Beam energy scan using a viscous hydro+cascade model
aa r X i v : . [ nu c l - t h ] O c t Beam energy scan using a viscous hydro+cascademodel
Iu.A. Karpenko , , M. Bleicher , , P. Huovinen and H. Petersen Frankfurt Institute for Advanced Studies, Ruth-Moufang-Straße 1, 60438 Frankfurt amMain, Germany Bogolyubov Institute for Theoretical Physics, 14-b, Metrolohichna str., 03680 Kiev, Ukraine Institute for Theoretical Physics, Johann Wolfgang Goethe Universit¨at, Max-von-Laue-Str.1, 60438 Frankfurt am Main, GermanyE-mail: [email protected]
Abstract.
Following the experimental program at BNL RHIC, we perform a similar “energyscan” using 3+1D viscous hydrodynamics coupled to the UrQMD hadron cascade, and studythe collision energy dependence of pion and kaon rapidity distributions and m T -spectra, as wellas charged hadron elliptic flow. To this aim the equation of state for finite baryon density froma Chiral model coupled to the Polyakov loop is employed for hydrodynamic stage. 3D initialconditions from UrQMD are used to study gradual deviation from boost-invariant scaling flow.We find that the inclusion of shear viscosity in the hydrodynamic stage of evolution consistentlyimproves the description of the data for Pb-Pb collisions at CERN SPS, as well as of the ellipticflow measurements for Au-Au collisions in the Beam Energy Scan (BES) program at BNL RHIC.The suggested value of shear viscosity is η/s ≥ . √ s NN = 6 . . . .
39 A GeV.
1. Introduction
During the recent years there has been considerable progress in modelling of bulk matterdynamics in ultrarelativistic heavy ion collisions at BNL Relativistic Heavy Ion Collider (RHIC)and CERN Large Hadron Collider (LHC). The state-of-the-art theoretical descriptions arebased on viscous hydrodynamics coupled to a hadron cascade. They successfully describe bulkobservables such as radial flow, elliptic and higher order flow harmonics [1], femtoscopy [2]etc. The idea of the present study is to apply a hybrid (or hydro+cascade) model to heavyion collisions at lower collision energies to study how well the model can describe the existingdata from CERN Super Proton Synchrotron (SPS) and recent results from Beam Energy Scanprogram at RHIC.
2. Model description
The lower the collision energy, the worse the approximation of boost-invariant scaling flow is.Thus it is important to employ initial conditions with non-trivial rapidity dependence. We usethe Ultrarelativistic Quantum Molecular Dynamics (UrQMD) model [3] for the description ofinitial stage dynamics. The two nuclei are initialized according to Wood-Saxon distributions andthe binary interactions are taken into account until a hypersurface at constant τ = √ t − z .We average over many UrQMD events to get smooth distribution of particles. The energyand momentum of particles is then converted to energy and momentum densities of the fluid.n addition to energy/momentum densities, initial baryon and charge densities are non-zeroand obtained from UrQMD, evolved in the hydro stage and accounted for in the particlizationprocedure. On the other hand, net strangeness density is set to zero.The hydrodynamic equations are solved in Milne ( τ − η ) coordinates. The starting time ofhydrodynamic evolution is chosen to be τ = 2 R/ p ( √ s/ m N ) −
1, where R is a radius ofnucleus and m N is a nucleon mass. This corresponds to the time when the two nuclei havepassed through each other.For beam energy scan, an equation of state (EoS) for finite baryon density must be used. Weemploy the Chiral model based EoS [5], which features correct asymptotic degrees of freedom,i.e. quarks and gluons at high temperature and hadrons in the low-temperature limits, crossover-type transition between confined and deconfined matter for all values of µ B and qualitativelyagrees with lattice QCD data at µ B = 0.The hydrodynamic code used solves the equations of relativistic viscous hydrodynamics inIsrael-Stewart framework [4]. In particular we solve the following equations for the shear stresstensor: h u γ ∂ ; γ π µν i = − π µν − π µν NS τ π − π µν ∂ ; γ u γ (1)where ∂ ; γ denotes covariant derivative and the brackets h A µν i denote the symmetric, tracelessand orthogonal to u µ part of A µν . For the purpose of the current study we consider onlythe effects of shear viscosity, fixing bulk viscosity to zero, ζ/s = 0. We do not include thebaryon/electric charge diffusion either. For viscous hydro simulations, we initialize the shearstress tensor to zero. The relaxation time for shear, τ π , is taken as τ π = 3 η/ ( sT ).The transition from fluid to particle description (so-called particlization) is made at theconstant energy density ǫ sw = 0 . when the medium has already hadronized (as inprevious studies at √ s NN = 200 A GeV RHIC and LHC energies [2]). Cornelius subroutine [6]is used to calculate the 3-volume elements dσ µ of the transition hypersurface. It is importantto note that while we use fixed energy density as transition criterion for all collision energies,the value of net baryon density on this surface is non-uniform and its average increases withdecreasing collision energy. This corresponds to the increase of average temperature and decreaseof baryon chemical potential with increasing collision energy, resembling the results for thecollision energy dependence of chemical freezeout parameters from thermal model studies [7].Also, since Chiral EoS deviates from free hadron-resonance gas (HRG) in hadronic phase, weswitch to free HRG EoS when sampling the particles according to Cooper-Frye prescription.We recalculate the energy density, pressure, flow velocity and corresponding thermodynamicalquantities from energy-momentum tensor using free HRG EoS, and employ them when sampling.Finally, we use the same corrections to the local equilibrium distribution functions for all hadronspecies: f i ( x, p ) = f i, eq ( x, p ) (cid:20) ∓ f i, eq ) p µ p ν π µν T ( ǫ + p ) (cid:21) (2)The scatterings and decays happening after particlization are then treated with UrQMDcode.
3. Results
First we simulate Pb-Pb collisions at energies E lab = 158 , , ,
30 and 20 A GeV (i.e. √ s NN = 17 . , . . . , . m T and dN/dy distributions with the data fromNA49 collaboration. Note that the only variable parameter is the shear viscosity to entropydensity ratio η/s in hydro stage, with the other parameters fixed as described above.With a given criterion for fluid to particle transition, the duration of hydro phase decreasesfrom about 6.5 fm/c (at √ s NN = 17 . √ s NN = 6 . √ s NN = 6 . y-4 -2 0 2 4 d N / d y ideal + UrQMD/S=0.1 + UrQMD η /S=0.2 + UrQMD η - π NA49 NA49 K+NA49 K- =160 A GeV lab E y-5 -4 -3 -2 -1 0 1 2 3 4 d N / d y ideal + UrQMD/S=0.1 + UrQMD η /S=0.2 + UrQMD η - π NA49, 40GeV NA49, 40GeV, K+NA49, 40GeV, K- =40 A GeV lab E Figure 1.
Rapidity distributions for π − , K + and K − for Pb-Pb collisions at E lab = 158 and40 A GeV (corresponding to √ s NN = 17 . . dN/dy profile narrower. The longitudinal expansion is weaker, andthe expansion tends to be more spherical, as seen in the comparison of the p T -spectra, Fig. 2: p T -spectra become flatter, which is due to stronger transverse expansion and larger radial flow.Next we calculate charged hadron elliptic flow at collision energies √ s NN = 7 . ,
27 and39 A GeV to compare with the results from RHIC BES, Fig. 3. Elliptic flow for the case of η/s = 0 overestimates the data, while choosing η/s = 0 . η/s = 0 , . .
2. So one can conclude that a consistent description of v , dN/dy , and p T -spectra requires avalue of η/s which is somewhat larger than 0 .
2, especially for lower energy points. The η/s = 0 . v ( p T ) almost independent of the collision energy.We conclude that the shear viscosity in hydrodynamic phase is a key component for a hybridmodel to better describe the data in the low energy region. The suggested value of the effectiveshear viscosity is η/s ≥ . η/s ≈ .
08 [11] obtained for √ s NN = 200 GeV RHIC energy and Monte Carlo Glauber initial state. Acknowledgements
IK and HP acknowledge the financial support by the ExtreMe Matter Institute EMMI andHessian LOEWE initiative. HP acknowledges funding by the Helmholtz Young InvestigatorGroup VH-NG-822. Computational resources have been provided by the Center for ScientificComputing (CSC) at the Goethe-University of Frankfurt.
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