of the various curves ) is lower for the cubic spline than any other kernel . this is caused by systematically too low @xmath294 in the rigidly rotating part at @xmath295 ( see leftmost panel if fig . [ fig : gresho : dv ] ) at all resolutions . the good performance of the quartic spline is quite surprising , in particular given the rather low @xmath0 . the quintic spline at @xmath299 and the wendland @xmath62 kernel at @xmath300 obtain very similar convergence , but are clearly topped by the wendland @xmath66 kernel at @xmath296 , demonstrating that high neighbour number is really helpful in strong shear flows . our final test is the classical @xcite shock tube , a 1d riemann problem , corresponding to an initial discontinuity in density and pressure . unlike most published applications of this test , we perform 3d simulations with glass - like initial conditions . our objective here is ( 1 ) to verify the ` e@xmath25-error ' reductions at larger @xmath0 and ( 2 ) the resulting trade - off with the resolution across the shock and contact discontinuities . other than for the vortex tests of [ sec : test : vortex ] , we only consider one value for the number @xmath2 of particles but the same six kernel-@xmath0 combinations as in fig . [ fig : gresho ] . the resulting profiles of velocity , density , and thermal energy are plotted in fig . [ fig : sod ] together with the exact solutions . note that the usual over - shooting of the thermal energy near the contact discontinuity ( at @xmath301 ) is prevented by our artificial conductivity treatment . this is not optimised and likely over - smoothes the thermal energy ( and with it the density ) . however , here we concentrate on the velocity . for the cubic spline with @xmath268 , there is significant velocity noise in the post - shock region . this is caused by the re - ordering of the particle positions after the particle distribution becomes anisotropically compressed in the shock . this type of noise is a well - known issue with multi - dimensional sph simulations of shocks ( e.g. * ? ? ? * ) . with increasing @xmath0 the velocity noise is reduced , but because of the smoothing of the velocity jump at the shock ( at @xmath302 ) the @xmath298 velocity error does not approach zero for large @xmath0 . instead , for sufficiently large @xmath0 ( @xmath303 in this test ) , the @xmath298 velocity error saturates : any ` e@xmath25-error ' reduction for larger @xmath0 is balanced by a loss of resolution . the only disadvantage of larger @xmath0 is an increased computational cost ( by a factor @xmath304 when moving from the quintic spline with @xmath299 to the wendland kernel @xmath66 with @xmath296 , see fig . [ fig : time ] ) . the wendland kernels have an inflection point and yet show no signs of the pairing instability . this clearly demonstrates that the traditional ideas for the origin of this instability ( la @xcite , see the introduction ) were incorrect . instead , our linear stability analysis shows that in the limit of large @xmath0 pairing is caused by a negative kernel fourier transform @xmath231 , whereas the related tensile instability with the same symptoms is caused by an ( effective ) negative pressure . while it is intuitively clear that negative pressure causes pairing , the effect of @xmath160 is less obvious . therefore , we now provide another explanation , not restricted to large @xmath0 . by their derivation from the lagrangian ( [ eq : l ] ) , the sph forces @xmath305 tend to reduce the estimated total thermal energy @xmath306 at fixed entropy is constant , but not the entropy @xmath13 , so that @xmath307 . ] . thus , hydrostatic equilibrium corresponds to an extremum of @xmath308 , and stable equilibrium to a minimum when small positional changes meet opposing forces . minimal @xmath308 is obtained for uniform @xmath169 , since a re - distribution of the particles in the same volume but with a spread of @xmath169 gives larger @xmath308 ( assuming uniform @xmath309 ) . an equilibrium is meta - stable , if @xmath308 is only a local ( but not the global ) minimum . several extrema can occur if different particle distributions , each obtaining ( near)uniform @xmath169 , have different average @xmath167 . consider , for example , particles in densest - sphere packing , replace each by a pair and increase the spacing by @xmath310 , so that the average density @xmath311 ( but not @xmath167 ) remains unchanged . this fully paired distribution is in equilibrium with uniform @xmath169 , but the _ effective _ neighbour number is reduced by a factor 2 ( for the same smoothing scale ) . now if @xmath312 , the paired distribution has lower @xmath308 than the original and is favoured . in practice ( and in our simulations in [ sec : test : noise ] ) , the pairing instability appears gradually : for @xmath0 just beyond the stability boundary , only few particle pairs form and the effective reduction of @xmath0 is by a factor @xmath313 . we conclude , therefore , that * pairing occurs if @xmath314 for some @xmath315 . * from fig . [ fig : rho ] we see that for the b - spline kernels @xmath316 always has a minimum and hence satisfies our condition , while this never occurs for the wendland or hoct4 kernels does not affect these arguments , because during a simulation @xmath171 in equation ( [ eq : rho : corr ] ) is _ fixed _ and in terms of our considerations here the solid curves in fig . [ fig : rho ] are simply lowered by a constant . ] . the stability boundary ( between squares and crosses in fig . [ fig : rho ] ) is towards slightly larger @xmath0 than the minimum of @xmath316 , indicating @xmath317 ( but also note that the curves are based on a regular grid instead of a glass as the squares ) . a disordered particle distribution is typically not in equilibrium , but has non - uniform @xmath169 and hence non - minimal @xmath308 . the sph forces , in particular their ` e@xmath25 errors ' ( which occur even for constant pressure ) , then drive the evolution towards smaller @xmath308 and hence equilibrium with either a glass - like order or pairing ( see also * ? ? ? thus , the minimisation of @xmath318 is the underlying driver for both the particle re - ordering capability of sph and the pairing instability . this also means that when operating near the stability boundary , for example using @xmath268 for the cubic spline , this re - ordering is much reduced . this is why in fig . [ fig : noise : qmin ] the transition between glass and pairing is not abrupt : for @xmath0 just below the stability boundary the glass - formation , which relies on the re - ordering mechanism , is very slow and not finished by the end of our test simulations . an immediate corollary of these considerations is that any sph - like method without ` e@xmath25 errors ' does not have an automatic re - ordering mechanism . this applies to modifications of the force equation that avoid the ` e@xmath25 error ' , but also to the method of @xcite , which employs a voronoi tessellation to obtain the density estimates @xmath169 used in the particle lagrangian ( [ eq : l ] ) . the tessellation constructs a partition of unity , such that different particle distributions with uniform @xmath169 have _ exactly _ the same average @xmath167 , i.e. the global minimum of @xmath308 is highly degenerate . this method has neither a pairing instability , nor ` e@xmath25 errors ' , nor the re - ordering capacity of sph , but requires additional terms for that latter purpose . neither the b - splines nor the wendland functions have been designed with sph or the task of density estimation in mind , but derive from interpolation of function values @xmath319 for given points @xmath4 . the b - splines were constructed to exactly interpolate polynomials on a regular 1d grid . however , this for itself is not a desirable property in the context of sph , in particular for 2d and 3d . the wendland functions were designed for interpolation of scattered multi - dimensional data , viz @xmath320 the coefficients @xmath321 are determined by matching the interpolant @xmath322 to the function values , resulting in the linear equations @xmath323 if the matrix @xmath324 is positive definite for _ any _ choice of @xmath121 points @xmath4 , then this equation can always be solved . moreover , if the function @xmath141 has compact support , then @xmath325 is sparse , which greatly reduces the complexity of the problem . the wendland functions were designed to fit this bill . as a side effect they have non - negative fourier transform ( according to * ? ? ? * ) , which together with their compact support , smoothness , and computational simplicity makes them ideal for sph with large @xmath0 . so far , the wendland functions are the only kernels which are stable against pairing for all @xmath0 and satisfy all other desirable properties from the list on page . in smooth flows , i.e. in the absence of particle disorder , the only error of the sph estimates is the bias induced by the smoothing operation . for example , assuming a smooth density field @xmath326 ( e.g. * ? ? ? * ; * ? ? ? * ) with @xmath327 defined in equation ( [ eq : sigma ] ) . since @xmath327 also sets the resolution of sound waves ( [ sec : stable : long ] ) , our definition ( [ eq : h ] ) , @xmath137 , of the sph resolution scale is appropriate for smooth flows . the result ( [ eq : rho : bias ] ) is the basis for the traditional claim of @xmath24 convergence for smooth flows . true flow discontinuities are smeared out over a length scale comparable to @xmath28 ( though we have not made a detailed investigation of this ) . in practice , however , particle disorder affects the performance and , as our test simulations demonstrated , the actual resolution of sph can be much worse than the smooth - flow limit suggests . there is no consensus about the best neighbour number in sph : traditionally the cubic spline kernel is used with @xmath270 , while @xcite favours @xmath328 ( at or even beyond the pairing - instability limit ) and @xcite use their hoct4 kernel with even @xmath329 ( corresponding to a @xmath330 times larger @xmath28 ) . from a pragmatic point of view , the number @xmath2 of particles , the neighbour number @xmath0 , and the smoothing kernel ( and between them the numerical resolution ) are _ numerical parameters _ which can be chosen to optimise the efficiency of the simulation . the critical question therefore is : * which combination of @xmath2 and @xmath0 ( and kernel ) most efficiently models a given problem at a desired fidelity ? * clearly , this will depend on the problem at hand as well as the desired fidelity . however , if the problem contains any chaotic or turbulent flows , as is common in star- and galaxy formation , then the situation exemplified in the gresho - chan vortex test of [ sec : test : vortex ] is not atypical and large @xmath0 may be required for sufficient accuracy . but are high neighbour numbers affordable ? in fig . [ fig : time ] , we plot the computational cost versus @xmath0 for different kernels . at @xmath331 the costs rise sub - linearly with @xmath0 ( because at low @xmath0 sph is data- rather than computation - dominated ) and high @xmath0 are well affordable . in the case of the vortex test , they are absolutely necessary as fig . [ fig : gresho : c ] demonstrates : for a given numerical accuracy , our highest @xmath0 makes optimal use of the computational resources ( in our code memory usage does not significantly depend on @xmath0 , so cpu time is the only relevant resource ) . particle disorder is unavoidable in strong shear ( ubiquitous in astrophysical flows ) and causes random errors of the sph force estimator . the good news is that particle disorder is less severe than poissonian shot noise and the resulting force errors ( which are dominated by the e@xmath25 term of * ? ? ? * ) are not catastrophic . the bad news , however , is that these errors are still significant enough to spoil the convergence of sph . in this study we investigated the option to reduce the ` e@xmath25 errors ' by increasing the neighbour number in conjunction with a change of the smoothing kernel . switching from the cubic to the quintic spline at fixed resolution @xmath28 increases the neighbour number @xmath0 only by a factor ) for the smoothing scale @xmath28 . the conventional factor is 3.375 , almost twice 1.74 , but formally effects to a loss of resolution , since the conventional value for @xmath28 of the b - spline kernels is inappropriate . ] 1.74 , hardly enough to combat ` e@xmath25 errors ' . for a significant reduction of the these errors one has to trade resolution and significantly increase @xmath0 beyond conventional values . the main obstacle with this approach is the pairing instability , which occurs for large @xmath0 with the traditional sph smoothing kernels . in [ sec : linear ] and appendix [ app : linear ] , we have performed ( it appears for the first time ) a complete linear stability analysis for conservative sph in any number of spatial dimensions . this analysis shows that sph smoothing kernels whose fourier transform is negative for some wave vector @xmath161 will inevitably trigger the sph pairing instability at sufficiently large neighbour number @xmath0 . such kernels therefore require @xmath0 to not exceed a certain threshold in order to avoid the pairing instability ( not to be confused with the tensile instability , which has the same symptoms but is caused by a negative effective pressure independent of the kernel properties ) . intuitively , the pairing instability can be understood in terms of the sph density estimator : if a paired particle distribution obtains a lower average estimated density , its estimated total thermal energy @xmath308 is smaller and hence favourable . otherwise , the smallest @xmath308 occurs for a regular distribution , driving the automatic maintenance of particle order , a fundamental ingredient of sph . the @xcite functions , presented in [ sec : kern : wend ] , have been constructed , albeit for different reasons , to possess a non - negative fourier transform , and be of compact support with simple functional form . the first property and the findings from our tests in [ sec : test : noise ] demonstrate the remarkable fact that these kernels are stable against pairing for _ all _ neighbour numbers ( this disproves the long - cultivated myth that the pairing instability was caused by a maximum in the kernel gradient ) . our 3d test simulations show that the cubic , quartic , and quintic spline kernels become unstable to pairing for @xmath332 , 67 , and 190 , respectively ( see fig . [ fig : noise : qmin ] ) , but operating close to these thresholds can not be recommended . a drawback of the wendland kernels is a comparably large density error at low @xmath0 . as we argue in [ sec : pairing : alt ] , this error is directly related to the stability against pairing . however , in [ sec : kern : dens ] we present a simple method to correct for this error without affecting the stability properties and without any other adverse effects . we conclude , therefore , that the wendland functions are ideal candidates for sph smoothing kernels , in particular when large @xmath0 are desired , since they are computationally superior to the high - order b - splines . all other alternative kernels proposed in the literature are computationally more demanding and are either centrally spiked , like the hoct4 kernel of @xcite , or susceptible to pairing like the b - splines ( e.g. * ? ? ? our tests of section [ sec : test ] show that simulations of both strong shear flows and shocks benefit from large @xmath0 . these tests suggest that for @xmath333 and 400 , respectively , the wendland @xmath62 and @xmath66 kernels are most suitable . compared to @xmath268 with the standard cubic spline kernel , these kernel-@xmath0 combinations have lower resolution ( @xmath28 increased by factors 1.27 and 1.44 , respectively ) , but obtain much better convergence in our tests . for small neighbour numbers , however , these tests and our linear stability analysis unexpectedly show that the quartic b - spline kernel with @xmath266 is clearly superior to the traditional cubic spline and can compete with the wendland @xmath57 kernel with @xmath271 . the reason for this astonishing performance of the quartic spline is unclear , perhaps the fact that near @xmath334 this spline is more than three times continuously differentiable plays a role . we note that , while the higher - degree wendland functions are new to sph , the wendland @xmath57 kernel has already been used ( @xcite , for example , employs it for 2d simulations ) . however , while its immunity to the pairing instability has been noted ( e.g. * ? ? ? * ) kernels in the context of 2d sph simulations . robinson refutes experimentally the traditional explanation ( la @xcite ) for the pairing instability and notices the empirical connection between the pairing instability and the non - negativity of the kernel fourier transform , both in excellent agreement with our results . ] , we are not aware of any explanation ( previous to ours ) nor of any other systematic investigation of the suitability of the wendland functions for sph . research in theoretical astrophysics at leicester is supported by an stfc rolling grant . we thank chris nixon and justin read for many stimulating discussions and the referee daniel price for useful comments and prompt reviewing . this research used the alice high performance computing facility at the university of leicester . some resources on alice form part of the dirac facility jointly funded by stfc and the large facilities capital fund of bis . we start from an equilibrium with particles of equal mass @xmath335 on a regular grid and impose a plane - wave perturbation to the unperturbed positions @xmath336 ( a bar denotes a quantity obtained for the unperturbed equilibrium ) : @xmath337 as in equation ( [ eq : x : pert : ] ) . we derive the dispersion relation @xmath338 by equating the sph force imposed by the perturbation ( to linear order ) to its acceleration @xmath339 to obtain the perturbed sph forces to linear order , we develop the internal energy of the system , and hence the sph density estimate , to second order in @xmath340 . if @xmath341 with @xmath342 and @xmath343 the first and second - order density corrections , respectively , then @xmath344 let us first consider the simple case of constant @xmath197 which remains unchanged during the perturbation . then , @xmath345 inserting @xmath346 into ( [ eq : varrho ] ) gives @xmath347 where ( assuming a symmetric particle distribution ) @xmath348 we can then derive @xmath349 with @xmath350 inserting these results into ( [ eq : hydro : p ] ) , we get @xmath351 if the @xmath8 are adapted such that @xmath352 remains a global constant @xmath353 , the estimated density is simply @xmath354 . we start by expanding @xmath355 to second order in both @xmath181 and @xmath356 . using a prime to denote differentiation w.r.t . @xmath357 , we have inserting these expressions into equation ( [ eq : hydro : p ] ) we find with relations ( [ eqs : derive:1 ] ) and ( [ eqs : derive:2 ] ) @xmath366 & & \label{eq : ddxi : ad } - \left ( \bar{c}^2-\frac{2\bar{p}}{\bar{\rho } } \right ) \frac{{\boldsymbol{a}}{\cdot}{\boldsymbol{t}}\,{\boldsymbol{t}}}{\bar{\rho}^2\bar{\omega}^2}\,\phi_i .\end{aligned}\ ] ] where @xmath367

the numerical convergence of smoothed particle hydrodynamics ( sph ) can be severely restricted by random force errors induced by particle disorder , especially in shear flows , which are ubiquitous in astrophysics . the increase in the number @xmath0 of neighbours when switching to more extended smoothing kernels _ at fixed resolution _ ( using an appropriate definition for the sph resolution scale ) is insufficient to combat these errors . consequently , trading resolution for better convergence is necessary , but for traditional smoothing kernels this option is limited by the pairing ( or clumping ) instability . therefore , we investigate the suitability of the wendland functions as smoothing kernels and compare them with the traditional b - splines . linear stability analysis in three dimensions and test simulations demonstrate that the wendland kernels avoid the pairing instability for _ all _ @xmath0 , despite having vanishing derivative at the origin ( disproving traditional ideas about the origin of this instability ; instead , we uncover a relation with the kernel fourier transform and give an explanation in terms of the sph density estimator ) . the wendland kernels are computationally more convenient than the higher - order b - splines , allowing large @xmath0 and hence better numerical convergence ( note that computational costs rise sub - linear with @xmath0 ) . our analysis also shows that at low @xmath0 the quartic spline kernel with @xmath1 obtains much better convergence then the standard cubic spline . [ firstpage ] hydrodynamics methods : numerical methods : @xmath2-body simulations

hard x - ray ( hxr ) and microwave emissions show fine structures both temporally and spatially during a solar flare , which revealed that a highly - fragmented and intermittent particle acceleration occurs ( e.g. * ? ? ? * ; * ? ? ? this fragmented structure of solar flares indicates that a flare is an ensemble of a vast amount of small scale energy release . statistical studies of solar flares have also shown that various kinds of physical parameters of flares , like peak intensity , flare duration , waiting time of soft x - ray ( sxr ) emissions between discrete events , are well described with power - law distributions ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? karlick et al . ( 2000 ) examined twelve flares , and showed that microwave spikes seen in each flare show power - law features in size and time scales ( i.e. scale of energy release ) . in addition , the occurrence of microflares and x - ray bright points is known to follow power - law distributions @xcite . recent development in magnetic reconnection theory also indicates that magnetic reconnection proceeds intermittently , involving repeated formation of magnetic islands and their subsequence coalescence @xcite . this process is known as the `` impulsive bursty '' regime of magnetic reconnection @xcite . as shibata @xmath3 tanuma ( 2001 ) showed , plasmoids of various scales are generated in the current sheet in a fractal manner . such fractal nature of magnetic reconnection might generate power - law characteristics that are observed in solar flares , as mentioned above . karlick et al . ( 2000 ) and kliem et al . ( 2000 ) discussed similar features seen in the hxr and microwave emissions , based on the theoretical view of dynamic magnetic reconnection . although the temporal resolutions of hxr and microwave observations were high enough to reveal fragmented features in the temporal scale , the time variability of flare kernels has not been discussed with two - dimensional images with high spatial and temporal resolutions . historically , the two ribbon structure has been observed in h@xmath2 and other wavelengths in solar flares . flare kernels inside the ribbons are well correlated with hxr and microwave emissions temporally and spatially in h@xmath2 @xcite . also in ultraviolets ( uvs ) , such as in 1550 taken with trace the same properties were observed @xcite , indicating that sudden plasma heating occurs in the upper chromosphere and the transition region by nonthermal particles or thermal conduction . hence h@xmath2 kernels and trace 1550 ( @xmath0 doublet emissions ) kernels can also be good tracers of hxr sources . in this paper , we examine the fine structures inside the flare ribbons seen in the uv images of the x2.5 flare that occurred on 2004 november 10 . we show the fragmented features of the bright emission sources , and that they follow a power - law distribution just in a single event . finally we discuss the fractal features of energy release region ( i.e. current sheet ) and the avalanching system of the flare to explain such fragmented structures . the large flare ( x2.5 in goes class ) occurred in the noaa active region 10696 ( n08@xmath5 , w50@xmath5 ) at 02:00 ut , 2004 november 10 . this flare was a long duration event that showed a typical two ribbon structure preceded by a filament eruption . the erupted filament showed a kinking structure @xcite , and a lot of attention has been paid to it because this is a candidate for the source of the geo - effective coronal mass ejection ( cme ; * ? ? ? * ) . we observed the flare with the sartorius 18 cm refractor telescope at kwasan observatory , kyoto university @xcite . the highest temporal and spatial resolutions of the sartorius data are 1 @xmath6 and 1.2 , respectively . figures 1(a)-1(c ) shows the images of the flare in h@xmath2 at 02:06 , 02:08 , and 02:10 ut , which correspond to the peak times of the hxr emission ( see also figure 2 ) . we can see some h@xmath2 flare kernels inside the ribbon structure . the uv images were taken by 1600 passband of trace as shown in figures 1(d)-1(f ) . they also show a two - ribbon structure . the trace 1600 data was obtained with the temporal and the spatial resolutions of 3 @xmath6 and 0.5 . during flares , the pair of @xmath0 doublet ( @xmath11550 ) in a broad response of the 1600 passband is strongly enhanced @xcite . the @xmath0 formation temperature is about 10@xmath7 k. therefore , the bright features in figures 1(d)-1(f ) observed in the impulsive phase are probably caused by the enhancement of the @xmath0 ( @xmath11550 ) emission line due to transition region heating . we call the bright features @xmath0 kernels . we overlaid hxr contour images ( 25 - 50 kev ) on trace 1600 images to compare the spatial distribution of radiation sources in h@xmath2 and hxr emissions ( see figures 1(d)-1(f ) ) . the hxr images were taken with rhessi . we synthesized the hxr image with the clean algorithm , which is the same method as is commonly used for analysis of radio data , and grids 3 - 9 , which give the spatial resolution ( fwhm ) of about 10 . the integration time is set to be 60 @xmath6 , and the total photon count was 3.8 - 7.7@xmath810@xmath9 counts for photons of 25 - 50 kev . these synthesizing tools are included in the solar software . we found that the hxr sources are associated with both h@xmath2 and @xmath0 kernels . the location of the hxr sources moves in the southeast direction as the flare progresses , that is , from a mixed polarity region to a strong magnetic field region , indicating a change in the site of the strong energy release . though the kernels are seen in the southeast of the h@xmath2 and trace images from 2:05 - 2:08 i.e. before the hxr sources have arrived there , this is probably because the hxr emissions are not large enough to be observed with the dynamic range of rhessi . actually small flare kernels in the southeast of figure 1(d ) at around 02:06 ut , which are the components of the ribbons , show small peaks of intensity less than 25 percent of the later impulsive burst at 02:10 ut , as well as in the h@xmath2 time profile ( see figure 2(b ) ) . we summarize the results of the comparison of the multi - wavelength observations in h@xmath2 , @xmath0 ( @xmath11550 ) and hxr emissions as follows : ( i ) there are not only good spatial but also temporal correlations among flare kernels observed in h@xmath2 and @xmath0 emissions , some of which are associated with the hxr emission . this implies that the @xmath0 and h@xmath2 kernels are caused by nonthermal electrons interacting with the ambient thick target plasma as well as hxrs . ( ii ) the @xmath0 flare ribbons are much thinner and sharper than the h@xmath2 ribbons . this is because the width of flare ribbons is typically determined by cooling time via thermal conduction and radiative cooling , and because thermal conduction time scale in the corona / transition region for @xmath0 is much shorter than that for h@xmath2 in the chromosphere . the ratio of the peak intensity of the h@xmath2 kernels to the background is not very large , as a result , the integrated h@xmath2 emission over the whole active region is similar to the soft x - ray emission ( see figure 2(b ) ) . on the other hand , the integrated @xmath0 emission is still similar to the hxr emission , showing corresponding peaks in their time profiles . as a result of the comparison of the multi - wavelength observations , we found that it is easier to identify peaks in the time profile of the @xmath0 emission , rather than in h@xmath2 . moreover , the seeing condition smeared the h@xmath2 images in the impulsive phase , and therefore we focused on the temporal variations of the @xmath0 flare kernels here . we measured the intensity , duration and time interval between each peak from the time profiles . we divided both flare ribbons into fine meshes . each mesh box is a square with size of 5 , 2 and 1 for comparison . although this is larger than the elemental h@xmath2 kernels , which are considered to be about 1 or even smaller @xcite , it is small enough for us to determine the essential structures inside the flare ribbons . next we examined time variations of the total intensity for each box in the meshes . as the mesh size becomes smaller and smaller , peaks in the time profile become isolated . this means that the light curves with a large ( 5 ) mesh possibly contain multiple flare kernels that are superposed over each other , while smaller size mesh can cover only single flare kernel . we found that a 2 mesh is enough to isolate most of the superposed peaks , though some peaks can not be separated even with a 1 mesh . this implies that the size of the heating source is comparable to or smaller than 2 . since a 1 mesh is too small and too noisy to analyze , we adopted a 5 and a 2 size mesh for our further analysis . we defined the maximum intensity of a light curve as the peak intensity ( @xmath10 ) , and determined the duration ( @xmath11 ) as the full width at the three fourths maximum intensity of each peak because not all of the peak durations can be measured with the full width at the half maximum ( fwhm ) . we identified 586 @xmath0 kernels using a 5 mesh in the impulsive phase , only with the requirement that the count rate of the detector exceeds 50 counts s@xmath12 to identify the peaks . figure 3(a ) , 3(b ) shows the frequency distributions of the peak intensity and the duration of each peak . we also recorded the peak time of the flare kernels across the whole active region . we determined the time interval of the peaks ( @xmath13 ) as the time difference between the peak times and show its frequency distribution in figure 3(c ) . the distribution of peak intensities , durations and time intervals reveal power - laws during the impulsive phase . from the slopes of the distribution , we obtain a power - law indices @xmath2 @xmath11.5 for the peak intensity , @xmath2 @xmath12.3 for the peak duration , and @xmath2 @xmath11.8 for the time interval between each peak . the lower limit of time duration of about 10 @xmath6 comes from the temporal resolution of trace such as 2 - 3 @xmath6 in a flare mode . when we change the mesh size from 5 to 2 , each peak became isolated and sharpened so that the number of the peaks with short duration increased . we found that the distributions of the peak intensity , duration and time interval well followed power - law distributions with the power - law indices of @xmath2 @xmath11.5 , 2.3 and 1.8 , respectively . these power - law indices remain unchanged , even if we change the size of the mesh box from 5 to 2 , and even if we change the threshold of the peak identification . in this individual event , we showed for the first time , the power - law behavior of flare kernels typically seen in studies of large numbers of flares , suggestive of a link between the observations and theoretical modeling of the fractal nature of magnetic reconnection in current sheets . if magnetic reconnection occurs in a fractal manner in the current sheet , one would expect energy release and particle heating / acceleration on a range of different sizes and time scales , such that power - law distributions could be expected in the size , duration , etc . of tracers of the energy release process . since flare kernels have been shown to be good proxies for the hxr energy release and , furthermore , trace @xmath0 kernels can also be good tracers of hxr sources , one would expect to see such behavior in their properties . in fact , the peak intensity and peak duration could be indicators of the released energy . the peak time also corresponds to the timing when heating of the foot point plasma occurs , that is the arrival time of released energy at the foot point . the duration of the transition region heating @xmath11 and the time interval @xmath13 are roughly characterized by alfvn time @xmath14 of the reconnection region , @xmath15 where @xmath16 is a characteristic length of the energy release region ( e.g. macroscopic length of a current sheet or plasmoid ) , @xmath17 is an alfvn velocity ( @xmath18 @xmath19 ) , and @xmath19 is a typical magnetic field strength in the corona . on the other hand , intensity of flare kernels @xmath10 is estimated as , @xmath20 so , if magnetic reconnection occurs in a fractal manner in the current sheet through the repeated formations of magnetic islands and their subsequence coalescence , current sheets become thinner and thinner and as a result , the self - similar structure of current sheet can be formed from macroscopic to microscopic scales . at that time , the size of energy release region @xmath16 can be expected to exhibit power - law behavior , so that power - law distributions can be expected in the energy , duration , etc . of tracers of the energy release process , such as @xmath10 and @xmath14 . these fractal structures mean that there are no characteristic scales of length , energy and time in the energy release process . our results also support the view of the impulsive bursty reconnection @xcite and the fractal features of the current sheet @xcite . a power - law distribution for magnetic energy of plasmoid is also reported in the magnetosphere by hoshino et al . ( 1994 ) . on the basis of the unified view suggested by shibata ( 1999 ) as the plasmoid - induced - reconnection model , plasmoid ejection plays a crucial role for energy storage and release , driving the inflow and the reconnection rate enhancement . on a large scale , the flare itself should exhibit these properties . our results are quite similar to the power - law behaviors typically seen in studies of large numbers of flares ( e.g. * ? ? ? * ) , which are often interpreted as evidence of an avalanching system of self - organized criticality ( soc ) . this suggests that the elemental energy release in this individual event may be similar to that in a typical x - ray flare and hence interpreted as an avalanching system of soc in a single event or by fractal current sheet in the impulsive reconnection region as discussed above . we first acknowledge an anonymous referee for his / her useful comments and suggestions . we wish to acknowledge all the members of kwasan observatory for their support during our observation , especially m. kamobe and a. edamura . we also thank a. hillier for his careful reading and correction of this letter . we would like to thank trace and rhessi data center for their extensive use . this work was supported in part by the grant - in - aid for creative scientific research `` the basic study of space weather prediction '' ( head investigator : k. shibata ) from the ministry of education , culture , sports , science , and technology of japan , and in part by the grand - in - aid for the global coe program `` the next generation of physics , spun from universality and emergence '' from the ministry of education , culture , sports , science , and technology ( mext ) of japan . + alexander , d. & coyner , a. j. 2006 , , 640 , 505 asai , a. , ishii , t. t. , kurokawa , h. , yokoyama , t. , & shimojo , m. , 2003 , , 586 , 624 aschwanden , m. j. 2002 , particle acceleration and kinematics in solar flares ( dordrecht : kluwer ) benz , a. o. & aschwanden , m. j. 1992 , in lecture notes in physics , vol . 399 : eruptive solar flares , eds . z. svestka , b. v. jackson , & m. e. machado ( new york : springer ) , 106 brekke , p. , rottman , g. j. , fontenla , j. & judge , p. g. , 1996 , , 468 , 418 dennis , b. r. 1985 , , 100 , 465 finn , j. m. & kaw , p. k. 1977 , phys . fluids , 20 , 72 handy , b. n. , et al . 1999 , , 187 , 229 harra l. k. , et al . 244 , 95 hoshino , m. , nishida , a. , yamamoto , t. , & kokubun , s. , 1994 , geophys . , 21 , 25 , 2935 karlick , m. , jiika , k. & sobotka , m. 2000 , , 195 , 165 kitahara , t. & kurokawa , h. 1990 , , 125 , 321 kliem , b. , karlick , m. & benz , a. o. 2000 , a&a , 360 , 715 kurokawa , h. 1986 , in proc . of nso / smm flare symp . , low atmosphere of solar flares , ed . d. neidig ( sunspot : nso ) , 51 kurokawa , h. , takahara , t. & ohki , k. 1988 , publ . japan , 40 , 357 lin , r. p. , et al . 2002 , , 210 , 3 priest , e. r. , 1985 , rep . prog 48 , 955 shibata , k. 1999 , astrophys . , 264 , 129 shibata , k. & tanuma , s. 2001 , earth , planets and space , 53 , 473 shimizu , t. 1996 , publ . japan , 47 , 251 shimizu , m. , et al . 2008 , , 683 , l203 shimojo , m. & shibata , k. 1999 , , 516 , 934 tajima , t. , sakai , j. , nakajima , h. , kosugi , t. , brunel , f. , kundu , m. r. , 1987 , , 321 , 1031 veronig , a. , temmer , m. , hanslmeier , a. , otruba , w. & messerotti , m. , 2002 , a&a , 382 , 1070 warren , h. p. & winebarger , a. r. , 2000 , , 535 , l63 warren , h. p. & warshall , a. d. 2001 , , 560 , l87 wheatland , m. s. 2000 , , 536 , l109 williams , d. r. , trk , t. , dmoulin , p. , van driel - gesztelyi , l. , kliem , b. , 2005 , , 628 , l163

we report a detailed examination of the fine structure inside flare ribbons and the temporal evolution of this fine structure during the x2.5 solar flare that occurred on 2004 november 10 . we examine elementary bursts of the @xmath0 ( @xmath11550 ) emission lines seen as local transient brightenings inside the flare ribbons in the ultraviolet ( 1600 ) images taken with transition region and coronal explorer , and we call them @xmath0 kernels . this flare was also observed in h@xmath2 with the sartorius 18 cm refractor telescope at kwasan observatory , kyoto university , in hard x - rays ( hxr ) with reuven ramaty high energy solar spectroscopic imager . many @xmath0 kernels , whose sizes were comparable to or less than 2 , were found to brighten successively during the evolution of the flare ribbon . the majority of them were well correlated with the h@xmath2 kernels in both space and time , while some of them were associated with the hxr emission . these kernels were thought to be caused by the precipitation of nonthermal particles at the foot points of the reconnecting flare loops . the time profiles of the @xmath0 kernels showed intermittent bursts , whose peak intensity , duration and time interval were well described by power - law distribution functions . this result is interpreted as an avalanching system of `` self - organized criticality '' of a single event or by fractal current sheets in the impulsive reconnection region .

quantum phase transitions , which occur when a driving parameter in the hamiltonian of the system changes across a critical point , play a central role in condensed matter physics @xcite . while most quantum phase transitions can be characterized by symmetry breaking , there is also an exception that can only be witnessed by topological order ( see , e.g. , @xcite ) . signatures of topological order in many - body quantum systems can characterize a topological quantum phase transition and include , e.g. , the existence of excitations obeying fractional statistics ( see , e.g. , @xcite ) , ground - state degeneracy related to the topology of the system ( instead of the symmetry ) ( see , e.g. , @xcite ) , and topological entanglement entropy @xcite . in particular , the spectral chern number @xcite serves as a topological number for characterizing a two - dimensional ( 2d ) system of noninteracting ( or weakly interacting ) fermions with an energy gap . without closing the gap , energy spectra with different chern numbers can not be deformed into each other @xcite . this is because a topological quantum phase transition occurs when changing the chern number . recently , it was shown @xcite that the topological quantum phase transition in the kitaev spin model can be characterized by nonlocal - string order parameters . in an appropriate dual representation , this order parameter can become local and the basic concept of landau theory of continuous phase transition is also applicable @xcite . in the kitaev model , a @xmath0-spin is placed at each site of a honeycomb lattice [ see fig . [ fig1](a ) ] and the interactions between nearest - neighbor spins are highly anisotropic with three types of bonds @xmath1 , and @xmath2 . to simplify the site - labelling of the honeycomb lattice , one can deform it to a topologically equivalent brick - wall lattice shown in fig . [ fig1](b ) . in @xcite , the topological quantum phase transition of the kitaev model on a brick - wall lattice was studied for the hamiltonian : @xmath3 where @xmath4 and @xmath5 are the pauli matrices at the site @xmath6 , with column index @xmath7 and row index @xmath8 . a nice jordan - wigner transformation was introduced @xcite to solve this model and the redundant gauge degrees of freedom were removed . the phase diagram of the kitaev model ( [ kitaev ] ) consists of two phases : a band insulator phase and a topologically non - universal gapless phase @xcite . the insulator phase , as kitaev has shown by using perturbation theory @xcite , is equivalent to a toric code model @xcite . while abelian anyons can be defined in the insulator phase , the vortices in the gapless phase do not have a well - defined statistics . applying an external magnetic field as a perturbation , which breaks the time - reversal symmetry in eq . ( [ kitaev ] ) , a gap opens in the gapless phase and the vortices then obey a well - defined non - abelian anyonic statistics @xcite . the third - order perturbation corresponds to exactly soluble models @xcite whose spectrum has recently been extensively studied @xcite . in this paper , we study the following hamiltonian @xcite : @xmath9 hereafter , we call the model in eq . ( [ hamiltonian ] ) an extended kitaev model . we solve this model on a torus and mainly focus on the quantum phase transition between the phase with abelian anyons and the phase with non - abelian anyons . we first apply the jordan - wigner transformation to the spin operators and then introduce majorana fermions to get the ground state of eq . ( [ hamiltonian ] ) in the vortex - free sector . we show that the third directional derivative of the ground - state energy is discontinuous at each point on the critical line separating the abelian and non - abelian phases , while its first and second directional derivatives are continuous at this point . this implies that the topological quantum phase transition is continuous in this extended kitaev model . moreover , at this critical point , we also study the nonanalyticity of the entanglement ( i.e. , the von neumann entropy ) between two nearest - neighbor spins and the rest of the spins in the system . we find that the second directional derivative of the von neumann entropy is closely related to the third directional derivative of the ground - state energy and it is also discontinuous at the critical point . our approach directly reveals that both the entanglement measure and the ground - state energy can be used to characterize the topological quantum phase transition in the extended kitaev model . and ( b ) the brick - wall lattice , which is deformed from the honeycomb lattice in ( a ) . this deformed lattice can be used to label the sites of the honeycomb lattice by column and row indices . [ fig1],width=316 ] ( color online ) phase diagram of the extended kitaev spin model , where @xmath10 and @xmath11 . the gray region corresponds to the non - abelian phase and the three triangular ( light gray ) regions correspond to the abelian phase . the thick solid , dashed and dotted lines are @xmath12 , @xmath13 , and @xmath14 , where @xmath15 and @xmath16 . these lines consist of the boundary of the gray region , which are the critical lines separating the abelian and non - abelian phases . the thin dotted line intersects the thick solid and dotted lines at the points @xmath17 and @xmath18 . the direction * _ l _ * has an inclination angle @xmath19 with respect to the horizontal axis and it indicates the direction along which the driving parameters @xmath20 and @xmath21 vary.,width=288 ] let us define the jordan - wigner transformation @xcite @xmath22 where @xmath23 if the integer @xmath24 is odd and @xmath25 if the integer @xmath24 is even . also , we introduce the following definitions for majorana fermions : @xmath26&=&c_{n , m}^{(1)},\nonumber\\~a_{n , m}^{(1)\dag}+a_{n , m}^{(1)}&=&d_{n , m}^{(1)}\end{aligned}\ ] ] for @xmath24 equal to an odd integer , and @xmath27&=&d_{n , m}^{(2)},\nonumber\\~a_{n , m}^{(2)\dag}+a_{n , m}^{(2)}&=&c_{n , m}^{(2)}\end{aligned}\ ] ] for @xmath24 equal to an even integer . when the phase ( arising from the jordan - wigner transformation ) related to each bond between the @xmath28th column and the zeroth column is chosen to be @xmath29 ( @xmath30 is an integer ) , the hamiltonian ( [ hamiltonian ] ) is reduced to @xmath31 in eq . ( [ hamilfermion ] ) , the @xmath32 operators @xmath33 , where @xmath24 is an even integer , commute with each other . the ground state is in the vortex - free sector @xcite with @xmath34 , which corresponds to the case with the eigenvalue of each plaquette operator @xcite @xmath35 equal to 1 . thus , we can set the @xmath32 operators @xmath33 all equal to 1 in eq . ( [ hamilfermion ] ) , in order to obtain the ground - state energy . for this quadratic hamiltonian , the fourier transformation of @xmath36 via @xmath37 gives rise to @xmath38 where @xmath39 and @xmath40 are pauli matrices , @xmath41 with @xmath42 , and @xmath43 let us define @xmath44c_{\mathbf{k}}^{(2)}}{\sqrt{|\alpha ( \mathbf{k})|^2+\left[\varepsilon(\mathbf{k})+2j\sin k_x\right]^2}},\end{aligned}\ ] ] where @xmath45 and @xmath46 it is straightforward to verify that @xmath47 i.e. , @xmath48 and @xmath49 are fermionic operators , and the hamiltonian ( [ hamil ] ) can be written as @xmath50.\ ] ] for hamiltonian ( [ realfermionhamiltonian ] ) , the ground - state energy is @xmath51 and the ground - state @xmath52 obeys @xmath53 . the energy spectrum @xmath54 is _ gapless _ @xcite only when @xmath55 , or @xmath56 , or @xmath57 , which corresponds to the thick solid , dashed , and dotted lines in fig . [ fig2 ] , respectively . when @xmath58 , the spectral chern number is @xmath59 if @xmath60 and @xmath61 , and 0 if @xmath62 or @xmath63 or @xmath64 ( see @xcite ) . these two cases correspond to the non - abelian and abelian phases in the kitaev model and both of them are _ gapped _ topological phases . the phase diagram is shown in fig . [ fig2 ] , where the gray area corresponds to the non - abelian phase and the critical lines ( denoted as thick solid , dashed and dotted lines ) separate the abelian and non - abelian phases . this indicates that the system can experience quantum phase transitions across these three thick lines . here we rescale the inter - spin coupling strengths by introducing @xmath65 , and @xmath66 , so as to conveniently characterize the quantum phase transition . to demonstrate the quantum phase transition , one may reveal the nonanalyticity of the ground - state energy . the ground - state energy per site @xmath67 ( in units of @xmath2 ) and its first , second , and third derivatives with respect to @xmath20 , where @xmath68 and @xmath69 ( which corresponds to the horizontal thin dotted line in fig . [ fig2 ] ) . it is clear that @xmath67 and @xmath70 are continuous functions , but @xmath71 is discontinuous at the transition points @xmath72 and @xmath73.,width=297 ] the ground - state energy per site is @xmath74 where @xmath75 denotes the first brillouin zone . its directional derivatives with respect to the driving parameter along any given direction @xmath76 ( see fig . [ fig2 ] ) are @xmath77 if the @xmath78th directional derivative @xmath79 ( @xmath78=1 , 2 , @xmath80 ) is nonanalytical at the critical point @xmath81 , and the directional derivatives @xmath82 with @xmath83 are analytical there , a topological quantum phase transition occurs at this critical point . it can be proved that ( see appendix a ) @xmath84 and @xmath85 where @xmath86 @xmath87 denotes @xmath88 with @xmath89 , and @xmath90 denotes @xmath91 with @xmath92 . in ( [ defineformarktext ] ) , @xmath93 equations ( [ 1and2ofground ] ) and ( [ analyticalproveground ] ) reveal that a continuous topological quantum phase transition occurs across the critical line @xmath12 ( denoted by the thick solid line in fig . [ fig2 ] ) . similarly , it can be shown that such a continuous topological quantum phase transition also occurs across the critical lines @xmath13 and @xmath94 ( denoted , respectively , by the thick dashed and dotted lines in fig . [ fig2 ] ) . as a numerical test , we choose @xmath95 and @xmath96 to show this quantum phase transition in fig . [ fig3 ] , where the range of @xmath20 is chosen by the thin dotted line in fig . it can be seen in fig . [ fig3 ] that the ground - state energy and its first and second directional derivatives are continuous for each @xmath20 , while its third directional derivative is nonanalytic at the points @xmath72 and @xmath97 . these two points satisfy the condition @xmath57 and @xmath55 , respectively . it is obvious that these two points are on the critical lines denoted by the thick dotted and solid lines in fig . it has been shown that the entanglement also exhibits critical behavior at the quantum phase transition point for both spin ( see , e.g. , @xcite ) and fermionic systems ( see , e.g. , @xcite ) . also , it was shown @xcite that there is a general relation between the bipartite entanglement and the quantum phase transition . in this section , we show that the nonanalyticity of the ground - state energy in the extended kitaev model results from the correlation functions [ see eqs . ( [ x - bond density ] ) , ( [ y - bond density ] ) and ( [ z - bond density ] ) for their definitions ] . furthermore , we show that the bipartite entanglement also exhibits nonanalyticity at the quantum phase transition point and its nonanalyticity is also due to the nonanalyticity of the same correlation functions . this reveals that both the ground - state energy and the bipartite entanglement can characterize the quantum phase transition in the kitaev model . from hellmann - feynman theorem @xcite , we have @xmath98 where @xmath67 is the ground - state energy per site given in eq . ( [ groundenergy ] ) , @xmath36 is the hamiltonian ( [ hamiltonian ] ) ( rescaled by @xmath2 ) , @xmath99 denotes the trace over the ground - state subspace , and @xmath100 is the density matrix of the system . when @xmath101 is traced over all spins except the two spins at @xmath102 and @xmath103 , the reduced density matrix is @xmath104 where @xmath105@xmath106@xmath107@xmath108 are pauli matrices @xmath109 and @xmath110 for @xmath111 @xmath106@xmath112@xmath108@xmath113 to 3 , and the unit matrix for @xmath111 @xmath106@xmath112@xmath108@xmath114 . when the two spins at @xmath102 and @xmath115 are linked by an @xmath116-type bond , the reduced density matrix becomes @xmath117 where @xmath24 is an odd integer , and @xmath118 is the unit operator . because of translational invariance , the correlation function @xmath119 is spatially invariant . thus , eq . ( [ xbond00 ] ) can be written as @xmath120 where @xmath24 is an odd integer . similarly , one has @xmath121 with @xmath122 where @xmath24 is an even integer for both @xmath123 and @xmath124 . here ( [ x - bond])-([z - bond ] ) are the results obtained for the reduced density matrix when the two spins at @xmath102 and @xmath103 are nearest - neighbors . when the two spins at @xmath102 and @xmath103 are not nearest - neighbors , the density matrix is @xmath125 using the jordan - wigner transformation ( [ phaseterm ] ) and the definitions ( [ majo001 ] ) and ( [ majo002 ] ) for the majorana fermions , we can derive that @xmath126 from eqs . ( [ realfermions ] ) , ( [ energyspectrum ] ) , and ( [ howtocorre ] ) we have @xmath127,\end{aligned}\ ] ] and @xmath128 which gives rise to @xmath129 . similarly , we have @xmath130,\end{aligned}\ ] ] with @xmath131 . from eqs . ( [ hf ] ) , ( [ x - bond density ] ) , and ( [ y - bond density ] ) , it follows that @xmath132 , \nonumber\\ \frac{\partial^3 e}{\partial l^3 } & = & \frac{1}{2}\cos^3\varphi\frac{\partial^2 \mathcal{g}_x}{\partial\lambda_x^2 } + \frac{1}{2}\sin^3\varphi\frac{\partial^2 \mathcal{g}_y}{\partial\lambda_y^2}\nonumber\\&&+\frac{1}{2}\cos^2\varphi\sin\varphi\left[2\frac{\partial^2 \mathcal{g}_x}{\partial\lambda_x\partial\lambda_y}+\frac{\partial^2 \mathcal{g}_y}{\partial\lambda_x^2}\right]\nonumber\\&&+\frac{1}{2}\cos\varphi\sin^2\varphi\left[2\frac{\partial^2 \mathcal{g}_y}{\partial\lambda_x\partial\lambda_y}+\frac{\partial^2 \mathcal{g}_x}{\partial\lambda_y^2}\right].\label{correforground}\end{aligned}\ ] ] equation ( [ correforground ] ) shows that the directional derivatives of the ground - state energy per site are determined by the correlation functions @xmath133 ( @xmath134 ) and their derivatives . section ii shows that @xmath135 and @xmath136 are continuous , while @xmath137 is _ discontinuous _ on the critical line , e.g. , @xmath12 ( i.e. , the thick solid line in fig . [ fig2 ] ) . equation ( [ correforground ] ) reveals that the nonanalyticity of @xmath67 on the critical line is due to the nonanalyticity of @xmath133 . as shown in appendix b , @xmath138 where @xmath139 , and @xmath140 where @xmath141 is given in eq . ( [ defineformarktext ] ) . equation ( [ disofthecorrtext ] ) shows that the spin - spin correlation function @xmath133 can signal the quantum phase transition , similar to the bond - bond correlation function in the original kitaev model @xcite . from eqs . ( [ correforground ] ) and ( [ disofthecorrtext ] ) , we have @xmath142 which is the same as in eq . ( [ analyticalproveground ] ) . this further reveals that the nonanalyticity of the ground - state energy results from the nonanalyticity of the correlation functions @xmath133 . we now focus on the bipartite entanglement of the ground state @xmath52 between two spins ( at @xmath102 and @xmath103 ) and the rest of the spins in the system . we use the von neumann entropy to measure the entanglement between these two spins and the rest of the spins in the system . the von neumann entropy can be defined by @xcite @xmath143,\end{aligned}\ ] ] where denotes the trace over the two - spin hilbert space , and @xmath144 if @xmath24 is an odd integer , and @xmath145 if @xmath24 is an even integer . also , this entropy can be written as @xmath146 where the sum runs over the four eigenvalues @xmath147 of the matrix @xmath148 . from eqs . ( [ x - bond density ] ) and ( [ y - bond density ] ) , it follows that @xmath149 thus , we have the entanglement measure @xmath150,\end{aligned}\ ] ] which is determined by the correlation function @xmath133 , similar to the thermal entanglement @xcite . the bipartite entanglement ( i.e. , the von neumann entropy ) @xmath151 and its first , and second derivatives with respect to @xmath20 , where @xmath68 and @xmath69 ( which corresponds to the horizontal thin dotted line in fig . [ fig2 ] ) . obviously , @xmath151 and @xmath152 are continuous functions , but @xmath153 is discontinuous at the transition points @xmath72 and @xmath73.,width=297 ] to see the relationship between the entanglement and the quantum phase transition , we analyze the directional derivatives of the von neumann entropy with respect to the driving parameters along any direction @xmath76 . the first and second directional derivatives of the bipartite entanglement are @xmath154 where @xmath155 from eqs . ( [ contcorrtext ] ) , ( [ disofthecorrtext ] ) , ( [ entropy ] ) , and ( [ derivative1and2 ] ) , we have @xmath156 and @xmath157 where @xmath158 equation ( [ anavon2 ] ) shows that the bipartite entanglement is nonanalytic with its second directional derivative @xmath159 discontinuous at the critical line @xmath12 ( denoted by the thick solid line in fig . [ fig2 ] ) . because @xmath160 , it follows from eq . ( [ derivative1and2 ] ) that the discontinuity of @xmath161 is due to the discontinuity of @xmath162 . similarly , it can be shown that @xmath161 also exhibits a discontinuity on the critical lines @xmath13 and @xmath94 ( denoted by the thick dashed and dotted lines in fig . [ fig2 ] ) which is due to the discontinuity of @xmath163 on these lines . as in fig . [ fig3 ] , we choose @xmath95 and @xmath96 as a typical example to show @xmath164 and its first and second derivatives with respect to @xmath20 ( see fig . [ fig4 ] ) . it is clear that @xmath164 and its first derivative @xmath165 are continuous as a function of @xmath20 , but its second derivative @xmath166 is discontinuous at the quantum phase transition points @xmath72 and @xmath97 . as shown above , both the nonanalyticity of the ground - state energy and that of the bipartite entanglement are due to the nonanalyticity of the spin - spin correlation functions . this reveals that the ground - state energy and the bipartite entanglement are closely related with each other , and both of them can be used to characterize the topological quantum phase transition in the extended kitaev spin model . in conclusion , we have studied the topological quantum phase transition between abelian and non - abelian phases in the extended kitaev spin model on a honeycomb lattice . from the ground - state energy , we show that this model displays a continuous quantum phase transition on the critical lines separating the abelian and non - abelian phases , where the third derivative of the ground - state energy is discontinuous . also , we use the von neumann entropy as a measure of bipartite entanglement to study this topological quantum phase transition . our results show that the bipartite entanglement is also nonanalytic on the same critical lines as the ground - state energy . moreover , we show that the discontinuity of the second derivative of the bipartite entanglement is related to the discontinuity of the third derivative of the ground - state energy . our approach directly reveals that both the entanglement and the ground - state energy can be used to characterize the topological quantum phase transition in this kitaev model . we thank z. d. wang and y. chen for useful discussions . j.q.y . and x.f.s . were supported in part by the national basic research program of china grant nos . 2009cb929300 and 2006cb921205 , the national natural science foundation of china grant nos . 10625416 , and the most international collaboration program grant no . 2008dfa01930 . y.y . was supported in part by the national natural science foundation of china , the national basic research program of china and a fund from cas . f.n . was supported in part by the u.s . national security agency , the laboratory for physical sciences , the u.s . army research office , and the national science foundation grant no . eia-0130383 . this appendix focuses on the analyticity of the first , second and third directional derivatives of the ground - state energy on the critical line denoted by the thick solid line in fig . [ fig2 ] . from eq . ( [ groundenergy ] ) , the derivative of the ground - state energy with respect to @xmath20 is @xmath167 where @xmath168 and @xmath169 denotes two small regions in the first brillioun zone , i.e. , half of the disk with radius @xmath170 , which is centered at @xmath171 , and half of the disk with radius @xmath170 , which is centered at @xmath172 , where @xmath173 . when @xmath174 , @xmath54 becomes zero only at the points @xmath175 , so @xmath176 is analytic because @xmath177 is the region excluding @xmath169 in the first brillioun zone . for the integral in the region @xmath169 , we can approximate it as @xmath178\frac{\delta}{s_2}\nonumber\\ & & + \int_0^{2\pi}d\theta \left[\frac{1}{3}\left(\delta^2+\epsilon^2s_2\right)^{\frac{3}{2}}-|\delta|^3\right]\frac{s_1 } { s_2 ^ 2}\nonumber\\ & & -\int_0^{2\pi}d\theta \frac{s_1\delta^2\sqrt{\delta^2+\epsilon^2s_2 } } { s_2 ^ 2 } \nonumber\\ & & + \int_0^{2\pi}d\theta|\delta|^3\frac{s_1 } { s_2 ^ 2},\label{a3}\end{aligned}\ ] ] where @xmath179 and @xmath180 from eq . ( [ a3 ] ) , we have @xmath181 where @xmath182 denotes @xmath183 with @xmath184 , and @xmath185 denotes @xmath183 with @xmath186 . when @xmath187 , it follows from eq . ( [ a5 ] ) that @xmath188 on the critical line @xmath12 ( i.e. , the thick solid line in fig . [ fig2 ] ) . thus , from eq . ( [ a1 ] ) , we have @xmath189 similarly , @xmath190 from eqs . ( [ directionalderivative ] ) , ( [ a6 ] ) , and ( [ a7 ] ) , it follows that @xmath191 using the same procedure as above , we can obtain @xmath192 where @xmath193 this appendix gives results regarding the analyticity of the correlation function @xmath133 ( @xmath134 ) and its first , and second directional derivatives on the critical line denoted by the thick solid line in fig . [ fig2 ] . similar to eq . ( [ a1 ] ) , one can divide the integral in ( [ correlation001 ] ) into two parts : @xmath194 where @xmath195 is given in eq . ( [ a2 ] ) . using the same procedure for eqs . ( [ a8 ] ) and ( [ a9 ] ) , we can derive from eq . 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we study the quantum phase transition between abelian and non - abelian phases in an extended kitaev spin model on the honeycomb lattice , where the periodic boundary condition is applied by placing the lattice on a torus . our analytical results show that this spin model exhibits a continuous quantum phase transition . also , we reveal the relationship between bipartite entanglement and the ground - state energy . our approach directly shows that both the entanglement and the ground - state energy can be used to characterize the topological quantum phase transition in the extended kitaev spin model .

the experimental realization of dilute degenerate bose and fermi gases has led to an explosion of activities in the field of cold atom gases . a particularly intriguing feature of atomic bose and fermi gases is that their interaction strengths can be tuned experimentally through the application of an external magnetic field in the vicinity of a feshbach resonance @xcite . this external knob allows dilute systems with essentially any interaction strength , including infinitely strongly attractive and repulsive interactions , to be realized . feshbach resonances have been experimentally observed for @xmath0- , @xmath6- and @xmath7-wave interacting gases @xcite and have been predicted to exist also for higher partial waves . a feshbach resonance arises due to the coupling of two born - oppenheimer potential curves coupled through a hyperfine hamiltonian , and requires , in general , a multi - channel description . for @xmath0-wave interacting systems , feshbach resonances can be classified as broad or narrow @xcite . whether a resonance is broad or narrow depends on whether the energy width of the resonance is large or small compared to the characteristic energy scale , such as the fermi energy or the harmonic oscillator energy , of the system . in contrast to @xmath0-wave resonances , higher partial wave resonances are necessarily narrow due to the presence of the angular momentum barrier @xcite . this paper uses an effective single channel description to investigate the behaviors of strongly - interacting bose and fermi systems with different orbital angular momenta . in dilute homogeneous bose and fermi gases with large @xmath0-wave scattering length @xmath1 , a regime has been identified in which the energy per particle takes on a universal value which is set by a single length scale , the average interparticle spacing @xmath2 @xcite . in this so - called unitary regime , the length scales of the @xmath0-wave interacting system separate according to @xmath8 , where @xmath9 denotes the range of the two - body potential . the energy per particle @xmath10 ( the subscripts `` @xmath11 '' and `` @xmath12 '' stand respectively for `` boson '' and `` @xmath0-wave interacting '' ) for a homogeneous one - component gas of bosons with mass @xmath13 in the unitary regime has been calculated to be @xmath14 using the lowest order constrained variational ( locv ) method @xcite . the energy @xmath10 at unitarity is thus independent of @xmath1 and @xmath9 , and depends on the single length scale @xmath2 through the boson number density @xmath15 , @xmath16 . however , bose gases in the large scattering length limit are expected to be unstable due to three - body recombination @xcite . on the other hand , the fermi pressure prevents the collapse of two - component fermi gases with equal masses and equal number of `` spin - up '' and `` spin - down '' fermions with large interspecies @xmath0-wave scattering length @xcite . at unitarity , the energy per particle is given by @xmath17 , where @xmath18 denotes the energy per particle of the non - interacting fermi gas @xcite . the fermi wave vector @xmath19 is related to the number density of the fermi gas by @xmath20 , which implies that @xmath21 depends on @xmath2 but is independent of @xmath1 and @xmath9 . we note that the inequality @xmath22 is equivalent to @xmath23 . this paper investigates bose and fermi systems with large generalized scattering lengths using the locv method . for @xmath6- and @xmath7-wave interacting bose systems , we define the unitary regime @xcite through the inequalities @xmath24 , where @xmath5 denotes a @xmath4-dependent length scale given by the geometric combination of @xmath2 and @xmath9 , i.e. , @xmath25 , and @xmath26 the relative scattering energy . the generalized energy - dependent scattering length @xmath27 @xcite characterizes the scattering strength ( see below ) . we find that the energy of @xmath6-wave interacting two - component bose gases and @xmath7-wave interacting one- and two - component bose gases at unitary is determined by the combined length @xmath5 . while bose gases with higher angular momentum in the unitary regime are of theoretical interest , they are , like their @xmath0-wave cousin , expected to be unstable . we comment that the energetics of two - component fermi gases with large generalized scattering length may depend on the same length scales . furthermore , we consider @xmath0-wave interacting bose systems over a wide range of densities . motivated by two recent studies by gao @xcite , we determine the energy per particle @xmath10 of the bose system characterized by two atomic physics parameters , the @xmath0-wave scattering lengh @xmath1 and the van der waals coefficient @xmath28 . our results lead to a phase diagram of liquid helium in the low - density regime that differs from that proposed in ref . @xcite . section [ sectionii ] describes the systems under study and introduces the locv method . section [ sectioniii ] describes our results for dilute @xmath0-wave interacting bose and fermi systems and for liquid helium . section [ sectioniv ] considers bose and fermi systems interacting through @xmath4-wave ( @xmath29 ) scattering . finally , section [ sectionv ] concludes . this section introduces the three - dimensional bose and fermi systems under study and reviews the locv method @xcite . the idea of the locv method is to explicitly treat two - body correlations , but to neglect three- and higher - body correlations . this allows the many - body problem to be reduced to solving an effective two - body equation with properly chosen constraints . imposing these constraints makes the method non - variational , i.e. , the resulting energy does not place an upper bound on the many - body energy . the locv method is expected to capture some of the key physics of dilute bose and fermi systems . the hamiltonian @xmath30 for a homogeneous system consisting of identical mass @xmath13 bosons is given by @xmath31 where the spherically symmetric interaction potential @xmath32 depends on the relative distance @xmath33 , @xmath34 . here , @xmath35 denotes the position vector of the @xmath36th boson . the hamiltonian @xmath37 for a two - component fermi system with equal masses and identical spin population is given by @xmath38 where the unprimed subscripts label spin - up and the primed subscripts spin - down fermions . throughout , we take like fermions to be non - interacting . our primary interest in this paper is in the description of systems for which many - body observables are insensitive to the short - range behavior of the atom - atom potential @xmath39 . this motivates us to consider two simple model potentials : an attractive square well potential @xmath40 with depth @xmath41 ( @xmath42 ) , @xmath43 and an attractive van der waals potential @xmath44 with hardcore @xmath45 , @xmath46 in all applications , we choose the hardcore @xmath45 so that the inequality @xmath47 , where @xmath48 , is satisfied . the natural length scale of the square well potential is given by the range @xmath9 and that of the van der waals potential by the van der waals length @xmath49 . the solutions to the two - body schrdinger equation for @xmath40 are given in terms of spherical bessel and neumann functions ( imposing the proper continuity conditions of the wave function and its derivations at those @xmath50 values where the potential exhibits a discontinuity ) , and those for @xmath44 in terms of convergent infinite series of spherical bessel and neumann functions @xcite . the interaction strength of the short - range square well potential can be characterized by the generalized energy - dependent scattering lengths @xmath51 , @xmath52 \left| \frac{\tan\delta_l(k)}{k^{2l+1 } } \right|^{1/(2l+1)},\end{aligned}\ ] ] where @xmath53 denotes the phase shift of the @xmath4th partial wave calculated at the relative scattering energy @xmath26 , @xmath54 . this definition ensures that @xmath51 approaches a constant as @xmath55 @xcite . for the van der waals potential @xmath44 , the threshold behavior changes for higher partial waves and the definition of @xmath51 has to be modified accordingly @xcite . in general , for a potential that falls off as @xmath56 at large interparticle distances , @xmath51 is defined by eq . ( [ eq_scatt1 ] ) if @xmath57 and by @xmath58 \left| \frac { \tan\delta_l(k)}{k^{n-2 } } \right|^{1/(n-2)}\end{aligned}\ ] ] if @xmath59 . for our van der waals potential , @xmath60 is equal to @xmath61 and @xmath51 is given by eq . ( [ eq_scatt1 ] ) for @xmath62 and by eq . ( [ eq_scatt2 ] ) for @xmath63 . the zero - energy generalized scattering lengths @xmath64 can now be defined readily through @xmath65 we note that a new two - body @xmath4-wave bound state appears at threshold when @xmath66 . the unitary regime for higher partial waves discussed in sec . [ sectioniv ] is thus , as in the @xmath0-wave case , closely related to the physics of extremely weakly - bound atom - pairs . to uncover the key behaviors at unitarity , we assume in the following that the many - body system under study is interacting through a single partial wave @xmath4 . while this may not be exactly realized in an experiment , this situation may be approximated by utilizing feshbach resonances . we now outline how the energy per particle @xmath67 of a one - component bose system with @xmath4-wave interactions @xcite is calculated by the locv method @xcite . the boson wave function @xmath68 is taken to be a product of pair functions @xmath69 , @xmath70 and the energy expectation value of @xmath30 , eq . ( [ hb ] ) , is calculated using @xmath68 . if terms depending on the coordinates of three or more different particles are neglected , the resulting energy is given by the two - body term in the cluster expansion , @xmath71 } f_l(r)\ , { \rm d}^3{\bf r}.\ ] ] the idea of the locv method is now to introduce a healing distance @xmath7 beyond which the pair correlation function @xmath69 is constant , @xmath72 to ensure that the derivative of @xmath69 is continuous at @xmath73 , an additional constraint is introduced , @xmath74 introducing a constant average field @xmath75 and varying with respect to @xmath69 while using that @xmath69 is constant for @xmath76 , gives the schrdinger - like two - body equation for @xmath77 , @xmath78}\left(rf_l(r)\right)= \lambda_l r f_l(r).\ ] ] finally , the condition @xmath79 enforces that the average number of particles within @xmath7 equals @xmath80 . using eqs . ( [ ebcluster ] ) , ( [ fdfdr_1 ] ) and ( [ schro ] ) , the energy per particle becomes , @xmath81 the second term on the right hand side of eq . ( [ ebn ] ) is identically zero for the square well potential @xmath40 but contributes a so - called tail or mean - field energy for the van der waals potential @xmath44 @xcite . we determine the three unknown @xmath15 , @xmath75 and @xmath7 by simultaneously solving eqs . ( [ schro ] ) and ( [ norm1 ] ) subject to the boundary condition given by eq . ( [ fdfdr_2 ] ) . note that @xmath15 and @xmath7 depend , just as @xmath69 and @xmath75 , on the angular momentum ; the subscript has been dropped , however , for notational convenience . in addition to one - component bose systems , sec . [ sectioniv ] considers two - component bose systems , characterized by @xmath4-wave interspecies and vanishing intraspecies interactions . the hamiltonian for the two - component bose system is given by eq . ( [ hf ] ) , with the sum of the two - body interactions restricted to unlike bosons . correspondingly , the product wave function is written as a product of pair functions , including only correlations between unlike bosons . the locv equations are then given by eqs . ( [ fdfdr_1 ] ) through ( [ norm1 ] ) with @xmath15 in eq . ( [ norm1 ] ) replaced by @xmath82 . next , we discuss how to determine the energy @xmath83 per particle for a two - component fermi system within the locv method @xcite . the wavefunction is taken to be @xmath84 where @xmath85 denotes the ground state wavefunction of the non - interacting fermi gas . the product of pair functions @xmath69 accounts for the correlations between unlike fermions . in accord with our assumption that like fermions are non - interacting , eq . ( [ eq_wavef ] ) treats like fermion pairs as uncorrelated . neglecting exchange effects , the derivation of the locv equations parallels that outlined above for the bosons . the boundary conditions , given by eqs . ( [ fdfdr_1 ] ) and ( [ fdfdr_2 ] ) , and the schrdinger - like differential equation for @xmath86 , eq . ( [ schro ] ) , are unchanged . the `` normalization condition , '' however , becomes @xmath87 where the left - hand side is the number of fermion pairs within @xmath7 . the fermion energy per particle is then the sum of the one - particle contribution from the non - interacting fermi gas and the pair correlation energy @xmath75 @xcite , @xmath88 this equation excludes the contribution from the tail of the potential , i.e. , the term analogous to the second term on the right hand side of eq . ( [ ebn ] ) , since this term is negligible for the fermion densities considered in this paper . the locv solutions for @xmath69 , @xmath75 and @xmath7 for the homogeneous one - component bose system and the two - component fermi system are formally identical if the boson density is chosen to equal half the fermion density , i.e. , if @xmath89 . this relation can be understood by realizing that any given fermion ( e.g. , a spin - up particle ) interacts with only half of the total number of fermions ( e.g. , all the spin - down fermions ) . consequently , the two - component fermi system appears twice as dense as the one - component bose system . the fact that the locv solutions for bosons can be converted to locv solutions for fermions suggests that some physics of the bosonic system can be understood in terms of the fermionic system and vice versa . in fact , it has been shown previously @xcite that the locv energy for the first excited gas - like state of @xmath0-wave interacting fermions at unitarity can be derived from the locv energy of the energetically lowest - lying gas - like branch of @xmath0-wave interacting bosons @xcite . here , we extend this analysis and show that the ground state energy of the fermi gas at unitarity can be derived from the energetically highest - lying liquid - like branch of the bose system . furthermore , we extend this analysis to higher angular momentum scattering . figure [ cap1 ] shows the energy per particle @xmath10 , eq . ( [ ebn ] ) , obtained by solving the locv equations for a one - component bose system interacting through the van der waals potential with @xmath0-wave scattering length @xmath90 . energy per particle @xmath10 as a function of the density @xmath15 , both plotted as dimensionless quantities , for a one - component bose system interacting through the van der waals potential with @xmath0-wave scattering length @xmath90 . the dotted line shows the gas branch and the dashed line the liquid branch . the minimum of the liquid branch is discussed in reference to liquid @xmath91he in the text . ] the dotted line in fig . [ cap1 ] has positive energy and increases with increasing density ; it describes the energetically lowest - lying `` gas branch '' for the bose system with @xmath90 and corresponds to the metastable gaseous condensate studied experimentally . the dashed line in fig . [ cap1 ] has negative energy at small densities , decreases with increasing density , and then exhibits a minimum ; this dashed line describes the energetically highest - lying `` liquid branch '' for a bose system with @xmath90 . within the locv framework , these two branches arise because the schrdinger - like equation , eq . ( [ schro ] ) , permits for a given interaction potential solutions @xmath92 with differing number of nodes , which in turn give rise to a host of liquid and gas branches @xcite . throughout this work we only consider the energetically highest - lying liquid branch with @xmath60 nodes and the energetically lowest - lying gas branch with @xmath93 nodes . to obtain fig . [ cap1 ] , we consider a class of two - body potentials with fixed @xmath94 , and decrease the value of the ratio @xmath95 till @xmath10 , eq . ( [ ebn ] ) , no longer changes over the density range of interest , i.e. , the number of nodes @xmath60 of the energetically highest - lying liquid branch is increased till convergence is reached . in fig . [ cap1 ] , the two - body van der waals potential is chosen so that the scattering length of @xmath90 coincides with that of the @xmath91he pair potential @xcite . the liquid branch in fig . [ cap1 ] can hence be applied to liquid @xmath91he , and has previously been considered in refs . the minimum of the liquid branch at a density of @xmath96 , or @xmath97 , agrees quite well with the experimental value of @xmath98 @xcite . the corresponding energy per particle of @xmath99 k deviates by 8.5 % from the experimental value of @xmath100 k @xcite . this shows that the locv framework provides a fair description of the strongly interacting liquid @xmath91he system , which is characterized by interparticle spacings comparable to the range of the potential . this is somewhat remarkable considering that the locv method includes only pair correlations and that the van der waals potential used here contains only two parameters . open circles connected by a dashed line in fig . [ cap2 ] show the liquid branch for @xmath90 in energy per particle @xmath10 as a function of the density @xmath15 for a one - component bose system interacting through the van der waals potential with @xmath0-wave scattering lengths @xmath90 ( open circles ) and @xmath101 ( filled circles ) . to guide the eye , dashed and dotted lines connect the data points of the liquid and gas branches , respectively . the liquid branches go to @xmath102 as the density goes to zero . the solid lines show @xmath10 at unitarity ; see text for discussion . compared to fig . [ cap1 ] , the energy and density scales are greatly enlarged . ] the small density region . as the density goes to zero , the energy per particle @xmath10 does not terminate at zero but , instead , goes to @xmath102 , where @xmath103 denotes the energy of the most weakly - bound @xmath0-wave molecule of @xmath44 . in this small density limit , the liquid branch describes a gas of weakly - bound molecules , in which the interparticle spacing between the molecules greatly exceeds the size of the molecules , and @xmath103 is to a very good approximation given by @xmath104 . as seen in fig . [ cap2 ] , we find solutions in the whole density range considered . in contrast to our findings , ref . @xcite reports that the locv solutions of the liquid branch disappear at densities smaller than a scattering length dependent critical density , i.e. , at a critical density of @xmath105 for @xmath90 . thus we are not able to reproduce the liquid - gas phase diagram proposed in fig . 2 of ref . @xcite , which depends on this termination of the liquid branch . we note that the liquid branch is , as indicated by its imaginary speed of sound , dynamically unstable at sufficiently small densities . the liquid of weakly - bound bosonic molecules discussed here can , as we show below , be related to weakly - bound molecules on the bec side of the bec - bcs crossover curve for two - component fermi gases . we now discuss the gas branch in more detail . open and filled circles connected by dotted lines in fig . [ cap2 ] show the energy per particle for @xmath90 and @xmath106 , respectively . these curves can be applied , e.g. , to @xmath107rb , whose scattering length can be tuned by means of a feshbach resonance and which has a @xmath49 value of @xmath108 , where @xmath109 denotes the bohr radius . for this system , a scattering length of @xmath110 corresponds to @xmath111 , a comparatively large value that can be realized experimentally in @xmath107rb gases . as a point of reference , a density of @xmath112 corresponds to a density of @xmath113 for @xmath107rb . the solid curve with positive energy in fig . [ cap2 ] shows the energy per particle @xmath10 at unitarity , @xmath114 @xcite . as seen in fig . [ cap2 ] , this unitary limit is approached by the energy per particle for the bose gas with @xmath101 ( filled circles connected by a dotted line ) . to illustrate this point , fig . [ cap3 ] shows the scaled average interparticle scaled interparticle spacing @xmath115 as a function of the scaled density @xmath116 for the gas branch of a one - component bose system interacting through the van der waals potential with @xmath101 . the horizontal lines show the scaled @xmath0-wave scattering length @xmath117 and the range of the van der waals potential , which is one in scaled units ( almost indistinguishable from the @xmath118-axis ) . this graph shows that the unitary inequalities @xmath119 hold for @xmath15 larger than about @xmath120 . ] spacing @xmath121 as a function of the scaled density @xmath116 for @xmath101 . this plot indicates that the unitary requirement , @xmath122 , is met for values of @xmath116 larger than about @xmath123 . similarly , we find that the family of liquid curves converges to @xmath124 ( see sec . [ sectioniv ] for details ) , plotted as a solid line in fig . [ cap2 ] , when the inequalities @xmath119 are fullfilled . we note that the unitarity curve with negative energy is also approached , from above , for systems with large negative scattering lengths ( not shown in fig . [ cap2 ] ) . aside from the proportionality constant , the power law relation for the liquid and gas branches at unitarity is the same . in addition to a bose system interacting through the van der waals potential , we consider a bose system interacting through the square well potential with range @xmath9 . for a given scattering length @xmath1 and density @xmath15 , the energy per particle @xmath10 for these two two - body potentials is essentially identical for the densities shown in fig . this agreement emphasizes that the details of the two - body potential become negligible at low density , and in particular , that the behavior of the bose gas in the unitary limit is governed by a single length scale , the average interparticle spacing @xmath2 . as discussed in sec . [ sectionii ] , the formal parallels between the locv method applied to bosons and fermions allows the energy per particle @xmath21 for a two - component fermi gas , eq . ( [ efn ] ) , to be obtained straightforwardly from the energy per particle @xmath10 of the bose system . figure [ cap4 ] shows the dimensionless energy scaled energy per particle @xmath125 as a function of @xmath126 for a two - component @xmath0-wave fermi gas interacting through the square well potential for @xmath127 . the combined dashed and dash - dotted curve corresponds to the bec - bcs crossover curve and the dotted curve corresponds to the first excited state of the fermi gas . the dashed and dotted linestyles are chosen to emphasize the connection to the gas and liquid branches of the bose system in figs . [ cap2 ] and [ cap3 ] ( see text for more details ) . ] @xmath125 as a function of the dimensionless quantity @xmath126 for the square well potential for @xmath128 . we find essentially identical results for the van der waals potential . the crossover curve shown in fig . [ cap4 ] describes any dilute fermi gas for which the range @xmath9 of the two - body potential is very small compared to the average interparticle spacing @xmath2 . in converting the energies for the bose system to those for the fermi system , the gas branches of the bose system ( dotted lines in figs . [ cap2 ] and [ cap3 ] ) `` turn into '' the excited state of the fermi gas ( dotted line in fig . [ cap4 ] ) ; the liquid branches of the bose system with positive @xmath1 ( dashed lines in figs . [ cap2 ] and [ cap3 ] ) `` turn into '' the part of the bec - bcs crossover curve with positive @xmath1 ( dashed line in fig . [ cap4 ] ) ; and the liquid branches of the bose system with negative @xmath1 ( not shown in figs . [ cap2 ] and [ cap3 ] ) `` turn into '' the part of the bec - bcs crossover curve with negative @xmath1 ( dash - dotted line in fig . [ cap4 ] ) . to emphasize the connection between the bose and fermi systems further , let us consider the bec side of the crossover curve . if @xmath129 , the fermion energy per particle @xmath21 is approximately given by @xmath102 , which indicates that the fermi gas forms a molecular bose gas . similarly , the liquid branch of the bose system with positive scattering length is made up of bosonic molecules as the density goes to zero . the formal analogy between the bose and fermi locv solutions also allows the energy per particle @xmath21 at unitarity , i.e. , in the @xmath130 limit , to be calculated from the energies for large @xmath1 of the gas and liquid branches of the bose system ( solid lines in fig . [ cap2 ] ) . for the excited state of the fermi gas we find @xmath131 , and for the lowest gas state we find @xmath132 . these results agree with the locv calculations of ref . @xcite , which use an attractive @xmath133-potential and a @xmath134-function potential . the value of @xmath135 is in good agreement with the energy of @xmath136 obtained by fixed - node diffusion monte carlo calculations @xcite . this section investigates the unitary regime of bose and fermi systems interacting through higher angular momentum resonances . these higher angular momentum resonances are necessarily narrow @xcite , and we hence expect the energy - dependence of the generalized scattering length @xmath51 to be particularly important in understanding the many - body physics of dilute atomic systems beyond @xmath0-wave . in the following we focus on the strongly - interacting limit . figure [ cap5 ] shows @xmath137 as a function of the relative scattering energy @xmath26 for the square - well potential @xmath138 as a function of the scaled relative scattering energy @xmath139 for the square well potential @xmath40 with infinite zero - energy scattering length @xmath64 , i.e. , @xmath140 , for three different partial waves [ @xmath141 ( solid line ) , @xmath142 ( dashed line ) , and @xmath143 ( dotted line ) ] . ] with infinite zero - energy scattering length @xmath64 for three different angular momenta , @xmath141 ( solid line ) , @xmath142 ( dashed line ) , and @xmath143 ( dotted line ) . figure [ cap5 ] shows that the energy - dependence of @xmath51 increases with increasing @xmath4 . our goal is to determine the energy per particle @xmath67 for bose systems with finite angular momentum @xmath4 in the strongly - interacting regime . for @xmath0-wave interactions , the only relevant length scale at unitarity is the average interparticle spacing @xmath2 ( see sec . [ sectioniii ] ) . in this case , the energy per particle at unitarity can be estimated analytically by evaluating the locv equations subject to the boundary condition implied by the zero - range @xmath0-wave pseudo - potential @xcite . unfortunatey , a similarly simple analysis that uses the boundary condition implied by the two - body zero - range pseudo - potential for higher partial waves fails . this combined with the following arguments suggests that @xmath67 depends additionally on the range of the underlying two - body potential for finite @xmath4 : i ) the probability distribution of the two - body @xmath4-wave bound state , @xmath29 , remains finite as @xmath64 approaches infinity and depends on the interaction potential @xcite . ii ) a description of @xmath4-wave resonances ( @xmath29 ) that uses a coupled channel square well model depends on the range of the square well potential @xcite . iii ) the calculation of structural expectation values of two - body systems with finite @xmath4 within a zero - range pseudo - potential treatment requires a new length scale to be introduced @xcite . motivated by these two - body arguments ( see also refs . @xcite for a treatment of @xmath6-wave interacting fermi gases ) we propose the following functional form for the energy per particle @xmath67 of a @xmath4-wave bose system at unitarity interacting through the square - well potential @xmath40 with range @xmath9 , @xmath144 here , @xmath145 denotes a dimensionless @xmath4-dependent proportionality constant . the dimensionless parameter @xmath146 determines the powers of the range @xmath9 and the density @xmath15 , and ensures the correct units of the right hand side of eq . ( [ ebpower ] ) . to test the validity of eq . ( [ ebpower ] ) , we solve the locv equations , eqs . ( [ fdfdr_2 ] ) through ( [ ebn ] ) , for @xmath141 to @xmath147 for the one - component bose system . note that the one - component @xmath6-wave system is unphysical since it does not obey bose symmetry ; we nevertheless consider it here since its locv energy determines the energy of two - component @xmath6-wave bose and fermi systems ( see below ) . figure [ cap7a ] scaled energy per particle @xmath148 for a one - component bose system for the energetically lowest - lying gas branch as a function of the scaled density @xmath149 obtained by solving the locv equations [ eqs . ( [ fdfdr_2 ] ) through ( [ ebn ] ) ] for @xmath40 for three different angular momenta @xmath4 [ @xmath141 ( crosses ) , @xmath142 ( asterisks ) and @xmath143 ( pluses ) ] . the depth @xmath150 of @xmath40 is adjusted so that @xmath151 . solid , dotted and dashed lines show fits of the locv energies at low densities to eq . ( [ ebpower ] ) for @xmath141 , @xmath142 and @xmath143 ( see text for details ) . note that the system with @xmath142 is of theoretical interest but does not describe a physical system . ] shows the energy per particle @xmath67 for a one - component bose system , obtained by solving the locv equations for the energetically lowest - lying gas branch , as a function of the density @xmath15 for @xmath141 ( crosses ) , @xmath142 ( asterisks ) , and @xmath143 ( pluses ) for the square well potential , whose depth @xmath150 is adjusted for each @xmath4 so that the _ energy - dependent _ generalized scattering length @xmath51 diverges , i.e. , @xmath152 . setting @xmath51 to infinity ensures that the @xmath4-wave interacting bose system is infinitely strongly interacting over the entire density regime shown in fig . [ cap7a ] . had we instead set the zero - energy scattering length @xmath64 to infinity , the system would , due to the strong energy - dependence of @xmath51 [ see fig . [ cap5 ] ] , `` effectively '' interact through a finite scattering length . table [ tab1 ] summarizes the values for @xmath146 and @xmath153 , .dimensionless parameters @xmath146 , @xmath154 and @xmath153 for @xmath141 to @xmath147 for a one - component bose system obtained by fitting the locv energies @xmath67 for small densities to the functional form given in eq . ( [ ebpower ] ) ( see text for details ) . [ cols="<,<,<,<",options="header " , ] in table [ tab2 ] . the energy per particle @xmath83 for @xmath4-wave interacting two - component fermi systems can be obtained from eq . ( [ efn ] ) using the locv solutions for the liquid and gas branches discussed above for @xmath4-wave interacting one - component bose systems . in the unitary limit , we find @xmath155 where @xmath156 and @xmath157 ( the @xmath158 are given in table [ tab1 ] ) . the first term on the right hand side of eq . ( [ efdwave ] ) equals @xmath3 , and the second term , which is obtained from the locv solutions , equals @xmath159 . the energy per particle @xmath83 at unitarity is positive for all densities for @xmath160 . for @xmath161 , however , the energy per particle @xmath83 at unitarity is negative for @xmath29 for small densities , and goes through a minimum for larger densities . this implies that this branch is always mechanically unstable in the dilute limit for @xmath29 . the locv treatment for fermions relies heavily on the product representation of the many - body wave function , eq . ( [ eq_wavef ] ) , which in turn gives rise to the two terms on the right hand side of eq . ( [ efdwave ] ) . it is the competition of these two energy terms that leads to the energy minimum discussed in the previous paragraph . future work needs to investigate whether the dependence of @xmath83 on two length scales as implied by eq . ( [ efdwave ] ) is correct . in contrast to the locv method , mean - field treatments predict that the energy at unitarity is proportional to @xmath162 , where @xmath163 denotes a range parameter that characterizes the underlying two - body potential @xcite . this paper investigates bose and fermi systems using the locv method , which assumes that three- and higher - order correlations can be neglected and that the behaviors of the many - body system are governed by two - body correlations . this assumption allows the many - body problem to be reduced to an effective two - body problem . besides the reduced numerical effort , this formalism allows certain aspects of the many - body physics to be interpreted from a two - body point of view . furthermore , it allows parallels between bose and fermi systems to be drawn . in agreement with previous studies , we find that the energy per particle `` corrected '' by the dimer binding energy , i.e. , @xmath164 , of dilute two - component @xmath0-wave fermi gases in the whole crossover regime depends only on the @xmath0-wave scattering length and not on the details of the underlying two - body potential . furthermore , at unitarity the energy per particle is given by @xmath165 . this locv result is in good agreement with the energy per particle obtained from fixed - node diffusion monte carlo calculations , which predict @xmath166 @xcite . this agreement may be partially due to the cancellation of higher - order correlations , and thus somewhat fortuitous . in contrast to ref . @xcite , we find that the liquid branch of bosonic helium does not terminate at low densities but exists down to zero density . for higher angular momentum interactions , we determine the energy per particle of one- and two - component bose systems with infinitely large scattering lengths . for these systems , we expect the locv formalism to predict the dimensionless exponent @xmath146 , which determines the functional dependenc of @xmath67 on the range @xmath9 of the two - body potential and on the average interparticle spacing @xmath2 , correctly . the values of the proportionality constants @xmath153 and @xmath154 , in contrast , may be less accurate . we use the locv energies to generalize the known unitary condition for @xmath0-wave interacting systems to systems with finite angular momentum . since higher angular momentum resonances are necessarily narrow , leading to a strong energy - dependence of the scattering strength , we define the universal regime using the energy - dependent scattering length @xmath51 . in the unitary regime , the energy per particle can be written in terms of the length @xmath5 , which is given by a geometric combination of @xmath2 and @xmath9 . the locv framework also allows a prediction for the energy per particle of two - component fermi gases beyond @xmath0-wave to be made [ see eq . ( [ efdwave ] ) ] . although the functional form of the many - body wave function for two - component fermi systems used in this work may not be the best choice , we speculate that the energy scales derived for strongly interacting bose systems are also relevant to fermi systems .

dilute fermi systems with large @xmath0-wave scattering length @xmath1 exhibit universal properties if the interparticle spacing @xmath2 greatly exceeds the range of the underlying two - body interaction potential . in this regime , @xmath2 is the only relevant length scale and observables such as the energy per particle depend only on @xmath2 ( or , equivalently , the energy @xmath3 of the free fermi gas ) . this paper investigates bose and fermi systems with non - vanishing angular momentum @xmath4 using the lowest order constrained variational method . we focus on the regime where the generalized scattering length becomes large and determine the relevant length scales . for bose gases with large generalized scattering lengths , we obtain simple expressions for the energy per particle in terms of a @xmath4-dependent length scale @xmath5 , which depends on the range of the underlying two - body potential and the average interparticle spacing . we discuss possible implications for dilute two - component fermi systems with finite @xmath4 . furthermore , we determine the equation of state of liquid and gaseous bosonic helium .

the massless gross - neveu ( gn ) model @xcite is the 1 + 1 dimensional quantum field theory of @xmath0 flavors of massless dirac fermions , interacting through a scalar - scalar contact interaction . suppressing flavor labels as usual , its lagrangian reads @xmath1 the physics phenomena inherent in this simple looking lagrangian are particularly rich and accessible in the t hooft limit ( @xmath2 const . ) , to which we restrict ourselves from here on . the gn model can be thought of as relativistic version of particles moving along a line and interacting via an attractive @xmath3-potential . however , it exhibits many non - trivial features characteristic for relativistic quantum fields such as covariance , renormalizability , asymptotic freedom , dimensional transmutation , spontaneous symmetry breaking , interacting dirac sea . it is also one of the few models known where most of the non - perturbative questions of interest to strong interaction physics can be answered in closed analytical form . such calculations have turned out to be both challenging and instructive , generating a continued interest in this particular toy model " over several decades , see e.g. the review articles @xcite . in the present paper we address the problem of time - dependent scattering of multi - fermion bound states in full generality . as will be recalled in more detail in the next section , the gn model possesses bound states which can be viewed as baryons " , with fermions bound in a dynamically created bag " of the scalar field @xmath4 @xcite . there are even multi - baryon bound states which might be identified with nuclei " @xcite . standard large @xmath0 arguments tell us that all of these bound states can be described adequately within a relativistic version of the hartree - fock ( hf ) approach . turning to the baryon - baryon scattering problem , the tool of choice is the time - dependent version of hartree - fock ( tdhf ) , as originally suggested by witten @xcite . the basic equations in that case are easy to state , @xmath5 but hard to solve , even in 1 + 1 dimensions . one of the reasons is the fact that the sum over occupied states includes the dirac sea , so that one is dealing with an infinite set of coupled , non - linear partial differential equations . no systematic , analytical method for solving such a complicated problem is known . nevertheless , the exact solution for the time - dependent scattering problem of two baryons has recently been found in closed analytical form by means of a joint ansatz for @xmath6 and @xmath7 @xcite . it provides us with a microscopic solution of the scattering of two composite , relativistic objects , exact in the large @xmath0 limit . the necessary details will be briefly summarized below . this result encourages us to go on and try to solve more ambitious scattering problems involving any number of bound states , including nuclei " in addition to the nucleons " considered so far . the paper is organized as follows . in sec . [ sect2 ] we briefly summarize what is known about multi - fermion bound states and their interactions in the gn model . we also remind the reader how the baryon - baryon scattering problem has been solved recently , since we shall use the same strategy in the present work . [ sect3 ] is devoted to the dirac equation and the ansatz for scalar potential and continuum spinors . [ sect4 ] and [ sect5 ] contain the central results of this work , namely the coefficients entering the ansatz , presented in the form of an algorithm . in sec . [ sect6 ] , we explain the extent to which the general result has been checked so far . [ sect7 ] deals with the bound state spinors which are then used in sec . [ sect8 ] to discuss the issue of self - consistency and the fermion density . [ sect9 ] addresses scattering observables like time delays or deformations of bound states . sec . [ sect10 ] contains a few illustrative examples , followed by a short summary and outlook in sec . [ sect11 ] . to put this study into perspective , we summarize what is known about multi - fermion bound states and their mutual interactions in the massless gn model , eq . ( [ 1.1 ] ) . static multi - fermion bound states have been derived systematically with the help of inverse scattering theory and resolvent methods @xcite . the best known examples are the callan - coleman - gross - zee kink ( cited in @xcite ) and the dashen - hasslacher - neveu ( dhn ) baryon @xcite , both of which can accommodate up to @xmath0 fermions . the kink is topologically non - trivial , reflecting the @xmath8 chiral symmetry of the massless gn model . its shape ( shown in fig . [ fig1 ] ) and mass are independent of its fermion content . the dhn baryon is topologically trivial and stabilized by the bound fermions which affect its shape and mass , as illustrated in fig . [ fig2 ] . multi - baryon bound states have been constructed systematically by feinberg @xcite . they possess continuous parameters related to the position of the baryon constituents on which the mass of the bound state does not depend ( moduli " ) . they may be topologically trivial like the dhn baryon or non - trivial like the kink , depending on the ( spatial ) asymptotic behavior of @xmath6 . some examples are shown in figs . [ fig3 ] and [ fig4 ] . a common feature of all static solutions is the fact that the scalar potential is transparent , i.e. , the fermion reflection coefficient vanishes for all energies . consequently the self - consistent , static solutions of the gn model coincide with the transparent scalar potentials of the dirac equation , investigated independently by nogami and coworkers @xcite . since the static dirac equation can be mapped onto a pair of ( supersymmetric ) schrdinger equations , this also yields a bridge between static , self - consistent dirac - hf solutions on the one hand and transparent potentials of the schrdinger equation on the other hand , a problem solved long ago by kay and moses @xcite . the non - relativistic limit of the topologically trivial , static gn solutions are well - known multi - soliton solutions of coupled non - linear schrdinger ( nls ) equations , arising in the hartree approximation to particles in 1d with attractive @xmath3-interactions @xcite . by boosting any static solution , one can trivially generate solutions of the tdhf equation @xcite . this kind of solution enters in the asymptotic states of the scattering problem which we are going to study . the breather is a time - dependent , oscillating solution of kink - antikink type . it was found by dhn , using the analogy with the sine - gordon breather @xcite . since it is neither a conventional bound state nor a scattering state , it has no analogue in real particle physics , but is reminiscent of collective , vibrational excitations of heavy nuclei or molecules . this underlines the classical character of the large @xmath0 limit . we shall not consider scattering of breathers in the present work . following a suggestion in ref . @xcite , kink - antikink scattering was solved in tdhf by analytic continuation of the breather @xcite . since the fermions do not react back , it is possible to map this problem rigorously onto the problem of kink - antikink scattering in sinh - gordon theory . if we set @xmath9 , then @xmath10 satisfies the classical sinh - gordon equation @xmath11 ( in natural units ) , as first noticed by neveu and papanicolaou @xcite . this mapping can be generalized . the known multi - soliton solutions of the sinh - gordon equation yield the self - consistent scalar potential for scattering of any number of kinks and antikinks @xcite . a poor man s simulation of nuclear interactions was the scattering of trains " of solitons moving with almost the same speed in ref . @xcite ( there are no multi - soliton bound states ) . kink dynamics has no non - relativistic analogue since the internal structure of kink is ultrarelativistic , as evidenced by a zero - energy bound state . time - dependent kink - antikink scattering is illustrated in fig . [ fig5 ] . a crucial ingredient in proving the correspondence between kink dynamics and sinh - gordon solitons is the fact that kink solutions satisfy the self - consistency mode - by - mode . they are of type i " in the classification of @xcite , i.e. , @xmath12 with constant @xmath13 for every single particle state @xmath14 . this is also the basis for an interesting geometrical interpretation of tdhf solutions , relating time - dependent solutions of the gn model to the embedding of surfaces of constant mean curvature into 3d spaces @xcite . scattering of dhn baryons is significantly more involved than kink - antikink scattering . presumably because the fermions react back , it does not seem possible to map this problem onto any known soliton equation . the exact tdhf solution for baryon - baryon scattering was found recently in a different way , namely by ansatz @xcite . a specific example is illustrated in fig . [ fig6 ] . since we shall follow the same strategy in the present paper , we briefly recall the main ideas behind the ansatz , referring the reader to ref . @xcite for technical details . the ansatz can best be described as follows . we start from the scalar mean field of a single ( boosted ) dhn baryon with label @xmath15 . it can be cast into the form of a rational function of an exponential @xmath16 , @xmath17 here , @xmath18 is a parameter governing the size of the baryon and related to its fermion number @xmath19 via @xmath20 @xmath21 denotes the baryon velocity , @xmath22 , and @xmath23 is an arbitrary real factor expressing the freedom of choosing the initial baryon position . the dirac components of the continuum spinor have the same rational form with different coefficients in the numerator only and an additional plane wave factor , @xmath24 the asymptotic behavior at fixed @xmath25 is @xmath26 for @xmath27 , showing that the potential is transparent . in order to solve the scattering problem for baryons @xmath15 and @xmath28 , we start by multiplying @xmath29 and @xmath30 and expand the numerator and denominator , @xmath31 this may be viewed as scalar potential for non - interacting baryons . the ansatz for interacting baryons proposed in @xcite now consists in assuming that the only effect of the interaction is to change the coefficients in the numerator and denominator of ( [ 2.5 ] ) , keeping the polynomial dependence on @xmath32 the same . likewise , the ansatz for the spinor is obtained by multiplying the rational factors of @xmath33 for baryons @xmath15 and @xmath28 and allowing for changes in the coefficients only . the overall exponential factor is kept unchanged , since it is expected that the potential is reflectionless also in the interacting case . it turns out that most of the coefficients in @xmath6 and @xmath33 are in fact determined by the asymptotic in- and out - states . only 4 coefficients remain to be determined , namely the factors in front of the monomials @xmath34 in the three numerators and the common denominator . inserting this ansatz into the dirac equation determines the missing coefficients and confirms that this simple idea yields the exact solution of the 2-baryon problem . so far , we have discussed only the fermion continuum states . bound states can be obtained by analytic continuation in a spectral parameter ( a function of @xmath35 ) and subsequent normalization . self - consistency can then be checked explicitly , confirming that the ansatz solves the tdhf problem . the solution is found to be of type iii , i.e. , the scalar density of any single particle orbit can be expressed as a linear combination of 3 distinct functions of ( @xmath36 ) . we have no a priori argument why the ansatz should be successful , but its simple form is most certainly a large-@xmath0 manifestation of the quantum integrability of finite-@xmath0 gn models . the result for the non - trivial coefficients is rather complicated , but by a proper choice of variables and light cone coordinates , one manages to keep all coefficients in rational form . unlike in the kink - antikink case , the non - relativistic limit is now accessible , since the dhn baryon goes over into the soliton of the nls equation in the limit of small fermion number . starting from the two - baryon solution , one then recovers the time - dependent solutions of the multi - component nls equation of nogami and warke for @xmath37 @xcite . this completes the overview of the present state of the art . here we propose to extend the two - baryon tdhf scattering solution of ref . @xcite to an arbitrary number of composite colliding particles , including multi - baryon bound states ( nuclei " ) in addition to baryons . the central idea is to use an ansatz for the scalar potential inspired by the product of @xmath0 single baryon potentials , assuming that only the coefficients of the resulting rational function of @xmath38 will be affected by the interactions . a convenient choice of the dirac matrices in 1 + 1 dimensions is @xmath39 together with light cone coordinates @xmath40 this simplifies the dirac - tdhf equation to @xmath41 here , @xmath42 is the upper , left - handed , @xmath43 the lower , right - handed spinor component . we posit the following ansatz for the scalar tdhf potential , @xmath44 as motivated in the preceding section , @xmath6 is assumed to be a rational function of @xmath0 exponentials @xmath16 , where @xmath0 is the number of baryons , @xmath45 each summation index @xmath46 runs over the values 0,1,2 , and the coefficients @xmath47 are real . the basic exponential @xmath16 has the form inferred from the single dhn baryon in flight , @xmath48 the parameter @xmath18 specifies the size ( or , equivalently , fermion number ) of the @xmath15-th baryon . @xmath49 is related to the baryon rapidity @xmath50 and velocity @xmath21 via @xmath51 for @xmath18 we shall use the parametrization @xmath52 to avoid the appearance of square roots . apart from the @xmath53 parameters @xmath54 , the baryon constituents are characterized by @xmath0 arbitrary , real scale factors @xmath23 needed to specify their initial positions . the @xmath16 must be ordered according to baryon velocities . we choose the convention that @xmath55 if @xmath56 . we now turn to the ansatz for the continuum spinors , assuming from the outset that the tdhf potential is reflectionless , @xmath57 here , @xmath58 denotes the light cone spectral parameter related to ordinary momentum and energy via @xmath59 @xmath60 are multivariate polynomials in the @xmath16 of the same degree as @xmath61 , @xmath62 but now with complex coefficients @xmath63 . in eq . ( [ 3.9 ] ) we have factored out the free dirac spinor @xmath64 to ensure that all polynomials start with a 1 " . the denominator @xmath65 in the spinor , eq . ( [ 3.9 ] ) , is assumed to be the same as in the scalar potential , eq . ( [ 3.4 ] ) . inserting this ansatz into the dirac equation ( [ 3.3 ] ) yields @xmath66 actually , we can eliminate the variable @xmath58 by rescaling @xmath67 via @xmath68 . this transforms @xmath16 into @xmath69 the final form of the dirac equation can then be obtained by setting @xmath70 in eq . ( [ 3.13 ] ) , @xmath71 the numerator and denominator functions ( @xmath72 ) are polynomials in the @xmath16 . since the @xmath16 are eigenfunctions of @xmath73 , the dirac equation ( [ 3.15 ] ) gets converted into the condition that 2 polynomials vanish identically . thus each coefficient of the monomials @xmath74 must vanish separately . the number of terms in each of the polynomials , eqs . ( [ 3.5 ] ) and ( [ 3.11 ] ) , is @xmath75 for @xmath0 baryons , as @xmath16 can appear with powers 0,1,2 . in the final dirac equation , @xmath16 appears with powers @xmath76 , so that eq . ( [ 3.15 ] ) is altogether equivalent to @xmath77 algebraic equations for the coefficients @xmath78 of our ansatz . in this and the following section , we present our results for the coefficients entering the scalar potential and the continuum spinors for @xmath0 baryons , i.e. , the coefficients of the polynomials @xmath72 introduced above . they fall naturally into 2 classes : reducible " coefficients which can be related to the @xmath79 baryon problem , and irreducible " ones which can not . the reducible coefficients are the subject of this section , the irreducible ones will be discussed in the next section . there are two distinct ways of reducing the @xmath0 baryon problem to the @xmath79 baryon problem , either by letting @xmath80 or by letting @xmath81 . in both cases , @xmath82 drops out of the expressions for @xmath6 and @xmath83 . since this can be done for any label @xmath84 , one gets a large number of recursion relations . as explained in greater detail in ref . @xcite , one has to take into account time delays and ( in the case of the spinors ) transmission amplitudes for final states , depending on whether the eliminated baryon @xmath84 has been scattered from the remaining @xmath79 baryons or not . let us consider the scalar potential first . starting point are the following basic relations , @xmath85 @xmath82 is missing on the right hand side , which therefore refers to @xmath79 baryons . the @xmath86 are ( real ) time delay factors satisfying @xcite @xmath87 it is important to keep track of the ordering of the baryon labels ( @xmath55 if @xmath56 ) when applying these formulas . relations ( [ 4.1 ] ) imply the following recursion relations for the coefficients in ( [ 3.5 ] ) , @xmath88 we use the convention that barred indices have to be omitted . the factors @xmath89 appear here because relations ( [ 4.1 ] ) determine only the ratio @xmath90 . they can be fixed as follows . we normalize the lowest and highest coefficients of @xmath61 to 1 for any number of baryons , @xmath91 this is always possible since we must recover the vacuum potential @xmath92 in the limit where all @xmath16 go to 0 or @xmath93 , and the @xmath16 contain arbitrary scale factors @xmath23 , see eq . ( [ 3.6 ] ) . specializing relations ( [ 4.3 ] ) to the cases where all indices are 0 or all indices are 2 and using eq . ( [ 4.2 ] ) , we then find @xmath94 this yields the following final recursion relations for the coefficients entering @xmath6 , @xmath95 they determine all @xmath0-baryon coefficients containing at least one 0 or one 2 in their subscripts in terms of @xmath96-baryon coefficients , leaving only the two irreducible coefficients @xmath97 in front of @xmath98 undetermined . for the spinors , we have to take into account transmission amplitudes in addition to the time delay factors . consequently the general reduction formulas ( [ 4.1 ] ) have to be replaced by @xmath99 where @xmath100 is the transmission amplitude of baryon @xmath84 @xcite @xmath101 it is unitary ( @xmath102 ) due to the reflectionless potential . using a normalization analogous to ( [ 4.4 ] ) , i.e. , @xmath103 we arrive at the recursion relations @xmath104 for the coefficients in @xmath105 . once again this leaves only the two irreducible coefficients @xmath106 of @xmath98 undetermined . altogether , there are @xmath107 coefficients in the ansatz for @xmath6 and @xmath83 for @xmath0 baryons . all but the 4 irreducible ones are determined by normalization and recursion relations . the first step towards solving the @xmath0 baryon problem is to eliminate all reducible coefficients , expressing the 4 polynomials in terms of irreducible coefficients , time delay factors and transmission amplitudes only . the above recursion scheme enables us to do just this . the result can most conveniently be cast into the form of an algorithm . we first formulate the algorithm and subsequently illustrate it with the explicit results for @xmath108 and point out its advantages . the algorithm will be stated separately for the 4 polynomials @xmath109 . 1 . denominator @xmath65 of @xmath6 1 . write down the product @xmath110 and expand it . 2 . if a term contains between 2 and @xmath0 factors @xmath111 , replace it by @xmath112 3 . substitute @xmath113 and expand again . 4 . if any term contains @xmath114 ( @xmath115 ) , replace it by @xmath116 5 . set @xmath117 2 . numerator @xmath118 of @xmath6 0.5 cm the numerator @xmath118 of @xmath6 can be obtained from the denominator @xmath65 of @xmath6 by replacing all @xmath119-coefficients by @xmath120-coefficients , @xmath121 3 . numerator @xmath122 of @xmath42 0.5 cm to get @xmath122 , start from @xmath65 and perform the following steps : 1 . replace @xmath123 where @xmath124 is the transmission amplitude of baryon @xmath15 . 2 . replace all @xmath119-coefficients by @xmath125-coefficients , @xmath126 4 . numerator @xmath127 of @xmath43 0.5 cm to get @xmath127 , start from @xmath122 and replace all @xmath125-coefficients by @xmath128-coefficients , @xmath129 to avoid misunderstandings , we illustrate the outcome of the algorithm with a few explicit examples . for @xmath37 ( 9 terms ) , one finds @xmath130 these results are fully consistent with ref . @xcite . for @xmath131 ( 27 terms ) the algorithm yields @xmath132 @xmath133 inspection of these examples shows the following advantages of presenting results in the form of an algorithm . first , the recursion relations relate @xmath0-baryon coefficients to @xmath96-baryon coefficients , cf . ( [ 4.6],[4.9 ] ) . the algorithm gives directly the iterated result where everything is expressed in terms of irreducible coefficients for 1,2, ... ,@xmath0 baryons . secondly , the number of terms in the explicit expressions increases like @xmath75 , so that writing down the explicit expressions like in ( [ 4.19],[4.20 ] ) becomes quickly prohibitive . the algorithm on the other hand has been stated concisely for arbitrary @xmath0 . it can also easily be implemented in maple , so that it is never necessary to deal manually with lengthy expressions . as a result of this section , we have reduced @xmath6 and @xmath83 to those coefficients @xmath78 whose subscripts contain only 1 s and which refer to 1,2, ... ,@xmath0 baryons with all permutations of labels . these irreducible coefficients have to be determined algebraically from the dirac equation ( [ 3.15 ] ) and are the subject of the following section . we denote those @xmath0-baryon coefficients of the polynomials @xmath72 which can not be determined recursively from the @xmath79 baryon problem as irreducible . as explained above , there are only 4 such coefficients for given @xmath0 , namely the coefficients of the monomials @xmath134 in each of the 4 polynomials , @xmath135 . they encode the dynamical information about the situation where all @xmath0 baryons overlap and have to be determined by means of the dirac equation . for reasons to be discussed later in more detail , this is a difficult task for computer algebra programs like maple , once the baryon number gets too large . we have therefore determined the irreducible coefficients for low baryon numbers analytically , analyzed their structure and extrapolated the formulas to arbitrary @xmath0 . in this section we present our conjectured results for the 4 irreducible coefficients and general @xmath0 . in the next section , we will describe in detail the extent to which these conjectured results have actually been checked so far . given the complexity of the coefficients , it is once again easier for us to communicate our results in the form of an algorithm , rather than a closed expression . the algorithm is actually a very simple one . let us define a combinatorial expression @xmath136 through the following two steps : 1 . write down the product @xmath137 where @xmath138 is a @xmath139 matrix , and expand it . 2 . for each of the @xmath140 terms in the sum and each index @xmath141 , denote by @xmath19 the number of indices @xmath15 appearing in this term ( @xmath142 ) . then , if @xmath143 is odd , multiply the term by @xmath144 by way of example , we write down the explicit result for @xmath37 ( 2 terms ) , @xmath145 and @xmath131 ( 8 terms ) , @xmath146 after this preparation , the irreducible coefficients can be expressed in compact form as follows , @xmath147 with @xmath148 all what remains to be done is to define exactly the various symbols appearing in ( [ 5.5],[5.6 ] ) . we divide them into two categories . the first category comprises those symbols which can be deduced from the single dhn baryon problem @xcite , @xmath149}{(z_i^2 + 1 ) ( \zeta_i - z_i)(\zeta_i z_i + 1 ) } , \nonumber \\ d_1^i & = & \frac{2 [ 2 z_i^2 - \zeta_i^2(z_i^4 + 1)]}{(z_i^2 + 1)(\zeta_i - z_i)(\zeta_i z_i + 1)}. \label{5.7}\end{aligned}\ ] ] they enter in the prefactor of the combinatorial expression @xmath136 in eq . ( [ 5.5 ] ) and are the same as in eqs . ( [ 2.2],[2.3 ] ) , up to trivial normalization factors in @xmath150 and @xmath151 . the 2nd category consists of symbols which can be deduced from the two - baryon problem if one applies these formulas to @xmath37 and compares them with the results of ref . @xcite , @xmath152 we have used everywhere the spectral parameter @xmath153 boosted into the rest frame of baryon @xmath15 , introduced in eq . ( [ 3.14 ] ) . note however that @xmath153 could be replaced by @xmath49 in @xmath154 and @xmath138 , so that the @xmath58-dependence of these quantities is spurious . by using the variable @xmath155 rather than @xmath18 and @xmath153 rather than @xmath21 and @xmath84 , we have achieved that all the basic expressions are rational functions of the 2@xmath0 arguments ( @xmath156 ) . the same holds true for @xmath86 , eq . ( [ 4.2 ] ) , and @xmath100 , eq . ( [ 4.7a ] ) . a noteworthy property of this construction is the fact that the algorithm leading to @xmath136 is based on a factorization in terms of quantities @xmath138 referring to 2 baryons @xmath157 only , see eq . ( [ 5.1 ] ) . this implies that the solution of the two - baryon scattering problem is sufficient to determine completely @xmath0 baryon scattering . this observation is behind the phrase evidence for factorized scattering " in the title of this paper . it goes beyond the usual factorization of the fermion scattering matrix , which holds trivially in our case ( see sec . [ sect9 ] ) . it teaches us that even when all @xmath0 baryons overlap , there is nothing new going on as compared to having two overlapping baryons only . in this sense , factorization does not only hold for the on - shell scattering matrix , but also off - shell . in the preceding sections , we have provided rules for explicitly constructing the scalar potential @xmath6 and the continuum spinors @xmath83 for the @xmath0-baryon tdhf problem in the gn model . let us summarize where we stand . the main ingredients in @xmath6 and @xmath83 are 4 polynomials in @xmath0 exponentials @xmath16 , consisting of @xmath75 terms each . the coefficients in these polynomials can all be expressed through a set of irreducible coefficients multiplying @xmath158 in the @xmath159 baryon problem , time delay factors @xmath86 and fermion transmission amplitudes @xmath124 , using the algorithm of sec . [ sect4 ] . the irreducible coefficients in turn can be constructed starting from 1- and 2-baryon input only , using the algorithm of sec . [ sect5 ] . since the dirac equation reduces to a set of algebraic equations and all ingredients are known rational functions , one would not expect any particular difficulties in checking that the spinor satisfies the dirac equation , using computer algebra programs like maple . however , the complexity of the resulting expressions increases rapidly with increasing baryon number , quickly exceeding the capabilities of maple due to storage and computation time problems . thus , for @xmath37 and @xmath131 , we could still check all @xmath160 algebraic equations analytically with maple in a straightforward way . for @xmath161 or larger , the maximum size of expressions which maple can handle is exceeded and we have only been able to check our results numerically , for random values of the input parameters @xmath156 . this test has been carried out successfully for @xmath162 . by increasing the number of digits , one can find out whether the floating point result is exact or approximate . since the number of operations increases faster than exponentially with @xmath0 , it is actually necessary to run maple with very high accuracy for large @xmath0 values . thus for example , during a full @xmath163 calculation , 40 digits get lost , so that one has to start out with 50 digits precision to be sure that the dirac equation is solved exactly . clearly , there must be a way of proving our results in full generality . the complexity of the solution and the intricate way in which @xmath0 baryon scattering is related to the scattering problem of fewer baryons have prevented us so far from finding such a proof . therefore , strictly speaking , our result still has the status of a conjecture . in the meantime , we shall restrict all applications shown below to problems with low values of @xmath0 for which we have established the validity beyond any doubt . we are confident that the results hold for arbitrary @xmath0 , but this has to await a complete mathematical proof . up to this point , we have only dealt with the dirac equation for continuum spinors . this still leaves open other aspects of the full tdhf problem like bound states , self - consistency , and fermion density . in some sense , all we have achieved so far is to find time - dependent , transparent potentials for the dirac equation , which look asymptotically like boosted static potentials . this solves in part another open problem which has been raised in the literature @xcite , namely to classify all time - dependent , transparent potentials of the 1 + 1 dimensional dirac equation . how general is our result in this respect ? all static transparent potentials are well known ( see the discussion in sec . [ sect2 ] ) . we can now construct all time - dependent transparent potentials which asymptotically consist of an arbitrary number of such static solutions , boosted to arbitrary velocities . this can not be the complete set of all transparent potentials though , as evidenced by the example of the breather which does not fit into this scheme . evidently , there must be another set of solutions where boosted breathers appear as asymptotic states , in addition to boosted static bound states . we do not know yet whether our ansatz will be capable of describing this more general class of solutions . all we have checked is that the single breather can indeed be reproduced with our ansatz , provided we allow for complex valued @xmath16 s . scattering problems involving breathers are interesting in their own right , but will be left for future studies . in the @xmath0 baryon problem , one expects @xmath0 positive and @xmath0 negative energy bound states . as discussed in ref . @xcite , the bound state spinors can be obtained from the continuum spinors by analytic continuation in the spectral parameter @xmath58 . to this end we first re - introduce the @xmath58 dependence of the coefficients ( [ 5.5][5.8 ] ) by using @xmath164 . only the coefficients @xmath165 are @xmath58-dependent . for positive energy bound states for example , @xmath166 develop a single pole at @xmath167 . the bound state spinor associated with baryon @xmath15 can then be obtained from the residue of @xmath83 at the pole , @xmath168 the result is a normalizable solution of the dirac equation . the normalization factor @xmath169 can readily be determined for times @xmath25 when the @xmath15-th baryon is isolated , with the result @xmath170 for this value of @xmath169 , the bound state spinor ( [ 7.1 ] ) is normalized according to @xmath171 this method has been checked analytically for @xmath37 in ref . @xcite and numerically for @xmath131 by us . the situation in the @xmath0-baryon problem is the same as in the 2-baryon problem @xcite . the scalar density for a continuum state can be decomposed as @xmath172 where @xmath173 is the perturbative piece which gives self - consistency by itself . the 2nd part is cancelled against the discrete state contribution , @xmath174 if one makes use of the self - consistency conditions in the asymptotic in- and out - states . we can deduce @xmath175 by subtracting the expression ( [ 8.2 ] ) from the full scalar density and can then check eq . ( [ 8.3 ] ) numerically , since we know the discrete state spinors and the integral is convergent . this test has been performed analytically for @xmath37 in ref . @xcite and numerically for @xmath131 in the present work . likewise , the fermion density can be dealt with in the same manner as for 1 or 2 baryons . the basic identity is @xmath176 relating the continuum and bound state densities @xcite . the integral is convergent owing to the vacuum subtraction . we have checked this identity here numerically for @xmath131 . from this and the self - consistency relation , one can again express the total , subtracted fermion density through the bound state densities as @xmath177 generalizing the @xmath37 results @xcite . the fermion transmission amplitude for the @xmath0-baryon problem factorizes , since it can be evaluated when all baryons are far apart , @xmath178 with @xmath100 from eq . ( [ 4.7a ] ) . this fact has actually already been used in the normalization conditions ( [ 4.8 ] ) . the more interesting question is how to characterize the outcome of the scattering process in terms of the baryon or multi - baryon bound states . comparing the asymptotics for @xmath179 , we find that the exponential @xmath16 acquires the following factor during an arbitrary @xmath0-baryon collision , @xmath180 the @xmath86 have been given in eq . ( [ 4.2 ] ) . if @xmath181 for one or several @xmath28 s , there is no shift factor because baryons @xmath15 and @xmath28 belong to the same compound state ( nucleus " ) and do not scatter from each other . how does this translate into observables ? the scattering process at the level of the tdhf potential is classical , so that the situation is analogous to classical soliton scattering . if a single baryon is involved in the scattering process , the situation is very simple . the incoming and outgoing baryons can be associated with straight - line space - time trajectories defined by @xmath182 they have the same slope in the ( @xmath36 ) diagram , since the velocity does not change . the factor @xmath183 given in eq . ( [ 9.2 ] ) then leads to a parallel shift of the outgoing space - time trajectory , which is usually interpreted as time delay ( or advance ) . if an @xmath159-baryon bound state ( nucleus " ) is scattered , the initial state contains @xmath159 baryon constituents @xmath184 moving with the same velocity @xmath185 on parallel straight - line trajectories . such a bound state depends on the scale factors @xmath23 of @xmath16 ( moduli " ) , cf . ( [ 3.6 ] ) , determining the relative positions and the shape of the bound state without affecting its energy . in the final state , the @xmath159 trajectories will be displaced laterally relative to the incoming trajectories . since all @xmath186 parameters within one composite state must be chosen differently , according to ( [ 9.2 ] ) , the displacement will be different for each trajectory . therefore the net result can not be interpreted anymore as a mere time delay , but is always accompanied by a change in moduli space , resulting in different relative baryon positions and a corresponding deformation of the scalar potential . in this sense , the scattering process is not really elastic and the composite bound states undergo a change in their internal structure . a time delay of the full composite object could be defined , but this is neither unambiguous , nor necessary . the full asymptotic information about the scattering process is contained in eq . ( [ 9.2 ] ) . since we have verified the above formulas analytically or numerically with high precision for up to 8 baryons , we now present some illustrative results for smaller values of @xmath0 . depending on the choice of velocity parameters , the same formalism can describe a variety of physical problems . for @xmath37 , there are two distinct possibilities . if the velocities are chosen to be equal , we obtain a boosted 2-baryon bound state , provided that the @xmath186 parameters are different . if the velocities are different , there is no restriction on the @xmath186 parameters and we describe scattering of baryon ( @xmath187 ) on baryon ( @xmath188 ) . in both cases , this yields nothing new as compared to refs . @xcite , but has been used to test our formulas . for @xmath131 , we have to distinguish 3 cases . if @xmath189 and all @xmath18 s are different , we are dealing with a boosted 3-baryon bound state . if two velocities are equal and the corresponding @xmath186-parameters are different , the formalism describes scattering of a baryon on a 2-baryon bound state , analogous to @xmath190-scattering in nature . an example of this process is shown in fig . [ fig7 ] , where the time evolution of the scalar tdhf potential during the collision is displayed . as announced above , the internal structure of the bound state necessarily changes during such a collision . to emphasize this point , we compare in fig . [ fig8 ] the first and last time slice of fig . [ fig7 ] , i.e. , the incoming and outgoing states . 1.0 cm if all 3 velocities are different , the formalism describes a 3-baryon scattering process with 3 baryons in the initial and final state . since scattering processes with more than 2 incident particles are somewhat academic from the particle physics point of view , we do not show any example . with increasing @xmath0 , the number of scattering channels increases . the next number of baryons is @xmath161 , describing one boosted 4-baryon bound state , scattering of a baryon on a 3-baryon bound state , scattering of two 2-baryon bound states , scattering of 3 particles ( 2 baryons and a 2-baryon bound state ) or of 4 particles ( 4 individual baryons ) . the most interesting and new process out of these is the scattering of 2 bound states , the analogue of @xmath191-scattering the simplest case of nucleus - nucleus scattering . this is illustrated in fig . the change in structure of the bound state is exhibited more clearly in fig . [ fig10 ] . 1.0 cm finally , we give an example with 5 baryons . out of the many possibilities , we have chosen scattering of a single baryon on a 4-baryon bound state , the analogue of @xmath192-scattering in the real world , see fig . [ fig11 ] . we refrain from showing any results with larger number of baryons , since we have not yet checked our formulas thoroughly beyond @xmath193 . however , we have no doubt that we could describe correctly scattering processes with any number of baryons . all of these examples involve topologically trivial bound states only . there is no difficulty in applying the same formulas to topologically non - trivial scatterers as well . as already demonstrated in ref . @xcite , all one has to do is let one or several @xmath186 s go to 1 . then , the corresponding baryon becomes a kink - antikink pair at infinite separation . this diverging separation has to be compensated by a change of the scale parameter @xmath23 in the @xmath16 factor , so that half of the baryon disappears at infinity . in this way one can describe scattering of any number of topologically trivial or non - trivial bound states , without need to derive separate formulas for this purpose . this paper has dealt with the large @xmath0 limit of the gn model , the quantum field theory of massless , self - interacting , flavored fermions in 1 + 1 dimensions . the fascinating aspect of lagrangian ( [ 1.1 ] ) is the fact that a single contact interaction term is able to generate a host of non - trivial phenomena . even more surprisingly , it seems that all of these can be worked out in closed analytical form , a rather exceptional situation in quantum field theory . the story begins with asymptotic freedom , the generation of a dynamical fermion mass , accompanied by spontaneous breakdown of the @xmath8 chiral symmetry , and a scalar fermion - antifermion bound state , in the original work @xcite . soon afterwards baryons were discovered @xcite , subsequently complemented by a whole zoo of multi - baryon bound states @xcite . as time evolved and computer algebra software became more powerful , ambitions were raised , leading to results like soliton crystals in the ground state and phase diagram of dense matter @xcite or time - dependent scattering processes of kinks and antikinks @xcite . the most recent result is the tdhf solution of time - dependent baryon - baryon scattering @xcite . in the present work , we have tried to add another chapter to this progress report . by generalizing the joint ansatz for the tdhf potential and the spinors recently proposed in ref . @xcite , we have most probably found the solution to a whole class of scattering problems , namely all those where the incoming and outgoing scatterers are boosted , static multi - fermion bound states of the gn model . the word probably " has to be used here because we have not yet been able to prove our results in full generality . the solution which we have presented is based on the analytical solution of the 2- and 3-baryon problems , followed by a tentative extrapolation to arbitrary @xmath0 . these results have then been checked numerically for @xmath162 , and all heralds well for their general validity . this method could only work because of a kind of factorization property which we have observed scattering of any number of baryons can apparently be predicted on the basis of 1- and 2-baryon input only . this holds not only for the asymptotic scattering data , but also during the entire time evolution , where more than 2 baryons can overlap at a time . we interpret these findings as a large-@xmath0 manifestation of the quantum integrability of the gn model . the solution which we have presented is relevant for yet another problem , namely how to find transparent , time - dependent scalar potentials for the dirac equation in 1 + 1 dimensions . it is clear that unlike in the static case , we have not yet arrived at the most general time - dependent solution . at least one time - dependent solution of the gn model is already known which does not belong to our class of solutions , the breather . it also yields a reflectionless potential . this suggests that a whole class of solutions is still missing , namely the tdhf potentials of scattering processes involving breathers in the initial and final states . we know already that the single breather can be obtained with our ansatz if one admits complex valued exponentials @xmath16 . it will be interesting to see whether breather - baryon or breather - breather scattering can be solved along similar lines . one other question which we have not been able to answer yet is whether our new solution is related to the solution of some known , classical non - linear equation or system of equations . this question is a natural one , given prior experience . thus for instance , all static baryons can be related to soliton solutions of the static nls equation . higher bound states are related to the static multi - channel nls equation . all dynamical kink solutions can be mapped onto multi - soliton solutions of the sinh - gordon equation . the non - relativistic limit of baryon - baryon scattering was shown to be equivalent to solutions of the time - dependent , multi - component nls equation . the advantage of such mappings is obvious . a lot of expertise and powerful techniques have been accumulated in the field of non - linear systems over the years , which can be helpful for finding new solutions of the gn model or proving certain results in full generality . a natural candidate for the present case would be the multi - component non - linear dirac equation , i.e. , the set of classical equations @xmath194 inspection of the various condensates in sec . [ sect8 ] shows that it is indeed possible to construct solutions of eq . ( [ 11.1 ] ) using our results . one needs @xmath195 components for @xmath0 baryons , since the solution is of type @xmath195 . however , it is not possible to restrict oneself to normalizable states as in the non - relativistic limit of the multi - component nls equation . one would have to invoke @xmath0 different bound states and one continuum state . hence , even if our results are related to the classical system ( [ 11.1 ] ) , it seems very unlikely that the solution presented here has already been given in the literature . keeping a continuum state as one of the components would be very hard to interpret classically . this is obviously a remnant of the dirac sea , without analogue in the classical fermion system . we thank gerald dunne and oliver schnetz for stimulating discussions . this work has been supported in part by the dfg under grant th 842/1 - 1 . 99 d. j. gross and a. neveu , phys . d * 10 * , 3235 ( 1974 ) . v. schn and m. thies , _ at the frontier of particle physics : handbook of qcd , boris ioffe festschrift _ , vol . m. shifman ( singapore : world scientific ) , ch . 33 , p. 1945 j. feinberg , ann . ( n.y . ) * 309 * , 166 ( 2004 ) . m. thies , j. phys . a : math . gen . * 39 * , 12707 ( 2006 ) . r. f. dashen , b. hasslacher and a. neveu , phys . d * 12 * , 2443 ( 1975 ) . e. witten , nucl . b * 160 * , 57 ( 1979 ) . g. v. dunne , c. fitzner , m. thies , phys . rev . d * 84 * , 105014 ( 2011 ) . d. j. gross , in les houches 1975 , proceedings , _ methods in field theory _ , amsterdam ( 1976 ) , p. 141 - 250 . y. nogami and f. m. toyama , phys . a * 45 * , 5258 ( 1992 ) . f. m. toyama , y. nogami , z. zhao , phys . a * 47 * , 897 ( 1993 ) . i. kay and h. e. moses , j. appl . * 27 * , 1503 ( 1956 ) . y. nogami and c. s. warke , phys . a * 59 * , 251 ( 1976 ) . w. brendel and m. thies , phys . d * 81 * , 085002 ( 2010 ) . a. klotzek and m. thies , j. phys . a * 43 * , 375401 ( 2010 ) . a. neveu and n. papanicolaou , commun . phys . * 58 * , 31 ( 1978 ) . c. fitzner and m. thies , phys . d * 83 * , 085001 ( 2011 ) . g. basar and g. v. dunne , jhep * 1101 * , 127 ( 2011 ) .

scattering of two baryons in the large-@xmath0 gross - neveu model via the time - dependent dirac - hartree - fock approach has recently been solved in closed analytical form . here , we generalize this result to scattering processes involving any number and complexity of the scatterers . the result is extrapolated from the solution of few baryon problems , found via a joint ansatz for the scalar mean field and the dirac spinors , and presented in analytical form . it has been verified numerically for up to 8-baryon problems so far , but a full mathematical proof is still missing . examples shown include the analogue of proton - nucleus and nucleus - nucleus scattering in this toy model . all the parameters of the general result can be fixed by one- and two - baryon input only . we take this finding as evidence for factorized scattering , but on the level of composite multi - fermion states rather than elementary fermions .

humankind has accrued _ a priori _ knowledge since the onset of _ homo sapiens_. from ancient cave paintings to modern research papers , the species desire toward sedimentation has been displayed as a documentary . an encyclopedia , a set of documents that contains a vast collection of information from the entire field of human knowledge , has played a pivotal role in disseminating these legacies @xcite . conventionally , a group of experts devote their expertise to these encyclopedias @xcite . taking advantage of technological developments , media that publish encyclopedias keep abreast of the times : handwriting , letterpress printing , and optical disks . the emergence of information technology has opened a new era of publishing traditional encyclopedias on the world wide web @xcite , which offers a variety of references and up - to - date information . although these new media can reduce the publication price , encyclopedia editing is still costly . besides the improvement of traditional encyclopedias , new media enable fresh challengers to participate in the competition . wikipedia @xcite , a representative player among the challengers , has proposed an entirely new manner : editing by volunteers with various backgrounds in a collective fashion . this new paradigm of sharing knowledge is one of the most famous examples of `` collective intelligence . '' however , due to the nature of open - edit policy , wikipedia does not guarantee that the contents are valid @xcite , thus it is regarded ambiguous and even inaccurate to utilize in scientific context @xcite . despite such a long - standing bias against the credibility of wikipedia , many studies suggest that wikipedia is more reliable than our prejudice ; wikipedia itself tends to refer reliable scientific sources @xcite . only 13% of wikipedia articles contain perceptible academic errors @xcite and the quantity of factual errors , omissions , or ambiguous statements in scientific context of wikipedia is comparable to traditional encyclopedias @xcite . gradually , prejudice against the quality of wikipedia s articles has been eroded and the number of citations to wikipedia in peer - reviewed scientific articles has increased over time @xcite . a bizarre gap between such prejudice and the actual situation appeals to the scholars , who have analyzed wikipedia s external characters and internal dynamics . for example , researchers have investigated editors of wikipedia and their editing patterns @xcite , and the occurrence and resolving of conflicts in wikipedia @xcite . despite the significant contributions of such endeavors , the previous studies mainly focus on the raw number of edits , and often neglect real time and the different editing patterns for articles with different sizes and ages . in this paper , we examine an exhaustive set of english wikipedia articles to understand how the article size and age displays external appearance in this open - editing encyclopedia . in particular , a simple time - rescaling method reveals articles belonging to various types , when we take account of the interrelation between observable parameters : the number of edits , the number of editors , and the article size . our analysis consists of both data analysis and modeling based on it . first , we use the entire edit history in wikipedia to inspect wikipedia s growth , mainly focusing on the number of edits , the number of editors , and the article size . in this process , we demonstrate that the consideration of real time is essential to understand the underlying dynamics behind the present wikipedia . second , to consider the formation of current wikipedia in more detail , we develop an agent - based model that imitates the interplay between an article and the editors in a society . our model shows inherent differences of articles belonging to different types of growth patterns . the results are consistent with real data , which suggests that a society s attitudes on wikipedia articles determine the growth pattern . we believe that this approach provides valuable insights for the formation of collective knowledge . we focus on the long - term formation of collective knowledge , which has significant effects on the progress of humankind over a variety of temporal scales . we hope that our work provides insights to solve some of the fundamental questions : why people collaborate , how the collective memory is formed , and how knowledge is spread and descended . the rest of the paper is organized as follows . in sec . [ sec : data_set ] , we introduce the wikipedia data that we use in our investigation . in sec . [ sec : data_analysis ] , we propose a time - rescaling method and show that the articles in wikipedia can be classified into multiple types based on their growth pattern . we present our model and results in sec . [ sec : model ] , including verification of our model with real - data . finally , we present our conclusions and discussion in sec . [ sec : conclusion ] . ) of editors are logged - in and 83.9% ( @xmath1 ) remain anonymous . , scaledwidth=50.0% ] for the analysis , we use the december 2014 dump of english wikipedia . this dump contains the complete copy of wikipedia articles from the very beginning up to december 8 , 2014 , including the raw text source and metadata source in the extensible markup language ( xml ) format @xcite . in this data set , there are a total of @xmath0 articles across all categories with the full history of edits . each article documents either the wikipedia account identification ( i d ) or internet protocol ( ip ) address of the editor for each edit , the article size and timestamp for each edit , etc . a single character in english takes 1 byte , so the article size is the direct measure of article length @xcite . there are @xmath2 editing events ( `` edits '' from now on ) for all wikipedia articles in total , where individual articles edit numbers range from @xmath3 to @xmath4 . previous studies tend to sample data sets for various reasons , and thus articles with small numbers of edits are necessarily filtered out @xcite . however , fig . [ numeditspdf ] suggests a fat - tailed distribution for the number of edits , so the majority of articles are not edited as many times as the articles in the tail part of the distribution and those articles should not be neglected . therefore , we consider all entries and use the entire set for analysis . additionally , we use the i d and ip address , for logged - in editors and unlogged - in editors , respectively , to identify distinct editors . in total , @xmath5 editors have participated in the establishment of the current wikipedia . among them , 83.9% ( @xmath1 ) of editors are unlogged - in and only 16.1% ( @xmath6 ) of editors are logged - in . interestingly , the absolute share of logged - in editors is rather smaller than that of unlogged - in editors ; most of the heavy editors tend to be logged - in ( fig . [ numeditslogged ] ) . specifically , logged - in editors have modified the articles 455397682 times ( 77.5% ) in total , meanwhile unlogged - in editors have modified the articles only 132475517 times ( 22.5% ) . considering the fact that the number of unlogged - in editors exceeds that of logged - in editors , the average influence of a single unlogged - in editor is much smaller than that of logged - in editors ( on average , 69.8 times per logged - in editor and 3.9 times per unlogged - in editor ) . there are possible biases for ip addresses when an ip address is shared , e.g. , editors who use a library , public wifi , virtual private network ( vpn ) , etc . , or move the locations . in those cases , there will be under- or overestimation of the number of distinct editors . additionally , several home internet connection methods allocate ip addresses dynamically , e.g. , digital subscriber line ( dsl ) and dial - up . for those cases , there might be overestimation and misidentification of distinct editors . however , it is reported that cable and fiber to the home ( ftth ) dominate the u.s . market share @xcite , which provide quasistatic ip addresses @xcite . considering both the market shares and modest impact of single unlogged - in editors on the current wikipedia , we believe that our analysis is robust . in fact , we actually check that even when we exclude unlogged - in editors , our results reported in sec . [ sec : data_analysis ] are not affected at all indeed . in addition , a small number of edits does not specify the editor , yet we use other information even for such cases based on the article size and timestamp . -th year '' corresponds to the edit event occurring between the first and the last day of the @xmath7-th year since the onset of the article . the time differences follow fat - tailed distribution , which is a sign of the burstiness , with a daily periodic pattern ( @xmath3 day = @xmath8 s).,scaledwidth=50.0% ] previous studies on the wikipedia data set did not use the information about article size changes after the edits @xcite or real timestamps of the edits @xcite . we combine such information together with conventional measures , such as the number of edits and the number of editors , to display the nature of wikipedia . our first analysis of time and size differences between two consecutive edits reveals regularity , regardless of an article age and size . the time between the consecutive edits follows a fat - tailed distribution with characteristic periodicity from the human circadian rhythm ( fig . [ deltatimeperage ] ) , which suggests that the editing timescale of wikipedia is intermittent or `` bursty , '' meaning that brief but intense activities are followed by much smaller activities for a long time @xcite . these intense activities in wikipedia are reported as `` wikipedia edit war , '' which refers to significantly rapid consecutive editing by various editors with conflicting opinions @xcite . our observation indicates that the `` edit number '' ( or the number of edits ) , which many studies use as the proxy of the real time @xcite , is not an unbiased proxy of the time . counterposed to the assumption that english wikipedia has already become global media , we observe strong periodicity for the time between the consecutive edits in fig . [ deltatimeperage ] . the peaks are located at every @xmath8 s or a single day , which implies that native english speakers ( mostly people in the united states because of the relative population , we presume ) still dominate english wikipedia even though there is no barrier to global access . such a circadian pattern in the frequency of editing events is mainly driven by editors with specific cultural backgrounds for the data until the beginning of 2010s , as reported in ref . our observation indeed shows that the circadian rhythm also affects the interediting time in a collective fashion , and this domination still remains in the current wikipedia . besides the time scale , we observe that an article s growth is mainly addition and subtraction with a characteristic size scale , which are rather independent of the current size ( fig . [ deltasizepersize ] ) . this observation is counterposed to the recent report that the growth of collaborated open - source software and mammalian body masses are proportional to their size @xcite , and implies that the influence of a single edit becomes smaller as article size is increased . most previous research @xcite does not take into account the degree of the influence for a single edit , and thus considers all of the edits as affecting the article of wikipedia equally . however , our observations propose the necessity of combining the time and size difference between the edits with the conventional measures . in sec . [ sec : edit_scale_of_wikipedia ] , we have shown that the time between two consecutive edits is quite heterogeneous ( fig . [ deltatimeperage ] ) . this global effect of various timescales itself makes it unfair to directly compare the characteristic parameters of articles : the number of editors , edits , and the article size for different articles . to compensate for such an effect , we employ rescaled measures for article @xmath9 as @xmath10 , @xmath11 , and @xmath12 , where @xmath13 , the age of article @xmath9 , is measured as the time between the moment of onset and that of the latest edit of article @xmath9 . @xmath14 , @xmath15 , and @xmath16 are the number of edits , the number of editors , and the article size for article @xmath9 , respectively . the rescaled measures are free from the temporal effects , making it possible to recruit myriad articles into the same ground for analysis in the sense of growth per unit time . for the number of edits , the number of editors , and the article size from the data , we hereafter use their rescaled values unless stated otherwise . a natural step forward is to search for any possible interplay between @xmath17 , @xmath18 , and @xmath19 in the formation of current wikipedia . one can suppose that the number of edits has varied gradually as a function of the number of editors , because both measures reflect the degree of interest in the article . unexpectedly , we discover that the articles show a peculiar bimodality in their number of editors across the entire value of the number of edits [ see fig . [ bimodality](a ) ] . the bimodality is characterized by the linear relation @xmath20 with two distinct proportionality constants , @xmath21 and @xmath22 , respectively . in other words , there are two groups of articles , determined by the proportion between the number of editors involved in the articles and the editors average activity ; one group is dominated by a relatively small number of enthusiasts who edit articles frequently , and the other is composed of a relatively large number of editors who seldom edit articles . besides the cases of edits and editors , wikipedia shows a similar division of article size for given numbers of editors [ see fig . [ bimodality](b ) ] . there are two types of articles determined by the average article size produced by an editor per unit time . this relation is also described by the linear dependency @xmath23 , where @xmath24 for the upper mode and @xmath25 for the lower mode . in other words , editors for some articles have generated about @xmath26 bytes on average , meanwhile the editors of the rest of the articles have generated only about @xmath27 bytes on average . our finding of bimodality in the two relations ( @xmath17 versus @xmath18 and @xmath18 versus @xmath28 ) triggers an interesting question : does each of the modes in one relation correspond to each mode in the other relation [ figs . [ bimodality](a ) and [ bimodality](b ) ] ? it seems natural to speculate that such modes have the counterparts in the other relation , or at least one is subordinative of the other . contrary to this speculation , our observation suggests that there is no visible relationship between the two different types of bimodality [ see figs . [ bimodality](c)[bimodality](f ) ] . the points in the figures are colored according to the modes to which the corresponding articles belong in the criteria based on @xmath29 or @xmath30 . we simply tear off the upper and lower modes by drawing a line [ the dashed lines in figs . [ bimodality](a ) and [ bimodality](b ) ] between the two modes and assign purple and blue colors for the points in the upper and the lower modes , respectively . those purple and blue points are totally mixed when the criterion is based on the other parameter relation . taken together , we conclude that there are at least four different groups of articles , which can be categorized by its growth per unit time . the possible mechanism behind the division is suggested based on our modeling study in sec . [ sec : model ] . to understand the underlying dynamics of the observed patterns , we develop a mechanistic model of editing dynamics by identifying two key factors that drive the evolution of wikipedia articles . we assume that there are two fundamental and inherent properties of an article reflecting the society s viewpoint on the article s topic : the preferences for referring wikipedia and the desires to edit ( namely , editability ) . in this section , we show that two such key drivers have elicited the wikipedia into its current state as shown in fig . [ bimodality ] . interestingly , each of those has a decisive effect on the distinct modality structure of @xmath29 and @xmath30 , respectively , and they have almost no impact on each other s modality . the preferences for referring wikipedia stems from its relative credibility compared to other conventional media . in other words , people tend to refer wikipedia more than other conventional media or opinion from others for certain topics . because of the nature of open - edit policy , there are long - lasting arguments of credibility , especially for the scientific contexts @xcite . as a result , people avoid referring wikipedia to reinforce their contention for scientific topics when they debate . nevertheless , several topics are almost free from the trust issue and wikipedia can be considered as a trustworthy source of knowledge . the subcultures such as animations , movies , and computer games are good examples , because the editors are not only a fan of the topic but also the creators of such cultures @xcite . in those cases , therefore , members of a society do not hesitate to utilize wikipedia as their grounds for the arguments . in addition , there are different levels of psychological barriers and desires in editing , depending on the topic . people tend to edit the article about which they have enough knowledge @xcite . thus , the average `` editability '' of articles , for members of a society , is diverse by its nature from the casual ones which are easily editable to the formal ones . this editability also depends on collective motives , which describe the significance of the topic as the common goal of social movements @xcite . therefore , the intrinsic rate of edit should be taken into account . besides these two key factors , editors are also engaged in articles when they have already given more effort to the articles by editing them @xcite , representing the feeling of attachment . additionally , it is hard to edit an article that already has a massive amount of information @xcite , so the motivation to edit will be reduced as the article size is increased . we describe how we implement the sociopsychological effects into our mathematical model in detail . by incorporating the aforementioned factors , we create a mechanistic model of the article growth . the model comprises @xmath31 agents where the individual agents represent members of a society and all of the agents are connected to a single wikipedia article . note that we take a single wikipedia article in our model , as we assume that different degrees of editability and credibility yield different types of articles in real wikipedia . to account for the modality shown in the interplays between three measures , @xmath28 , @xmath18 , and @xmath17 in fig . [ bimodality ] , we introduce corresponding model parameters . first , the article has its own length @xmath32 corresponding to @xmath33 in our data analysis . at the beginning , the length is assigned as @xmath34 , where @xmath3 is the minimum length to which agent can reduce the article , so @xmath35 always . the number of edits at time @xmath36 , denoted as @xmath37 , is also defined as the total number of article updates until @xmath36 , under the update rules described in sec . [ sec : agent_wiki_dynamics ] . additionally , @xmath38 corresponds to the number of distinct agents who edited the article at least once . besides the quantities explicitly measured in data analysis , we also adopt internal parameters for the agents and the article . the agents are connected to each other with the erds - rny random network @xcite . such connections between agents stand for various relationships in society : friends , co - workers , even enemies . every agent @xmath9 has its own opinion @xmath39 $ ] ( real numbers between @xmath40 and @xmath3 ) at time @xmath36 , where @xmath40 and @xmath3 are the two extremes of conflicting opinions , e.g. , conservatism and progressivism . one should be aware that this number does not correspond to a certain merit or superiority . initially , @xmath41 is assigned as a randomly generated number from the uniform distribution in the interval @xmath42 $ ] . the wikipedia article also has its own opinion @xmath43 at time @xmath36 , which is the overall stance of wikipedia on the topic . we set @xmath44 , to get the insights of the situation that agents and the wikipedia article adjust their opinions to the most radical one . similar to the fact that it is impossible to gauge the `` stance '' of the article and agents to the topic , we do not explicitly display the values and those are used only for stochastic simulation . for each time step @xmath36 , the agent - agent interaction described in sec . [ sec : agent_agent_dynamics ] and the agent - wikipedia dynamics described in sec . [ sec : agent_wiki_dynamics ] occur in turn . our model colligates resolving of conflicts between agents with the contribution of agents to modify wikipedia . in our model , all members of society are open - minded and they can change their mind . for each timestep @xmath36 , a pair of agents @xmath9 and @xmath45 , which are neighbors in a preassigned network , is chosen and they try to convince each other for the topic of the wikipedia article . we assume that agents rely on references to reinforce their opinion . for simplicity , we consider only two major types of references : wikipedia and general media . general media , denoting the entire set of references other than wikipedia , represent the ordinary viewpoint of the society toward the topic . as we described above , wikipedia is a more reliable source for certain topics . hence , we set a probability @xmath46 with which agents choose wikipedia as their reference , and this probability corresponds to the reliability of wikipedia [ see fig . [ modeldescription_fig](a ) ] . otherwise , agents decide to follow the standards of society by following general media s opinion , which is defined as the average opinion of entire agents in the society . in other words , the reference opinion @xmath47 once we choose the reference , an agent whose opinion is closer to the reference always succeeds in convincing the other agent . for the convenience , we call the agent as @xmath9 whoever s opinion is closer to the reference than the other , i.e. , @xmath48 [ see fig . [ modeldescription_fig](a ) ] . agent @xmath45 changes its opinion toward @xmath9 s , while agent @xmath9 keeps its opinion . people tend to minimize the amount of changing @xcite ; thus the agent @xmath45 sets his / her target as @xmath49 or @xmath50 , depending on which one is closer . as a result , the opinions of agents @xmath9 and @xmath45 at the next step @xmath51 are given by @xmath52 and @xmath53 respectively . the parameter @xmath54 $ ] represents the tolerance of agents , which indicates the psychological limit to change the opinion , as discussed in the introductory part of sec . [ sec : model ] . the value of @xmath55 affects mainly the timescale of simulation , yet does not have a large impact on our model conclusions ; thus we fix this value as @xmath56 to set a moderate time scale . a distinct character of our model , compared to opinion spreading models with external field @xcite , is that agents can also modify the media , which corresponds to wikipedia in our model . one additional key difference from the previous models of wikipedia @xcite is that we introduce the _ length _ @xmath32 into our model . this length not only has an impact on the edits but also changes by the edits . we focus on the length of the article instead of the specific opinion values . as we described in the introductory part of sec . [ sec : model ] , we assume that there are fundamental differences in `` editablity '' among articles , thus we set the base activity of edit @xmath57 as the control parameter . for every timestep , a randomly chosen agent attempts to edit the article in wikipedia . suppose that agent @xmath58 is chosen , then it edits the wikipedia article with the probability @xmath59 where @xmath60 is base activity , and @xmath61 is attachment of agent @xmath58 for the wikipedia article @xcite , which is assigned as @xmath40 at the beginning and increased by unity every time an agent edits the wikipedia article [ see fig . [ modeldescription_fig](b ) ] . naturally , the term @xmath62 / [ \sum_{s}{a_s(t ) + 1}]$ ] accounts for the fact that people tend to edit more frequently when they have contributed to establishing the current state of the article more @xcite . the term @xmath63 in eq . represents the reduced motivation as the article size is increased , due to the amount of information @xcite . a recent report that the growth of wikipedia has slowed down supports this factor @xcite . if an agent decides to edit an article , wikipedia s opinion changes as @xmath64 / l(t ) \,.\end{gathered}\ ] ] in our model , the amount of change is inversely proportional to @xmath32 . figure [ deltasizepersize ] indicates that the impact of a single edit event should be decreased as the article size is increased , because the absolute amount of change is preserved . additionally , @xmath65 represents the physical and psychological limit for editing @xcite . the value of @xmath65 affects also mainly the timescale of simulation similar to @xmath55 . we fix this value as @xmath56 to set a moderate time scale , and this value does not have a large impact on our model conclusions . finally , the length parameter @xmath32 is changed after the update of the article s opinion , as follows : @xmath66 where the random variable @xmath67 is chosen according to the following rule . if the agent has modified the article toward an extreme position ( @xmath40 or @xmath3 ) , we suppose that the agent tend to append new contents to the article . in contrast , agents are likely to replace the contents to neutralize the article s opinion . specifically , we divide the update into the two following cases : ( i ) @xmath68 or @xmath69 ( toward an extreme ) and ( ii ) any other cases ( neutralize ) . for ( i ) , the article size is increased by @xmath67 drawn from the interval @xmath70 $ ] uniformly at random , to reinforce the argument . otherwise , @xmath67 is drawn from the interval @xmath71 $ ] uniformly at random , which implies replacement of arguments . the fixed parameter @xmath72 is related to the physical limit in fig . [ deltasizepersize ] . the value of @xmath72 affects mainly the length . however , in this study , we use the ratio of the length to other measures rather than the absolute length of the article . we display the result with @xmath73 ( fig . [ modelresults ] ) , yet we verify that our conclusions are robust for other values of @xmath72 because the parameter governs only the overall length scale [ see fig . [ modeldescription_fig](b ) for the illustration on the @xmath67 criterion ] . in sec . [ sec : model_results ] , we discuss how the modes in fig . [ bimodality ] are formulated during the evolution of wikipedia in our model . and @xmath46 using the page view statistics of wikipedia ( 2014 ) and google @xmath7-gram data set ( 2008 ) , which are the latest data sets for both . ( a ) the average value of estimated @xmath57 , which is calculated by dividing the number of edits in 2014 by the page view statistics in 2014 @xcite , as a function of @xmath74 . as expected , the estimated @xmath57 decreases for larger @xmath74 values . ( b ) the average value of estimated @xmath46 : the ratio of the page view statistics in 2014 @xcite to the google 1-gram frequency in 2008 @xcite , as a function of @xmath75 . both plots are drawn from the same sampled set of @xmath76 articles , with the conditions described in the text . ] for both @xmath77 versus @xmath78 and @xmath78 versus @xmath79 relations , our mechanistic model captures the essential features of the observed empirical relations reported in sec . [ sec : time_rescaled ] with proper parameter values . as we have shown in sec . [ sec : data_analysis ] , the proportionality coefficients between characteristic parameters are classified into two modes : @xmath80 in particular , both @xmath46 for the agent - agent interaction ( in sec . [ sec : agent_agent_dynamics ] ) and @xmath57 for the agent - wikipedia interaction ( in sec . [ sec : agent_wiki_dynamics ] ) are crucial to generate the splits of modes into different groups : @xmath57 is essential to reproduce a separation of @xmath81 [ see fig . [ modelresults](a ) ] and @xmath46 is indispensable for the division of @xmath82 [ see fig . [ modelresults](b ) ] . in the early stage , @xmath81 is almost unity across the systems with the entire parameter space composed of @xmath46 and @xmath57 , which corresponds to the single ( or unimodal , in contrast to the bimodal pattern shown in real data ) linear relation . while this single linear relation is characterized at the early stage , as time goes by , we observe the decreasing trend of @xmath81 . despite the fact that the @xmath81 decrement over time occurs for the entire parameter space , the pace of decreasing is determined by @xmath57 , the base rate for editing an article . @xmath81 drops much slower for smaller @xmath57 values , which leads systems to fall into two different regimes : @xmath83 and @xmath84 [ fig . [ modelresults](a ) ] . interestingly , this divarication solely depends on the value of @xmath57 . on the other hand , @xmath82 also shows unimodality in the early stage , but it is suddenly increased with time only for @xmath85 [ fig . [ modelresults](b ) ] . analogous to @xmath81 , @xmath82 is also almost solely driven by @xmath46 , but there also exists a small amount of influence by @xmath57 ; small values of @xmath57 do not guarantee the large article size across all @xmath46 values , but low @xmath57 yields large article size at similar values of @xmath46 . based on our model results , we suggest a possible mechanism that yields the bimodality in fig . [ bimodality ] , which encourages us to verify the model results compared to the real data : either parameter @xmath57 or @xmath46 should be a decreasing function of @xmath86 and @xmath87 , respectively . however , we can not extract simulation parameters @xmath46 and @xmath57 directly from the data . we therefore use a bypass to estimate @xmath57 and @xmath46 . fortunately , wikipedia offers page view statistics of articles that can be used for estimating such parameters @xcite . we assume that this page view in a certain period reflects the degree of interest of wikipedia users in the articles , and the number of edits in the same period naturally displays the editing frequency . thus , the ratio of the number of edits to this page view for a certain period can be related to the base edit rate @xmath57 . analogous to our presumption , this ratio is a decreasing function of @xmath86 ( see fig . [ modelverify ] ) . to treat the other parameter @xmath46 , we should employ the proxy that can reflect the general interest of the entire society in the topic . we suggest that the google books @xmath7-gram , a vast digitized collection of documents produced in the world is a suitable choice @xcite . google books @xmath7-gram is a database containing about 6% of english books ever published . this data set offers a yearly number of occurrences for any phrase less than six words from 1800 to 2008 , and this number of occurrences can be considered as the proxy of interests in society for a certain phrase . in our model , @xmath46 is the proportion of degree of interest in wikipedia versus that of the entire society . in other words , wikipedia page view on a certain topic versus its @xmath7-gram frequency can be the estimator of @xmath46 . for fair comparison , we also only take the wikipedia articles that satisfy the following conditions . first , the title of article should exist in google 1-gram data set in 2008 , the latest year of the data set . second , the article should be visited at least once in 2014 . to avoid the effect of inflectional variation of words , we use the stem of wikipedia articles title and google 1-gram data set @xcite , instead of using the word directly . after this filtering process , @xmath76 articles are left among the total set of @xmath0 articles . this estimator of @xmath46 also decays , as @xmath87 is increased as we expected . both figs . [ modelverify](a ) and [ modelverify](b ) indeed show the behaviors expected from their estimators ( @xmath57 and @xmath46 , respectively ) , which indicates that our model is suitable to describe the real wikipedia . note that the estimators of @xmath57 and @xmath46 should not be taken as the exact face values of model parameter values for a real article in wikipedia , and the results should be understood as a proxy of statistical properties of articles . first , page view statistics might be affected by the number of hyperlinks pointed to the article . such relative importance within the network topology may increase the page view by random visits , yet there is a positive feedback between the page view and the number of hyperlinks . an article also tends to have connections to popular articles @xcite , which eventually yields disproportionally many hyperlinks for popular items ; thus there could be overestimation of page views for the popular articles . moreover , there is a recent report that warns of the possible bias of google @xmath7-gram as the proxy of real popularity in our society @xcite . this year - by - year level fluctuation may give unfairness to compare the frequencies many years apart . to avoid such fluctuation , we restrict our results for the year 2008 . additionally , word - by - word fluctuation should be canceled during the averaging process , because each data point corresponds to a massive number of articles . as a result , we believe that our observation is still valid , in spite of such fluctuations that might cause some degree of bias . the heterogeneity for the ratio of the number of editors to that of edits , @xmath86 , leads us to the eventual question : is this heterogeneity from structural inequality ? in other words , does the existence of dictatorship or monopoly of small group editors , or super editors @xcite , make it difficult for others to participate in editing processes ? to find the answer , we use the gini coefficient , which is a common measure for inequality in economics @xcite ranging from @xmath40 for the minimal inequality ( or the maximal equality ) to @xmath3 as the maximal inequality . we consider the number of edits for individual editors as the wealth variable in the gini coefficient . the trend of the gini coefficient as a decreasing function of @xmath86 shown in fig . [ gini](a ) suggests the modes with slope @xmath88 and @xmath89 in fig . [ bimodality](a ) are in equilibrium and non - equilibrium states , respectively . additional analysis of the gini coefficient in terms of the @xmath57 estimator ( the ratio of the number of edits in 2014 to the page view statistics in 2014 ) also indicates that the larger @xmath57 induces more severe inequality for editing [ see fig . [ gini](b ) ] . this is counterintuitive because it actually means that articles inducing larger motivation to edit eventually set a larger barrier to participate in editing . it is doubtful that the phenomenon is caused by the amount of information @xcite , since the gini coefficient does not vary much according to its amount of information [ see fig . [ gini](c ) ] . similar to the real wikipedia , our model also supports the observed inequality . although we use a simplified estimator of @xmath57 in our real data , the ratio of the number of edits in 2014 to the page view statistics , the gini coefficient is an increasing function of @xmath57 in the model as in the real data [ see fig . [ gini](d ) ] . additionally , since @xmath57 has a limited effect on the article size ( see fig . [ bimodality ] ) , the model observation of the gini coefficient is compatible to our observation that article size does not have a large effect on the gini coefficients . such logical elimination suggests that a few engaged and dominating editors make it indeed hard for laypeople to participate in editing processes . there are `` democratic '' articles ( with slope of @xmath88 in fig . [ bimodality ] ) and `` dictatorial '' articles ( with slope of @xmath89 in fig . [ bimodality ] ) . in short , inequality exists indeed . traditionally , collaboration used to be mainly regional and face - to - face interactions were demanded , which had prevented the world - wide formation of collective intelligence . nowadays , improvements of modern information technology bring us a whole new stage of online collaboration . in this study , we have examined such a new passion of collective intelligence through long - term data from wikipedia @xcite . people believe that such a new paradigm will eventually yield democratization of knowledge @xcite . as a representative medium , wikipedia is also considered as a spearhead of such pro - democracy movements @xcite . however , our observation suggests that the current status of wikipedia is still apart from the perfect world - wide democracy . the observed periodicity for the time between edits alludes that the english wikipedia is still regional for english natives ( see fig . [ deltatimeperage ] ) . bimodality and its inequality index suggest that there are articles dominated by a small number of super editors ( figs . [ bimodality][gini ] ) . notwithstanding the fact that there is no explicit ownership for wikipedia articles , some kind of privatization by dedicated editors for given topics is happening in reality . the value of such dedicated editors should not be depreciated , of course . their dedication has indeed played the main role in keeping the current state - of - the - art accuracy in the current wikipedia @xcite . however , in the long run , knowledge can not survive without collaboration between experts and society @xcite . although most advanced knowledge is invented by experts , such experts occupy a rather small proportion in a society ; thus , knowledge without support from other members of the society will lose its dynamic force to sustain . additionally , despite our findings that the amount of contents created by an editor ( @xmath87 ) mainly depends on the degree of referring wikipedia ( namely @xmath46 ) , an equitable opportunity for participation also increases such individual productivity ( see fig . [ modelresults ] ) . our study not only gives significant insight into the formation and current state of wikipedia , but also offers the future direction of wikipedia . our simulation results suggest that such inequality is increased with time , which may result in less productivity and less accuracy as by - product in the future than now [ see figs . [ modelresults](a ) and [ gini](e ) ] . it is indeed already reported that the growth of wikipedia is slowing down @xcite and our observation suggests that it will become even slower if we do not take any active action . to sustain collaborating environments , it is worth giving more motivation and incentives to the newbies to reduce the monopolized structure in wikipedia . we hope that extending our approach to various collaboration environments such as open - source movement @xcite might give us the insight for the future investment that brings us a new level of collaborating environments . finally , we would like to emphasize that the results and implications of our study are not restricted to the wikipedia or online collaboration systems , but have much wider applications in human or nonhuman interactions in the world . we are grateful to beom jun kim ( ) , pan - jun kim ( ) , and hyunggyu park ( ) for insightful comments . this work was supported by the national research foundation of korea through grant no . 2011 - 0028908 ( j.y . and h.j . ) . a. kittur , b. suh , and e.h . chi , _ can you ever trust a wiki ? : impacting perceived trustworthiness in wikipedia _ , proceedings of the 2008 acm conference on computer supported cooperative work ( cscw 08 ) , p. 477 ( 2008 ) . adler , k. chatterjee , l. de alfaro , m. faella , i. pye , and v. raman , _ assigning trust to wikipedia content _ , proceedings of the 4th international symposium on wikis ( wikisym 08 ) , article no . 26 ( 2008 ) . bould , e. s. hladkowicz , a .- a.e . pigford , l .- a . ufholz , t. postonogova , e. shin , and s. boet , _ references that anyone can edit : review of wikipedia citations in peer reviewed health science literature _ , bmj * 348 * , g1585 ( 2014 ) . a. kittur , b. suh , b. a. pendleton , and e.h . chi , _ he says , she says : conflict and coordination in wikipedia _ , proceedings of the sigchi conference on human factors in computing systems ( chi07 ) , ( 2007 ) . bryant , a. forte , and a. bruckman , _ becoming wikipedian : transformation of participation in a collaborative online encyclopedia _ , proceedings of the 2005 international acm siggroup conference on supporting group work ( group 05 ) ( 2005 ) . lakhani and r. wolf , why hackers do what they do : understanding motivation and effort in free / open source software projects , _ perspectives on free and open source software _ ( mit press , cambridge , ma , 2005 ) . a. capocci , v.d.p . servedio , f. colaiori , l.s . buriol , d. donato , s. leonardi , and g. caldarelli , _ preferential attachment in the growth of social networks : the internet encyclopedia wikipedia _ , phys . e * 74 * , 036116 ( 2006 ) . h. hasan and c. pfaff , _ emergent conversational technologies that are democratising information systems in organisations : the case of the corporate wiki _ , proceedings of the information systems foundations ( isf ) : theory , representation and reality conference ( 2006 ) .

wikipedia is a free internet encyclopedia with an enormous amount of content . this encyclopedia is written by volunteers with various backgrounds in a collective fashion ; anyone can access and edit most of the articles . this open - editing nature may give us prejudice that wikipedia is an unstable and unreliable source ; yet many studies suggest that wikipedia is even more accurate and self - consistent than traditional encyclopedias . scholars have attempted to understand such extraordinary credibility , but usually used the number of edits as the unit of time , without consideration of real - time . in this work , we probe the formation of such collective intelligence through a systematic analysis using the entire history of @xmath0 english wikipedia articles , between 2001 and 2014 . from this massive data set , we observe the universality of both timewise and lengthwise editing scales , which suggests that it is essential to consider the real - time dynamics . by considering real time , we find the existence of distinct growth patterns that are unobserved by utilizing the number of edits as the unit of time . to account for these results , we present a mechanistic model that adopts the article editing dynamics based on both editor - editor and editor - article interactions . the model successfully generates the key properties of real wikipedia articles such as distinct types of articles for the editing patterns characterized by the interrelationship between the numbers of edits and editors , and the article size . in addition , the model indicates that infrequently referred articles tend to grow faster than frequently referred ones , and articles attracting a high motivation to edit counterintuitively reduce the number of participants . we suggest that this decay of participants eventually brings inequality among the editors , which will become more severe with time .

the anisotropic flow of particles has been an interesting observable , since data from the first heavy ion collisions became available at the bevalac . the deflection of the produced particles in the reaction plane ( defined as the plane between impact parameter and beam direction ) can be quantified by the so called directed flow , @xmath0 . at very low beam energies of @xmath2 gev per nucleon , the rotation of the system will lead to a strong overall directed flow coefficient , that has been observed and understood within fluid dynamical calculations @xcite . at very high beam energies , as they are achieved at the large hadron collider ( lhc ) and the relativistic heavy ion collider ( rhic ) , the slope of the traditional directed flow is close to zero at midrapidity due to the almost perfect transparency of the colliding nuclei . the small negative slope of charged particles ( mostly pions ) at top rhic energy can be explained within a fluid dynamical model and a slightly tilted initial state @xcite as well as a hadronic transport model @xcite . in the last 3 years more studies where focused on odd flow coefficients related to initial state fluctuations . the so called rapidity - even @xmath0/directed flow was defined to quantify the dipole moment generated by fluctuations in the initial transverse density profile . in the present study , we are solely interested in the traditional rapidity - odd directed flow , that forms independent of initial fluctuations . at intermediate colliding energies , studied at the beam energy scan program at rhic , the future facility for antiproton and ion research ( fair ) and the former ags - sps experiments , the systematic study of directed flow is thought to be more interesting . within fluid dynamical calculations , it has been predicted that the slope of the directed flow of baryons will turn negative and then positive again as a function of energy if a first order phase transition is present . this means that more protons ( most of the baryons at lower beam energies are protons ) are emitted in direction opposite to the spectators than aligned with them . this effect , called `` antiflow '' or `` collapse of flow '' , has been attributed to a softening of the equation of state ( eos ) , during the expansion , due to a first order phase transition @xcite , leading to a rotation or tilt of the fireball in the reaction plane @xcite . the corresponding measurements of the na49 collaboration @xcite for the directed flow of protons had insufficient statistics to draw definite conclusions . recently , the star collaboration has measured the predicted qualitative behavior of the slope of the net - proton directed flow as a function of beam energy which turns negative and then positive again @xcite . since the early predictions were made with exclusively fluid dynamical models , which over predicted all other flow components , the goal of our study is to understand the eos dependence of directed flow within more modern transport approaches . first , we validate the qualitative predictions within a pure fluid dynamical calculation and confirm that with a first order phase transition the proton @xmath0 slope has the expected qualitative behavior , including a dip below zero . as in the previous studies , this sign change happens at much lower beam energies than what star has measured . in section [ section_purehydro ] we explore the influence of the freeze - out criterion on this result ( isochronous compared to iso - energy density ) and show the relation to the time evolution of the directed flow . then we perform the calculation within the ultrarelativistic quantum molecular dynamics ( urqmd ) hybrid approach with a more realistic treatment of the initial state and final stages employing non - equilibrium hadron - string transport . in this calculation the sensitivity of the directed flow to the equation of state is less obvious . finally , in section [ section_discussion ] we point out additional issues that need to be addressed , before a clear conclusion can be drawn . in the following we will study the effect of the equation of state of hot and dense nuclear matter on the directed flow measures in relativistic nuclear collisions . in particular we want to know whether the slope of the directed flow , as function of rapidity , is sensitive to the order of the qcd phase transition . we therefore have to compare two different scenarios . one where the qcd transition is of first order and one where it is a crossover . for the first order transition scenario we will employ a well known maxwell construction which has been used in several investigations on the effect of the eos @xcite . the maxwell construction is used to connect a mean field type su(2)@xmath3 hadronic model ( hm ) and a bag model eos ( bm ) that consists of free quarks and gluons . the conditions for a maxwell construction are the equality of the thermodynamic variables temperature @xmath4 , baryochemical potential @xmath5 and pressure @xmath6 . as a result of the construction one obtains a single phase system inside the coexistence region of the transition . for simplicity , in the following , we will refer to the constructed eos only as the bag model eos ( bm ) . due to the maxwell construction the iso - thermal speed of sound @xmath7 essentially vanishes , and also the isentropic speed of sound @xmath8 drops considerable inside inside the coexistence region . note that the maxwell construction only accounts for the so called softening of the equation of state , due to the phase transition , and it lacks important features associated with a first order phase transition , e.g. a region of mechanical instability or the surface tension @xcite . however since we are only interested in the effect of the softening on the bulk dynamics the maxwell constructed eos will suffice for our current investigations . alternatively we will use an equation of state which follows from the combination of a chiral hadronic model with a constituent quark model @xcite , later referred to as the @xmath9-over equation of state . this eos gives a reasonable description of lattice qcd results at vanishing net baryon density , including a smooth crossover from a confined hadronic phase to a deconfined quark phase . this crossover continues into the finite density region of the phase diagram for essentially all densities relevant for the present investigations . we therefore are able to compare the fluid dynamics of systems which always evolve through a first order transition to those which always evolve through a smooth crossover . early studies on the directed flow in relativistic nuclear collisions suggested the collapse of flow to be a possible signal for a first order phase transition in dense nuclear matter . in particular one extracted the net x - momentum per nucleon @xmath10 , defined in the direction of the impact parameter , in a given rapidity window from fluid dynamical simulations as : @xmath11 where @xmath12 and @xmath13 are the local net baryon density and fluid velocity and @xmath14 is the nucleon mass . it was found that , when the bag model equation state , with a strong first order phase transition , was used , @xmath15 , i.e. the slope of the directed net momentum with rapidity , would be negative for collisions where the system remains in the mixed phase for a considerable time @xcite . as a first step we want to reproduce these results with the above described bag model eos , that includes a maxwell construction from a hadronic to a quark gluon plasma phase . furthermore we also use the @xmath9-over eos that only shows a crossover and is consistent with lattice data at @xmath16 , to see if the observed anti - flow is unique for a first order transition , i.e. a very strong softening . we use the 1fluid shasta algorithm , which is an ideal 3 + 1d fluid dynamic code described in @xcite for all calculations . the time dependence of @xmath17 at fixed rapidity window @xmath18 from the ideal 1-fluid calculations with a bag model and crossover eos . we show two different beam energies , @xmath19 and @xmath20 gev , solid and dashed lines respectively . there is an evident non - monotonic time dependence of @xmath17 . , scaledwidth=50.0% ] in early investigations @xcite the full collision was simulated within ideal fluid dynamics . as a consequence the two colliding nuclei have to be described as two homogeneous density distributions colliding head on . in this so called cold nuclear matter initialization no distinct nucleons exist , but two distributions of cold , locally equilibrated , nuclear matter . we therefore initialize two energy- and baryon density distributions , according to boosted woods - saxon profiles with a central density of saturated nuclear matter @xmath21 , corresponding to the two au nuclei with a given center of mass energy , at impact parameter @xmath22 fm . the simulation is started at a point in time just before the two nuclei first make contact . in the early stage of the collision the kinetic energy of the nuclei is then rapidly stopped and transformed into large local densities . from the consecutive fluid dynamical simulation we can extract @xmath17 , according to eq . ( [ pxdir ] ) as a function of time and at fixed rapidity , in the center of mass frame . figure [ f1 ] shows a comparison of the time dependent directed momentum per nucleon as extracted from the pure 1-fluid simulation at two different beam energies . a noticeable non - monotonic dependence of the directed flow on time can be observed , due to the angular momentum of the fireball , and we expect the final result to depend considerably on the decoupling time of the evolution . a typical transition point , from the fluid dynamical phase to the final hadronic decoupling , used in most recent simulations @xcite is roughly four times the nuclear ground state energy density @xmath23 . the slope of @xmath10 is obtained by a linear fit , to the rapidity dependent @xmath24 between @xmath25 , at the time when all cells of the calculation are below that criterion . in figure [ f2 ] we compare the beam energy dependence of the slopes from both possible equations of state , the first order transition and crossover scenarios . as can be seen , we reproduce the predicted negative slope of the directed flow around @xmath26 gev when a first order transition is present @xcite . the crossover eos also shows a minimum over a range of @xmath27 gev , however the slope always remains positive . beam energy dependence of the directed flow slope around mid rapidity . extracted from the ideal 1-fluid calculations with a bag model ( black ) and @xmath9-over eos ( red ) for au+au collisions , with an impact parameter of @xmath22 fm . , scaledwidth=50.0% ] already in the early studies it was noted that the quantity extracted with equation ( [ pxdir ] ) is not directly comparable to experimental measured , identified particle @xmath0 , defined as : @xmath28 where @xmath29 is the reaction plane angle and @xmath30 the transverse angle of a particular particle . the average is usually performed over all particles in all events , in a given rapidity bin . in order to transform the fluid dynamical fields into real particles we will use the cooper frye prescription @xcite on a pre defined hypersurface . the hypersurface , extracted from the unique fluid dynamical final state , is then used to sample a large number of hadronic final states which are independently evolved in the urqmd transport model . because the slope of the directed x - momentum was extracted from the fluid dynamical simulation at a fixed time we will first use a isochronous hypersurface for our particle production . the resulting slopes of the directed flow around mid rapidity ( fitted for @xmath25 ) , for different particle species , are shown in figure [ f3 ] . again , the negative slope is observed in the first order transition scenario around @xmath26 gev for protons and pions . the calculation with the @xmath9-over eos shows only a broad minimum in the @xmath31 slope for protons and pions . however it always remains positive . the softening of the two eos therefore leads to a minimum of the directed flow slope , but not always to a negative anti - flow . also the position of the minimum in beam energy varies with the eos , as the crossover leads to a softening also at larger densities , resulting in a very shallow minimum at larger beam energy . in figure [ f4 ] we show the same quantity as in figure [ f3 ] , but this time we use an iso - energy density hypersurface for the transition in the subsequent hadronic afterburner . to construct this hypersurface we employ the cornelius hypersurface finder @xcite , which has been already successfully used in previous studies @xcite . the minimum in the extracted @xmath32 slopes occurs again at the same beam energies as with the isochronous freeze out , however the proton @xmath0 slope remains now always positive , even when we use the eos with the large softening due to the phase transition . as shown in figure [ f1 ] the directed flow , at a given rapidity , shows a non monotonic time dependence . the positive proton @xmath0 slope in the iso - energy density freeze out scenario can therefore be regarded as the result of an effective shortening of the fluid dynamical low viscosity phase as compared to the iso - chronous freeze out . it is noteworthy that in both discussed cases the slope of the proton @xmath0 always turns positive again once the beam energy is increased above the softest point of the eos and that the pion directed flow shows always the same qualitative behavior as the proton flow . beam energy dependence of the @xmath0 slope of protons and negatively charged pions around mid rapidity extracted from the ideal 1-fluid calculations with a bag model ( black ) and @xmath9-over eos ( red ) . for particle production we applied a cooper - frye prescription on a iso - chronous hypersurface . , scaledwidth=50.0% ] until now we have assumed that the colliding systems are in local equilibrium from the beginning of the collision , in order to apply ideal fluid dynamics also for the initial interpenetration phase . this had the advantage that we could use different equations of state also for the initial phase , which leads to different compression dynamics and subsequently has an impact on the directed flow . however , the assumption of local equilibrium is certainly not justified for the very early stage of a nucleus - nucleus collision . in this stage a non - equilibrium approach is better suited to describe the early time dynamics . one example of such an approach is studied in this section . the urqmd hybrid approach , described in detail in @xcite , was introduced to combine the advantages of a boltzmann transport approach with fluid dynamics . because the urqmd model is used for the initial interpenetration stage of the collision , the effective equation of state during that stage is defined by the microscopic properties of the model , i.e. a purely hadronic phase . as stated in section [ section_purehydro ] , the fluid dynamical evolution is realized using the shasta algorithm , while the initial state before equilibrium and the final state with hadronic rescatterings and decays are computed using urqmd . slope of @xmath0 of protons and pions around mid rapidity extracted from the ideal 1-fluid calculations with a bag model ( black ) and @xmath9-over eos ( red ) . for particle production we applied a cooper - frye prescription on a iso - energy density hypersurface . , scaledwidth=50.0% ] in the hybrid simulations , the transition from initial transport to fluid dynamics happens when the two colliding nuclei have passed through each other : @xmath33 , where @xmath34 represents the nuclear radius and @xmath35 is the lorentz factor in the center - of - mass frame of the colliding nuclei . the transition from fluid dynamics back to transport happens on an iso - energy density @xmath36 surface , which is constructed using the cornelius hypersurface finder . the directed flow was calculated using events with impact parameter @xmath37 fm , to approximate the @xmath38 centrality range of the star data . as seen in figure [ f5 ] ( b ) , the hybrid model overestimates the directed flow as function of rapidity for protons at @xmath39 gev , in comparison to the experimental data and the standard urqmd result . however , for charged pion @xmath0 at the same collision energy the hybrid model results agree with experimental data better than standard urqmd or the pure fluid dynamical simulation ( fig . [ f5 ] ( a ) ) . all the hybrid model calculations , up to @xmath40 gev , reproduce the qualitative feature observed at lower sps energy , that the proton @xmath0 has the opposite sign of the pion @xmath0 . comparison of pion ( a ) and proton ( b ) @xmath41 from the various models , for a beam energy of @xmath42 gev , compared with data @xcite . here we always used the @xmath9-over eos in the fluid dynamical phase . , scaledwidth=50.0% ] the full beam energy dependence for the hybrid model results of the midrapidity @xmath0 slope for negatively charged pions and protons / antiprotons is shown in figure [ f7 ] ( a ) and ( b ) respectively . both proton and antiproton slopes are overestimated for the whole examined collision energy range , while @xmath43 for pions agrees with the data at lower collision energies but changes sign at @xmath44 - @xmath45 gev , which is not supported by the star data . while the difference between the investigated equations of state was already rather small in the pure fluid results ( fig . [ f4 ] ) , the two eos are completely indistinguishable in the hybrid simulations . for comparison we also present the standard urqmd results as grey lines . the qualitative behavior is very similar to the hybrid model results . the standard urqmd appears to better describe the experimental proton data , however . slope of @xmath0 of negatively charged pions ( a ) and protons and anti - protons ( b ) around mid rapidity extracted from the hybrid model calculations with a bag model and crossover eos . we compare with standard urqmd and experimental data @xcite . , scaledwidth=50.0% ] figure [ f9 ] shows the comparison of the hybrid and the pure hydro model results for the midrapidity @xmath46 for protons and antiprotons , where the bag model equation of state is used . both approaches give significantly too large slopes compared to the star data . as noted already in section [ section_purehydro ] , the proton @xmath46 changes sign only for the model with isochronous cooper - frye hypersurface . comparison of results for pure hydro and hybrid model calculations with a first order eos . shown is the slope of @xmath0 of protons and anti protons around mid rapidity compared to experimental data . , scaledwidth=50.0% ] no model calculation seems to capture the qualitative experimental trend showing the directed flow slope for all particles turning negative at some point and approaching 0 from below . also the overall magnitude seems to be strongly overestimated . as observed in previous studies @xcite , the hybrid model ( and to some extend also standard urqmd ) usually shows a reasonable agreement with experimental particle spectra , as well as the second and third order flow coefficients , at the energies investigated in this paper . furthermore it has been shown @xcite that fluid dynamical simulations can quite successfully account for the rapidity even @xmath0 moment , which is caused by initial state fluctuations . the strong deviation of the directed flow of all models , as compared to data , noted in the previous section is therefore surprising and requires a further discussion about the possible sources of differences in the @xmath0 extracted from the model as compared to the experiment . one particular difference for example lies in the determination of the reaction plane angles @xmath47 used in equation ( [ v1 ] ) . in our study the reaction plane ( rp ) angle is always defined to be zero along the impact parameter axis ( x - axis ) . experiments determine a @xmath0 event plane ( ep ) using certain assumptions . usually the ep for the directed flow is defined along the vector of the projectile and target spectator transverse momentum motion ( defined also by measurement ) @xcite . in an ideal scenario a model study would also define the ep in such a way . however , this is not possible here , as the spectators in the hybrid model , by definition , do not obtain a momentum correlated with the rp and in the pure hydro calculation are @xmath48 correlated with the true reaction plane . in the hybrid model , as in standard urqmd , the particles usually defined as spectators obtain only a random total momentum , caused by the finite net fermi momentum of the spectators . as a result both spectators have essentially uncorrelated transverse momenta and are not correlated to the rp . furthermore this spectator momentum should be balanced by momentum transfered to the fireball , in the initial state . due to our definition of @xmath0 with respect to the true rp this `` conserved '' momentum does not contribute to the extracted @xmath0 as it would if we used a ep defined as in experiment . finally a more realistic scenario should not only consider the momentum transfer from the spectator to the fireball ( and vice versa ) from the random fermi momenta but also from likely correlations and binding of the cold nuclei @xcite . we know experiment measures a finite @xmath49 for the spectators , but not it s origin . this momentum must be balanced , so the fireball should have a momentum contributing opposite to the `` naive '' @xmath0 by definition , which is lacking in our model study . future quantitative investigations on the correlations between the spectators as well as their average transverse momentum might help to constrain model uncertainties arising from the incomplete description of spectator - fireball momentum transfer . we have presented model simulation results on the directed flow of identified particles in nuclear collisions of beam energies ranging from @xmath50-@xmath51 gev . to describe the strongly interacting systems created in these collisions we used different approaches , combining hadronic transport and ideal fluid dynamics . we find that the pure fluid dynamical approach can reproduce older findings @xcite which predicted a negative slope of the proton directed flow if a strong first order phase transition is present in the eos . however we also find that this anti - flow is only observed if , and only if the full dynamics , expansion and initial compression , is treated fully in the ideal fluid dynamics model . in the idealized 1-fluid case the slope of the directed flow becomes positive at beam energies above the softest point of the eos , just as observed in the 3-fluid simulations @xcite and hybrid model . when we apply a more realistic freeze out procedure and a hadronic transport model for the initial state we observe a positive slope of the proton directed flow for all beam energies under investigation . comparing our results to experimental data we find that essentially all models , including the standard hadronic transport urqmd , can not even describe the qualitative behavior , observed by experiment , of the proton directed flow . all models severely overestimate the data , even though other observables , like the radial or elliptic flow , are usually well described within these models . therefore the measured negative slope of the directed flow is not explained . the calculated directed flow is very sensitive to details in the description of the initial state , the freeze out prescription ( and f.o . time ) as well as the method of determining the event plane . the definition of the event plane in experiment , as well as the momentum transfer between the spectators and the fireball is not properly treated in the present model calculations . these issues need to be addressed , before definite conclusions about the relation between the slope of directed flow and the eos ( including a phase transition ) can be made . we would like to thank declan keane , mike lisa , paul sorensen , huan huang and nu xu for interesting discussions about the star data . this work was supported by gsi and the hessian initiative for excellence ( loewe ) through the helmholtz international center for fair ( hic for fair ) . h. p. and j. a. acknowledge funding by the helmholtz association and gsi through the helmholtz young investigator grant vh - 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the sign change of the slope of the directed flow of baryons has been predicted as a signal for a first order phase transition within fluid dynamical calculations . recently , the directed flow of identified particles has been measured by the star collaboration in the beam energy scan ( bes ) program . in this article , we examine the collision energy dependence of directed flow @xmath0 in fluid dynamical model descriptions of heavy ion collisions for @xmath1 gev . the first step is to reproduce the existing predictions within pure fluid dynamical calculations . as a second step we investigate the influence of the order of the phase transition on the anisotropic flow within a state - of - the - art hybrid approach that describes other global observables reasonably well . we find that , in the hybrid approach , there seems to be no sensitivity of the directed flow on the equation of state and in particular on the existence of a first order phase transition . in addition , we explore more subtle sensitivities like e.g. the cooper - frye transition criterion and discuss how momentum conservation and the definition of the event plane affects the results . at this point , none of our calculations matches qualitatively the behavior of the star data , the values of the slopes are always larger than in the data .

the unconventional superconductors have a rich phase diagram determined by the interplay of multiple competing , or coexisting , types of order . nematic order ( which breaks the c@xmath0 symmetry of the underlying square lattice down to c@xmath1 ) has been shown to emerge in certain regimes of the phase diagrams of the copper - oxide @xcite and the iron - based @xcite superconductors . in the latter case , the nematic order accompanies ( and in some cases , precedes ) the magnetic order which occurs at a wavevector that breaks the lattice rotational symmetry . recently , the structure of the vortex cores in the mixed state of clean fese films was studied by means of scanning tunneling microscopy ( stm ) @xcite . strong anisotropy was observed in the zero bias conductance map around the cores , which have an eccentricity of the order of unity . although the lattice structure of fese at low temperature is orthorhombic@xcite , it has been claimed @xcite that the crystalline anisotropy ( of the order of a few tenths of a percent ) is too small to explain the large anisotropy of the vortex cores , which is likely to have an electronic origin . this experiment raises several questions , some of which we address in this paper : assuming that there is an electronic nematic order in superconducting fese , what is its microscopic origin ? what is its relation to superconductivity - _ , are these two types of order competing ? is the nematic order localized in the vortex cores ( and hence stabilized by the application of the magnetic field ) , or does it extend throughout the system ( and is only apparent in the stm spectrum near the cores ) ? here , we study the structure of the vortex core using a phenomenological landau - ginzburg ( lg ) theory in terms of two competing order parameters . using our lg analysis we have calculated the structure of an isolated vortex in the presence of the nematic order . our main result is that by looking at the profile of the gap near the vortex core , it is possible to distinguish between two different configurations of the nematic order , namely the presence of a localized nematic order within the superconducting vortex as opposed to the presence of a long range nematic order in the system . if the nematic order is localized at the core , the superconducting gap should be anisotropic only near the core and the anisotropy decays exponentially as we move away from the core . on the other hand , if the nematic order is long - ranged , the superconducting gap should exhibit an anisotropy which decays as a power law . if the nematic order is near its critical point , there is a large region in which the anisotropy of the gap depends logarithmically on the distance , eventually crossing over to a power law . moreover , we find qualitative differences in the shape of the contours of constant gap around the core in the different cases . if the nematic order exists only in the cores , the equal - gap contours tend to be elliptical ; if the nematic order is long - ranged , we find that the gap function tends to develop a `` four - lobe '' structure , with more pronounced higher harmonics . these features can be sought in stm experiments by mapping the magnitude of the gap around the core as a function of position . the paper is organized as follows : in section [ mod ] we introduce the lg functional with the two competing order parameters and carry out a preliminary analysis in the absence of the anisotropy . in section [ pd ] , we investigate the mean - field phase diagram of a single vortex . in section [ min ] , we introduce the anisotropy and perform a numerical minimization of the functional , commenting on the interesting features . finally , in section [ anis ] , we present our analytical results explaining the various interesting features observed by minimizing the free energy . we consider a lg type free energy for two competing order parameters : a complex field @xmath2 , describing the superconducting order parameter , and a real field @xmath3 , which describes a nematic order that competes with the superconducting order parameter . the form of the free energy density is given by @xmath4+\frac{\lambda_{2}}{2}\phi\bigg[(\partial_{x}\phi)^{2}-(\partial_{y}\phi)^{2}\bigg].\label{glfun}\end{aligned}\ ] ] apart from the standard free energy contributions arising due to @xmath3 and @xmath2 , we have a competition term , controlled by @xmath5 ( @xmath6 ) , and a term that gives rise to different effective masses for @xmath2 in the two directions , which is controlled by @xmath7 . @xmath8 is invariant under a rotation by 90 degrees , represented by @xmath9 we will be interested in the limit of @xmath10 , where @xmath11 is the london penetration depth , so that we can neglect the coupling to the electromagnetic field . at the outset , we set @xmath12 , since the @xmath13 term is small compared to the @xmath7 term in the limit where @xmath3 is small . it is convenient to define the coherence length of @xmath2 and the healing length of @xmath3 as @xmath14 taking the unit of distance to be @xmath15 , we can recast the above free energy in a more transparent form as follows , @xmath16+\frac{\gamma^{2}}{2\gamma_{s}}|\tilde{\psi}|^{2}\tilde{\phi}^{2}\nonumber \\ & + & \lambda\tilde{\phi}[(\partial_{\tilde{x}}\tilde{\psi}^{*})(\partial_{\tilde{x}}\tilde{\psi})-(\partial_{\tilde{y}}\tilde{\psi}^{*})(\partial_{\tilde{y}}\tilde{\psi})],\label{lgf}\end{aligned}\ ] ] where @xmath17 , @xmath18 , @xmath19 , and @xmath20 . from now on , we will drop the tilde symbols . for @xmath21 , a short - distance cutoff has to be imposed on eq . [ lgf ] . otherwise , the system is unstable towards developing modulations of @xmath22 with sufficiently short wavelength . we discuss the instability in appendix [ appins ] . in practice , we will mostly ignore this issue , assuming that there is a short - distance cutoff ( which is provided by the finite grid used in our numerical calculations ) . before we begin our analysis , let us comment about the choice of parametrization in this problem . we would like to think of this problem in terms of a fixed @xmath23 . then on choosing a particular ratio of the length scales of @xmath3 and @xmath22 , we still have one degree of freedom left in terms of the masses or the stiffnesses of the two order parameters , which is fixed by tuning @xmath24 . if we assume that @xmath25 , then the uniform ground state is given by @xmath26 and @xmath27 . this also constrains @xmath3 to be localized around the vortex cores , by making the mass term for @xmath3 positive deep inside the superconducting region . if we further assume that the nematic order is small , such that @xmath28 , then we can essentially ignore the feedback of @xmath3 on @xmath22 . therefore , we will first find the full profile of @xmath29 , the isolated vortex solution , in the absence of the nematic order and use that to find the form of the nematic order . then @xmath30 satisfies the following asymptotic relations : where @xmath32 , @xmath33 being the radius in the original coordinate system , and @xmath34 is a dimensionless constant . in general , it is difficult to find the solution of the full lg equation for @xmath30 for all @xmath35 analytically . therefore we obtain the vortex solution @xmath36 for all @xmath35 by minimizing the functional in eqn . [ lgf ] numerically in the absence of @xmath3 . the numerical solution conforms to the two asymptotic expressions above . it is interesting to note that @xmath30 does not recover from the vortex core to its bulk value exponentially , but rather as a power law @xcite . the behavior of @xmath37 in the vicinity of a vortex with @xmath38 was studied by ref . . we will now describe the mean - field phase diagram of a single vortex in the presence of a competing nematic order . there are three possible phases : in phase i , @xmath27 everywhere ; in phase ii , @xmath3 vanishes at large distance from the vortex core but becomes non - zero near the vortex core due to the suppression of the competing @xmath22 field to zero at the core ; and in phase iii , @xmath39 even far away from the core . a non - zero solution for @xmath3 is favored whenever the smallest eigenvalue @xmath40 of the following eigenvalue problem @xcite : @xmath41^{2}\bigg]\phi\left(x\right)=\epsilon\phi\left(x\right ) , \label{eigeqn}\ ] ] satisfies @xmath42 . in order to find the phase diagram , we solve this eigenvalue problem numerically on a discrete grid . the boundary between phases i and ii is the locus of points at which the smallest eigenvalue satisfies @xmath43 . for @xmath44 , @xmath3 becomes long - ranged , corresponding to phase iii . the resulting phase diagram is shown in fig . [ lvsg ] . this phase diagram is strictly valid as long as @xmath45 . if this is not the case , then the state with uniform nematic background and no superconductivity is energetically favorable over any other state . plane obtained by solving eqn . [ eigeqn ] numerically on a grid with @xmath46 and @xmath47 . the regions with qualitatively different solutions for @xmath3 are marked . phase i has no nematic order with @xmath27 everywhere , phase ii has nematic order localized in the vortex core , and phase iii has long - range nematic order away from the core . the blue squares correspond to points which we explore in more detail later . the dashed line @xmath48 represents the boundary between phases ii and iii . ] the physics behind the phase diagram can be understood as follows . when @xmath49 we are forcing the nematic order to coexist with superconductivity in a large region . this is unfavorable energetically due to the competition term @xmath50 . when @xmath25 , there is no @xmath39 solution . if @xmath44 , @xmath3 becomes non - zero even far away from the vortex core . in the opposite limit of @xmath51 , the nematic order exists deep within the superconducting vortex . since there is very little overlap between the two order parameters , the system can afford to have a higher value of critical @xmath24 below which there is a nontrivial nematic order . this explains the increasing trend of the critical @xmath24 for decreasing @xmath52 . + in the @xmath51 case , it is possible to give an analytical expression for the phase boundary between regions i and ii . the equation for this curve is given by , @xmath53 where @xmath34 is the constant which appears in eqn . [ psi0as ] . the details of this computation are diskussed in section [ phases ] . we are now in a position to include the effect of the anisotropy and investigate the structure of the vortex cores in different regions of the phase diagram described above . we now turn to diskuss the characteristics of the vortex profile in the different regimes shown in fig . [ lvsg ] . to solve for the vortex profile , we minimize the free energy ( [ lgf ] ) with respect to @xmath22 and @xmath3 numerically on a disk geometry . this is equivalent to solving the coupled landau - ginzburg equations with neumann boundary conditions , as we diskuss in appendix [ boundary ] . many of the features found in the numerical solution can be understood analytically , as we diskuss in the next section . we can expand both @xmath22 and @xmath3 in terms of the different angular momentum channels ( @xmath54 ) . the term proportional to @xmath55 only couples angular momentum channels that differ by 2 units of angular momentum in @xmath22 . therefore , in the presence of @xmath3 , the bare vortex solution ( @xmath56 ) gives rise to components of the form @xmath57 , @xmath58 , etc . similarly , the feedback of the superconducting order on @xmath3 gives rise to the generation of the even harmonics , i.e. @xmath59 gives rise to terms proportional to @xmath60 and @xmath61 . it is also possible to have a solution with only the even harmonics of @xmath22 , in which case , the vortex is absent . these two solutions do not mix with each other and therefore we shall focus on the solution in the presence of the vortex . + in light of this , we expand the order parameters as @xmath62 in terms of the expansions in eqn . [ expan ] , the free energy density can be written as , @xmath63-\frac{\psi_{n}^{2}}{2}+\frac{1}{4}\sum_{n , p , q}\psi_{n}\psi_{n+p - q}\psi_{p}\psi_{q}\nonumber \\ & + & \bigg(\frac{\gamma}{\gamma_{s}}\bigg)^{2}\bigg[\frac{1}{2}\bigg(\sum_{n}\bigg(\frac{\partial\phi_{n}}{\partial\rho}\bigg)^{2}+\frac{(2n)^{2}}{\rho^{2}}\phi_{n}^{2}-\phi_{n}^{2}\bigg)+\frac{1}{4}\sum_{n , p , q}\phi_{n}\phi_{n+p - q}\phi_{p}\phi_{q}\bigg]\nonumber \\ & + & \frac{\lambda}{2}\sum_{m , p}\phi_{p}\bigg[\bigg(\frac{\partial\psi_{m}}{\partial\rho}-(2m+1)\frac{\psi_{m}}{\rho}\bigg)\bigg(\frac{\partial\psi_{m+p+1}}{\partial\rho}+(2(m+p+1)+1)\frac{\psi_{m+p+1}}{\rho}\bigg)\nonumber \\ & + & \bigg(\frac{\partial\psi_{m}}{\partial\rho}+(2m+1)\frac{\psi_{m}}{\rho}\bigg)\bigg(\frac{\partial\psi_{m+p-1}}{\partial\rho}-(2(m+p-1)+1)\frac{\psi_{m+p-1}}{\rho}\bigg)\bigg]\nonumber \\ & + & \frac{\gamma^{2}}{2\gamma_{s}}\sum_{n , p , q}\psi_{n}\psi_{n+p - q}\phi_{p}\phi_{q},\label{lghar}\end{aligned}\ ] ] and we are interested in minimizing @xmath64 . we shall minimize the above free energy for a given system size and for only a fixed number of harmonics at a time . we have kept @xmath65 harmonics for @xmath22 and @xmath3 , where for any given @xmath65 ( odd ) we take all the harmonics @xmath66 to @xmath67 , @xmath68 , and similarly for @xmath3 . we have tried @xmath69 and found no substantial qualitative change in the results that we shall quote here , indicating that the results converge even with only 3 harmonics . we consider a system on a disk of radius @xmath70 . below , we describe the results in regions ii and iii of the phase diagram , and on the critical line dividing them ( in region i , where there is no nematic order , we get the regular circularly symmetric vortex ) . the specific values of @xmath71 which were used are marked by blue squares in the phase diagram in fig . [ lvsg ] . in this region , we expect to obtain a solution with a non - zero uniform @xmath72 away from the vortex core and a non - zero @xmath3 localized near the vortex core , decaying exponentially away from the core ( given that @xmath73 and @xmath25 ) . the contour plot for @xmath74 is shown in fig . [ gam2l0p1 ] . the parameters used here are @xmath75 . as can be seen in the figure , the core has an elliptical shape because of the interaction with the nematic order which coexists with superconductivity in the core region . as we go away from the core , the contours of equal @xmath74 become more and more isotropic , due to the rapid decay of the nematic order away from the core . this region in the phase diagram corresponds to the case where there is a uniform nematic background coexisting with superconductivity , even away from the vortex core . in this regime , as we move away from the core , @xmath3 goes to a constant and @xmath22 remains anisotropic . in fig . [ gam0p5l0p5 ] , the harmonics @xmath76 and @xmath77 are almost constant for large @xmath35 . moreover , @xmath78 for large @xmath35 . the contour plot of @xmath74 reveals a large anisotropic `` halo '' around the core , with a non - elliptical shape . far away from the core , where @xmath3 is constant , the landau - ginzburg equations can be solved analytically , showing that the anisotropy in @xmath74 decays as a power law in this case . we diskuss this solution in the next section . finally , we diskuss the critical line separating regions ii and iii in fig . [ lvsg ] , in which the @xmath3 field is critical far away from the core . naively , one would expect @xmath3 to go as @xmath79 asymptotically in this regime . however , depending on the details of the solution at small @xmath35 , there may be an intermediate regime in which @xmath80 , eventually crossing over to @xmath79 at a larger distance . this feature is diskussed in more detail in the next section . [ gam1l3]a shows @xmath22 and @xmath3 in the critical regime , with @xmath81 and @xmath82 . indeed , we observe that @xmath3 decays slowly away from the core . @xmath83 also have long tails . the contour plot for @xmath74 shares features that are similar to the behavior in region iii , namely a long - range , non - elliptical anisotropic halo . it is shown in fig . [ gam1l3]b . in the next section , we analyze the asymptotic behavior of the solution in the critical case , showing that the anisotropic component of @xmath74 falls off as @xmath84 at intermediate @xmath35 , crossing over to @xmath85 at sufficiently large @xmath35 . in this section , we propose various analytical arguments to explain the different features that were observed above by carrying out the minimization numerically . in subsection [ phases ] , we diskuss the solution for the nematic order in the presence of superconductivity . then in subsection [ coex ] , we analyze region iii of the phase diagram ( fig . [ lvsg ] ) . finally , in subsection [ lingl ] , we study the linearized gl equations in order to explain some of the other interesting features that were observed earlier . here we will briefly review the solution for @xmath3 , and supplement it with some further details . the lg - equation for @xmath37 , assuming that @xmath38 , is given by , @xmath41^{2}+\phi^{2}\bigg]\phi=0,\label{scaledphi}\ ] ] we shall now be interested in solving the linearized version of the above equation , which is justified for @xmath25 . for @xmath86 , this becomes equivalent to solving the problem @xmath87\phi=0.\label{phismalldist}\ ] ] this is identical to solving the schrdinger equation for the 2d quantum harmonic oscillator . we know that @xmath88 . the solution for this equation is given by @xmath89 where @xmath90 are the laguerre polynomials . the profiles of @xmath37 for a few different values of @xmath91 are shown in fig.[phis ] . as a function of @xmath35 for different values of @xmath91 over a distance of one correlation length of @xmath3 , @xmath15 . ] . [ phis ] at this point , we can also describe how we obtained the equation for the phase boundary between regions i and ii in the phase diagram ( fig . [ lvsg ] ) . in this case , @xmath3 is non - zero only very close to the center of the core . we can therefore expand @xmath22 around @xmath92 and keep only the leading order term ( eq . [ asymp ] ) . eqn.[phismalldist ] can be re - written as , @xmath93\phi=(\frac{1}{2}+\epsilon)\phi.\ ] ] non - trivial solutions exist for @xmath94 . the above equation is the schrdinger equation for a quantum harmonic oscillator in two dimensions with @xmath95 . then the smallest eigenvalue which corresponds to the zero point energy of the oscillator leads to the following equation for the curve @xmath96 in the limit of small @xmath52 . + on the other hand , for @xmath97 , we have to solve @xmath98\phi=0,\label{philargedist}\ ] ] from which we see that @xmath99 . however , there is a _ fine - tuned _ point at @xmath48 , at which the field @xmath3 is critical far away from the vortex core . the full equation for @xmath3 becomes @xmath100\phi=0,\label{finephi}\ ] ] @xmath101 . note that the @xmath79 solution can be obtained only for specific boundary conditions . for generic boundary conditions , with @xmath102 as @xmath103 , the solution is nevertheless asymptotic to @xmath104 at large @xmath35 . if @xmath105 , for instance , then @xmath106 at intermediate values of @xmath35 , where @xmath107 and @xmath108 are constants . @xmath37 crosses over to @xmath109 at radii of the order of @xmath110 ( see appendix [ appasym ] ) . this behavior reflects itself in the asymptotic decay of the anisotropy of the field @xmath22 away from the vortex core , as we saw in sec . [ min ] . in this subsection , we are interested in analyzing region iii of the phase diagram , in which superconductivity and nematicity coexist even far away from the core . let us assume , for simplicity , that far away from the core @xmath3 can be replaced by a constant . the effect of a constant @xmath3 is to render the effective masses in the two directions different . therefore , if we re - scale the coordinates as @xmath111 where @xmath112 , then this problem now becomes identical to the isotropic problem we had solved in the beginning of section [ mod ] . the solution for @xmath30 can then be written in terms of the new coordinates as , @xmath113 note that due to the presence of the background nematic order , @xmath30 does not tend to @xmath114 asymptotically . we now go back to our original coordinate system @xmath115 by expanding the above result to linear order in @xmath116 . then we get , @xmath117 in the above expression , the first bracket corresponds to @xmath30 , the second bracket corresponds to @xmath77 while the last one represents @xmath76 . it is interesting to observe that asymptotically , @xmath76 and @xmath118 approach the same constant value . we observe this feature in fig.([gam0p5l0p5]a ) . however , the harmonics do not recover to their asymptotic value as a power law , which is a result of the boundary conditions that were imposed while minimizing the free energy in the disk geometry ( see appendix [ boundary ] ) . from eqn . [ psireg3 ] , we can evaluate the form of @xmath74 and find that , @xmath119 therefore , asymptotically , @xmath74 is isotropic and the anisotropy decays as @xmath120 . in this section , we shall carry out an analysis of the linearized lg equations , to give an analytical explanation for some of the features that we have observed by carrying out the full minimization . for the sake of simplicity , let us ignore the feedback on @xmath3 resulting in the generation of the higher harmonics and assume that @xmath3 is isotropic ( i.e. @xmath121 ) . then , the linearized lg equations for the harmonics of @xmath22 can be written as , @xmath122\nonumber \\ + \lambda\partial_{\rho}\phi(\rho)\bigg[\bigg(\partial_{\rho}-\frac{2n-1}{\rho}\bigg)\psi_{n-1}(\rho ) & + & \bigg(\partial_{\rho}+\frac{2n+3}{\rho}\bigg)\psi_{n+1}(\rho)\bigg]\label{harmonics_nd}\end{aligned}\ ] ] there are some features of the problem that can not be deduced from a study of the linearized version of the problem , which include the overall scale and sign of @xmath3 and the signs of the different harmonics of @xmath22 . in the limit of @xmath86 , i.e. inside the vortex core , at leading order @xmath123 , where @xmath124 . this is a necessary condition for the harmonics to be well behaved in the limit of @xmath125 . on the other hand , in the limit of @xmath97 , i.e deep inside the superconducting region , the homogenous solution for the above equation gives exponentially damped solutions for all the @xmath126 , i.e. the anisotropy is short ranged . moreover , the source term , which is proportional to @xmath127 and is itself exponentially damped ( region ii ) , is also not strong enough to give rise to any long ranged solution . however , when @xmath3 is critical ( i.e. @xmath48 ) , the source term leads to the presence of long tails in the harmonics . in the regime where @xmath3 falls off logarithmically while @xmath30 is a constant , at leading order @xmath83 just follow @xmath3 , i.e. they also fall off logarithmically ( with prefactors of equal magnitude but opposite sign ) and have a correction of the form @xmath128 . on the other hand , when @xmath3 crosses over to the power law form , at leading order @xmath83 also fall off as @xmath129 with a correction of order @xmath130 . we have studied the interplay between nematic order and superconductivity in the presence of a vortex . if the nematic order coexists with superconductivity in the vicinity of a vortex core , the coupling between the two order parameters leads to an elongated shape of the core . we diskuss two distinct scenarios : in one the nematic order coexists with superconductivity everywhere ( i.e. , even far away from the vortex core ) , whereas in the other the competition between the two order parameter suppresses the nematic order in the bulk , and nematicity only exists close to the core where the superconducting order parameter is diminished . both scenarios lead to an anisotropic core . however , we show that they can , in principle , be distinguished by the way the anisotropy of the superconducting gap decays away from the core . if the nematicity exists only near the core , the anisotropy in the superconducting gap decays exponentially ; if it exists throughout the sample , we expect the gap anisotropy to decay as @xmath131 , where @xmath33 is the distance from the core . moreover , there are qualitative differences in the shape of the core in the two cases . in the former case , in which only the core region is nematic , the contours of equal gap tend to be more or less elliptical . in the latter case , the contours of equal gap tend to develop non - elliptical shapes with a four - petal pattern . therefore , analyzing the gap profiles measured by stm around a vortex could reveal the nature of the nematic ordering - whether it is localized at the vortex core , or coexists with superconductivity in the bulk . so far , we have diskussed the structure of an isolated vortex at the mean - field level . however , if the nematic ordering is favored only within a vortex core , an isolated vortex can not have static nematic order , since either thermal or quantum fluctuations would destroy such order . static nematic order is only possible when the density of vortices is finite . the coupling between the nematic halos of different vortices scales as @xmath132 $ ] , where @xmath133 is the inter - vortex distance ( @xmath134 is the applied magnetic field ) . the system can be described by an effective two - dimensional transverse field ising model with a spin - spin interaction @xmath135 and a @xmath134-independent transverse field . ( note that , unlike ref . @xcite , we are considering a thin film , rather than a three - dimensional system . ) this model has a nematic transition at a certain critical @xmath134 , which should be seen , e.g. , by measuring the anisotropy of the vortex cores as a function of @xmath134 . if an external rotational symmetry breaking field exists , as is presumably the case in fese due to the small orthorhombic lattice distortion@xcite , the electronic nematic transition is smoothed out . however , one still expects a sharp crossover as a function of magnetic field if the orthorhombic distortion is sufficiently weak . the microscopic origin of the anisotropic vortex cores observed in fese@xcite remains to be understood . it is likely that it originates from electronic nematicity rather than from the lattice distortion , since the experimentally reported orthorhombic distortion seems too small to produce such a large effect . the electronic nematic order could have an orbital character@xcite . alternatively , it could arise from a field - induced magnetic ordering@xcite at a wavevector @xmath136 or @xmath137 in the one iron unit cell , which is necessarily accompanied by a nematic component ( similar to the ordering in the iron arsenides ) . although static ordering of this type has not been observed in the iron selenides@xcite , it remains to be seen if they develop a static ordering in the presence of an applied magnetic field . neutron scattering experiments revealed a magnetic resonance at this wavevector in the superconducting state of fetese@xcite . moreover , ordering at such wavevectors nearly nests the electron and hole pockets , and therefore it is expected to couple strongly to superconductivity , explaining why the resulting anisotropy of the vortex cores is so large . _ note added : _ after this work was submitted for publication , another manuscript@xcite that studied the experimental features observed in fese @xcite came to our attention . in this paper , the authors study the effect of orbital ordering on the vortex structure in a two band model , by solving the bogoliubov - de gennes equations . this study is complementary to our phenomenological ginzburg - landau approach . this research was supported by the national science foundation under grants dmr-1103860 , dmr-0705472 and by a muri grant from afosr . d.c . thanks gilad ben - shach for a critical reading of the manuscript and for his comments . d.c . also thanks the physics department at harvard university for an e.m . purcell fellowship during 2010 - 11 . an interesting feature associated with the lg functional introduced in section [ mod ] is that there is an instability to a state with modulated @xmath22 . this arises due to a competition between two terms in the free energy , namely the @xmath3 and @xmath138 terms . let us suppose that @xmath3 does not vary spatially and @xmath139 . then at leading order , the contribution to the free energy from @xmath3 is of the form @xmath140 from the above expression , we see that for a sufficiently large @xmath141 , it becomes energetically favorable to gain energy from the second term by condensing a large negative value of @xmath3 . by extremizing the above with respect to @xmath3 , we obtain @xmath142 . hence , the contribution to free energy from @xmath143 is @xmath144 . this energy gain from a non - zero @xmath145 always dominates over the energy cost of order @xmath146 for a sufficiently large @xmath145 . in order to prevent this instability , we have to add a term of the form @xmath147 to the free energy , which is an allowed term from the underlying symmetry of the problem . we now want to obtain some restrictions on @xmath148 . + first of all , @xmath148 should be such that it prevents the instability . this gives us a lower bound on the value of @xmath148 . at the same time , @xmath148 should be small enough so that it should not change the physics significantly . this gives us an upper bound on the value of @xmath148 . therefore , we obtain , @xmath149 the above expression is not valid when @xmath3 becomes critical , i.e. when @xmath150 . in this case , we have to compare @xmath3 with @xmath151 . + however , when we minimized the free energy in section [ min ] , we did not have to include the above term with a finite @xmath148 as for a sufficiently small @xmath55 , the cutoff in @xmath145 arising from the discrete lattice prevented this instability from showing up . in general , when we derive the gl equations from the free energy , there is a surface term arising from the gradient terms in the energy which can be ignored in the limit of an infinite system size . however , for a finite sized system , the boundary term does play an important role . let us consider only the contribution of the gradient term of the superconducting order parameter in the free energy , in the absence of any nematic order . then we have , @xmath152 where @xmath153 contains the usual @xmath154 terms . on varying @xmath155 by @xmath156 in @xmath157 , we obtain for a finite system ( up to other variations due to @xmath153 denoted by @xmath158 ) , @xmath159 where @xmath160 is the area element , normal to the boundary . when we solve for @xmath22 in the interior of the region , only the first term contributes and the boundary term can be ignored . however , when we solve for @xmath22 on the boundary , only the surface term plays a role , since it can be thought of as appearing with an infinite weight of the form @xmath161 where @xmath162 is the radius of the disk on which we are minimizing the free energy and @xmath163 is the dirac - delta function . therefore , in order to solve for @xmath22 , we have to solve for @xmath164 in the interior of the region subject to the boundary condition @xmath165 ( neumann boundary conditions ) . + now in the presence of a constant nematic background ( @xmath166 ) , the gradient term in the free energy is , @xmath167,\ ] ] where @xmath168 . as we did earlier , on carrying out the variation over @xmath169 this amounts to solving for @xmath170 subject to the boundary condition , @xmath171 , where @xmath172 in polar coordinates , this condition can be written as , @xmath173=0.\ ] ] these boundary conditions mean , in particular , that the current perpendicular to the boundary is zero . in our numerical calculations , we have used a disk geometry ; therefore the boundaries are found to have a significant effect whenever we are considering a non - circularly symmetric solution , in particular in the regime where the nematic order is non - zero even far away from the core . we circumvent this problem , however , by taking a sufficiently large system and considering the solution only close to the vortex core , where the boundary effects are small . in this appendix , we analyze the asymptotics of the field @xmath3 far away from the vortex core in the case @xmath48 , in which the nematic order is critical . in this case , and for @xmath174 , the landau - ginzburg equation for @xmath3 ( eq . [ scaledphi ] ) becomes @xmath175\phi=0.\label{eq : phi_critical}\ ] ] this non - linear equation admits the solution @xmath176 @xcite . this solution is valid , however , for specific initial conditions , e.g. , @xmath177 , @xmath178 . physically , the initial conditions for eq . [ eq : phi_critical ] are determined by the details of the vortex profile at short distances , determined by eq . [ scaledphi ] . nevertheless , one can make some general statements about the asymptotic behavior of the solution . if , for some arbitrary @xmath108 such that @xmath179 ( far from the core ) , @xmath3 satisfies @xmath180 , then it is justified to neglect the @xmath138 term in eq . [ eq : phi_critical ] . then , the solution close to @xmath108 behaves as @xmath181 , where @xmath107 , @xmath134 are determined by the initial conditions . this can only be valid , however , up to a point @xmath182 at which @xmath183 , i.e. , at distances which are much smaller than the length scale set by the initial condition of eq . [ eq : phi_critical ] . at longer distances , we expect a crossover to @xmath184 . in fig . [ fig : phi_r ] , we present a numerical solution of eq . [ eq : phi_critical ] with boundary conditions @xmath185 and various values for @xmath186 . when @xmath177 , we get @xmath187 ( where the deviations are due to the boundary condition at @xmath188 ) . for smaller @xmath186 , there is an intermediate region where @xmath3 does not follow a power law , eventually crossing over to @xmath79 at larger @xmath35 . @xmath37 is approximately logarithmic in the intermediate region , as shown in fig . [ fig : phi_r]b . physically , we expect that @xmath189 ( since @xmath190 corresponds to the equilibrium value of @xmath3 in the absence of superconductivity ) . therefore , there is an intermediate logarithmic region , which becomes parametrically large in the limit of small @xmath3 . m. yi , d. lu , j .- h chu , j. g. analytis , a. p. sorini , a. f. kemper , b. moritz , s .- k . mo , r. g. moore , m. hashimoto , w .- s . lee , z. hussain , t. p. devereaux , i. r. fisher , and z .- x . shen , pnas * 108 * , 6878 ( 2011 ) . can - li song , yi - lin wang , peng cheng , ye - ping jiang , wei li , tong zhang , zhi li , ke he , lili wang , jin - feng jia , hsiang - hsuan hung , congjun wu , xucun ma , xi chen , and qi - kun xue , science * 332 * , 1410 ( 2011 ) . similarly to the magnetic field induced antiferromagnetic order seen in certain cuprate superconductors . see , e.g. , b. lake , h.m . rennow , n.b . christensen , g. aeppli , k. lefmann , d.f . mcmorrow , p. vorderwisch , p. smeibidl , n. mangkorntong , t. sasagawa , m. nohara , h. takagi , and t.e . mason , nature(london ) * 415 * , 299 ( 2002 ) . s. medvedev , t.m . mcqueen , i.a . troyan , t. palasyuk , m.i . eremets , r.j . cava , s. naghavi , f. casper , v. ksenofontov , g. wortmann and c. felser , nat . mater . * 8 * , 630 ( 2009 ) . y. qiu , w. bao , y. zhao , c. broholm , v. stanev , z. tesanovic , y. c. gasparovic , s. chang , j. hu , bin qian , minghu fang , and zhiqiang mao , phys . lett . * 103 * , 067008 ( 2009 ) ; h. a. mook , m.d . lumsden , a.d . christianson , brian c. sales , rongying jin , michael a. mcguire , athena sefat , d. mandrus , s.e . nagler , t. egami and c. de la cruz , arxiv:0904.2178 ( unpublished ) .

we present a phenomenological theory of the interplay between nematic order and superconductivity in the vicinity of a vortex induced by an applied magnetic field . nematic order can be strongly enhanced in the vortex core . as a result , the vortex cores become elliptical in shape . for the case where there is weak bulk nematic order at zero magnetic field , the field - induced eccentricity of the vortex core has a slow power - law decay away from the core . conversely , if the nematic order is field - induced , then the eccentricity is confined to the vortex core . we discuss the relevance of our results to recent scanning tunneling microscopy experiments on fese ( song _ et al . _ , science * 332 * , 1410 ( 2011 ) ) .

since the invention of neutrino beams at accelerators and the consequent discovery of the two flavors of neutrinos@xcite , the reactions @xmath3 and @xmath4 , which are the dominant reactions of muon and electron neutrinos with energies from @xmath5 mev to @xmath6 gev , have played an important role in studies of neutrino flavor . these charged - current quasi - elastic ( ccqe ) interactions are important not only because they identify the flavor of the neutrino through the production of the lepton in the final state , but also because the two body kinematics permit a measurement of the neutrino energy from only the observation of the final state lepton . accelerator neutrino experiments like t2k@xcite , nova@xcite and a number of proposed experiments seek to make precision measurements of the neutrino flavor oscillations @xmath0 or @xmath7 in order to determine the mass hierarchy of neutrinos and to search for cp violation in neutrino oscillations . uncertainties on differences between these cross - sections for muon and electron neutrinos will contribute to experimental uncertainties in these flavor oscillation measurements . interactions of the charged - current with fundamental fermions , such as @xmath8 , have no uncertainties in the differences between the reactions for muon and electron neutrino interactions because the weak interaction is experimentally observed to be flavor universal . in particular , the effect of the final state lepton mass in this two body reaction of fundamental fermions can be unambiguously calculated . one such calculable difference occurs because of radiative corrections to the tree - level ccqe process . radiative corrections from a particle of mass @xmath9 in a process with momentum transfer @xmath10 are of order @xmath11 , which implies a significant difference due to the mass of the final state lepton@xcite . although this effect is calculable , it is not accounted for in neutrino interaction generators used in recent analysis of experimental data , such as genie@xcite , neut@xcite and nuance@xcite . there are , however , cross - section differences due to lepton mass which can not be calculated from first principles with current theoretical tools . the presence of the target quarks inside a strongly bound nucleon lead to a series of _ a priori _ unknown form factors in the nucleon level description of the reaction , e.g. , @xmath1 . it is the uncertainties on these form factors combined with the alteration of the kinematics due to lepton mass that leads to uncertainties , and that is the focus of the results of this paper . there is also a modification of the reaction cross - sections due to the effects of the nucleus in which the target nucleons are bound . the model incorporated in genie@xcite , neut@xcite and nuance@xcite is a relativistic fermi gas model@xcite which provides a distribution of nucleon kinematics inside the nucleus . a more sophisticated description from a nuclear spectral function model@xcite is implemented in the nuwro generator@xcite . we do not consider the effect of the nucleus in this work , although it may be important in the relative weighting of nucleon kinematics at low energy . however , this work provides a blueprint for studying the effect of the final state lepton mass in different nuclear models . the cross section for quasi - elastic scattering of neutrinos at energies relevant for oscillation experiments may be calculated from the fermi theory of weak interactions with the effective lagrangian , @xmath12 where @xmath13 is the fermi constant and the @xmath14 are the leptonic and hadronic currents . the form of the leptonic current is specified by the theory to be @xmath15 because the leptons are fundamental fermions . however , as mentioned above the hadronic current for quasi - elastic scattering depends on unknown form factors of the nucleons . the hadronic current can be decomposed into vector and axial components , @xmath16 @xmath17 contains three terms related to the vector form factors @xmath18 , @xmath19 and @xmath20 , and @xmath21 contains three terms related to the axial form factors @xmath22 , @xmath23 and @xmath24 . a description the the bilinear covariant structure of the currents is given in several standard texts and review papers@xcite . we follow most closely the notation of ref . . from the effective lagrangian of eq [ eq : efflagranian ] and currents above in eqs . [ eq : leptoniccurrent ] and [ eq : hadroniccurrent ] , the quasi - elastic cross section on free nucleons is : @xmath25 \nonumber\\ & & \times\frac{m^2 g_f^2 \cos^2 \theta_{c}}{8 \pi e_{\nu}^2}\end{aligned}\ ] ] where the invariant @xmath26 and @xmath27 is the four momentum transfer from the leptonic to hadronic system , @xmath28 is the mass of the nucleon , @xmath29 is the cabibbo angle , and @xmath30 is the neutrino energy in the lab . the combination of mandelstam invariants @xmath31 and @xmath32 can be written as , @xmath33 where @xmath9 is the mass of the final state lepton . the functions a(@xmath34 ) , b(@xmath34 ) and c(@xmath34 ) depend on the nucleon form factors and @xmath35 , the difference between the anomalous magnetic moment of the proton and the neutron : @xmath36 , \\ % \end{split } % \end{equation } % \begin{equation } \label{eq : bfunc } % \begin{split } b(q^2 ) % & & = & \frac{q^2}{m^2 } re f_a^ * \left ( f_v^1 + \xi f_v^2\right ) % & - \frac{m^2}{m^2 } re \left [ \left ( f_v^1 -\frac{q^2}{4 m^2 } \xi f_v^2 \right ) ^ * f_v^3 \right . % & \left . - \left ( f_a - \frac{q^2 f_p}{2 m^2 } \right)^ * f_a^3 \right ] { \rm\textstyle and}\\ % \end{split } % \end{equation } % \begin{equation } \label{eq : cfunc } c(q^2 ) & = & \frac{1}{4 } \left ( \vert f_a \vert ^2 + \vert f_v^1 \vert ^2 + \frac{q^2}{m^2 } \left| \frac{\xi f_v^2}{2 } \right| ^2 + \frac{q^2}{m^2 } \vert f_a^3 \vert ^2 \right ) . % \end{equation}\end{aligned}\ ] ] note that the form factors themselves are functions of @xmath34 in eqs . [ eq : afunc][eq : cfunc ] . @xmath18 and @xmath19 are the vector and @xmath22 and @xmath37 the axial form factors of the first class currents . first class currents conserve both time and charge symmetry . in addition , first class vector currents commute with the g - parity operator while first class axial currents anti - commute with it@xcite . the terms associated with @xmath18 and @xmath22 are considered the leading terms in the hadron current since they have no dependence on the four - momentum transfer , excepting that of the form factors , and they are not suppressed by powers of the final state lepton mass as @xmath37 is . vector elastic form factors are precisely known at @xmath38 from the nucleon electric charges and magnetic moments and are precisely measured over a wide range of @xmath34 in charged - lepton elastic scattering from protons and deuterium . the axial nucleon form factor at zero @xmath34 is precisely measured in studies of neutron beta decay . at higher @xmath34 , much of our knowledge of the axial form factors comes from muon neutrino quasi - elastic scattering measurements . for @xmath39 ( gev / c)@xmath40 , the vector form factors and the axial form factors are observed to follow a dipole form , that is @xmath41 where the constant @xmath42 is often expressed as a mass of the same order of magnitude as the mass of the nucleon . at high @xmath34 the vector form factors do not follow the dipole structure@xcite . the neutrino scattering data contains conflicting results among different experiments@xcite which limit our ability to effectively use that information to constrain the axial form factor . pion electroproduction experiments@xcite have also measured the axial form factor at @xmath43 0.2 ( gev / c)@xmath40 . the form factor @xmath37 is determined from pcac which , under minimal assumptions , states that@xcite : @xmath44 where @xmath45 is the renormalized field operator that creates the @xmath46 and @xmath42 is a constant which may be computed at @xmath38 . pcac gives the following relation between @xmath37 and the pion nucleon form factor , @xmath47 , @xmath48 where @xmath49 is the pion mass . the goldberger - treiman relation@xcite predicts @xmath50 where @xmath51 is the pion decay constant . under the assumption that the goldberger - treiman relation holds for all values of @xmath34 , then @xmath37 is @xmath52 this is the relationship that is used in all modern neutrino generators@xcite . @xmath20 and @xmath23 are form factors associated with the second class current ( scc ) . the existence of such currents requires charge or time symmetry violation , and current measurements show the size of these violations to be small . additionally a nonzero @xmath20 term would violate conservation of the vector current ( cvc ) . both @xmath53 and @xmath54 can be limited experimentally in studies of beta decay . almost all current analyses of neutrino data assume that the sccs are zero . the vector sccs only enter the cross - section in terms suppressed by @xmath55 , but there are unsuppressed terms involving the axial scc form factor . in this section , we will study the dependence of the muon - electron cross - section differences as a function of @xmath56 and @xmath34 . differences arise due to the variation of kinematic limits due to the final state lepton mass , different radiative corrections to the tree level process and uncertainties in nucleon form factors . equations [ eq : afunc ] and [ eq : bfunc ] contain explicitly the dependence of the ccqe cross - section in terms of the form factors . lepton mass , @xmath9 , enters in both @xmath57 and @xmath58 and these terms involve all the form factors discussed above . note that @xmath37 and @xmath20 _ only _ appear in terms multiplied by @xmath55 and therefore are negligible in the electron neutrino cross - section , but not in the muon neutrino cross - section . as metrics , we define the fractional differences between the muon and electron neutrino ccqe cross - sections : @xmath59 the integrals over @xmath34 in eqs . [ eq : diff ] and [ eq : intdiff ] are taken within the kinematic limits of each process , and those limits depend on lepton mass as discussed in the next section . another useful metric is the difference between a cross - section in a model with a varied assumption from that of a reference model . our reference model derives @xmath60 and @xmath61 from the electric and magnetic vector sachs form factors which follow the dipole form of eq . [ eq : dipole ] with @xmath62 ( gev / c)@xmath40 , and it assumes @xmath22 is a dipole with @xmath63 ( gev / c)@xmath40 . the reference model uses the derived @xmath37 from eq . [ eqn : fp ] , and assumes that @xmath64 at all @xmath34 . we then define : @xmath65 where @xmath66 is the reference model for @xmath67 or its anti - neutrino analogue and @xmath68 is the model to be compared to the reference . large differences between the electron and muon neutrino quasi - elastic cross - sections exist at low neutrino energies from the presence of different kinematic limits due to the final state lepton mass and due to the presence of the pseudoscalar form factor , @xmath37 , derived from pcac and the goldberger - treiman relation . these differences are typically accounted for in modern neutrino interaction generators . there are also significant differences due to radiative corrections , particularly in diagrams that involve photon radiation attached to the outgoing lepton leg which are proportional to @xmath69 . these differences are calculable , but are typically not included in neutrino interaction generators employed by neutrino oscillation experiments . if our estimate of these differences , of order @xmath70 , is confirmed by more complete analyses , then this is a correction that needs to be included as it is comparable to the size of current systematic uncertainties at accelerator experiments@xcite . modifications of the assumed @xmath37 from pcac and the goldberger - treiman relation and the effect of the form factors @xmath20 and @xmath23 corresponding to second class vector and axial currents , respectively , are not included in neutrino interaction generators . a summary of the possible size of these effects , as we have estimated them , is shown in fig . [ fig : summary ] . these differences , particularly from the second class vector currents , may be significant for current@xcite and future@xcite neutrino oscillation experiments which seek precision measurements of @xmath71 and its anti - neutrino counterpart at low neutrino energies . previous work@xcite has demonstrated sensitivity to these second class currents in neutrino and anti - neutrino quasi - elastic muon neutrino scattering , and future work with more recent data@xcite and newly analyzed data@xcite may help to further limit uncertainties on possible second class currents . the suggestion for this work came out of conversations with alain blondel about systematics in future oscillation experiments and we thank him for inspiring this work . we are grateful to ashok das , tamar friedmann and tom mcelmurry for their clear and patient explanations of the bilinear covariant structure of weak interactions . we thank arie bodek for a helpful discussion of available tests of the cvc hypothesis . we thank gabriel perdue and geralyn zeller for helpful comments on a draft of this manuscript . we are grateful to bill marciano for his helpful insights into the radiative corrections after the initial draft of this paper appeared online . a. de rujula , r. petronzio and a. savoy - navarro , nucl . b * 154 * , 394 ( 1979 ) . c. andreopoulos [ genie collaboration ] , acta phys . b * 40 * , 2461 ( 2009 ) . y. hayato , nucl . suppl . * 112 * , 171 ( 2002 ) . y. hayato , acta phys . polon . b * 40 * , 2477 ( 2009 ) . d. casper , nucl . suppl . * 112 * , 161 ( 2002 ) . smith and e.j . moniz nucl . * b43 * 605 ( 1972 ) . o. benhar , a. fabrocini , s. fantoni and i. sick , nucl . a * 579 * , 493 ( 1994 ) . j. sobczyk , pos nufact * 08 * , 141 ( 2008 ) . r.e marshak , riazuddin and c.p . ryan , _ theory of weak interactions in particle physics _ , wiley - interscience ( 1969 ) . a. bodek , s. avvakumov , r. bradford and h. s. budd , j. phys . conf . ser . * 110 * , 082004 ( 2008 ) . v. lyubushkin _ et al . _ [ nomad collaboration ] , eur . j. c * 63 * , 355 ( 2009 ) . j. l. alcaraz - aunion _ et al . _ [ sciboone collaboration ] , aip conf . proc . * 1189 * , 145 ( 2009 ) . m. dorman [ minos collaboration ] , aip conf . * 1189 * , 133 ( 2009 ) . a. a. aguilar - arevalo _ et al . _ [ miniboone collaboration ] , phys . d * 81 * , 092005 ( 2010 ) . s. choi , v. estenne , g. bardin , n. de botton , g. fournier , p. a. m. guichon , c. marchand and j. marroncle _ et al . _ , phys . lett . * 71 * , 3927 ( 1993 ) . a. liesenfeld _ et al . _ [ a1 collaboration ] , phys . b * 468 * , 20 ( 1999 ) . stephen l. adler , phys . rev . * 137 * , 10221033 ( 1964 ) . m. l. goldberger and s. b.treiman , phys . rev . * 5 * , 1178 - 1184 ( 1958 ) . k. kubodera , j. delorme and m. rho , nucl . phys . * b66 * , 253 - 292 ( 1973 ) . m. oka and k. kubodera , phys . * b90 * 45 ( 1980 ) . k. minamisono _ et al . _ , phys . rev . * c65 * , 015501 ( 2001 ) . k. minamisono _ et al . _ , phys . * c84 * , 055501 ( 2011 ) . d.h . wilkinson , eur . j. * a7 * 307 ( 2000 ) .

accelerator neutrino oscillation experiments seek to make precision measurements of the neutrino flavor oscillations @xmath0 in order to determine the mass hierarchy of neutrinos and to search for cp violation in neutrino oscillations . these experiments are currently performed with beams of muon neutrinos at energies near 1 gev where the charged - current quasi - elastic interactions @xmath1 and @xmath2 dominate the signal reactions . we examine the difference between the quasi - elastic cross - sections for muon and electron neutrinos and anti - neutrinos and estimate the uncertainties on these differences .

graphical models @xcite are a class of statistical models which combine the rigour of a probabilistic approach with the intuitive representation of relationships given by graphs . they are composed by a set @xmath15 of _ random variables _ describing the data @xmath4 and a _ graph _ @xmath16 in which each _ vertex _ or _ node _ @xmath17 is associated with one of the random variables in @xmath18 . nodes and the corresponding variables are usually referred to interchangeably . the _ edges _ @xmath19 are used to express the dependence relationships among the variables in @xmath18 . different classes of graphs express these relationships with different semantics , having in common the principle that graphical separation of two vertices implies the conditional independence of the corresponding random variables @xcite . the two examples most commonly found in literature are _ markov networks _ @xcite , which use undirected graphs ( ugs , see * ? ? ? * ) , and _ bayesian networks _ @xcite , which use directed acyclic graphs ( dags , see * ? ? ? * ) . in the context of bayesian networks , edges are often called _ arcs _ and denoted with @xmath20 ; we will adopt this notation as well . the structure of @xmath0 ( that is , the pattern of the nodes and the edges ) determines the probabilistic properties of a graphical model . the most important , and the most used , is the factorisation of the _ global distribution _ ( the joint distribution of @xmath18 ) into a set of lower - dimensional _ local distributions_. in markov networks , local distributions are associated with _ cliques _ ( maximal subsets of nodes in which each element is adjacent to all the others ) ; in bayesian networks , each local distribution is associated with one node conditional on its _ parents _ ( nodes linked by an incoming arc ) . in markov networks the factorisation is unique ; different graph structures correspond to different probability distributions . this is not so in bayesian networks , where dags can be grouped into _ equivalence classes _ which are statistically indistinguishable . each such class is uniquely identified by the underlying ug ( i.e. in which arc directions are disregarded , also known as _ skeleton _ ) and by the set of _ v - structures _ ( i.e. converging connections of the form @xmath21 , @xmath22 , in which @xmath23 and @xmath24 are not connected by an arc ) common to all elements of the class . as for the global and the local distributions , there are many possible choices depending on the nature of the data and the aims of the analysis . however , literature have focused mostly on two cases : the _ discrete case _ @xcite , in which both the global and the local distributions are multinomial random variables , and the _ continuous case _ @xcite , in which the global distribution is multivariate normal and the local distributions are univariate ( in bayesian networks ) or multivariate ( in markov networks ) normal random variables . in the former , the parameters of interest @xmath1 are the _ conditional probabilities _ associated with each variable , usually represented as conditional probability tables . in the latter , the parameters of interest @xmath1 are the _ partial correlation coefficients _ between each variable and its neighbours in @xmath0 . conjugate distributions ( dirichlet and wishart , respectively ) are then used for learning and inference in a bayesian setting . the choice of an appropriate probability distribution for the set @xmath11 of the possible edges is crucial to make the derivation and the interpretation of the properties of @xmath11 and @xmath25 easier . we will first note that a graph is uniquely identified by its edge set @xmath26 ( or by its arc set @xmath27 for a dag ) , and that each edge @xmath28 or arc @xmath29 is uniquely identified by the nodes @xmath23 and @xmath30 , @xmath31 it is incident on . therefore , if we model @xmath11 with a random variable we have that any edge set @xmath26 ( or arc set @xmath27 ) is just an element of its sample space ; and since there is a one - to - one correspondence between graphs and edge sets , probabilistic properties and inferential results derived for traditional graph - centric approaches can easily be adapted to this new edge - centric approach and vice versa . in addition , if we denote @xmath32 , we can clearly see that @xmath33 . on the other hand , @xmath34 for ugs and even larger for dags @xcite and their equivalence classes @xcite . we will also note that an edge or an arc has only few possible states : * an edge can be either present ( @xmath35 ) or missing from an ug ( @xmath36 ) ; * in a dag , an arc can be present in one of its two possible directions ( @xmath37 or @xmath38 ) or missing from the graph ( @xmath39 and @xmath40 ) . this leads naturally to the choice of a bernoulli random variable for the former , @xmath41 and to the choice of a trinomial random variable for the latter , @xmath42 where @xmath43 is the arc @xmath44 and @xmath45 is the arc @xmath46 . therefore , a graph structure can be modelled through its edge or arc set as follows : * ugs , such as markov networks or the skeleton and the moral graph of bayesian networks @xcite , can be modelled by a _ multivariate bernoulli random variable _ ; * directed graphs , such as the dags used in bayesian networks , can be modelled by a _ multivariate trinomial random variable_. in addition to being the natural choice for the respective classes of graphs , these distributions integrate smoothly with and extend other approaches present in literature . for example , the probabilities associated with each edge or arc correspond to the _ confidence coefficients _ from @xcite and the _ arc strengths _ from @xcite . in a frequentist setting , they have been estimated using bootstrap resampling @xcite ; in a bayesian setting , markov chain monte carlo ( mcmc ) approaches @xcite have been used instead . let @xmath47 , @xmath48 be bernoulli random variables with marginal probabilities of success @xmath49 , that is @xmath50 , @xmath51 . then the distribution of the random vector @xmath52^t$ ] over the joint probability space of @xmath47 is a _ multivariate bernoulli random variable _ @xcite , denoted as @xmath53 . its probability function is uniquely identified by the parameter collection @xmath54 which represents the _ dependence structure _ among the @xmath55 in terms of simultaneous successes for every non - empty subset @xmath56 of elements of @xmath57 . other characterisations and fundamental properties of the multivariate bernoulli distribution can be found in @xcite . @xcite focus on the bivariate models specific to @xmath58 . additional characterisations and results specific to particular applications can be found in ( * ? ? ? * variable selection ) , ( * ? ? ? * longitudinal studies ) , ( * ? ? ? * combinatorial optimisation ) and ( * ? ? ? * clinical trials ) , among others . from literature we know that the expectation and the covariance matrix of @xmath57 are immediate extensions of the corresponding univariate bernoulli ones ; @xmath59^t & & \text{and } & & { \mathsf{cov}}(\mathbf{b } ) = [ \sigma_{ij } ] = p_{ij } - p_{i}p_{j}.\end{aligned}\ ] ] in particular , the covariance matrix @xmath60 $ ] has some interesting numerical properties . from basic probability theory , we know its diagonal elements @xmath61 are bounded in the interval @xmath62 $ ] ; the maximum is attained for @xmath63 , and the minimum for both @xmath64 and @xmath65 . for the cauchy - schwarz theorem then @xmath66 $ ] . as a result , we can derive similar bounds for the eigenvalues @xmath67 of @xmath68 , as shown in the following theorem . [ thm : mvebereigen ] let @xmath69 , and let @xmath68 be its covariance matrix . let @xmath70 , @xmath51 be the eigenvalues of @xmath68 . then @xmath71 see appendix [ app : proofs ] . these bounds define a closed convex set in @xmath72 , described by the family @xmath73\right\}\ ] ] where @xmath74 is the non - standard @xmath75 simplex @xmath76 construction and properties of the multivariate trinomial random variable are similar to the ones illustrated in the previous section for the multivariate bernoulli . for this reason , and because it is a particular case of the multivariate multinomial distribution , the multivariate trinomial distribution is rarely the focus of research efforts in literature . some of its fundamental properties are covered either in @xcite or in monographs on contingency tables analysis such as @xcite . let @xmath77 , @xmath48 be trinomial random variables assuming values @xmath78 and denoted as @xmath79 with @xmath80 . then the distribution of the random vector @xmath81^t$ ] over the joint probability space of @xmath77 is a _ multivariate trinomial random variable _ , denoted as @xmath82 . the parameter collection @xmath83 which uniquely identifies the distribution is @xmath84 and the reduced parameter collection we will need to study its first and second order moments is @xmath85 from the definition , we can easily derive the expected value and the variance of @xmath86 , @xmath87 ^ 2 \end{aligned}\ ] ] and the covariance between two variables @xmath86 and @xmath88 , @xmath89 + \left[\ , p_{ij(-1,-1 ) } - p_{i(-1)}p_{j(-1 ) } \,\right ] - \notag \\ & \qquad - \left[\ , p_{ij(-1,1 ) } - p_{i(-1)}p_{j(1 ) } \,\right ] - \left[\ , p_{ij(1,-1 ) } - p_{i(1)}p_{j(-1 ) } \,\right].\end{aligned}\ ] ] again , the diagonal elements of the covariance matrix @xmath68 are bounded . this can be proved either by solving the constrained maximisation problem @xmath90 or as an application of the following theorem by @xcite . if a discrete random variable @xmath91 can take values only in the segment @xmath92 $ ] of the real axis , the maximum standard deviation of @xmath91 equals @xmath93 . the maximum is reached if @xmath91 takes the values @xmath94 and @xmath95 with probabilities @xmath96 each . see @xcite . in both cases we obtain that the maximum variance is achieved for @xmath97 and is equal to @xmath98 , so @xmath99 $ ] and @xmath100 $ ] . furthermore , we can also prove that the eigenvalues of @xmath68 are bounded using the same arguments as in lemma [ thm : mvebereigen ] . [ thm : dirlambda ] let @xmath101 , and let @xmath68 be its covariance matrix . let @xmath70 , @xmath51 be the eigenvalues of @xmath68 . then @xmath102 see the proof of lemma [ thm : mvebereigen ] in appendix [ app : proofs ] . these bounds define again a closed convex set in @xmath72 , described by the family @xmath103\right\},\ ] ] where @xmath74 is the non - standard @xmath75 simplex from equation [ eq : simplex ] . another useful result , which we will use in section [ sec : dagprop ] to link inference on ugs and dags , is introduced below . [ thm : triber2 ] let @xmath101 ; then @xmath104 and + @xmath105 . see appendix [ app : proofs ] . it follows that the variance of each @xmath86 can be decomposed in two parts : @xmath106 the first is a function of the corresponding component @xmath107 of the transformed random vector , while the second depends only on the probabilities associated with @xmath108 and @xmath98 ( which correspond to @xmath45 and @xmath43 in equation [ eqn : tridef ] ) . the results derived in the previous section provide the foundation for characterising @xmath13 and @xmath14 . to this end , it is useful to distinguish three cases corresponding to different configurations of the probability mass among the graph structures @xmath109 : * _ minimum entropy _ : the probability mass is concentrated on a single graph structure . this is the best possible configuration for @xmath14 , because only one edge set @xmath26 ( or one arc set @xmath27 ) has a non - zero posterior probability . in other words , the data @xmath4 provide enough information to identify a single graph @xmath0 with posterior probability @xmath98 ; * _ intermediate entropy _ : several graph structures have non - zero probabilities . this is the case for informative priors @xmath13 and for the posteriors @xmath14 resulting from real - world data sets ; * _ maximum entropy _ : all graph structures in @xmath9 have the same probability . this is the worst possible configuration for @xmath14 , because it corresponds to the non - informative prior from equation [ eqn : flatprior ] . in other words , the data @xmath4 do not provide any information useful in identifying a high - posterior graph @xmath0 . clearly , _ minimum _ and _ maximum entropy _ are limiting cases for @xmath14 ; the former is non - informative about @xmath110 , while the latter identifies a single graph in @xmath9 . as we will show in sections [ sec : ugprop ] ( for ugs ) and [ sec : dagprop ] ( for dags ) , they provide useful reference points in determining which edges ( or arcs ) have significant posterior probabilities and in analysing the variability of the graph structure . in the _ minimum entropy _ case , only one configuration of edges @xmath26 has non - zero probability , which means that @xmath111 the uniform distribution over @xmath9 arising from the _ maximum entropy _ case has been studied extensively in random graph theory @xcite ; its two most relevant properties are that all edges @xmath28 are independent and have @xmath112 . as a result , @xmath113 ; all edges display their maximum possible variability , which along with the fact that they are independent makes this distribution non - informative for @xmath11 as well as @xmath25 . the _ intermediate entropy _ case displays a middle - ground behaviour between the _ minimum _ and _ maximum entropy _ cases . the expected value and the covariance matrix of @xmath11 do not have a definite form beyond the bounds derived in section [ sec : mvber ] . when considering posteriors arising from real - world data , we have in practice that most edges in @xmath11 represent conditional dependence relationships that are completely unsupported by the data . this behaviour has been explained by @xcite with the tendency of `` good '' graphical models to represent the causal relationships underlying the data , which are typically sparse . as a result , we have that @xmath114 and @xmath115 for many @xmath28 , so @xmath68 is almost surely singular unless such edges are excluded from the analysis . edges that appear with @xmath116 have about the same marginal probability and variance as in the _ maximum entropy _ case , so their marginal behaviour is very close to random noise . on the other hand , edges with probabilities near @xmath117 or @xmath98 can be considered to have a good support ( against or in favour , respectively ) . as @xmath118 approaches @xmath117 or @xmath98 , @xmath28 approaches its _ minimum entropy_. the closeness of a multivariate bernoulli distribution to the _ minimum _ and _ maximum entropy _ cases can be represented in an intuitive way by considering the eigenvalues @xmath119^t$ ] of its covariance matrix @xmath68 . recall that the @xmath120 can assume values in the convex set @xmath121 defined in equation [ eq : simplex ] , which corresponds to the region of the first orthant delimited by the non - standard simplex @xmath122 . in the _ minimum entropy _ case we have that @xmath123 , so @xmath124 , and in the _ maximum entropy case _ @xmath113 , so @xmath125 ; both points lie on the boundary of @xmath121 , the first in the origin and the second in the middle of @xmath122 . the distance between @xmath120 and these two points provides an intuitive way of measuring the variability of @xmath11 and , indirectly , the entropy of the corresponding probability distributions @xmath14 and @xmath13 . it is important to note , however , that different distributions over @xmath9 may have identical first and second order moments when modelled through @xmath11 . such distributions will have the same @xmath120 and will therefore map to the same point in @xmath121 . a simple example comprising three different distributions over a set of two edges is illustrated below . [ ex : base ] consider three multivariate bernoulli distributions @xmath126 , @xmath127 , @xmath128 over two edges ( denoted with @xmath129 and @xmath130 for brevity ) with covariance matrices @xmath131 and eigenvalues @xmath132 their positions in @xmath121 are shown in figure [ fig : base ] . @xmath126 is the closest to @xmath133 , the point corresponding to the maximum entropy case , while @xmath127 and @xmath128 are farther from @xmath133 than @xmath126 due to the increasing correlation between @xmath134 and @xmath135 ( which are independent in the _ maximum entropy _ case ) . the correlation coefficients for @xmath126 , @xmath127 and @xmath128 are @xmath136 , @xmath137 , @xmath138 , and they account for the increasing difference between the eigenvalues of each covariance matrix . in fact , @xmath139 is nearly singular because of the strong linear relationship between @xmath134 and @xmath135 , and it is therefore very close to one of the axes delimiting the first quadrant . , @xmath140 and @xmath139 from example [ ex : base ] represented as functions of their eigenvalues in the convex set @xmath121 . the points @xmath141 and @xmath142 correspond to the _ minimum entropy _ and _ maximum entropy _ cases . ] if we denote with @xmath143 , @xmath144 , @xmath145 , and @xmath146 all possible edge sets and with @xmath147 , @xmath148 , @xmath149 and @xmath150 the associated probabilities , for @xmath126 we have @xmath151 this is indeed close to a uniform distribution . the probability of both @xmath134 and @xmath135 is @xmath152 and the variance is @xmath153 , which are again similar to the reference values for the _ maximum entropy _ case . on the other hand , for @xmath127 we have @xmath154 these probabilities are markedly different from a uniform distribution ; the probabilities of @xmath134 and @xmath135 are respectively @xmath155 and @xmath156 . considering also the correlation between @xmath134 and @xmath135 , it is intuitively clear why @xmath140 is not as close as @xmath157 to @xmath133 . this is also true for @xmath128 , which has the same marginal distributions as @xmath127 but with a much stronger correlation . the behaviour of the multivariate trinomial distribution in the _ minimum _ and _ intermediate entropy _ cases is similar to the one of the multivariate bernoulli in many respects , but presents profound differences in the _ maximum entropy _ case . the reason for these differences is that the structure of a bayesian network is assumed to be acyclic . therefore , the state of each arc ( i.e. whether is present in the dag and its direction ) is influenced by the state of all other possible arcs even in the _ maximum entropy _ case , when otherwise they would be independent . furthermore , the acyclicity constraint can not be written in closed form , making the derivation of exact results on the moments of the distribution of @xmath11 particularly difficult . to obtain some simple expressions for the expected value and the covariance matrix , we will first prove a simple theorem on dags , which essentially states that if we reverse the direction of every arc the resulting graph is still a dag . [ thm : acyclic ] let @xmath158 be a dag , and let @xmath159 another directed graph such that @xmath160 for every @xmath161 . then @xmath162 is also acyclic . see appendix [ app : proofs ] . an immediate consequence of this theorem is that for every dag including the arc @xmath43 there exists another dag including the arc @xmath45 . since all dags have the same probability in the _ maximum entropy case _ , this implies that both directions of every arc have the same probability , @xmath163 then the expected value of each marginal trinomial distribution is equal to @xmath164 and its variance is equal to @xmath165 the joint probabilities associated with each pair of arcs also symmetric in the maximum entropy case , again due to theorem [ thm : acyclic ] . denote with @xmath166 the event that arc @xmath29 is not present in the dag . if we consider that both directions of every arc have the same probability and that there is no explicit ordering among the arcs , we have @xmath167 then the expression for the covariance simplifies to @xmath168,\ ] ] which can be interpreted as the difference in probability between a _ serial connection _ ( i.e. @xmath169 , if @xmath170 ) and a _ converging connection _ ( i.e. @xmath171 ) if the arcs are incident on a common node @xcite . this is interesting because v - structures are invariant within equivalence classes , while other patterns of arcs are not @xcite ; indeed , equivalence classes are usually represented as _ partially directed acyclic graphs _ ( pdags ) in which only arcs belonging to v - structures are directed . all other arcs , with the exclusion of those which could introduce additional v - structures ( known as _ compelled arcs _ ) , are replaced with the corresponding ( undirected ) edges . therefore , the combination of high values of @xmath172 and @xmath173 is indicative of the belief that the corresponding arcs are directed in the pdag identified by the equivalence class . along with with @xmath174 and @xmath175 , it is also indicative of the stability of the graph structure , both in the arcs and their directions . in an uninformative prior , such as the distribution we are now considering in the _ maximum entropy _ case , we expect all covariances to be small ; we will show this is the case in theorem [ thm : hoeffding ] . on the other hand , in an informative distribution such as the ones considered in the _ intermediate entropy _ case , we expect covariances to be closer to their upper bounds for arcs that are compelled or part of a converging connection , and closer to zero for arcs whose direction is not determined in the equivalence class . note that the sign of @xmath176 depends on the way the two possible directions of each arc are associated with @xmath98 and @xmath108 ; a simple way to obtain a consistent parameterisation is to follow the natural ordering of the variables ( i.e. if @xmath177 then the arc incident on these nodes is taken to be @xmath178 , @xmath43 is associated with @xmath98 and @xmath45 with @xmath108 ) . , @xmath179 , @xmath180 , @xmath181 , and @xmath182 nodes . the dotted line represents the limiting value in the number of nodes.,scaledwidth=90.0% ] to @xmath183 nodes . the dotted line represents the limiting value in the number of nodes.,scaledwidth=90.0% ] the equalities in equations [ eqn:1storder ] and [ eqn:2ndorder ] drastically reduce the number of free parameters in the _ maximum entropy _ case . the marginal distribution of each arc now depends only on @xmath184 , whose value can be derived from the following numerical approximation by @xcite . [ thm : idecozman ] the average number of arcs in a dag with @xmath185 nodes is approximately @xmath186 in the _ maximum entropy _ case . see @xcite . [ thm : corollary ] let @xmath158 be a dag with @xmath185 nodes . then for each possible arc @xmath187 we have that in the maximum entropy case @xmath188 see appendix [ app : proofs ] . the quality of this approximation is examined in figure [ fig : exactarcprob ] and figure [ fig : mtarcprob ] . in figure [ fig : exactarcprob ] , the values provided by theorem [ thm : corollary ] for dags with @xmath189 , @xmath179 , @xmath180 , @xmath181 and @xmath182 nodes are compared to the corresponding true values . the latter have been computed by enumerating all possible dags of that size ( i.e. the whole population ) and computing the relative frequency of each possible arc . in figure [ fig : mtarcprob ] , the values provided by theorem [ thm : corollary ] for dags with @xmath190 to @xmath183 nodes are compared with the corresponding estimated values computed over a set of @xmath191 dags of the same size . the latter have been generated with uniform probability using the algorithm from @xcite as implemented in the bnlearn package @xcite for r @xcite . we can clearly see that the approximate values are close to the corresponding true ( in figure [ fig : exactarcprob ] ) or estimated ( in figure [ fig : mtarcprob ] ) values for dags with at least @xmath181 nodes . this is not a significant limitation ; the true values can be easily computed via exhaustive enumeration for dags with @xmath189 , @xmath179 and @xmath180 nodes ( they are reported in appendix [ app : numbers ] , along with other relevant quantities ) . furthermore , it is evident both from theorem [ thm : corollary ] and from figures [ fig : exactarcprob ] and [ fig : mtarcprob ] that , as the number of nodes diverges , @xmath192 if we take the absolute value of this asymptotic trinomial distribution , the resulting random variable is @xmath193 with @xmath112 , which is the marginal distribution of an edge in an ug in the _ maximum entropy _ case . the absolute value transformation can be interpreted as ignoring the direction of the arc ; the events @xmath37 and @xmath38 collapse into @xmath35 , while @xmath194 maps to @xmath195 . as a result , the marginal distribution of an arc is remarkably similar to the one of the corresponding edge in an undirected graph for sufficiently large dags ; in both cases , the nodes @xmath23 and @xmath30 are linked with probability @xmath96 . no result similar to theorem [ thm : idecozman ] has been proved for arbitrary pairs of arcs in a directed acyclic graph ; therefore , the structure of the covariance matrix @xmath68 can be derived only in part . variances can be approximated using the approximate probabilities from theorem [ thm : corollary ] : @xmath196 to @xmath183 nodes . the dotted lines represent the respective limiting values.,scaledwidth=90.0% ] to @xmath183 nodes . the dotted lines represent the respective limiting values.,scaledwidth=90.0% ] therefore , maximum variance ( of each arc ) and maximum entropy ( of the graph structure ) are distinct , as opposed to what happens in ugs . however , we can use the decomposition of the variance introduced in equation [ eqn : vardecomp ] to motivate why the _ maximum entropy _ case is still a `` worst case '' outcome for @xmath14 . as we can see from figure [ fig : vardecomp ] , the contributions of the presence of an arc ( given by the transformation @xmath197 ) and its direction ( given by the @xmath198 term ) to the variance are asymptotically equal . this is a consequence of the limits in equation [ eqn : limits ] , which imply that an arc ( modulo its direction ) has the same probability to be present in or absent from the dag and that its directions also have the same probability . as a result , we are not able to make any decision about either the presence of the arc or its direction . on the contrary , when @xmath174 reaches it maximum at @xmath98 we have that @xmath199 and @xmath200 , so we are sure that the arc will be present in the dag in one of its two possible directions . as for the covariances , it is possible to obtain tight bounds using _ hoeffding s identity _ @xcite , @xmath201 and the decomposition of the joint distribution of dependent random variables provided by the _ farlie - morgenstern - gumbel _ ( fmg ) family of distributions @xcite , which has the form @xmath202 , & & |\varepsilon| \leqslant 1.\end{aligned}\ ] ] in equations [ eqn : hoeffding ] and [ eqn : fmg ] , @xmath203 , @xmath204 and @xmath205 denote the cumulative distribution functions of the joint and marginal distributions of @xmath91 and @xmath206 , respectively . [ thm : hoeffding ] let @xmath207 be a dag , and let @xmath29 , @xmath31 and @xmath208 , @xmath209 be two possible arcs . then in the _ maximum entropy _ case we have that @xmath210 ^ 2 \left[\frac{1}{4 } + \frac{1}{4(n - 1)}\right]^2\ ] ] and @xmath211 ^ 2 \left[\frac{1}{4 } + \frac{1}{4(n - 1)}\right].\ ] ] see appendix [ app : proofs ] . the bounds obtained from this theorem appear to be tight in the light of the true values for the covariance and correlation coefficients ( computed again by enumerating all possible dags of size @xmath189 to @xmath182 ) . figure [ fig : approxcor ] shows the bounds for dags with @xmath181 to @xmath183 nodes ; for dags with @xmath189 , @xmath179 and @xmath180 nodes the approximation of @xmath184 the bounds are based on is loose , and the true values of covariance and correlation are known . non - null covariances range from @xmath212 ( for dags with @xmath189 nodes ) to @xmath213 ( for dags with @xmath182 nodes ) , while non - null correlation coefficients vary from @xmath214 ( for dags with @xmath189 nodes ) to @xmath215 ( for dags with @xmath182 nodes ) . both covariance and correlation appear to be strictly increasing in modulus as the number of nodes increases , and converge to the limiting values of the bounds ( @xmath216 and @xmath217 , respectively ) from below . ( solid line ) and @xmath218 ( dashed line ) for dags with @xmath190 to @xmath183 nodes . the dotted line represents their asymptotic value.,scaledwidth=90.0% ] some other interesting properties are apparent from true values of the covariance coefficients reported in appendix [ app : numbers ] . they are reported below as conjectures because , while they describe a systematic behaviour that emerges from the dags whose sizes we have a complete enumeration for , we were not able to substantiate them with formal proofs . [ conj : unc ] arcs that are not incident on a common node are uncorrelated . this is a consequence of the fact that if we consider @xmath178 and @xmath219 with @xmath220 , we have @xmath221 . therefore @xmath222 . this property seems to generalise to dags with more than @xmath182 nodes . figure [ fig : approxp ] shows approximate estimates for @xmath223 and @xmath224 for dags with @xmath190 to @xmath183 nodes , obtained again from @xmath191 dags generated with uniform probability . the curves for the two probabilities are overlapping and very close to each other for all the considered dag sizes , thus supporting conjecture [ conj : unc ] . the covariance matrix @xmath68 is sparse . the proportion of arcs incident on a common node converges to zero as the number of nodes increases ; therefore , if we assume conjecture [ conj : unc ] is true , the proportion of elements of @xmath68 that are equal to @xmath117 has limit @xmath225 furthermore , even arcs that are incident on a common node are not strongly correlated . [ conj : increasing ] both covariance and correlation between two arcs incident on a common node are monotonically increasing in modulus . the covariance between two arcs incident on a common node takes values in the interval @xmath226 @xmath227 $ ] in modulus , while the correlation takes values in @xmath228 $ ] in modulus . these intervals can be further reduced to @xmath229 $ ] and @xmath230 @xmath231 $ ] for dags larger than @xmath182 nodes due to conjecture [ conj : increasing ] . as far as the other two cases are concerned , in the _ minimum entropy _ case we have that @xmath232 as in the _ minimum entropy _ case of ugs . the _ intermediate entropy _ case again ranges from being very close to the _ minimum entropy _ case ( when the graph structure displays little variability ) to being very close to the _ maximum entropy _ case ( when the graph structure displays substantial variability ) . the bounds on the eigenvalues of @xmath68 derived in lemma [ thm : dirlambda ] allow a graphical representation of the variability of the network structure , equivalent to the one illustrated in example [ ex : base ] for ugs . several functions have been proposed in literature as univariate measures of spread of a multivariate distribution , usually under the assumption of multivariate normality ; for some examples see @xcite and @xcite . three of them in particular can be used as descriptive statistics for the multivariate bernoulli and trinomial distributions : the _ generalised variance _ , @xmath233 the _ total variance _ , @xmath234 and the squared _ frobenius matrix norm _ of the difference between @xmath68 and a target matrix @xmath235 , @xmath236 both generalised variance and total variance associate high values of the statistic to unstable network structures , and are bounded due to the properties of the multivariate bernoulli and trinomial distributions . for total variance , it is easy to show that either @xmath237 $ ] ( for the multivariate bernoulli ) or @xmath238 $ ] ( for the multivariate trinomial ) , due to the bounds on the variances @xmath61 and on the eigenvalues @xmath70 derived in sections [ sec : mvber ] and [ sec : mvtri ] . generalised variance is similarly bounded due to hadamard s theorem on the determinant of a non - negative definite matrix @xcite : @xmath239 $ ] for the multivariate bernoulli distribution and @xmath240 $ ] for the multivariate trinomial . they reach the respective maxima in the _ maximum entropy _ case and are equal to zero only in the _ minimum entropy _ case . generalised variance is also strictly convex , but it is equal to zero when @xmath68 is rank deficient . for this reason it may be convenient to reduce @xmath68 to a smaller , full rank matrix ( say @xmath241 ) and consider @xmath242 instead of @xmath243 ; using a regularised estimator for @xmath68 such as the one presented in @xcite is also a viable option . the behaviour of the squared frobenius matrix norm , on the other hand , depends on the choice of the target matrix @xmath235 . for @xmath244 ( the covariance matrix arising from the _ minimum entropy _ case for both the multivariate bernoulli and the multivariate trinomial ) , @xmath245 associates high values of the statistic to unstable network structures , like @xmath246 and @xmath243 ; however , @xmath247 does not have a unique maximum and none of its maxima corresponds to the _ maximum entropy _ case , making its interpretation unclear . a better choice seems to be a multiple of the covariance matrix arising from the _ maximum entropy _ case , say @xmath248 , associating high values of @xmath249 to stable network structures . for the multivariate bernoulli , if we let @xmath250 , @xmath251 can be rewritten as @xmath252 it has both a unique global minimum ( because it is a convex function ) , @xmath253 and a unique global maximum , @xmath254 which correspond to the _ maximum _ and _ minimum entropy _ covariance matrices , respectively . similar results can be derived for the multivariate trinomial distribution , using an approximate estimate for @xmath255 based on the results presented in section [ sec : dagprop ] . all the descriptive statistics introduced in this section can be normalised as follows : @xmath256 these normalised statistics vary in the @xmath257 $ ] interval and associate high values to graphs whose structures display a high variability . since they vary on a known and bounded scale , they are easy to interpret as absolute quantities ( i.e. goodness - of - fit statistics ) as well as relative ones ( i.e. proportions of total possible variability ) . they also have a clear geometric interpretation as distances in @xmath121 , as they can all be rewritten as function of the eigenvalues @xmath67 . this allows , in turn , to provide an easy interpretation of otherwise complex properties of @xmath13 and @xmath14 and to derive new results . first of all , the measures introduced in equation [ eqn : normalised ] can be used to select the best learning algorithm @xmath258 in terms of structure stability for a given data set @xmath4 . different algorithms make use of the information present in the data in different ways , under different sets of assumptions and with varying degrees of robustness . therefore , in practice different algorithms learn different structures from the same data and , in turn , result in different posterior distributions on @xmath9 . if we rewrite equation [ eqn : structlearn ] to make this dependence explicit , @xmath259 and denote with @xmath260 the covariance matrix of the distribution of the edges ( or the arcs ) induced by @xmath261 , then we can choose the optimal structure learning algorithm @xmath262 as @xmath263 or , equivalently , using @xmath264 or @xmath265 instead of @xmath266 . such an algorithm has the desirable property of maximising the information gain from the data , as measured by the distance from the non - informative prior @xmath13 in @xmath121 . in other words , @xmath262 is the algorithm that uses the data in the most efficient way . furthermore , an optimal @xmath262 can be identified even for data sets without a `` golden standard '' graph structure to use for comparison ; this is not possible with the approaches commonly used in literature , which rely on variations of hamming distance @xcite and knowledge of such a `` golden standard '' to evaluate learning algorithms ( see , for example * ? ? ? * ) . similarly , it is possible to study the influence of different values of a tuning parameter for a given structure learning algorithm ( and again a given data set ) . such parameters include , for example , restrictions on the degrees of the nodes @xcite and regularisation coefficients @xcite . if we denote these tuning parameters with @xmath267 , we can again choose an optimal @xmath268 as @xmath269 another natural application of the variability measures presented in equation [ eqn : normalised ] is the study of the consistency of structure learning algorithms . it has been proved in literature that most of structure learning algorithms are increasingly able to identify a single , minimal graph structure as the sample size diverges ( see , for example * ? ? ? therefore , @xmath14 converges towards the _ minimum entropy _ case and all variability measures converge to zero . however , convergence speed has never been analysed and compared across different learning algorithms ; any one of @xmath266 , @xmath264 or @xmath265 provides a coherent way to perform such an analysis . lastly , we may use the variability measures from equation [ eqn : normalised ] as basis to investigate different prior distributions for real - world data modelling and to define new ones . relatively little attention has been paid in literature to the choice of the prior over @xmath9 , and the uniform _ maximum entropy _ distribution is usually chosen for computational reasons . its only parameter is the _ imaginary sample size _ , which expresses the weight assigned to the prior distribution as the size of an imaginary sample size supporting it @xcite . however , choosing a uniform prior also has some drawbacks . firstly , @xcite and @xcite have shown that both large and small values of the imaginary sample size have unintuitive effects on the sparsity of a bayesian network even for large sample sizes . for instance , large values of the imaginary sample size may favour the presence of an arc over its absence even when both @xmath13 and @xmath4 imply the variables the arc is incident on are conditionally independent . secondly , a uniform prior assigns a non - null probability to all possible models . therefore , it often results in a very flat posterior which is not able discriminate between networks that are well supported by the data and networks that are not @xcite . following @xcite s suggestion that `` good '' graphical models should be sparse , sparsity - inducing priors such as the ones in @xcite and @xcite should be preferred to the _ maximum entropy _ distribution , as should informative priors @xcite . for example , the prior proposed in @xcite introduces a prior probability @xmath270 to include ( independently ) each arc in a bayesian network with a given topological ordering , which means @xmath271 and @xmath272 for all @xmath273 in @xmath13 . thus , @xmath274 , @xmath275 and @xmath276 . the prior proposed in @xcite , on the other hand , controls the number of parents of each node for a given topological ordering . therefore , it favours low values of @xmath277 in @xmath13 and again @xmath272 for all @xmath273 . clearly , the amount of sparsity induced by the hyperparameters of these priors determines the variability of both the prior and the posterior , and can be controlled through the variability measures from equation [ eqn : normalised ] . furthermore , these measures can provide inspiration in devising new priors with the desired form and amount of sparsity . bayesian inference on the structure of graphical models is challenging in most situations due to the difficulties in defining and analysing prior and posterior distributions over the spaces of undirected or directed acyclic graphs . the dimension of these spaces grows super - exponentially in the number of variables considered in the model , making even map analyses problematic . in this paper , we propose an alternative approach to the analysis of graph structures which focuses on the set of possible edges @xmath11 of a graphical model @xmath278 instead of the possible graph structures themselves . the latter are uniquely identified by the respective edge sets ; therefore , the proposed approach integrates smoothly with and extends both frequentist and bayesian results present in literature . furthermore , this change in focus provides additional insights on the behaviour of individual edges ( which are usually the focus of inference ) and reduces the dimension of the sample space from super - exponential to quadratic in the number of variables . for many inference problems the parameter space is reduced as well , and makes complex inferential tasks feasible . as an example , we characterise several measures of structural variability for both bayesian and markov networks using the second order moments of @xmath13 and @xmath14 . these measures have several possible applications and are easy to interpret from both an algebraic and a geometric point of view . the author would like to thank to adriana brogini ( university of padova ) and david balding ( university college london ) for proofreading this article and providing many useful comments and suggestions . furthermore , the author would also like to thank giovanni andreatta and luigi salce ( university of padova ) for their assistance in the development of the material . since @xmath68 is a real , symmetric , non - negative definite matrix , its eigenvalues @xmath70 are non - negative real numbers ; this proves the lower bound in both inequalities . the upper bound in the first inequality holds because @xmath279 as the sum of the eigenvalues is equal to the trace of @xmath68 . this in turn implies @xmath280 which completes the proof . it is easy to show that each @xmath107 , with @xmath281 and @xmath282 . it follows that the parameter collection @xmath83 of @xmath283 reduces to @xmath284 after the transformation . therefore , @xmath285 is a uniquely identified multivariate bernoulli random variable according to the definition introduced at the beginning of section [ sec : mvber ] . let s assume by contradiction that @xmath162 is cyclic ; this implies that there are one or more nodes @xmath286 such that @xmath287 for some @xmath288 . however , this would mean that in @xmath289 we would have @xmath290 which is not possible since @xmath289 is assumed to be acyclic . each possible arc can appear in the graph in only one direction at a time , so a directed acyclic graph with @xmath185 nodes can have at most @xmath291 arcs . therefore @xmath292 but in the _ maximum entropy _ case we also have that @xmath293 , so @xmath294 which completes the proof . in the maximum entropy case , all arcs have the same marginal distribution function , @xmath295 \\ & \frac{1}{4 } + \frac{1}{4(n - 1 ) } & & \text{in } ( -1 , 0 ] \\ & \frac{3}{4 } - \frac{1}{4(n - 1 ) } & & \text{in } ( 0 , 1 ] \\ & 1 & & \text{in } ( 1 , + \infty ) \end{aligned } \right.,\ ] ] so the joint distribution of any pair of arcs @xmath29 and @xmath208 can be written as a member of the farlie - morgenstern - gumbel family of distribution as @xmath296 . \end{aligned}\ ] ] then if we apply hoeffding s identity from equation [ eqn : hoeffding ] and replace the joint distribution function @xmath297 with the right hand of equation [ eqn : this ] we have that @xmath298 - f_a(a_{ij})f_a(a_{kl } ) \right| \\ & = \sum_{\{-1 , 0\}}\sum_{\{-1 , 0\ } } ( 1 - f_a(a_{ij}))(1 - f_a(a_{kl } ) ) . \end{aligned}\ ] ] we can now compute the bounds for @xmath299 and @xmath300 using only the marginal distribution function @xmath301 from equation [ eqn : distrfun ] and the variance from equation [ eqn : approxvar ] , thus obtaining the expressions in equation [ eqn : boundcov ] and equation [ eqn : boundcor ] . below are reported the exact values of the parameters of the marginal trinomial distributions and of the first and second order moments of the multivariate trinomial distribution in the maximum entropy case . all these quantities have been computed by a complete enumeration of the directed acyclic graphs of a given size ( @xmath189 , @xmath179 , @xmath180 , @xmath181 and @xmath182 ) .

graphical model learning and inference are often performed using bayesian techniques . in particular , learning is usually performed in two separate steps . first , the graph structure is learned from the data ; then the parameters of the model are estimated conditional on that graph structure . while the probability distributions involved in this second step have been studied in depth , the ones used in the first step have not been explored in as much detail . in this paper , we will study the prior and posterior distributions defined over the space of the graph structures for the purpose of learning the structure of a graphical model . in particular , we will provide a characterisation of the behaviour of those distributions as a function of the possible edges of the graph . we will then use the properties resulting from this characterisation to define measures of structural variability for both bayesian and markov networks , and we will point out some of their possible applications . marco scutari + genetics institute , university college london , united kingdom + m.scutari@ucl.ac.uk graphical models @xcite stand out among other classes of statistical models because of their use of graph structures in modelling and performing inference on multivariate , high - dimensional data . the close relationship between their probabilistic properties and the topology of the underlying graphs represents one of their key features , as it allows an intuitive understanding of otherwise complex models . in a bayesian setting , this duality leads naturally to split model estimation ( which is usually called _ learning _ ) in two separate steps @xcite . in the first step , called _ structure learning _ , the graph structure @xmath0 of the model is estimated from the data . the presence ( absence ) of a particular edge between two nodes in @xmath0 implies the conditional ( in)dependence of the variables corresponding to such nodes . in the second step , called _ parameter learning _ , the parameters @xmath1 of the distribution assumed for the data are estimated conditional to the graph structure obtained in the first step . if we denote a graphical model with @xmath2 , so that @xmath3 , then we can write graphical model estimation from a data set @xmath4 as @xmath5 furthermore , following @xcite , we can rewrite structure learning as @xmath6 the prior distribution @xmath7 and the corresponding posterior distribution @xmath8 are defined over the space of the possible graph structures , say @xmath9 . since the dimension of @xmath9 grows super - exponentially with the number of nodes in the graph @xcite , it is common practice to choose @xmath10 as a non - informative prior , and then to search for the graph structure @xmath0 that maximises @xmath8 . unlike such a _ maximum a posteriori _ ( map ) approach , a full bayesian analysis is computationally unfeasible in most real - world settings @xcite . therefore , inference on most aspects of @xmath7 and @xmath8 is severely limited by the nature of the graph space . in this paper , we approach the analysis of those probability distributions from a different angle . we start from the consideration that , in a graphical model , the presence of particular edges and their layout are the most interesting features of the graph structure . therefore , investigating @xmath7 and @xmath8 through the probability distribution they induce over the set @xmath11 of their possible edges ( identified by the set of unordered pairs of nodes in @xmath0 ) provides a better basis from which to develop bayesian inference on @xmath0 . this can be achieved by modelling @xmath11 as a multivariate discrete distribution encoding the joint state of the edges . then , as far as inference on @xmath0 is concerned , we may rewrite equation [ eqn : structlearn ] as @xmath12 as a side effect , this shift in focus reduces the effective dimension of the sample space under consideration from super - exponential ( the dimension of @xmath9 ) to polynomial ( the dimension @xmath11 ) in the number of nodes . the dimension of the parameter space for many inferential tasks , such as the variability measures studied in this paper , is likewise reduced . the content of the paper is organised as follows . basic definitions and notations are introduced in section [ sec : definitions ] . the multivariate distributions used to model @xmath11 are described in section [ sec : distributions ] . some properties of the prior and posterior distributions on the graph space , @xmath13 and @xmath14 , are derived in section [ sec : properties ] . we will focus mainly on those properties related with the first and second order moments of the distribution of @xmath11 , and we will use them to characterise several measures of structural variability in section [ sec : variability ] . these measures may be useful for several inferential tasks for both bayesian and markov networks ; some will be sketched in section [ sec : variability ] . conclusions are summarised in section [ sec : conclusion ] , and proofs for the theorems in sections [ sec : distributions ] to [ sec : variability ] are reported in appendix [ app : proofs ] . appendix [ app : numbers ] lists the exact values for some quantities of interest for @xmath13 , computed for several graph sizes .

changes of the vector meson properties in strongly interacting matter at finite baryon density and temperature are presently of great interest , both theoretically and experimentally . in particular , the current heavy - ion experiments with the detector hades @xcite at the heavy - ion synchrotron sis18 ( gsi , darmstadt ) are mainly aimed at measuring in - medium modifications of light vector meson via the @xmath6 decay channel with high accuracy . one of the primary goals of the future experiments planned at sis100/200 is also to study very dense baryon matter and the expected strong changes of the in - medium hadrons . it is widely believed that the in - medium spectral change of the light mesons is related to the chiral symmetry restoration at finite temperature and baryon density . there are indeed various theoretical indications concerning an important sensitivity of the meson spectral density on the partial restoration of the chiral symmetry in a hot / dense nuclear medium . for instance , at finite temperature the vector and axial - vector meson correlators become mixed in accordance with in - medium weinberg sum rules @xcite . such a mixing causes an increasing degeneracy of vector and axial - vector spectral functions which would manifest themselves as a decrease of the @xmath0 and @xmath7 meson mass splitting . similarly , the degeneracy of scalar ( @xmath8 channel ) and pseudo - scalar ( @xmath9 channel ) correlators found in lattice qcd @xcite can lead to a considerable enhancement of the @xmath8 meson spectral function at finite temperature and density @xcite . in spite of substantial efforts undertaken to understand the nature of vector mesons in a dense medium there is so far no unique and widely accepted quantitative picture of their in - medium behavior . the brown and rho conjecture @xcite on the direct interlocking of vector meson masses and chiral quark condensate @xmath10 supplemented by the `` vector manifestation '' of chiral symmetry in medium @xcite predict a strong and quantitatively the same decrease of the in - medium @xmath0 and @xmath1 meson masses . at the same time , model calculations based on various effective lagrangians ( cf . @xcite ) predict rather moderate and different mass shifts for @xmath0 and @xmath1 mesons in a dense medium . in order `` to match '' both sets of predictions one has to go beyond simplifications made in the above mentioned approaches : the in - medium vector meson modification is governed not only by @xmath11 but also by condensates of higher order to be evaluated beyond mean - field approximation . further , effective lagrangians are dealing with the scattering amplitudes in free space , but effects related to the in - medium change of the qcd condensates should be included @xcite . the very consistent way to incorporate in - medium qcd condensates is through qcd sum rules ( qsr ) . the qsr for vector mesons in nuclear matter were first developed in @xcite , where within a simple parameterization of the spectral density in terms of a delta function at the resonance peak an agreement with the brown - rho scaling , i.e. the same dropping of the @xmath0 and @xmath1 meson masses , in nuclear matter was obtained . while the zero - width approximation for the resonance spectral density is successful in vacuum @xcite , such an approximation is not well grounded for the in - medium mesons which can undergo rather strong inelastic scatterings off the surrounding nucleons . for realistic in - medium qsr evaluations one needs to take into account the finite meson widths including collision broadening effects . the important impact of the finite width was studied , e.g. , in @xcite using a plausible ansatz for the in - medium spectral density . as shown in this qsr analysis , there is no inevitable necessity for in - medium dropping of the vector meson masses , but the global changes of mesons like mass shift and width broadening turn out to be correlated in nuclear matter . to avoid too many unknown parameters in the qsr equation and to make more definite predictions one has to specify in a detailed manner the ansatz for the hadron spectral density . as we show below such a specification for @xmath0 and @xmath1 vector mesons can be done basing on an effective lagrangian approach which gives a realistic behavior of the @xmath4 and @xmath5 scattering amplitudes . as well known , qsr in nuclear matter contain also an uncertainty related to the poorly known density dependence of the four - quark condensate . the majority of the qsr evaluations employs mean - field approximations for the in - medium 4-quark condensate , i.e. its density dependence is simply governed by the chiral condensate squared . at the same time , as pointed out in @xcite the in - medium mass shift of the @xmath0 and @xmath1 mesons is dominated by the dependence of the 4-quark condensate on density . in particular , the sign of the @xmath1 meson mass shift is changed by the variation of the strength of the density dependence of the 4-quark condensate beyond mean - field approximation . this result was confirmed in @xcite , where the @xmath1 meson spectral density was constrained within a general form of the in - medium @xmath1 meson propagator including collision broadening via the imaginary part of the @xmath5 scattering amplitude delivered by an effective chiral lagrangian @xcite . a direct observation of the @xmath1 meson spectral change via the e@xmath12 decay channel appears to be an experimental challenge in heavy - ion collisions at sis18 energies . both transport code simulations @xcite and a hydrodynamical model approach @xcite point to a considerable contribution of the reaction @xmath13 into dilepton spectra in the wanted region . a chance to separate e@xmath12 pairs from in - medium @xmath0 and @xmath1 mesons crucially depends on the quantitative details of their mass shift and width broadening in nuclear matter . this gives rise to a strong request from the experimental side to find out the @xmath0 and @xmath1 meson in - medium spectral changes simultaneously on a unique basis including self - consistently effects of the qcd condensates and collision broadening in nuclear matter . in the present paper we study systematically within the borel qsr the important role of the 4-quark condensate for spectral modifications of the @xmath0 and @xmath1 mesons in baryon matter . being still within the low - density expansion we go beyond the mean - field approximation and vary the strength of the density dependence of the 4-quark condensate . concerning the in - medium meson spectral density entering the hadronic part of the qsr evaluation we use a constraint motivated by the general structure of the vector meson propagator with finite in - medium width of @xmath0 and @xmath1 mesons reflecting the scattering of vector mesons off nucleons the in nuclear medium . seeking realistic @xmath4 and @xmath5 scattering amplitudes we employ the results of the recent covariant unitarized coupled channel approach @xcite which satisfactorily describes the experimental pion- and photon - nucleon scattering data . we find that in - medium modifications of the @xmath0 and @xmath1 mesons are indeed dominated by the dependence of the 4-quark condensate on density . in particular , the numerical value of a parameter , which describes the strength of the linear density dependence of the 4-quark condensate , governs the decrease of the @xmath0 meson mass as a function of density . for the @xmath1 meson the sign of the in - medium mass shift is changed by variations of this parameter . since the difference of the vector and axial - vector correlators is proportional to the 4-quark condensate the sign of the vector meson mass shift , measured via the @xmath14 channel , can serve as a tool for determining how fast nuclear matter approaches the chiral symmetry restoration with increasing baryon density . our paper is organized as follows . in section ii we recapitulate the necessary equations and formulate the borel qcd sum rule . the systematic evaluation of this sum rule is presented in section iii for @xmath0 and @xmath1 mesons . as supplement , we consider in section iv the case of the @xmath2 meson . the summary and a discussion can be found in section v. appendices a and b summarize the vacuum @xmath0 self - energy and the @xmath15 meson - nucleon scattering amplitudes , respectively . in appendix c we report on some technical details . for the sake of self - containment we list here the relevant equations for the borel qcd sum rule which our evaluations are based on . within qcd sum rules the in - medium vector mesons @xmath16 are considered as resonances in the current - current correlation function _ ( q , n ) = i d^4 x ^i q x j_(x ) j_(0)_n , [ eq_5 ] where @xmath17 is the meson four momentum , @xmath18 denotes the time ordered product of the respective meson current operators @xmath19 , and @xmath20 stands for the expectation value in medium . in what follows , we focus on the ground state of low - density baryon matter approximated by a fermi gas with nucleon density @xmath21 . we consider isospin symmetric nuclear matter , where the @xmath22 mixing effect is negligible @xcite . in terms of quark field operators , the vector meson currents are given by @xmath23 , where the negative ( positive ) sign is for the @xmath0 ( @xmath1 ) meson . the correlator ( [ eq_5 ] ) can be reduced to @xmath24 for a vector meson at rest , @xmath25 , in the rest frame of matter . in each of the vector meson channels the corresponding correlator @xmath26 satisfies the twice subtracted dispersion relation , which can be written with @xmath27 as = - ^(v ) ( 0 ) - q^2 _ 0^ ds , [ eq_10 ] with @xmath28 and @xmath29 as subtraction constants , and @xmath30 . as usual in qcd sum rules @xcite , for large values of @xmath31 one can evaluate the r.h.s . of eq . ( [ eq_5 ] ) by the operator product expansion ( ope ) leading to = - c_0(q^2 ) + _ i=1^ , [ eq_15 ] where the coefficients @xmath32 include the wilson coefficients and the expectation values of the corresponding products of the quark and gluon field operators , i.e. condensates . performing a borel transformation of the dispersion relation ( [ eq_10 ] ) with appropriate parameter @xmath33 and taking into account the ope ( [ eq_15 ] ) one gets the basic qsr equation ^(v ) ( 0,n ) + _ 0^ d s r^(v ) ( s ) e^-s / m^2 = c_0 m^2 + _ i=1^ . [ eq_20 ] the advantage of the borel transformation is ( i ) the exponential suppression of the high - energy part of @xmath34 , and ( ii ) the possibility to suppress higher - order contributions to the r.h.s . sum . choosing sufficiently large values of the internal technical parameter @xmath35 one can truncate the sum in controlled way , in practice at @xmath36 . the general structure of the coefficients @xmath32 up to @xmath37 is given , for instance , in @xcite . in order to calculate the density dependence of the condensates entering the coefficients @xmath32 we employ the standard linear density approximation , which is valid for not too large density . this gives for the chiral quark condensate @xmath38 , where we assume here isospin symmetry for the light quarks , i.e. @xmath39 mev and @xmath40 . the nucleon sigma term is @xmath41 mev . the gluon condensate is obtained as usual employing the qcd trace anomaly @xmath42 where @xmath43 is the qcd coupling constant and @xmath44 mev is the nucleon mass in the chiral limit . the vacuum gluon condensate is @xmath45 . the coefficient @xmath46 in eq . ( [ eq_20 ] ) contains also the mass dimension-6 4-quark condensates ( cf . @xcite for a recent calculation of corresponding matrix elements ) @xmath47 , @xmath48 , @xmath49 , and @xmath50 which are common for @xmath0 and @xmath1 mesons . on this level , @xmath0 and @xmath1 mesons differ only by the condensate @xmath51 ( cf . @xcite ) , causing the small @xmath22 mass splitting in vacuum @xcite . the standard approach to estimate the density dependence of the 4-quark condensates consists in the use of the mean - field approximation . within such an approximation the 4-quark condensates are proportional to @xmath52 and their density dependence is actually governed by the square of the chiral quark condensate . keeping in mind the important role of the 4-quark condensate for the in - medium modifications of the vector mesons , we go beyond this approximation and employ the following parameterization @xcite ( |q _ ^5 ^a q)^2 _ n = q_0 ^ 2 _ 0 . [ eq_35 ] in vacuum , @xmath53 , the parameter @xmath54 reflects a deviation from the vacuum saturation assumption . the case @xmath55 corresponds obviously to the exact vacuum saturation @xcite as used , for instance , in @xcite . to control the deviation of the in - medium 4-quark condensate from the mean - field approximation we introduce the parameter @xmath56 . the limit @xmath57 recovers the mean - field approximation , while the case @xmath58 ( @xmath59 ) is related to the stronger ( weaker ) density dependence compared to the mean - field approximation . an analog procedure applies for the other 4-quark condensates , each with its own @xmath54 and @xmath56 , which sum up to a parameter @xmath60 and a parameter @xmath3 . below we vary the poorly constrained parameter @xmath3 to estimate the contribution of the 4-quark condensates to the qsr with respect to the main trends of the in - medium modification of the vector meson spectral function . as seen in eq . ( [ eq_35 ] ) and eq . ( [ eq_40 ] ) below , @xmath3 parameterizes the density dependence of the summed 4-quark condensates ; @xmath60 is adjusted to the vacuum masses . strictly speaking , @xmath60 and @xmath3 differ for @xmath0 and @xmath1 mesons due to contributions of the above mentioned flavor - mixing condensate ; in addition , in medium a twist-4 condensate make further @xmath0 and @xmath1 to differ @xcite . however , the differences can be estimated to be sub - dominant . therefore , we use in the present work one parameter @xmath3 , keeping in mind that it may slightly differ for different light vector mesons . using the above condensates and usual wilson coefficients one gets as relevant terms for mass dimension @xmath61 and twist @xmath62 c_0 & = & ( 1 + ) , [ eq_2.6 ] + c_1 & = & - , + c_2 & = & m_q q_0 + n + + a_2 m_n n , + c_3 & = & - _ s _ 0 q_0 ^ 2 - a_4 m_n^3 n. [ eq_40 ] the last terms in @xmath63 correspond to the derivative condensates from non - scalar operators as a consequence of the breaking of lorentz invariance in the medium . these condensates are proportional to the moments @xmath64 $ ] of quark and anti - quark distributions inside the nucleon at scale @xmath65 ( see for details @xcite ) . our choice of the moments @xmath66 and @xmath67 is @xmath68 and @xmath69 , respectively . the value of @xmath60 in eq . ( [ eq_35 ] ) is related to such a choice of the chiral condensate @xmath70 to adjust the vacuum vector meson masses . in our qsr we have used @xmath71 , obtaining @xmath72 mev close to the nominal vacuum values . the ratio @xmath73 in the parameterization ( [ eq_35 ] ) is restricted by the condition @xmath74 , so that one gets @xmath75 as reasonable numerical limits when considering @xmath76 , as dictated by our low - density approximation . the case of finite baryon density temperature has been considered in @xcite . here we focus on density effects with the reasoning that temperature effects below 100 mev are negligible . to model the hadronic side of the qsr ( [ eq_20 ] ) we make the standard separation of the vector meson spectral density @xmath77 into resonance part and continuum contribution by means of the threshold parameter @xmath78 r^(v)(s , n)= f_v ( s_v - s ) + c_0 ( s - s_v ) , [ eq_45 ] where @xmath79 stands for the resonance peak in the spectral function ; the normalization @xmath80 is unimportant for the following consideration . in the majority of the previous qcd sum rule evaluations , the zero - width approximation @xcite or some parameterization of @xmath81 @xcite are employed . in contrast to this , we use here a more realistic ansatz for the resonance spectral density @xmath81 based on the general structure of the in - medium vector meson propagator s^(v ) ( s , n ) = - , [ eq_50 ] with @xmath82 and @xmath83 as real and imaginary part of the in - medium vector meson self - energy . an important point of our approach is that the meson mass parameter @xmath84 becomes density dependent in nuclear matter . this dependence is determined by the qcd sum rule eq . ( [ eq_20 ] ) and mainly governed by the qcd condensates . as a result ( see below ) the in - medium change of the qcd condensates causes global modifications of the vector meson spectral function , in addition to the collision broadening . ( an analogous approach was used in @xcite . ) the in - medium vector meson mass is determined by the pole position of the meson propagator m^2_v ( n ) = _ v^2 ( n ) + _ v ( s = m_v^2 ( n ) , n ) , [ eq_55 ] which looks similar to the vacuum case , where @xmath85 . the difference @xmath86 can be associated with the in - medium vector meson mass shift that is widely used to characterize the spectral change of mesons in matter . within the linear density approximation the vector meson self energy is given by _ v ( e , n ) = ^vac_v ( e ) - nt_v n ( e ) , [ eq_60 ] where @xmath87 is the meson energy , @xmath88 , and @xmath89 is the ( off - shell ) forward meson - nucleon scattering amplitude in free space . the renormalized quantity @xmath90 is summarized in the appendix a ; for the @xmath1 meson we absorb as usual @xmath91 in @xmath92 ( cf . @xcite for details ) and put @xmath93 with the vacuum values of mass @xmath94 and width @xmath95 . the described framework is well defined , supposed @xmath96 is reliably known . unfortunately , the determination of @xmath96 is hampered by uncertainties ( cf . results in @xcite and @xcite ) . @xmath97 is more directly accessible , while @xmath98 follows by a dispersion relation with sometimes poorly known subtraction coefficients . since our emphasis here is to include the collision broadening and finite width effects in the spectral function , we absorb in the following @xmath98 in @xmath99 thus neglecting a possible strong energy dependence . in such a way , the uncertainties of @xmath98 become milder since @xmath100 is then mainly determined by the qsr . we take the needed @xmath101 for @xmath0 and @xmath1 mesons from results of the detailed analysis of pion- and photon - nucleon scattering data performed recently in @xcite on the footing of the bethe - salpeter equation approach with four - point meson - baryon contact interactions and a unitary condition for the coupled channels . because of the presence of dynamically generated nucleon resonances , like the s - waves n@xmath102 , n@xmath103 and d - wave n@xmath104 resonances , the vector meson - nucleon scattering amplitudes obtained in @xcite exhibit rapid variations with energy ( see appendix b , figs . [ fig_b1 ] and [ fig_b2 ] ) . for the @xmath0n channel , the dominant contribution in @xmath105 comes from the resonances n@xmath102 and n@xmath104 . due to the rather moderate coupling of the @xmath4 channel to n@xmath104 , the value of the inelastic @xmath4 scattering amplitude is comparatively small and , therefore , the @xmath0 meson width is not significantly increased . at the same time , n@xmath104 is coupled strongly to the @xmath5 channel . this causes the pronounced peak in the subthreshold region of @xmath106 . such a peak like energy dependence differs even qualitatively from results of the chiral lagrangian approach @xcite . we do not advocate here a particular effective lagrangian approach for the vector meson - nucleon scattering amplitudes in vacuum . our aim is rather to demonstrate the impact of the qcd side , in particular of the in - medium 4-quark condensate , on the global vector meson spectral change in nuclear matter . for the subtraction constants @xmath107 in eq . ( [ eq_10 ] ) we use @xmath108 , @xmath109 , which are actually the thomson limit of the @xmath110 scattering processes , but also coincide with landau damping terms elaborated in @xcite for the hadronic spectral function entering the dispersion relation without subtractions . for details about the connection of subtraction constants and landau damping term we refer the interested reader to @xcite . taking the ratio of the eq . ( [ eq_20 ] ) to its derivative with respect to @xmath33 , and using ( [ eq_45 ] ) one gets @xmath111 - \frac{c_2}{m^2 } - \frac{c_3}{m^4}}{\displaystyle c_0\,\left(1-{\rm e}^{-s_v / m^2}\right ) + \frac{c_1}{m^2 } + \frac{c_2}{m^4 } + \frac{c_3}{2 m^6 } - \frac{\pi^{(v ) } ( 0,n)}{m^2 } } \label{eq_70}\end{aligned}\ ] ] with the coefficients @xmath112 from eqs . ( [ eq_2.6 ] @xmath113 [ eq_40 ] ) and the resonance spectral function @xmath79 from ( [ eq_50 ] ) . eq . ( [ eq_70 ] ) determines the mass parameter @xmath114 being here the subject of the qcd sum rule . before coming to the results we have to specify the numerical evaluation of the qcd sum rule ( [ eq_70 ] ) . at a given baryon density @xmath21 the continuum threshold @xmath78 is determined by requiring maximum flatness of @xmath115 as a function of @xmath33 within the borel window @xmath116 . the minimum borel parameter @xmath117 is determined such that the terms of order @xmath118 on the ope side eq . ( [ eq_20 ] ) contribute not more that 10% @xcite . selecting such sufficiently large values of @xmath117 suppresses higher - order contributions in the ope eq . ( [ eq_20 ] ) and justifies the truncation . typically , @xmath119 is in the order of 0.6 gev@xmath120 . the values for @xmath121 are roughly determined by the @xcite , i.e. , the continuum part of the hadronic side must not contribute more than 50% to the total hadronic side . according to our experience @xcite , @xmath122 is not very sensitive to variations of @xmath121 . we can , therefore , fix the maximum borel parameter by @xmath123 for the @xmath1 ( @xmath0 ) meson , in good agreement with the 50% rule . the sensitivity of the results on these choices of the borel window is discussed in appendix c. two examples of @xmath124 as a function of the borel parameter @xmath33 are displayed in figs . [ fig_1 ] and [ fig_2 ] for our default parameters and for @xmath125 @xmath126 . one observes , indeed , flat curves @xmath127 within the borel window . this is a prerequisite for the stability of the following analyses . to get finally the vector meson mass parameter @xmath128 we average the quantity @xmath114 within the above borel window to get @xmath129 which is used in eqs . ( [ eq_50 ] , [ eq_55 ] ) . the results of our qsr evaluations for the density dependence of the vector meson masses @xmath130 and @xmath131 , defined in eq . ( [ eq_55 ] ) , for @xmath132 are exhibited in figs . [ fig_3 ] and [ fig_4 ] , respectively . as seen in fig . [ fig_3 ] the @xmath0 meson mass drops with increasing nucleon density . the value of the @xmath0 meson mass shift at given density is directly governed by the parameter @xmath3 , i.e. the strength of the density dependence of the 4-quark condensate . some qualitative arguments to understand such an important role of the 4-quark condensate for the in - medium @xmath0 meson mass shift are given in @xcite . the impact of the 4-quark condensate is more pronounced for the isoscalar channel . in fig . [ fig_4 ] one can observe that such a global characteristic as the sign of the @xmath1 meson mass shift is changed by a variation of the parameter @xmath3 . similar to the @xmath0 meson the density dependence of the @xmath1 meson mass @xmath131 is mainly governed by the qcd mass - parameter @xmath133 in accordance with the in - medium change of the 4-quark condensate . ( this confirms previous results obtained within the zero - width approximation , which is equivalent to an evaluation of a normalized moment of the spectral function @xcite , and the finite width treatment in @xcite based on an effective chiral lagrangian @xcite . ) in particular , for a weak dependence of the 4-quark condensate on density @xmath134 the @xmath1 meson mass @xmath131 is increased , while for a greater value of @xmath3 the @xmath1 meson mass decreases with density . the sign of the @xmath1 meson mass shift is important with respect to the expectation to produce nuclear bound states of @xmath1 meson using suitable projectiles impinging on a nuclear target @xcite . from our study one can conclude that a negative @xmath1 meson mass shift , corresponding an effective attractive potential , is caused by a strong dependence of the 4-quark condensate on density , i.e. for @xmath135 . the different behavior of @xmath136 and @xmath137 can be traced back , to some extent , to different values of the subtraction constants @xmath138 , as emphasized in @xcite . the strikingly different vacuum widths , @xmath139 , cause further differences , in medium additionally amplified by different shapes of @xmath140 . our calculations also show that the main pattern of the behavior of @xmath100 plotted in figs . [ fig_3 ] and [ fig_4 ] remains stable even for the extreme cases when including @xmath141 or discarding @xmath142 at all . this still points to the crucial role of the 4-quark condensate for the @xmath15 meson in - medium mass shifts . the robustness of the pattern of @xmath100 as a function of the density under variations of @xmath96 can be interpreted as stringent impact of the density dependence of the condensates , while the influence of the strong interaction encoded in @xmath96 is , within the qcd sum rule approach , of sub - leading order , for the given examples . while the global mass shifts of the in - medium @xmath0 and @xmath1 mesons are governed mainly by the strength of the 4-quark condensate density dependence , the details of the vector meson spectral functions depend also on the meson - nucleon scattering amplitude @xmath96 . in fig . [ fig_5 ] we plot the @xmath0 meson spectral density for @xmath143 at normal nuclear density . ( note that the spectral functions determine the emission of di - electrons from the vector meson decays @xcite . ) the main trend of the down shift of the @xmath0 meson spectral function peak position is in accordance with the dropping @xmath0 meson mass obtained above by a stronger density dependence , parameterized by larger values of @xmath3 . when the peak of @xmath144 is in the interval @xmath145 gev , i.e. for @xmath146 , the width of the spectral function decreases as the peak moves to the smaller values of @xmath147 . this is not a surprise if one takes into account the energy dependence of @xmath148 in the same ( subthreshold ) energy interval ( see fig . [ fig_b1 ] ) , where @xmath148 also drops with decreasing energy . from this one can also conclude that in a wide region of @xmath3 the @xmath0 meson does not undergo drastic collision broadening at normal nuclear density , in contrast to earlier expectations but in line with @xcite . in fig . [ fig_6 ] we display the change of the @xmath1 meson spectral function in nuclear matter at normal nuclear density for the same parameters @xmath3 as for the @xmath0 meson ( see fig . [ fig_5 ] ) . the in - medium spectral change is still seen to be dominated by the density dependence of the 4-quark condensate . the dependence of the peak position on the parameter @xmath3 is similar to @xmath131 , namely , for a weak density dependence of the 4-quark condensate @xmath149 the peak is up - shifted compared to vacuum , while for @xmath135 the peak moves to a smaller value of the energy . for @xmath150 with up - shifted peak positions the width remains almost constant . this is in agreement with the approximately constant value of @xmath151 in the region @xmath152 gev ( see fig . [ fig_b2 ] ) . when the peak of @xmath150 moves to a smaller energy ( for @xmath135 ) the width of the @xmath1 meson increases moderately , which is caused by the increase of @xmath151 ( see fig . [ fig_b2 ] ) in the corresponding interval of energy . the pure hadronic calculation in @xcite predicts a slight up - shift of the original @xmath1 peak . this case is reproduced in our approach by @xmath153 . however , such a value of @xmath3 delivers a strong down - shift of the original @xmath0 peak ( see fig . [ fig_5 ] ) , at variance to the results in @xcite . otherwise , in contrast to @xcite , but in agreement with @xcite , the @xmath0 width is less affected by in - medium effects ; rather for a strongly decreasing @xmath0 mass the width may even become smaller , as discussed above . the treatment of the @xmath2 meson proceeds along the same strategy as presented above . the corresponding current operator in eq . ( [ eq_5 ] ) is @xmath154 which renders the coefficients @xmath155 to be used in eq . ( [ eq_70 ] ) into c_1 & = & - , + c_2 & = & m_s s_0 + y n + + 12 a_2^s m_n n , + c_3 & = & - _ s _ 0 s _ 0 ^ 2 - 56 a_4^s m_n^3 n , [ eq_444 ] where @xmath156 is not changed . at the scale @xmath157 gev@xmath120 the condensates are @xmath158 and @xmath159 @xcite . @xmath160 is the poorly known strangeness content of the nucleon which may vary from 0 to 0.25 @xcite . we utilize here @xmath161 , as in @xcite . further parameters are @xmath162 with @xmath163 and @xmath164 mev . the subtraction constant is negligible , i.e. @xmath165 @xcite . @xmath166 is absorbed again in @xmath167 , while @xmath168 with vacuum parameters @xmath169 . @xmath170 gev@xmath120 is dictated by the 50% rule . for @xmath171 we employ the previous estimates @xcite ( see solid curve in figure 8 in first reference of @xcite ) . since this @xmath171 is comparatively large we find some weak dependence of @xmath172 on @xmath3 , see fig . [ fig_7 ] . the pattern of the @xmath3 dependence resembles the one of the @xmath0 meson but is much more moderate . ( note that the slope of the curves @xmath173 scale with @xmath174 @xcite . ) since the used amplitude @xmath175 shows minor variations at @xmath176 , the widths of the shifted spectral functions is quite independent of @xmath3 , see fig . [ fig_8 ] . ( when using the amplitude of @xcite the width would become larger with increasing values of @xmath3 . ) in summary we present a systematic evaluation of the borel qcd sum rule for @xmath0 and @xmath1 mesons . we go beyond the often employed zero - width approximation and use a realistic ansatz for the spectral function . a crucial element for our analysis is the use of the recent @xmath15 meson - nucleon scattering amplitudes adjusted to a large data basis . these differ noticeably from earlier employed amplitudes . despite of such differences , the results of our analysis are robust : the @xmath0 meson suffers a down shift by an amount determined by the yet poorly known density dependence of the 4-quark condensates . the latter ones determine also whether the @xmath1 meson suffers an up - shift or a down - shift . one consequence of the scattering amplitudes @xcite is a moderate in - medium broadening of the @xmath15 spectral functions , in contrast to earlier predictions . we focus on the region in the vicinity of the @xmath15 peaks in vacuum . therefore , we do not address such problems as the development of second , low - energy peak in the @xmath1 strength , as found in @xcite . besides the exploration of the importance of the density dependence of the 4-quark condensates , the determination of the in - medium modification of @xmath0 and @xmath1 on a common footing is the main objective of the present paper . this is highlighted in fig . [ fig_9 ] , which points to drastic shifts of either the @xmath1 meson or the @xmath0 meson , or to still noticeable shifts of both . fairly independent of @xmath3 is the @xmath22 mass splitting of about 200 mev at normal nuclear matter density . ( using @xmath177 from @xcite results in a smaller @xmath22 mass splitting which even disappears for @xmath178 . ) it should be stressed , however , that the use of a common parameter @xmath3 for the light vector mesons is an approximation , since actually @xmath0 , @xmath1 and @xmath2 mesons have their own @xmath3 s . the detailed analysis deserves a separate investigation . it turns out that the in - medium cross properties of the @xmath0 and @xmath1 mesons are determined , to a large extent , by the condensates , while the meson - nucleon scattering amplitudes are important for the quantitative behavior . ( e.g. @xmath179 } > \im t_{\rho n}^{[19]}$ ] causes more support of @xmath180})$ ] than @xmath181})$ ] at smaller values of @xmath147 , which is compensated by a stronger down - shift of @xmath136 when using @xmath182}$ ] . otherwise , @xmath183 } < \im t_{\omega n}^{[19]}$ ] for @xmath184 mev and @xmath183 } > \im t_{\omega n}^{[19}]$ ] for @xmath185 mev which explains the somewhat larger up - shift of @xmath137 when using @xmath186}$ ] and small values of @xmath3 . directly evident is that different @xmath187 can cause different shapes of the spectral function . ) basing on this observation we consider also the @xmath2 meson using estimates of the @xmath2 meson - nucleon scattering amplitude . in contrast to the @xmath15 mesons , the in - medium modification of the @xmath2 meson is determined by the strangeness chiral condensate and depends essentially on the strangeness content of the nucleon . in our approach we rely on the linear density approximation . there are examples in the literature ( e.g. @xcite ) which show that , e.g. , the chiral condensate begins to deviate from the linear density behavior at normal nuclear matter density . resting on this argument one can expect the quantitative validity of our results up to @xmath188 . we have truncated , according to the common praxis , the ope at order 3 . higher - order terms are not yet calculated in a systematic way . this issue needs further consideration , as also the case of a finite spatial momentum of the vector mesons @xcite . concerning an experimental opportunity to observe both @xmath0 and @xmath1 mesons in - medium mass shifts simultaneously in heavy - ion collisions , our analysis still shows the crucial importance of the in - medium density dependence of the 4-quark condensate . in particular , if the 4-quark condensate density dependence is not too strong ( i.e. for @xmath189 ) there is a chance to observe the up - shifted peak of the @xmath1 resonance , while the @xmath0 meson is down - shifted . the measurements with hades , once the @xmath0 and @xmath1 peaks are identified , will constrain the mentioned important density dependence of the 4-quark condensates and , consequently , the strength of approaching chiral symmetry restoration . _ acknowledgments : _ we thank e.g. drukarev , r. hofmann , s. leupold , v.i . zakharov , and g.m . zinovjev for useful discussions . we are especially grateful to m.f.m . lutz and gy . wolf for discussions on the results of @xcite and for supplying the information of @xmath96 . o.p.p . acknowledges the warm hospitality of the nuclear theory group in the research center rossendorf . this work is supported by bmbf 06dr121 , stcu 15a , cern - intas 2000 - 349 , nato-2000-pst clg 977 482 . the self - energy of @xmath0 meson in vacuum is @xmath190 where the self - energy tensor @xmath191 within an effective lagrangian for the @xmath192 interaction is given by @xcite i ^ ( q ) = g^2 _ ( - 2 g^ ) [ ap2 ] with the coupling constant @xmath193 and @xmath194 gev . using the renormalization scheme of @xcite we get for the meson at rest , i.e. @xmath195 , ^vac _ ( e ) & = & ( 4 m_^2 - e^2 ) \ { ( ) . ( ) } , [ ap3 ] where @xmath196 gev is the vacuum mass of @xmath0 meson . in this scheme , @xmath197 follows from eq . ( [ eq_55 ] ) . for definiteness we plot in figs . [ fig_b1 ] and [ fig_b2 ] the spin and isospin averaged amplitudes for @xmath0 and @xmath1 mesons , respectively , which are employed in our qsr evaluations and not explicitly given in @xcite . here we would like to report a few technical details of our sum rule evaluation . let us first consider the density dependence of the continuum threshold , see figs . [ fig_c1 ] and [ fig_c2 ] . when changing the density , but keeping the rules for the borel window as described above , the continuum thresholds @xmath78 change . the overall pattern resembles the behavior of @xmath124 : a decreasing ( increasing ) @xmath124 implies a decreasing ( increasing ) @xmath78 . if one would freeze the continuum thresholds to the vacuum values , i.e. , @xmath198 , the @xmath22 mass splitting at normal nuclear matter density is reduced to about 100 mev and the dependence on @xmath3 becomes much weaker . next we consider the stability of our results with respect of the choice of the borel window at normal nuclear matter density . [ fig_c3 ] and [ fig_c4 ] exhibit the change of the parameter @xmath124 when changing @xmath117 . as expected , @xmath124 slightly increases with decreasing @xmath117 ( compare also with figs . [ fig_1 ] and [ fig_2 ] ) . the change is fairly moderate but points to some dependence of the absolute values of @xmath124 and @xmath199 on the borel window . this , however , is not important since our focus here is the pattern of the in - medium modification and not absolute predictions , which are hampered anyhow by the uncertainty related with the 4-quark condensate . a similar statement holds for changes of @xmath121 , see figs . [ fig_c5 ] and [ fig_c6 ] . with virtue to figs . [ fig_1 ] and [ fig_2 ] the decrease of @xmath124 with increasing @xmath121 is counter - intuitive . the explanation of this behavior comes from the change of @xmath78 when changing @xmath121 . when directly determining @xmath121 by the 50 % rule @xcite we arrive at a sliding borel window where the borel sum rule ( [ eq_20 ] ) is explicitly solved . the results displayed in fig . 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within the borel qcd sum rule approach at finite baryon density we study the role of the four - quark condensates for the modifications of the vector mesons @xmath0 , @xmath1 and @xmath2 in nuclear matter . we find that in - medium modifications of the @xmath0 and @xmath1 mesons are essentially dominated by the dependence of the 4-quark condensate on the nucleon density . in particular , the numerical value of a parameter ( @xmath3 ) , which describes the strength of the density dependence of the 4-quark condensate beyond the mean - field approximation , governs the decrease of the @xmath0 mass as a function of the density . for the @xmath1 meson the sign of the in - medium mass shift is changed by variations of @xmath3 . to study consistently the in - medium broadening of the light vector mesons we employ @xmath4 and @xmath5 scattering amplitudes derived recently from a covariant unitary coupled channel approach adjusted to pion- and photo - induced reactions . in contrast to the @xmath0 and @xmath1 mesons , the in - medium mass of the @xmath2 meson is directly related to the chiral ( strange ) quark condensate . measurements of the vector meson spectral change in heavy - ion collisions with hades can shed light on the yet unknown density dependence of the 4-quark condensate .

measurements of the integrated light from a galaxy at 2000 provides a fairly direct measure of the instantaneous rate of star formation , since the massive stars that provide most of this radiation are short - lived compared with the age of the galaxy . knowledge of the star formation rate also gives a measure of the rate of heavy element production in a galaxy , or in the universe when a large sample of galaxies are measured ( @xcite ) . the integrated light from these galaxies contributes to the extragalactic background light at ultraviolet wavelengths , whose main sources are hot stars and active galactic nuclei . measurements of galaxy number counts in the ultraviolet have been made by @xcite using the foca balloon - borne uv telescope , @xcite and @xcite using hst archival fields . these data have been interpreted with models that predict number counts based on galaxy spectral energy distributions ( sed s ) and luminosity functions , such as those of @xcite and @xcite . the total far - ultraviolet extragalactic background has been measured to be as high as 500 ph @xmath2 s@xmath3 @xmath3 and as low as 30 ph @xmath2 s@xmath3 @xmath3 ( see review by @xcite ) . predictions for the number of galaxies that might be detected in deep ultraviolet optical monitor ( om ) images are given by @xcite . in this paper , we detect galaxies in a deep uv image taken with the optical monitor ( om ) and use these galaxy number counts to place constraints on galaxy luminosity evolution via a a galaxy evolution model similar to that of @xcite . we also find a lower limit to the galaxy contribution to the extragalactic uv background . the om 13 hr deep field ( at j2000.0 13 34 37.00 , + 37 54 44.0 ) was observed for approximately 200 ks with xmm - newton around june 22 , 2001 . details of the om exposures used in this study are shown in table [ tab : tab1 ] . lcl + filter & central wavelength & exposure time + & ( ) & ( ksec ) + + b & 4200 & 10 + u & 3900 & 10 + uvw1 & 3000 & 20 + uvm2 & 2500 & 31.5 + uvw2 & 2000 & 30 + + several exposures of typically 7 ks were brought to a common astrometric reference frame and coadded . we searched each image for sources using sextractor and made a catalog of the sources we found . we concentrate here on sources in the uvw2 image ( fig . [ tsasseen - f7_fig1 ] ) and use measurements in the other filters to differentiate between stars , galaxies and qso s . we also use a deep r band image ( to r@xmath427 ) of this field taken with the 8 m subaru telescope on mauna kea ( fig . [ tsasseen - f7_fig2 ] ) to check for source shape and possible confusion . we perform two checks to discriminate stars from galaxies . first , we compare the sed of each uvw2 source ( determined from om photometry ) against stellar templates . second , we compute an inferred distance , as if the source were a main sequence star , from u - b color and b magnitude , as shown in fig . [ tsasseen - f7_fig3 ] . we find these checks form reliable stellar discriminators for more than 90% of the sources brighter than ab=22 . where @xmath5 is given in ergs @xmath2 s@xmath3 hz@xmath3 ( @xcite ) . ] we also find a number of qso s in the field that show uv excess and appear point - like in the om and subaru images . we categorize these separately in our galaxy number counts . further work remains to completely discriminate any remaining stellar content and the qso populations . we plot the detected galaxy counts as a function of magnitude in fig . [ tsasseen - f7_fig4 ] . our counts are in approximate agreement with that of @xcite ( also shown in fig . [ tsasseen - f7_fig4 ] ) in the range of overlap , and we extend these counts to ab=22 . we have constructed a model is similar to that of @xcite and use it to predict galaxy counts at 2000 as a function of apparent magnitude . the model uses a schechter absolute luminosity distribution function for 6 different galaxy types at redshifts between zero and 1.2 , along with k - corrections and a single parameter luminosity evolution factor for each galaxy type . we have normalized the schechter function using observed counts at bj=17 , and set our evolution parameters to agree with the modeled galactic evolution of @xcite , following @xcite . our model implicitly includes the effects of dust absorption and scattering because it is based on observed uv sed s . like armand & milliard , our model predicts fewer galaxies in each magnitude band than our measured number counts , as shown in figure [ tsasseen - f7_fig4 ] . we also compare the observed counts with the model of @xcite , whose model explicitly includes expected contributions to the observed galaxy counts from starburst galaxies and dust . our model agrees well with the granato et al . model that includes dust , but our observed counts are higher than both models that include dust . the summed the flux from non - stellar sources detected in the uvw2 image totals 3236 ph @xmath2 s@xmath3 sr@xmath3 @xmath3 , with the higher limit including the contribution from qso s and active galaxies . the integrated far - ultraviolet light from discrete galaxies has been measured recently by @xcite to be 144195 ph @xmath2 s@xmath3 sr@xmath3 @xmath3 , based on galaxies detected in the range ab = 24 to 29.5 and a model to infer the flux from brighter galaxies . these authors claim there appears to be a break in the slope of the galaxy number counts that occurs around ab = 24 , with substantial flattening of function at fainter magnitudes . our measurements show an intriguing downturn in galaxy counts at the faint end , which may indicate the start of the change in the slope of the number counts . there still remains some uncertainty in the number counts in the gap between our measurements and those of @xcite , which indicates the total integrated flux of galaxies is still uncertain . the discrepancy between the models shown in fig . [ tsasseen - f7_fig4 ] and both our data and that of @xcite may indicate that we are missing some components in our understanding of how galaxies evolve . some possible reasons for the descrepancy between their model and measurements are given by @xcite , including faster evolution of the star formation rate or the possiblity that there is a population of blue galaxies that is substantially more numerous at z = 0.7 than they are today . there are a number of effects we have not yet evaluated in detail that may affect our measurement and conclusions . these include the effects of galaxy inclination , morphology and apertures on our photometry , the effects of comparing measurements made in slightly different bandpasses , and the detailed effects of dust absorption and possible evolution in galaxies ( @xcite ) . our simple model assumes a smooth evolution in star formation , but there is evidence that star formation may be espisodic or occur in bursts , possibly because of merger activity , _ e.g. _ @xcite . galaxies change over time in many ways and our model predicts only one facet of these changes , namely an evolving star formation rate . the full picture of galaxy evolution is certainly more complicated . it remains to explore further the connections between changes in the star formation rate and changes in galaxy appearance and morphology , metallicity , gas content , spectral energy output , and merger activity that have been discussed at length by other researchers . \1 ) we have obtained galaxy counts at 2000 to a magnitude of ab = 22 in deep images from the optical monitor on xmm - newton . the long om exposure allows us to measure galaxy counts 1.5 magnitudes fainter than @xcite , and we find similar counts in range of overlap . 2 ) two evolutionary models underpredict the observed galaxy counts , and may indicate that several process may be at work , including episodic star formation , changes in the optical depth within galaxies to 2000 radiation , or a new population of galaxies that is less visible in the present epoch . 3 ) the total integrated flux from the galaxies we detect to ab=22 is 3236 ph @xmath2 s@xmath6 @xmath3 sr@xmath3 . this flux is a lower limit to the integrated extragalactic background light at 2000 , and represents about 2025% of the integrated , far - ultraviolet flux from galaxies inferred from the deep hst measurements of @xcite . hill , r. , gardener , j. , heap , s. malamuth , e. & collins , n. , 1997 in the ultraviolet universe at low and high redshift : probing the progress of galaxy evolution , ed . w. h. waller et al . ( new york : american institute of physics ) , p. 21 sasseen , t. p. , crdova , f. , ho , c. & priedhorsky , w. , 1997 , in the ultraviolet universe at low and high redshift : probing the progress of galaxy evolution , ed . w. h. waller et al . ( new york : american institute of physics ) , p. 21 this research was supported by nasa grant nag5 - 7714 . we would also like to thank the optical monitor team , and esa for their successful program to produce the first rate space observatory , xmm - newton ,

we use galaxies detected in a deep ultraviolet xmm - newton optical monitor image and a model that predicts uv galaxy counts based on local counts and evolution parameters to constrain galaxy evolution to z=1.2 . the 17 square 2000 ( uvw2 filter ) image was taken as part of the xmm - om team s guaranteed time program . we detect sources in this image to a flux limit of 2.7 @xmath0 10@xmath1 ergs @xmath2 s@xmath3 @xmath3 ( ab magnitude = 22 ) . since some of the sources may be stars , we perform a number of checks , including shape , color and implied distance to remove stars from the detected counts . we find galaxy number counts as a function of magnitude roughly in agreement @xcite , but again find these counts are in excess of evolution models . the excess counts at faint magnitudes may provide evidence for either a new population of galaxies emerging around z=0.7 or more dramatic evolution than some earlier predictions . the integrated light from the detected galaxies totals 3236 ph @xmath2 s@xmath3 @xmath3 sr@xmath3 , placing a firm lower limit on the integrated uv light from galaxies .

at large temperatures or large densities hadronic matter is expected to undergo two phase transitions : one which deconfines quarks ( and gluons ) and one which restores chiral symmetry . up to now it is an unsettled issue whether these two phase transitions are distinct or coincide . the more , it is even unclear whether there are real phase transitions or only rapid crossover transitions . such transitions have received much attention in heavy ion physics as well as in the context of neutron stars which provide a unique environment to study cold matter at supernuclear densities @xcite . even though a deconfinement phase transition seems intuitively evident at large enough densities , from a theoretical point of view a confirmation of the existence of a deconfined quark phase in neutron stars is so far limited by the uncertainties in modeling qcd at large densities . all the more it is important to study and compare different available models to shed some light on similarities and differences with respect to the behavior of matter at large densities as well as on the corresponding predictions of neutron star properties like e.g. its mass and radius . in the future such experience may prove to be useful if either an improved understanding of matter under extreme conditions provides a more exclusive selection between the various models or new experimental results on neutron star properties are available to set more stringent constraints . usually the quark matter phase is modeled in the context of the mit bag model @xcite as a fermi gas of @xmath0 , @xmath1 , and @xmath2 quarks . in this model the phenomenological bag constant @xmath3 is introduced to mimic qcd interactions to a certain degree . the investigation of such a phase was furthermore stimulated by the idea that a quark matter phase composed of almost an equal amount of the three lightest quark flavors could be the ground state of nuclear matter @xcite . indeed , for a wide range of model parameters such as the bag constant , bag models predict that the quark matter phase is absolutely stable i.e. its energy per baryon at zero pressure is lower than the one of @xmath4fe . if this is true , this has important consequences in physics and astrophysics @xcite leading e.g. to the possibility of so called `` strange stars '' @xcite which are neutron stars purely consisting of quark matter in weak equilibrium with electrons . of course , to check the model dependence of such findings it is important to perform the corresponding calculations also in models different from the mit bag model . in a recent work by buballa and oertel @xcite the equation of state ( eos ) of quark matter was investigated in the framework of the nambu jona - lasinio ( njl ) model with three quark flavors . applying this model it was found that strange quark matter is not absolutely stable . this would rule out the existence of strange stars . on the other hand , the possibility of quark phases in the interior of neutron stars is in principle not excluded by this result even though this possibility gets energetically less likely . only a detailed phase transition calculation can answer the question which effect the findings in @xcite have on the existence of quark phases inside neutron stars . this is what we are aiming at in the present work . in principle , for the description of a neutron star which consists of a quark phase in its center and a surrounding hadronic phase ( and , as we shall discuss below , a mixed phase in between ) we need models for both phases . the most favorite case would be to have one model which can reliably describe both phases . so far , there are no such models . therefore , we will use various versions of the relativistic mean field model to parametrize the hadronic phase . for the quark phase we follow buballa and oertel @xcite in using the three - flavor version of the njl model . the njl model has proved to be very successful in the description of the spontaneous breakdown of chiral symmetry exhibited by the true ( nonperturbative ) qcd vacuum . it explains very well the spectrum of the low lying mesons which is intimately connected with chiral symmetry as well as many other low energy phenomena of strong interaction @xcite . at high enough temperature and/or density the njl model predicts a transition to a state where chiral symmetry becomes restored . despite that promising features which at first sight might suggest the njl model as a good candidate for modeling both the low and high density region of a neutron star this model has one important shortcoming , namely it does not confine quarks . at low densities , however , the bulk properties of strongly interacting matter are significantly influenced by the fact that quarks are confined there . therefore , we can not expect that the njl model gives reliable results for the eos at low densities . thus we will use the relativistic mean field model to describe the confined phase . at higher densities , however , the quarks are expected to be deconfined . there we expect the njl model to be applicable since the lack of confinement inherent to this model is irrelevant in that regime . the interesting feature of the njl model is that it reflects the chiral symmetry of qcd . clearly , it would be preferable to have a lagrangian for the hadronic phase which also respects chiral symmetry like e.g. the one constructed in @xcite for the two - flavor case and the su(3 ) generalizations @xcite . such lagrangians , however , are more complicated to deal with . first applications to neutron star matter seem to indicate that the modifications are rather small as compared to the relativistic mean field models used here @xcite . for simplicity , we therefore will restrict our considerations to the much simpler extensions of the walecka model which include hyperonic degrees of freedom ( relativistic mean field models ) . the paper is organized as follows : in sec . [ sec : hp ] we discuss how the eos for the hadronic phase of a neutron star is calculated within several variants of relativistic mean field models . we keep brief here since such models are frequently used and well documented in the literature ( cf . e.g. @xcite ) . in sec . [ sec : qp ] we apply the njl model to the description of the possible quark phase of the neutron star . here we present much more details as compared to sec . [ sec : hp ] since to the best of our knowledge it is the first time that the njl model is applied to the description of the quark phase in a neutron star . [ sec : pt ] is devoted to the construction of the phase transition and to the application of the complete eos to the internal structure of the neutron star . finally we summarize and discuss our results in sec . [ sec : sum ] . neutron stars cover a wide range of densities . from the surface of the star which is composed of iron with a density of @xmath5g/@xmath6 the density can increase up to several times normal nuclear matter density in the center of the star . since there is no single theory that covers this huge density range , we are forced to use different models to meet the requirements of the various degrees of freedom opened up at different densities . for subnuclear densities we apply the baym - pethick - sutherland eos @xcite . the degrees of freedom in this eos are nuclei , electrons and neutrons . the background of neutrons appears above neutron drip density ( @xmath7 ) when the most weakly bound neutrons start to drip out of the nuclei which themselves get more and more neutron rich with increasing density . for a detailed discussion of the baym - pethick - sutherland eos see also @xcite . we also refer to @xcite where a relativistic mean field model is extended to also describe this low density range . at densities of about normal nuclear density @xmath8 the nuclei begin to dissolve and merge together and nucleons become the relevant degrees of freedom in this phase . we want to describe this phase in the framework of the relativistic mean field ( rmf ) model which is widely used for the description of dense nuclear matter @xcite . for an introduction to the rmf model see e.g.@xcite . we use three eos s calculated by schaffner and mishustin in the extended rmf model @xcite ( denoted as tm1 , tm2 , gl85 ) and one by gosh , phatak and sahu @xcite . for the latter one we use gps as an abbreviation . these models include hyperonic degrees of freedom which typically appear at @xmath9 @xmath8 . table [ nuclprops ] shows the nuclear matter properties and the particle composition of the four eos s . ccccc hadronic eos & tm1 & tm2 & gl85 & gps + reference & @xcite & @xcite & @xcite & @xcite + @xmath10[@xmath11 & @xmath12 & @xmath13 & @xmath12 & @xmath14 + @xmath15[mev ] & @xmath16 & @xmath17 & @xmath18 & @xmath19 + @xmath20[mev ] & @xmath21 & @xmath22 & @xmath23 & @xmath24 + @xmath25 & @xmath26 & @xmath27 & @xmath28 & @xmath29 + @xmath30[mev ] & @xmath31 & @xmath32 & @xmath33 & @xmath34 + composition & a & a & a & b + + + the rmf eos s are matched to the baym - pethick - sutherland eos at densities of @xmath35g/@xmath36 . even if the relevant degrees of freedom are specified ( in the rmf case basically nucleons and hyperons ) the high density range of the eos is still not well understood . the use of different hadronic models should reflect this uncertainty to some degree . in the following we denote the phase described by the baym - pethick - sutherland eos and by the rmf model as the _ hadronic phase _ ( hp ) of the neutron star . to describe the deconfined _ quark phase _ ( qp ) we use the nambu jona - lasinio ( njl ) model @xcite with three flavors @xcite in hartree ( mean field ) approximation ( for reviews on the njl model cf . the lagrangian is given by ( cf . @xcite ) @xmath37 \nonumber \\ & & \phantom{mmmmmm } - k \ , [ \,{\rm det}_f ( \bar q \ , ( 1+\gamma_5 ) \ , q ) + { \rm det}_f ( \bar q \ , ( 1-\gamma_5 ) \ , q ) \ , ] \label{eq : nambulag}\end{aligned}\ ] ] where @xmath38 denotes a quark field with three flavors , @xmath0 , @xmath1 , and @xmath2 , and three colors . @xmath39 is a @xmath40 matrix in flavor space . for simplicity we use the isospin symmetric case , @xmath41 . the @xmath42 matrices act in flavor space . for @xmath43 they are the generators of @xmath44 while @xmath45 is proportional to the unit matrix in flavor space ( see @xcite for details ) . the four - point interaction term @xmath46 is symmetric in @xmath47 . in contrast , the determinant term @xmath48 which for the case of three flavors generates a six - point interaction breaks the @xmath49 symmetry . if the mass terms are neglected the overall symmetry of the lagrangian therefore is @xmath50 . in vacuum this symmetry is spontaneously broken down to @xmath51 which implies the strict conservation of baryon and flavor number . the full chiral symmetry which implies in addition the conservation of the axial flavor current becomes restored at sufficiently high temperatures and/or densities . the finite mass terms introduce an additional explicit breaking of the chiral symmetry . on account of the chiral symmetry breaking mechanism the quarks get constituent quark masses which in vacuum are considerable larger than their current quark mass values . in media with very high quark densities constituent and current quark masses become approximately the same ( concerning the strange quarks this density regime lies far beyond the point where chiral symmetry is restored ) . the coupling constants @xmath52 and @xmath53 appearing in ( [ eq : nambulag ] ) have dimension energy@xmath54 and energy@xmath55 , respectively . to regularize divergent loop integrals we use for simplicity a sharp cut - off @xmath56 in 3-momentum space . thus we have at all five parameters , namely the current quark masses @xmath57 and @xmath58 , the coupling constants @xmath52 and @xmath53 , and the cut - off @xmath56 . following @xcite we use @xmath59mev , @xmath60 , @xmath61 , @xmath62mev , and @xmath63mev . these parameters are chosen such that the empirical values for the pion decay constant and the meson masses of pion , kaon and @xmath64 can be reproduced . the mass of the @xmath65 meson is underestimated by about 6% . we treat the three - flavor njl model in the hartree approximation which amounts to solve in a selfconsistent way the following gap equations for the dynamically generated constituent ( effective ) quark masses : @xmath66 with @xmath67 being any permutation of @xmath68 . at zero temperature but finite quark chemical potentials the quark condensates are given by @xmath69 where we have taken the number of colors to be @xmath70 . @xmath71 denotes the fermi momentum of the respective quark flavor @xmath72 . it is connected with the respective quark chemical potential @xmath73 via @xmath74 the corresponding quark particle number density is given by @xmath75 for later use we also introduce the baryon particle number density @xmath76 the eqs . ( [ eq : gap ] ) , ( [ eq : cond ] ) serve to generate constituent quark masses which decrease with increasing densities from their vacuum values of @xmath77mev and @xmath78mev , respectively . before calculating the eos we would like to comment briefly on the hartree approximation to the njl model which we use throughout this work . this treatment is identical to a leading order calculation in the inverse number of colors @xmath79 @xcite . in principle , one can go beyond this approximation by taking into account @xmath79 corrections in a systematic way . this amounts in the inclusion of quark - antiquark states ( mesons ) as rpa modes in the thermodynamical calculations @xcite . several things then change : first of all , these mesons might contribute to the eos . we are not aware of a thorough discussion of such an eos for three flavors with finite current quark masses . the two - flavor case is discussed in @xcite . qualitatively the masses of the meson states rise above the chiral transition point . therefore , they should become energetically disfavored and thus less important . an additional technical complication arises due to the fact that the relation between the fermi energy and the chemical potential becomes nontrivial . instead of ( [ eq : fermimom ] ) one has to solve an additional gap equation for each flavor species . these gap equations are coupled to the gap equations for the constituent quark masses given in ( [ eq : gap ] ) . we refer to @xcite for details . for simplicity we will restrict ourselves in the following to the hartree approximation and comment on the possible limitations of that approach in the last section . coming back to the eos we also need the energy density and the pressure of the quark system . in the hartree approximation the energy density turns out to be @xcite @xmath80 while pressure and energy density are related via @xmath81 where the effective bag pressure @xmath82 is given by @xmath83 with @xmath84 \nonumber \\ & & { } + 4 k \langle \bar u u \rangle \langle \bar d d \rangle \langle \bar s s \rangle \label{eq : defbag}\end{aligned}\ ] ] and @xmath85 note that @xmath86 depends implicitly on the quark densities via the ( density dependent ) constituent quark masses . the appearance of the density independent constant @xmath87 ensures that energy density and pressure vanish in vacuum . we note here that this requirement fixes the density independent part of @xmath82 which influences the eos via ( [ eq : endensq ] ) , ( [ eq : pressq ] ) and therefore the possible phase transition to quark matter . we will come back to this point in the last section . in what follows we shall frequently compare the results of the three - flavor njl model with the simpler mit bag model @xcite . for that purpose it is important to realize that the njl model predicts a ( density dependent ) bag pressure @xmath82 while in the mit bag model the bag constant @xmath3 is a density independent free parameter . there usually also the quark masses @xmath88 are treated as density independent quantities . ( an exception is the model discussed in @xcite which uses density dependent effective quark masses caused by quark interactions in the high density regime . ) in the bag model energy density and pressure of the quark system are given by @xmath89 and @xmath90 suppose now that the densities are so high that in the three - flavor njl model the effective quark masses have dropped down to the current quark masses . in this case , energy density and pressure take the form of the respective expressions in the mit bag model with @xmath91 and @xmath92 . however , a word of caution is in order here . for _ very high _ quark particle number densities the corresponding fermi momenta become larger than the momentum cut - off @xmath56 introduced to regularize the njl model . in this case the results of the njl model become unreliable . e.g. the upper limit of the momentum integration in ( [ eq : numbdens ] ) would be no longer given by the fermi momentum but by the cut - off @xmath56 which would be clearly an unphysical behavior of the model . thus for all practical purposes one should always ensure that in the region of interest the fermi momenta are smaller than the momentum cut - off @xmath56 . [ fig : pfrho ] shows the fermi momenta of the quarks as a function of the baryon particle number density ( for a charge neutral system of quarks and electrons in weak equilibrium ; cf . next paragraph for details ) . obviously all fermi momenta stay below the cut - off @xmath56 for the region of interest . we will come back to that point at the end of this section . the qp which might be found in the center of a neutron star consists of @xmath0 , @xmath1 , and @xmath2 quarks and electrons in weak equilibrium , i.e. the weak reactions @xmath93 imply relations between the four chemical potentials @xmath94 , @xmath95 , @xmath96 , @xmath97 which read @xmath98 since the neutrinos can diffuse out of the star their chemical potentials are taken to be zero . the number of chemical potentials necessary for the description of the qp in weak equilibrium is therefore reduced to _ two _ independent ones . for convenience we choose the pair ( @xmath99 , @xmath97 ) with the neutron chemical potential @xmath100 in a pure qp ( in contrast to quark matter in a mixed phase which we will discuss later ) we can require the qp to be charge neutral . this gives us an additional constraint on the chemical potentials via the following relation for the particle number densities : @xmath101 where @xmath102 denotes the electron particle number density . neglecting the electron mass it is given by @xmath103 utilizing the relations ( [ eq : weakeq ] ) and ( [ eq : chargeneu ] ) the eos can now be parametrized by only _ one _ chemical potential , say @xmath99 . at this point it should be noted that the arguments given here for the qp also holds for the hp . there one also ends up with _ two _ independent chemical potentials ( e.g. @xmath99 and @xmath97 ) if one only requires weak equilibrium between the constituents of the hp and with _ one _ chemical potential ( e.g. @xmath99 ) if one additionally requires charge neutrality . as we will discuss later , the number of independent chemical potentials plays a crucial role in the formulation of the gibbs condition for chemical and mechanical equilibrium between the hp and the qp . in the pure qp total energy density and pressure are given by the respective sums for the quark and the electron system , i.e. @xmath104 and @xmath105 where the system of electrons is treated as a massless ideal gas . one obtains the analogous expressions for the mit bag model if @xmath106 and @xmath107 are replaced by the respective mit expressions ( [ eq : endensmit ] ) and ( [ eq : pressmit ] ) . demanding weak chemical equilibrium ( [ eq : weakeq ] ) and charge neutrality ( [ eq : chargeneu ] ) as discussed above all thermodynamic quantities as well as quark condensates , effective quark masses etc . can be calculated as a function of one chemical potential @xmath99 . the curves in fig . [ fig : pfrho ] as well as in figs . [ fig : qqrho]-[fig : peps ] which we shall discuss in the following are obtained by varying @xmath99 while obeying simultaneously the constraints ( [ eq : weakeq ] ) and ( [ eq : chargeneu ] ) . [ fig : qqrho ] and [ fig : mrho ] show the quark condensates and the effective quark masses , respectively , as a function of the baryon particle number density . note that we start already at a density as high as two times nuclear saturation density @xmath108@xmath109 since we want to describe only the high density regime of the neutron star with quark degrees of freedom while for low densities we use the hadronic eos described in the previous section . concerning the low density regime of the three - flavor njl model we refer to @xcite for details . there it was shown that the energy per baryon of a charge neutral system of quarks and electrons in weak equilibrium ( described by the njl model and a free electron gas ) shows a minimum somewhat above two times @xmath110 . this implies that in the density region below this minimum the pressure is negative . we are not interested in the ( low density ) part of the eos with negative pressure since it can not be realized in a neutron star . in the region of interest figs . [ fig : qqrho ] and [ fig : mrho ] show that the strange quark condensate and the effective strange quark mass stay constant until the strange chemical potential @xmath96 overwhelms the strange quark mass . only then according to ( [ eq : fermimom ] ) the strange quark particle number density @xmath111 and the corresponding fermi momentum @xmath112 ( cf . [ fig : pfrho ] ) become different from zero causing a decrease of @xmath113 and @xmath114 . note that all condensates have negative values ( cf . ( ) ) . one might wonder why the dropping of the condensates of the light up and down quarks does not decrease the strange quark mass ( and condensate ) due to the last coupling term in ( [ eq : gap ] ) . indeed , strange quark mass and condensate have dropped in the low density region ( not shown here ) from their vacuum values down to the plateaus shown in figs . [ fig : qqrho ] , [ fig : mrho ] due to their coupling to the up and down quark condensates . in the plateau region , however , these condensates have already decreased so much that their influence on the strange quark mass is diminished . we refer to @xcite for details . as we shall see below , the large plateau value of the strange quark mass will have considerable influence on the phase structure in the interior of neutron stars . [ fig : brho ] shows the bag pressure @xmath82 as a function of the baryon particle number density . after staying more or less constant up to roughly 5 times nuclear saturation density it starts to increase towards @xmath87 which , however , it will reach only very slowly . again the rising of @xmath82 can be traced back to the strange quarks which come into play at high densities . thermodynamic quantities are shown in figs . [ fig : earho]-[fig : peps ] . for comparison various curves calculated within the mit bag model are added . the curves labeled with a specific value of the bag pressure are obtained from ( [ eq : endensmit],[eq : pressmit ] ) in weak equilibrium where the respective value of @xmath3 and the current quark masses @xmath115mev and @xmath116mev are used . in contrast to that for the curve labeled with `` mit '' we have used the plateau values of the bag pressure @xmath117mev ( cf . [ fig : brho ] ) and of the strange quark mass @xmath118mev ( cf . fig . [ fig : mrho ] ) . for up and down quarks we have used the current quark mass values also here . fig . [ fig : earho ] shows the energy per baryon as a function of the baryon particle number density . we find that the results of the njl model calculation can not be reproduced by a bag model using the current strange quark mass no matter which bag pressure is chosen . as already discussed above , the reason simply is that in the njl model up to four times @xmath110 there are no strange quarks in a system which is in weak equilibrium ( cf . fig . [ fig : pfrho ] ) . on the other hand , in bag models using the much lower current strange quark mass one finds a reasonable amount of strange quarks already at vanishing pressure which typically corresponds to @xmath119 times @xmath110 @xcite . in contrast to that , a bag model with the plateau values for bag pressure and strange quark mass ( denoted as `` mit '' in the figures ) yields a very good approximation to the njl result for the energy per baryon up to @xmath120 times @xmath110 . for higher particle number densities the njl result bends over and can be better described by bag models using the current strange quark mass and higher bag pressures ( roughly @xmath87 ) . all these findings also apply to the interpretation of fig . [ fig : pmub ] which shows the total pressure of the system versus the baryon chemical potential . comparing the two curves with the same bag constant labeled with `` @xmath121mev '' and with `` mit '' , respectively , one observes that the latter one has a significantly lower pressure . this is due to the use of the much larger effective strange quark mass of @xmath118mev in the latter case as compared to the current strange quark mass of @xmath122mev used in the former . the @xmath123 versus @xmath99 relation is an important ingredient for the construction of the phase transition from hadronic to quark matter inside a neutron star . we note already here , however , that we need in addition the thermodynamical relations also for a quark - electron system away from the charge neutral configuration to describe correctly the phase transition ( see below ) . the outlined picture concerning the comparison of njl and bag models is somewhat modified when looking at fig . [ fig : peps ] which shows the total pressure as a function of the energy density . eos s in the form @xmath124 enter the tolman - oppenheimer - volkoff @xcite equation which in turn determines the mass - radius relation of neutron stars . we see that in the lower part of the plotted energy density range the eos in fig . [ fig : peps ] is reasonably well described by mit bag models with the plateau value @xmath121mev no matter which quark masses are chosen ( current or effective quark masses ) . the reason is that the @xmath124 relation is not very sensitive to the quark masses . this has already been observed in a somewhat different context in @xcite . going to higher densities the strange quarks enter the game and the eos in fig . [ fig : peps ] obtained from the njl model starts to deviate from the eos of the mit bag models with the plateau value @xmath121mev . for very high densities the pressure determined from the njl model becomes comparable to the one calculated in the bag model with a high bag constant ( roughly @xmath87 ) . it is interesting to note that the deviation between the njl curve and the `` mit '' curve starts to increase in fig . [ fig : peps ] much earlier than in figs . [ fig : earho ] and [ fig : pmub ] . this shows that the pressure versus energy density relation is much more sensitive to the detailed modeling than the relations shown in figs . [ fig : earho ] and [ fig : pmub ] . before constructing the phase transition inside the neutron star let us briefly discuss the limitations of the njl model in the form as we have treated it here . as a typical low energy theory the njl model is not renormalizable . this is not an obstacle since such theories by construction should be only applied to low energy problems . in practice the results depend on the chosen cut - off or , to turn the argument around , the njl model is only properly defined once a cut - off has been chosen . this cut - off serves as a limit for the range of applicability of the model . here we have used one cut - off @xmath56 for the three - momenta of all quark species . concerning the discussion of other cut - off schemes and their interrelations we refer to @xcite . when the density in the quark phase gets higher the fermi momenta of the quarks rise due to the pauli principle . eventually they might overwhelm the cut - off of the njl model . at least beyond that point the model is no longer applicable . we have made sure in our calculations that this point is never reached ( cf . [ fig : pfrho ] ) . in addition , at very high densities one presumably enters a regime which might be better described by ( resummed ) perturbation theory . while the nonperturbative features of the njl model vanish with rising density , medium effects as mediated e.g. by one - gluon exchange grow with the density @xcite . to summarize , concerning the calculation of the eos it turns out that the njl model should neither be used at low densities where confinement properties are important nor at very high densities where the njl model as a low energy theory leaves its range of applicability . however , the njl model might yield reasonable results in a window of the density range where confinement is no longer crucial but chiral symmetry as a symmetry of full qcd remains to be important . in the previous sections we have discussed the underlying eos s thought to reflect the properties of confined hadronic matter ( hp ) and deconfined quark matter ( qp ) in its particular regime of applicability . applying these eos s we want to calculate in this section the phase transition from the hp to the qp to see which phase is the favored one at which densities . ( the existence of a qp inside the neutron star of course requires the phase transition density to be smaller than the central density of the star . ) it is worth to point out which phase structure is in principle possible if a hadronic model and the njl model are connected at a certain density value @xmath125 ( which is dynamically determined in the present work by a gibbs construction as we shall discuss below ) . at density @xmath125 we assume a first order phase transition from confined hadronic to deconfined quark matter . even without a matching to a hadronic model the njl model already exhibits a transition , namely from a low density system with broken chiral symmetry to a high density system where chiral symmetry is restored . the respective density is denoted by @xmath126 . for densities larger than @xmath126 the goldstone bosons which characterize the chirally broken phase are no longer stable but can decay into quark - antiquark pairs . if @xmath126 was larger than @xmath125 the following scenario would be conceivable : there would be three phases , namely ( i ) a hadronic , i.e. confined phase at low densities , ( ii ) a phase where quarks are deconfined but massive ( in this phase e.g. pions would still appear as bound states ) , and ( iii ) a high density phase where quarks are deconfined and their masses are so low that all mesons can decay into quarks . had we neglected all current quark masses , the quarks in the third phase would be exactly massless . with finite current quark masses , however , the constituent quark masses keep on dropping with rising density in the third phase ( cf . [ fig : mrho ] ) . this definitely interesting scenario with three phases is not realized in our model . it turns out that the deconfinement phase transition happens far beyond the chiral transition , i.e. @xmath127 . therefore , only the phases ( i ) and ( iii ) appear here . in principle , since we assume the deconfinement phase transition to be of first order these two phases can coexist in a mixed phase . indeed , it was first pointed out by glendenning that beside a hp and a qp also this mixed phase ( mp ) of quark and hadronic matter may exist inside neutron stars @xcite . ( for a discussion of the geometrical structure of the mp and its consequences for the properties of neutron stars see @xcite . ) this possibility was not realized in previous calculations due to an inadequate treatment of neutron star matter as a one - component system ( one which can be parameterized by only one chemical potential ) . as we have already discussed , the treatment of neutron star matter as a charge neutral phase in weak equilibrium indeed reduces the number of independent chemical potentials to one . but the essential point is that - if a mp exists - charge neutrality can be achieved in this phase e.g. with a positively charged amount of hadronic matter and a negatively charged amount of quark matter . therefore it is not justified to require charge neutrality in both phases separately . in doing so we would `` freeze out '' a degree of freedom which in principle could be exploited in the mp by rearranging electric charge between both phases to reach `` global '' charge neutrality . a correct treatment of the phase transition therefore only requires both phases to be in weak equilibrium , i.e. both phases still depend on two independent chemical potentials . we have chosen the pair ( @xmath99 , @xmath97 ) . such a system is called a two - component system . the gibbs condition for mechanical and chemical equilibrium at zero temperature between both phases of the two - component system reads @xmath128 using eq . ( [ gibbscondition ] ) we can calculate the equilibrium chemical potentials of the mp where @xmath129 holds . fig.[p3d ] illustrates this calculation . the hp@xmath130mp phase transition takes place if the pressure of the charge neutral hp ( white line ) meets the pressure surface of the qp ( njl ) . up to this point the pressure of the qp is below the pressure of the hp making the hp the physically realized one . at higher pressure the physically realized phase follows the mp curve which is given by the gibbs condition ( [ gibbscondition ] ) . finally the mp curve meets the charge neutral qp curve ( white line ) and the pressure of the qp is above the pressure of the hp , making the qp the physically realized one . for every point on the mp curve one now can calculate the volume proportion @xmath131 occupied by quark matter in the mp by imposing the condition of global charge neutrality of the mp @xmath132 here @xmath133 and @xmath134 denote the respective charge densities . from this , the energy density @xmath135 of the mp can be calculated by @xmath136 along the mp curve the volume proportion occupied by quark matter is monotonically increasing from @xmath137 to @xmath138 where the transition to the pure qp takes place . taking ( i ) the charge neutral eos of the hp at low densities ( sec . [ sec : hp ] ) , ( ii ) eq . ( [ gibbscondition ] ) , ( [ globalcharge ] ) , and ( [ epsilonqp ] ) for the mp , and ( iii ) the charge neutral eos of the qp ( sec . [ sec : qp ] ) we can construct the full eos in the form @xmath139 . for simplicity we denote this eos as the _ hybrid star _ eos . fig . [ pepsgps ] shows this eos if we apply gps for the hp eos . the hybrid star eos consists of three distinct parts . at low densities ( @xmath140 ) matter is still in its confined hp . at ( @xmath141 ) the first droplets of deconfined quark matter appear . above this density matter is composed of a mixed phase of hadronic and quark matter . this mp part of the eos is shaded gray . only at unaccessible high densities ( @xmath142 ) matter consists of a pure qp . the preceding statements refer to the use of gps for the eos of the hp . concerning all the other variants of rmf used here ( tm1 , tm2 , gl85 ) we have found that the hp@xmath130mp transition does not appear below @xmath143 . as we will discuss below such high energy densities can not be reached inside a stable neutron star which is described by one of these eos . at this point we should note the essential difference between the treatment of neutron star matter as a one- and a two - component system ( cf . while the former one leads to the well known phase transition with a constant pressure mp ( like in the familiar liquid - gas phase transition of water ) , we can see in fig.[pepsgps ] that the pressure is monotonically increasing even in the mp if we apply the correct two - component treatment . this has an important consequence on the structure of the neutron star . since we know from the equations of hydrostatic equilibrium the tolman - oppenheimer - volkoff ( tov ) equations @xcite that the pressure has to increase if we go deeper into the star , a constant pressure mp is strictly excluded from the star while a mp with increasing pressure can ( in principle ) occupy a finite range inside the star . to see if the densities inside a neutron star are high enough to establish a mp or a qp in its center we have to solve the tov equations with a specified hybrid star eos following from our phase transition calculation . from the solutions of the tov equations we get a relation between the central energy density ( or central pressure ) and the mass of the neutron star ( cf . [ fig : meps ] ) . the maximum possible central energy density ( the critical energy density @xmath144 ) is reached at the maximum mass that is supported by the fermi pressure of the particular hybrid star eos . above this critical density the neutron star gets instable with respect to radial modes of oscillations @xcite . we have applied the four hp eos s ( denoted by gps , tm1 , tm2 and gl85 ) to calculate the four corresponding hybrid star eos s . ( the one for gps is shown in fig.[pepsgps ] . ) we found that in _ no _ eos the central energy density of a typical @xmath145 neutron star is large enough for a deconfinement phase transition . ( here @xmath146 denotes the mass of the sun . ) the corresponding neutron stars are purely made of hadronic matter ( hp ) . in fig . [ pepsgps ] where gps is used also the neutron star masses are shown as a function of the central energy density . there the central energy density of a @xmath147 is about @xmath148 which is clearly below @xmath141 which is at least necessary to yield a mp core . ( this is also shown in the context of the gibbs construction in fig.[p3d ] where the central pressure of a @xmath147 and of a @xmath149 neutron star is marked . ) in fig . [ pepsgps ] we can see that only near the maximum mass of @xmath150 neutron stars with a mp core are possible . this , however , only holds for the gps hybrid star eos and only in a quite narrow mass range from @xmath151 . in the density range up to the critical density all other eos s ( tm1 , tm2 , gl85 ) do not show a phase transition at all . the critical energy densities for these eos s are in the range of @xmath152 while the densities for the hp@xmath130mp transition are above @xmath143 . ( the corresponding maximum masses are @xmath153 . ) from this we conclude that within the model constructed here the appearance of deconfined quark matter in the center of neutron stars turns out to be very unlikely . we have studied the possible phase transition inside neutron stars from confined to deconfined matter . for the description of the quark phase we have utilized the njl model which respects chiral symmetry and yields dynamically generated quark masses via the effect of spontaneous chiral symmetry breaking . we found that the appearance of deconfined quark matter in the center of a neutron star appears to be very unlikely , for most of the studied hadronic eos s even impossible . the ultimate reason for that effect is the high value of the effective strange quark mass which turns out to be much higher than its current mass value in the whole relevant density range ( cf . [ fig : mrho ] ) . this finding , of course , is based on several assumptions which need not necessarily be correct . in lack of an eos based on a full qcd calculation at zero temperature and finite nuclear density , we had to rely on simpler models for the eos in different density regimes . concerning the low density regime we have used various relativistic mean field ( rmf ) models . to some degree the use of different variants of the rmf model should reflect the uncertainties of this approach . these models are generalizations of the walecka model @xcite which describes the hadronic ground state of nuclear matter at density @xmath110 quite successfully . at somewhat higher densities the used rmf models deal with hyperons as additional degrees of freedom . since these rmf models do not have any explicit quark degrees of freedom we expect them to become unreliable at high densities where the confinement forces are screened and the hadrons dissolve into quarks . to describe this high density regime we have utilized the nambu jona - lasinio ( njl ) model in its three - flavor extension . the merits of the njl model are ( at least ) twofold : for the vacuum case , it gives a reasonable description of spontaneous chiral symmetry breaking and of the spectrum of the low lying mesons . for sufficiently high density and/or temperature , the njl model exhibits the restoration of chiral symmetry . a shortcoming of the njl model is that it does not confine quarks , i.e. there is no mechanism which prevents the propagation of a single quark in vacuum . therefore in an njl model calculation the quarks significantly contribute to the eos also at low densities . this was the ultimate reason why we considered the njl model only in the high density regime where confinement is supposed to be absent anyway while utilizing the hadronic rmf models to describe the confined phase . on the other hand , we should recall that energy density and pressure of the njl model were determined such that both vanish at zero density , i.e. in a regime where we have not utilized the njl model afterwards . this procedure determines the effective bag pressure @xmath82 given in ( [ eq : defeffbag ] ) by fixing @xmath87 ( [ eq : defb0 ] ) to @xmath154 . clearly , this procedure is somewhat unsatisfying since the effective bag pressure @xmath82 influences the eos and therefore the onset of the phase transition . indeed , if we reduce @xmath87 by only @xmath155% by hand from its original value of @xmath154 we already observe drastic changes in the phase structure of the neutron star favoring deconfined quark matter . on the other hand , the physical requirement that any model should yield vanishing energy density and pressure in vacuum is the only way to uniquely determine the eos of the njl model without any further assumptions . possible alternatives to the use of the njl model for the description of the deconfined quark matter are the mit bag model @xcite and the extended effective mass bag model @xcite . the latter includes medium effects due to one - gluon exchange which rise with density ( while the effective masses of the njl model decrease ) . as already discussed at the end of sec . [ sec : qp ] for the regime of very high densities such a resummed perturbation theory might be more adequate . furthermore , mit bag models can be very useful in interpreting more involved models like the njl model in terms of simple physical quantities like the bag constant and the quark masses . therefore we have frequently compared our njl model results with the mit bag model in sec . [ sec : qp ] . for a further discussion of the mit bag model and its application to neutron stars see @xcite . the distinct feature of the njl model is that nonperturbative effects are still present beyond the phase transition point . it is reasonable to consider such effects since it was found in lattice calculations @xcite that for qcd at finite temperature the eos beyond the phase transition point can neither be properly described by a free gas of quarks and gluons nor by qcd perturbation theory @xcite . presumably this holds also for the finite density regime . on the mean field level the most prominent nonperturbative feature of the njl model which remains present beyond the phase transition point is the constituent strange quark mass which is much larger than the current strange quark mass in the whole relevant density regime ( cf . [ fig : mrho ] ) . this high strange quark mass has turned out to be crucial for the phase transition . as we have shown above the eos of the qp can be reasonably well approximated by an mit bag eos up to 5 times @xmath110 using a comparatively low bag constant of @xmath121mev and an effective strange quark mass of @xmath118mev . these are the plateau values of the corresponding quantities in the njl model calculations shown in figs . [ fig : mrho ] and [ fig : brho ] . in figs . [ fig : earho]-[fig : peps ] the curves labeled by mit use these values for the bag constant and the effective strange quark mass . using such a bag constant in connection with the _ current _ strange quark mass would allow the existence of a qp inside a neutron star @xcite . this , however , does not remain true once a much higher effective strange quark mass is used . qualitatively , the chain of arguments is that a higher mass leads to a lower pressure ( cf . [ fig : pmub ] ) . this disfavors the quark phase in the gibbs construction , i.e. shifts the phase transition point to higher densities . this is the reason why in our calculations the existence of quark matter in the center of a neutron star is ( nearly ) excluded . especially for typical neutron stars with masses @xmath156 the central energy density is far below the deconfinement phase transition density ( cf . [ fig : peps ] ) . this finding is independent of the choice of the version of the rmf model . this suggests that it is the njl model with its large strange quark mass which defers the onset of the deconfinement phase transition rather than the modeling of the hadronic phase . throughout this work we have used the njl parameter set of @xcite . we have also explored the set given in @xcite with @xmath157mev , @xmath158 , @xmath159 , @xmath160mev , and @xmath161mev . the results are very similar to the ones presented here . for simplicity we have treated in the present work the njl model in the hartree approximation . in principle , going beyond the mean field approximation might influence the order of the chiral phase transition ( for related work towards that direction for the two - flavor case cf . if it turned out that this would result in a strong first order phase transition then the effective strange quark mass might change more drastically and in the region of interest would be perhaps much lower than in the case studied in the present work . this would favor the appearance of quark matter in the interior of neutron stars . clearly , it would be interesting to study how a more involved treatment of the njl model beyond the hartree approximation would influence our findings presented here . this , however , is beyond the scope of the present work . p. papazoglou , s. schramm , j. schaffner - bielich , h. stcker , and w. greiner , phys . c57 ( 1998 ) 2576 ; + p. papazoglou , d. zschiesche , s. schramm , j. schaffner - bielich , h. stcker , and w. greiner , phys . c59 ( 1999 ) 411 . g. baym , c.j . pethick , and p. sutherland , astrophys . j. 170 ( 1971 ) 299 ; + r.p . feynman , n. metropolis , and e. teller , phys . rev . 75 ( 1949 ) 1561 ; + g. baym , h.a . bethe , and c.j . pethick , nucl . a175 ( 1971 ) 225 . glendenning , f. weber , and s.a . moszkowski , nucl . a572 ( 1994 ) 693 ; + j.i . kapusta and k.a . olive , phys . lett . 64 ( 1990 ) 13 ; + j. ellis , j.i . kapusta , and k.a . olive , nucl . b348 ( 1991 ) 345 .

we study the possible existence of deconfined quark matter in the interior of neutron stars using the nambu jona - lasinio model to describe the quark phase . we find that typical neutron stars with masses around 1.4 solar masses do not possess any deconfined quark matter in their center . this can be traced back to the property of the njl model which suggests a large constituent strange quark mass over a wide range of densities .

let @xmath5 be a compact torus with lie algebra @xmath6 and lattice @xmath7 . suppose that @xmath5 acts on a compact symplectic manifold @xmath8 with isolated fixed points and moment map @xmath9 , where @xmath10 is dual to @xmath6 . then @xmath11 where @xmath12 denotes the vector field on @xmath2 generated by the action and @xmath13 is defined by @xmath14 . here , @xmath15 is the natural pairing between @xmath10 and @xmath6 . if @xmath16 is * generic * , that is , if @xmath17 for each weight @xmath18 in the symplectic representation @xmath19 for every @xmath20 in the fixed set @xmath21 , then @xmath22 is a morse function with critical set @xmath21 . given @xmath23 , the negative tangent bundle @xmath24 is a representation with no fixed sub - bundle . hence , the index of @xmath0 at @xmath20 is even ; let @xmath25 denote half the index of @xmath0 at @xmath20 . the individual weights of this representation are well defined and non - zero ; our convention for the moment map implies that these weights are exactly the * positive weights * of the @xmath5 action on @xmath19 , that is , the weights @xmath26 such that @xmath27 . let @xmath28 denote the product of these weights . ( conversely , the weights in the positive tangent bundle are the _ negative weights _ of the @xmath5 action on @xmath29 . ) finally , for all @xmath30 the inclusion @xmath31 induces a map @xmath32 in equivariant cohomology ; let @xmath33 denote the image of a class @xmath34 under this map . [ de : canonical ] let a torus @xmath5 act on a compact symplectic manifold @xmath8 with isolated fixed points and moment map @xmath35 . let @xmath36 be a generic component of the moment map . a cohomology class @xmath37 is the * canonical class * at a fixed point @xmath20 with respect to @xmath0 if 1 . @xmath38 2 . @xmath39 for all @xmath40 such that @xmath41 . is stronger than the frequently encountered condition that @xmath42 for all @xmath43 such that @xmath44 . see lemmas [ le : pclass ] and [ le:2prime ] . ] moreover , we say that the canonical class @xmath1 is * integral * if @xmath45 is torsion free ; see lemma [ le : pclass ] . therefore , we can naturally identify @xmath46 with a subgroup of @xmath47 . ] we can not always find canonical classes ; see example [ ex : cp2 ] . however , each canonical class is unique and can be thought of as an equivariant poincar dual to the closure of the stable manifold . if @xmath1 exists for all @xmath23 , then @xmath48 forms a basis of @xmath49 as a module over @xmath50 . since the fixed set is isolated , the natural restriction map @xmath51 is surjective ; under this map , the canonical classes also define a basis for the ordinary cohomology @xmath52 . in the case that @xmath53 , where @xmath54 is a complex semi - simple lie group ( of any type ) and @xmath55 is a borel subgroup , the equivariant schubert classes are canonical classes . under the map to ordinary cohomology , they are exactly the poincar duals to schubert varieties in ordinary cohomology . hence , our work is a direct generalization of that setting . this paper is concerned with a new formula for how to restrict canonical cohomology classes to fixed points . since the fixed points are isolated , the inclusion of the fixed point set @xmath21 into @xmath2 induces an injection @xmath56 , where the latter ring is a direct sum of polynomials rings . thus each cohomology class on @xmath2 may be described by an integral polynomial associated to each fixed point . once the restriction of canonical classes is known at each fixed point , one can easily derive a formula for the structure constants in the ( equivariant ) cohomology ring . ( see @xcite . ) recall that the structure constants for @xmath49 are the set @xmath57 given by @xmath58 conversely , the structure constants also provide a formula for the restrictions . our formulas have some echoes in the literature ; s. billey @xcite found a different manifestly positive formula for the restriction of equivariant schubert classes when @xmath59 . v. guillemin and c. zara @xcite found a non - positive path formula for the restrictions in the case of gkm graphs , which we discuss in more detail below . our main contribution in this article can be seen as an inductive formula for the restriction of canonical classes to fixed points ; we prove this in section [ se : induction ] . the formula depends on only the values of the moment map and @xmath60 , where @xmath61 and @xmath62 are fixed points whose indices differ by two . given a directed graph with vertex set @xmath63 and edge set @xmath64 , a * path * from a vertex @xmath20 to a vertex @xmath4 is a @xmath65-tuple @xmath66 so that @xmath67 , @xmath68 , and @xmath69 for all @xmath70 ; let @xmath71 denote the * length * of @xmath72 . [ th : pathformula ] let a torus @xmath5 act on a compact symplectic manifold @xmath8 with isolated fixed points and moment map @xmath9 . let @xmath73 be a generic component of the moment map . assume that there exists a canonical class @xmath74 for all @xmath23 . define an oriented graph with vertex set @xmath75 and edge set @xmath76 given @xmath20 and @xmath4 in @xmath21 , let @xmath77 denote the set of paths from @xmath20 to @xmath4 in @xmath78 ; then @xmath79 [ * positivity * ] [ positivity ] we say that @xmath80 is * positive * if @xmath81 and * negative * if @xmath82 . in some cases , the restriction @xmath83 is itself negative ; see example [ ex : nonkahler ] . a fortiori , in these examples some of the summands in are negative . however , whenever @xmath84 for all @xmath20 and @xmath30 such that @xmath85 , our formula is * manifestly positive * , in the sense that each summand is positive . to see this , note that @xmath86 and @xmath87 are positive by definition , @xmath88 and @xmath89 are positive by corollary [ co : increasing ] , and @xmath90 is positive by assumption . for example , for flag varieties @xmath91 of semisimple lie groups the canonical classes are schubert classes ; see @xcite . in this case , the restriction @xmath83 is positive for all @xmath20 and @xmath4 by @xcite . alternatively , it is very easy to check this directly when @xmath85 ; see section [ se : examples ] for the case @xmath92 and @xcite for the general case . consider the situation described in theorem [ th : pathformula ] . if there is no path in @xmath78 from a fixed point @xmath20 to a fixed point @xmath4 , then @xmath93 . moreover , if @xmath84 for all @xmath20 and @xmath4 in @xmath21 such that @xmath94 , then @xmath84 for all @xmath20 and @xmath4 in @xmath21 and @xmath95 exactly if there is at least one path from @xmath20 to @xmath4 . we now restrict our attention to an important special case where it is especially easy to make these calculations : gkm spaces . let a torus @xmath5 act on a compact symplectic manifold @xmath8 with moment map @xmath96 . we say that @xmath97 is a * gkm space * if @xmath2 has isolated fixed points and if , for every codimension one subgroup @xmath98 , every connected component of the fixed submanifold @xmath99 has dimension two or less . [ gkmgraph ] let @xmath97 be a gkm space . we define the * gkm graph * to be the labelled directed graph @xmath100 given as follows . the vertex set @xmath63 is the fixed set @xmath21 ; we label each @xmath23 by its moment image @xmath101 . the edge set @xmath102 consists of pairs of distinct points @xmath103 such that there exists a codimension one subgroup @xmath98 so that @xmath20 and @xmath4 are contained in the same component @xmath104 of @xmath99 . we label the edge @xmath105 by the weight @xmath106 associated to the representation of @xmath5 on @xmath107 . let @xmath108 be a generic component of the moment map . note that @xmath25 is the number of edges @xmath109 such that @xmath110 ; moreover , @xmath111 , where the product is over all such edges . we say that @xmath0 is * index increasing * if @xmath112 implies that @xmath113 for all @xmath114 . see example [ ex : indexincreasing ] and remark [ rm : increasing ] . given any weight @xmath115 , the projection which takes @xmath116 to @xmath117 naturally induces a endomorphism @xmath118 of @xmath119 , the symmetric algebra on @xmath10 . since @xmath2 is a gkm space , the weights at each fixed point are pairwise linearly independent ; hence , @xmath120 and @xmath121 for all @xmath122 . following @xcite , we define @xmath123 where @xmath124 is the field of fractions of @xmath119 . [ th : gkmpathformula ] let @xmath97 be gkm space . let @xmath73 be a generic component of the moment map ; assume that @xmath0 is index increasing . define an oriented graph with vertex set @xmath125 and edge set @xmath126 where @xmath100 is the gkm graph associated to @xmath2 . then * there exists a canonical class @xmath127 for all @xmath23 . * given @xmath20 and @xmath4 in @xmath21 , let @xmath77 denote the set of paths from @xmath20 to @xmath4 in @xmath128 ; then @xmath129 * @xmath130 for all @xmath131 . [ re : equivalencetheta ] a straightforward calculation shows that , since @xmath132 is an integer , @xmath133 moreover , since @xmath134 is not a multiple of @xmath135 , equation has a unique solution and so provides an alternative definition of @xmath136 . in fact , since @xmath137 is a positive multiple of @xmath135 for all @xmath122 , the formula is a manifestly positive exactly if @xmath138 for all @xmath122 ; cf . remark [ positivity ] . however , @xmath132 is not always positive ; see example [ ex : nonkahler ] . [ cpn ] @xmath139 denote the standard basis for @xmath140 , and @xmath141 denote the dual basis for @xmath142 . let @xmath143 be the diagonal circle , and @xmath144 the quotient torus . the standard action of @xmath145 on @xmath146 induces a symplectic action of @xmath5 on complex projective space @xmath147 with moment map @xmath148 , where @xmath149 and @xmath150 ) = \sum_{i=1}^{n+1 } \left ( \frac{1}{n+1 } - \frac { |z_i|^2 } { \sum_j |z_j|^2 } \right ) x_i.\ ] ] it is straightforward to check that @xmath151 is a gkm space , and that the associated gkm graph is the complete directed graph on @xmath152 vertices @xmath153 , where @xmath154 $ ] . moreover , @xmath155 for all @xmath156 , and @xmath157 for all @xmath158 . let @xmath73 , where @xmath159 . then @xmath160 is positive exactly if @xmath161 , and so @xmath162 and @xmath163 for all @xmath156 . therefore , @xmath0 is index increasing and @xmath164 . in particular , there is exactly one path @xmath165 from @xmath166 to @xmath167 in @xmath168 if @xmath169 ; otherwise , there is none . finally , since @xmath170 , @xmath171 thus , by theorem [ th : gkmpathformula ] , @xmath172 guillemin and zara also give a formula for @xmath83 for gkm spaces as a sum over paths in @xcite . in fact , their formula is identical to ours in the case that @xmath173 , and also works in a slightly broader context . however , in general the formulas are quite different . for example , their formula for @xmath83 includes a contribution for each path @xmath61 from @xmath20 to @xmath4 in @xmath100 such that @xmath174 for each edge @xmath175 ; our formula only includes a contribution from a subset of such paths those such that @xmath176 . in practice , this means that we sum over many fewer paths . for example , if @xmath177 their formula for @xmath83 contains @xmath178 terms , whereas ours contains just one term ; see example [ cpn ] . moreover , their formula is almost never manifestly positive , in the sense described above . in @xcite , knutson gives a positive formula for the duistermaat - heckman measure of a torus action on a smooth algebraic variety with an invariant palais - smale metric , and suggests a technique for computing the duistermaat - heckman measure of certain subvarieties . in fact , in the case that @xmath2 is an algebraic variety and there exists an invariant palais - smale metric , it is possible to use the results of @xcite to give an alternate proof of theorem [ th : pathformula ] . we hope to do this in our next paper ; we also plan to use theorem [ th : pathformula ] to extend his formula for the duistermaat - heckman measure to the non - algebraic case . however , in this greater generality , the summands in the formula are _ not _ always positive . this occurs , for example , in the manifold considered in example [ ex : nonkahler ] . after we initially announced these results , knutson showed that he could extend his results in @xcite by dropping the condition that there exist an invariant palais - smale metric ; see @xcite . finally , several techniques have recently been discovered which use the ideas in this paper to find a positive integral formula in certain important cases , including when @xmath2 is a flag manifold ( @xcite,@xcite ) . we would like to thank victor guillemin , whose questions inspired this project . we would also like to thank sara billey , allen knutson , catalin zara , and silvia sabatini for many helpful discussions . in this section , we demonstrate some properties of canonical classes . in particular , we show that if they exist then they form a natural basis for @xmath179 as a @xmath180 module . additionally , they do exist in a number of important cases . for this purpose , it is natural to work in a slightly more general context . let a torus @xmath5 act on a compact oriented manifold @xmath2 with isolated fixed points . an invariant morse function @xmath181 is a * formal moment map * if the critical set of @xmath0 is exactly the fixed point set @xmath21 . as we saw in the introduction , if @xmath2 is symplectic and the action is hamiltonian , then any generic component of the moment map is a formal moment map . the cohomological properties of symplectic manifolds with hamiltonian actions described in the introduction continue to hold in the more general case of formal moment maps ; see appendix g of @xcite . in particular , the restriction map from @xmath46 to @xmath182 is injective in this case . let @xmath20 be a critical point for a formal moment map @xmath183 . since @xmath24 is a real representation with no fixed subbundle , the index of @xmath0 at @xmath20 is even ; let @xmath25 denote half the index of @xmath0 at @xmath20 . the signs of the individual weights of this representation are not well defined . however , if we fix an orientation on the negative normal bundle @xmath24 then the product of the weights which we will denote @xmath28 is well - defined . [ de : canclass ] let a torus @xmath5 act on a compact oriented manifold @xmath2 with isolated fixed points , and let @xmath184 be a formal moment map . we say that a cohomology class @xmath37 is a * canonical class * ( with respect to @xmath0 ) at a fixed point @xmath20 if there exists an orientation on @xmath24 such that 1 . @xmath38 , and 2 . @xmath39 for all @xmath40 such that @xmath41 . moreover , the canonical class is * integral * if @xmath185 . canonical classes do not always exist . [ ex : cp2]let the torus @xmath186 act on @xmath187 by @xmath188 = [ t_1 z_1 : t_2 z_2 : z_3].\ ] ] let @xmath2 be the blow - up of @xmath187 at @xmath189 $ ] , and let @xmath190 be the moment map for the induced @xmath5 action . let @xmath191 where @xmath192 . label the four fixed points @xmath193 so that @xmath194 . there exists a basis @xmath195 for @xmath179 as a @xmath180 module so that @xmath196 and @xmath197 a straightforward calculation shows that there is no canonical class for @xmath198 . ( although this example is a gkm space , it is consistent with theorem [ th : gkmpathformula ] because @xmath0 is not index increasing . ) however , if canonical classes do exist , they give a natural basis for @xmath49 . [ prop : basis ] let a torus @xmath5 act on a compact oriented manifold @xmath2 with isolated fixed points , and let @xmath199 be a formal moment map . fix an orientation on @xmath24 for all @xmath23 . if there exists a canonical class @xmath200 for all @xmath23 , then the classes @xmath201 are a natural basis for @xmath202 as a module over @xmath203 . moreover , if the canonical classes are integral then they are a natural basis for @xmath179 as a module over @xmath180 . this is an immediate consequence of lemmas [ le : pclass ] , [ le : canprop ] , and [ le:2prime ] below . [ sensitive ] this basis does not depend very sensitively on the choices that we have made ; it only depends on the @xmath28 s at each fixed point . for example , let a torus @xmath5 act on a compact symplectic manifold @xmath8 with isolated fixed points and moment map @xmath35 . let @xmath73 be a generic component of the moment map . let @xmath204 be a canonical class at @xmath20 for @xmath0 . if @xmath205 is another generic component , then @xmath1 is also a canonical class for @xmath206 , as long as @xmath207 where @xmath208 denotes the set of weights at @xmath20 . similarly , if @xmath209 is another invariant symplectic form with moment map @xmath210 , then @xmath1 is also a canonical class for @xmath211 as long as @xmath212 and @xmath213 are deformation equivalent . the following lemma is a key ingredient in our proof that canonical classes exist in certain cases . in particular , the existence of these closely related classes is guaranteed by straightforward morse theoretic arguments . [ kirwan ] [ le : pclass ] let a torus @xmath5 act on a compact oriented manifold @xmath2 with isolated fixed points , and let @xmath183 be a formal moment map . for every fixed point @xmath20 and for each orientation on @xmath24 , there exists an integral cohomology class @xmath214 so that 1 . @xmath215 , and 2 . @xmath216 for every @xmath40 such that @xmath44 . moreover , for any such classes , the @xmath217 are a basis for @xmath218 as a module over @xmath180 . kirwan proved this result for rational cohomology classes on compact symplectic manifolds with hamiltonian actions @xcite ( see also @xcite ) . the proof generalizes easily to the case of formal moment maps , and to integral cohomology when the fixed points are isolated . [ co : towardindexstatement ] let a torus @xmath5 act on a compact oriented manifold @xmath2 with isolated fixed points , and let @xmath183 be a formal moment map . given a point @xmath23 and a class @xmath219 such that @xmath220 whenever @xmath30 satisfies @xmath221 , * the restriction @xmath222 where @xmath223 . * in particular , if @xmath224 then @xmath225 . * if @xmath226 is integral then @xmath227 , where @xmath228 is integral . by lemma [ le : pclass ] for each @xmath30 we can fix an orientation on @xmath229 and choose a class @xmath230 which satisfies properties @xmath231 and @xmath232 moreover , we can write @xmath233 , where @xmath234 lies in @xmath203 for all @xmath4 . if @xmath235 is integral , then each @xmath234 lies in @xmath180 . if @xmath236 for all @xmath30 so that @xmath221 , then properties @xmath231 and @xmath232 together imply that @xmath237 . otherwise , there exists @xmath30 so that @xmath221 and @xmath238 , but @xmath239 for all @xmath61 such that @xmath240 . hence @xmath220 _ and _ @xmath241 , which is impossible . this corollary leads to the following properties of canonical classes . [ le : canprop ] let a torus @xmath5 act on a compact oriented manifold @xmath2 with isolated fixed points , and let @xmath199 be a formal moment map . for all @xmath23 , the canonical class @xmath1 is uniquely determined by the orientation on @xmath24 . fix an orientation on @xmath24 and let @xmath1 and @xmath242 be canonical classes for @xmath23 . consider the class @xmath243 . assume @xmath244 . then , since the restriction map from @xmath179 to @xmath245 is injective , there exists @xmath30 such that @xmath246 but @xmath247 for all @xmath248 satisfying @xmath240 . by the definition of canonical class , @xmath249 for all @xmath250 such that @xmath251 ; therefore @xmath252 . but this contradicts corollary [ co : towardindexstatement ] . [ le:2prime ] let a torus @xmath5 act on a compact oriented manifold @xmath2 with isolated fixed points , and let @xmath199 be a formal moment map . if @xmath253 is a canonical class for a fixed point @xmath20 , then @xmath1 also satisfies the property : * @xmath39 for all @xmath40 such that @xmath254 . there exists a point @xmath30 so that @xmath255 but @xmath256 for all @xmath248 so that @xmath240 . by the definition of canonical classes , the fact that @xmath257 implies that either @xmath258 or @xmath259 . in the latter case , corollary [ co : towardindexstatement ] implies that @xmath39 . thus @xmath258 . this has the following important consequence . [ co : increasing ] let a torus @xmath5 act on a compact oriented manifold @xmath2 with isolated fixed points , and let @xmath199 be a formal moment map . define an oriented graph with vertex set @xmath75 and edge set @xmath76 if there exists a path in @xmath260 from a point @xmath20 to a different point @xmath4 , then @xmath261 . finally , canonical classes do exist in a number of important special cases . for example , the following lemma is a special case of lemma 1.13 in @xcite . [ mcduff - tolman ] let the circle @xmath262 act on a compact oriented manifold @xmath2 with isolated fixed points , and let @xmath199 be a formal moment map . then there exists a canonical class @xmath263 for all @xmath23 . note that if @xmath0 is a formal moment map then @xmath264 is also a formal moment map . if the index of @xmath0 at @xmath265 is @xmath266 , the index of @xmath264 at @xmath4 is @xmath267 . finally , the tangent bundle at @xmath4 is an oriented real representation of @xmath5 ; let @xmath268 denote the product of the weights of this representation . as one might expect , canonical classes for @xmath0 exist for all points @xmath269 exactly when they exist for @xmath264 for all points . [ le : beta ] let a torus @xmath5 act on a compact oriented manifold @xmath2 with isolated fixed points and let @xmath184 be a formal moment map . if there exists a canonical class @xmath253 with respect to @xmath0 for each @xmath269 , then there exists a canonical class @xmath270 with respect to @xmath264 for each @xmath271 moreover , @xmath272 is integral for all @xmath273 if and only if @xmath1 is integral for all @xmath269 . to begin , fix an orientation on @xmath274 for all @xmath248 , and consider the formal moment map @xmath264 . we need to find a class @xmath275 such that 1 . @xmath276 , and 2 . @xmath277 for all @xmath278 such that @xmath279 . by lemma [ le : pclass ] , for all @xmath248 there exists a cohomology class @xmath280 such that 1 . @xmath281 , and 2 . @xmath282 for all @xmath283 so that @xmath284 . now let @xmath272 be any class which satisfies @xmath285 and @xmath286 for @xmath4 . define @xmath287 if @xmath288 , we are done . otherwise , let @xmath289 be an element which maximizes @xmath290 . then @xmath291 where @xmath28 is the product of the weights on @xmath24 with respect to the orientation compatible with @xmath1 . now consider any fixed point @xmath292 . if @xmath293 , then @xmath294 and so @xmath256 . similarly , if @xmath295 then @xmath256 by lemma [ le : canprop ] . on the other hand , if @xmath296 and @xmath297 then @xmath298 since @xmath20 maximizes @xmath290 . thus the product @xmath299 we now integrate using the atiyah - bott - berline - vergne localization formula to obtain @xmath300 this expression is integral if @xmath1 and @xmath272 are both integral . hence , the class @xmath301 satisfies * @xmath302 , * @xmath303 for all @xmath304 such that @xmath305 , and * in particular , @xmath306 also satisfies @xmath285 and @xmath286 for @xmath4 . the result now follows by induction . assume that for each @xmath23 , there exists a canonical class @xmath307 with respect to @xmath0 which is compatible with a chosen orientation on @xmath24 . let @xmath308 be a canonical class with respect to @xmath264 for each @xmath30 described in lemma [ le : beta ] . clearly , @xmath272 can be chosen to be compatible with the orientation of @xmath309 induced by the orientations on @xmath2 and on @xmath229 . then , since @xmath310 , the sets @xmath48 and @xmath311 are dual basis for @xmath49 as an @xmath49-module under the intersection pairing . to see this , note that if @xmath259 , then @xmath312 by degree considerations . if @xmath313 and @xmath314 , then @xmath315 for all @xmath248 by the definition of canonical class , and so @xmath316 by the atiyah - bott - berline - vergne localization formula . finally , by a similar argument , @xmath317 we are now ready to prove theorem [ th : pathformula ] . let a torus @xmath5 act on a compact symplectic manifold @xmath8 with isolated fixed points and moment map @xmath9 . let @xmath73 be a generic component of the moment map . assume that there exists a canonical class @xmath74 for all @xmath23 . define an oriented graph with vertex set @xmath75 and edge set @xmath318 we need to show that for all @xmath20 and @xmath4 in @xmath21 , @xmath319 where @xmath77 denotes the set of paths from @xmath20 to @xmath4 in @xmath128 . by lemma [ le : beta ] , for all @xmath30 , there exists a class @xmath320 satisfying 1 . @xmath276 , and 2 . @xmath277 for all @xmath278 such that @xmath279 . in our proof of , we will also show that @xmath321 for all fixed points @xmath20 and @xmath4 . consider first the case that @xmath322 . if @xmath323 , then @xmath39 by definition . moreover , there are no paths from @xmath20 to @xmath4 in @xmath128 , and so the right hand side of vanishes , as required . if @xmath324 , then @xmath325 . in this case , the right hand side of is a sum over one degenerate path @xmath326 , and the product is the empty product , so the total contribution is @xmath327 . thus also holds in this case . a nearly identical argument proves that is satisfied . next , suppose that @xmath328 . if @xmath257 , then there is one path @xmath105 from @xmath20 to @xmath4 , and the right hand side of is @xmath83 . on the other hand , if @xmath39 , then @xmath329 , and so the right hand side of vanishes . to prove , note that by the definition of canonical class , @xmath330 now consider any fixed point @xmath61 which is not @xmath20 or @xmath4 . if @xmath294 then @xmath256 , while if @xmath296 , then @xmath298 . therefore , @xmath331 since @xmath332 , the integral of @xmath333 over @xmath2 is zero . thus , by the atiyah - bott - berline - vergne localization theorem , @xmath334 therefore , @xmath335 fix @xmath336 , and assume that and hold for all fixed points @xmath20 and @xmath4 so that @xmath337 . consider fixed points @xmath20 and @xmath4 so that @xmath338 . we will prove that and follow for this @xmath20 and @xmath4 . suppose first that @xmath339 a fortiori @xmath340 . then the left hand sides of and vanish by lemma [ le:2prime ] . since there is no path from @xmath20 to @xmath4 by lemma [ co : increasing ] , the right hand sides also vanish . so assume instead that @xmath341 . let @xmath342 be an equivariant extension of @xmath212 . since @xmath343 for all @xmath248 , @xmath344 since @xmath336 , @xmath345 and so the integral of @xmath346 over @xmath2 is zero . therefore , by the atiyah - bott - berline - vergne localization formula , @xmath347 since @xmath341 , we can solve the above equation for @xmath83 ; @xmath348 consider any fixed point @xmath292 or @xmath4 . assume first that @xmath349 , and let @xmath350 . by the inductive assumption @xmath351 therefore , if @xmath352 denotes the set of paths in @xmath353 from @xmath20 to @xmath4 that pass through @xmath61 , then @xmath354\frac { \alpha_{s_{i-1}}(s_i ) } { \lambda^-_{s_i } } } { \prod_{i \in \{0,\ldots , k\ } \smallsetminus \{l\ } } \phi(r)-\phi(s_i ) } \notag \\ \label{restriction } & = \lambda_r \sum_{{\bf s } \in \sigma_p^q(r ) } \prod_{i=1}^k [ \phi(s_i)-\phi(s_{i-1 } ) ] \frac { \alpha_{s_{i-1}}(s_i ) } { \lambda^-_{s_i } } \prod_{i\neq l } \frac{1}{\phi(r)-\phi(s_i)},\end{aligned}\ ] ] where we use the expression @xmath355 as shorthand for @xmath356 . on the other hand , if @xmath294 or @xmath41 , then @xmath357 . by lemma [ co : increasing ] , the right hand side of also vanishes . therefore , holds for all @xmath358 . substituting into , we see that @xmath359 \frac { \alpha_{s_{i-1}}(s_i ) } { \lambda^-_{s_i } } \prod_{i\neq l}\frac{1 } { \phi(r)-\phi(s_i ) } \right ) \\ & = \frac{- \lambda_q^-}{\phi(q ) - \phi(p ) } \sum _ { { \bf s } \in\ , \sigma_p^q } \prod_{i=1}^k [ \phi(s_i ) - \phi(s_{i-1 } ) ] \frac { \alpha_{s_{i-1}}(s_i ) } { \lambda^-_{s_i } } \left ( \sum_{l=1}^{k-1 } \prod_{i \neq l } \frac{1 } { \phi(s_l ) - \phi(s_i ) } \right ) \\ & = \frac{- \lambda_q^-}{\phi(q ) - \phi(p ) } \sum _ { { \bf s } \in\ , \sigma_p^q } \prod_{i=1}^k [ \phi(s_i ) - \phi(s_{i-1 } ) ] \frac { \alpha_{s_{i-1}}(s_i ) } { \lambda^-_{s_i } } \left ( - \prod_{i = 1}^{k-1 } \frac{1 } { \phi(s_k ) - \phi(s_i ) } \right ) \\ & = \lambda_q^- \sum _ { { \bf s } \in\ , \sigma_p^q } \prod_{i=1}^k [ \phi(s_i ) - \phi(s_{i-1 } ) ] \frac { \alpha_{s_{i-1}}(s_i ) } { \lambda^-_{s_i } } \prod_{i = 1}^{k } \frac{1 } { \phi(q ) - \phi(s_{i-1 } ) } , \\\end{aligned}\ ] ] where the third equality is by lemma [ le : cpncancellation ] . the proof of is nearly identical . in fact , the same proof works if a torus @xmath5 acts on a manifold @xmath2 with isolated fixed points , @xmath184 is a formal moment map , and @xmath360 is any closed equivariant @xmath361-form ( not necessarily symplectic ) so that @xmath341 for every pair of fixed points @xmath20 and @xmath4 so that there is a path ( of length two of more ) in @xmath78 from @xmath20 to @xmath4 . the following corollary is immediate . consider the situation described in theorem [ th : pathformula ] . if @xmath73 achieves its minimum value at @xmath20 , then for any fixed point @xmath4 , @xmath362 1 = _ q^- _ * r * _ p^q _ i=1^|*r*| , @xmath362 where @xmath77 denotes the paths in @xmath78 from @xmath20 to @xmath4 . in particular , every fixed point is connected by a path in @xmath78 to the minimum ( and to the maximum ) . since @xmath363 , @xmath364 ; hence , @xmath365 for all fixed points @xmath4 . our proof of theorem [ th : pathformula ] relies on the following fact , which was also proved in @xcite using different techniques . [ le : cpncancellation ] given @xmath336 distinct vectors @xmath366 in a vector space @xmath63 , @xmath367 the @xmath368 action on @xmath369 induces a symplectic action of on @xmath370 with fixed points @xmath371 . so that the weights at @xmath372 are @xmath373 $ ] . since @xmath374 , the integral of @xmath375 over @xmath370 is @xmath376 . therefore , by the atiyah - bott - berline - vergne localization theorem , the following equation holds in the field of rational functions @xmath377 : @xmath378 since the @xmath379 s are distinct , the claim follows easily . [ ex : indexincreasing ] here generic component of how example 2.1 has an index - increasing component of moment map . i think that we can skip this ! the main goal of this section is to prove theorem [ th : gkmpathformula ] . in fact , this theorem is an immediate consequence of theorem [ th : pathformula ] and the theorem below . [ th : gkmonestep ] let @xmath97 be gkm space , and let @xmath100 be the associated gkm graph . let @xmath73 be a generic component of the moment map ; assume that @xmath0 is index increasing . then * there exists a canonical class @xmath380 for all @xmath23 . * given @xmath20 and @xmath4 in @xmath21 such that @xmath173 , @xmath381 * @xmath130 for all @xmath382 . _ conversely , if there exists a canonical class @xmath200 for all @xmath23 , then @xmath0 is index increasing . to see this , suppose that there exists an edge @xmath114 so that @xmath261 and @xmath279 . on the one hand , since @xmath1 is a canonical class this implies that @xmath39 . on the other hand , compatibility along @xmath105 guarantees that @xmath383 is a multiple of @xmath384 . sine @xmath385 is not a multiple of @xmath384 , this implies that @xmath257 . _ [ rm : increasing]_there are several situations where we can immediately conclude that @xmath0 is index increasing . _ for example , if there is an @xmath5-invariant palais - smale metric @xmath386 on @xmath2 , then @xmath0 is index increasing . to see this , consider an edge @xmath114 such that @xmath261 . there exists a codimension one subgroup @xmath387 so that @xmath20 and @xmath4 are contained in the same component @xmath104 of @xmath99 . since the metric is @xmath5-invariant , @xmath388 must be contained in both the flow up from @xmath20 and the flow down from @xmath4 . since these flows intersect transversally , this implies that the intersection has dimension @xmath389 , which proves @xmath390 . similarly , if @xmath391 for all @xmath156 such that @xmath392 , then proposition 3.4 in @xcite implies that every generic component of the moment map is index increasing . our proof of the theorem above relies heavily on a technical proposition , proposition [ pr : gkm ] , for which we need a few definitions . let @xmath97 be a gkm space , and let @xmath100 be the associated gkm graph . fix a generic @xmath16 and consider the morse function @xmath393 a path @xmath61 in @xmath100 is * ascending * if @xmath394 for all @xmath156 such that @xmath395 ; it is * descending * if @xmath396 for all such @xmath156 . given @xmath23 , the * stable set * of @xmath20 , denoted @xmath397 , is the set of @xmath398 such that there exists an ascending path from @xmath20 to @xmath4 in @xmath100 ; the * unstable set * of @xmath20 , denoted @xmath399 , is the set of @xmath398 such that there exists an descending path from @xmath20 to @xmath4 . note that @xmath20 itself lies in both @xmath397 and @xmath399 . moreover , since @xmath122 exactly if @xmath400 , @xmath401 exactly if @xmath402 . [ pr : gkm ] let @xmath97 be a gkm space , and let @xmath73 be a generic component of the moment map . * for every fixed point @xmath20 , there exists a class @xmath403 such that * * @xmath385 , and * * @xmath42 for all @xmath404 . * given a class @xmath405 and point @xmath30 , @xmath406 proposition [ pr : gkm](a ) is proved for _ rational _ classes in the more general setting of gkm graphs in @xcite . part ( a ) of this proposition is exactly what geometric intuition leads you to expect . to see this , fix a generic @xmath5-invariant metric on @xmath2 . as the name suggests , the stable set @xmath397 should be the set of vertices which are in the closure of the stable manifold of @xmath20 . moreover , one should be able to adapt the morse theoretic proof of lemma [ le : pclass ] to directly prove that the class @xmath1 which you construct is supported on the set of vertices in the closure of the stable manifold of @xmath20 . part ( b ) is slightly more subtle . given a class @xmath407 and an edge @xmath122 , @xmath408 is a rational multiple is in the image of the restriction map @xmath409 exactly if @xmath410 is a multiple of @xmath384 for every edge @xmath411 ( see @xcite ) . ] of @xmath135 . since the weights at each fixed point are pairwise linearly independent , this immediately implies that for any fixed point @xmath4 , @xmath412 however , the same statement is not true integrally . although @xmath413 must be an integral multiple of @xmath414 for each @xmath415 such that @xmath416 , it might not be an integral multiple of the product of these weights because the weights might not be pairwise relatively prime ; see example [ ex : notprime ] . notice that expression has fewer terms in the product . [ ex : notprime]let @xmath417 act on @xmath418 by @xmath419 , [ w_0:w_1 ] \right ) = \left ( [ t_1 ^ 2\ z_0 : z_1 ] , [ t_2 ^ 2\ w_0:w_1 ] \right);\ ] ] let @xmath9 be the moment map . let @xmath420 and let @xmath73 . the associated gkm graph has four vertices , @xmath421,[1:0 ] ) , \ sn = ( [ 1:0],[0:1 ] ) , \ ns = ( [ 0:1],[1:0 ] ) , \ \mbox{and } nn = ( [ 0:1],[0:1]).\ ] ] there are four ascending edges , @xmath422 , @xmath423 , @xmath424 , and @xmath425 ( and hence also four descending edges ) . the first two have weight @xmath426 , the latter two have weight @xmath427 . there exists a class @xmath428 so that @xmath429 although @xmath430 , @xmath431 because @xmath432 . similarly , @xmath433 therefore , even though @xmath434 is not a multiple of @xmath435 , this example does satisfy in proposition [ pr : gkm ] . we begin by showing that theorem [ th : gkmonestep ] follows from proposition [ pr : gkm ] . fix @xmath23 . by proposition [ pr : gkm ] , there exists a class @xmath436 which satisfies properties @xmath231 and @xmath437 . since @xmath0 is index increasing , @xmath252 for all @xmath438 . hence , @xmath1 is a canonical class . consider @xmath30 such that @xmath85 . if @xmath439 , then @xmath440 , and so @xmath39 . now assume that @xmath114 . there are @xmath441 _ other _ edges @xmath442 of the form @xmath443 with @xmath444 . since @xmath0 is index increasing , @xmath445 and so @xmath446 for all @xmath156 . thus @xmath447 for all @xmath156 . by proposition [ pr : gkm ] , @xmath83 is an integral multiple of @xmath448 ; by degree considerations this implies that @xmath449 on the other hand , compatibility along @xmath105 guarantees that @xmath450 is a multiple of @xmath384 , and hence @xmath451 satisfies @xmath452 since @xmath28 is not a multiple of @xmath384 , the integer @xmath451 is nonzero . finally , a straightforward calculation shows that @xmath453 hence , @xmath454 . we now turn to the proof of proposition [ pr : gkm ] . we begin by establishing some terminology , after which we prove two key lemmas . finally we use these lemmas and an inductive argument on the dimension of @xmath2 to complete the proof . [ de : propq ] let @xmath97 be a gkm space , and let @xmath73 be a generic component of the moment map . we say that @xmath226 is * robustly zero * at @xmath455 if @xmath456 . we say that @xmath235 is * robustly integral * at @xmath273 if @xmath457 thus proposition [ pr : gkm](b ) is the statement that every class @xmath405 is robustly integral at every point @xmath273 . we first show that proposition [ pr : gkm](b ) implies [ pr : gkm](a ) . [ le : gkm1b ] let @xmath97 be a gkm space , and let @xmath73 be a generic component of the moment map . assume that every class @xmath458 is robustly integral at every @xmath30 . then , for every fixed point @xmath20 , there exists a class @xmath403 such that 1 . @xmath385 , and 2 . @xmath42 for all @xmath404 . by lemma [ le : pclass ] , for all @xmath30 there exists a class @xmath459 1 . @xmath461 for all @xmath462 such that @xmath463 . let @xmath436 be any class which satisfies @xmath231 and @xmath232 for @xmath20 . define @xmath464 if @xmath288 , we are done . otherwise , let @xmath465 be an element which minimizes @xmath466 . consider any @xmath467 . since @xmath240 and @xmath4 is minimal , @xmath468 . moreover , since @xmath469 but @xmath440 , we conclude that @xmath470 ; hence @xmath471 . therefore , @xmath472 for each edge @xmath473 such that @xmath474 . since @xmath1 is robustly integral at @xmath4 this implies that there exists @xmath475 so that @xmath476 . the difference @xmath477 satisfies * @xmath478 , * @xmath479 for all @xmath462 such that @xmath463 , and * in particular , @xmath242 also satisfies @xmath231 and @xmath480 for @xmath20 . the result now follows by induction . now we show that proposition [ pr : gkm](a ) plus the assumption that all the @xmath1 are robustly integral at all fixed points implies proposition [ pr : gkm](b ) . [ le : gkm2b ] let @xmath97 be a gkm space , and let @xmath73 be a generic component of the moment map . assume that , for every fixed point @xmath20 , there exists a class @xmath403 such that 1 . @xmath385 , and 2 . @xmath42 for all @xmath404 . @xmath1 is robustly integral at every point @xmath273 . then every class @xmath405 is robustly integral at every point @xmath273 . fix a class @xmath458 . by lemma [ le : pclass ] , we can write @xmath481 fix @xmath248 such that @xmath482 . we claim that @xmath483 for all @xmath484 . if not , then there exists @xmath484 so that @xmath485 but @xmath236 for all @xmath486 such that @xmath221 . but then @xmath487 , which contradicts the assumption that @xmath482 . therefore , if @xmath485 , then @xmath488 , and so @xmath489 ; hence @xmath490 . since each @xmath1 is robustly integral at every point @xmath30 , this completes the proof . assume that proposition [ pr : gkm ] is true for all manifolds of dimension less than @xmath491 ; we will prove that it is true for @xmath2 . the result will then follow by induction . we first consider the case when @xmath5 acts on @xmath2 effectively . choose any @xmath226 and fixed point @xmath273 . by lemma [ le : gkm1b ] , it is enough to prove that @xmath235 is robustly integral at @xmath4 . the proposition is obvious when @xmath492 . hence , @xmath413 is an integer multiple of @xmath414 for every @xmath493 with @xmath494 . since these weights are pairwise linearly independent , this implies that @xmath413 is a _ rational _ multiple of @xmath495 now consider any prime @xmath496 and natural number @xmath497 such that @xmath498 divides the product . let @xmath499 and let @xmath500 be the component of @xmath501 containing @xmath4 . then @xmath502 is also a gkm space , and @xmath503 is a generic component of the moment map . let @xmath504 be the associated gkm graph . given any edge @xmath493 , the weight @xmath414 is a multiple of @xmath496 exactly if @xmath505 . therefore , since @xmath498 divides the product , it divides the ( smaller ) product @xmath506 since @xmath507 , if @xmath226 is robustly zero at @xmath508 , then @xmath509 is robustly zero at @xmath61 , where @xmath510 is the restriction map . moreover , since the @xmath5 action on @xmath2 is effective , @xmath511 . by the inductive hypothesis , this implies that @xmath512 is an integral multiple of the product . therefore , since @xmath498 divides this product , @xmath498 also divides @xmath413 . this proves the proposition when @xmath5 acts effectively . we now consider the general case . by lemma [ le : gkm2b ] , it is enough to show that for each @xmath20 there exists a class @xmath1 which satisfies @xmath231 and @xmath437 and is robustly integral at every point @xmath273 . let @xmath513 and @xmath514 . let @xmath515 be the natural projection , and let @xmath516 be the induced map on the lie algebras . notice that @xmath517 takes the lattice @xmath7 to a sublattice of @xmath518 . since @xmath519 acts naturally on @xmath2 , the map @xmath515 induces maps in equivariant cohomology and @xmath520 be contractible spaces on which @xmath5 and @xmath519 , respectively , act freely . then @xmath5 acts on @xmath521 by @xmath522 . the projection from @xmath523 to @xmath524 descends to a map from @xmath525 to @xmath526 . this induces a map from @xmath527 to @xmath528 . ] @xmath529 since @xmath530 , these fit together into the following commutative diagram . @xmath531 moreover , for each @xmath23 , in degree two the map @xmath532 is identified with the dual map @xmath533 under the natural identification of @xmath534 and @xmath535 with the lattices @xmath536 and @xmath537 , respectively . clearly , there exists a moment map @xmath538 so that @xmath539 is a gkm space . since @xmath540 , @xmath541 is a generic component of the moment map . note that the vertices and edges of the graphs for the @xmath5 action and the @xmath519 action are naturally identical , and that @xmath542 takes the @xmath519-weight of each edge to the @xmath5-weight of the same edge . in particular , @xmath543 , where @xmath544 denotes the @xmath519-equivariant euler class of the negative normal bundle of @xmath0 at @xmath20 . since @xmath519 acts effectively on @xmath2 , the first part of this proof implies there exists a class @xmath545 satisfying 1 . @xmath478 for all @xmath547 3 . @xmath242 is robustly integral at every point @xmath30 . let @xmath548 . then @xmath1 has the desired properties . we conclude our paper with two explicit examples . @xmath549 . the standard action of @xmath550 induces an action on the complete flag manifold @xmath549 whose fixed points are given by flags in the coordinate lines @xmath551 . these are indexed by permutations @xmath552 on @xmath553 letters ; the fixed point corresponding to @xmath554 is given by @xmath555 where the brackets indicate the span of the vectors and @xmath556 is the standard basis of @xmath551 . by abuse of notation , we will denote both the permutation and the corresponding fixed point by @xmath554 . let @xmath557 denote the length of @xmath554 , and choose a generic @xmath558 such that @xmath559 implies @xmath560 for all @xmath561 . note that @xmath562 if there exists a transposition @xmath563 such that @xmath564 and @xmath565 ( in contrast , @xmath102 consists of all @xmath566 such that @xmath567 for some @xmath563 ) . if @xmath568 is the transposition switching @xmath156 and @xmath569 with @xmath570 , then @xmath571 and by our convention , this is considered a _ positive _ weight . we begin by showing that for all edges @xmath562 , @xmath572 . recall that the weights that occur in the representation of @xmath5 on the negative normal bundle at a point @xmath554 are positive weights . the length @xmath557 is also the number of positive weights at @xmath554 . the weight @xmath573 is a positive weight at @xmath574 if and only if @xmath575 is a positive weight @xmath554 . thus there is a bijection of positive weights at @xmath554 and positive weights at @xmath574 excepting @xmath576 . moreover , for each weight @xmath26 at @xmath554 , the weight @xmath575 at @xmath574 has the property that @xmath577 . it follows that @xmath578 . now consider any fixed point @xmath554 , and let @xmath579 be the associated canonical class . theorem [ th : gkmpathformula ] says for any @xmath580 , @xmath581 note that @xmath582 is positive . similarly , @xmath583 is a product of positive weights . finally , @xmath584 and @xmath585 are positive for each @xmath156 . more precisely , @xmath586 where @xmath587 and @xmath588 , where @xmath569 indexes the vertices in a path @xmath589 from @xmath590 to @xmath591 . thus every term in the expression is positive . [ ex : nonkahler ] the integer @xmath592 is not always positive ; when it is negative , the canonical class @xmath1 restricts to a negative value at @xmath4 . in @xcite , the second author demonstrated the existence of a gkm space that does not have a @xmath5-invariant khler metric . the corresponding gkm graph @xmath102 can be expressed as the image of the singular set under the moment map @xmath593 , pictured in figure [ fi : nonkahler ] , where we have represented each pair of edges @xmath105 and @xmath594 by one drawn edge . let @xmath595 be as indicated , and note that @xmath596 is an index increasing component of the moment map . consider the edge corresponding to @xmath105 . indicated on the figure are the positive weights ( excluding @xmath384 ) of the @xmath5 action on the negative normal bundles at @xmath20 and @xmath4 , according to the choice of @xmath558 indicated . under the map @xmath597 , these vectors project to vectors opposite in sign ( and of equal magnitude ) . . it immediately follows from theorem [ th : gkmpathformula ] that @xmath599 . a. knutson , a compactly supported formula for equivariant localization and simplicial complexes of biaynicki - birula decompositions , to appear in _ pure app . q , _ special issue in honor of m. atiyah . s. tolman and j. weitsman , on the cohomology rings of hamiltonian @xmath5-spaces . _ northern california symplectic geometry seminar , _ 251258 , amer ser . 2 , * 196 * , _ amer . soc . , providence , ri , _ 1999 .

the main purpose of this article is to extend some of the ideas from schubert calculus to the more general setting of hamiltonian torus actions on compact symplectic manifolds with isolated fixed points . given a generic component @xmath0 of the moment map , which is a morse function , we define a canonical class @xmath1 in the equivariant cohomology of the manifold @xmath2 for each fixed point @xmath3 . when they exist , canonical classes form a natural basis of the equivariant cohomology of @xmath2 ; in particular , when @xmath2 is a flag variety , these classes are the equivariant schubert classes . we show that the restriction of a canonical class @xmath1 to a fixed point @xmath4 can be calculated by a rational function which depends only on the value of the moment map , and the restriction of other canonical classes to points of index exactly two higher . therefore , the structure constants can be calculated by a similar rational function . our restriction formula is _ manifestly positive _ in many cases , including when @xmath2 is a flag manifold . finally , we prove the existence of integral canonical classes in the case that @xmath2 is a gkm manifold and @xmath0 is _ index increasing_. in this case , our restriction formula specializes to an easily computable rational sum which depends only on the gkm graph .

the cosmological constant problem ( ccp ) is a biggest puzzle in particle physics @xcite , and consists of several pieces . the first piece is that the vacuum energy density @xmath4 can be the cosmological constant @xmath5 and various sources of @xmath4 exist , e.g. , a zero point energy of each particle and potential energies accompanied with phase transitions such as the breakdown of electroweak symmetry via higgs mechanism and the chiral symmetry breaking due to quark condensations . the second one is that @xmath4 can receive large radiative corrections including a cutoff scale . the third one is that the experimental value of @xmath5 is estimated as @xmath6gev@xmath7 where @xmath8 is the newton constant , and @xmath9 . ] from the observation that the expansion of present universe is accelerating @xcite . the energy density defined by @xmath10 is referred as @xmath11dark energy density , and its existence has been a big mystery . the pieces of puzzle are not fitted in the framework of the einstein gravity and the standard model of particle physics . because a fundamental theory including gravity has not yet been established , it would be meaningful to give a suggestion based on an effective description of various experimental results concerning the vacuum energy . it might be necessary to introduce a principle , assumptions and/or a framework beyond common sense of an accepted physics , and then the ccp is replaced by the problem to disclose the essence of new staffs . in this article , we study physics on the ccp in the framework of effective field theory and suggest that a dominant part of dark energy can originate from gravitational corrections of vacuum energy under the assumption that the classical gravitational fields do not couple to a large portion of the vacuum energy effectively , in spite of the coupling between graviton and matters at a microscopic level . the content of our article is as follows . in the next section , we explain the pieces of puzzle on the ccp and their implications . in sect . 3 , we give an effective description for physics concerning the ccp and predict that gravitational corrections of vacuum energy can be a candidate of dark energy . in the last section , we give conclusions and discussions . _ the vacuum energy density @xmath4 can be the cosmological constant @xmath5 . _ the energy - momentum tensor of perfect fluids is given by @xmath12 where @xmath13 , @xmath14 and @xmath15 are an energy density , a pressure and a four - velocity of fluids , respectively . the energy - momentum tensor of vacuum is of the form , @xmath16 where @xmath17 is a constant . from ( [ t - pf ] ) and ( [ t - v ] ) , we obtain the relations , @xmath18 where @xmath19 and @xmath20 are an energy density and a pressure of vacuum , respectively . the vacuum has a negative pressure with @xmath21 . the einstein equation is given by @xmath22 where @xmath23 is a bare cosmological constant . from ( [ t - v ] ) and ( [ e - eq ] ) , the cosmological constant is given by @xmath24 , effectively . _ there can be various sources of @xmath4 . _ the first one is a zero point energy of each particle . for a relativistic bosonic particle with a mass @xmath25 , the energy density @xmath26 and the pressure @xmath27 due to zero point fluctuations are given by @xmath28 respectively . here @xmath29 is a momentum of particle . the second one is the energy density from the higgs potential after the breakdown of electroweak symmetry , and its absolute value is estimated as @xmath30 where @xmath31gev ) is the vacuum expectation value on the neutral component of higgs doublet . the third one is the energy density accompanied with the chiral symmetry breaking due to quark condensations , and its absolute value is estimated as @xmath32 where @xmath33 is the qcd scale . _ the vacuum energy density can receive large radiative corrections including a cutoff scale . _ the zero point energy density is calculated by using an effective potential at the one - loop level , and it naively contains quartic , quadratic and logarithmic terms concerning an ultra - violet ( uv ) cutoff parameter @xmath34 . by imposing the relativistic invariance of vacuum ( [ rel - v ] ) on @xmath26 and @xmath27 , @xmath26 in ( [ zero - point ] ) should be of the form , @xmath35 up to some finite terms . note that the terms proportional to @xmath36 and @xmath37 do not satisfy @xmath38 , and they can be regarded as artifacts of the regularization procedure . after the subtraction of logarithmic divergence , @xmath26 is given by @xmath39 where @xmath40 is a renormalization point . for the higgs boson , its zero point energy is estimated as @xmath41 where we use @xmath42gev for the higgs boson mass and take @xmath43gev corresponding to the temperature of present universe @xmath44k . _ the experimental value of cosmological constant is estimated as @xmath45gev@xmath7 , from the observation that the expansion of our present universe is accelerating . _ the vacuum energy density of universe is theoretically given by @xmath46 where @xmath47 is the zero point energy density due to a particle labeled by @xmath48 and the ellipsis stands for other contributions containing unknown ones from new physics . from ( [ rho - higgs ] ) , ( [ rho - qcd ] ) and ( [ rho - zero - higgs ] ) , we have the inequalities , @xmath49 where @xmath50 is a dark energy density defined by @xmath10 . there is a possibility that the magnitude of @xmath4 becomes the 4-th power of the terascale through a cancellation among various contributions from a higher energy physics based on a powerful symmetry such as supersymmetry . because supersymmetry can not work to reduce @xmath4 close to @xmath50 , an unnatural fine - tuning is most commonly required to realize @xmath50 . from ( [ rho - th ] ) and ( [ rho - ineq ] ) , we have a puzzle that consists of unfitted pieces . nature proposes us a big riddle @xmath51why is the observed vacuum energy density so tiny compared with the theoretical one? and a big mystery @xmath51what is an identity of dark energy density? to uncover a clue of ccp and probe into an identity of @xmath50 , let us start with the question whether @xmath4 in ( [ rho - th ] ) exists in physical reality , it gravitates or the classical gravitational field feels @xmath4 . in the absence of gravity , the vacuum energy from matters @xmath52 itself is not observed directly because there is a freedom to shift the origin of energy . here , matters mean various fields including radiations such as photon except for graviton . for instance , the zero point energy of free fields is removed by taking a normal ordering in the hamiltonian . only energy differences can be physically meaningful , as suggested by the casimir effect . in the presence of gravity , if @xmath4 gravitates , the motion of the planets in our solar system can be affected by @xmath4 @xcite . from the non - observation of such an effect for mercury , we have a constraint , @xmath53 as seen from ( [ rho - higgs ] ) , ( [ rho - qcd ] ) and ( [ rho - zero - higgs ] ) , the existence of @xmath54 , @xmath47 ( for particles heavier than 10ev ) and/or @xmath55 threatens the stability of our solar system . hence , it seems to be natural to suppose that the classical gravitational field does not feel a large portion of @xmath52 . in contrast , the ratios of the gravitational mass to the inertial mass stay for heavy nuclei , and hence it is reasonable to conclude that the equivalence principle holds with accuracy at the atomic level and the gravitational field couples to every process containing radiative corrections , accompanied by an emission and/or an absorption of matters . more specifically , the external matter - dependent part of energies must gravitate in both macroscopic and microscopic world . now , let us move to the next step , as phenomenological ingredients of ccp are already on the table . first , based on a standpoint that the einstein gravity is a classical effective theory , physics on the ccp can be described by @xmath56,~~ \rho_{\rm de } \equiv \varlambda_{\rm c(exp)}/(8 \pi g ) = 2.4 \times 10^{-47 } { \rm gev}^4 , \label{scl}\end{aligned}\ ] ] where @xmath57 is the ricci scalar made of the classical gravitational field @xmath58 , @xmath59 , @xmath60 is the lagrangian density of matters as classical objects and @xmath60 does not contain a constant term . because an effective theory is , in general , an empirical one , it would not be so strange even if it can not answer the questions why a large portion of @xmath52 does not gravitate and what the identity of dark energy is . those questions remain as subjects in a fundamental theory . second , we explain why it is difficult to derive ( [ scl ] ) in the framework of ordinary quantum field theory , starting from the action , @xmath61 , \label{s}\end{aligned}\ ] ] where @xmath62 is the ricci scalar made of the graviton @xmath63 , @xmath64 , and @xmath65 and @xmath66 are the lagrangian densities for the standard model particles and other particles beyond the standard model , respectively . the amplitudes representing the coupling between gravitons and the vacuum energy are evaluated by calculating green s functions , @xmath67 for example , on the background minkowski spacetime , two - point function is written as @xmath68 where @xmath69 is the quantum part of graviton in the interaction picture and @xmath70 is the interaction hamiltonian density . the transition amplitude is obtained by removing the propagators on the external lines . the vacuum expectation value @xmath71 corresponds to the vacuum energy density , and then a large cosmological constant term is derived after the identification of classical gravitational fields for external gravitons . here , external gravitons mean gravitons in real states represented by wave functions . third , to reconcile ( [ scl ] ) and ( [ g2 ] ) , we need a radical idea and take a big assumption that _ the classical gravitational fields do not couple to a large portion of the vacuum energy effectively , in spite of the coupling between graviton and matters at a microscopic level . _ we expect that it stems from unknown features of external gravitons . for example , if a kind of exclusion principle works , as a bold hypothesis , that _ external gravitons can not take the same place in the zero total four - momentum state _ , external gravitons would not feel the vacuum energy . however , if it holds in the strong form , we would arrive at undesirable conclusions such as the violation of the equivalence principle for external matter fields with the zero total four - momentum , the vanishing scattering amplitudes among only gravitons and the absence of dark energy . to improve them , we need another assumption such that _ the exclusive attribute of external gravitons is violated by the coupling of external matter fields ( at the same point and/or different ones ) , gravitons with derivatives or internal gravitons . _ here , internal gravitons mean gravitons in virtual states represented by propagators . note that the exclusion principle is merely an example of reasoning to justify the first assumption . the point of the second one is that _ external gravitons can couple to gravitational corrections of vacuum energy involving internal gravitons . _ under the above assumptions , we give a conjecture on a candidate of dark energy for the case that the minkowski spacetime is taken as a background one , i.e. , @xmath72 . in this case , the full propagator of graviton is proportional to @xmath73 . here , @xmath74 is the gravitational scale ( the reduced planck scale ) defined by @xmath75gev ) . using the propagator , we obtain the gravitational corrections of @xmath76 , @xmath77 at one - loop level . by replacing @xmath76 into @xmath52 , we obtain the zero point energy density of @xmath78 , @xmath79 after the subtraction of logarithmic divergence . if @xmath80 dominates @xmath50 , the magnitudes of @xmath81 and @xmath52 are estimated as @xmath82 and @xmath83 respectively . here we take @xmath43gev . then , we have a conjecture that physics around the terascale is relevant to the dark energy of our universe . if the zero point energy of some scalar field dominates @xmath52 , such scalar field has a mass of @xmath84tev and become a candidate of dark matter called @xmath11wimp ( weakly interacting massive particle ) . if superpartners appear around the terascale , they can produce zero point energies of @xmath84tev@xmath7 . in this case , we obtain an interesting scenario that a vacuum acquires an energy of @xmath84tev@xmath7 from zero point energies of dark matter and/or superpartners , it does not gravitate directly and the zero point energy of graviton becomes a source of dark energy . let us study the evolution of @xmath50 in the case with @xmath85 and @xmath86 . if we identify @xmath40 with a temperature of the universe , @xmath50 is not constant but varying logarithmically such that @xmath87 , \label{rho - de}\end{aligned}\ ] ] where @xmath44k and @xmath88 @xmath89 is a scale factor of the universe ( the present one ) . here , we also use the fact that the temperature is inversely proportional to the radius of our universe . the evolution of several energy densities are depicted in figure [ f1 ] . , @xmath90 and @xmath50 , respectively . here , @xmath91 and @xmath90 are energy densities of radiations and non - relativistic matters ( except for dark matter ) , respectively.,title="fig:",width=377 ] - in the appendix , we derive ( [ rho - de ] ) and show that the logarithmically changing energy density is described by the equation of state @xmath92 . if @xmath50 evolves as ( [ rho - de ] ) , there are several possibilities for physics beyond the terasclae . first one is that there is no sensitive physics to contribute the vacuum energy beyond the terascale , i.e. , no processes associated with a huge vacuum energy such as the breakdown of grand unified symmetry and no superheavy particles generating huge zero point energies . second one is that there is a higher energy physics , but a miraculous cancellation can occur among various contributions based on a powerful symmetry such as supersymmetry . then , the vacuum energies can be diminished in supersymmetric grand unified theories and/or supergravity theories , and the vacuum energy due to inflaton can vanish at the end of inflation in early universe . third one is that there survives a huge vacuum energy originated from a higher energy physics in the absence of a powerful mechanism , but even virtual gravitons do not couple to such a vacuum energy beyond the terascale . then , another graviton could be required to realize a higher energy physics . finally , we pursue a last possibility in the presence of the zero point energy of inflaton with a mass of @xmath93gev . we consider the action , @xmath94 , \label{tildes}\end{aligned}\ ] ] where @xmath95 is a coupling constant , @xmath96 is the ricci scalar made of another graviton @xmath97 , @xmath98 , and @xmath99 is the lagrangian density of inflaton . we assume that the ordinary graviton does not couple to inflaton directly . in the same way as the ordinary graviton , we obtain the zero point energy of @xmath97 , @xmath100 where @xmath101 . if @xmath102 dominates @xmath50 with @xmath103gev@xmath7 , the magnitude of @xmath104 is estimated as @xmath105gev . we have studied physics on the ccp in the framework of effective field theory and suggested that a dominant part of dark energy can originate from gravitational corrections of vacuum energy of @xmath84tev@xmath7 , under the following assumptions . * the graviton @xmath78 couples to the potential of matter fields with the strength of @xmath106 , and couples to the vacuum energy of matters with the same strength in the virtual state . * the classical gravitational fields do not couple to a large portion of the vacuum energy effectively . * external gravitons can couple to gravitational corrections of vacuum energy involving internal gravitons . the second one is beyond our comprehension , because it is difficult to understand it in the framework of ordinary local quantum field theory . however , if the ccp is a highly non - trivial problem that can not be solved without a correct theory of gravity in a proper manner , it would not be so strange that it is not explained from the present form of quantum gravity theory . it ease a major bottleneck of the ccp if realized with unknown features of gravity at a more fundamental level , and hence we have tried to step boldly from common sense . as an example of reasoning , we have presented a kind of exclusion principle that external gravitons can not take the same place in the zero total four - momentum state , unless they couple to external matter fields ( at the same point and/or different ones ) , gravitons with derivatives or internal gravitons . it may provide a useful hint to disclose physics behind the ccp . the universe dominated by our dark energy might cause instability , because it does not satisfy the energy condition such as @xmath107 for perfect fluids . a phenomenologically viable model must be fulfilled the requirements that matters are stable and a time scale of instability should be longer than the age of universe . if our speculation were correct , the ccp is replaced by the challenge to construct a microscopic theory of gravity compatible with the above assumptions . we might need a new ingredient or a novel formalism . it would be important to pursue much more features of graviton and the relationship between the classical gravitational field and the quantum one . in such a case , the theory of fat graviton may provide a helpful perspective and clue . we would like to thank prof . t. inami , prof . k. izumi and prof . m . ho for useful comments on the bimetric theory and the area - metric theory . according to them , we have returned the title and contents of our article to the original one . we discuss the evolution of dark energy ( [ rho - de ] ) . the effective potential including contributions of graviton at one - loop level is given by @xmath108 where @xmath76 is the potential of matters . from the feature that @xmath109 is independent of the renormalization point @xmath40 , i.e. , @xmath110 , we obtain the relation , @xmath111 from ( [ dv - eff ] ) , we find that the magnitude of @xmath112 is much less than @xmath52 if @xmath52 is much less than @xmath113 . then , the zero point energy density @xmath114 given by ( [ gr - loop ] ) varies on @xmath40 almost logarithmically . hence , the relation ( [ rho - de ] ) is derived after @xmath114 is regarded as the dark energy density @xmath50 and @xmath40 is identified with the temperature , that is proportional to the radius of our universe . the potential @xmath76 changes after the incorporation of zero point energies of particles and the breakdown of symmetry , and contains @xmath4 given in ( [ rho - th ] ) at present . from the above observation , the magnitude of @xmath76 is almost constant except for the period of the early universe . next , we show that the logarithmically changing energy density is described by the van der waals type equation of state , @xmath115 where @xmath116 is a positive constant . the energy conservation @xmath117 is rewritten as @xmath118 where @xmath88 is the radius of our universe . from ( [ eq - state ] ) and ( [ drho ] ) , we derive the differential equation @xmath119 and obtain the solution , @xmath120 where @xmath121 is a constant . by comparing ( [ rho - de ] ) with ( [ rho - b ] ) , @xmath116 and @xmath121 are determined as @xmath122 , \label{b - b0}\end{aligned}\ ] ] respectively . it is pointed out that , for models of dark energy characterized by @xmath123 , the system can be unstable if @xmath124 , which is realized by a negative kinetic term . although our effective theory of dark energy is different from the case with @xmath124 , @xmath14 and @xmath13 yielding ( [ eq - state ] ) do not satisfy the condition @xmath107 derived from various energy conditions and might cause instability . it is not clear whether they are phenomenologically viable or not , because it depends on the details of model at a microscopic level .

we study physics concerning the cosmological constant problem in the framework of effective field theory and suggest that a dominant part of dark energy can originate from gravitational corrections of vacuum energy , under the assumption that the classical gravitational fields do not couple to a large portion of the vacuum energy effectively , in spite of the coupling between graviton and matters at a microscopic level . our speculation is excellent with terascale supersymmetry . dark energy from gravitational corrections .45em 1.5 cm yugo abe@xmath0 , masaatsu horikoshi@xmath1 and yoshiharu kawamura@xmath1 1.5em @xmath2_graduate school of science and engineering , shimane university , _ + matsue 690 - 8504 , japan + @xmath3_department of physics , shinshu university , _ + matsumoto 390 - 8621 , japan 4.5em = 1.0em = 1.0em = 0.5em = 0.5em

we would like to study unflavoured decays of light neutral pseudoscalar mesons . this reduces the particle content to @xmath1 , @xmath2 and eventually @xmath3 , ruling out @xmath4 decays that violate hypercharge conservation and are suppressed by @xmath5 ( two - photon decays are further suppressed by @xmath6 compared to hadronic ones ) . standard model is thus reduced to qcd ( extended eventually only by qed corrections ) which is successfully described by an effective theory known as chiral perturbation theory ( chpt ) . the @xmath1 meson being the lightest meson can not decay to other hadronic states . its dominant decay mode ( with more than 98% probability ) is @xmath7 and is connected with the adler - bell - jackiw triangle anomaly @xcite . the @xmath8 vertex is closely connected with other allowed @xmath1 decay modes : @xmath9 , @xmath10 , @xmath11 ( with branching ratios @xcite : @xmath12 , @xmath13 , @xmath14 , respectively ) . in order to describe these processes with sufficient precision one can employ two - flavour chpt at appropriate order . this can simply incorporate corrections to the current algebra result attributed either to @xmath15 masses or electromagnetic corrections with other effects hidden in the low energy constants ( lecs ) . naively , two - flavour chpt should converge very fast and next - to - leading order ( nlo ) should be sufficient from the point of view of today s experiments . however , as we are exploring the anomalous sector which is poorly known , phenomenologically richer @xmath0 chpt must be also used in order to obtain numerical prediction for low energy constants . this on the other hand enables to describe @xmath16 in the same framework . the motivation for our study is both theoretical and experimental . as mentioned , @xmath17 represents ( probably ) the most important example of the triangle anomaly in quantum field theory . it is interesting that at nlo the amplitude gets no chiral corrections from the so - called chiral logarithms @xcite and this motivate the calculation at nnlo even for @xmath18 chpt as was done in @xcite . it was found that there are indeed chiral logarithms generated by two - loop diagrams , but they are relatively small . it turns one s attention back to nlo order and contributions proportional to lecs . to this end the phenomenology of @xmath16 and inevitably @xmath19 mixing must be employed . we intend to do the full two - loop calculation of both @xmath20 and @xmath16 in three - flavour chpt . as a first step we will present here the calculation and result in the @xmath0 limit , i.e. for @xmath21 . from the experimental side let us mention the primex experiment at jlab . it is designed to perform the most precise measurement of the neutral pion lifetime using the primakoff effect ( for first run results see @xcite ) . after jlab s 12 gev upgrade the extension of the experiment for @xmath2 and @xmath3 radiative width measurements is planned . neutral pion decay modes were studied with interesting results at ktev and it is promising to measure them in forthcoming na62 at cern . let us briefly summarize main points of chpt , for details see @xcite . starting point is the chiral symmetry of qcd , called chiral because it acts differently on left and right - handed quarks , which is exact for @xmath22 : @xmath23 where we dropped @xmath24 which is not a good symmetry due the anomaly . however , this anomaly is proportional to a divergence which must thus vanish in any order of perturbation theory . we are touching the problem referred as @xmath25 problem and we will avoid further discussion assuming that the ninth axial current is really not conserved and a possible divergence term is not present in qcd lagrangian ( referred itself as strong cp problem ) . assuming further confinement it can be proven that the axial subgroup of @xmath26 is spontaneously broken and the associated 8 goldstone bosons can be identified with pions , kaons and eta . the real non - zero masses of @xmath27 quarks , explicit symmetry breaking , are added as a perturbation and this expansion around the chiral limit together with the momentum expansion is referred to as chpt . standard power counting assumes that @xmath28 , and lorentz invariance implies that only even powers of derivatives ( @xmath29 ) can occur . the leading order ( lo ) thus starts at @xmath30 and one can have only tree diagrams . the next - to - leading order ( nlo ) is @xmath31 and can include one - loop contribution and similarly next - to - next - to - leading order ( nnlo ) is @xmath32 and can have up to two - loop diagrams . the last important point to be discussed here is the so - called chiral or external anomaly which would correctly incorporate the full symmetry pattern of qcd . it is connected with the fact that quarks carry also electromagnetic charge . in fact some green functions of qcd ( e.g. @xmath33 ) are not invariant under chiral symmetry , the difference was calculated first by bardeen @xcite and incorporated to the action by wess , zumino and witten ( wzw ) @xcite . this action starts at @xmath31 and thus the anomalous vertex shifts our counting by one order ( i.e. nnlo here is @xmath34 ) . we are primarily interested now in two - photon decays of @xmath1 and @xmath2 . nevertheless let us summarize shortly their `` spin - off '' products , namely * @xmath35 so called dalitz decay is important in normalization of rare pion and kaon decays . this was supported by its precise and stable prediction : for 30 years its official pdg value was same ( based on lampf experiment ) . however the last edition changed this number , based on aleph results and so it will have impact in other measurements via the normalization . the differential decay rate is discussed in @xcite . * @xmath36 or double dalitz decay enables experimental verification of @xmath1 parity . ktev set recently new limits on parity and cpt violation @xcite * @xmath37 depends directly only on fully off - shell @xmath38 vertex . ktev measurement @xcite is off by @xmath39 from the existing models . it can set valuable limits on models beyond sm * @xmath40 , exotics and violation processes were also studied in @xmath1 decays . it includes mainly decay to neutrinos but is also interesting in beyond sm scenarios ( neutralinos , extra - light neutral vector particle , etc . ) ( for more references cf . the same modes are also possible in @xmath2 decays , see e.g. @xcite . in the chiral limit the decay width is fixed by axial anomaly with the result @xmath41 it is in excellent agreement with experiment , which is the opposite situation to two - photon @xmath2 decay . in @xmath0 limit ( and also in chiral limit ) the two studied amplitudes are connected by wigner - eckart theorem @xmath42 , i.e. @xmath43 which is far from experiment @xmath44 @xcite . ( note that using @xmath45 instead of @xmath46 makes this difference even larger . ) the difference is attributed to @xmath19 mixing . at nlo order , apart from tree diagrams coming from wzw and @xmath32 odd - parity lagrangian , we should include two one - loop topologies ( depicted in fig.[oneloop ] ) . the full one - loop calculation based on wave function renormalization and chiral expansion of masses and decay constants leads to : @xmath47 ^ 2,\notag\\ & \gamma(\eta \to \gamma\gamma)^{nlo } = \gamma(\eta \to \gamma\gamma)^{ca } \times \bigr[\frac{f_\pi^2}{f_\eta^2 } + \frac{256\pi^2}{9}\label{gnlo}\\ & \qquad\times\bigl((4 m_k^2 - 7 m_\pi^2 ) c^{wr}_7 + 24 ( m_k^2 - m_\pi^2 ) c^{wr}_8 \bigr ) \bigl]^2.\notag\end{aligned}\ ] ] note , as anticipated , the very simple , polynomial form of the results without logarithms . this is especially accomplished by correct replacement of @xmath48 , i.e. @xmath49 @xmath46 and @xmath45 in @xmath1 and @xmath2 decay respectively . it is clear from ( [ gnlo ] ) that @xmath19 mixing must be hidden in @xmath50 lec . a rough estimate using resonance saturation suggests that @xmath50 must be much bigger than @xmath51 . for further discussion see @xcite and @xcite . the @xmath34 , ( or equivalently nnlo , or two - loop ) calculation was already performed for @xmath17 in two - flavour chpt . natural extension for @xmath0 will supply us with both @xmath1 and also @xmath16 and enable to test and verify chiral expansion in odd intrinsic sector ( cf . study for even sector @xcite ) . it is , however , clear that this calculation will be difficult : we are facing instead of one , three different scales in overlapping two - loop diagrams ( sunset and vertex ) . big effort was already given in the simpler two - point ( sunset ) case , and we still lack general analytic form . we plan to calculate it using method described in @xcite but we need to go beyond the loop integrals computed there . there exists , however , apart from chiral limit , one non - trivial limit which can be used to obtain analytical result as it depends again only on one scale . it is an @xmath0 limit , where we set @xmath53 . this we can simply connect with @xmath30 mass : @xmath54 . the current algebra prediction , fixed by the anomaly , is free from any mass contribution . the mass enters explicitly at nlo order only , and therefore to obtain nnlo order we need to connect @xmath30 parameter with physical ( referring to a world where @xmath21 ) @xmath0 mass : @xmath55 + o(m_\pi^4)\ ] ] with chiral logarithm defined as @xmath56 . on the other hand connection of @xmath48 with physical @xmath0 decay constants is needed up to nnlo order @xmath57 the nnlo part was already calculated in general @xmath58 in @xcite and for our @xmath59 is given by @xmath60 with @xmath61 and @xmath62 using renormalization coefficients taken from @xcite . as already mentioned , for @xmath0 limit @xmath1 and @xmath2 decays are related by wigner - eckart theorem and we thus need to calculate only one of these processes . following weinberg power - counting at nnlo we need to consider @xmath63 ) tree graphs with either @xmath64 ) one vertex from odd @xmath34 sector or @xmath65 ) one from odd ( even ) @xmath32 and second from even ( odd ) @xmath31 ; @xmath66 ) one - loop diagrams with one vertex with nlo coupling ( even or odd ) and @xmath67 ) the two - loop graphs with one vertex taken from the wzw lagrangian . all other vertices should be generated by the @xmath30 chiral lagrangian . case @xmath65 ) is treated via wave function renormalization . however , the odd - sector lagrangian at @xmath34 for three flavours has not yet been studied . the connected lec will be denoted as @xmath68 and set only a posteriori to cancel all local divergences . concerning one - loop feynman diagrams , we have already summarized them in fig.[oneloop ] , for nnlo the topology stays the same , we need just to insert higher - order vertices . non - trivial part of calculation is hidden in two loops . the feynman diagrams to deal with are summarized in fig.[twoloop ] . corrections ( tadpoles ) to propagators are not depicted . note that the most of diagrams are the same as in the two - flavour case . as anticipated by the nature of the anomaly there is one new topology ( the last one diagram in fig.[twoloop ] ) with anomalous vertex without direct photon insertion ( so - called chesire - cat smile ) . of course , into these graphs one should insert all possible combinations of pions , kaons and eta ( fortunately in @xmath0 limit with identical masses ) . we summarize the preliminary result in the following form ( @xmath69 is normalized as @xmath70 at lo , cf . eqs ( [ gnlo ] ) ) . @xmath71 with @xmath72 and @xmath73 using renormalization coefficients taken from @xcite . the @xmath34 chiral coupling which would cancel local divergences in @xmath0 limit is denoted by @xmath74 and our exact calculation fixes its decomposition @xmath75.\end{aligned}\ ] ] we have summarized here our preliminary results concerning a two - loop calculation of @xmath76 in @xmath0 limit ( where @xmath77 ) . the word preliminary refers also to the fact that independent calculation with physical masses is in progress @xcite and it should allow us to crosscheck here presented result in this limit . the possibility of studying two - photon decays of light - meson on lattice was very recently demonstrated in @xcite . the simple analytical result can be very useful in this direction as one can vary masses without changing lecs . k.k . would like to thank the organizers for a very enjoyable conference . 999 s. l. adler , phys . * 177 * ( 1969 ) 2426 , j. s. bell and r. jackiw , nuovo cim . a * 60 * ( 1969 ) 47 , w. a. bardeen , phys . rev . * 184 * ( 1969 ) 1848 . k. nakamura _ et al . _ [ particle data group ] , j. phys . g * 37 * ( 2010 ) 075021 . j. f. donoghue , b. r. holstein and y. c. r. lin , phys . * 55 * ( 1985 ) 2766 [ erratum - ibid . * 61 * ( 1988 ) 1527 ] ; 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e. shintani , s. aoki , s. hashimoto , t. onogi and n. yamada [ jlqcd collaboration ] , arxiv:0912.0253 [ hep - lat ] .

present and planned experiments motivate new theoretical study of properties of light unflavoured pseudoscalar meson decays . an overview including details on two - loop calculation in @xmath0 limit is given . chiral perturbation theory , radiative decay of @xmath1 , higher - order correction

one of the most profound discoveries of observational physics is that the universe is accelerating in its expansion @xcite . there have been many attempts to explain this late - time acceleration , for example , a pure cosmological constant , dark energy associated with some new scalar field and modified gravitational theories , although all current models require some level of fine - tuning and none are considered to be a complete explanation . whatever is responsible for the current acceleration may arise from some completely new physical principle . this is the possibility we consider in this paper . our goal is to construct a toy model that represents a late - time accelerating universe using a new , possibly fundamental , principle . as our guiding principle , we hypothesize the existence of a _ minimal curvature _ scale in gravity . in a friedmann , robertson - walker ( frw ) space - time , without cosmological constant @xmath0 and with only standard matter sources such as dust and radiation , the universe will always decelerate as it expands . one way to avoid this is to add matter to the system that violates the strong energy condition ( sec ) . in a cosmological context this violation constitutes the addition of matter sources satisfying the equation of state @xmath1 . a second possibility is to explicitly remove flat space - time as a solution to the theory . in this case the vacuum of the theory , which is approached at late times as the energy density in matter fields becomes more and more dilute , is not minkowski space - time , but instead an accelerating universe @xcite . to remove flat spacetime as a solution we hypothesize the existence of a minimal curvature in our underlying fundamental theory . the simplest example of this is , of course , to introduce a bare cosmological constant into general relativity . however , in principle there may exist many other ways to achieve this result . indeed , it appears that many accelerating cosmological models derived from modified gravity theories contain such a minimal curvature @xcite . the idea of a minimal curvature scale in gravity mirrors that of a maximal curvature scale . in the literature many authors have considered this possibility and used it to remove the curvature singularities of general relativity by bounding curvature invariants from above at the level of the classical action @xcite-@xcite . in the case of singularity removal , it is necessary to bound _ all _ curvature invariants in order to cover all possible physical situations in which such a singularity may occur . by contrast , in the case of a minimal curvature approached at late times in a homogeneous , isotropic universe , symmetry implies that it is only necessary to bound the ricci scalar @xmath2 from below . hence , unlike in the case of a maximal curvature hypothesis , we shall see that one may implement a minimal curvature by using a modified brans - dicke theory where the brans - dicke field couples non - minimally to the matter lagrangian . within this context we demonstrate that the existence of the minimal curvature ( mc ) produces a universe that evolves from a matter dominated period to an accelerating phase mimicking the @xmath0-cold - dark - matter ( @xmath0cdm ) model . we emphasize that the model presented here is only a _ toy construction of the late universe . the model is not intended to provide a consistent cosmology from the time of big - bang nucleosynthesis ( bbn ) until today . it is unlikely that the precise model presented here is compatible with solar system experiments and the tight constraints on the time variation of newton s constant . however , the model _ does provide an example of how the postulated existence of a minimal curvature scale in gravity can provide a new mechanism to generate cosmological acceleration of the late universe . furthermore , the model may capture features of a possibly more fundamental theory that admits a minimal curvature scale . _ _ in section [ sec : mc ] , we describe the minimal curvature construction , first by using a toy example and then by using a class of modified brans - dicke theories . we solve the equations of motion for this example and demonstrate how the universe evolves from a matter dominated phase to an accelerating period as the curvature approaches its minimal value . in section [ sec : comp ] , we compare the mc model with @xmath0cdm and to the supernovae ( sneia ) gold " sample of @xcite . finally , we comment on the possibility of constructing more realistic models that satisfy the limiting curvature hypothesis and offer our conclusions and speculations in section [ sec : conclusions ] . in appendix a , we provide a detailed analysis of the vacuum mc theory . in appendix b , we construct an einstein frame description of the vacuum theory and compare it to the mc vacuum . our goal is to construct theories in which a certain physical quantity is bounded from below . before leaping directly into our model , it is instructive to consider an example of how a similar effect may be achieved in a simpler theory - the bounding of velocities from above in special relativity by the speed of light @xcite . the newtonian action for a free particle of mass @xmath3 in motion is [ old ] s = dt m x^2 . in this classical theory the velocity of the particle is _ without bound_. now let us implement one of the fundamental consequences of special relativity : to ensure that the speed of this particle is _ limited _ by the speed of light we introduce a field @xmath4 which couples to the quantity in the action that we want to bound ( @xmath5 ) and has a potential " @xmath6 . the resulting action is [ newa ] s = m dt . the variational equation with respect to @xmath7 [ bit ] x^2 = , ensures that @xmath8 is bounded , provided @xmath9 is bounded . note the absence of a kinetic term for @xmath10 in the action , and hence , the reason the word _ potential appears in quotes above . in order to obtain the correct newtonian limit for small @xmath8 and small @xmath7 we take @xmath6 proportional to @xmath11 . in the newtonian limit the action ( [ newa ] ) reduces to ( [ old ] ) . a simple potential satisfying the above asymptotics is u ( ) = . integrating out @xmath7 yields ( up to an irrelevant constant ) the action for relativistic particle motion : s_sr = m dt . the above model provides a powerful example of how a toy construction based on a fundamental principle the existence of a universal speed limit " can capture features of a more fundamental theory . _ we now use a similar construction to model the existence of a _ minimal _ curvature ( mc ) scale in gravity . because we are interested in late time cosmology , we need only be concerned with bounding one curvature invariant , the ricci scalar @xmath2 . in direct analogy with our example from special relativity , we introduce a scalar field @xmath12 that couples to the quantity we wish to bound , @xmath2 , and a potential " function @xmath13 [ act2 ] s_mc = d^4 x ( r - r + v ( ) ) , where @xmath14 is the reduced planck mass and both @xmath12 and @xmath15 posses dimensions of mass , @xmath16 = [ \g]=m$ ] . the vacuum theory is equivalent to a brans - dicke theory with brans - dicke parameter @xmath17 . this is seen by re - writing the action ( [ act2 ] ) in terms of a new field [ newp ] = ( - ) , so that the action becomes [ actbd ] s_bd = d^4 x ( r + v ( ) ) . ordinarily , such a theory can be re - cast as a purely gravitational theory with lagrangian @xmath18 ( see , e.g. @xcite ) ; however , this is not possible for all forms of the potential @xmath19 . for reasons that will become clear shortly , we allow the field @xmath12 to couple non - minimally to matter . the non - minimal coupling yields the matter action , [ actmat ] s_m = d^4 x _ m(_i , , f(/m_pl ) | ) , where @xmath20 is the lagrangian made up of whatever matter fields @xmath21 are in the theory . in the case were @xmath22 represents a dark matter dirac spinor , the field @xmath12 couples non - minimally _ only to the dark matter sector and need not couple to baryons . in this case it is possible to avoid constraints on such a coupling from solar - system and table - top tests of gravity . in string theory , non - perturbative string loop effects do not generically lead to universal couplings , allowing the possibility that the dilaton decouples more slowly from dark matter than ordinary matter ( see , e.g. @xcite ) . _ this coupling can be used to address the coincidence problem , since the acceleration is triggered by the coupling to matter . for the purposes of our toy construction , we do not distinguish between baryonic and dark matter in the remainder of our discussion . note that in the case of @xmath23 theories which are conformally identical to models of quintessence in which matter is coupled to dark energy with a large coupling , this strong coupling induces a cosmological evolution radically different from standard cosmology @xcite . similar difficulties may arise in the model presented here , however , we have yet to investigate this issue . the matter stress - energy tensor is given by t _ - . we assume a perfect fluid t _ = ( _ m + p_m ) u_u_+p_m g _ , where @xmath24 is the fluid rest - frame four - velocity , and the energy density @xmath25 and pressure @xmath26 are related by the equation of state @xmath27 . because we are focusing on the late universe we shall ignore the presence of radiation and consider only a matter density @xmath25 , which redshifts with expansion in the usual manner ( with the exception of the non - trivial @xmath12 dependence ) _ m . in the above , @xmath28 is a function describing the non - minimal coupling of the field @xmath12 to the matter lagrangian . such couplings in the context of ordinary scalar - einstein gravity were studied in @xcite where it was found that this coupling can be made consistent with all known current observations , the tightest constraint coming from estimates of the matter density at various redshifts . this coupling plays a critical role in modified source gravity , introduced in @xcite . variation of the total action @xmath29 with respect to the metric tensor , @xmath30 yields the modified einstein equations [ eomg ] ( 1- ) r _ & - & ( 1- ) r g _ + & + & _ _ - g _ - v g _ = 8 g t _ where @xmath31 and we have introduced @xmath32 . variation with respect to the field @xmath12 gives [ eomp ] r = + . equation ( [ eomp ] ) is the key to imposing the limiting curvature construction . it is clear that the curvature @xmath2 will remain bounded and approach a constant curvature at late times ( @xmath33 ) as long as @xmath34 constant ; this is the essence of the construction . we assume a flat ( @xmath35 ) friedmann - robertson - walker metric ds^2=-dt^2+a^2(t)\{dr^2 + r^2 d^2 } , [ frwm ] where @xmath36 is the scale factor of the universe and @xmath37 is the line - element of the unit @xmath38sphere . defining the hubble parameter by @xmath39 and substituting the metric ansatz ( [ frwm ] ) into ( [ eomg ] ) and ( [ eomp ] ) gives the generalized friedmann equation ( the @xmath40-component of ( [ eomg ] ) ) and the equation of motion for @xmath12 : [ eom1 ] 3 ^ 2 h^2 - 6 h^2 - 6 h + v ( ) = , & & + 6(2h^2 + h ) - v ( ) = f ( ) , where a prime denotes differentiation with respect to @xmath12 , @xmath41 , @xmath42 and @xmath43 are the values of @xmath12 and @xmath44 today . by considering the asymptotics of our cosmology at early and late times , we can constrain the forms of the functions @xmath13 and @xmath45 . we require that the effective newton s constant for our theory remain positive definite so that gravity is always attractive . this imposes a constraint on @xmath46 in equation ( [ actbd ] ) . there are rather strong constrains on the time variation of newton s constant from the period of nucleosynthesis until today ( roughly , @xmath47 @xcite ) . for the time being , we will allow ourselves to ignore this constraint in order to produce a toy model capable of realizing the mc conjecture . furthermore , because we are only interested in the behavior of the universe from the matter dominated epoch until today , we have ignored the presence of radiation . the absolute earliest our theory is valid is up to the period of equal matter and radiation domination @xmath48 . for specificity , by _ early times we refer to times near the time of photon decoupling at a redshift of @xmath49 , during which the universe is typically already well into the matter dominated regime . _ at late times ( and low curvatures ) we want to bound the ricci scalar @xmath2 from below . this will constitute a successful example of a model obeying the minimal curvature hypothesis . to bound the curvature we use equation ( [ eomp ] ) . it is clear that the curvature @xmath2 will remain bounded if we bound @xmath50 , where the prime denotes differentiation with respect to @xmath12 . hence , we require @xmath51 at late times , where we denote the hypothesized minimal curvature scale by @xmath52 . we anticipate that , by construction , there will be a late - time attractor that is a constant curvature space - time with @xmath53 . this attractor is not an actual solution to the equations of motion . the above considerations restrict the functional forms of potential @xmath19 and the non - minimal coupling function @xmath28 . integration singles out a class of theories that must obey @xmath54 as @xmath55 . the simplest forms for @xmath19 and @xmath28 obeying the above constraint are the linear functions [ vandf ] v ( ) = ^3 , f ( ) = , where @xmath56 is , in principle , another free parameter with dimensions of mass . however , we will take @xmath57 , so that @xmath58 , and @xmath15 is the only free parameter in the theory and then take the limit that @xmath59 the theory resembles the action of the _ modified source gravity models studied in @xcite . _ ] . we now rescale time @xmath60 so that today @xmath61 and introduce the following dimensionless quantities [ dimen ] a , h , [ dimen2 ] _ m = , r = , and take @xmath62 , where @xmath63 is the value of the scale factor today . in terms of the dimensionless quantities the eom become [ eomf ] h^2 - 2 h^2 - 2 h + = , & & + ( 2h^2 + h ) - = f ( ) [ rdshift ] , the successful implementation of the minimal curvature hypothesis is now apparent . recasting equation ( [ rdshift ] ) in terms of the curvature scalar @xmath64 , and substituting in our choices for @xmath19 and @xmath28 ( [ vandf ] ) : [ simple ] r = ^2 + 3 . we see that , as the universe expands and the matter term dilutes , we asymptotically approach the minimal value of the curvature @xmath65 . it is both interesting and surprising that the solution to eq . ( [ simple ] ) reduces to the simple case of @xmath0cdm plus an arbitrary amount of _ dark radiation _ which may have either positive or negative effective energy density . most notably , this model arises in randall - sundrum brane cosmology which has been extensively studied in the literature . the friedmann equation derived from the randall - sundrum model for a flat universe is [ bwf ] h^2 = + + , where @xmath66 is the five dimensional planck mass and @xmath67 is the so called dark radiation term , since it scales like radiation , but it s origins are purely gravitational and it does not interact with standard matter @xcite . at low energies ( when the energy density is much less than the critical brane tension ) , the @xmath68 term can be safely neglected . the main observational restrictions on the dark radiation term come from the acoustic scale at recombination ( see , e.g. @xcite ) , and from the amount of total growth of density perturbations in the non - relativistic matter component from the time of equal matter and radiation until the present day @xcite . as a result , the density of dark radiation can not be significantly larger than the present cmb energy density . making use of ( [ dimen ] ) and ( [ dimen2 ] ) , eq . ( [ bwf ] ) may be recast ( neglecting the @xmath68 term ) as [ hbw ] h^2 = + + , where we have included a cosmological constant term @xmath69 , and @xmath70 includes contributions from both the dark and ordinary radiation . from ( [ hbw ] ) we find h = - . constructing the ricci scalar @xmath71 from the above expressions yields eq . ( [ simple ] ) , with @xmath72 . in [ app : vac ] , we provide a detailed analysis of the vacuum mc equations with @xmath73 . although the presence of matter plays an important role in our minimal curvature construction , an analysis of the vacuum theory provides valuable insight into the solutions we are interested in studying . in [ app : einst ] , we transform the vacuum mc theory into an einstein frame and relate quantities of physical interest in both frames . for general solutions to the equations of motion ( [ eomf ] ) and ( [ rdshift ] ) with functions ( [ vandf ] ) we solve for the hubble parameter @xmath74 and scalar field @xmath75 . we plot the relevant portion of the @xmath76 phase space in fig . [ phase1.eps ] . vs. @xmath77 . all solutions asymptotically approach a late - time accelerating phase with constant @xmath77 at the minimal curvature @xmath78 denoted by the red line.,width=5 ] to solve the equation we integrate from the past to today @xmath79 and then from today into the future and patch the solutions together . in the plots we take the conditions @xmath80 , @xmath81 , @xmath82 and let the values of @xmath42 vary . at late times , the solutions approach the constant @xmath77 attractor when the universe is well into the accelerating epoch . we now compare our model with @xmath0cdm . to do so , we must enter reasonable initial conditions into our numerical study and solve the equations of motion ( [ eomf ] ) , ( [ rdshift ] ) together with ( [ vandf ] ) and the equation @xmath83 . let us begin by considering the value of the minimal curvature . physically the minimal curvature corresponds to an emptying of the matter in the universe due to cosmological expansion . in our model the value of the minimal curvature is approached asymptotically . today , the value of the curvature is given by [ rprox ] r_0 = 6 ( 2 h_0 ^ 2 + h_0 ) 12 h_0 ^ 2 . because we observe a significant amount a matter in the universe today , we know that we have not yet reached the minimal curvature . however , because we are accelerating , we know that we are _ near the minimal curvature ( i.e. the first and second terms on the right hand side of equation ( [ simple ] ) must be comparable ) . hence , from equation ( [ rprox ] ) , we expect the value of the minimal curvature @xmath78 to be close to but less than @xmath84 . therefore , in terms of our dimensionless quantities ( [ dimen ] ) , @xmath85 and the free parameter in our theory @xmath86 . for the solutions considered below , we choose a value of @xmath81 meeting the above requirements and that follows the @xmath0cdm model to our satisfaction for our toy construction ( i.e. a value leading to a matter dominated cosmology followed by a jerk " near a redshift of @xmath87 into an accelerating phase ) . _ in the action ( [ act2 ] ) , the effective newton s constant is @xmath88 is given by ( 16 g_n_eff)^-1 = - . to ensure that the effective newton s constant remains positive definite over the history of the universe ( @xmath89 ) we must have @xmath90 . we are almost in a position to compare our model both with @xmath0cdm , which has the friedmann equation [ lcdm ] h = , and with the observational data provided by the sneia gold sample . to make a comparison with the observational data we require an understanding of the luminosity distance formula in the context of modified gravity models . an important consideration arises when using the formula for the luminosity distance in theories of the form : [ modact ] s = d^4 x , where the @xmath91 represent the different types of possible matter lagrangians present . such a theory arises as the low - energy effective action for the massless modes of dilaton gravity in string theory , and our model is an action of just this sort @xcite ; albeit , with an unusual choice for the functions @xmath92 . as we have already discussed , these theories typically lead to time variation in newton s gravitational constant . the time variation can affect the way one should compare the theory to observations @xcite . in particular , the time - evolution can alter the basic physics of supernovae . for example , the time variation in @xmath93 leads to different values of the chandrasekhar mass at different epochs , and hence , a supernova s peak luminosity will vary depending on when the supernova exploded . this makes treating the supernovae ia as standard candles difficult @xcite . specifically , the peak luminosity of sneia is proportional to the mass of nickel synthesized which is a fixed fraction of the chandrasekhar mass @xmath94 . hence , the luminosity peak of sneia varies as @xmath95 and the corresponding absolute magnitude evolves as m = m_0 + , where the subscript zero indicates the local values of the quantities . therefore , the magnitude - redshift relation of sneia in modified gravity theories of the type given by ( [ modact ] ) is related to the luminosity distance via @xcite : m(z ) = m_0 + 5 + . even if gravitational physics is described by some theory other than general relativity the standard formula for the luminosity distance applies as long as one is considering a metric theory of gravity @xcite : [ lumd ] d_l(z ; h(z ) , h_0 ) = _ 0^z . for @xmath0cdm , the luminosity distance ( [ lumd ] ) can be written @xcite : [ lumlcdm ] d_l(z ; _ m , _ , h_0 ) = & & + s ( _ 0^z ^- ) where , @xmath96 and @xmath97 for @xmath98 while @xmath99 with @xmath100 for @xmath101 while @xmath102 and @xmath103 , for @xmath104 . here and throughout , @xmath105 . given the above considerations we take the following values of parameters today : [ initc ] a_0 = 1.0 , h_0 = 1.0,_0 = 0.1 , = 3.15 , _ m^(0 ) = 0.25 . we then integrate our equations from the past to today and then from today into the future and patch the solutions together . while we do not provide an exhaustive study of the parameter space for the mc model , the parameters given above provide a successful example of the construction which fulfills our rather modest goals . ] . it is quite possible that the parameters ( [ initc ] ) can be tuned to achieve an even better agreement with @xmath0cdm . the history and future of the curvature @xmath2 together with the co - moving hubble radius are plotted in fig . [ crvandcmh2.eps ] . . the curvature decreases from the matter - dominate past ( red curve ) to the constant minimal value at @xmath106 indicated by the green line . today , @xmath107 and the blue curve indicates the future evolution of the curvature . while the universe is accelerating the co - moving hubble radius decreases . ] in the figure the past history of the curvature is plotted in red while the future is plotted in blue . today we are at the value @xmath108 . the minimal curvature is given by the green line at @xmath109 . as expected , the curvature is large in the past when the universe is matter dominated and then decreases , approaching the accelerating late - time de - sitter attractor with constant minimal curvature @xmath53 . the second plot in the figure shows the evolution of the co - moving hubble radius @xmath110 . the phenomenologically desired cosmological transition from matter domination to late - time acceleration of the universe is clearly indicated by the decreasing of the co - moving hubble radius , when < 0 . in some modified gravity models it can be difficult to achieve this transition ( see , e.g. @xcite ) . the majority of our results are presented in figures [ vslcdm3.eps ] and [ mcfit4.eps ] . in figure [ vslcdm3.eps ] we plot the scale factor @xmath111 as a function of cosmic time @xmath112 , the hubble parameter @xmath77 as function of redshift @xmath113 , where z + 1 , the deceleration parameter q(z ) = - , and the luminosity distance , @xmath114 . quantities plotted in red are the mc model while those in blue are for @xmath0cdm . today we are at the values @xmath115 , @xmath116 . plot a. ) shows the time evolution of the scale factor . plot b. ) shows the hubble parameter as a function of redshift . plot c. ) compares the deceleration parameters as a function of redshift . plot d. ) shows the luminosity distance , @xmath114 , as a function of redshift . the mc model is clearly a good fit with @xmath0cdm . ] for the particular set of parameters considered , the scale factor of our model differs from @xmath0cdm most strongly in the far future , although the difference in the expansion rates of the mc construction with @xmath0cdm is apparent in the plot of the hubble parameters at high redshifts . the transition from matter domination to acceleration ( the jerk ) occurs at a slightly lower redshift than @xmath0cdm . there is only a slight difference in @xmath114 that occurs at high redshifts ( @xmath117 ) , although the difference is not significant enough to distinguish our model from @xmath0cdm using only the current supernova data . cdm models with observational data . the supernovae data points are plotted with error bars and the data is taken from @xcite . the luminosity distance @xmath114 for the mc model is plotted by the dashed ( blue ) curve . the various theoretical predictions for @xmath0cdm are represented by the solid curves and were examined in @xcite.,width=480 ] in figure [ mcfit4.eps ] , we compare the luminosity distance predicted by our model with several versions of @xmath0cdm and with the observational data . the luminosity distances are plotted out to a redshift of @xmath118 ( the highest redshift supernova data is from @xmath119 ) . the theoretical predictions of the minimal curvature construction and @xmath0cdm are compared with the gold " supernovae sample of @xcite . the particular choice of @xmath0cdm models shown are from @xcite . the luminosity distance for the minimal curvature construction is denoted by the blue dashed line and fits the supernova data extremely well . the luminosity distance @xmath114 is virtually indistinguishable from @xmath0cdm with @xmath120 and @xmath121 ( there is a small discrepancy at high redshift as can be seen from figure [ vslcdm3.eps ] ) . we have shown that a period of late - time cosmic acceleration can follow directly from a simple minimal curvature conjecture ( mcc ) . the model fits the sneia data exceptionally well . while the specific formulation considered here is only a toy construction , unlikely to be compatible with constraints from solar system and table top test of the equivalence principle , it may capture phenomenologically interesting features of a more fundamental theory that admits a limiting minimal curvature . furthermore , the construction successfully demonstrates the possibility that a new fundamental physical principle may ultimately be responsible for the recent period of cosmological acceleration . it is possible that experimentally viable models based on the minimal curvature conjecture exist . the search for such models within the context of scalar - gauss - bonnet gravity is currently underway . despite the tight theoretical and experimental constraints on scalar - gauss - bonnet cosmologies @xcite , we remain optimistic that an experimentally and theoretically viable model based on the minimal curvature construction can be discovered . it is a pleasure to thank r. brandenberger , r. gregory , v. jejjala , i. moss , r. myers and t. underwood for helpful discussions . i am especially grateful to m. trodden for numerous useful discussions over the course of this work . this work is supported in part by pparc and by the eu 6th framework marie curie research and training network universenet " ( mrtn - 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however , once the universe becomes sufficiently large and dilute the dynamics will resemble those of the vacuum theory . as we shall see the vacuum mc theory is significantly richer than that of @xmath0cdm . the equations describing the vacuum are given by the equations ( [ eom1 ] ) with @xmath122 . the dimensionless forms are [ eomvac ] h^2 - 2 h^2 - 2 h + = 0 , & & + ( 2h^2 + h ) - = 0 , where we have re - introduced the parameter @xmath56 from equation ( [ vandf ] ) ( previously set equal to @xmath15 ) for completeness and the potential is @xmath123 . in the case of the vacuum we may solve exactly for @xmath74 . we focus on two types of solutions of particular interest . in the first , the hubble parameter is given by [ htanh ] h(t ) = , where@xmath124 is the minimal curvature and @xmath125 is a constant . in this case h = , and the scale factor evolves as a(t ) = a_0 the quantities mentioned above , along with the field @xmath12 are plotted in fig . ( [ vacplot.eps ] ) . . in the plots we take @xmath126 . ] in this case the universe undergoes a bounce as indicated in plots a. ) and b. ) in fig . ( [ vacplot.eps ] ) . after the bounce the hubble parameter is pulled up to the value at the minimum curvature @xmath127 . this is a de sitter solution of the vacuum theory with [ julia ] h_ds = , while @xmath12 continues to evolve exponentially @xmath128 . interestingly , @xmath77 and @xmath129 evolve in such a way that the curvature @xmath130 is constant throughout the evolution of the solution , fixed at the minimal value @xmath131 . this behavior is not surprising as the constraint on @xmath2 comes from the second equation in the eom ( [ eomvac ] ) . the second family of solutions are of greater relevance . they are the vacuum analogs of the solutions plotted in fig . ( [ phase1.eps ] ) and discussed at the end of section [ sec : mc ] . these solution are pulled _ down to the value of @xmath77 at the minimal curvature . the entire phase space of solutions to the vacuum theory is shown in fig . ( [ vacphase.eps ] ) . representative solutions of the two families of solutions discussed above are plotted by the solid curves . the red curve is the solution given by equation ( [ htanh ] ) . the second set of solutions ( relevant to section [ sec : mc ] ) are plotted in green in the upper right quadrant of the phase space . the system defined by the vacuum equations ( equation ( [ eomvac ] ) ) has two unstable saddle equilibrium points at the values [ eqpts ] ( , h ) = ( - , ) . the equilibrium points are marked by the purple dots in fig . ( [ vacphase.eps ] ) . _ is plotted on the vertical axis and @xmath12 is plotted on the horizontal axis . the vector field for solutions is drawn in black . solutions of particular interests are plotted by the solid curves . the constant de sitter value of @xmath77 at the minimal curvature is indicated by the dashed ( blue ) line . the two unstable saddle equilibrium points are designated by the purple dots . in the plot we take @xmath132.,width=5 ] it is not our intent to provide a complete einstein frame analysis of the full mc system with matter ; however , it is instructive to consider the vacuum theory in an einstein frame . to move to the einstein frame we begin with the brans - dicke frame defined by the action ( [ actbd ] ) . passage to the einstein frame is achieved via a conformal transformation of the form [ ctrans ] g _ = ^2 g _ , where @xmath133 is the conformal factor which must be positive to leave the signature of the metric unaltered . from this point forward a tilde shall denote a quantity built out of the einstein - frame metric tensor @xmath134 . under this transformation the infinitesimal line element and the determinant of the metric transform as ds^2 = ^2 ds^2 , = ^4 , respectively . the ricci scalar transforms as r = ^2 ( r + 6 ( ) - 6 g^ ) . omitting the ordinary divergence @xmath135 , the action in the brans - dicke frame transforms to [ actcf ] s = d^4 x ( r - 6 g^ _ _ + ) , where the potential in terms of @xmath136 is given by ( [ newp ] ) together with ( [ vandf ] ) : [ vofbdp ] v ( ) = ( - ) . here we have re - introduced the parameter @xmath56 from equation ( [ vandf ] ) ( previously set equal to @xmath15 ) for completeness . by choosing our conformal factor to be @xmath137 , and performing a field redefinition to define the einstein frame field @xmath138 , = , the action ( [ actcf ] ) becomes the einstein frame action [ acteinst ] s_ef = d^4 x ( r - g^ _ _ - ( ) ) , where we have defined the potential ( ) = - ( ) ^2 . the einstein frame potential is given by [ einstv ] ( ) = ( e^/ - 1 ) e^-2 / , where @xmath124 is the minimal curvature and @xmath139 . in this frame the field @xmath138 has a canonically normalized kinetic term making the interpretation of solutions more simple due to our familiarity with minimally coupled scalar field to ordinary einstein gravity . it is important to note , however , that so far we have only considered the vacuum theory . even in the einstein frame the einstein field @xmath138 will couple non - minimally to matter and therefore , when matter is present in significant amounts , the simple einstein frame vacuum solutions will not be an accurate description of the theory . under the conformal transformation ( [ ctrans ] ) , the cosmic time coordinate transforms as [ ttran ] d t^2 = e^/ dt^2 . taking the einstein - frame friedmann - robertson - walker flat metric ds^2 = - d t^2 + a^2(t ) d**x^2 , leads to the familiar equations of motion [ sasha ] h^2 = ( ^2 + v ( ) ) , and + 3 h + = 0 , where prime denotes differentiation with respect to the einstein frame cosmic time coordinate @xmath140 and @xmath141 . using the conformal transformation ( [ ctrans ] ) , we find the hubble parameters in the einstein and minimal curvature frames are related by [ relate ] h = e^ ( h - ) . * * let us examine the first possibility in greater detail . the unstable de sitter solution with constant @xmath144 sits at the value at the maximum of the potential ( [ einstv ] ) . the maximum is at constant value [ dspt ] = _ ds = ^-1 , from equation ( [ sasha ] ) we find the corresponding @xmath145 h_ds = . from equation ( [ relate ] ) we see that the corresponding value in the minimal curvature frame is h(h_ds , _ ds ) = , which is the exact de sitter solution we found in our analysis of the vacuum of the minimal curvature theory ( [ julia ] ) . the minimal curvature frame field is related to the einstein frame field via [ vpofphi ] ( ) = ( 1 - e^/ ) , and consequently , ( ) = . using the relation ( [ vpofphi ] ) , we find the value of @xmath12 at the de sitter point ( [ dspt ] ) : ( _ ds ) = - . hence we conclude that the unstable de sitter solution in the einstein frame is mapped to one of the unstable saddle critical points in the vacuum of the minimal curvature frame ( see equation ( [ eqpts ] ) ) . we now examine case iii ) : when @xmath138 rolls to large positive values the einstein frame potential may be approximated by [ vprox ] ( ) e^-/ , leading to the exact solutions = + h(t ) = , where @xmath146 , corresponding to power - law acceleration in the einstein frame with [ ltpl ] a(t ) = a_0 t^3 . to find the corresponding behavior in the minimimal curvature frame we use the conformal transformation ( [ ctrans ] ) along with the transformation for the cosmic time coordinate ( [ ttran ] ) . we find the late - time power law attractor in the einstein frame ( [ ltpl ] ) is mapped to the asymptotic de sitter attractor in the mc frame at the minimal curvature @xmath78 ( depicted by the dashed blue line in fig . ( [ vacphase.eps ] ) ) , a(t ) = a_0 e^ t .

we consider the hypothesis of a limiting minimal curvature in gravity as a way to construct a class of theories exhibiting late - time cosmic acceleration . guided by the minimal curvature conjecture ( mcc ) we are naturally lead to a set of scalar tensor theories in which the scalar is non - minimally coupled both to gravity and to the matter lagrangian . the model is compared to the lambda cold dark matter concordance model and to the observational data using the gold " sneia sample of riess et . al . ( 2004 ) . an excellent fit to the data is achieved . we present a toy model designed to demonstrate that such a new , possibly fundamental , principle may be responsible for the recent period of cosmological acceleration . observational constraints remain to be imposed on these models . dcpt-06/17 s l h

the turn of the century is a good time to assess the importance and impact of three - nucleon forces ( 3nfs ) on the development of the field of few - nucleon physics . it has been 67 years since wigner@xcite first raised the possibility that three - nucleon forces might be significant in the triton : `` @xmath0 one must assume a certain potential energy @xmath0 or a three - body force . '' it is significant that the triton had not yet been discovered , although he predicted it would be bound by nucleon - nucleon ( nn ) forces alone . since that time we have relied on field - theoretic techniques , phenomenology , and sophisticated symmetry arguments to construct 3nfs , and the most modern and advanced experimental facilities have recently been used to validate these forces . in a very real sense we are fortunate that three - nucleon potentials are not too strong or too weak . indeed , i would nt be giving this talk if they were . imagine these forces to be @xmath1 orders of magnitude stronger than they actually are . in that case 3nfs would be comparable to nn forces and ( without stretching the imagination ) 4nfs , @xmath0 could also be comparable . in this scenario nuclear physics would be intractable , and in all likelihood this conference would not be held . on the other hand we could imagine such forces to be @xmath1 orders of magnitude weaker than they actually are . in this case they would play almost no role in nuclear physics , and although few - nucleon physics would be healthy and this conference would be held , the topic would be vacuous and there would be no such talk . what sets the scales such that our universe lies between these limits , where 3nfs are weak but significant ? the answer lies in the scales associated with qcd , which i will introduce later . for better or worse , these scales allow me to stand before you today and discuss these most interesting of forces . indeed , these scales allow a qualitative discussion of many aspects of few - nucleon physics , and i will rely on this approach to find common ground . my first task is to estimate the size of the effect of three - nucleon forces using scales . this can be achieved by a handwaving argument that is nevertheless correct in its essence . figure ( 1a ) shows two nucleons interacting via an nn potential , @xmath2 ( dashed line ) . adding another nucleon makes this a three - body system ( fig . ( 1b ) ) and , in addition to the normal nn interaction between the original two nucleons , that force @xmath2 will somehow feel the effect of the additional nucleon ( wavy line ) , and the size of this additional effect on the energy should scale as @xmath3 , since all of the nucleons are the same . this quantity unfortunately no longer has the dimensions of energy and we need to divide by an additional energy scale in order to obtain a final estimate . we motivate this scale in fig . ( 1c ) by showing a virtual pion ( with four - momentum @xmath4 ) emitted by a nucleon . normally we ignore the time component of @xmath5 , which scales as the difference of kinetic energies of the initial and final nucleon ; that is , it scales as @xmath6 , where @xmath7 is the nucleon mass . thus we might suspect that the * additional * effect of the third nucleon on the potential energy of the original pair of nucleons scales as @xmath8 where we have placed a factor of @xmath9 to make the dimensions correct . this simple result , which in chiral perturbation theory has @xmath10 replaced by @xmath11 gev ( a generic large - mass qcd scale ) gives us a quick estimate of the energy shift . using @xmath12 mev / pair we find @xmath13 mev , for what is either a three - nucleon force effect , a relativistic effect ( because of the @xmath14 ) , or an off - shell effect ( this latter is not obvious , but is intimately related to the @xmath15 in our `` derivation '' ; it is an essential part of the `` quasipotential '' @xcite problem ) . one important and obvious caveat for theorists is that it will be difficult to interpret calculations that have numerical errors greater than @xmath16 mev , if our goal is to understand three - nucleon forces . indeed , we should do much better than that and restrict triton errors to @xmath17 mev @xmath18 of the triton binding energy . in addition , @xmath19 absolute experiments are extremely difficult and rare . calculations with numerical errors @xmath20 have consequently become the standard and are called `` exact , '' `` complete , '' or `` rigorous '' . they are one of the biggest success stories in our field in the past 50 years . a bit of history is always a good way to start a discussion about the future . as scientists we naturally tend to concentrate on our unsolved problems , and successes are often overlooked . in the process of giving my views on where the field is going , i will also enumerate a few of the many successes in our business , which highlight the progress that we have made@xcite . one can conveniently categorize few - nucleon calculations as follows : ( a ) bound states ( i.e. , @xmath21h and @xmath21he ) ; ( b ) nd scattering below deuteron - breakup threshold ; ( c ) nd scattering above deuteron - breakup threshold ; ( d ) transitions between bound and continuum states . all of these types of calculations have been performed , and benchmarked comparisons between different methods exist for all categories except pd scattering ( i.e. , including a coulomb force between protons@xcite ) at finite energies . the ability to perform these extremely difficult calculations , especially the scattering calculations , has been one of the major successes in few - nucleon physics . when one considers this together with the incredible accomplishments of vijay pandharipande@xcite and his collaborators ( including my colleague , joe carlson ) for @xmath22 , this area is one of the most successful in all of nuclear physics , and goes far beyond even the dreams that theorists had 25 years ago . i summarize this part of the talk by noting that @xmath19 calculations are needed in order to disentangle systematically the relatively small effects of three - nucleon forces ( or relativistic effects , off - shell effects , @xmath0 ) . such calculations are now possible using many different techniques . most observables agree very well with experiment ; indeed , most are insensitive to 3nfs . the trick is to find the proper observables to investigate . wigner s mention of three - nucleon forces in his calculation of @xmath21h was subsequently ignored . there was no hope then ( and little now ) of being able to calculate and interpret results for a strongly interacting system dominated by such forces . while we have significant and extensive experimental information on the nn force , we have very little knowledge with which to constrain three - nucleon forces . we are forced to rely almost entirely on theory , particularly on field theory . early efforts involved primitive models that have not left their mark on the field . one calculation that has left an indelible mark is the august fujita - miyazawa model@xcite , based entirely on isobar intermediate states and pion propagation within nuclei . this was motivated by the dominant role of nuclear resonances in some processes . examples both familiar and unfamiliar are shown in fig . ( 2 ) . in fig . ( 2a ) a pion emitted by one nucleon interacts in a complicated way with a second nucleon and then is absorbed by a third nucleon . the fujita - miyazawa ( fm ) approximation to the entire process is shown in fig . one can also replace one ( or both ) pions by a heavy meson ( as shown in fig . a variety of such short - range mechanisms are depicted in chiral perturbation theory by fig . ( 2c ) , where all of the short - range processes ( induced by heavy - mass particles ) are shrunk to a point . we will first discuss the @xmath23-exchange forces . the second noteworthy calculation was performed by one of our conference organizers , shin nan yang@xcite . the yang model was the first three - nucleon potential model based on chiral - symmetry considerations , although there were previous calculations of effects based on pcac . this model was used in a variational calculation to estimate @xmath24 mev . this set the stage for the development of the most widely used 3nf , the tucson - melbourne ( tm ) model@xcite , which exploited current algebra and pcac ( and treated off - shell effects in a serious manner ) in deriving that force . the various parameters of this model incorporate phenomenology ( including isobars ) in a more meaningful way than the fm model . in addition to the models already mentioned , there are the urbana - argonne ( ua)@xcite [ an offshoot of fm ] , the ruhrpot@xcite , the brazil@xcite , and the [ @xmath25pt - based ] texas models@xcite . it is not a good thing to have so many different models for what should be a single correct physical force . indeed , one of our tasks for the new millennium is to force a `` convergent evolution '' on these models by incorporating proper amounts of the correct physics . recent progress has been made in that direction . the ua model now incorporates a chiral - symmetry - breaking term ( the `` a '' term of tm ) while a direct appeal to @xmath25pt@xcite leads to the elimination of a rather unimportant term of short - range + pion - range character ( the `` c '' term ) in the tm model , leaving the @xmath23-exchange parts of the two models essentially equivalent except for minor differences in parameter values . thus @xmath25pt has produced a recent unification of @xmath23-exchange forces . we emphasize that in order to accomplish this we must incorporate two key pieces of physics : * adequate phenomenology ( such as isobars ) * chiral constraints . before discussing the road to the future for 3nfs that have a short - range component , it is necessary to implement an organizational scheme . there are simply too many possible operators that can contribute . chiral perturbation theory fortunately allows us to sort 3nfs into classes and identify the terms that should be dominant . indeed , this sorting process is the organizational scheme of @xmath25pt . how does it work ? the `` natural '' degrees of freedom of qcd are quarks and gluons , whose interactions manifestly reflect the symmetries of qcd . we are not required to use these degrees of freedom , however , and the traditional degrees of freedom of nuclear physics , namely nucleons and pions , are the most effective ones . if we imagine somehow mapping qcd expressed in terms of quarks and gluons onto the hilbert space of all particles , and then freezing out the effect of the heavy particles ( e.g. , all nucleon resonances and all mesons with mass @xmath27 gev ) we arrive at chiral perturbation theory@xcite . the freezing - out process is familiar in nuclear physics as feshbach [ p , q ] reaction theory@xcite , and is known in that context to lead to complicated operators . it is nevertheless possible to implement the important chiral - symmetry constraints of qcd in this `` qcd in disguise '' theory . even more important is the power counting that makes this scheme work as a field theory@xcite . power counting is a kind of dimensional analysis based on ( only ) two energy scales associated with qcd . one scale is @xmath28 , the pion ( beta-)decay constant ( @xmath29 mev ) that controls the goldstone bosons ( such as the pion ) , while the second is @xmath11 gev , the scale of qcd bound states ( the @xmath30 and @xmath31 mesons , nucleon resonances , etc . ) , above which we agree to freeze out all excitations . that these scales are all that is needed is not only not obvious , but it s a little bit miraculous ! using these scales it can be shown that a given ( lagrangian ) building block should scale as @xmath32 two vital properties of this simple construction are : ( 1 ) @xmath33 because of chiral symmetry ; ( 2 ) c is a dimensionless constant that satisfies @xmath34 , which is the condition of `` naturalness '' . the latter is also not obvious , but is extremely important . if @xmath35 could vary over many orders of magnitude in a problem , this organizational scheme would be useless . moreover , unless positive powers of @xmath36 exist in the denominator ( even in the presence of vacuum fluctuations ) this would not lead to an expansion in powers of ( small / large ) . this formal scheme can be implemented in nuclei to estimate the sizes of various types of operators in the nuclear medium@xcite . an additional nuclear scale is needed in order to characterize the medium , and this is given by the * effective * nuclear momentum ( or inverse correlation length ) : @xmath37 , where @xmath38 is the pion mass . then it is possible to show that the one - pion - exchange nn potential satisfies@xcite @xmath39 and @xmath40 and we have reproduced our previous result with @xmath41 . note that this size estimate applies only to the leading - order terms ; smaller terms exist that might be significant in special situations . there are 7 basic types of 3nfs in leading order . four are of two - pion range , two are of mixed pion - range + short - range , and there is a class of short - range + short - range forces . figure 2 shows several examples . the generic two - pion - exchange force is given by fig . ( 2a ) and can be broken down into the `` a '' , `` b '' , and `` d '' terms of the tm force@xcite , plus the so - called born terms . the latter have been derived@xcite but have never been used in their entirety ( there are many terms ) in any @xmath21h calculation . the two mixed terms are those represented generically in fig . ( 2c ) , one specific mechanism in this category being that of fig . ( 2d ) ( the so - called @xmath42-term ) . these terms have been investigated only once or twice@xcite . finally , there are purely short - range terms of the type incorporated in the ua 3nf . in the near future it will be necessary to investigate thoroughly the importance of this set . most urgent are the born terms . the local terms are almost certainly unimportant compared to the isobar mechanism , but none of the nonlocal terms have been incorporated into existing calculations . a preliminary and not wholly satisfactory set of calculations@xcite exists for the mixed short - range + pion - range forces . these need to be extended and refined . finally , it is by no means certain that the effects of these different forces are entirely linear when added together ( as indicated in ref.@xcite ) . thus a lot of different calculations need to be performed in different combinations and for as many different observables as possible . completion of this exercise will indeed provide us with a solid base in this area from which we can extend few - nucleon physics into the new millennium . we have postponed until the end a discussion of evidence for these forces , both direct and indirect . the indirect evidence is strong but not compelling . with all modern `` second - generation '' nn forces the triton is underbound by roughly @xmath43 mev , in agreement with our earlier estimate , and @xmath44he , @xmath0 are also underbound . unless our understanding of these nn forces is badly deficient , such nn forces require an additional three - nucleon force . better evidence is provided by a recent analysis of the tail of the @xmath45 potential@xcite , which generated very strong support for calculations of two - pion - exchange forces obtained from @xmath25pt . several effective coupling constants ( for pion - nucleon scattering operators ) were fit in that analysis , and they agree with the same couplings that are used in two - pion - exchange 3nfs , validating the latter mechanisms . in other words , once the building blocks have been established it makes little difference what those blocks are used to construct . direct evidence is available in the sagara discrepancy , which is shown in fig . ( 3 ) in the differential cross section for pd scattering at 65 mev@xcite . if we ignore the effects of the coulomb interaction in the forward direction , agreement between the experimental data and the calculation with nn forces alone ( dashed line ) is very good , except in the diffraction minimum where the data lie above the calculation . if one adds a 3nf the solid curve results , which nicely fills in the minimum , and agreement with the data is quite good this is rather strong evidence for 3nfs and it exists at other energies . an estimate of the effect of the 3nf alone that is based on dwba is shown in the long - dashed curve . this general behavior is very familiar in glauber scattering at high energy , where a dominant single - scattering term falls rapidly with increasing angle , until the double - scattering amplitude ( which decreases more slowly with angle ) becomes dominant , and so on . finally , we discuss the effects of the short - range + pion - range terms , and in particular the one depicted in fig . this is the so - called @xmath42-term that has recently been shown@xcite to have a nonnegligible effect on the nucleon asymmetry @xmath46 scattering observable . that observable has been a serious problem for theorists for a long time . calculations using nn forces alone are significantly smaller than the data , particularly at low energies ( e.g. , @xmath47 mev ) , both for the pd and nd scattering . a variety of explanations have been proposed ( this observable is very sensitive to spin - orbit interactions ) , but the most plausible is the 3nf mechanism@xcite . an example of this is nd scattering at 3 mev . calculations that incorporate only an nn force are about 30% smaller than the experimental data at the maximum . incorporating the tm 3nf removes about @xmath48 of the discrepancy . adding the @xmath42-term ( that we discussed before ) with a dimensionless strength coefficient @xmath49 removes another @xmath48 of the discrepancy . technical problems prevented calculations with stronger versions of this 3nf . it is nevertheless clear that 3nfs of various types make significant contributions to this observable . much more work is needed on this problem . we note that discrepancies also exists for electromagnetic reactions and in the four - nucleon problem . special circumstances may dictate that classes of 3nf operators smaller than leading order will play a role . an example of this is neutron matter ( or neutron - rich nuclei ) . the isospin dependence of three - nucleon forces takes 3 forms : @xmath50 , @xmath16 , and @xmath51 . the first vanishes for three neutrons . because 3 neutrons exist in a @xmath52 state , only the projection @xmath53 contributes to that state , while the projection @xmath54 vanishes . some mechanisms ( such as isobar configurations ) that prefer large isospins may be enhanced , as shown by vijay pandharipande and his collaborators . in these special circumstances the dimensionless isospin factors ( which typically average to about 1 ) can conspire to give enhanced coupling strengths . this has not yet been investigated in detail , but it should be . we have made great advances in the past 25 years in our understanding of both three - nucleon systems and three - nucleon forces . we stand poised to make further advances , based on recent technical developments . hopefully we will soon be able to develop a consensus `` standard model '' of 3nfs with all significant features incorporated , which will then allow us to pursue three - nucleon physics into the new millennium . we summarize this section as follows . * most three - nucleon observables are insensitive to 3nfs . * 3nfs are small in size but appear necessary for the @xmath21h binding energy , the sagara discrepancy , and the @xmath55 puzzle . * chiral symmetry provides a unified approach to 3nfs ; power counting identifies dominant mechanisms . * the leading - order ( dominant ) 2@xmath56-exchange 3nfs have been calculated ; they have large isobar contributions . * new short - range plus pion - range mechanisms may help resolve the @xmath55 puzzle . * although much remains to be investigated , a consensus appears to be developing for the bulk of 3nf terms . the work of j.l.f . was performed under the auspices of the united states department of energy .

most nuclear physics ranges from insensitive to relatively insensitive to many - nucleon forces . the dominant ingredient in calculations of nuclear properties is the nucleon - nucleon potential . three - nucleon forces nevertheless play an important role in nuclear physics because of the great precision of modern calculational methods for systems of relatively few nucleons . we explore the reasons why many - body forces are weak in nuclei by using a classification scheme for such forces that is based on dimensional power counting , which is used to organize chiral perturbation theory . an assessment will be made of how close we are to a `` standard '' three - nucleon force . recent advances in determining the significance of three - nucleon forces will also be discussed .

the application of cross - correlation techniques to measure velocity shifts has a long history ( simkin 1972 , 1974 ; lacy 1977 ; tonry & davis 1979 ) , and with the advent of massive digital spectroscopic surveys of galaxies and stars , the subject has renewed interest . the recently completed sloan digital sky survey ( sdss ) has collected spectra for more than 600,000 galaxies and 90,000 quasars ( adelman - mccarthy et al . 2007 , york et al . 2000 ) . the sdss has also obtained spectra for about 200,000 galactic stars , and it is now being extended at lower galactic latitudes by segue with at least as many spectra ( rockosi 2005 , yanny 2005 ) . another ongoing galactic survey , rave , is expected to collect high - resolution spectra for a million stars by 2011 ( steinmetz et al . 2006 ) , and the plans for the gaia satellite include measuring radial velocities for 10@xmath0 stars by 2020 ( katz et al . 2004 ) . extracting the maximum possible information from these spectroscopic surveys requires carefully designed strategies . cross - correlation has been the target of numerous developments in recent years ( see , e.g. , mazeh & zucker 1994 , statler 1995 , torres , latham & stefanik 2007 , zucker 2003 ) , but several practical aspects of its implementation would benefit from further research . these include the selection of templates ( e.g. , observed vs. synthetic libraries ) , how to combine measurements from multiple templates , the method to determine the maximum of the cross - correlation function , data filtering , and error determination . some of these issues are briefly addressed in this paper , but our focus is on how the requirement of coherence among all entries in a radial velocity data base can be used to improve the original measurements . a different but plausible approach has been recently proposed by zucker & mazeh ( 2006 ) . the doppler shifts of targets in a spectroscopic survey are determined one at a time . each object s projected velocity is measured independently , not counting a possible common set of cross - correlation templates . for a given template , from any pair of ( projected ) velocity measurements , we can derive a relative velocity between the two objects involved . however , that figure will likely be numerically different from the value inferred from the direct cross - correlation between their spectra , even if the two objects are of the same class . in this paper , we argue that it is possible to improve the original determinations by imposing consistency among all available measurements . our discussion is oriented to the case of a homogeneous sample : multiple observations of the same or similar objects . in the following section i introduce cross - correlation , with a brief discussion about error evaluation . section [ basic ] presents the notion of _ self - improvement _ and section [ general ] extends the method to the more realistic scenario in which the spectra in a given data set have varying signal - to - noise ratios . in [ sdss ] we explore an application of the proposed technique involving low - resolution spectra , concluding the paper with a brief discussion and reflections about future work . the most popular procedure for deriving relative velocities between a stellar spectrum and a template is the cross - correlation method ( tonry & davis 1979 ) . this technique makes use of all the available information in the two spectra , and has proven to be far superior than simply comparing the doppler shifts between the central wavelengths of lines when the signal - to - noise ratio is low . the cross - correlation of two arrays ( or spectra ) * t * and * s * is defined as a new array * c * @xmath1 if the spectrum * t * is identical to * s * , but shifted by an integer number of pixels @xmath2 , the maximum value in the array * c * will correspond to its element @xmath3 . cross - correlation can be similarly used to measure shifts that correspond to non - integer numbers . in this case , finding the location of the maximum value of the cross - correlation function can be performed with a vast choice of algorithms . the most straightforward procedure to estimate realistic uncertainties involves an accurate noise model and monte - carlo simulations , and that is the method we use in section [ sdss ] . we employ gaussians and low - order polynomials to model the peak of the cross - correlation function . for these simple models , implemented in a companion idl code , it is possible to derive analytical approximations that relate the uncertainty in the location of the maximum of the cross - correlation function to the covariance matrix [ u@xmath4 . digital cross - correlation , introduced in section [ xcorr ] , is commonly employed to derive doppler radial velocities between two spectra . the discussion in this section is , nonetheless , more general , and deals with the statistical improvement of a set of relative velocity measurements . if three spectra of the same object are available and we refer to the relative radial velocity between the first two as @xmath5 , an alternative estimate of @xmath5 can be obtained by combining the other relative velocity measurements , @xmath6 . assuming uniform uncertainties , the error - weighted average of the two values is @xmath7 . for a set of @xmath8 spectra , we can obtain an improved relative radial velocity determination between the pair @xmath9 by generalizing this expression @xmath10 it can be seen from eq . [ ci ] that the correlation of * t * and * s * is equal to the reverse of the correlation between * s * and * t*. thus , when the relative velocities between two spectra is derived from cross - correlation and the spectra have a common sampling , it will be satisfied that @xmath11 , but this will not be true in general . for example , if we are dealing with grating spectroscopy in air , changes in the refraction index with time may alter the wavelength scale and the spectral range covered by any particular pixel , requiring interpolation . if our choice is to interpolate the second spectrum ( * s * ) to the scale of the first ( * t * ) , this may introduce a difference between @xmath12 and @xmath13 due to different interpolation errors . we can accommodate the general case by writing @xmath14 note that this definition ensures that @xmath15 , and @xmath16 . if the quality of the spectra is uniform , and all measured radial velocities @xmath12 have independent uncertainties of the same size @xmath17 , the primed values would have an uncertainty @xmath18 . despite @xmath12 may be numerically different from @xmath13 , @xmath19 will be highly correlated with @xmath20 , and thus the uncertainty in the primed velocities will not be reduced that fast . in addition , all @xmath12 are also correlated with all @xmath21 , driving the improvement farther away from the ideal @xmath22 behavior . we can expect that after a sufficient number of spectra are included , either random errors will shrink below the systematic ones or all the available information will already be extracted , and no further improvement will be achieved . the case addressed in section [ basic ] corresponds to a set of spectra of the same quality . if the uncertainties in the measured relative radial velocities differ significantly among pairs of spectra , eq . [ vprime ] can be generalized by using a weighted average @xmath23 where @xmath24 and the uncertainty is @xmath25 in the common case in which @xmath26 , the counterpart of eq . [ symmetry ] for dealing with spectra of varying signal - to - noise ratios reduces to @xmath27 where @xmath28 in the next section we use simulated spectra for a case study : multiple observations of the same object or massive surveys involving large numbers of very similar objects at intermediate spectral resolution . the sdss spectrographs deliver a resolving power of @xmath29 , over the range 381910 nm . these two fiber - fed instruments are attached to a dedicated 2.5 m telescope at apache point observatory ( gunn et al . each spectrograph can obtain spectra for 640 targets simultaneously . as a result of a fixed exposure time in sdss spectroscopic observations , the flux in a stellar spectrum at a reference wavelength of 500 nm , @xmath30 , correlates well with the @xmath31 magnitude of the star and with the signal - to - noise ratio at 500 nm ( @xmath32 ) . on average , we find @xmath33 at @xmath34 mag . to build a realistic noise model , we used the fluxes and uncertainties for 10,000 spectra publicly released as part of dr2 ( abazajian et al . 2004 ) to derive , by least - squares fitting , a polynomial approximation . when @xmath30 is expressed in erg @xmath35 s@xmath36 @xmath36 , which are the units used in the sdss data base , we can write @xmath37 where @xmath38 . this relationship holds in the range @xmath39 . the uncertainties in the sdss fluxes for stars mostly relatively bright calibration sources are not dominated by photon noise , but by a _ floor _ noise level of 23% associated with a combination of effects , including imperfect flat - fielding and scattered light corrections . errors are highly variable with wavelength , but the noise at any given wavelength depends linearly on the signal . based on the same set of sdss spectra used for eq . [ snr500 ] , we determine the coefficients in the relation @xmath40 which we use here for our numerical experiments . for a given choice of @xmath32 , we interpolate the table of coefficients @xmath41 and @xmath42 derived from sdss data , and by inverting eq . [ snr500 ] we derive the flux at 500 nm . finally , we scale the spectrum fluxes and calculate the expected errors at all wavelengths using eq . gaussian noise is introduced for each pixel position , according to the appropriate error , simulating multiple observations of the same star to create an entire library of spectra . we employed a spectrum of hd 245 , a nearby g2 star@xmath43 ) , surface gravity ( @xmath44 ) , and kinematics , make this object a prototypical thick - disk turn - off star . ] , to produce spectra that resemble sdss observations with various signal - to - noise ratios . radial velocities are also artificially introduced . the spectrum of hd 245 used here has a resolving power of @xmath45 at 660 nm to 7700 at 480 nm . this variation is , however , irrelevant when smoothing the data to @xmath46 as we do in these experiments . ] and is included in the elodie.3 database ( moultaka et al . 2004 , prugniel & soubiran 2001 ) . as the rest of the library , this spectrum was obtained with the 1.9 m telescope and the elodie spectrograph at haute provence . the original fluxes are resampled to @xmath47 , and then smoothed to @xmath46 by gaussian convolution . the output fluxes are sampled with 12 pixels per resolution element . the doppler shift due to the actual radial velocity of hd 245 has already been corrected in the elodie library . new values for the radial velocity in the library of simulated sdss observations are drawn from a normal distribution with a @xmath48 km s@xmath36 , as to approximate the typical range found in f- and g - type stellar spectra included in the sdss ( mostly thick - disk and halo stars ) . the wavelength scale is then doppler shifted , changed to vacuum ( @xmath49 ) , and the spectrum resampled with a step of @xmath50 in @xmath51 ( approximately 2.17 pixels per resolution element ) . the elodie spectra only cover the range @xmath52 nm , and therefore a similar range is finally kept for the sdss - style files , which include 2287 pixels . we employed a set of 40 test spectra with @xmath53 , measuring the relative radial velocities for all possible pairs . [ xcf ] illustrates two sample spectra and their cross - correlation function . to avoid very large or small numbers , the input arrays are simply divided by their maximum values before cross - correlation . we used second and third order polynomials , as well as a gaussian to model the cross - correlation function and estimate the location of its maximum by least - squares fitting . the solid line in the lower panels of the figure are the best - fitting models . we experimented varying the number of pixels involved in the least - squares fittings ( @xmath54 ) . with the sampling used , the measured relative shifts in pixel space ( @xmath55 ) correspond to a velocity @xmath56 , where @xmath57 is the speed of light in vacuum ; one pixel corresponds to 69 km s@xmath36 . we compare the relative velocities between all pairs of spectra derived from the measurement of the location of the cross - correlation peaks with the _ known _ , randomly drawn , relative velocities . the average difference for the 1600 velocities ( 40 spectra ) @xmath58 and the rms scatter ( @xmath59 ) are used to quantify systematic and random errors , respectively . our experiments exhibit no systematic errors in the derived velocity when the number of points entering the fit @xmath54 was an odd number , i.e. , when we use the same number of data points on each side from the pixel closest to the peak of the cross - correlation function . modest offsets ( @xmath60 ) , however , are apparent when fitting polynomials to an even number of data points , despite we enforce the maximum to be bracketed by the two central data points . random errors increase sharply with the number of data points involved in the fittings for the polynomial models , but not for the gaussian model . the best results for the polynomials are obtained when the lowest possible orders are used . using less than 11 points for the gaussian did not produce reliable results , as there was not enough information to constrain all the parameters of the model , which includes a constant base line . the best performance @xmath61 km s@xmath36 was obtained using a second order polynomial and @xmath62 . using a gaussian model achieved a minimum @xmath63 km s@xmath36 , fairly independent of @xmath54 . the third order polynomial provided the poorest performance , @xmath64 km s@xmath36 at best . the cross - correlation can be computed in fourier space , taking advantage of the correlation theorem ( brigham 1974 ) . this fact is usually exploited to speed up the calculation dramatically , as fast fourier transforms can be calculated with a number of operations proportional to @xmath65 , compared to @xmath66 required by eq . note , however , that for medium - resolution surveys of galactic stars , the velocity offsets , limited by the galactic escape velocity , usually correspond to a limited number of pixels . therefore , it is only necessary to compute the values of * c * in the vicinity of the center of the array , rendering the timing for a direct calculation similar to one performed in transformed space ) took @xmath67 seconds in fourier space ( arrays padded to @xmath68 or @xmath69 ) , while in pixel space , with a lag range restricted to @xmath70 pixels ( @xmath71 km s@xmath36 ) , it took @xmath72 seconds . ] . spectra and @xmath33 . the solid line represents the original distribution , and the dashed line the result after applying self - improvement . the error distributions are symmetric because the array * v * is antisymmetric . , width=317 ] to test the potential of the proposed self - improvement technique we repeat the same exercise described in [ sdss1 ] , but using increasingly larger datasets including up to 320 spectra , and adopting three different values for the @xmath73 per pixel at 500 nm : 50 , 25 , and 12.5 . we calculated the cross - correlation between all pairs of spectra ( matrix * v * ) , and performed quadratic fittings to the 3 central data points , cubic polynomial fittings to the central 4 points , and gaussian fittings involving the 11 central points . we estimated the uncertainties in our measurements by calculating the rms scatter between the derived and the known relative velocities for all pairs . then we applied eq . [ symmetry ] to produce a second set of _ self - improved _ velocities . ( because the array of wavelengths , @xmath74 , is common to all spectra , the matrix * v * is antisymmetric and we can use eq . [ symmetry ] instead of eq . [ vprime ] . ) a first effect of the transformation from * v * to * v * , is that the systematic offsets described in [ sdss1 ] when using polynomial fittings with even values of @xmath54 disappear ( the same systematic error takes place for measuring @xmath12 and @xmath75 , canceling out when computing @xmath76 ) . more interesting are the effects on the width of the error distributions . fig . [ dist ] illustrates the case when a quadratic model is used for @xmath77 and @xmath33 . the solid line represents the original error distribution and the dashed line the resulting distribution after self - improvement . [ sigma ] shows the rms scatter as a function of the number of spectra for our three values of the @xmath73 ratio at 500 nm . the black lines lines show the original results , and the red lines those obtained after self - improvement . each panel shows three sets of lines : solid for the quadratic model , dotted for the cubic , and dashed for the gaussian . extreme outliers at @xmath78 km s@xmath36 , if any , were removed before computing the width of the error distribution ( @xmath59 ) . note the change in the vertical scale for the case with @xmath79 . for the experiments with @xmath80 , several runs were performed in order to improve the statistics , and the uncertainty ( standard error of the mean ) is indicated by the error bars . these results are based on the gaussian random - number generation routine included in idl 6.1 , but all the experiments were repeated with a second random number generator and the results were consistent . as described in [ sdss1 ] , the quadratic model performs better on the original velocity measurements for @xmath81 and @xmath82 . at the lowest considered @xmath73 value of 12.5 , however , the gaussian model delivers more accurate measurements . self - improvement reduces the errors in all cases . although a second order polynomial fitting works better than third order for the original measurements , the two models deliver a similar performance after self - improvement . interestingly , the impact of self improvement is smaller on the results from gaussian fittings than on those from polynomial fittings . as expected , the errors in the original measurements are nearly independent of the number of spectra in the test , but there is indication that at low signal - to - noise the errors after self - improvement for the polynomial models decrease as the sample increases in size , until they plateau for @xmath83 . from these experiments , we estimate that the best accuracy attainable with the original cross - correlation measurements are about 3 , 6 , and 15 km s@xmath36 at @xmath84 , 25 , and 12.5 , respectively . our results also indicate that by applying self - improvement to samples of a few hundred spectra , these figures could improve to roughly 2.5 , 4 , and 9 km s@xmath36 at @xmath84 , 25 , and 12.5 , respectively . we obtained an independent estimate of the precision achievable by simply measuring for 320 spectra the wavelength shift of the core of several strong lines ( h@xmath85 , h@xmath86 , h@xmath87 , and h@xmath88 ) relative to those measured in the solar spectrum ( see allende prieto et al . 2006 ) , concluding that radial velocities can be determined from line wavelength shifts with a @xmath89 uncertainty of 3.8 km s@xmath36 at @xmath90 , 7.2 km s@xmath36 at @xmath91 , and 15.9 km s@xmath36 at @xmath92 only 1020% worse than straight cross - correlation but these absolute measurements can not take advantage of the self - improvement technique . allende prieto et al . ( 2006 ) compared radial velocities determined from the wavelength shifts of strong lines for sdss dr3 spectra of g and f - type dwarfs with the sdss pipeline measurements based on cross - correlation . the derived @xmath89 scatter between the two methods was 12 km s@xmath36 or , assuming similar performances , a precision of 8.5 km s@xmath36 for a given method . the spectra employed in their analysis have a @xmath32 distribution approximately linear between @xmath93 and 65 , with @xmath94 and with mean and median values of 22 and 18 , respectively . their result is in line with the expectations based on our numerical tests that indicate a potential precision of 67 km s@xmath36 at @xmath95 . independent estimates by the sdss team are also consistent with these values ( rockosi 2006 ; see also www.sdss.org ) . after correcting for effects such as telescope flexure , the wavelength scale for stellar spectra in dr5 is accurate to better than 5 km s@xmath36 ( adelman - mccarthy et al . 2007 ) . this value , derived from the analysis of repeated observations for a set of standards and from bright stars in the old open cluster m67 , sets an upper limit to the accuracy of the radial velocities from sdss spectra , but random errors prevail for @xmath96 . provided no other source of systematic errors is present , our tests indicate that self - improvement could reduce substantially the typical error bars of radial velocities from low signal - to - noise sdss observations . this paper deals with the measurement of relative doppler shifts among a set of spectra of the same or similar objects . if random errors limit the accuracy of the measured relative velocity between any two spectra , there is potential for improvement by enforcing self - consistency among all possible pairs . this situation arises , for example , when a set of spectroscopic observations of the same object are available and we wish to co - add them to increase the signal - to - noise ratio . the spectra may be offset due to doppler velocity offsets or instrumental effects , the only difference being that in the former case the spectra should be sampled uniformly in velocity ( or @xmath97 ) space for cross - correlation , while in the latter a different axis may be more appropriate . another application emerges in the context of surveys that involve significant numbers of spectra of similar objects . radial velocities for individual objects can be derived using a small set of templates and later _ self - improved _ by determining the relative velocities among all the survey targets and requiring consistency among all measurements . the potential of this technique is illustrated by simulating spectra for a fictitious survey of g - type turn - off stars with the sdss instrumentation . our simulations show that applying self - improvement has a significant impact on the potential accuracy of the determined radial velocities . the tests performed dealt with relative velocities , but once the measurements are linked to an absolute scale by introducing a set of well - known radial velocity standards in the sample , the relative values directly translate into absolute measurements . the ongoing segue survey includes , in fact , large numbers of g - type stars , and therefore our results have practical implications for this project . the proposed scheme handles naturally the case when multiple templates are available . templates and targets are not treated differently . relative velocities are measured for each possible pair to build @xmath12 , and consistency is imposed to derive @xmath76 by using eqs . [ vprime ] or [ vprimegen ] . if , for example , the templates have been corrected for their own velocities and are the first 10 spectra in the sample , the velocity for the @xmath98th star ( @xmath99 ) can be readily obtained as the weighted average of the @xmath76 elements , where @xmath100 runs from 1 to 10 . the final velocities would take advantage of all the available spectra , not just the radial velocity templates , with differences in signal - to - noise among spectra already accounted for automatically . very recently , zucker & mazeh ( 2006 ) have proposed another approach with the same goals as the method discussed here . their procedure determines the relative velocities of a set of @xmath8 spectra by searching for the doppler shifts that maximize the value of the parameter @xmath101 , where @xmath102 is the maximum eigenvalue of the correlation matrix a two - dimensional array whose @xmath9 element is is the cross - correlation function between spectra @xmath100 and @xmath103 . zucker & mazeh s algorithm is quite different from the self - improvement method presented here . it involves finding the set of velocities that optimally aligns the sample spectra , whereas self - improvement consists on performing very simple algebraic operations on a set of radial velocities that have already been measured . self - improvement is obviously more simple to implement , but a detailed comparison between the performance of the two algorithms in practical situations would be very interesting . this paper also touches on the issue of error determination for relative radial velocities derived from cross - correlation , and convenient analytical expressions are implemented in an idl code available online . we have not addressed many other elements that can potentially impact the accuracy of doppler velocities from cross - correlation , such as systematic errors , filtering , sampling , or template selection . the vast number of spectra collected by current and planned spectroscopic surveys should stimulate further thought on these and other issues with the goal of improving radial velocity determinations . there is certainly an abundance of choices that need to be made wisely .

the measurement of doppler velocity shifts in spectra is a ubiquitous theme in astronomy , usually handled by computing the cross - correlation of the signals , and finding the location of its maximum . this paper addresses the problem of the determination of wavelength or velocity shifts among multiple spectra of the same , or very similar , objects . we implement the classical cross - correlation method and experiment with several simple models to determine the location of the maximum of the cross - correlation function . we propose a new technique , _ self - improvement _ , to refine the derived solutions by requiring that the relative velocity for any given pair of spectra is consistent with all others . by exploiting all available information , spectroscopic surveys involving large numbers of similar objects may improve their precision significantly . as an example , we simulate the analysis of a survey of g - type stars with the sdss instrumentation . applying _ self - improvement _ refines relative radial velocities by more than 50% at low signal - to - noise ratio . the concept is equally applicable to the problem of combining a series of spectroscopic observations of the same object , each with a different doppler velocity or instrument - related offset , into a single spectrum with an enhanced signal - to - noise ratio .

the motivation behind studying the structure of the photon results from the interest in understanding the formation of hadronic matter . permitted by the heisenberg uncertainty relation , the photon can fluctuate for some time into a quark anti - quark state . this fluctuation can be disturbed , e.g. , by an electron or proton probe which allows the density of quarks and gluons of the partonic state of the photon to be determined . at the lep @xmath0 and hera @xmath1 colliders , photons are emitted by the leptons which gives access to the partonic structure of almost real photons @xcite as well as highly virtual photons . the measurements to obtain information on the partonic state of the photons discussed here are 1 . the photon structure function from deep inelastic electron photon scattering ( fig . [ fig : diseg ] ) , 2 . jet and particle cross sections ( e.g. fig . [ fig : jetgp ] ) , and 3 . the total photon photon cross section . new @xmath6 structure function measurements have been performed in the interesting region of small parton momenta @xmath7 by the l3 collaboration @xcite . @xmath6 is determined from the measurement of the double differential inclusive cross section @xmath8 where @xmath9 is the electro - magnetic coupling constant , @xmath5 denotes the virtuality of the probing photon and gives the resolution scale of the process , and @xmath10 is the inelasticity @xmath11 . in fig . [ fig : f2-x ] , the @xmath12 dependence of @xmath6 is shown in two bins of @xmath5 . a major challenge in this analysis is the determination of @xmath12 : since the lepton that emitted the target photon remains undetected , the energy of the target has to be determined from the hadronic final state . using a new improved reconstruction method for @xmath12 , two results for @xmath6 are presented by the l3 collaboration using two different monte carlo generators for the correction of detector effects ( phojet @xcite , twogam @xcite ) . these two data sets demonstrate that over a large region in @xmath12 the structure function result does not depend on the details of simulating the hadronic final state . only below @xmath7 this limitation becomes sizable . in the same figure , previous results of the opal collaboration are shown @xcite . within the errors , good agreement is observed between the two experiments . also shown are different parameterizations of the quark density in the photon demonstrating that the data give new information on the quark distributions at low @xmath12 ( lac @xcite , grv @xcite , sas @xcite ) . scaling violations caused by gluon emission off the quark before the scattering process occurs results in a rise of @xmath6 below a small value of @xmath12 . the data are not yet precise enough to confirm or reject such a rise at @xmath7 . in the momentum region around @xmath13 , where the quark and the anti - quark each carry half of the photon energy , results on the structure function @xmath6 exist from many experiments . a compilation of these measurements is shown in fig . [ fig : f2-q2-g ] as a function of the resolution scale @xmath5 @xcite . the data are compatible with an increasing quark density in the photon as @xmath5 increases . this @xmath5 dependence is very different from that of hadronic structure functions at large @xmath12 and is expected by perturbative qcd ( fig . [ fig : f2-q2-p ] and discussion in section [ subsec : cse ] ) : the splitting of the photon into a quark - anti - quark pair gives rise to the probability @xmath14 of finding a quark in the photon to increase as @xmath15 in leading order . in the same figure an effective parton distribution @xmath16 of the photon is shown which has been extracted from di - jet measurements in photon proton collisions by the h1 collaboration @xcite . this effective parton distribution combines the quark and the gluon densities of the photon with a weight of color factors @xcite : @xmath17 the vertical scale for @xmath18 on the right side of fig.[fig : f2-q2-g ] has been adjusted relative to the @xmath19 scale , since in contrast to the @xmath6 measurements the jet processes are independent of the electric charges of the quarks . the relevant resolution scale is the transverse momentum @xmath20 of the scattered partons which is here taken to have the same resolution power as @xmath5 . the results of the di - jet measurements are in good agreement with the @xmath6 data . the jet data probe the partons of the photon at large resolution scales and compete well in precision with the @xmath6 measurements . the quark density close to the kinematic limit @xmath21 is analysed in photoproduction of two jets . here the contributions of the direct and resolved photon proton processes need to be understood ( fig . [ fig : jetgp ] ) . they differ in their matrix elements and therefore in the distribution of the parton scattering angle @xmath22 . in fig . [ fig : costh ] , a new di - jet cross section measurement of the zeus collaboration is shown differentially in @xmath23 for large di - jet masses and correspondingly large @xmath12 @xcite . also shown are next - to - leading order qcd calculations @xcite using two different parton parameterizations of the photon ( grv @xcite , gs @xcite ) . the direct photon contribution ( not shown in the figure ) is not sufficient to describe the measured jet cross section either in shape or in the absolute normalization . contributions of resolved photon processes are required to describe the data which are sufficiently precise to discriminate different parton parameterizations of the photon at large @xmath12 . ( 5.0,10.0 ) ( 0.5,-0.9 ) new measurements of the inclusive charm production cross section at the large lep beam energies are shown in fig . [ fig : charm ] by the l3 collaboration @xcite . the cross section has been determined using semi - leptonic charm decays in the electron and muon channels . in the same figure , next - to - leading order qcd calculations @xcite using two different charm masses and the grv parameterization @xcite of the parton distributions in the photons are shown . the dominant contribution to the cross section results from gluon induced processes with an average gluon momentum as small as @xmath24 @xcite . also di - jet data are used to access the low-@xmath12 gluon distributions of the photon . in fig . [ fig : xgamma ] , a new measurement of the di - jet cross section is shown as a function of the parton momentum @xmath12 by the h1 collaboration @xcite . the histograms represent a leading - order qcd calculation @xcite showing the contributions of the direct photon - proton interactions and quark and gluon induced processes using the grv parton parameterizations for the photon and the proton . both the charm and di - jet measurements give compatible conclusions on the low-@xmath25 gluon density of the photon and are precise to the level of @xmath26 . new information on the gluon distribution of the photon results from di - jet production in photon - photon collisions which has been measured by the opal collaboration @xcite . in fig . [ fig : opal - pt ] , the cross section is shown differentially in the transverse jet energy @xmath27 . at sufficiently large @xmath27 the measurement can well be described by a next - to - leading order qcd calculation @xcite using the parton distribution function of grv @xcite . in fig . [ fig : opal - eta ] , the di - jet cross sections are shown differentially in the jet rapidity @xmath28 . the data explore different regions of the parton fractional momentum @xmath29 with a precision of @xmath30 . they are compared to leading order qcd calculations ( phojet @xcite , pythia @xcite ) and discriminate different parameterizations of the gluon distributions of the photon ( lac @xcite , grv @xcite , sas @xcite ) . ( 5.0,10.0 ) ( 0.0,0.0 ) the fluctuation of a virtual photon into a quark - anti - quark pair is suppressed by the photon virtuality @xmath5 . in comparison with real photons one therefore expects a smaller probability of finding the virtual photon in a partonic state . also , there is less time to develop from the @xmath31 pair a vector meson bound state such that the hadronic contributions to the virtual photon structure should be small . in fig . [ fig : virtual ] , the first triple - differential di - jet cross section is shown as a function of the photon virtuality @xmath5 in two bins of the parton momentum @xmath12 for a fixed resolution scale @xmath32 gev@xmath33 @xcite . the cross section measurement at @xmath21 ( fig . [ fig : virtual]b ) is well described by a leading order qcd calculation using the direct photon - proton interaction processes only ( dashed curve @xcite ) . at @xmath13 ( fig . [ fig : virtual]a ) the absolute cross section is found to be smaller compared to the measurement at @xmath21 as expected from the short fluctuation time of the photon . here the direct photon contributions are not sufficient to describe the data at small @xmath34 gev@xmath33 : the di - jet process is able to resolve the partonic structure of the virtual photon . as @xmath5 approaches the squared transverse energy of the jets of @xmath35 gev@xmath33 , the resolution power of the di - jet process becomes insufficient for detecting the fluctuations of the virtual photons . ( 5.0,9.0 ) ( 1.,2 . ) a ) ( 5.,2 . ) b ) ( 0.,0 . ) in analogy to the real photon case , eq . ( [ eq : effpdf ] ) , an effective parton distribution for virtual photons @xmath36 has been extracted from the data and is shown in fig . [ fig : effpdf]a in the interval @xmath37 gev@xmath33 for @xmath38 and @xmath39 gev@xmath33 . the partonic structure of the virtual photon is only slowly suppressed with the photon virtuality @xmath5 . such a dependence is predicted by perturbative qcd : in the region of @xmath40 the probability of finding a quark in the virtual photon decreases logarithmically as @xmath5 approaches the jet resolution scale : @xmath41 the formation of a hadronic bound state from the @xmath31 pair of the photon can be studied with the production of @xmath42 mesons . in fig . [ fig : effpdf]b , the @xmath5 dependence of the @xmath42 cross section is shown which exhibits a fast decrease proportional to @xmath43 with @xmath44 @xcite . as expected from the short photon fluctuation time into a quark - anti - quark pair , the probability to develop a hadronic bound state from the quark - anti - quark pair is highly suppressed . at sufficiently large @xmath5 , the partonic structure of the virtual photon can therefore be predicted by perturbative qcd . in fig . [ fig : effpdf]a , the full curve represents a qcd inspired model of the effective parton distribution of the virtual photon ( sas1d @xcite ) which is in agreement with the measurement within the experimental errors . ( 5.0,10.0 ) ( 12.2,8.4 ) b ) ( 4.5,8.4 ) a ) ( 0.,0 . ) ( 8.,0 . ) the total photon - photon cross section @xmath45 is dominated by soft scattering processes in which the photons develop a hadronic structure before the interaction occurs . a major challenge of this measurement is the understanding of the different contributions , the elastic , diffractive and non - diffractive processes . the visibility of the first two contributions in the detectors is small and requires reliable monte carlo generator calculations . progress has recently been made by the l3 experiment which succeeded in collecting a few hundred events of exclusive four pion production which contains contributions of elastic double-@xmath42 production at center of mass energies below @xmath46 gev ( fig . [ fig : sigma - rr ] ) @xcite . these data test the two generator calculations shown ( phojet @xcite , pythia @xcite ) . a new measurement of the total photon - photon cross section is shown in fig . [ fig : sigma - gg ] using the two different monte carlo generators ( l3 collaboration @xcite ) . the data show a rise above @xmath47 gev and are compatible within errors with the preliminary measurement of the opal collaboration @xcite . this observed rise can be described by a power law @xmath48 with the rise has the tendency to be stronger than expected from soft pomeron exchange which successfully describes all hadron hadron and photon proton total cross sections with ( 10.0,8.4 ) ( 4.,0 . ) improved knowledge on the partonic structure of real photons results from * new structure function @xmath49 measurements at low parton fractional momenta @xmath50 , * di - jet cross section measurements at @xmath12 values down to @xmath51 and high @xmath52 in photon - proton and photon - photon interactions , and * charm production in photon photon processes at low @xmath50 . for the first time the partonic structure of highly virtual photons @xmath53 gev@xmath33 has been investigated in @xmath1 collisions . the fluctuations of the virtual photon into a quark - anti - quark pair is only slowly suppressed with @xmath5 and is compatible with a logarithmic decrease as predicted by perturbative qcd . the understanding of the total photon - photon cross section has improved by the detection of elastic @xmath42 production . overall , the results on the photon obtained in @xmath0 and @xmath1 collisions complement each other and are well compatible . the precision of the measurements remains a challenge for the next few years in order to be well prepared for the linear collider . a sizable fraction of strong interaction processes includes the exchange of colour singlet objects . at the hera @xmath1 and tevatron @xmath2 colliders , these objects are emitted by the hadrons and may involve the exchange of quantum numbers ( meson exchange ) or may not ( diffractive processes ) . a handle on the type of the interaction process is given , e.g. , by the observation of a fast baryon in the proton beam direction . detection of energetic neutrons indicate that isospin-1 exchanges are present , in particular charged pion exchange . protons are sensitive to both isoscalar and isovector exchanges . where the leading proton is close to the beam energy , diffractive scattering is expected to be dominant . partonic scattering processes in such diffractive interactions give access to quark - gluon configurations that are colour neutral but different from the well known hadrons . the following measurements to obtain information on colour singlet exchange are discussed here : 1 . the @xmath1 structure function with a tagged baryon ( fig . [ fig : feynman - lb ] ) , 2 . the @xmath1 structure function of diffractive exchange ( figs . [ fig : feynman - lb ] `` @xmath54 '' and [ fig : feynman - disrg ] ) , 3 . di - jet and w - boson production in diffractive @xmath2 scattering ( fig . [ fig : feynman - tev ] ) , and 4 . vector meson production in @xmath1 interactions ( fig . [ fig : feynman - vm ] ) . the production of protons and neutrons is studied in both the h1 and zeus experiments . a hadron calorimeter detects neutrons scattered at zero angle with respect to the proton direction . a series of roman pot stations between the beam magnets serves as a proton spectrometer . in fig . [ fig : h1-lb ] , new measurements of the structure function from @xmath1 collisions with a tagged baryon @xmath55 are shown as a function of the baryon fractional energy @xmath56 for fixed photon virtuality @xmath57 gev@xmath33 and parton fractional momentum @xmath58 ( h1 collaboration @xcite ) . @xmath55 was determined from cross section measurements which were integrated over the baryon transverse momenta in the range @xmath59 gev : @xmath60 here @xmath9 is the electro - magnetic coupling constant and @xmath10 denotes the inelasticity @xmath61 . the proton tagged structure function is found to be larger than that of the neutron tagged data . the curves are predictions of model calculations inspired by regge phenomenology ( fig . [ fig : feynman - lb ] ) . in this picture , the proton data can not be explained by @xmath62 exchange alone , since from the @xmath63 and @xmath64 isospin @xmath65 states one would expect the proton measurement to be a factor two below the neutron data . instead , the proton data can be explained by an admixture of @xmath62 , reggeon ( @xmath66 , @xmath67 ) and pomeron exchange . the neutron data can be explained by charged pion exchange alone and demonstrate the potential access to the pion structure function @xcite in the new kinematic domain at small parton momenta around @xmath68 . ( 10.0,4.0 ) ( 9.5,0.1 ) p , n ( 8.05,0.72 ) , @xmath69 , @xmath70 ( 6.5,0.72 ) @xmath71 ( 6.,0 . ) further information on the type of the interaction process comes from a new measurement of tagged baryons with the coincident formation of a large rapidity gap between the systems @xmath72 and @xmath73 ( fig . [ fig : feynman - disrg ] ) where @xmath73 may or may not be observed in the main detector ( zeus collaboration @xcite ) . in fig . [ fig : zeus - lb ] , the rate of events with a large rapidity gap is shown as a function of the baryon fractional energy @xmath74 . for @xmath75 , the tagged proton production ( full circle ) is dominated by diffractive processes . for @xmath76 , the minimum gap size chosen for the analysis implies that @xmath77 . in this kinematic region , the rate of events with a large rapidity gap is small and shows that diffraction is not the main mechanism for the production of the baryons . ( 5.0,4.5 ) ( 5.,4 . ) evidence for diffractive scattering processes in @xmath1 interactions can be obtained from different methods : a : : tagging of highly energetic protons in the proton spectrometers ( section [ subsec : lb ] ) , b : : from analysis of rapidity regions which are free of hadronic activity ( `` rapidity gap '' , fig . [ fig : feynman - disrg ] ) , or c : : from the mass distribution of the hadronic final state which is observed in the main detector . the structure function @xmath78 for diffractive exchange @xmath79 has been measured by the zeus collaboration using the tagged proton method a @xcite as a function of the following four variables : 1 . the virtuality @xmath5 of the exchanged photon . 2 . the squared four - momentum transfer @xmath80 from the proton side , 3 . the momentum fraction @xmath81 ( @xmath82 ) , with @xmath83 being the mass of the diffractive system observed in the main detector , and 4 . the fractional momentum @xmath84 . when interpreting this process in terms of the exchange of a colour singlet object ( fig [ fig : feynman - lb ] ) , @xmath85 gives the fractional momentum that this object takes from the proton , and @xmath86 is the fractional momentum of the quark involved in the electron - quark scattering process . therefore , this deep inelastic scattering measurement gives access to the partonic structure of diffractive color singlet exchange and provides information on the corresponding @xmath87 distribution of this process . a new measurement of the @xmath87 distribution is shown in fig . [ fig : zeus - t ] ( zeus collaboration @xcite ) . the methods b ( fig . [ fig : feynman - disrg ] ) and c of measuring the deep inelastic scattering of diffractive exchange have to integrate over some @xmath87-range and can here take advantage of the knowledge of the @xmath87 distribution of the proton tagged data . the two methods also do not include the detection of proton remnant particles at small masses @xmath88 of the dissociated proton system and integrate over a small range of this mass ( typically 1 - 4 gev ) . since the acceptance of proton - tagged events is at the percent level , the statistics using methods b , c are much larger by far . in fig . [ fig : f2d3 ] , a new triple differential structure function measurements @xmath89 of the zeus collaboration ( method c @xcite ) are compared with previous measurements by the h1 collaboration ( method b @xcite ) . the data are shown in a small selection of the large phase space covered as a function of the fractional momentum @xmath85 , which the colour singlet object takes from the proton , in two bins of the parton momentum observable @xmath86 and the photon virtuality @xmath5 . at small @xmath85 , they are consistent in these and surrounding phase space bins with a power law @xmath90 and therefore are compatible with factorization of the @xmath85 dependence . the measured value by the zeus collaboration @xcite is @xmath91 and is compatible with the result of the h1 collaboration @xcite . the measured value of @xmath92 is slightly larger than the value expected for soft pomeron exchange in regge inspired models ( @xmath93 ) . the @xmath86 and @xmath5 dependence of @xmath89 at fixed small value of @xmath85 therefore gives the partonic structure of colour singlet exchange . the results of the two collaborations are consistent in most of the 15 phase space regions commonly covered , for example in fig . [ fig : f2d3]b , and call in a few of them for homework ( fig . [ fig : f2d3]a ) , especially in an understanding of slightly different kinematic regions covered in the squared momentum transfer @xmath87 and the mass of the diffractive system @xmath88 . ( 16.,9 . ) ) ( 0.,0 . ) ( 7.3,0 . ) in fig . [ fig : f2-q2-d ] , the resolution scale @xmath5 dependence of @xmath89 of colour singlet exchange at large parton momenta @xmath94 is shown . this measurement has been newly extended to large @xmath5 up to @xmath95 gev@xmath33 by the h1 collaboration @xcite . the data are at relatively large values of @xmath96 and can be described by a dominant diffractive exchange ( pomeron exchange ) together with meson contributions ( reggeon exchange ) . the @xmath5 dependence of @xmath89 is found to be consistent with flat which is very different from the structure function measurements of hadrons , e.g. , the proton structure function ( fig . [ fig : f2-q2-p ] @xcite ) . it is also different from the @xmath5 dependence of the photon structure function ( fig . [ fig : f2-q2-g ] ) . the different distributions can be understood from the qcd evolution equations : the probability @xmath97 of finding a quark in the proton , color singlet exchange , or photon depends logarithmically on @xmath5 : @xmath98 the @xmath99 denote the splitting functions convoluted with the parton densities . the first term @xmath100 represents the contribution of quarks after radiating a gluon . the second term @xmath101 gives the contributions of gluons that split into a quark anti - quark pair . the third term @xmath102 adds the quarks resulting from the photon splitting into a quark anti - quark pair ( relevant for photon only ) . the proton structure function falls at large @xmath103 with increasing resolution scale @xmath5 : the probability of finding a parton in the proton above the average valence quark momentum decreases with increasing resolving power @xmath5 ( first term of eq . ( [ eq : dglap ] ) ) . the logarithmic increase of the photon structure function with @xmath5 is caused by the third term of eq . ( [ eq : dglap ] ) which is to first approximation independent of @xmath5 . the structure function of diffractive exchange differs from those of the proton and the photon : the flat shape makes it distinct from a quark dominated object . the large rate of diffractive exchange excludes an explanation by photon exchange . instead , a large gluon density in the exchanged diffractive object can explain the observed @xmath5 dependence of the structure function which is driven by the second term of eq . ( [ eq : dglap ] ) . therefore the structure function measurement mainly probes the gluon splitting into a quark anti - quark pair and reflects the structure of the strong interactions . this partonic structure of colour singlet exchange has been quantified by extracting gluon and quark distributions from the diffractive data using the structure function measurements alone ( h1 collaboration @xcite ) or in combination with jet cross section measurements ( zeus collaboration @xcite ) . different final state observables have been measured in diffractive @xmath1 scattering by the h1 and zeus collaborations , e.g. , thrust @xcite , di - jet cross sections @xcite , energy flow @xcite , multiplicity @xcite , and charm production @xcite . a large fraction of the measurements have been compared to monte carlo generators which simulate diffractive @xmath1 scattering processes by the emission of colour singlet objects with the parton distributions as extracted from the fits to @xmath89 mentioned above . overall , the data are well described by such simulations which demonstrates a consistently working framework for understanding diffractive parton scattering processes in @xmath1 collisions . a deviation of this good description of the data may be seen in the photoproduction of di - jets which is discussed below in the comparison of the rates of diffractive processes at the hera and tevatron colliders . note that the picture of exchanging a colour singlet object with a partonic structure is not the only one to describe the data : interesting alternative approaches exist which need fewer parameters and describe certain aspects of the data well . examples are electron scattering off a quark or a gluon of the proton with colour neutralization by the exchange of a second parton that cancels the colour charge , or models that predict the @xmath86 dependence of @xmath89 , or the concept of fracture functions @xcite . for reviews of the different approaches refer to , e.g. , @xcite . the methods used by the tevatron experiments cdf and d0 to select diffractive scattering processes are detection of leading protons ( method a ) or measurement of rapidity gaps ( method b ) . the observables used to analyse the diffractive exchange are di - jet formation and the production of w - bosons . both experiments have observed events involving the exchange of one or - as a new result - two colour singlet objects ( fig . [ fig : feynman - tev]a , c @xcite ) . in the latter process , the jets are produced centrally and in each beam direction a large rapidity gap or a tagged proton is observed . in fig . [ fig : cdf - diff ] , the shapes of transverse energy @xmath27 distributions of the leading jets in di - jet events are compared for single and double colour singlet exchange and non - diffractive data ( cdf collaboration ) . the @xmath27 range covered and the similarity of these distributions give several interesting observations : the diffractive di - jet production results from the same type of parton parton scattering processes as the non - diffractive data . in the latter case , the fractional momenta of the partons from the proton are small @xmath104 and therefore likely to come from gluon gluon scattering processes . in the diffractive case with the exchange of one or two colour singlet objects , the center - of - mass energy of the scattering process is much smaller than that of the @xmath2 beams since these objects carry only a fraction of the beam proton energy . nevertheless , the jet transverse energy reaches out to @xmath105 gev such that almost the full energy of these objects is involved in the hard parton parton scattering process . ( 5.0,6.0 ) ( -0.8,0 . ) both tevatron experiments have observed events with a rapidity gap between two jets ( fig . [ fig : feynman - tev]b @xcite ) . in these events , the full energy of the exchanged object is involved in the jet production process and the object is probed at very large squared four - momentum transfer @xmath106 of the order of @xmath107 . in fig . [ fig : d0 ] , the rate of events with such a colour singlet exchange relative to non - diffractive events is shown from the d0 collaboration . the distribution of the size of the rapidity gap is shown to be within errors independent of the jet transverse energy ( fig . [ fig : d0]b , c ) . in fig . [ fig : d0]a , the rate is given as a function of the jet transverse energy which has a tendency to rise with increasing @xmath27 . the data are sufficiently precise to discriminate different models of colour singlet exchange : they exclude the exchange of a photon ( dotted curves in fig . [ fig : d0 ] ) and a calculation using two hard gluons ( `` bfkl '' , dashed curves @xcite ) . the data can be consistently described by a model calculating the exchange of one energetic gluon with an additional parton to ensure colour neutrality ( full curve @xcite ) . the cdf experiment has observed the production of w - bosons in diffractive scattering processes @xcite . these events have essentially one lepton and missing transverse energy and can be interpreted as resulting from quark anti - quark fusion . a comparison of the diffractive w - boson rate with that of the di - jet production is shown in fig . [ fig : cdf - gluon ] as a function of the relative gluon contribution in the colour singlet object and is expressed as a momentum sum rule . the gluon contribution is found to be large @xmath108 which is well compatible with previous ( shown in the figure ) and new fits of the zeus collaboration @xcite ( not shown ) and previous results of the h1 collaboration @xcite . while the large gluon component is consistently observed in diffractive processes at hera and the tevatron , the rate of such events is found to be largely different : at hera , the rate of diffractive deep inelastic scattering events is of the order of @xmath109 . in the phase space regions covered so far , the hera final state data are overall consistently described when compared to calculations that use the parton distributions resulting from the diffractive structure function measurements . using the same parton distributions for the tevatron diffractive data , the predicted rate is much larger than the observed rate of the order of @xmath110 @xcite . this discrepancy can , e.g. , be expressed in terms of a momentum sum rule as shown in fig . [ fig : cdf - gluon ] . the inconsistency is a puzzle which is under lively discussion . instructive measurements have been made , allowing the energy @xmath111 involved in the interaction to be measured relative to the total hadronic center - of - mass energy @xmath112 . in fig . [ fig : hera - tev ] , rates of diffractive events are shown as a function of the ratio @xmath113 . for the tevatron jet results @xcite , the jet transverse energy @xmath27 at the threshold has been used as a measure of @xmath111 ( fig . [ fig : hera - tev]a - c ) . the rate of diffractive events appears to decrease as the total center - of - mass energy @xmath114 becomes large relative to the energy involved in the scattering process . such dependence can , e.g. , be explained by the increased potential of destroying the rapidity gap by beam remnant interactions which may be formulated in a reduced survival probability for the rapidity gap . different other explanations have been suggested , key words are here absorption corrections , flux renormalization , or other means of factorization breaking @xcite . for the hera data , two measurements are discussed here : in the case of deep inelastic scattering data , the mass @xmath83 of the diffractive system has been taken as a measure of @xmath111 . the data in fig . [ fig : hera - tev]d are consistent with being flat as a function of @xmath115 @xcite and show no indication of a decreasing survival probability . in photoproduction of di - jets , the fractional momentum @xmath12 of the parton from the photon is related to the ratio @xmath116 . in fig . [ fig : h1-diffjet ] , the di - jet cross section from diffractive scattering processes is shown as a function of @xmath12 from h1 data @xcite . at large @xmath117 , where the direct photon contribution dominates , the data are described by the calculations of the pompyt generator @xcite when using the parton distribution functions for the colour singlet exchange as extracted from the @xmath89 measurements . however , at @xmath118 the data are better described , if an overall reduction factor of @xmath119 is applied to the calculation of the resolved photon proton interactions . this observation hints for a reduced survival probability of the rapidity gap in resolved photon proton processes . owing to the presence of a proton and a photon remnant , these @xmath120 processes are similar to that of diffractive processes in @xmath2 collisions . in the future , more extended and precise measurements of the photoproduction of jets may help in the understanding of the different diffractive rates observed by the hera and tevatron experiments . in elastic vector meson production from @xmath1 collisions , the full energy of the colour singlet object is involved in the scattering process ( fig . [ fig : feynman - vm ] ) . of special interest are processes with a hard scale such as 1 . the mass @xmath121 of a heavy vector meson , 2 . the virtuality @xmath5 of the photon in a deep inelastic scattering process , or 3 . the squared four - momentum transfer @xmath87 of the colour singlet exchange . such processes allow perturbative qcd calculations to be compared with the measurements and therefore give additional information on colour singlet exchange as well as on the proton and the vector meson @xcite . in this context , the following measurements of vector meson production in @xmath1 collisions at hera are discussed here : 1 . vector meson cross sections and their dependencies on the center - of - mass energy @xmath122 , the photon virtuality @xmath5 , and the squared momentum transfer @xmath87 , and 2 . photoproduction of @xmath123 mesons from proton and nuclear targets . in fig . [ fig : sigma - vm ] , a compilation of the measurements of the total photoproduction cross section @xmath124 and elastic vector meson cross sections @xmath125 up to the production of @xmath126 @xcite is shown as a function of the photon proton center - of - mass energy @xmath127 . the measured total cross section is at large center - of - mass energies compatible with a slowly rising distribution as @xmath128 with @xmath129 @xcite . the optical theorem relates the total cross section to the imaginary part of the amplitude of forward elastic scattering . therefore , elastic vector meson cross sections should rise with approximately twice the power : @xmath130 photoproduction of light vector mesons ( @xmath42 , @xmath67 , @xmath131 ) show an increase in the production that is compatible with this prediction . however , photoproduction of the heavy @xmath123 mesons exhibit a stronger dependence on the center - of - mass energy with @xmath132 . a steeper energy dependence is also observed for light vector meson production in deep inelastic scattering processes : in fig . [ fig : lambda]a , the energy dependence of new @xmath125 measurements by the h1 and zeus collaborations @xcite was again expressed in terms of the fit parameter @xmath133 using eq . ( [ eq : sigma - vm ] ) and is shown as a function of the scale . the scale was here chosen to be the sum of the photon virtuality and the vector meson squared mass @xmath134 . the parameter @xmath133 is found to increase with increasing scale . ( 5.0,9.0 ) ( 10.5,8.1 ) b ) ( 5.5,8.1 ) a ) ( 0.0,0.0 ) ( 8.0,0.0 ) such energy dependence is similar to that observed in inclusive deep inelastic scattering cross sections ( fig . [ fig : lambda]b @xcite ) . at fixed @xmath5 and small parton momenta @xmath135 , the total photon proton cross section @xmath136 is directly related to the large gluon density observed in the proton which gives rise to the @xmath137 dependence of the proton structure function @xmath138 . using the relation @xmath139 gives an energy dependence of the cross section as @xmath140 . the similar energy dependencies observed in vector meson production ( fig . [ fig : lambda]a ) and inclusive deep inelastic scattering processes ( fig . [ fig : lambda]b ) is suggestive of sensitivity of the vector meson data to the gluon distribution of the proton . in fig . [ fig : rho ] , the longitudinal component of the @xmath42 meson production cross section is shown as a function of @xmath135 in four bins of @xmath5 @xcite . similarly , the @xmath123 production cross section is shown in fig . [ fig : psi ] as a function of the photon proton center - of - mass energy @xmath141 @xcite . the measurements can be described by perturbative qcd calculations which use existing parameterizations of the gluon distributions in the proton ( curves @xcite ) . the calculations use the square of the gluon density to account for the colour neutrality of the exchanged object ( e.g. fig . [ fig : feynman - psi ] ) . therefore , the comparisons of the data with the calculations give a highly sensitive measure of the gluons in the proton . a further component of the calculations is the mechanism for formation of the vector meson such that the comparisons to the data will give new information also on this part of the process . in elastic vector meson production , the squared momentum transfer @xmath87 , which is exchanged between the vector meson and the proton , gives information on the size of the interaction region . such @xmath87 distributions can be fitted for small values of @xmath87 using an exponential distribution @xmath142 . in fig . [ fig : b - slope ] , a compilation of the fitted @xmath143 parameters is shown for the hera data for @xmath42 , @xmath131 , and @xmath123 meson production as a function of the scale ( new measurements : @xcite ) . the scale has here again been chosen to be the sum of the photon virtuality @xmath5 and the vector meson squared mass @xmath144 . with increasing scale , the data tend to approach a constant value of @xmath145 gev@xmath146 which corresponds to the size of the proton . the size of the @xmath31 state is therefore small compared to that of the proton and probes the proton at small distances . the @xmath143 parameter measured for the photoproduction of @xmath123 mesons indicates the small size of the charm anti - charm object in the interaction with the proton . further information on this @xmath147 configuration results from nuclear dependencies of non - diffractive @xmath123 meson production in comparison to that of protons @xcite : in fig . [ fig : dy - proton ] , the shapes of proton nucleus cross sections @xmath148 @xcite are shown as a function of the rapidity change @xmath149 . here @xmath149 denotes the rapidity difference between the beam proton and the most energetic tagged proton . these distributions can be described by an exponential form @xmath150 . the fitted slopes decrease as the nuclear mass increases , i.e. , the protons are on average more decelerated with a heavier target . ( 5.0,13.5 ) ( 0.0,-0.8 ) ( 8.0,-0.8 ) it is interesting to compare the deceleration process of the protons with that of @xmath123 mesons resulting from non - diffractive photoproduction off nuclei @xmath151 . here the rapidity difference between the photon and the @xmath123 is used as a measure of the deceleration process ( fig . [ fig : dy - proton ] , data of the emc @xcite , h1 @xcite , and zeus @xcite experiments ) . also these distributions in the rapidity difference can be described by an exponential form . in contrast to the proton data , the slope of the @xmath123 production does not decrease with increasing mass of the nucleus which implies that the iron target does not decelerate the @xmath147 object better than the proton target . the slight increase in the slope with @xmath152 can be explained , according to monte carlo generator studies , by the different center - of - mass energies of the emc and the hera experiments . the absence of a nuclear deceleration effect for the @xmath123 mesons may be interpreted as resulting from nuclear transparency . for a discussion of nuclear transparency effects refer , e.g. , to @xcite . in this interpretation , the colour charges of the small quark anti - quark configuration are sufficiently screened to penetrate a nucleus without further interactions . the hera and tevatron experiments have measured different observables that can be related to the parton distributions of diffractive exchange . using structure function measurements in @xmath1 collisions , di - jet production in @xmath1 and @xmath2 scattering , and w - boson production in @xmath2 collisions , they consistently find a large gluon component in this colour singlet state . the overall rate of diffractive processes observed in @xmath1 and @xmath2 collisions , however , is found to be different and challenges explanation . measurements of elastic vector meson production involving a hard scale provide an alternative approach to understanding colour singlet exchange . comparisons of the data with qcd calculations that rely on two - gluon exchange give new information on the gluon distribution of the proton and on the vector meson states . at sufficiently large scales , the spatial extension of the quark anti - quark states appears to be small . photoproduction of @xmath123 mesons indicates that the @xmath147 state penetrates a nuclear environment essentially undisturbed . overall , diffractive physics is a very active field of research and is developing away from a soft interaction language to the understanding of a fundamental process of strong interactions within the framework of qcd . i wish to thank a. astbury for providing a very positive conference atmosphere for the exchange of the new scientific results . i wish to thank for kind help in preparing the talk h. abramowicz , m. albrow , v. andreev , k. borras , a. brandt , j. dainton , t. doyle , k. freudenreich , c. glasman , b. heinemann , m. kienzle , p. newman , r. nisius , g. snow , and s. sldner - 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energy physics , vancouver , canada ( 1998 ) d.s barton et al . , emc collaboration , j.j . aubert et al . , h1 collaboration , s. aid et al . , zeus collaboration , j. breitweg et al . , l.l . frankfurt , g.a . miller and m. strikman , _ ann . nucl . part . * 45 * , 501 ( 1994 )

recent experimental results on the partonic structure of the photon and on the color singlet exchange in strong interaction processes are reviewed . at the lep @xmath0 and hera @xmath1 colliders , complementary and consistent measurements have been achieved on the quark - gluon structure of quasi - real and virtual photons . at the hera @xmath1 and tevatron @xmath2 colliders , the quark - gluon configuration of the diffractive exchange is consistently found to have a large gluon component . the rate of diffractive interactions observed by the hera and tevatron experiments , however , is largely different and challenges explanation . # 1#2#3#4#1 * # 2 * ( # 3 ) # 4 @xmath3 @xmath4 desy-98 - 209b issn 0418 - 9833 + december 1998 * diffraction and low-@xmath5 physics + including two - photon physics * martin erdmann + universitt karlsruhe , engesserstr . 7 , d-76128 karlsruhe + e - mail : martin.erdmann@desy.de _ invited plenary talk at the xxix international conference on high energy physics , vancouver , b.c . canada ( 1998 ) _

broad emission line regions ( blrs ) in active galactic nuclei ( agns ) have been the subject of extensive studies for more than two decades . such regions are not spatially resolved , and all the available information about their geometry is obtained from analysis of variable lines . it is well established that photoionization by the central radiation source is the main source of ionization and excitation of the blr gas . indeed , photoionization calculations , when applied to time - averaged spectra , can reasonably explain most of the observed line ratios ( for review and references see ferland , in these proceedings , and netzer 1990 ) . however , such time - averaged calculations contain little information about the spatial distribution of the gas . extensive monitoring campaigns , during the last decade , have produced several high quality data sets . they include the time dependence of the multi - wavelength continuum , as well as the change in line fluxes and line profiles as a function of time ( for a review see , horne these proceedings , peterson 1993 , netzer & peterson 1997 ) . excellent data sets are now available for half a dozen low luminosity agns . less complete ( in terms of wavelength coverage ) yet very detailed data sets , are available on a dozen or so more sources . unfortunately , theoretical understanding lags behind and there are few , if any , systematic attempts to produce complete blr models that reproduce the new light curves . most recent studies focused on obtaining transfer functions , and little effort has been devoted to reconstruct the physical conditions in the gas . in particular , only one or two emission lines have been considered while many more lines , and thus more information and constraints , are available , at least in some data sets . this work , as well as the more detailed results in kaspi and netzer ( 1999 ) , present an attempt to investigate one of the best data sets in a new way . the goal is to reconstruct the observed light curves of as many emission lines as possible in the seyfert 1 galaxy ngc 5548 . as shown below , the observational constraints on the line intensity and line ratios as a function of time , enable us to deduce the run of density , column density and cloud distribution across the blr in this source . below we demonstrate how the time dependent relative and absolute line intensities , and their relationship to the variable continuum , leave little freedom in modeling the blr . previous attempts to model the blr differ in the method used to reconstruct the gas distribution and the assumptions made about the blr properties . we distinguish between direct and indirect methods . direct methods involve initial guess of the gas distribution and other properties . these are later checked by calculating the predicted emission line light curves , assuming the above properties and the given variable continuum . indirect methods attempt to obtain the gas distribution by computing transfer functions for various emission lines . this is somewhat ill - defined since it produces emissivity maps , rather than mass - distribution maps . it therefore requires an additional confirmation that the so - obtained emissivity maps are consistent with photoionization calculations . while there were several attempts to produce transfer functions for various lines , we are not aware of any successful mapping that is consistent with full photoionization calculations ( see also maoz 1994 ) . the first systematic attempt to reconstruct the blr in ngc 5548 is by krolik et al . these authors used the maximum entropy method to reconstruct the transfer function and to model the blr as a spherically symmetric system of isotropically emitting clouds . the krolik et al . blr is divided into two distinct zones : one emitting the high - ionization lines ( column density of @xmath1110@xmath12 @xmath5 and ionization parameter of 0.3 ) and the other emitting the low - ionization lines ( column density of @xmath1110@xmath13 @xmath5 and ionization parameter of 0.1 ) . later , obrien , goad , & gondhalekar ( 1994 ) have combined photoionization and reverberation calculations ( [ formalism ] ) . their study , and also the one by prez , robinson & de la funte ( 1992 ) , focused on the shape of the transfer function under different conditions and on the line emissivity for a few generic models . they did not attempt any detailed reconstruction of a specific data set . bottorff et al . ( 1997 ) presented a detailed kinematic model , combined with photoionization calculations . this was applied to only _ ( civ@xmath01549 ) in the spectrum of ngc 5548 . dumont , collin - suffrin , & nazarova , ( 1998 ) modeled the blr in ngc 5548 as a 3 zones region where the various component locations are determined by _ average _ line - to - continuum lags . much of their conclusions regarding the required density and column density are based on the relative strength of the balmer lines . finally , goad & koratkar ( 1998 ) re - examined the same ngc 5548 data set and deduced a strong radial dependence ( @xmath14 over the density range of @xmath15 @xmath10 ) . here again , the main assumption is of a simple two zone model and the photoionization calculations are not applied to all the lines . none of the above models presents a complete study or attempts full recovery of all light curves . hence , global consistency checks are missing . our work relies heavily on the direct approach . we prefer to avoid , as much as possible , ill - defined transfer functions and unreliable emissivity distributions . instead , we make a large number of initial assumptions ( `` guesses '' ) and check them , one by one , against the complete data set . this makes the numerical procedure more time - consuming but is more robust because of the direct relationship between the assumed geometry and the resulting light - curves . we follow the formalism presented in kaspi & netzer ( 1999 ) which is first described in netzer ( 1990 ) . the reader is referred to those references for more information . in summary , we consider a spherical blr consisting of numerous small spherical clouds and a point - like ionizing source . all important physical properties are represented by simple power - laws in @xmath16 , the distance from the central source . a possible justification for this may be a radial dependent external pressure that determines the cloud properties . the case presented here assumes isotropically emitting clouds . we are fully aware of the potential complications due to this assumption , in particular for lines like @xmath17 . the particle density in the model , @xmath18 ( assumed to be constant within each cloud ) , is given by @xmath19 the cloud column density , @xmath20 , is computed by considering spherical clouds , of radius @xmath21 . the mass of the individual clouds is conserved , but it is not necessarily the same for all clouds , thus , @xmath22 . the typical cloud column density is @xmath23 and the geometrical cross - section is @xmath24 the number density of such clouds per unit volume is @xmath25 the clouds are illuminated by a central source whose ionizing luminosity , @xmath26 , varies in time . designating @xmath27 as the flux emitted by the cloud in a certain emission - line @xmath28 per unit projected surface area ( @xmath29 ) , we find the following relation for a single cloud emission : @xmath30 assuming the system of clouds extends from @xmath31 to @xmath32 we integrate over @xmath16 to obtain cumulative line fluxes : @xmath33 having determined the properties of the emission line clouds , and having assumed a spectral energy distribution ( sed ) for the ionizing source , we now calculate @xmath27 using a photoionization code and follow the formalism to obtain @xmath34 . we also consider the changes of @xmath35 , and possibly also the sed , in time . we take this into account by calculating @xmath36 for the entire range of continuum luminosity applicable to the source under discussion . the calculated line fluxes are the results of integrating eq . [ el ] , using , at each radius , the relevant ionizing flux , i.e. the one obtained with the ionizing luminosity @xmath37 . a model is specified by the source luminosity and sed , the radial parameter @xmath38 , and the normalization of the various free parameters . these include @xmath39 , @xmath40 and the density and column density at a fiducial distance which we take to be one light - day . the comparison with observations further requires the normalization of the total line fluxes and hence the integrated number of clouds ( an alternative way of presenting this normalization is by defining a radial dependent covering fraction ) . we have calculated a large grids of photoionization models covering the entire range of density , column density , and incident flux applicable for this source . the calculations were performed using ion97 , the 1997 version of the code @xmath41 ( see netzer 1996 , and references therein ) . there are several limitations for such codes which should be considered . most important ( and crucial for any blr model ) is the transfer of the optically thick lines . this is treated with a simple , local escape probability method which has long been suspected to be inadequate for the balmer lines ( see netzer 1990 ) . the problem is not yet solved and is common to most detailed photoionization models similar to @xmath41 , like @xmath42 by g. ferland . there is no simple solution for this problem and we prefer , at this stage , not to consider blamer lines in this work . there is a similar problem for several other low ionization lines , like mgii@xmath02798 and the feii lines . on the other hand , the transfer of lines like ly@xmath43 and civ@xmath01549 is much better understood . the seyfert 1 galaxy ngc 5548 is one of the best studied agn . it was monitored , for 8 months , in the optical - uv , in 1989 , and was also the subject of an intensive optical spectroscopic monitoring for 8 years ( peterson et al , 1999 ) . several shorter monitoring campaigns , each with a duration of several months , took place in various wavelength bands . this makes ngc 5548 an excellent choice for testing our model . in this study we have concentrated on the @xmath44 1989 campaign ( clavel et al . 1991 ) and used the resulting light curves of the uv continuum ( at @xmath01337 ) and the following emission lines : ly@xmath451216 , civ@xmath01549 , ciii]@xmath01909 , heii@xmath01640 , and mgii@xmath02798 . ( all diagram in this paper show light curves of only the first three lines . ) several additional observational constraints have been considered : 1 . the @xmath46 spectra clearly show that the ciii]@xmath01909 line is blended with si@xmath01895 . clavel et al . ( 1991 ) have measured the combined flux of both lines , and listed it as ciii]@xmath01909 . hence , our calculated flux for both lines will be combined ( see kaspi and netzer 1999 for more details regarding the use of this ratio as a density diagnostics ) . the observed sed of ngc 5548 is reviewed by dumont et al . we found a strong dependence of our results on the ratio of the uv to x - ray continuum flux . we have chosen a typical seyfert 1 sed with @xmath47 similar to the one measured for ngc 5548 . 3 . using the observed @xmath01337 continuum flux , an assumed cosmology ( @xmath48= 75 kms@xmath49mpc@xmath49 and @xmath50=0.5 ) , and assumed sed , we have estimated a time averaged ionizing luminosity of @xmath51 erg s@xmath49 . during the @xmath46 campaign , the uv continuum flux varied by a factor of @xmath52 , hence we choose our grid to cover the ionizing luminosity range of @xmath53 to @xmath54 erg s@xmath49 . 4 . based on reverberation mapping results , we have defined our grid of distances to cover the range of 1 to 100 light days . model calculations proceed in two stage . first we produce a two dimensional grid of @xmath36 for several values of the parameter @xmath38 . we have considered @xmath38=1 , 1.5 , and 2 , with the additional normalizations of the density at 10 light day ( @xmath8=10 ) ) in the range @xmath55 to @xmath56 @xmath10 , and column densities ( @xmath3=10 ) ) of @xmath57 to @xmath58 @xmath5 . in the second stage we calculate theoretical light curves by integrating over eq . [ el ] with the given @xmath38 , the chosen @xmath31 and @xmath32 , and several values of the parameter @xmath39 . we have examined models with @xmath39=1 , 1.5 , and 2 for each grid . in each model we vary both @xmath31 and @xmath32 to minimize @xmath59 . this score is calculated by comparing the theoretical and observed lines fluxes for _ all chosen lines_. the smallest @xmath59 determines our best parameters . to illustrate our model , we present the results of two sets of calculations . first we consider a model with @xmath38=2 and @xmath39=1.5 ( this particular choice results in @xmath60 ) . we chose @xmath3=10)=10@xmath61 @xmath5 and study the full density range . we note again that is referred to here as _ a single model _ covers in fact a large range of density and column density . for example , choosing @xmath8=10)=10@xmath62 @xmath10 means a full density range of 10@xmath63 @xmath10 and a full column density range of 10@xmath64 @xmath5 . examining this case we note some obvious features . first , for @xmath8=10)@xmath65 10@xmath66 @xmath10 the response of the modeled emission lines is reversed , i.e. , increasing continuum flux results in decreasing line flux . this is demonstrated in fig . [ la_m2 ] ( check the doted line in the diagram ) . this is a clear sign of optically thin material in the inner blr . it is caused by a combination of large incident fluxes and small densities ( i.e. large ionization parameters ) with relatively small column density . such models are in poor agreement with observations and smaller ionization parameters and/or larger columns must be considered . the problem can be cured by raising the density to @xmath8=10)@xmath6710@xmath66 @xmath10 . the resulting light curves is in much better agreement with the observations ( fig . [ la_m2 ] : dashed and solid lines ) . the reversed response in ly@xmath43 has disappeared and the theoretical line light curves nicely fit the observed ones with @xmath31=3 ld and @xmath32=25 ld assuming @xmath8=10)=10@xmath68 @xmath10 . however , applying this geometry to the other emission lines , we get unsatisfactory results . this is illustrated in fig . [ test3]a ( solid lines ) that show the improved fit to the ly@xmath43 , and civ data and the very poor agreement for ciii ] . the above example demonstrate the difficulty in fitting , simultaneously , line responses and line ratios . the example illustrates why a particular radial dependence , like the one used here ( @xmath69 ) , can not adequately fit the observed variable emission line spectrum of ngc 5548 . our simulations show that no normalization or a choice of @xmath39 can cure this problem . in the second example @xmath38=1 with the following normalization : @xmath8=10)= 10@xmath68 @xmath10 , @xmath3=10)=10@xmath70 @xmath5 , @xmath31=3 ld and @xmath32=100 ld . the light curves are presented in fig . [ test3]b . this set of models give much better fit to the data ( note in particular the ciii ] light - curve ) . comparing both models demonstrate the usefulness of this approach in constraining the parameter space . we have checked a large range of model parameters and comment , below , on several of the more obvious trends . we first note that as the column density grows from @xmath3=10)=10@xmath71 to @xmath58 @xmath5 , the agreement with the observations is much better . thus , models with @xmath3=10)=10@xmath71 @xmath5 do not fit the observations while @xmath3=10)@xmath7210@xmath12 @xmath5 already results in a reasonable agreement . photoionization codes like the one used here , are limited to column density less than about @xmath73 ( i.e. compton thin clouds ) . hence we are unable to put an upper limit on the column density . using models with @xmath38=1.5 and @xmath38=1 , we can obtain lower and upper limits on the gas density . models with @xmath8=10)@xmath6510@xmath9 @xmath10 do not fit the observation ( in the @xmath38=1.5 models there is a reverse response of the lines and in the @xmath38=1 models the line ratios do not agree with the observations ) . for @xmath8=10)@xmath7210@xmath6 @xmath10 , the line ratios for both value of @xmath38 do not agree with the observations . hence we find the blr density of ngc 5548 , in our model , to be in the range of 10@xmath6@xmath7@xmath8=10)@xmath74 @xmath10 . a general trend for all values of @xmath38 is that as @xmath39 increases from 1 to 2 ( i.e. , more weight is given to clouds closer to the central source ) the amplitude of the modeled light curves is in better agreement with the observations and so are most line ratios . hence , models with higher @xmath39 are preferred . a common problem for all models is the weak mgii@xmath02798line . while we do not have a complete explanation for this , we suspect it is due to one of two reasons : either the transfer of this line is inaccurate , similar to the case of the balmer lines , or else it is caused by the thousands of highly broadened feii lines , in that part of the spectrum , that make the measured line intensity highly uncertain ( see for example the discussion in wills , wills and netzer , 1985 ; maoz et al . 1993 ) ( a third possibility of enhanced metallicity is discussed in kaspi and netzer 1999 ) . our @xmath59 evaluation does not include the mgii line . this is a definite failure of the model . another common trend is the improvement of the @xmath75 score with increasing @xmath32 . this is most noticeable as @xmath32 increases from 50 to 100 light days . considering all the above trends and limitations , and using the @xmath59 score , we find that models with @xmath38=1 best fit the observed spectra and models with @xmath76 give somewhat inferior fits . an example of one of our best models is shown in fig . [ test3]b . in this model the reduced @xmath59 score for the four lines is 4.5 for the @xmath39=1 model , 3.1 for the @xmath39=1.5 model , and 2.2 for the @xmath39=2 model . the total covering factors found for these models are 0.25 , 0.28 . and 0.30 , respectively . our direct method allow us to investigate , in a critical way , a large variety of blr models . unlike indirect methods that are based on transfer function of individual lines , and make no use of their relative or absolute intensity , we are able to introduce many more observational constraints . the @xmath75 minimization applied to 4 emission lines , at _ all times _ , enable us to choose among various models and to rule out cases of unsuitable density , column density and covering fraction . in particular we were able to show that : 1 . there is a narrow range of density and density dependence that fit the observed light cures . using our parameters we find that @xmath38 is in the range of 11.5 and the largest density ( at one light day ) is about 10@xmath77 @xmath10 . the simulations rule out steep density laws like @xmath69 . this is in disagreement with the results of goad & koratkar ( 1998 ) despite of the fact that their range of acceptable densities ( @xmath7810@xmath79 @xmath10 in the inner blr and @xmath7810@xmath80 @xmath10 in the outer blr ) is similar to ours and their deduced lower limit on the column density is also in agreement with ours . we suspect that goad & koratkar s conclusion about the good fit of the @xmath69 case is related to their use of mean time lags rather than compared with our detailed fit to the light curves . detailed emission line variability can be used to put useful constraints on the column density of clouds across the blr . realistic models require large enough columns to avoid optically thin clouds at small distances . simple two or three zone models contain too little information and hence can not constrain , accurately enough , the physical conditions in the gas . for example dumont et al . ( 1998 ) have used a three zone model and reached several conclusions based on time - averaged properties . they noted three problems arising from their modeling : an energy budget problem , a line ratio problem , and a line variation problem . the first two are most probably related to the balmer line intensity and , as explained , we suspect this to be a general limitation of current photoionization models . regarding the line intensity , we find good agreement for both high and low ionization lines and suspect that even a three - zone model is highly simplified for the purpose of realistic reconstruction of the blr . several recent ideas about the blr can be tested against real observations by using an approach similar to ours . ` locally optimally - emitting clouds ' ( locs ) models ( baldwin et al . 1995 , korista et al . 1997 ) have been suggested to explain the broad line spectrum of agns . the models assume that there are clouds with a range of density and column density at each distance . locs must be put into a real test by checking whether they result in light curves that are in better agreement with the observations , compared with the simpler models assumed here . + alexander & netzer ( 1997 ) have suggested that the blr clouds may be bloated stars ( bss ) with extended envelopes . in their work they fitted the emission - line intensities , profiles and variability to mean observed properties of agns . one of the conclusion is that the density at the external edge of the bss ( the part emitting the lines ) falls off like @xmath81 , and the number density of bss falls off like @xmath1 . these two trends are in good agreement with our preferred values of @xmath38 and @xmath39 . while the model is consistent with mean time - lags of intermediate luminosity agns , it remains to be seen whether it can fit , in detail , the time dependent spectrum of objects like ngc 5548 . there are obvious limitations and several ways to improve our models . first , different abundances ought to be considered . second , line beaming ( anisotropy in the emission line radiation pattern ) must be considered . unfortunately , similar to the balmer line problem , the radiation pattern can not be accurately calculated in present - day escape - probability based codes . multi - component models , with a range of density and column density at each radius ( not necessarily similar to the loc distribution ) , must be tested too . finally , a variable shape sed ( romano and peterson 1998 ) is a likely possibility that may affect the outcome of such models . some of these additional factors are addressed in kaspi and netzer ( 1999 ) . others must await future work . alexander , t. , & netzer , h. 1997 , mnras , 284 , 967 baldwin , j. , ferland , g. , korista , k. , & verner , d. 1995 , apjl , 455 , l119 bottorff , m. , korista , k.t . , shlosman , i. , & blandford , d.r . 1997 , apj , 479 , 200 clavel , j. , et al . 1991 , apj , 366 , 64 dumont , a .- m . , collin - suffrin , s. , & nazarova , l. 1998 , a&a , 331 , 11 goad , m. , & koratkar , a. 1998 , apj , 495 , 718 kaspi , s. , & netzer , h. 1999 , in preparation korista , k. , baldwin , j. , ferland , g. , & verner , d. 1997 , apjs , 108 , 401 krolik , j.h . , horne . k. , kallman , t.r . , malkan , m.a . , edelson , r.a . , & kriss , g.a . 1991 , apj , 371 , 541 maoz , d. 1994 , in reverberation mapping of the broad - line region in agn , eds . p.m. gondhalekar , k. horne , & b.m . peterson ( san francisco : asp ) , 95 maoz , d. et al . 1993 , apj , 404 , 576 netzer , h. 1990 , in _ active galactic nuclei _ eds . , courvoisier and m. mayor ( berlin : springer - verlag ) netzer , h. 1996 , apj , 473 , 781 netzer , h. , & peterson , b. m. 1997 , in _ astronomical time series _ eds . , d. maoz , a. sternberg , and e. leibowitz ( dordrecht : kluwer academic publishers ) obrien , p.t . , goad , m.r . , & gondhalekar , p.m. 1994 , mnras , 268 , 845 prez , e. , robinson , a. , & de la funte , l. 1992 , mnras , 256 , 103 peterson , b.m . 1993 , pasp , 105 , 247 peterson , b.m . , et al . 1999 , apj ( in press ) . romano , p. , & peterson , b.m . 1998 , in _ structure and kinematics of quasars broad line regions _ , eds . gaskell , c.m . , brandt , w.n . , dietrich , m. , dultzin - hacyan , d. , and eracleous , m. ( san francisco : asp ) wills , b.j . , wills , d. , and netzer , h. 1985 , apj , 288 , 94

we present a new scheme for modeling the broad line region in active galactic nuclei . it involves photoionization calculations applied to a number of variable emission lines at _ all times_. we demonstrate how fitting all lines simultaneously provide strong constraints on several of the more important parameters , such as the density and column density , and the radial distribution of the emission line clouds . when applying the model to the seyfert 1 galaxy ngc 5548 , we are able to reconstruct the light curves of four emission - lines , in time and in absolute flux . we argue that the balmer line light curves , and possibly also the mgii@xmath02798 light curve , do not fit this scheme because of the limitations of present - day photoionization codes . we rule out models where the particle density scales as @xmath1 and favor models where it scales as @xmath2 . we can place lower limits on the column density at a distance of 10 ld , of @xmath3=10)@xmath4 @xmath5 , and limit the particle density to be in the range of 10@xmath6@xmath7@xmath8=10)@xmath710@xmath9 @xmath10 .

observational determinations of the history of star formation in early - type galaxies ( hereafter etgs ) are of great importance because hierarchical models of galaxy formation make firm predictions for the relation between age , metallicity and @xmath0-enhancement as a function of mass . these scaling relations , plotted against the observational proxy for mass , the velocity dispersion ( hereafter @xmath4 ) , have been the focus of many recent studies of etgs ( annibali et al . 2007 , bernardi et al . , 2006 , de la rosa et al . 2007 , gallazzi et al . 2006 , jimenez et al . 2007 , kuntschner et al . 2001/2 , lucey et al . 2007 , mateus et al . 2007 , nelan et al . 2005 , proctor et al . 2004/8 , snchez - blzquez et al . 2006 , smith et al . 2007 , terlevich & forbes 2002 , thomas et al . 2005 ) . these studies clearly show that the simple picture of early formation of low mass galaxies , which then merge to form more massive systems is incorrect . the oldest stellar populations are found in the most massive galaxies one aspect of so - called `` downsizing '' . however , the observational constraint is sensitive to the epoch at which star formation ceased , not when it started , so that low mass etgs can still be `` old '' , as dynamically bound objects , but have a _ mean _ stellar age that is much younger . however , this is not sufficient to save the simple hierarchical picture because the @xmath0-enhancement is also seen to increase with mass - implying more rapid formation for massive objects . this is a prediction of monolithic collapse models . it is important to remember that both the star formation history and mass assembly history of etgs determine their evolution . the most common method of determining the age , metallicity and @xmath0-enhancement of etgs is by comparison of the narrow band absorption line indices with simple stellar population ( ssp ) models . it is well - known that stellar population parameters based on these indices are also sensitive to minor episodes of recent star formation . luminosity - weighted , ssp equivalent stellar population parameters , such as those discussed here , do not therefore , distinguish between a genuinely young galaxy and an old galaxy that has experienced a `` rejuvenation '' event . some recent studies , based on the colour - magnitude relation of etgs using colours that are extremely sensitive to recent star formation , in fact paint a surprising picture of etg evolution . schawinski et al . ( 2007 ) have used galex ultraviolet imaging to show that 30% of massive etgs show _ ongoing _ star formation and that this fraction is higher in low - density environments . a similar picture is given by mid - infrared spitzer data . both clemens et al . ( 2008 ) and bressan et al . ( 2006 ) find that @xmath5 of etgs in the coma and virgo clusters have experienced some star formation in the recent past . a recent study by rogers et al . ( 2007 ) has combined sdss spectra and galex data to conclude that `` weak episodes of recent star formation '' are a phenomenon more commonly associated with etgs in the _ cluster _ environment , a result seemingly , but not necessarily , inconsistent with several studies that find older ssp - equivalent ages in denser environments . here we repeat the analysis carried out in clemens et al . ( 2006 , hereafter paper i ) , which used 3614 objects selected from data release 3 ( dr3 ) of the sdss . applying the same selection criteria to dr6 we define a sample of 14353 etgs , four times as many objects . sample selection is identical to that described in paper i. the local environmental density is defined as the inverse of the distance to the fifth nearest neighbour , @xmath6 , corrected for the redshift dependent effect of survey boundaries . we measure the 21 line - strength indices of the original lick - ids system plus the additional indices h@xmath7f , h@xmath8f , b4000 and hk . however , before measuring the narrow - band indices from each of the sdss , spectra we smooth the spectra to the wavelength dependent resolution of the lick - ids spectra . this step is essential as the models that we use to derive the age , metallicity and @xmath0enhancement are based on the lick system . the index values are then corrected for the smoothing effects of the galaxy s velocity dispersion and aperture corrected to a standard normalized radius ( a fraction of the half - light radius ) . see paper i for a more detailed description . besides being based on dr6 , the present work differs from that of paper i only in 2 respects . firstly , in paper i we chose to correct for the fixed angular diameter of @xmath9 sampled by the sdss fibre using the radial index measurements of 50 nearby e and s0 galaxies ( rampazzo et al . 2005 ) . here , we make use of the r - band effective radii provided in the sdss catalogue to derive the aperture correction directly . statistically , a more reliable correction should be obtained in this way . however , this also means that we must choose a larger standard radius to which to correct ( @xmath10/10 was used in paper i ) . typically , the @xmath9 diameter sdss fibre samples @xmath11 , with very few objects being so large that the fibre samples @xmath12 . to avoid extrapolating beyond measured radii we therefore aperture correct to a standard radius of either @xmath13 or @xmath14 . a plot of index value versus the fraction of @xmath10 sampled by the fibre will show a gradient . however , the gradient is due not only to the radial gradients within each galaxy but also to the correlation between @xmath10 and @xmath4 . because index values are correlated with @xmath4 this effect increases the magnitude of the gradient . to determine the aperture effect we therefore consider the variation of index value as a function of the fraction of @xmath10 sampled by the fibre in restricted bins of @xmath4 . in this way we minimize the effect of the index-@xmath4 relations and determine the radial index gradients as a function of @xmath4 . we plot index values as a function of @xmath15 in 5 bins , for 8 separate bins of @xmath4 . the bins are chosen to maintain a large number of objects in each bin and gradients are then derived by weighting each point by @xmath16 where n is the number in the bin . the values of the radial index gradients and their variation with @xmath4 are shown in table [ tab : radgrad ] . we use the radial index gradients to correct our index values for aperture effects . firstly , we use the measured value of @xmath4 for a given galaxy to determine the value of the radial index gradient @xmath17 : @xmath18 where @xmath19 is the velocity dispersion in units of @xmath20 , and @xmath21 is the value of the index gradient for @xmath22 . values for @xmath23 and @xmath21 are given in table [ tab : radgrad ] . this value of the radial index gradient is then used with the measured value of @xmath10 of the galaxy to correct the index value to the equivalent radius , @xmath24 or @xmath25 : @xmath26 where @xmath27 is the corrected index , @xmath28 is the measured index value , @xmath10 is the effective radius of the galaxy in arcsec and @xmath29 is the standard radius expressed as a fraction of @xmath10 ( 0.5 or 0.25 here ) . the use the the effective radii provided in sdss ( the petrosian half - light radius ) to effect the aperture correction has one caveat . that is , that these radii are not seeing corrected . therefore for galaxies with a small angular diameter @xmath10 is over - estimated . median r - band seeing for sdss imaging is @xmath30 ( adelman - mccarthy et al . 2007 , fig . 4 ) and the median effective diameter for our sample is @xmath31 . we believe , however , that this is not a serious problem because the seeing also alters the light entering the spectroscopic fibre . the considerable agreement we find in values for the indices as a function of @xmath4 with other studies ( see below ) reinforces this view . the second difference from paper 1 is the removal of one index , g4300 , from the fitting procedure used to derive the age , metallicity and @xmath0enhancement as a function of @xmath4 and environment . this was done because the index seems less well modeled than previously thought . here we will use the index values to derive various evolutionary parameters as a function of @xmath4 , environment and galaxy radius . before that , however , we briefly evaluate various trends seen in the fully @xmath4 and aperture corrected index values . in table [ tab : grad ] we show the corrected index values as a function of @xmath4 . because the values refer to indices aperture corrected to @xmath13 , they are not directly comparable to those of paper i , where corrections were made to @xmath12 . nonetheless , most indices show similar behaviour to those of paper i. we briefly note here some of the larger differences seen in important indices ( we refer to gradients expressed as @xmath32 as @xmath33 and those as @xmath34 as @xmath35 ) . * c4668 * : the gradient of @xmath36 much larger than that of paper i ( @xmath37 ) . this value is closer to the value of @xmath38 found by nelan et al . ( 2005 , hereafter n05 ) . * h@xmath39 * : the gradient of @xmath40 ( @xmath41 ) is shallower ( @xmath42 in paper i ) . this value is in excellent agreement with bernardi et al . ( 2003 , hereafter b03 ) ( @xmath41 ) and n05 ( @xmath43 ) . * fe5015 * : the gradient of @xmath44 in paper i contrasts to the present value of @xmath45 . this is much more consistent with n05 who find @xmath46 . * mg2 * : the value of @xmath47 is similar to that of paper i , but is now more consistent with both b03 and kuntschner et al . * mgb * : the gradient of @xmath48 ( @xmath49 ) compares with 3.7 in paper i. this is closer to that of b03 ( @xmath50 ) and n05 ( @xmath51 ) . * fe5270 * : in paper i a null gradient was found . the present value of @xmath52 is consistent with n05 who find an identical value . we note , that globally , the new index gradients are much closer to those derived by n05 despite the fact that these authors aperture corrected index values to a fixed physical radius , rather than to a fixed fraction of @xmath10 as done here . the spatial gradients we measure here describe the _ mean index value in apertures of varying radii _ , at fixed @xmath4 . this is in contrast with ` true ' spatial index gradients , which are measured in increasing annuli projected on the galaxy . as a result , the values we measure are smaller in magnitude than the true gradients . our values , which we give in table [ tab : radgrad ] , are , however , directly applicable to aperture correction . all the narrow line indices , with the exception of hk , show a radial gradient . the gradients are negative except for the hydrogen line indices and the @xmath53 break , b4000 . additionally , some indices show a well defined trend of index gradient with @xmath4 ( these can be quickly identified in the last column of table [ tab : radgrad ] ) . in all cases ( except cn1 and cn2 ) the sense of this variation is that the index gradient becomes less steep with increasing @xmath4 . in some cases , including @xmath54 , a significant index gradient at low values of @xmath4 disappears completely for @xmath55 . we find no dependence on the radial index gradients with density of environment at fixed @xmath4 . the fact that some indices show gradients which decrease with increasing @xmath4 does not necessarily imply that some process has acted to mix the stellar populations in more massive systems . we return to this below . .radial index gradients as a function of velocity dispersion , @xmath4 . the radial index gradients are expressed as , @xmath56 , where @xmath28 is the value of the index . the third column gives values for the dependence of the radial index gradients on @xmath4 in units of @xmath20 , @xmath19 . @xmath21 is the value of the index gradient at @xmath57 . these parameters have been used in the aperture correction of all indices . most indices show a well - defined radial gradient , but rather few show a convincing trend of this gradient on @xmath4 . the last column shows the ratio of the gradient in the third column and its error and so is an estimate of the statistical significance of the gradient as a function of @xmath4 . for those indices whose radial gradients show little trend with @xmath4 , the value of @xmath21 is a good measure of the radial gradient for galaxies of any @xmath4 . [ cols="<,>,>,^",options="header " , ] we now make use of the index values to derive the age , metallicity and @xmath0- enhancement of the galaxy population in our sample . we repeat the multiple linear regression procedure described in paper i ( to which the reader is referred for a detailed description ) . we briefly summarize the procedure here . firstly , because our index values are not calibrated to the lick system ( due to the lack of lick standard star spectra in sdss ) we consider index variations relative to the mean value at a @xmath4 of @xmath57 . by working with these differential index values we avoid both the problem of absolute calibration to the lick system and potential problems in the absolute calibration of the ssp models . we therefore perform a multiple linear regression according to equation 5 of paper i. the linear regression is performed on the whole sample and on 2 subsets of environmental density , @xmath58 ( typical of the field ) and @xmath59 ( more typical of a cluster ) . we also perform the analysis on indices aperture corrected to 2 different radii , @xmath13 and @xmath14 , to investigate radial trends within the individual galaxies . the results of this regression analysis showed that the carbon abundance did not depend on @xmath4 , having a constant offset as a function of environment , in contrast to paper i. this difference is probably due to the better aperture correction used here and/or the exclusion of the g4300 index . we therefore remove the explicit carbon abundance from the regression analysis , allowing the carbon abundance to be included in the metallicity term . the _ simultaneous a posteriori fit _ of the model to three example indices is compared with the median of the data in the different bins of @xmath4 , in fig . [ fig : model_fits ] . in the left panel of figure [ fig : results ] we show the results for the entire sample for two different radii . for @xmath14 the trend of age with @xmath4 is very similar to that seen in paper i with a steady rise in age from the lowest mass systems and an approximately constant age for galaxies with @xmath60 . for the larger aperture , however , the trend is slightly different , with a less pronounced flattening towards high values of @xmath4 . for @xmath61 the mean age is @xmath62 dex older for @xmath13 compared to @xmath14 . this corresponds to an age difference of @xmath63 for a galaxy of age @xmath64 . the cross - over point of the 2 lines in the top panel of fig . [ fig : results ] shows that galaxies with @xmath65 have positive radial age gradients . neither snchez - blzquez et al . ( 2007 ) nor mehlert et al . ( 2003 ) find evidence of radial age gradients in etgs . we derive an age gradient in massive galaxies _ despite the absence of a gradient of the h@xmath39 index_. the increase of metallicity with @xmath4 is less strong than that found in paper i. the gradient for the indices , corrected to @xmath14 , for @xmath66 is @xmath67 . this value is similar to that found by nelan et al . ( 2005 ) , thomas et al . ( 2005 ) and smith et al . ( 2007 ) but smaller than that of kuntschner et al . ( 2001 ) and graves et al . ( 2007 ) . for @xmath68 , however , there is no significant trend of index value with @xmath4 . for the @xmath13 aperture , the metallicity is @xmath62 dex lower so that early - type galaxies are less metal rich at larger radii . negative metallicity gradients have also been reported by proctor et al . ( 2008 ) , annibali et al . ( 2007 ) , snchez - blzquez et al . ( 2007 ) and harris & harris ( 2002 ) . there is also evidence that the metallicity gradients are steeper for more massive galaxies as found by forbes et al . the metallicity gradient likely compensates the age gradient to remove radial gradients in indices like h@xmath39 . the trend of @xmath0-enhancement with @xmath4 is also slightly less steep than found in paper i , with @xmath69 / d\,\log(\sigma)\simeq 0.55 $ ] , similar to that found by annibali et al . ( 2007 ) but steeper than the @xmath70 found by several authors ( thomas et al . , 2005 , kuntschner et al . , 2001 , n05 , bernardi et al . , 2006 , smith et al . , 2007 ) . although most similar studies refer to a smaller radius , we see that the larger aperture , @xmath13 , has a marginally shallower gradient . the @xmath0-enhancement within this larger aperture is slightly lower , with the difference being largest ( @xmath71 dex ) for the most massive galaxies . this does not support the ` outside - in ' etg formation scenario ( pipino , matteucci , & chiappini , 2006 ) . a negative @xmath0-enhancement gradient is also seen in the halo stars of the galaxy ( fulbright , 2000 ) . in the right panel of figure [ fig : results ] we show the variation of evolutionary parameters as a function of the density of environment for index values aperture corrected to @xmath14 . in both environments the increase in age with @xmath4 is similar , with a flattening above @xmath72 . objects in dense environments ( @xmath59 ) , however , are @xmath73 dex older than those in less dense environments ( @xmath58 ) . this is a difference of 2 gyr if the ages are close to 10 gyr and is consistent with several earlier studies ( terlevich & forbes 2002 , kuntschner et al . 2002 , de la rosa et al . 2007 , snchez - blzquez et al . 2006 ) . the flattening of the age-@xmath4 relation is also slightly more pronounced for the lower density environment . similar age trends were seen in paper i. there is marginal evidence that the metallicity is lower in high density environments . formally the difference is @xmath74 dex . thomas et al . ( 2005 ) also found a small environmental dependence on the metallicity in the same sense and de la rosa ( 2007 ) finds a difference of @xmath75 dex between hickson compact groups and the field . other authors have found both larger differences in the same sense ( proctor et al . 2004 , kuntschner et al . 2002 ) , no effect ( bernardi et al . , 2006 , annibali et al . 2007 ) and the opposite effect ( gallazzi et al . 2006 , mateus et al . 2007 ) . in paper i no difference was found . no environmental effect is found for the @xmath0-enhancement , in agreement with kuntschner et al . ( 2002 ) , thomas et al . ( 2005 ) , annibali et al . ( 2007 ) and gallazzi et al . . however , proctor et al . ( 2004 ) , bernardi et al . ( 2006 ) and lucey et al . ( 2007 ) all find increased @xmath0-enhancement in denser environments . we find positive correlations between age , metallicity and @xmath0-enhancement and the velocity dispersion , @xmath4 , in etgs . galaxies in dense environments are @xmath76 older than those in low density environments for all @xmath4 ( @xmath1 for an age of @xmath64 ) . the trend with age flattens above @xmath72 , especially for galaxies in low density environments . we find a marginally significant trend towards higher metallicities in low density environments but the environment has no effect on the @xmath0-enhancement . apart from the marginal metallicity difference between field and cluster the results are very similar to those of paper i. there we concluded that an anti - hierarchical scenario , in which star formation lasts longer but with lower efficiency in lower mass objects ( see granato et al . 2004 ) was consistent with the data . here the additional determination of ssp parameters as a function of galactic radius places additional constraints on the evolutionary scenario . massive etgs ( @xmath65 ) have positive radial age gradients , negative metallicity gradients and marginally significant negative @xmath0-enhancement gradients . when a massive halo becomes non - linear it accretes smaller halos which started to collapse at earlier times . the radial trends suggest that these halos do not contain only gas , but also pristine stars . the gaseous component falls dissipatively into the potential well of the massive ( proto-)spheroid , fueling rapid star formation . the increase in mass increases the rate and efficiency of star formation , driving the main correlations with galaxy mass . the pristine stellar component of each sub - halo , however , being dissipationless , is deposited at a radius consistent with the angular momentum of the encounter . these stars , which are slightly older , more metal poor and have moderate @xmath0-enhancement will therefore be spread over larger radii . at early times , rapid gas - rich mergers lead to an almost monolithic formation , at later times mergers become increasingly `` dry '' . very similar scenarios have been proposed to explain both the bi- modal metallicity distribution of globular clusters and the greater radial scale length of metal - poor relative to metal - rich globular clusters in elliptical galaxies ( ct , marzke & west , 1998 , bekki et al . our results imply that the metal - poor globular cluster population in etgs should be older , less metal - rich and slightly less @xmath0-enhanced than the metal - rich clusters . because the age difference at larger radii is actually the luminosity weighted ssp equivalent age in a larger aperture ( not an annulus ) , the value of 0.05 dex , @xmath63 , is a lower limit to the real age difference at larger radii . this time difference limits the assembly redshift of massive etgs simply due to the lack of time to accommodate the formation of stars in the lower mass halos . our limit translates into an upper limit to the assembly redshift of massive etgs of @xmath3 ; in which case the stars in low mass halos formed at @xmath77 , for a standard cosmology ( @xmath78 , @xmath79 , @xmath80 ) . this also provides an estimate of the star formation rate in the assembled spheroid . if the final stellar mass were @xmath81 then stars must have formed at a rate @xmath82 . we conclude by stressing the statistical nature of our results . because a galaxy s velocity dispersion is a function of both the halo mass and virialization redshift , variations in these parameters may render small samples insensitive to the trends we find . the catalogue on which this article is based can be found at , www.mrao.cam.ac.uk/@xmath83bn204/galevol/clemensetal08.html . 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 web site is http://www.sdss.org/. we acknowledge a financial contribution from contract asi - inaf i/016/07/0 . adelman - mccarthy j. k. , et al . , 2007 , apjs , 172 , 634 annibali f. , bressan a. , rampazzo r. , zeilinger w. w. , danese l. , 2007 , a&a , 463 , 455 bekki k. , yahagi h. , nagashima m. , forbes d. a. , 2008 , mnras , 387 , 1131 bernardi m. , nichol r. c. , sheth r. k. , miller c. j. , brinkmann j. , 2006 , aj , 131 , 1288 bernardi m. , et al . , ( b03 ) 2003 , aj , 125 , 1882 bressan , a. et al . , 2006 , apj , 639 , l55 clemens m. s. , bressan a. , nikolic b. , alexander p. , annibali f. , rampazzo r.,(paper i ) 2006 , mnras , 370 , 702 clemens m. s. , bressan a. , panuzzo p. , rampazzo , r. , silva l. , buson l. , granato g. l. , arxiv:0808.2899 ct p. , marzke r. o. , west m. j. , 1998 , apj , 501 , 554 de la rosa i. g. , de carvalho r. r. , vazdekis a. , barbuy b. , 2007 , aj , 133 , 330 forbes d. a. , snchez - blzquez p. , proctor r. , 2005 , mnras , 361 , l6 fulbright j. p. , 2000 , aj , 120 , 1841 gallazzi a. , charlot s. , brinchmann j. , white s. d. m. , 2006 , mnras , 370 , 1106 granato g. l. , de zotti g. , silva l. , bressan a. , danese l. , 2004 , apj , 600 , 580 harris w. e. , harris g. l. h. , 2002 , aj , 123 , 3108 jimenez r. , bernardi m. , haiman z. , panter b. , heavens a. f. , 2007 , apj , 669 , 947 kuntschner h. , smith r. j. , colless m. , davies r. l. , kaldare r. , vazdekis a. , 2002 , mnras , 337 , 172 kuntschner h. , lucey j. r. , smith r. j. , hudson m. j. , davies r. l. , 2001 , mnras , 323 , 615 lucey j. r. , smith r. j. , hudson m. j. , nelan j. e. , wegner g. a. , 2007 , aspc , 379 , 117 mateus a. , sodr l. , cid fernandes r. , stasiska g. , 2007 , mnras , 374 , 1457 mehlert d. , thomas d. , saglia r. p. , bender r. , wegner g. , 2003 , a&a , 407 , 423 nelan j. e. , smith r. j. , hudson m. j. , wegner g. a. , lucey j. r. , moore s. a. w. , quinney s. j. , suntzeff n. b. , ( n05 ) 2005 , apj , 632 , 137 pipino a. , matteucci f. , chiappini c. , 2006 , apj , 638 , 739 proctor r. n. , lah p. , forbes d. a. , colless m. , couch w. , 2008 , mnras , 386 , 1781 proctor r. n. , forbes d. a. , hau g. k. t. , beasley m. a. , de silva g. m. , contreras r. , terlevich a. i. , 2004 , mnras , 349 , 1381 rampazzo , r. , annibali , f. , bressan , a. , longhetti , m. , padoan , f. , zeilinger , w. w. , 2005 , a&a , 433 , 479 rogers b. , ferreras i. , lahav o. , bernardi m. , kaviraj s. , yi s. k. , 2007 , mnras , 382 , 750 snchez - blzquez p. , gorgas j. , cardiel n. , gonzlez j. j. , 2006 , a&a , 457 , 809 snchez - blzquez p. , forbes d. a. , strader j. , brodie j. , proctor r. , 2007 , mnras , 377 , 759 schawinski k. , et al . , 2007 , apjs , 173 , 512 smith r. j. , lucey j. r. , hudson m. j. , 2007 , mnras , 381 , 1035 terlevich a. i. , forbes d. a. , 2002 , mnras , 330 , 547 thomas d. , maraston c. , bender r. , mendes de oliveira c. , 2005 , apj , 621 , 673

we define a volume limited sample of over 14,000 early - type galaxies ( etgs ) selected from data release six of the sloan digital sky survey . the density of environment of each galaxy is robustly measured . by comparing narrow band spectral line indices with recent models of simple stellar populations ( ssps ) we investigate trends in the star formation history as a function of galaxy mass ( velocity dispersion ) , density of environment and galactic radius . we find that age , metallicity and @xmath0-enhancement all increase with galaxy mass and that field etgs are younger than their cluster counterparts by @xmath1 . we find negative radial metallicity gradients for all masses and environments , and positive radial age gradients for etgs with velocity dispersion over @xmath2 . our results are qualitatively consistent with a relatively simple picture for etg evolution in which the low - mass halos accreted by a proto - etg contained not only gas but also a stellar population . this fossil population is preferentially found at large radii in massive etgs because the stellar accretions were dissipationless . we estimate that the typical , massive etg should have been assembled at @xmath3 . the process is similar in the cluster and the field but occurred earlier in dense environments . [ firstpage ] galaxies : elliptical and lenticular , cd , galaxies : formation , galaxies : abundances

let @xmath3 be a closed manifold and @xmath4 be a diffeomorphism on @xmath3 . a periodic point @xmath5 is said to be _ hyperbolic _ , if the linearization @xmath6 does nt admit any eigenvalue of norm 1 . associated to a hyperbolic periodic orbit are the _ stable _ and _ unstable manifolds _ @xmath7 of @xmath8 . a point in the intersection @xmath9 for another hyperbolic periodic point @xmath10 is called a _ heteroclinic intersection _ ( a _ homoclinic intersection _ if @xmath11 ) . note that the intersection @xmath9 may not be _ transverse _ ( even when @xmath8 and @xmath10 have the same stable dimension ) . poincar was the first one to consider the phase portrait when there exists a transverse homoclinic intersections during his study of the @xmath12-body problem around 1890 . later in @xcite , poincar described the phenomenon that any transverse homoclinic intersection is accumulated by many other homoclinic points and this mechanism generates various complicated dynamical behaviors . this mechanism was developed by birkhoff , who showed in @xcite the persistent existence of _ infinitely many _ hyperbolic periodic points whenever there is a transverse homoclinic intersection , and by smale , who introduced in @xcite the geometric model , now called _ smale horseshoe _ , for the dynamics around a transverse homoclinic intersection , and started a systematic study of general hyperbolic sets . melnikov developed in @xcite a method for detecting homoclinic intersections in dynamical systems ( this has also been used by poincar ) . poincar conjectured in @xcite that for a generic @xmath13 , and for every hyperbolic periodic point @xmath8 of @xmath14 , 1 . the set of ( hyperbolic ) periodic points is _ dense _ in the space @xmath3 ; 2 . @xmath15 ( weak version ) ; 3 . @xmath16 is _ dense _ in @xmath17 ( strong version ) . these conjectures are closely related to the _ closing lemma _ and _ connecting lemma _ , see @xcite for a historic account of these terminologies . poincar s conjectures have been one of the main motivations for the recent development in dynamical systems . all three parts have been proved for @xmath18 : ( p1 ) follows from pugh s closing lemma @xcite , ( p2a ) was proved by takens in @xcite , ( p2b ) was proved by xia @xcite . there are some special classes of maps that ( p1)(p2b ) hold everywhere ( not only generically ) : anosov s _ uniformly hyperbolic _ systems @xcite , and pesin s _ nonuniformly hyperbolic _ systems @xcite . there are some partial results for ( p1 ) and ( p2a ) for systems beyond hyperbolicity when @xmath19 ( mainly in 2d ) . robinson proved in @xcite that on two - sphere , if the unstable manifold of a hyperbolic fixed point accumulates on its stable manifold , then a @xmath0 small perturbation can create a homoclinic intersection . pixton @xcite extended robinson s result to periodic orbits , and proved that ( p2a ) holds on @xmath1 . that is , for a @xmath0 generic area - preserving diffeomorphism on @xmath1 , there exist some homoclinic orbits for every hyperbolic periodic point . using some topological argument in @xcite , oliveira showed the generic existence of homoclinic orbits on @xmath20 . his result was extended in @xcite to any compact surface ( among those whose induced actions @xmath21 on the first homology group @xmath22 are irreducible ) . then xia proved in @xcite the generic existence of homoclinic orbits on general compact surface among the class of systems homotopic to identity , and among the class of hamiltonian diffeomorphisms . recently in @xcite , we proved the existence of homoclinic intersections for every hyperbolic periodic point for generic convex billiards on @xmath23 and @xmath1 , respectively . in this paper we study the homoclinic intersections of some symplectic partially hyperbolic systems . as an illustration of our main result , let s start with a special case . let @xmath24 be a closed symplectic manifold , @xmath25 be a symplectic anosov diffeomorphism . let @xmath26 be a closed surface with an area - form @xmath27 on @xmath26 , and @xmath28 such that @xmath29 is partially hyperbolic with center bundle @xmath30 . replacing @xmath14 by @xmath31 for large enough @xmath12 if necessary , we may assume @xmath29 is @xmath32-normally partially hyperbolic . let @xmath33 and @xmath34 . then @xmath35 , and there exists a @xmath36 open neighborhood @xmath37 of @xmath29 such that each @xmath38 is @xmath32-normally partially hyperbolic with stably integrable center bundle . moreover , the foliation @xmath39 is leaf conjugate to the trivial foliation @xmath40 . therefore , each center leaf @xmath41 is diffeomorphic to the surface @xmath26 . this class of maps in @xmath42 have been studied in @xcite , where they proved ( p1 ) . that is , @xmath0-generically in @xmath42 , the set of ( hyperbolic ) periodic points are dense in @xmath3 . in this paper , we show [ thm : prod ] let @xmath26 be diffeomorphic to either the 2-sphere @xmath1 or 2-torus @xmath2 . then there is a small neighborhood @xmath37 of @xmath29 , such that for a @xmath0-generic @xmath38 , @xmath43 for each hyperbolic periodic point @xmath8 of @xmath44 . more generally , let s consider a skew product system . that is , let @xmath25 be a symplectic anosov diffeomorphism , and @xmath45 be a @xmath0 smooth cocycle over @xmath3 . let @xmath46 , @xmath47 be the induced skew product of @xmath48 over @xmath14 . then the subbundle @xmath49 with @xmath30 is @xmath50-invariant . replacing @xmath14 by @xmath31 for large enough @xmath12 if necessary , we may assume @xmath51 is @xmath32-normally partially hyperbolic . similarly , we show that [ thm : skew ] let @xmath26 be diffeomorphic to either @xmath1 or @xmath2 . let @xmath14 and @xmath48 be given as above , and @xmath51 be the skew product of @xmath48 over @xmath14 . then there is a small neighborhood @xmath37 of @xmath51 , such that for a @xmath0-generic @xmath38 , @xmath43 for each hyperbolic periodic point @xmath8 of @xmath44 . now let s state our main result . let @xmath24 be a closed symplectic manifold , @xmath52 be the set of partially hyperbolic symplectic diffeomorphisms on @xmath3 , and @xmath53 be the set of partially hyperbolic maps such that the partially hyperbolic splitting is _ @xmath32-normally hyperbolic _ and the center bundle is _ stably integrable_. let @xmath54 be those map in @xmath52 with @xmath55-dimensional center , and @xmath56 be the set of partially hyperbolic maps in @xmath57 with 2d center bundles . [ main ] there is a residual subset @xmath58 such that for each @xmath59 , for each hyperbolic periodic point @xmath8 of @xmath14 , if @xmath60 is homeomorphic to either @xmath1 or @xmath2 , then @xmath43 . note that the center foliation @xmath61 may not be ( uniformly ) compact , even there are some compact leaves . an intuitive example is the orbit foliation of a anosov flow : a leaf is compact if and only if it is a periodic orbit . it is clear that theorem [ thm : prod ] and [ thm : skew ] follow directly from the above theorem . now let s sketch the proof of theorem [ main ] . we start with a result on kupka smale properties of generic symplectic systems proved by robinson , then make a local perturbation on the center leaf around each nonhyperbolic periodic point such that the restricted dynamics in the corresponding center leaf is moser stable . next we use the generating function to lift the local center - leaf perturbation to a local perturbation of the ambient manifold . then we apply the prime - end theory developed by mather to deduce the recurrence of all stable and unstable manifolds restricted on the center leaf . simple topology assumption , that is , the center leaf is homeomorphic to either @xmath1 or @xmath2 , ensures the existence of homoclinic intersections for all hyperbolic periodic points . let @xmath3 be a closed manifold endowed with some riemannian metric , @xmath62 be the set of @xmath0 diffeomorphisms on @xmath3 . let @xmath63 be a splitting of @xmath64 into two @xmath65-invariant subbundles . then we say that @xmath66 is _ dominated _ by @xmath51 , if there exists @xmath67 such that for any @xmath68 , * @xmath69 for any unit vectors @xmath70 and @xmath71 . note that both @xmath66 and @xmath51 are continuous subbundles of @xmath64 . then @xmath14 is said to be _ partially hyperbolic _ , if there exist a three - way splitting @xmath72 , such that 1 . @xmath73 is dominated by @xmath74 , and @xmath75 is dominated by @xmath76 ; 2 . there exists @xmath77 such that @xmath78 and @xmath79 . in particular , @xmath14 is said to be _ anosov _ ( or equivalently , uniformly hyperbolic ) , if @xmath80 . the above definition of partially hyperbolic maps is elegant . in the following we give an equivalent , but more easy - to - use definition for later convenience . recall that a function @xmath81 induces a _ multiplicative cocycle _ @xmath82 on @xmath3 , where @xmath83 , and @xmath84 for each @xmath68 and for all @xmath67 . [ de : ph ] the map @xmath14 is said to be _ partially hyperbolic _ , if there exist a three - way splitting @xmath72 , a constant @xmath85 , four continuous functions @xmath86 , @xmath87 , @xmath88 and @xmath89 with @xmath90 and @xmath91 , such that for any @xmath68 , and for any unit vector @xmath92 , @xmath93 generally speaking , the partially hyperbolic splitting of @xmath14 may not be unique . note that the stable bundle @xmath73 is uniquely integrable . let @xmath94 be the stable foliation of @xmath14 , whose leaves @xmath95 are @xmath0 immersed submanifolds . so is the unstable one , and denote the unstable foliation by @xmath96 . however , the center bundle @xmath97 may _ not _ be integrable , and when integrable , the center leaves may not be @xmath0 . a partially hyperbolic map @xmath14 is said to be _ dynamically coherent _ , if there exists an @xmath14-invariant foliation @xmath61 such that @xmath98 for every @xmath68 . then @xmath14 is said to be _ stably dynamically coherent _ , if there is an open neighborhood @xmath99 , such that every @xmath100 is dynamically coherent . note that there are several versions of definitions of dynamical coherence in the literature . see @xcite for more details . let @xmath101 such that the center foliation @xmath102 is @xmath36 . then @xmath14 is stably dynamically coherent . as a direct corollary , a product system @xmath103 is stably dynamically coherent . plaque expansiveness , a condition weaker than @xmath102 being @xmath36 , is introduced in @xcite . they showed that if @xmath102 is plaque expansive , then @xmath14 is stably dynamically coherent . let @xmath104 , and @xmath105 and @xmath106 be the functions given in definition [ de : ph ] . then @xmath14 is said to be @xmath32-normally hyperbolic , if @xmath107 and @xmath108 . it follows from the definition that every partially hyperbolic diffeomorphism is @xmath32-normally hyperbolic , for some @xmath109 . let @xmath104 such that @xmath97 is integrable . if @xmath14 is @xmath32-normally hyperbolic , then all center leaves of @xmath102 are @xmath0 . a @xmath110-dimensional manifold @xmath3 is said to be _ , if there exists a closed nondegenerate 2-form @xmath111 on @xmath3 . let @xmath112 be a continuous subbundle such that @xmath113 for any @xmath68 . in this case we also denote it by @xmath114 . the _ symplectic orthogonal complement _ of @xmath66 , denoted by @xmath115 , is given by @xmath116 . clearly @xmath117 . a subbundle @xmath66 is said to be _ isotropic _ , if @xmath118 ; is said to be _ coisotropic _ , if @xmath119 ; is said to be _ symplectic _ , if @xmath120 ; and is said to be _ lagrangian _ , if @xmath121 . let @xmath122 be the set of symplectic diffeomorphisms @xmath4 , that is , @xmath123 . similarly , let @xmath124 be the set of symplectic partially hyperbolic diffeomorphisms on @xmath3 . note that for a given map @xmath125 , the partially hyperbolic splitting of @xmath14 may not be unique . however , the center bundle can always be chosen to be a symplectic subbundle of @xmath64 . let @xmath126 , and @xmath63 be a @xmath65-invariant splitting of @xmath14 with @xmath127 such that @xmath66 is dominated @xmath51 . then @xmath14 is partially hyperbolic , where @xmath128 , @xmath129 and @xmath130 . moreover , @xmath73 and @xmath76 are isotropic , @xmath131 and @xmath97 are symplectic and are skew - orthogonal to each other . in the following the partially hyperbolic splitting @xmath72 for @xmath104 will be fixed such that the center bundle @xmath97 and the combined bundle @xmath131 are symplectic . in particular , we have let @xmath132 , and @xmath102 is an @xmath14-invariant foliation tangent to the center bundle @xmath97 of @xmath14 . then the center leaves @xmath133 are symplectic ( possibly immersed ) submanifolds with respect to the restricted symplectic form @xmath134 . moreover , the restrictions of @xmath14 from @xmath135 are symplectic diffeomorphisms . moreover , it is proved in @xcite that symplectic partially hyperbolic maps are symmetric . that is , one can take @xmath136 and @xmath137 in definition [ de : ph ] . [ pr : de ] if @xmath104 , then there exist a constant @xmath85 , two continuous functions @xmath88 , @xmath138 such that for any @xmath68 , and for any unit vector @xmath92 , @xmath139 the normal hyperbolicity condition defined in [ sec : nor ] for general partially hyperbolic maps admits a simpler form in the symplectic case . that is , a map @xmath104 is said to be @xmath32-normally hyperbolic , if the two functions @xmath86 and @xmath88 in proposition [ pr : de ] satisfy @xmath107 . let @xmath26 be a 2d surface , and @xmath140 be a @xmath0 symplectic map fixing a point @xmath141 . let @xmath142 and @xmath143 be the two eigenvalues of @xmath144 . then @xmath8 is said to be _ hyperbolic _ if @xmath145 , be _ parabolic _ if @xmath146 , and be _ elliptic _ if otherwise . [ mos1 ] suppose @xmath8 is an elliptic fixed point of @xmath147 such that @xmath148 for each @xmath149 . then there exists a real - analytic symplectic diffeomorphism @xmath44 , defined on a neighborhood of @xmath150 in @xmath151 with @xmath152 , such that in the complex coordinate @xmath153 , one has : @xmath154 where @xmath155 depends continuously on @xmath147 . for a proof of above theorem , see @xcite . the formulation on the right side of is called the _ birkhoff normal form _ of @xmath147 at @xmath8 , and @xmath156 is called the ( first ) _ birkhoff coefficient _ of @xmath147 at @xmath8 . an elliptic fixed point @xmath8 of a surface map @xmath140 is said to be _ moser stable _ , if there is a fundamental system @xmath157 of nesting neighborhoods in @xmath26 around @xmath8 , where each @xmath158 is an invariant closed disk surrounding the point @xmath8 , such that the restriction of @xmath147 on @xmath159 is transitive ( minimal ) . note that moser stable periodic points are isolated from the dynamics in the sense that it can not be reached from any invariant ray whose starting point lies outside some @xmath158 . the following is moser s _ twisting mapping theorem _ ( see @xcite ) : [ moser ] let @xmath8 be an elliptic fixed point of @xmath147 and @xmath156 be the birkhoff coefficient of @xmath147 at @xmath8 . if @xmath160 , then @xmath8 is moser stable . let @xmath161 , @xmath67 and @xmath162 be the set of points fixed by @xmath31 . clearly @xmath162 is a closed set . let @xmath8 be a periodic point of @xmath14 of period @xmath163 . then @xmath8 is said to be _ hyperbolic _ if @xmath164 for any eigenvalue of the linearization @xmath165 of @xmath14 at @xmath8 . given a hyperbolic periodic point @xmath8 of @xmath14 , let @xmath166 and @xmath167 be the stable and unstable manifolds of @xmath8 . more generally , a periodic point @xmath8 of minimal period @xmath12 is said to be _ @xmath168-elementary _ , if @xmath169 for each @xmath170 and for each eigenvalue @xmath171 of @xmath6 . then @xmath8 is said to be _ elementary _ , if it is @xmath168-elementary for any @xmath172 . robinson proved in @xcite the following property : [ rob - ks ] there exists an open and dense subset @xmath173 , such that for each @xmath174 , 1 . @xmath162 is finite and depends continuously on @xmath14 , and each periodic point in @xmath162 is @xmath12-elementary ; 2 . for any two @xmath175 , @xmath176 . let @xmath177 : then @xmath178 contains a @xmath0-residual subset of @xmath179 , and each @xmath180 is kupka smale . that is , 1 . each periodic point of @xmath14 is elementary ; 2 . @xmath181 for any two hyperbolic periodic points @xmath182 . the second item of the above property says that , _ @xmath166 and @xmath183 have a nontrivial intersection , the intersection is actually transverse . however , it does not address the question whether @xmath166 and @xmath183 can have any nontrivial intersection . theorem [ main ] confirms the existence of homoclinic intersections of every hyperbolic periodic point generically . it is proved in @xcite on @xmath1 , and @xcite on @xmath2 that @xmath0 generically , every hyperbolic periodic point admits transverse homoclinic points . for the maps @xmath184 , the center - leaf maps @xmath185 ( counting to periods ) may or may not be the generic ones . it is not clear if one can tell whether such a given center - leaf diffeomorphism satisfies their genericity requirement . so we need to a _ handy _ criterion for proving existence of homoclinic points for the center - leaf maps , see [ sec : homo ] . in this section we will give some perturbation results about partially hyperbolic symplectic diffeomorphisms with 2d center . let @xmath54 be the set of partially hyperbolic maps @xmath186 with center dimension @xmath187 . let @xmath188 be the set of partially hyperbolic maps @xmath189 that are @xmath32-normally hyperbolic whose center bundles are stably integrable . it is evident that @xmath188 is an open subset of @xmath54 . let @xmath8 be a periodic point of @xmath14 of period @xmath12 . then the splitting @xmath190 are @xmath191-invariant , and the eigenvalues of @xmath191 along the two hyperbolic directions have modulus different from @xmath192 . the two eigenvalues of @xmath191 along the center direction @xmath193 satisfy @xmath194 . therefore , * either @xmath195 for both @xmath196 . in this case @xmath8 is a hyperbolic periodic point of @xmath14 ; * or @xmath197 for both @xmath196 . in this case @xmath8 is nonhyperbolic with a 2d neutral direction . note that for @xmath104 , for any _ hyperbolic _ periodic point @xmath8 , any eigenvalue @xmath171 in with eigenvector @xmath198 has norm different from 1 , and @xmath199 , respectively . however , @xmath200 is _ always _ strictly contained in @xmath7 , since there are contraction / expansion along the center direction @xmath193 [ pro : bir ] there exists an open and dense subset @xmath201 such that for each @xmath202 and each periodic point @xmath203 , either @xmath8 is hyperbolic , or the center - leaf birkhoff coefficient @xmath204 , where @xmath163 is the minimal period of @xmath8 . let @xmath205 , where @xmath206 is the open and dense subset given in proposition [ rob - ks ] . let @xmath207 , and @xmath203 be a point fixed by @xmath31 , and @xmath163 be the minimal period of @xmath8 . then @xmath208 . in this case , the center leaf @xmath60 of @xmath8 is also invariant under @xmath209 , and we can consider the restriction of @xmath209 on @xmath60 , which is a symplectic surface diffeomorphism . since @xmath210 , we see that @xmath8 is @xmath211-elementary , and hence the center eigenvalue @xmath212 for each @xmath149 . then we can define the birkhoff normal form around @xmath8 in the center leaf @xmath60 , and let @xmath213 be the first birkhoff coefficient of this center - leaf bikhoff normal form at @xmath8 . let @xmath42 be an open neighborhood of @xmath14 in @xmath214 such that @xmath215 has constant cardinality and varies continuously on @xmath42 . * claim . * if @xmath204 , then there exists an open neighborhood @xmath216 of @xmath14 such that @xmath217 for all @xmath218 . firstly note that @xmath8 is nondegenerate . let @xmath219 be the continuation of @xmath8 for maps @xmath147 close to @xmath14 . moreover , the partially hyperbolic splitting on the maps @xmath147 depends continuously on @xmath147 , and @xmath147 admits a @xmath147-invariant center foliation @xmath220 . therefore , the system @xmath221 varies continuously , so is the birkhoff coefficient @xmath222 . this completes the proof of claim . in the following we assume @xmath223 . then we make a @xmath0-small local perturbation on the center leaf , say @xmath224 , supported on a small neighborhood @xmath225 of @xmath8 , such that @xmath226 , @xmath227 and the birkhoff coefficient @xmath228 . note that @xmath163 is the period of @xmath8 , not the center leaf @xmath60 . in particular it is possible that @xmath229 for some @xmath230 . in this case the intersection @xmath231 is a finite set , and the support of @xmath232 can be made small enough such that it does not interfere with the intermediate returns of @xmath8 to @xmath60 . however , note that the map @xmath232 hasnt been defined on @xmath233 . next we will extend @xmath232 to the whole manifold @xmath3 . by darboux s theorem , there exists a local coordinate system @xmath234 on @xmath235 around @xmath8 such that the restriction @xmath236 . let @xmath237 . then @xmath238 is a _ close 1-form _ on @xmath235 and hence also _ exact _ ( since @xmath235 is simply connected ) . so @xmath239 for some function @xmath240 which is identically 0 on @xmath241 . note that @xmath240 is @xmath242-small , and is called a _ generating function _ of @xmath232 . note that all iterates @xmath243 are compact leaves . using darboux s theorem again , one can extend the local coordinate system @xmath234 on @xmath244 to a local neighborhood @xmath245 containing @xmath235 , say @xmath246 , such that @xmath247 . then we extend the above _ center - leaf _ generating function @xmath240 to a generating function @xmath26 supported on @xmath248 such that @xmath249 . let @xmath44 be the corresponding symplectic diffeomorphism generated by @xmath26 . note that @xmath250 on @xmath251 , @xmath44 is @xmath0-close to identity and @xmath252 . let @xmath253 . then we have @xmath254 for each @xmath255 , @xmath256 and @xmath257 . note that any invariant @xmath32-normally hyperbolic manifold is isolated and persists under perturbations . then the fact @xmath60 is an @xmath32-normally hyperbolic manifold of @xmath258 implies that @xmath259 . therefore , we can rewrite the above conclusion as @xmath260 . applying the previous claim again , we have that there is an open neighborhood @xmath261 of @xmath147 such that for any @xmath262 , the continuation @xmath263 satisfies @xmath264 . let @xmath265 , which is a constant on @xmath42 . then by induction , we can find an open subset @xmath266 , such that for each @xmath267 and each periodic point @xmath268 , either @xmath219 is hyperbolic , or the center - leaf birkhoff coefficient @xmath269 , where @xmath163 is the minimal period of @xmath219 . note that our @xmath14 is chosen arbitrarily in @xmath270 , and @xmath271 contains an open set in an arbitrarily small open neighborhood @xmath42 of @xmath14 . putting these sets @xmath271 together , we get an open and dense subset in @xmath270 , say @xmath272 , such that for each @xmath202 and each periodic point @xmath203 , either @xmath8 is hyperbolic , or the center - leaf birkhoff coefficient @xmath273 , where @xmath163 is the minimal period of @xmath8 . then it follows that @xmath272 is an open and dense subset of @xmath57 . the perturbation @xmath44 constructed in the proof is localized around @xmath8 , and does not interfere with the dynamics around the other iterates @xmath274 . therefore , @xmath275 for all @xmath276 . however , the partially hyperbolic splitting of @xmath147 and the center foliation @xmath220 are not the same after the perturbation @xmath14 . in particular , most of the center leaves @xmath277 are slightly deformed comparing to @xmath278 . let @xmath272 be the open set given in proposition [ pro : bir ] , and @xmath279 . then @xmath280 contains a residual subset of @xmath188 . [ pro : r ] let @xmath59 . we have that 1 . @xmath162 is finite , and each periodic point is elementary ; 2 . @xmath281 for any two hyperbolic periodic points @xmath182 ; 3 . the center birkhoff coefficient @xmath282 for each nonhyperbolic periodic point @xmath8 . let @xmath283 be the residual subset given by proposition [ pro : r ] , @xmath284 and @xmath8 be a hyperbolic periodic point of @xmath14 whose center leaf @xmath60 is diffeomorphic to either @xmath1 or @xmath2 . in the following we denote @xmath285 , and @xmath286 , where @xmath163 is the minimal period of @xmath8 . then we list some properties of this new map : 1 . @xmath287 , and every periodic point of @xmath140 is elementary ; 2 . @xmath288 for any hyperbolic periodic points @xmath289 of @xmath147 ; 3 . each nonhyperbolic periodic point has nonzero birkhoff coefficient and is moser stable . an important method , the _ prime - end extension _ , was first used by mather @xcite in the study of general surface dynamics . let @xmath248 be a bounded , simply connected domain on the plane . note that the set - theoretic boundary @xmath290 may be very complicate . however , there always exists a conformal map @xmath291 , where @xmath292 is the open unit disk . the prime - end compactification of @xmath248 is obtained by attaching to @xmath248 an ideal boundary @xmath293 via the conformal map @xmath44 . more precisely , each point @xmath294 is specified by a nested sequence of open arcs @xmath295 with @xmath296 such that two endpoints of @xmath297 lie on both sides of @xmath294 and the sequence @xmath297 are nested in @xmath298 and converge to @xmath299 ( see @xcite ) . the equivalent class of this nested sequence defines a _ prime point _ , say @xmath300 . denote by @xmath301 the prime - end compactification of @xmath248 , whose topology is uniquely determined by the extended homeomorphism @xmath302 , such that @xmath303 and @xmath304 . it is important to note the relations between @xmath305 and the set - theoretic closure @xmath306 : * a prime - end point @xmath307 may cover a set of points in @xmath290 ( * ? ? ? * theorem 17.7 ) . * a point @xmath308 may be lifted to a set of points in the prime - ends @xmath309 ( ( * ? ? ? * figure 37-(b ) ) ) . see also @xcite for various examples . prime - end compactifications can also be defined for any connected open subset on a closed surface @xmath26 . let @xmath310 be an open connected subset on @xmath26 , whose boundary consists of a finite number of connected pieces , each of these boundary pieces has more than one point . then we can attach to @xmath248 a finite number of circles , to get its prime - end compactification @xmath305 . see also @xcite . let @xmath311 be a homeomorphism . then there exists uniquely an extension of @xmath147 to @xmath305 , say @xmath312 . if @xmath147 is orientation - preserving , then the restriction @xmath313 is an orientation - preserving circle homeomorphism . the rotation number @xmath314 is called the _ carathodory rotation number_of the map @xmath147 on @xmath248 , see @xcite . it is well known that @xmath313 has periodic orbits if and only if @xmath314 is rational . moreover , we have [ downfixed ] let @xmath147 be a symplectic diffeomorphism on a closed surface @xmath26 , such that each fixed point of @xmath147 is either hyperbolic , or elliptic and moser stable . let @xmath310 be an open connected , @xmath147-invariant subset , and @xmath312 be the induced prime - end extension of @xmath147 on @xmath248 . if @xmath315 has a fixed point on @xmath316 , then @xmath147 has a hyperbolic fixed point on @xmath290 . see @xcite for a proof of the above lemma when @xmath317 . their proof actually works for any surface , see @xcite . let @xmath8 be a hyperbolic periodic point of @xmath147 , @xmath167 be the unstable manifold of @xmath8 , which is an immersed curve passing through @xmath8 . let @xmath318 be a branch of the unstable manifold . without loss of generality , we assume @xmath147 also fixes @xmath319 ( otherwise , let s consider @xmath320 ) . pick @xmath321 . then @xmath322 $ ] forms a fundamental interval of @xmath319 , in the sense that the intervals @xmath323 , @xmath324 have mutually disjoint interiors , and the union @xmath325 . then @xmath319 is said to be _ recurrent _ , if @xmath326 . note that this definition is independent of the choices of @xmath321 . there are cases when some branch of the unstable / stable manifolds is not recurrent . in particular , a branch @xmath319 of an unstable manifold is said to form a _ saddle connection _ if @xmath319 is also a branch of the stable manifold of a hyperbolic fixed point @xmath10 . in the case @xmath11 , @xmath319 is said to be a _ homoclinic loop_. [ recurrent ] let @xmath140 be a symplectic diffeomorphism such that each periodic point of @xmath147 is either hyperbolic , or elliptic and moser stable . then for any branch @xmath319 of the invariant manifolds of any hyperbolic periodic point @xmath8 , we have the following dichotomy : 1 . either @xmath327 : then @xmath319 is recurrent . 2 . or @xmath328 : then @xmath319 forms a saddle connection between @xmath8 and @xmath10 . the proof relies on the study of the prime - end compactification of a connected component of @xmath329 , see @xcite for more details . as a corollary , we have the following characterization of the closure of branches of stable and unstable manifolds : [ closure ] let @xmath140 be a symplectic diffeomorphism such that each periodic point of @xmath147 is either hyperbolic , or elliptic and moser stable . if @xmath147 has no saddle connection , then all four branches of the stable and unstable manifolds of any hyperbolic periodic point @xmath299 of @xmath147 are recurrent and have the same closure . let @xmath330 be the set of maps in @xmath331 that @xmath32-normally hyperbolic ( with center dimension @xmath55 ) and are stably dynamically coherent . let @xmath280 be the set given by proposition [ pro : r ] , which contains residual subset of @xmath188 . let @xmath59 , and @xmath8 be a hyperbolic periodic point of @xmath14 with period @xmath163 . we will prove that @xmath332 in the case that the center leaf @xmath333 or @xmath2 . in the following we denote @xmath334 and @xmath335 for short . note that it suffices to show that being a hyperbolic fixed point of the surface diffeomorphism @xmath140 , @xmath336 . then @xmath337 , and the intersection must be transverse due to the choice of @xmath59 . we first assume @xmath26 is diffeomorphic to @xmath1 . our proof of the existence of homoclinic intersections in this spherical case follows the same _ closing gate _ approach used in @xcite , see also @xcite . we argue by contradiction . suppose there is no homoclinic intersection of @xmath8 . let @xmath319 be a branch of the unstable manifold of @xmath8 . pick a local coordinate system @xmath338 around @xmath8 such that @xmath319 leaves @xmath8 along the positive @xmath299-axis and is recurrent through the first quadrant , and the stable manifold of @xmath8 moves along the @xmath339-axis . pick @xmath340 sufficiently small , and let @xmath341 . let @xmath10 be the first moment on @xmath319 that @xmath319 intersects the set @xmath342 . adjusting @xmath343 if necessary , we may assume @xmath344 . let @xmath345 be the closed curve that starts from @xmath8 , first travels along @xmath319 to the point @xmath10 , and then slide from @xmath10 to @xmath8 along the closing segment @xmath346 . then @xmath345 is a simple closed curve , see fig . [ adjacent ] . adjacent.pdf ( 13,8)@xmath8 ( 33,7)@xmath319 ( 3,35)@xmath347 ( 83,35)@xmath348 ( 98,35)@xmath345 ( 20,25)@xmath342 ( 25,60)@xmath10 ( 65,21)@xmath349 let @xmath347 be a branch of the stable manifold of @xmath8 moving along the positive @xmath339-axis . since the closure of @xmath347 contains @xmath319 , @xmath347 also intersects @xmath342 . let @xmath348 be the corresponding simple closed curve by closing the first intersection @xmath349 of @xmath347 with @xmath342 , see also fig . [ adjacent ] . note that @xmath350 , since our hypothesis is that there is no homoclinic point . clearly @xmath351 and @xmath352 , since we cut @xmath319 and @xmath347 at their first intersection points with @xmath353 . then we see that @xmath354 , and this intersection is a topological crossing . so the algebraic intersection number @xmath355 . however , the algebraic intersection number between any two closed curves on @xmath1 must be 0 , and we arrive at a contradiction . therefore , the hypothesis that @xmath8 has no homoclinic intersection must be false . this completes the proof when @xmath26 is diffeomorphic to a sphere . note that the same argument applies to general surface , as long as the algebraic intersection number of two closing curves @xmath356 and @xmath357 is not 1 . see lemma [ tor-0 ] for the toric case . in this subsection we assume @xmath358 . we will argue by contradiction . beside the above closing gate technique , oliveira @xcite took advantage of the property that the homotopy group of @xmath2 is commutative . our proof uses the same idea of oliveira , but much shorter . for example , it is not necessary to consider the lift of the diffeomorphism @xmath147 to @xmath23 . we will show that the geometric picture of the lifts of branches is sufficient to derive a contradiction . assume there is no homoclinic intersection of @xmath8 . we take a local coordinate system around @xmath8 such that the local unstable and stable manifolds are along @xmath299-axis and @xmath339-axis , respectively . consider canonical projection @xmath359 from its universal cover with @xmath360 . then we lift the stable and unstable manifolds @xmath7 of @xmath8 to @xmath23 , and denote them by @xmath361 for each @xmath362 . it is easy to see that none of branches of the lifted stable and unstable manifolds in @xmath23 intersect with each other , since the intersection , if exists , would induce a homoclinic intersection on @xmath2 . [ lem : bound ] let @xmath363 be the lift of a branch @xmath364 at @xmath365 , and @xmath366 be the omega limit set of @xmath363 in @xmath23 . then we have : 1 . if @xmath367 for some @xmath368 , then @xmath369 for all @xmath77 . if @xmath370 is bounded , then @xmath371 . \(1 ) note that if @xmath367 , then @xmath372 . therefore , @xmath373 . by an induction on @xmath163 , we see that @xmath374 for all @xmath77 . note that @xmath370 is unbounded . \(2 ) it follows from ( 1 ) that @xmath375 for any @xmath376 . the recurrence of @xmath319 on @xmath2 and the boundedness of @xmath370 implies @xmath377 for some @xmath362 . putting them together , we have @xmath378 . since our hypothesis is that @xmath8 has no homoclinic intersection , we have the following result : [ tor-0 ] all four branches of @xmath379 are unbounded in @xmath23 . we argue by contradiction . assume one of the branches of @xmath379 , say @xmath370 , is bounded . then any other branch , say @xmath380 , is also bounded , since @xmath381 for some @xmath362 . then it follows from lemma [ lem : bound ] that @xmath371 for any branch @xmath370 of @xmath382 . as in [ spherical ] , we can find two adjacent branches of @xmath379 that they accumulate to themselves via the quadrant between them on @xmath23 . then we construct on @xmath23 the two closed curves @xmath356 and @xmath357 crossing each other at @xmath150 , and deduce that their intersection number is also @xmath192 . however , this contradicts the fact that the algebraic intersection number of two simple closed curves in @xmath23 must be zero . this completes the proof . we start with two branches on @xmath2 that accumulate to themselves via the quadrant between them , and consider their lift at @xmath383 . without loss of generality , we assume they are @xmath384 and @xmath385 , and put them on positive @xmath299-axis and positive @xmath339-axis locally . let @xmath386 ( for some small @xmath343 ) , and @xmath387 . we also denote @xmath388 for short . let @xmath389 be the place where the first intersection of @xmath390 happens , and @xmath391 be the place where the first intersection of @xmath392 happens . assume that there is no homoclinic point . then @xmath393 . let @xmath394 be the point of the first intersection @xmath395 , respectively . let @xmath396 be the projection of @xmath397 $ ] to @xmath2 , where @xmath398 $ ] is the interval connecting @xmath399 and @xmath400 . similarly , we define @xmath401 and its projection @xmath402 . non - existence of homoclinic intersection implies that the intersection number @xmath403 . therefore @xmath404 and @xmath393 . now let s consider the curve @xmath405 obtained by uniting @xmath406 , @xmath407 , @xmath408 and @xmath409 . this is a simply closed curve in @xmath23 , which bounds a simply connected domain , say @xmath410 . moreover , the translations of @xmath410 by @xmath393 are disjoint and their union covers the whole plane . in other words , @xmath410 is a fundamental domain . [ funda ] for two illustrations of @xmath411 . q1.pdf ( 33,25)@xmath150 ( 40,20)@xmath412 ( 32,5)@xmath400 ( 37,9)@xmath399 ( 17,32)@xmath401 ( 3,25)@xmath413 ( 8,30)@xmath414 ( 3,6)@xmath415 ( 25,20)@xmath410 ( 62,24)@xmath150 ( 77,29)@xmath412 ( 89,24)@xmath400 ( 95,27)@xmath399 ( 52,26)@xmath401 ( 61,4)@xmath413 ( 65,9)@xmath414 ( 90,4)@xmath415 ( 77,20)@xmath410 now let s describe the 4 corners of @xmath410 . it is easy to see @xmath410 contains a vertical thin wedge in @xmath416 , a horizontal thin wedge in @xmath417 , and an acute wedge in @xmath418 , where @xmath419 . the projection of the union of these three wedges covers only the first quadrant at @xmath8 . therefore , @xmath410 has to contain all three quadrants from the second to the forth quadrant around @xmath150 . in particular , it contains two local branches : @xmath420 and @xmath421 . both branches @xmath422 and @xmath423 are unbounded , and they ca nt stay in @xmath410 forever . let @xmath424 and @xmath425 be the points of the first intersections of @xmath422 and @xmath423 with @xmath426 , respectively . these two points must lie on the gates of @xmath405 , since all other parts of @xmath405 are on @xmath427 or @xmath428 . let @xmath429 $ ] is the closing gate starting at @xmath424 , and @xmath430 be the projection of @xmath431 $ ] to @xmath2 . similarly we define @xmath432 , @xmath433 and its projection @xmath434 . assume that there is no homoclinic point . then @xmath435 . in particular , the homotopy class of both @xmath436 is @xmath415 . first let s observe that * @xmath424 lies either on the gate @xmath437 or on @xmath438 , since it is on an unstable branch ; * @xmath425 lies either on the gate @xmath439 or on @xmath440 , since it is on a stable branch . suppose @xmath424 lies on @xmath437 . there are two cases when computing the intersection number of @xmath430 with @xmath434 : * @xmath425 lies on @xmath439 : then @xmath434 is in the class of @xmath413 , and @xmath441 ; * @xmath425 lies on @xmath440 : then @xmath434 is in the class of @xmath415 , and @xmath442 . on the other hand , @xmath443 , since @xmath444 is the first intersection of @xmath445 with @xmath405 , respectively . moreover , the intersection at @xmath8 is not a topological crossing . then the intersection number @xmath446 , a contradiction . therefore , @xmath424 lies on @xmath438 . similarly , one can prove @xmath425 lies on @xmath440 . this completes the proof . next we consider another curve @xmath447 obtained by uniting the following * @xmath406 , @xmath448 , and @xmath449 ; * @xmath409 , @xmath450 , and @xmath451 . this is a closed curve in @xmath23 , since both @xmath452 and @xmath433 end at the point @xmath415 . moreover , @xmath447 is a simply closed curve since we are working under the hypothesis that there is no homoclinic point . let @xmath453 be the simply connected domain in @xmath23 bounded by @xmath447 . note that @xmath453 also contains the two local branches : @xmath420 and @xmath421 . again none of the branches @xmath422 and @xmath423 ca nt stay in @xmath410 forever since they are unbounded . let @xmath454 and @xmath455 be the point of the first intersection of @xmath422 and @xmath423 with @xmath456 , respectively . note that each of the six components used to define @xmath447 contains a closing segment . following a similar reasoning for determining the locations of @xmath444 , we have 1 . @xmath454 lies on either @xmath437 , or @xmath457 , or @xmath458 ; 2 . @xmath455 lies on either @xmath439 , or @xmath459 , or @xmath460 . now consider the projection @xmath461 of @xmath462 $ ] , where @xmath463 ( depending on the location of @xmath454 ) . similarly we define @xmath464 $ ] and its projection @xmath465 , where @xmath466 ( depending on the location of @xmath455 ) . in any combination of the possible locations of the two points @xmath454 and @xmath455 , we always have @xmath467 on the other hand , @xmath468 , since @xmath469 is the first intersection of @xmath445 with @xmath447 , respectively . moreover , the intersection at @xmath8 is not a topological crossing . then the intersection number @xmath470 , which contradicts eq . . this completes the proof . the author thanks sam lisi and jeff xia for useful discussions .

we study some generic properties of partially hyperbolic symplectic systems with 2d center . we prove that @xmath0 generically , every hyperbolic periodic point has a transverse homoclinic intersection for the maps close to a direct / skew product of an anosov diffeomorphism with a map on @xmath1 or @xmath2 .