Cooper-Frye Negative Contributions in a Coarse-Grained Transport Approach
CCooper-Frye Negative Contributions in aCoarse-Grained Transport Approach
D. Oliinychenko , , P. Huovinen , , H. Petersen , Frankfurt Institute for Advanced Studies, D-60438 Frankfurt am Main, Germany Institut f¨ur Theoretische Physik, Goethe-Universit¨at, D-60438 Frankfurt am Main, Germany Bogolyubov Institute for Theoretical Physics, Kiev 03680, UkraineE-mail: [email protected]
Abstract.
Many models of heavy ion collisions employ relativistic hydrodynamics to describethe system evolution at high densities. The Cooper-Frye formula is applied in most ofthese models to turn the hydrodynamical fields into particles. However, the number ofparticles obtained from the Cooper-Frye formula is not always positive-definite. Physicallynegative contributions of the Cooper-Frye formula are particles that stream backwards into thehydrodynamical region.We quantify the Cooper-Frye negative contributions in a coarse-grained transport approach,which allows to compare them to the actual number of underlying particles crossing thetransition hypersurface. It is found that the number of underlying inward crossings is muchsmaller than the one the Cooper-Frye formula gives under the assumption of equilibriumdistribution functions. The magnitude of Cooper-Frye negative contributions is also investigatedas a function of hadron mass, collision energy in the range E lab = 5 − A GeV, and collisioncentrality. The largest negative contributions we find are around 13% for the pion yield atmidrapidity at E lab = 20 A GeV collisions.
1. Introduction
In the hot and dense system of strongly interacting matter created in heavy ion collisions themean free path of the particles is much smaller than the size of the fireball. This fact togetherwith the assumption of fast thermal equilibration allows to apply relativistic hydrodynamicsfor the dynamical description of heavy ion collisions. At later times of the evolution and at itsedges the system is dilute, the mean free path is larger than system size, and hydrodynamicsis not applicable. An adequate description of the system in those circumstances is providedby kinetic (transport) equations, such as the Boltzmann equation or its modifications for thequantum case. State of the art simulations of heavy ion collisions couple hydrodynamics for theearly stage of the evolution to hadron transport for the late stage. Such approaches are calledhybrid approaches [1].The transition from hydrodynamics to transport (so-called particlization) is non-trivial,because hydrodynamics contains no microscopic information that is needed for transport.Currently, most of the models implement particlization in the following way. Hydrodynamicalequations are solved in the whole forward light cone, including those regions, wherehydrodynamics is not applicable. Then the particlization hypersurface is found from thehydrodynamical evolution. Usually a hypersurface of constant time, temperature or energy a r X i v : . [ nu c l - t h ] D ec ensity is taken. Particle distributions to be fed into the kinetic model are generated on thishypersurface according to the Cooper-Frye formula [2]: p dNd p = p µ dσ µ f ( p ) (1)Here dNd p is a spectrum of particles emerging from an element of the hypersurface, dσ µ is thenormal four-vector to the hypersurface, its length being equal to the area of hypersurface element.In ideal fluid calculations the distribution function f ( p ) is taken as an equilibrium Fermi or Bosedistribution: f ( p ) = (cid:104) exp (cid:16) p µ u µ − µT (cid:17) ± (cid:105) − . An advantage of the Cooper-Frye formula is that itrespects conservation laws. The disadvantage is that for space-like elements of the hypersurface( i.e. where dσ µ dσ µ <
0) there exist particle momenta p µ such that p µ dσ µ < dNd p < dNd p = (cid:90) σ p µ p dσ µ f ( p )Θ( p µ dσ µ ) + (cid:90) σ p µ p dσ µ f ( p )Θ( − p µ dσ µ ) . (2)The second term is called Cooper-Frye negative contributions and the first one - positive. Forthe applicability of the Cooper-Frye formula the negative contributions must be much smallerthan the positive ones. This is usually true for simulations of high energy collisions. Negativecontributions of around 9% relative to the positive ones were reported for RHIC top energy [4].At the same time negative contributions at E lab = 160 A GeV were found to be around 13%.This suggests that at lower collision energies negative contributions might become large andCooper-Frye formula will be inapplicable. Does it really happen? To answer this question wesystematically investigate the negative Cooper-Frye contributions differentially in rapidity andtransverse momentum against collision energy, centrality and particle species.Several ways to circumvent the problem of negative contributions were suggested (see [5]for a short summary). One practical way to do it in hybrid models would be to simultaneouslyneglect negative contributions and remove the particles from the transport calculation, if theyfly to the hydrodynamic region. If the distributions in hydrodynamics and cascade are the same,then conservation laws will be fulfilled. Such an approach has never been implemented. To checkif it is feasible, we compare negative contributions from the Cooper-Frye formula to distributionsof backscattered particles from the cascade. We take advantage of the coarse-grained approach,which allows to calculate both within one framework.
2. Methodology
We aim at two goals: systematic estimation of Cooper-Frye negative contributions andcomparing them to distributions of transport particles flying inwards to the hydrodynamicsregion. As a transport model we take Ultra-Relativistic Quantum Molecular Dynamics (UrQMD3.3p2) [3]. UrQMD allows to simulate heavy ion collisions as a sequence of elementary particlecollisions. Included processes are 2 to 2 scattering, resonance formation and decays, stringexcitation and fragmentation.We generate an ensemble of UrQMD Au+Au collision events and average them on arectangular grid to obtain the energy momentum tensor T µν and the baryon current j µ ineach cell. In each cell we find the Landau rest frame (LRF, a frame, where energy flow is zero: T iLRF = 0). We obtain the energy density (cid:15) LRF = T LRF , flow velocity u µ and baryon densityin the LRF n LRF = j µB u µ . Knowing (cid:15) LRF for each grid cell we construct the hypersurface of constant (cid:15)
LRF and find the normal vectors dσ µ for each piece of Σ. The latter is doneusing the Cornelius subroutine [4], that provides a continuous surface without holes and avoidsdouble counting of hypersurface pieces. The hypersurface Σ mimics the transition hypersurfacein hybrid models. When Σ is obtained we perform a Cooper-Frye calculation on it and compareto distributions of underlying UrQMD particles that cross Σ.We perform our calculations with time step ∆ t = 0.1 fm/c, grid spacing in the beam direction∆ z =0.3 fm, and grid spacings in transverse direction ∆ x = ∆ y = 1 fm. For collision energy E lab = 160 A GeV we take ∆ z =0.1 fm, and ∆ x = ∆ y = 0.3 fm. We have checked that forsuch a choice of grid spacing conservation laws on the surface are fulfilled with an accuracybetter than 1%. In other words (cid:82) Σ T µ dσ µ and total energy flowing out of the hypersurfacecalculated by particles differ by no more than 1%. To create a smooth hypersurface andobtain reproducible results we employ a Gaussian smearing procedure. For the constructionof the hypersurface every UrQMD particle with coordinates ( t p , x p , y p , z p ) and 4-momentum p µ is substituted by 300 marker particles with coordinates distributed with the probability density 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 ) − / . In this wayevery particle contributes to T µν and j µ not only of the cell, where it is located, but also tothe adjacent cells. We take as number of events N = 1500 and the Gaussian width σ = 1 fm.Choice of N , σ , grid spacing and sensitivity of results to these choices are discussed in [5].
3. Results
If the distribution of UrQMD particles is thermal on some closed hypersurface and the systemis in chemical equilibrium, then the Cooper-Frye formula should give results identical to explicitparticle counting. This would allow to compensate negative contributions by removing particlesfrom UrQMD, if they cross the hypersurface inwards. In such a treatment conservation laws onthe surface would be respected. We check, if this can be done on the hypersurface Σ of constantenergy density in Landau frame, (cid:15) = 0 . . π negative d N / dy y -3 -2 -1 1 2 3 π positive d N / dy y -3 -2 -1 1 2 3 Figure 1.
Cooper-Frye rapidity spectra for pions onΣ (red circles) are compared to distribution of UrQMDpions crossing Σ (blue crosses). Left panel: negativecontributions and inward crossings. Right panel: positivecontributions and outward crossings. Collision energy E lab = 40 A GeV, central collisions. Note the very differentscale of negative and positive contributions.
Cooper-Frye πK + ρNΔ E = 40 AGeV, b=0 fm ( d N - / dy ) / ( d N + / dy ) [ % ] y -3 -2 -1 1 2 3 Figure 2.
Rapidity distributionof the ratio of negative to pos-itive contributions for differenthadrons: pions (red circles), K + (blue crosses), ρ (grey bars), nu-cleons (magenta rectangles) and∆ (green triangles). Cooper-Fryecalculation in central Au+Au col-lisions at E lab = 40 A GeV.
When calculating the net number of pions passing through the surface, one finds that thenumber of pions in UrQMD is larger than in the equilibrated Cooper-Frye calculation [5]. Itight be possible to explain this as a sign of chemical non-equilibrium in UrQMD, but whenone looks at the positive and negative contributions to the pion distributions shown in Fig. 1,one sees that a difference in the pion density only is not sufficient to explain the differences inthe contributions. The positive contribution is much larger in UrQMD than in the Cooper-Fryecalculation, whereas the negative contribution depicts the opposite behaviour: in UrQMD it ismuch smaller than in the Cooper-Frye scenario. This kind of distributions may indicate that thecollective flow velocity of pions is much larger than the collective velocity of other particles [6, 7],or that the dissipative corrections are very large.Since neither negative contributions coincide with inward crossings, nor positive contributionscoincide with outward crossings, the above mentioned idea of compensating negativecontributions will not work in our case. Instead we concentrate on finding when negativecontributions play the most prominent role, and evaluate the ratio of negative contributionsto positive ones to estimate the error they bring into hybrid calculations. For that we varyhadron sort, collision energy and centrality. As a relevant variable we consider the ratio ofnegative to positive contributions integrated over the hypersurface, ( dN − /dy ) / ( dN + /dy ).From Fig. 2 one can see that negative contributions become smaller for larger particle mass.It is simple to understand this result in the rest frame of the fluid element where the surfaceis moving. If the hypersurface Σ moves inwards, as is usually the case in fluid-dynamicalcalculations, a particle must be faster than the surface to cross it inwards. For larger particlemass the average velocities of the thermal motion are smaller and the probability to catch up thehypersurface is also smaller. Therefore, to find maximal negative contributions it is sufficient toconsider pions only. For them we find out that if negative contributions are binned accordingto rapidity, then the largest contribution occurs at midrapidity.We show the negative to positive contribution ratio for the pion yield at midrapidity forcollision energies E lab = 5–160 A GeV in Fig. 3. The ratio in UrQMD calculation is much smallerthan in the equilibrated Cooper-Frye calculation at all collision energies, but the maximum liesin the region of 20 A GeV in both approaches. The value of the maximum is about 13% inCooper-Frye calculation, and about 4% in UrQMD calculation. (dN π- /dy)/(dN π+ /dy) @ |y| < 0.05Cooper-Fryeby particles [ % ] E lab [AGeV] Figure 3.
The ratio of negative to positivecontributions on the (cid:15) ( t, x, y, z ) = (cid:15) c = 0 . surface for pions at midrapidity. Redcircles depict the ratio in a Cooper-Frye calculationassuming thermal equilibrium, and blue crosses inexplicit calculation of UrQMD particles. π Cooper-Frye, b = 0 fmCooper-Frye, b = 6 fmCooper-Frye, b = 12 fmE = 40 AGeV, ε c = 0.3 GeV/fm ( d N - π / dy ) / ( d N + π / dy ) [ % ] y -3 -2 -1 1 2 3 Figure 4.
Rapidity distribution of the ratioof negative to positive contributions for pions inAu+Au collisions at E lab = 40 A GeV at variouscentralities: b = 0 (red circles), b = 6 fm (bluecrosses) and b = 12 fm (green rectangles). The dependency of negative to positive contribution ratio on collision energy is non-monotonous. This is not obvious, because several factors influence the ratio: the temperatureon the hypersurface, the relative velocities between the flow and the surface, and the relativemounts of volume and surface emission, i.e. emission from the time- and space-like parts ofthe surface. Larger temperature results in larger negative contributions because the thermalvelocities increase. Larger relative velocity leads to smaller negative contributions. Thelarger the relative amount of volume-emission, the smaller the negative contributions. Withincreasing collision energies the temperature saturates and thus the changes in the last twofactors make negative contributions fall with increasing collision energy. The decrease of negativecontributions at higher energy is predictable, because the relative amount of volume emission andrelative velocity between flow and surface increase with collision energy. However, the behaviourof negative contributions at lower energy is caused by the interplay of all three factors.We plot negative contributions for pions versus collision centrality in Fig. 4, and find that forperipheral collisions negative contributions are smaller than for central collisions. The relativeamount of surface and volume emission plays the most prominent role here. For peripheralcollisions volume emission dominates due to the short lifetime of the system and negativecontributions are small.After varying the collision energy and the centrality, we can conclude that in the worstscenario negative contributions can hardly exceed 15%. Changing the criterion of hypersurfaceto (cid:15) c = 0 . does not change this conclusion (see Ref. [5]).
4. Conclusions
We have investigated negative Cooper-Frye contributions and backscattering using a coarse-grained molecular dynamics approach. Au+Au collisions at E lab = 5–160 A GeV energies havebeen simulated using UrQMD, and a hypersurface Σ of constant Landau rest frame energydensity has been constructed. On this surface we have calculated two quantities: The ratio ofCooper-Frye negative to positive contributions ( r eq ), which assumes local thermal equilibrium,and the ratio of UrQMD particles crossing Σ inward to crossing Σ outward ( r neq ), which doesnot assume equilibrium.We found that at all collision energies r eq (cid:29) r neq . We explain this by a deviation of pions inUrQMD simulation from equilibrium. A non-monotonous dependency of r eq and r neq on collisionenergy was found with a maximum at 10-20 A GeV, maximal r eq being around 13%. The sizeof the negative contributions is a result of an interplay of several factors: the temperature onthe hypersurface, the relative velocities between flow and surface, and the relative amounts ofvolume and surface emission. Acknowledgments
This work was supported by the Helmholtz International Center for the Facility for Antiprotonand Ion Research (HIC for FAIR) within the framework of the Landes-Offensive zur EntwicklungWissenschaftlich-Oekonomischer Exzellenz (LOEWE) program launched by the State of Hesse.DO and HP acknowledge funding of a Helmholtz Young Investigator Group VH-NG-822 fromthe Helmholtz Association and GSI, and PH by BMBF under contract no. 06FY9092. DOacknowledges support of HGS-HIRe. Computational resources have been provided by the Centerfor Scientific Computing (CSC) at the Goethe University of Frankfurt.
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