Systematic Investigation of Negative Cooper-Frye Contributions in Heavy Ion Collisions Using Coarse-grained Molecular Dynamics
SSystematic Investigation of Negative Cooper-Frye Contributions in Heavy IonCollisions Using Coarse-grained Molecular Dynamics
D. Oliinychenko,
1, 2, ∗ P. Huovinen,
1, 3, † and H. Petersen
1, 3, ‡ Frankfurt Institute for Advanced Studies, D-60438 Frankfurt am Main, Germany Bogolyubov Institute for Theoretical Physics, Kiev 03680, Ukraine Institut f¨ur Theoretische Physik, Goethe-Universit¨at, D-60438 Frankfurt am Main, Germany
In most heavy ion collision simulations involving relativistic hydrodynamics, the Cooper-Fryeformula is applied to transform the hydrodynamical fields to particles. In this article the so-callednegative contributions in the Cooper-Frye formula are studied using a coarse-grained transportapproach. The magnitude of negative contributions is investigated as a function of hadron mass,collision energy in the range of E lab = 5–160 A GeV, collision centrality and the energy densitytransition criterion defining the hypersurface. The microscopic results are compared to negativecontributions expected from hydrodynamical treatment assuming local thermal equilibrium.The main conclusion is that the number of actual microscopic particles flying inward is smallerthan the negative contribution one would expect in an equilibrated scenario. The largest impactof negative contributions is found to be on the pion rapidity distribution at midrapidity in centralcollisions. For this case negative contributions in equilibrium constitute 8–13% of positive contri-butions depending on collision energy, but only 0.5–4% in cascade calculation. The dependence onthe collision energy itself is found to be non-monotonous with a maximum at 10-20 A GeV.
I. INTRODUCTION
Relativistic hydrodynamics is nowadays the standardapproach for modeling ultrarelativistic heavy-ion colli-sions at highest RHIC (Relativistic Heavy Ion Collider)and LHC (Large Hadron Collider) energies. These dy-namical descriptions are either based on ideal [1, 2] ordissipative hydrodynamics [3, 4] and describe the en-tire expansion fluid dynamically. In so called hybrid ap-proaches [5, 6] only the early hot and dense stage of theexpansion is described using hydrodynamics and the laterdilute stage by hadron transport.Most of these models use a conceptually similar proce-dure: Given an initial condition, the hydrodynamic equa-tions are solved in the whole forward light cone. Nearthe boundary of vacuum and at the late times of evolu-tion hydrodynamics is not applicable any more, whenthe density is small and the mean free path is largerthan the system size. Therefore, models switch to anoff-equilibrium microscopic description in terms of par-ticles in this region. In hybrid approaches particles canscatter, while other models allow only free-streaming andresonance decays. In any case, the most commonly usedway to convert the fluid-dynamical fields to particles, aprocess that we call here ’particlization’, is by using theCooper-Frye formula.The Cooper-Frye formula assumes particlization totake place on infinitesimally thin three-dimensional hy-persurface in four-dimensional space-time. This hyper-surface Σ is usually determined a posteriori from hydro-dynamical solution in the whole forward light cone, usu-ally as a hypersurface of constant time, energy density, ∗ oliiny@fias.uni-frankfurt.de † [email protected] ‡ petersen@fias.uni-frankfurt.de temperature, or Knudsen number. Particle distributionson an infinitesimal element of hypersurface, d Σ, are cal-culated using the following formula: p d Nd p = p µ dσ µ f ( p ) , (1)where f ( p ) is a distribution function and dσ µ a normalfour-vector of hypersurface with length equal to the areaof the infinitesimal surface element. This formula wasobtained by Cooper and Frye [7] with the main featurethat it respects four-momentum conservation. Thoughformula (1) is valid for any f ( p ), the distribution func-tion is usually assumed to be either the boosted ther-mal distribution f ( p ) = f ( p ) = (cid:104) exp (cid:16) p µ u µ − µT (cid:17) ± (cid:105) − (ideal fluid), or a distribution close to the boosted ther-mal distribution f ( p ) = f ( p ) + δf ( p ) (viscous fluid),where δf ( p ) is the dissipative correction. Here T , µ and u µ = γ (1 , v ) are temperature, chemical potential and theflow velocity of the fluid, respectively.There is, however, a conceptual problem with theCooper-Frye formula. Where the surface is space-like, i.e. , its normal vector dσ µ is space-like, and p µ dσ µ < p . Thus if f ( p ) > p , as is the casefor the thermal distribution, d Nd p < p . Thiscan be easily seen in the local rest frame of a space-likesurface (which always exists since v surf < c for space-like surfaces), where p µ dσ µ = p · n and thus d Nd p < i.e. , dσ > d Nd p > p . This can be also understood as follows: surface is”escaping” faster than the speed of light, so no particlecan cross it inward. (For a summary of the properties oftime-like and space-like surfaces, see Table I).If d Nd p is interpreted as a phase-space density, negative a r X i v : . [ nu c l - t h ] M a r time-like surface space-like surfacetime-like normal space-like normal dσ µ dσ µ > dσ µ dσ µ < v surf > c v surf < c ∃ RF: dσ µ = ( ± dx dy dz, , , ∃ RF: dσ µ = (0 , , , dt dx dy ) dσ > ⇒ ∀ p µ : p µ dσ µ > ∃ p µ : p µ dσ µ < dσ > ⇒ ∀ p µ : d N CF d p > ∃ p µ : d N CF d p < g µν = (1 , − , − , − d N CF d p denotes particle distributionfrom the hypersurface element calculated using the Cooper-Frye formula. values of it are clearly unphysical, but instead of givinga literal phase-space density, Cooper-Frye formula rathercounts the world lines of particles crossing the surfaceelement d Σ, and gives positive weight to particles mov-ing “outward” and negative weight to particles moving“inward”. Thus the negative values of d Nd p , the so-callednegative Cooper-Frye contributions, refer to particles fly-ing inward toward the hydrodynamical region, and whichshould thus be absorbed back to the fluid.In pure hydrodynamical models, this poses a problem:Particlization takes place at freeze-out when rescatter-ings cease, and particles stream free. Thus, once par-ticles cross the particlization surface, there is nothingfrom where particles could scatter back toward the sur-face, and thus there should be no particles flying back.To avoid this problem, one could choose a completelytime-like particlization hypersurface, for example a hy-persurface of a constant time without any negative con-tributions. However, it was shown [8] that particle spec-tra obtained in such an approach are dramatically dif-ferent from spectra on a constant temperature hypersur-face. Another way is to consider cut-off distribution [9]: p d Ndp = p µ dσ µ f ( p )Θ( p µ dσ µ ). Such a prescription vio-lates conservation laws, unless one adjusts temperature,chemical potentials, and flow velocity in the particle dis-tribution f ( p ) [10, 11].On the other hand, there is no such a problem in hy-brid models. Particlization takes place where rescatter-ings are abundant, and thus it is natural to have particlesflying back to the fluid-dynamical region. The problem israther a practical one: What does the negative weight ofa particle mean when one samples the particle distribu-tions at particlization surface to create an initial state forthe hadron transport? Usually one simply ignores them(see e.g. Ref. [12]), which violates conservation laws. Anattempt to include these negative weights to the hadrontransport was recently made in Ref [13]. Alternatively, ifthe transition from fluid to transport takes place in a re-gion where hydrodynamics and transport are equivalent,the negative Cooper-Frye contributions coincide with thedistribution of particles that backscatter to hydrodynam-ical region. Thus all one needs to do is to remove theseparticles from the cascade, but such removing is techni- cally challenging, and the problem remains how to findthe region where hydrodynamics and transport lead toequal solutions—assuming that such a region exists atall! Thus the ultimate solution to the problem would beto construct a model, solving coupled hydrodynamicaland kinetic equations with the kinetic model providingboundary condition for hydrodynamics. An attempt inthis direction was taken by Bugaev [14–16], but theseideas have not yet been implemented in practice.Fortunately, at high collision energies, the explosiveexpansion dynamics keeps the negative contributions onthe level of a few percent. Emission of particles fromtime-like areas of surface where no negative contribu-tions appear (so-called volume emission) is much largerthan emission from space-like areas (so-called surfaceemission), and as we will discuss later, large flow veloc-ity reduces negative contributions from space-like sur-faces. Nevertheless, there are very few studies that ac-tually quote the values of negative contributions, and in-vestigations at lower collision energies are lacking com-pletely. In this article the negative contributions arisingon the Cooper-Frye transition surface assuming distri-bution functions in local equilibrium are compared tothe actual underlying microscopic dynamics to investi-gate the systematic differences between a transport anda hybrid approach.Therefore, the aim of the current study is to comparethe expected negative contributions in a locally equili-brated hydrodynamical approach with the actual numberof particles that scatter back through a hypersurface ina coarse-grained microscopic transport approach. A con-stant energy density transition surface is constructed andnegative Cooper-Frye contributions are compared to ac-tual backscattered particles. In addition, the magnitudeof negative contributions is calculated in a systematicway depending on hadron sort, collision energy, central-ity, and choice of the transition surface. In Section II theframework for the calculation is explained. Section IIIshows results of tests of the numerical setup and sensi-tivity to internal parameters of the calculation. Finally,Section IV contains physical results: the quantification ofCooper-Frye negative contributions and their comparisonto backscattered particles.
II. METHODOLOGY
Our calculation is based on the hadronic transport ap-proach - Ultrarelativistic Quantum Molecular Dynamics(UrQMD 3.3p2) [17]. The degrees of freedom in UrQMDare hadrons, resonances up to a mass of 2.2 GeV andstrings and the implemented processes include binaryelastic and inelastic scatterings which mainly proceed viaresonance formation and decays or string excitation andfragmentation at higher collision energies. The UrQMDparticles move along classical trajectories and scatter ac-cording to their free-particle cross-sections. In our stud-ies there are no long range potentials and particle trajec-tories between collisions are always straight lines. UsingUrQMD we simulate Au + Au collisions at laboratoryframe energies E lab = 5, 10, 20, 40, 80 and 160 A GeV.This energy region is chosen because we expect UrQMDto provide a reasonable description of the collision dy-namics at those energies, and the Cooper-Frye negativecontributions to become significant in this energy range.The general procedure for our calculations is:1. Generate many UrQMD events and coarse-grainthem using a 3+1D space-time grid.2. Find the local energy density in the Landau restframe of each grid cell, (cid:15)
LRF ( t, x, y, z ), and the col-lective flow velocity in each cell, v ( t, x, y, z ).3. Construct the hypersurface Σ of a constant energydensity (cid:15) LRF ( t, x, y, z ) = (cid:15) c .4. Calculate the particle spectra on Σ by using theCooper-Frye formula and by counting the actualUrQMD particles that cross Σ. To obtain thesespectra and to compare them to each other is thegoal of the current work.This procedure mimics switching from hydrodynamics totransport in a hybrid model, but here the ”hydrodynam-ical” picture is obtained by averaging over particle distri-butions on a space-time grid. Since all the information isstill available in the underlying microscopic approach weare able to compare the negative Cooper-Frye contribu-tions to the spectrum of actual backscattered particles.In the following we explain all necessary details for eachof these steps of the calculation. A. Calculating physical quantities on a grid
To obtain the energy density in the Landau rest frameas a function of space-time, that is necessary to constructthe Cooper-Frye transition surface, the energy momen-tum tensor and the net baryon current in the computa-tional frame are calculated T µν ( t, x, y, z ) = 1∆ x ∆ y ∆ z (cid:28)(cid:88) p µ p ν p (cid:29) N (2) j µB ( t, x, y, z ) = 1∆ x ∆ y ∆ z (cid:28)(cid:88) p µ p B (cid:29) N , (3)where the sum is over all particles in each grid cell at themoment t , and B is the baryon number of each particle.Angular brackets denote averages over N UrQMD events.The cell sizes need to be small enough so that gradients ofall relevant physical quantities within the cell are small.On the other hand, if the cell sizes are too small oneneeds to generate infeasibly many events to damp statis-tical fluctuations of T µν components from cell to cell, andobtain a smooth surface Σ. To satisfy these conditionsand to ensure energy conservation precisely we choose∆ x = ∆ y = 1 fm, ∆ z = 0.3 fm and time step ∆ t = 0.1 fm. For the highest collision energy, E lab = 160 A GeV,the gradients are larger, so even smaller grid sizes weretaken: ∆ x = ∆ y = 0.3 fm and ∆ z = 0.1 fm. This choiceis further discussed in the Section III, where the sensi-tivity of results to the grid size is studied. Since even N = 10 000 events do not provide enough statistics toobtain a smooth hypersurface, and increase of N is notfeasible due to limited storage capacities, the individualparticles are smeared by marker particles distributed ac-cording to a Gaussian distribution.Every UrQMD particle with coordinates ( t p , x p , y p , z p )and four-momentum p µ is substituted by N split parti-cles with coordinates distributed with probability den-sity f ( x, y, z ) ∼ exp (cid:16) − ( x − x p ) σ − ( y − y p ) σ − γ z ( z − z p ) σ (cid:17) ,where γ z = (1 − p z /p ) − / . These marker particles areattributed the 4-momentum and quantum numbers ofthe original particle divided by N split . In our calcula-tion N split = 300 and σ = 1 fm. The sensitivity of ourresults to the width of the Gaussian is discussed in Sec-tion III. When this Gaussian smearing is applied, stableresults are obtained with only N = 1500 events, whichwe employ for our calculations. B. The hypersurface construction
After obtaining T µν in the computational frame, ithas to be transformed to the Landau rest frame (LRF)in each cell. By definition, T iLRF = 0, i.e. , the energyflow in the LRF is zero. To find the LRF we solvethe generalized eigenvalue problem ( T µν − λg µν ) h ν = 0.The eigenvector corresponding to the largest eigenvalueis proportional to the 4-velocity of the LRF and theproportionality constant is fixed by the constraint that √ u µ u µ = 1. After finding T µνLRF the hypersurface of con-stant Landau rest frame energy density is constructedwhere T LRF ≡ (cid:15) LRF ( t, x, y, z ) = (cid:15) c , with (cid:15) c a parame-ter that characterizes the hypersurface. In such a waywe mimic the transition surface in hybrid models, whichtypically use (cid:15) c = 0 . [12]. The isosurfaceis constructed using the Cornelius subroutine [12], thatprovides a continuous surface without holes and avoidsdouble counting of hypersurface pieces. The subroutineprovides the normal four-vectors dσ µ of the hypersurface.The physical quantities on the grid, i.e. , the energy, netbaryon density and the flow velocity, are linearly inter-polated to the geometrical centers of the hypersurfaceelements. C. Thermodynamic quantities
To apply the Cooper-Frye formula one needs the tem-perature T and chemical potentials on the surface, whichdo not exist in the microscopic picture. Strictly speak-ing they make sense only in the vicinity of thermal andchemical equilibrium, which may not be the case in ourUrQMD simulation. Nevertheless, we take the LRF en-ergy density and net baryon density to mean equilibriumdensities—as is the case when deviations from equilib-rium are small—and obtain temperature and chemicalpotentials from an ideal hadron resonance gas (HRG)equation of state (EoS) containing the same hadronsand resonances as UrQMD. Since our EoS assumes zerostrangeness density, we impose this constraint as well,even if UrQMD itself allows local non-zero strangeness.In practice, this means solving the following coupledequations to find the temperature T , baryon chemicalpotential µ B and strangeness chemical potential µ S : (cid:15) LRF = (cid:88) p g p (2 π ) (cid:90) d k √ k + m e ( √ k + m p − µ B B p − µ S S p ) /T ± n LRFB = (cid:88) p g p B p (2 π ) (cid:90) d ke ( √ k + m p − µ B B p − µ S S p ) /T ± n LRFS = (cid:88) p g p S p (2 π ) (cid:90) d ke ( √ k + m p − µ B B p − µ S S p ) /T ± (cid:15) LRF = T is the energy density in LRF, n LRFB isthe baryon density in LRF, n s is the strangeness density,and the sum runs over all hadron species that appear inUrQMD; m p is the mass of a hadron p , g p is its spin andisospin degeneracy factor, and B p and S p are its baryonnumber and strangeness, respectively. D. Cooper-Frye and ”by particles” calculations
After the hypersurface of constant LRF energy densityΣ is obtained and T and µ are evaluated using the EoS,the Cooper-Frye formula is applied on the hypersurface.The spectrum from the Cooper-Frye formula is split intopositive and negative parts: dN + CF p T dp T dϕdy = g (2 π ) (cid:90) σ Θ( p µ dσ µ ) p µ dσ µ e ( p ν u ν − µ ) /T ± dN − CF p T dp T dϕdy = − g (2 π ) (cid:90) σ Θ( − p µ dσ µ ) p µ dσ µ e ( p ν u ν − µ ) /T ± dN/dy or dN/p T dp T the integrations are per-formed numerically, applying the 36 ×
36 points Gauss-Legendre method to integrals transformed to finite lim-its.For comparison with the Cooper-Frye calculation wecount the actual microscopic (not marker) particles cross-ing the same hypersurface Σ that is used for Cooper-Fryecalculations. Inward and outward crossings are countedseparately. To find the point, where a particle trajectorycrosses Σ we use the fact that by construction the energydensity (cid:15) > (cid:15) c inside the surface and (cid:15) < (cid:15) c outside of it.The energy density is interpolated to the particle trajec-tory to find the point where (cid:15) − (cid:15) c changes sign. Eachof these crossings is counted as positive, if the particlestreams outward and negative, if the particle flies towardhigher energy densities. Both Cooper-Frye calculation and particle countingstart at the same time t start , which depends on the colli-sion energy. Following the prescription from hybrid mod-els, we take t start = Rvγ . This is the time two nuclei needto pass through each other. Numerical values are 8 fm/cfor 5 A GeV, 5.6 fm/c for 10 A GeV, 4 fm/c for 20 A GeV,2.8 fm/c for 40 A GeV, 2 fm/c for 80 A GeV and 1.4 fm/cfor 160 A GeV. The same t start is used for all centralities. III. SENSITIVITY TO INTERNALPARAMETERS AND FULFILLMENT OFCONSERVATION LAWS (a)
E = 10 A GeV, b = 0 fm, 1500 events, σ = 1 fm
CF positive/10outward crossings/10CF negativeinward crossings d N / dy | y =
05 dz [fm]0 0.2 0.8 1 (b)
E = 10 A GeV b = 0 fm σ = 1 fm
CF positive/10outward crossings/10CF negativeinward crossings d N / dy | y =
025 N10 (c) E = 10 A GeV b = 0 fm 1500 events
CF positive/10outward crossings/10CF negativeinward crossings d N / dy | y =
05 σ [fm]0 0.5 1.5 2
Figure 1. Sensitivity of results to internal parameters ofthe simulation: grid spacing along z axis, dz (a), numberof events, N (b) and the width σ of Gaussian smearing (c). Besides physical parameters like the beam energy, E lab ,and centrality of the collision controlled by the impactparameter b , our simulation contains internal parameterslike grid spacing, the width of the smearing Gaussian σ , % − % − t [fm/c] [ % ] − (c) E lab = 160 A GeV b = 0 fmGeV (b) E lab = 40 A GeV b = 0 fm
Δ E (t)/dtΔ E (t)/dt GeV (a) E lab = 10 A GeV b = 0 fm [ G e V / (f m / c )] Figure 2. (Color online) Energy flux through the surface at different times evaluated as actual flow, ∆ E ( t ) /dt = (cid:82) tt − dt T µ dσ µ /dt (circles), and as a difference in energy within the surface at different times, ∆ E ( t ) /dt = ( E in ( t ) − E in ( t − dt )) /dt (rectangles). Lower panel shows the relative difference between these two measures in %, and thus the conservation of energyin the calculation. and the number of events N . Ideally, we should work insuch a region of internal parameters, that our results areindependent of them. To see how sensitive our results areto these internal parameters, the positive and negativecontributions to the pion yield at midrapidity, dNdy | y =0 ,at different values of these parameters are evaluated.The calculation is more sensitive to the grid spacing inz direction, dz , than to the spacings in x and y directions, dx and dy , since gradients of T µν are largest in the longi-tudinal direction. Although, as shown in Fig. 1 a), eventhe sensitivity to dz is weak over a reasonable range ofvalues. The main motivation for choosing the grid spac-ing and time step comes in fact from the requirement ofenergy conservation discussed later.The results are very sensitive to the small number ofevents (see Fig. 1 b), but already N = 500 events pro-vides sufficient statistics for stable results. To be on thesafe side, we have analyzed N = 1500 events for our finalresults. Unfortunately, our results are not completelyindependent of the width σ of the Gaussian smearing,as shown on Fig. 1 c). The number of inward crossingUrQMD pions is most sensitive to σ . Two effects playa role here: at small σ the surface still has large sta-tistical fluctuations and small scale structures, “lumps”(See Fig. 2 of Ref [18]), whereas at large σ Gaussiansmearing pushes transition surface further out in space.Further out the densities are smaller, and the UrQMDparticle distributions are further away from equilibriumso that especially the number of particles moving towardthe center is strongly reduced. We choose σ = 1 fmas a reasonable value for our calculations, but keep inmind that varying σ in the range from 0.6 fm to 1.4 fmcauses ∼
20 % difference in the number of inward cross-ings. We consider this a systematic error in our analysis, but fortunately this uncertainty does not affect our mainconclusions.To check that energy is conserved in the coarse-graining procedure, we evaluate the energy flow throughthe surface during the time step dt , ∆ E ( t ) = (cid:82) tt − dt T µ dσ µ , and compare it to the change in energywithin the surface during the same time step, ∆ E ( t ) = E in ( t ) − E in ( t − dt ), where E in is total energy of par-ticles inside the surface. Ideally ∆ E ( t ) = ∆ E ( t ) forany dt , but finite cell sizes limit the precision and breakthe conservation of energy. The accuracy of ∆ E ≈ ∆ E improves when grid spacing and time step are decreased.Fig. 2 shows the energy flux through the surface and therelative difference between ∆ E ( t ) and ∆ E ( t ) in centralcollisions at energies E lab = 10, 40, 160 A GeV. To achievebetter than 5% percent accuracy at all times, we usesmall grid spacing with ∆ x = ∆ y = 1 fm, ∆ z = 0.3 fm,and time step ∆ t = 0.1 fm/c in collisions with E lab ≤ A GeV, and even finer grid with ∆ x = ∆ y = 0 . z = 0 . E lab = 160 A GeV. Whenintegrated over the whole collision time, the violation ofenergy conservation is less than 1% at all collision ener-gies. We have done a similar check for the net baryoncharge, and obtained similar results.
IV. RESULTS AND DISCUSSION
Let us start by investigating the properties of the tran-sition hypersurface itself as a function of beam energy.Figure 3 depicts the surface Σ in longitudinal directionalong the x axis. We see that with increasing energy, thelifetime of the system increases. This indicates longerlasting surface emission (from space-like parts of the sur- (a) A GeV10 A GeV20 A GeV 40 A GeV80 A GeV160 A GeV t = |z| t [f m / c ] z [fm] -20 -10 10 20 (b) [ G e V - ] N T d N A GeV10 A GeV20 A GeV40 A GeV80 A GeV160 A GeV
T [GeV]
Figure 3. (Color online) Upper panel: Hypersurface of con-stant LRF energy density (cid:15) ( t, , , z ) = (cid:15) c = 0 . .Lower panel: The fraction of hypersurface elements with (ap-parent) temperature T in central Au+Au collisions at thecollision energy of E lab = 5, 10, 20, 40, 80, 160 A GeV. face), which might lead to larger negative contributions.On the other hand, with increasing energy the longitudi-nal expansion leads to larger volume of the final volumeemission (from time-like parts of the surface), which indi-cates smaller negative contributions. Thus we have twocompeting effects, and one has to carry out the actualcalculation to find out how the negative contributionsdepend on energy.Distributions of the (apparent) temperature of thehypersurface elements are shown on the right panel ofFig. 3. At each collision energy temperature distributionis rather narrow, which means that constant energy den-sity surface approximately coincides with constant tem-perature surface. As well, the average temperature in-creases with increasing collision energy as expected fromthermal model fits to particle yields [19].In Fig. 4 we compare rapidity spectra of identified par-ticles in E lab = 40 A GeV Au+Au collisions obtained byCooper-Frye calculation and by counting of the micro-scopic particles. Even though, we are showing the re-sults only for one collision energy, all results are qualita- (a) π Cooper-Frye positiveCooper-Frye totalparticles outward crossingsparticles outward - inward d N π / dy y -3 -2 -1 1 2 3 (b) K - Cooper-Frye positiveCooper-Frye totalparticles outward crossingsparticles outward - inward d N K / dy y -3 -2 -1 1 2 3 (c) N Cooper-Frye positiveCooper-Frye totalparticles outward crossingsparticles outward - inward d N N / dy y -3 -2 -1 1 2 3 Figure 4. (Color online) Rapidity distribution of identifiedparticles obtained from Cooper-Frye formula on the surfaceΣ and from explicit counting of particles that cross the samesurface. Positive contributions and the net distribution, i.e. ,positive-negative, are shown separately. E lab = 40 A GeV,central Au+Au collisions. tively the same at all other energies. If UrQMD is closeto equilibrium on a surface at (cid:15) c = 0 . , bothapproaches should yield similar distributions. At midra-pidity this is the case for nucleons, and with a lesseraccuracy for kaons. ∆’s, Λ’s, ρ ’s and η ’s which are notshown in the figure depict a behavior similar to nucle-ons. However, the pion yields are wildly different indi-cating that pions are—and thus the entire system is—far away from chemical equilibrium at least. To cancelthe effect of non-equilibrium and to make the differencesin momentum distributions visible we consider not theabsolute value of the negative contributions, but the ra-tio of negative to positive ones, ( dN − /dy ) / ( dN + /dy ) or( dN − /dp T ) / ( dN + /dp T ). From Fig. 4 it is also appar-ent that the magnitude of the negative contributions isalways small compared to the positive ones as expected. Cooper-Frye πK + ρNΔ E = 40 A GeV, b=0 fm ( d N - / dy ) / ( d N + / dy ) [ % ] y -3 -2 -1 1 2 3 Figure 5. (Color online) Rapidity distribution of the ratio ofnegative to positive contributions for different hadron species:pions (circles), K + (crosses), ρ (bars), nucleons (rectangles),and deltas (triangles). Cooper-Frye calculation in centralAu+Au collisions at E lab = 40 A GeV.
The dependence of the ratio ( dN − /dy ) / ( dN + /dy ) onthe hadron type is illustrated in Fig. 5 by the Cooper-Frye results. Since for all cases, the microscopic nega-tive contributions of backstreaming particles are muchsmaller than the Cooper-Frye ones we concentrate onshowing the maximal effect. Surface temperature andvelocity profiles are identical for all hadrons, so the plotdemonstrates first of all the effect of particle mass. Onecan see that the average value of ( dN − /dy ) / ( dN + /dy )decreases with particle mass. This can be understoodby considering a small volume of fluid in its rest frame,and a space-like surface moving through it with a veloc-ity 0 < v surf < c so that lower density, i.e. , outside, is inthe negative direction. To be counted as a negative con-tribution, a particle must enter the fluid, and thus havea larger velocity than the surface. Average thermal ve-locity decreases with increasing mass, and therefore theheavier the particle, the fewer of them cross the surfaceinward. Since relative negative contributions for pionsare several times larger than for other hadrons we willconsider only pions in the following.As could be seen in Fig. 4, imposing equilibrium forCooper-Frye calculation leads to significantly larger neg-ative to positive contribution ratio at midrapidity thanthe counting of UrQMD particles. As shown in Fig. 6this holds for all the energies we have considered, show-ing that the system is out of not only chemical, but alsoof kinetic equilibrium. Either the collective flow veloc-ity of pions is different from the collective velocity of (dN - /dy)/(dN + /dy) @ |y| < 0.05 Cooper-Fryeby particles [ % ] E lab [GeV/nucleon] Figure 6. (Color online) The ratio of negative to positivecontributions on the (cid:15) ( t, x, y, z ) = (cid:15) c = 0 . surfacefor pions at midrapidity in central Au+Au collisions at vari-ous collision energies. Circles depict Cooper-Frye result andrectangles the explicit counting of UrQMD particles. other particles [20, 21] or the dissipative corrections topion distribution are very large. We have also checkedthat the relative microscopic negative contributions aremuch smaller in UrQMD at all centralities, for all parti-cle species, and on isosurfaces of energy density (cid:15) c = 0 . .On the other hand, the trend as a function of collisionenergy in Cooper-Frye and UrQMD calculations is thesame: both curves have a maximum at 10-20 A GeV andthen decrease with increasing energy. This behavior is aresult of a complicated interplay of several factors: tem-perature, relative velocities between surface and fluid,and relative amounts of volume and surface emission, i.e. ,emission from the time- and space-like parts of the sur-face. To gain some insight we consider all these factorsseparately. The same argument used to explain the sensi-tivity of negative contributions to particle mass, explainswhy larger temperature leads to larger negative contri-butions. Temperature on the constant density surfacegrows with increasing collision energy (see Fig. 3), whichwould lead one to expect an increase of negative contri-butions with increasing collision energy. On the otherhand, larger relative velocity between the fluid and sur-face reduces the negative contributions (again the sameargument), and we see that the average relative veloc-ity increases with increasing collision energy. Finally, asargued when discussing Fig. 3, we have seen that thelarger the collision energy, the larger the fraction of vol-ume emission. Which, as mentioned, reduces the nega-tive contributions.It is instructive to evaluate the negative contributionsas function of transverse momentum p T as well, as shownin Fig. 7 for Cooper-Frye calculation and ”by particles”. Cooper-Frye, 10 A GeVCooper-Frye, 40 A GeVby particles, 10 A GeVby particles, 40 A GeV ( d N - π / dp T ) / ( d N + π / dp T ) [ % ] p T [GeV] Figure 7. (Color online) The ratio of negative to positivepion contributions as a function of transverse momentum atmidrapidity in central Au+Au collisions at E lab = 5, 10, 20,40, 80 A GeV.
One can see that the largest negative contributions arelocated at small p T , which means that one can reduce theuncertainty caused by the negative contributions by a low p T cut. Also as a function of transverse momentum, theamount of microscopically backward streaming particlesis much smaller than in an equilibrium scenario.When discussing Fig. 6 we mentioned that, indepen-dent of the energy density of the surface, the nega-tive contributions are much smaller when counting theUrQMD particles. Furthermore, in Cooper-Frye calcula-tions the strength of the negative contributions dependson the value of (cid:15) c where the distributions are evaluatedas shown in Fig. 8. Larger (cid:15) c leads to larger negativecontribution at midrapidity and lower at back- and for-ward rapidities. This result arises from interplay of twofactors: larger temperature and smaller average v rel forlarger energy density. Quite surprisingly the negativecontributions evaluated by counting the UrQMD parti-cles is almost independent of the value of (cid:15) c . This indi-cates that even in much higher temperature T ∼ π Cooper-Frye, ε c = 0.3 GeV/fm Cooper-Frye, ε c = 0.6 GeV/fm by particles, ε c = 0.3 GeV/fm by particles, ε c = 0.6 GeV/fm E = 40 A GeV b = 0 fm ( d N - π / dy ) / ( d N + π / dy ) [ % ] y -3 -2 -1 1 2 3 Figure 8. (Color online) Rapidity distribution of the ratio ofnegative to positive contributions for pions on (cid:15) ( t, x, y, z ) = (cid:15) c = 0 . (circles) and (cid:15) c = 0 . (crosses)surfaces in central Au+Au collisions at E lab = 40 A GeV.Full symbols correspond to Cooper-Frye calculation and opensymbols to explicit counting of UrQMD particles. ies. In [12] negative contributions were evaluated on the (cid:15) = 0 . transition surface of a hybrid modelat SPS and RHIC energies— E lab = 160 A GeV and √ s NN = 200 GeV, respectively—and found to be around( dN − π /dy ) / ( dN + π /dy ) (cid:39)
13% and 9% at y = 0. Thenegative contributions for 160 A GeV are slightly largerthan in our calculation. The reason for this discrepancylies in the difference of the velocity profiles on the hy-persurfaces: In hydrodynamics the average relative ve-locity between flow and surface is smaller than in ourtransport-based approach, which leads to larger negativecontributions.
V. CONCLUSIONS
We have investigated negative Cooper-Frye contribu-tions and backscattering using a coarse-grained molecu-lar dynamics approach. Au+Au collisions at E lab = 5–160 A GeV energies have been simulated using UrQMD,and a hypersurface Σ of constant Landau rest frame en-ergy density has been constructed. On this surface wehave calculated two quantities: The ratio of Cooper-Fryenegative to positive contributions, which assumes localthermal equilibrium, and the ratio of UrQMD particlescrossing Σ inward to crossing Σ outward, which assumesno equilibrium.We found that at all collision energies the ratio of in-ward to outward moving particles calculated counting theUrQMD particles is much smaller than the same ratiocalculated assuming equilibrium, i.e. , the Cooper-Fryenegative to positive ratio. This finding poses a question (a) π Cooper-Frye, b = 0 fmCooper-Frye, b = 6 fmCooper-Frye, b = 12 fm
E = 40 A GeV ε c = 0.3 GeV/fm ( d N - π / dy ) / ( d N + π / dy ) [ % ] y -3 -2 -1 1 2 3 (b) E = 40 A GeV ε c = 0.3 GeV/fm b = 0 fmb = 6 fmb = 12 fmt = |z| t [fm/c] z [fm] -20 -10 10 20 Figure 9. (Color online) Upper panel: Rapidity distributionof the ratio of negative to positive contributions for pions inAu+Au collisions at E lab = 40 A GeV at various centralities: b = 0 (circles), b = 6 fm (crosses) and b = 12 fm (rectangles).Lower panel: hypersurfaces along the z axis in the same col-lisions at the same centralities. to the construction of hybrid models, and the treatmentof freeze-out in hydrodynamical models: If the cascade leads to distributions nowhere near equilibrium, how arethe hydrodynamical and cascade stages to be connectedin a consistent fashion? On the other hand, this resultshows that an ideal fluid dynamics hybrid approach con-tains the worst case scenario for negative contributionsand even then they are on the order of max. 15% forthe pion yield at midrapidity. What remains to be seen,however, is whether we could get closer to the UrQMDresult if we allowed dissipative corrections to the distribu-tion function of Cooper-Frye, or whether the deviationsfrom equilibrium are so large that dissipative expansionis not feasible.The largest observed impact of negative contributionsis to pion rapidity spectrum at midrapidity in centralcollisions. In thermally equilibrated Cooper-Frye calcula-tions it constitutes 8–13%, but only 0.5–4% in the count-ing of UrQMD particles. The Cooper-Frye value roughlyagrees with the values obtained previously for hydrody-namics at 160 GeV. We found several systematic featuresin these ratios. They are smaller for larger hadron massand therefore largest for pions. The relative negativecontributions decrease as a function of collision energyand by going from central to peripheral collisions. Onthe other hand, they increase if a higher energy densityis chosen as a surface criterion. The small scale struc-tures on the surface, its “lumpiness”, play a significantrole: If the surface is not smooth enough both ratios canincrease dramatically. Therefore, an interesting futurestudy could be to compare single fluctuating events tothe averaged result. ACKNOWLEDGMENTS
This work was supported by the Helmholtz Inter-national Center for the Facility for Antiproton andIon Research (HIC for FAIR) within the framework ofthe Landes-Offensive zur Entwicklung Wissenschaftlich-Oekonomischer Exzellenz (LOEWE) program launchedby the State of Hesse. DO and HP acknowledge fund-ing of a Helmholtz Young Investigator Group VH-NG-822 from the Helmholtz Association and GSI, and PHby BMBF under contract no. 06FY9092. Computationalresources have been provided by the Center for ScientificComputing (CSC) at the Goethe University of Frankfurt. [1] P. F. Kolb and U. Heinz, in
Quark-Gluon Plasma 3 ,edited by R. C. Hwa and X.-N. Wang (World Scientific,Singapore, 2004), p. 634.[2] P. Huovinen and P. V. Ruuskanen, Ann. Rev. Nucl. Part.Sci. , 163 (2006).[3] U. W.Heinz and R. Snellings, Annu. Rev. Nucl. Part. Sci. , 123 (2013).[4] C. Gale, S. Jeon and B. Schenke, Int. J. Mod. Phys. A , 1340011 (2013).[5] T. Hirano, P. Huovinen, K. Murase and Y. Nara, Prog. Part. Nucl. Phys. , 108 (2013).[6] H. Petersen, arXiv:1404.1763 [nucl-th].[7] F. Cooper and G. Frye, Phys. Rev. D , 186 (1974).[8] D. H. Rischke, Lect. Notes Phys. , 21 (1999).[9] K. A. Bugaev, Nucl. Phys. A , 559 (1996).[10] C. Anderlik, Z. I. Lazar, V. K. Magas, L. P. Csernai,H. Stoecker and W. Greiner, Phys. Rev. C , 388 (1999).[11] C. Anderlik et al. , Phys. Rev. C , 3309 (1999).[12] P. Huovinen and H. Petersen, Eur. Phys. J. A , 171(2012). [13] S. Pratt, Phys. Rev. C , 024910 (2014).[14] K. A. Bugaev and M. I. Gorenstein, nucl-th/9903072.[15] K. A. Bugaev, Phys. Rev. Lett. , 252301 (2003).[16] K. A. Bugaev, Phys. Rev. C , 034903 (2004).[17] S. A. Bass et al. , Prog. Part. Nucl. Phys. (1998) 255;M. Bleicher et al. , J. Phys. G G (1999) 1859. [18] P. Huovinen, M. Belkacem, P. J. Ellis and J. I. Kapusta,Phys. Rev. C , 014903 (2002).[19] A. Andronic, P. Braun-Munzinger and J. Stachel, Phys.Lett. B , 142 (2009).[20] H. Sorge, Phys. Lett. B , 16 (1996).[21] S. Pratt and J. Murray, Phys. Rev. C57