What the collective flow excitation function can tell about the quark-gluon plasma
WWhat the collective flow excitation function can tell about thequark-gluon plasma
Jussi Auvinen ∗ Frankfurt Institute for Advanced Studies (FIAS),Ruth-Moufang-Strasse 1, D-60438 Frankfurt am Main, Germany
Jan Steinheimer † and Hannah Petersen ‡ Frankfurt Institute for Advanced Studies (FIAS),Ruth-Moufang-Strasse 1, D-60438 Frankfurt am Main, Germany andInstitut f¨ur Theoretische Physik, Goethe Universit¨at,Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany
Abstract
Recent STAR data from the RHIC beam energy scan (BES) show that the midrapidity slope dv /dy of the directed flow v of net-protons changes sign twice within the collision energy range 7.7- 39 GeV. To investigate this phenomenon, we study the collision energy dependence of v utilizing aBoltzmann + hydrodynamics hybrid model. Calculations with dynamically evolved initial and finalstate show no qualitative difference between an equation of state with a cross-over and one with afirst-order phase transition, in contrast to earlier pure fluid predictions. Furthermore, our analysisof the elliptic flow v shows that pre-equilibrium transport dynamics are partially compensatingfor the diminished elliptic flow production in the hydrodynamical phase at lower energies, whichleads to a qualitative agreement with STAR BES results in midcentral collisions. No compensationfrom transport is found in our model for integrated v , which decreases from ≈ .
02 at √ s NN = 27GeV to ≈ .
005 at √ s NN = 7 . PACS numbers: 24.10.Lx,24.10.Nz,25.75.Ld ∗ Electronic address: auvinen@fias.uni-frankfurt.de † Electronic address: steinheimer@fias.uni-frankfurt.de ‡ Electronic address: petersen@fias.uni-frankfurt.de a r X i v : . [ nu c l - t h ] A ug . INTRODUCTION In 2010, a beam energy scan program was launched at the Relativistic Heavy Ion Collider(RHIC) to study the features of the QCD phase diagram, in particular the location andthe order of the phase transition between the hadronic and QCD matter in the plane ofbaryochemical potential µ B and temperature T . The main observables are the coefficients v n of the Fourier expansion of the azimuthal angle distribution of final-state particle momenta,which are typically associated with collective flow.Fluid dynamical calculations have predicted that the slope of the directed flow v ofbaryons will turn negative and then positive again as a function of energy if there is afirst order phase transition between hadronic and QCD matter [1–4]. Qualitatively similarbehavior of the midrapidity slope of the net-proton directed flow, dv /dy , has been found bySTAR experiment in the RHIC beam energy scan, with a minimum in the energy interval √ s NN = 11 . − . v is one of the key observables supportingthe formation of a strongly coupled quark-gluon plasma (QGP) at the highest energies ofRHIC and the Large Hadron Collider (LHC). However, the measured differential ellipticflow v ( p T ) for charged hadrons remains nearly unchanged from the collision energy range √ s NN = 39 GeV down to 7.7 GeV [6], where the formation of hydrodynamically evolvingQCD matter is expected to be considerably diminished compared to top RHIC energies. Inaddition, the preliminary STAR data suggests that the magnitude of the triangular flow v remains constant at lower collision energies [7].We study the collision energy dependence of the collective flow in the RHIC BES rangewith a hybrid approach, where the non-equilibrium phases at the beginning and in theend of a heavy-ion collision event are described by a transport model, while a hydrodynamicdescription is used for the intermediate hot and dense stage and the phase transition betweenthe QGP and hadronic matter. This provides a consistent framework for investigating bothhigh-energy heavy ion collisions with negligible net-baryon density and notable quark-gluonplasma phase, and the collisions at smaller energies with finite net-baryon density, whereless QGP is expected to form. This approach should thus be ideal for beam energy scanstudies. 2 I. HYBRID MODEL
In this study, the transport + hydrodynamics hybrid model by Petersen et al. [8] isutilized. The transport model describing the initial and final state is the UltrarelativisticQuantum Molecular Dynamics (UrQMD) string / hadron cascade [9, 10]. The transitiontime to hydrodynamics is defined as the moment when the two colliding nuclei have passedthrough each other: t start = 2 R/ (cid:112) γ CM −
1, where R represents the nuclear radius and γ CM = 1 / (cid:112) − v CM is the Lorentz factor.The (3+1)-D ideal hydrodynamics evolution equations are solved with the SHASTA al-gorithm [11, 12]. The equation of state (EoS) which is utilized most of this study is fromSteinheimer et al. [13]. It is a combination of a chiral hadronic and a constituent quarkmodel and has the important feature of being applicable also at finite net-baryon densitiesfound at lower collision energies.The transition from hydro to transport (aka ”particlization”), is done when the energydensity (cid:15) reaches the critical value (cid:15) C = n(cid:15) , where (cid:15) = 146 MeV/fm is the nuclear groundstate energy density. In this study, the values n = 2 and n = 4 are used. The particledistributions are generated according to the Cooper-Frye formula from the iso-energy densityhypersurface, which is constructed using the Cornelius hypersurface finder [14]. The finalrescatterings and decays of these particles are then computed in the UrQMD. III. RESULTS
To investigate the sensitivity of the directed flow v ( y ) = (cid:104) p x p T (cid:105) y i = y on the order of thephase transition, we run the simulations with a first-order phase transition ”Bag model”EoS as an alternative to the above described chiral model EoS which has a cross-over phasetransition. The transition from fluid dynamics back to transport happens on an iso-energydensity (cid:15) C = 4 (cid:15) ≈ . / fm hypersurface.To emulate the earlier fluid calculations, we first utilize a cold nuclear matter initializa-tion, where the colliding nuclei are represented by two distributions of energy and baryondensity, which respect boosted Woods-Saxon profiles with a central density of saturatednuclear matter ρ ≈ .
16 fm − . The starting point of the simulation is just before the twonuclei first make contact; in the early stage of the collision the kinetic energy of the nuclei is3hen transformed into large local densities. Figures 1a and 1b show the difference in dv /dy between the two equations of state with a cold nuclear matter initialization and the UrQMDafterburner for Au+Au collisions at impact parameter b = 8 fm. The predicted minimumin dv /dy as a function of √ s NN with a first-order phase transition is clearly observed whenusing isochronous particlization condition; however, the difference between the two equa-tions of state diminishes greatly when using iso-energy density fluid-to-particles switchingcondition instead.Figure 1c shows the result of the full hybrid simulation with the initial non-equilibriumtransport phase for the energy dependence of midrapidity slopes of proton and antiproton v . The directed flow was calculated using events with impact parameter b = 4 . − . − v as a signal of the first-order phase transition.Some possible sources for the difference between the model and the experimental data arethe momentum transfer between the spectator particles and the fireball, which is not ac-counted for in this study, and the method used to determine the event plane, as here v wascalculated with respect to the reaction plane of the simulation. These uncertainties needfurther investigation before drawing definite conclusions.Utilizing the hybrid model with the crossover EoS, the flow coefficients v and v arecalculated from the particle momentum distributions using the event plane method [18, 19].Figure 2a shows the elliptic flow v produced in Au+Au -collisions, integrated over the p T range 0.2 - 2 GeV, compared with the STAR data for the (0-5)%, (20-30)% and (30-40)%centrality classes. In the model these are respectively represented by the impact parameterintervals b = 0 − . b = 6 . − . b = 8 . − . (cid:15) C = 2 (cid:15) is used here, as this value has been found to give a reasonable agreement withthe experimental data for particle m T spectra at midrapidity | y | < . E lab = 40 AGeV to √ s NN = 200 GeV [21, 22]. Figure 2b demonstrates the magnitudeof v at three different times: just before the hydrodynamical evolution, right after theparticlization, and the final result after the hadronic rescatterings have been performed inthe UrQMD model.In the impact parameter range b = 8 . − . ProtonsBM χ -over Center of Mass Energy √ s NN [GeV] d v / d y | y = IC-hypersurface a) IE-hypersurface
ProtonsBM χ -over Center of Mass Energy √ s NN [GeV] d v / d y | y = b) FIG. 1: a) and b) Slope of v of protons around midrapidity | y | < . b = 8 fm. c) Midrapidity slope dv /dy of protons (solid symbols)and anti-protons (open symbols) for impact parameter range b = 4 . − .
10% on the final result. The hydrodynamics produce very little elliptic flow at √ s NN ≤ . v below √ s NN = 10 GeV is in practice completely produced by the transportdynamics (resonance formations and decays, string excitations and fragmentation). Thisinitial transport gains importance at lower energies due to the prolonged pre-equilibriumphase. On the other hand, above √ s NN = 19 . v .The simulation results overshoot the experimental data for all collision energies. Thissuggests that the viscous corrections should be included – indeed, good results have alreadybeen achieved using similar hybrid approach with viscous hydro [23]. In the most centralcollisions below √ s NN = 11 . v has weaker collision energy dependence than one would have naively5 n t e g r a t e d v √s NN [GeV]
10 100
STAR v {EP}( 0- 5)% Hybridv {EP}b=0- 3.4fmSTAR v {EP}(20- 30)% Hybridv {EP}b=6.7- 8.2fmSTAR v {EP}(30- 40)% Hybridv {EP}b=8.2- 9.4fm Charged hadrons|η| < 1.00.2 GeV < p T < 2.0 GeV a) Charged hadrons, b =8.2 - 9.4 fm I n t e g r a t e d v √s NN [GeV]
10 100
Before hydroAfter hydroAfter hadronic rescattering
Eventplaneanalysis b) I n t e g r a t e d v −0.00500.0050.010.0150.020.025 √s NN [GeV]
10 100
Hybrid v {EP} b = 0 - 3.4 fmHybrid v {EP} b = 6.7 - 8.2 fmCharged hadrons|η| < 1.00.2 GeV < p T < 2.0 GeV c) FIG. 2: a) Integrated v at midrapidity | η | < . √ s NN = 7 . −
200 GeV at three impact parameter ranges, compared to the STAR data [6, 20](stars). b) Integrated v for impact parameter range b = 8 . − . √ s NN = 5 −
200 GeV, atthe beginning of hydrodynamical evolution (diamonds), immediately after particlization (squares)and after the full simulation (circles). c) Integrated v at midrapidity | η | < . b = 0 − . b = 6 . − . √ s NN = 5 −
200 GeV. Plots from [21]. expected. To study this phenomenon further, we do the same analysis for the triangularflow v , which originates purely from the event-by-event variations in the initial spatialconfiguration of the colliding nucleons.As illustrated by Figure 2c, the p T -integrated v increases from ≈ .
01 to above 0.015with increasing collision energy in the most central collisions. However, in midcentrality b = 6 . − . ≈ √ s NN = 5 GeV to the value of ≈ . √ s NN = 27 GeV, after which the magnitude remains constant. The energy dependenceof v in midcentral collisions qualitatively resembles the hydrodynamically produced v inFigure 2b; the viscous medium described by transport smears the anisotropies in the initialenergy density profile instead of converting them into momentum anisotropy, and is thusunable to compensate for the lack of ideal fluid described by hydrodynamics. This makes v the clearer signal of the presence of low-viscous medium. IV. CONCLUSIONS
Contrary to the earlier fluid calculations, we have found no difference between an equationof state with a first order phase transition and one with a cross-over phase transition forthe midrapidity slope of the directed flow v , when utilizing the full hybrid model withan iso-energy density switching criterion between hydrodynamics and final transport. Thus6 v /dy cannot currently be considered as a good signal for the existence of a first-order phasetransition. However, there is currently a notable discrepancy between the model results andthe experimental data which necessitates further investigation.We have demonstrated that a hybrid transport + hydrodynamics approach can qualita-tively reproduce the experimentally observed behavior of v as a function of collision energy √ s NN . While the v production by hydrodynamics is diminished at lower collision energies,this is partially compensated by the pre-equilibrium transport dynamics. Same does notapply to triangular flow v , which decreases considerably faster, reaching zero in midcentralcollisions at √ s NN = 5 GeV. Thus v is the better signal for the formation of quark-gluonplasma in heavy ion collisions. V. ACKNOWLEDGEMENTS
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. ac-knowledge funding by the Helmholtz Association and GSI through the Helmholtz YoungInvestigator Grant No. VH-NG-822. The computational resources were provided by theLOEWE Frankfurt Center for Scientific Computing (LOEWE-CSC). [1] D. H. Rischke, Y. P¨urs¨un, J. A. Maruhn, H. St¨ocker and W. Greiner, Heavy Ion Phys. 1(1995) 309.[2] L. P. Csernai and D. R¨ohrich, Phys. Lett. B 458 (1999) 454.[3] J. Brachmann et al. , Phys. Rev. C 61 (2000) 024909.[4] H. St¨ocker, Nucl. Phys. A 750 (2005) 121.[5] L. Adamczyk et al. [STAR Collaboration], arXiv:1401.3043 [nucl-ex].[6] L. Adamczyk et al. [STAR Collaboration], Phys. Rev. C 86 (2012) 054908.[7] Y. Pandit [STAR Collaboration] Talk at Quark Matter 2012.[8] H. Petersen, J. Steinheimer, G. Burau, M. Bleicher and H. St¨ocker, Phys. Rev. C 78 (2008)044901.[9] S. A. Bass, M. Belkacem, M. Bleicher, M. Brandstetter, L. Bravina, C. Ernst, L. Gerland and . Hofmann et al. , Prog. Part. Nucl. Phys. 41 (1998) 255.[10] M. Bleicher, E. Zabrodin, C. Spieles, S. A. Bass, C. Ernst, S. Soff, L. Bravina and M. Belkacem et al. , J. Phys. G 25 (1999) 1859.[11] D. H. Rischke, S. Bernard and J. A. Maruhn, Nucl. Phys. A 595 (1995) 346.[12] D. H. Rischke, Y. Pursun and J. A. Maruhn, Nucl. Phys. A 595 (1995) 383 [Erratum Nucl.Phys. A 596 (1996) 717].[13] J. Steinheimer, S. Schramm and H. St¨ocker, Phys. Rev. C 84 (2011) 045208.[14] P. Huovinen and H. Petersen, Eur. Phys. J. A 48 (2012) 171.[15] C. Alt et al. [NA49 Collaboration], Phys. Rev. C 68 (2003) 034903.[16] H. Liu et al. [E895 Collaboration], Phys. Rev. Lett. 84 (2000) 5488.[17] J. Steinheimer, J. Auvinen, H. Petersen, M. Bleicher and H. St¨ocker, Phys. Rev. C 89 (2014)054913.[18] A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C 58 (1998) 1671.[19] J. Y. Ollitrault, Preprint nucl-ex/9711003 (1997).[20] J. Adams et al. [STAR Collaboration], Phys. Rev. C 72 (2005) 014904.[21] J. Auvinen and H. Petersen, Phys. Rev. C 88 (2013) 064908.[22] J. Auvinen and H. Petersen, PoS CPOD 2013 (2013) 034.[23] I. .A. Karpenko, M. Bleicher, P. Huovinen and H. Petersen, J. Phys. Conf. Ser. 503 (2014)012040.[STAR Collaboration], Phys. Rev. C 72 (2005) 014904.[21] J. Auvinen and H. Petersen, Phys. Rev. C 88 (2013) 064908.[22] J. Auvinen and H. Petersen, PoS CPOD 2013 (2013) 034.[23] I. .A. Karpenko, M. Bleicher, P. Huovinen and H. Petersen, J. Phys. Conf. Ser. 503 (2014)012040.