Far From Equilibrium Hydrodynamics and the Beam Energy Scan
FFar From Equilibrium Hydrodynamics and the BeamEnergy Scan
Travis Dore
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Emma McLaughlin
Department of Physics, Columbia University, 538 West 120th Street, New York, NY 10027,USA
Jacquelyn Noronha-Hostler
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Abstract.
The existence of hydrodynamic attractors in rapidly expanding relativistic systemshas shed light on the success of relativistic hydrodynamics in describing heavy-ion collisions atzero chemical potential. As the search for the QCD critical point continues, it is important toinvestigate how out of equilibrium effects influence the trajectories on the QCD phase diagram.In this proceedings, we study a Bjorken expanding hydrodynamic system based on DMNRequations of motion with initial out of equilibrium effects and finite chemical potential in asystem with a critical point. We find that the initial conditions are not unique for a specificfreeze-out point, but rather the system can evolve to the same final state freeze-out point witha wide range of initial baryon chemical potential, µ B . For the same initial energy density andbaryon density, depending on how far out of equilibrium the system begins, the initial µ B canvary by ∆ µ B ∼
350 MeV. Our results indicate that knowledge of the out-of-equilibrium effectsin the initial state provide vital information that influences the search for the QCD criticalpoint.
1. Introduction
Determining the initial conditions for heavy-ion collisions is an ongoing problem in the field,and no less so as the Beam Energy Scan program at RHIC continues its search for the QCDcritical point [1]. Studies of hydrodynamic attractors [2, 3, 4, 5, 6, 7, 6, 8, 9, 10, 11, 12, 13, 14,15, 16, 17, 18, 19, 20, 21, 22, 23, 24] provide evidence that the effectiveness of hydrodynamicsin describing heavy-ion data may come from the fact that the system quickly forgets about theinitial conditions and collapses on to a universal attractor on short time scales. At LHC energies,where µ B ∼
0, the story is simpler as one does not need to worry about a multi-dimensionalphase diagram. However, at lower beam energies where baryon stopping is relevant, as thesystem evolves it will traverse a path in the T − µ B plane. If the system were to evolve entirely inequilibrium, it would simply follow isentropes (entropy over baryon number would be conservedi.e. S/N B = const ) in the phase diagram (see a number of recent studies relying on isentropes[25, 26, 27, 28, 29, 30]). However, the system evolves out of equilibrium, and depending on a r X i v : . [ nu c l - t h ] J un ow far the system begins away from equilibrium, one would expect the deviations from theisentropic trajectories to be large.Here, we study the effects of initializing the full T µν , specifically the initial shear stresstensor π µν and initial bulk pressure Π, in an out of equilibrium state on the trajectories throughthe phase diagram. Previous studies of the full T µν on collective flow observables were onlyperformed only at µ B = 0 [31, 32, 33] and, in this context, new causality conditions derived in[34] will be useful to constrain the far from equilibrium behavior. Here we run different initialconditions covering a wide range of initial shear and bulk viscous effects with a non-conformalequation of state that has a parametrized critical point. We find that far-from-equilibrium initialconditions can lead to significant deviations from isentropes, which implies connecting the initialstate and final state can be significantly complicated at the beam energy scan. We also studywhether or not hydrodynamic attractors persist in the presence of shear and bulk viscous termsthat depend on both temperature and baryon chemical potential.
2. Hydrodynamic Model
Relativistic viscous hydrodynamics has enjoyed much success in predicting many observables inthe low transverse momentum sector of heavy-ion collisions. As the Beam Energy Scan programat RHIC continues, a number of new developments concerning the hydrodynamic modeling ofheavy-ion collisions have been performed to study a baryon-rich environment [35, 36, 37, 38]and transport coefficients that are also dependent on { T, µ B } [39, 40, 41, 42, 43, 43, 44, 45].However, in order to find the critical point in the QCD phase diagram (if it exists) it will benecessary to have a thorough understanding of how a dynamical evolution past the critical pointcan influence final state observables [46]. Moreover, an understanding of how out of equilibriumeffects may influence criticality as well as system dynamics given two thermodynamic degreesof freedom, T and µ B , is essential.For the purpose of this proceedings, we will discuss only results obtained using DNMRhydrodynamic equations of motion [47]. As a starting point for future work, we use a simplifiedBjorken expanding model [48] to obtain qualitative results. To our knowledge, this is the firstwork to study out of equilibrium effects on trajectories in the phase diagram, especially inregards to their impact on the search for the critical point. To that end, a simplified Bjorkenpicture is ideal for obtaining a basic understanding of the sought after effects. For a Bjorkenexpanding fluid (where τ = √ t − z is the propertime) the DNMR equations of motion become˙ (cid:15) = − τ (cid:2) e + p + Π − π ηη (cid:3) (1) τ π ˙ π ηη + π ηη = 1 τ (cid:20) η − π ηη ( δ ππ + τ ππ ) + λ π Π Π (cid:21) (2) τ Π ˙Π + Π = − τ (cid:18) ζ + δ ΠΠ Π + 23 λ Π π π ηη (cid:19) (3)˙ ρ = ρ τ (4)where (cid:15) is the energy density (with ˙ (cid:15) = d(cid:15)/dτ ), p is the equilibrium pressure, π ηη is the nonzerocomponent of the shear stress tensor, Π is the bulk pressure, ρ is the baryon density ( ρ isthe initial baryon density), η is the shear viscosity, ζ is the bulk viscosity, and the remaining a) (b) Figure 1: Fig (a) shows the time evolution of ηT /w for hydrodynamic runs with different initialconditions. Fig (b) shows the time evolution of ζT /w for hydrodynamic runs with differentinitial conditions having some critical point sensitivity.transport coefficients are given by [36, 49] τ π = 5 η(cid:15) + p (5) τ Π = ζ e + p ) (cid:0) − c s (cid:1) (6) λ π Π = 65 τ π (7) δ ππ = 43 τ π (8) τ ππ = 107 τ π (9) λ Π π = 85 (cid:18) − c s (cid:19) τ Π (10) δ ΠΠ = 23 τ Π (11)where c s is the isentropic speed of sound. We use a temperature dependent shear viscosity( ηT /w where w = e + p is the enthalpy), given by an excluded volume calculation below thetransition temperature [50] (using the PDG16+ [51]) that is matched onto a QCD motivatedparameterization in the deconfined phase [52, 53]. The transition between the hadron gas phaseand the QGP phase is matched using a tanh function in order to ensure a smooth matching.Additionally, we ensure here that the transition line matches that in the equation of state (EOS).At µ B = 0 we ensure that the shear viscosity to entropy density ratio has a minimum of 0 . T η/s,min = 196 MeV, the µ B dependence of the minimum ηT /w is determined bythe change in ηT /w with µ B within the excluded volume HRG model. The minimum of ηT /w converges to the transition line from the EOS at the critical point. Further work will extendthis viscosity to the full 4 dimensional phase diagram of { T, µ B , µ S , µ Q } taking into accountstrangeness and electric charge chemical potentials.For the bulk viscosity, we use the parameterization ζTw = 92 π (cid:18) − c s (cid:19) (12)igure 2: The T − µ B trajectories of multiple hydrodynamic runs with different initial conditions,which all go through the same freeze-out region (green circle). The green line is the isentrope,for reference. For the isentrope away from the critical point, S/N B = 65, and for the other S/N B = 21.which has a maximum ζTw ∼ . c s allows the bulk viscosity to have some sensitivity to criticaldynamics, as it is expected to drop considerably at the critical point [27] while the bulk viscosityis expected to diverge [42].In this work we use an equation of state that captures effects of criticality at finite baryonchemical potential. To this end, we use the most up to date equation of state reconstructed fromLattice QCD results, which are then mapped to a parameterized 3D Ising Model [27]. Here weplace a critical point at ( T, µ B ) = (143 , α = 3, α = 93, ω = 1, and ρ = 2, which was a default EOS setting from [27]. We note that the current version of this EOSis only reliable up to µ B ∼
400 MeV. We, however, extend this out to µ B ∼
600 MeV in the hightemperature region. That being said, we find limitations due to the maximum µ B ∼
600 MeVwhen studying the critical point. Finally, at this time we assume µ S = µ Q = 0 but in futurestudies plan to incorporate the full 4D EOS in { T, µ B , µ S , µ Q } from [28] but we note that the3D Ising critical point has not yet been implemented in the 4D EOS.
3. Trajectories in the QCD Phase Diagram
In Fig. 2 we display the thermodynamic trajectories generated by hydrodynamics for a range ofdifferent initial conditions where the ratio between the dissipative stresses and the enthalpy χ ≡ π ηη / ( (cid:15) + p ) (13)Ω ≡ Π / ( (cid:15) + p ) (14)are varied between ± .
5. The initial energy density for the trajectories away from the criticalpoint is 7 . and 1 . for those that pass through the critical region. We place constraint on the trajectories produced by imposing that all trajectories must pass througha freeze-out region defined along an isentrope close to where one would expect the system tofreeze-out at given thermal fits [55, 56, 57, 28, 58, 59, 60]. The initial baryon density is varieduntil trajectories are found that pass through either side of the freeze-out region. The regionwas chosen sufficiently small to mimic the uncertainty in the freeze-out temperature and isrepresented in Fig. 2 by a small green circle which trajectories pass through. If it were true thatthe system evolved isentropically, then one could uniquely determine the path traversed by thesystem given its freeze-out temperature and chemical potential. This path would correspond tothe solid and thicker, green line (also shown). In this work, we argue that when consideringviscous corrections, the unique correspondence between a freeze-out state and a trajectory inthe QCD phase diagram no longer exists.It can be seen in Eq. 3 that shear and bulk out of equilibrium effects both contributelinearly for determining the dynamic behavior of (cid:15) , albeit with opposite sign (see Table 3 forthe different initial conditions used). Directly, one can notice that positive shear and negativebulk contributions work to slow down the rate at which energy density decreases. With this inmind, when looking at Fig. 2, it is consistent that those trajectories with the largest positiveshear and negative bulk initial contributions start out the flattest in temperature and thereforemust start at larger chemical potential in order to make it to the same freeze-out point as theother trajectories. We note that limitations in the EOS for producing data at large chemicalpotential ( >
600 MeV) prevented us from being able to explore the full range of initial positiveshear and negative bulk, as was done for trajectories away from the critical point at low µ B .Again in Fig. 2, one can see a spread in initial µ B of about 350 MeV and a spread in initialtemperature of about 10 MeV. In order to better constrain the trajectory path, it is importantto be able to calculate the initial conditions for the system (which include the initial values forthe dissipative stresses) precisely. Currently there exists some models for hydrodynamic initialconditions at both zero [61, 62, 32] and finite chemical potential [63, 62, 64]. However, the initialcondition models at finite µ B do not incorporate a fully initialized T µν such that we have noway to estimate how wide of fluctuations to anticipate in π µν and Π on an event-by-event basis.
4. Hydrodynamic Attractor
In recent studies, it has been suggested that much of the success of relativistic viscoushydrodynamics in describing the system with such little knowledge of the initial conditionscan be attributed to the existence of an attracting solution in the equations of motion of theviscous fields. In this work, we expand on this by studying a non-conformal hydrodynamicsystem with a finite chemical potential.A natural question to ask when using more realistic transport coefficients that depend ontemperature and chemical potential is whether or not the hydrodynamic attractor persists.Typically, one would expect that when scaling the time evolution of these quantities by theirrespective relaxation times of τ π and τ Π , these quantities would collapse onto a non-zero andnon-trivial attractor and then eventually evolve towards zero (i.e., equilibrium). However, pastwork has primarily only focused on shear viscosity, which then only has one characteristic timescale. Here we have two and a nontrivial equation of state. In this work, we vary the initialvalues χ and Ω between ± . χ approaches a constant (cid:54) = 0 at latetimes). The added help of the arrows pointing in the direction of the χ trajectory at the finaltime in Fig. 4a allows one to see that while the curves have not collapsed onto each other theydo appear to be converging to a single line. At infinitely large times, it is likely that the χ trajectories under this rescaling would also collapse on to each other. Interesting to note, is thepresence of weak attraction in the unscaled evolution, Fig. 6a. There has not yet been muchigure 3: Reference for identifying which lines correspond to which initial conditions. Not shownhere is the legend for χ = Ω = 0, which is represented by solid black curves in other figures.Note that not all combinations of χ and Ω are used, especially not in those trajectories thatpass through the critical region. (a) (b) Figure 4: Fig. (a) shows time evolution of χ rescaled by its relaxation time. The end arrowspoint along the instantaneous directional derivative. Fig. (b) shows time evolution of χ rescaledby the bulk relaxation time.evidence to support the existence of an attractor in the unscaled time evolution. One shouldalso note that the weak attraction seemingly present in Fig. 4b may simply be due to the factthat τ Π is of O (1).This explanation seems consistent with the fact that the attracting behavior for both χ andΩ in their unscaled evolution appears stronger than when their evolution is scaled by the bulkrelaxation time, τ Π . This seems especially noticeable in comparison between Figs. 5b and 6b.However, this notion goes against the intuition that the scaled time should be controlling theattracting behavior for its relevant quantity. Here we offer no solution to this slight paradox,but only point out its existence.Either way, Fig. 6b shows a clear non-trivial and non-zero attractor for Ω. Even as the systemgoes through the critical point, which the bulk viscosity is sensitive to via c s , it is about an orderof magnitude smaller than the shear viscosity. The implications are interesting given that thesystem exhibits memory effects of the initial state and never actually relaxes to equilibrium.While we are still missing critical fluctuations in this approach, it is interesting to see attractor-like behavior in the bulk viscosity near the critical point. Given that there is no reason to believeinitial viscous effects in heavy-ion collisions at BES energies should be small, theoretical models a) (b) Figure 5: Fig. (a) shows time evolution of Ω rescaled by the shear relaxation time. Fig (b)shows time evolution of Ω rescaled by its relaxation time. (a) (b)
Figure 6: Fig (a) shows time evolution of χ without rescaling. Fig (b) shows time evolution ofΩ without rescalingshould take far-from-equilibrium initial conditions into consideration.
5. Conclusions
Opening up of the new degree of freedom, µ B , in conjunction with exploring hydrodynamicnon-equilibrium effects, lead to system dynamics that must be more thoroughly understood.In connection with the Beam Energy Scan program at RHIC, there are many importantimplications. For instance, given an initial (cid:15) and ρ , widely different trajectories may be seenthroughout the phase diagram depending on how far-from-equilibrium the system begins. Forthese reasons, it is crucial to begin studying more deeply how to properly initialize the full T µν as input to hydrodynamics, compatible with event-by-event fluctuations, non-conformal systemdynamics, and a finite chemical potential. Ongoing efforts such as [65] make it possible to studydiffusive dynamics in the more heavily studied µ B = 0 regime by taking into consideration localfluctuations of the conserved charges. Moving forward, this seems to be a promising frameworkto begin asking the same kinds of questions asked here, but in a more realistic scenario. cknowledgements J.N.H. acknowledges support from the US-DOE Nuclear Science Grant No. de-sc0019175, theAlfred P. Sloan Foundation, and the Illinois Campus Cluster, a computing resource that isoperated by the Illinois Campus Cluster Program (ICCP) in conjunction with the NationalCenter for Supercomputing Applications (NCSA) and which is supported by funds from theUniversity of Illinois at Urbana-Champaign.
References [1] Bzdak A, Esumi S, Koch V, Liao J, Stephanov M and Xu N 2020
Phys. Rept.
Preprint )[2] Heller M P and Spalinski M 2015
Phys. Rev. Lett.
Preprint )[3] Buchel A, Heller M P and Noronha J 2016
Phys. Rev.
D94
Preprint )[4] Heller M P, Kurkela A, Spali´nski M and Svensson V 2018
Phys. Rev.
D97
Preprint )[5] Spali´nski M 2018
Phys. Lett.
B776
Preprint )[6] Romatschke P 2017
JHEP
079 (
Preprint )[7] Romatschke P 2018
Phys. Rev. Lett.
Preprint )[8] Behtash A, Cruz-Camacho C N and Martinez M 2018
Phys. Rev.
D97
Preprint )[9] Strickland M, Noronha J and Denicol G 2018
Phys. Rev.
D97
Preprint )[10] Denicol G S and Noronha J 2018
Phys. Rev.
D97
Preprint )[11] Blaizot J P and Yan L 2018
Phys. Lett.
B780
Preprint )[12] Casalderrey-Solana J, Gushterov N I and Meiring B 2018
JHEP
042 (
Preprint )[13] Heller M P and Svensson V 2018
Phys. Rev.
D98
Preprint )[14] Rougemont R, Critelli R and Noronha J 2018
Phys. Rev.
D98
Preprint )[15] Denicol G S and Noronha J 2019
Phys. Rev.
D99
Preprint )[16] Almaalol D and Strickland M 2018
Phys. Rev.
C97
Preprint )[17] Casalderrey-Solana J, Herzog C P and Meiring B 2019
JHEP
181 (
Preprint )[18] Behtash A, Cruz-Camacho C N, Kamata S and Martinez M 2019
Phys. Lett.
B797
Preprint )[19] Behtash A, Kamata S, Martinez M and Shi H 2019
Phys. Rev.
D99
Preprint )[20] Strickland M 2018
JHEP
128 (
Preprint )[21] Strickland M and Tantary U 2019
JHEP
069 (
Preprint )[22] Kurkela A, van der Schee W, Wiedemann U A and Wu B 2019 (
Preprint )[23] Jaiswal S, Chattopadhyay C, Jaiswal A, Pal S and Heinz U 2019
Phys. Rev.
C100
Preprint )[24] Denicol G S and Noronha J 2019 (
Preprint )[25] G¨unther J, Bellwied R, Borsanyi S, Fodor Z, Katz S D, Pasztor A and Ratti C 2017
EPJ Web Conf.
Phys. Rev.
C99
Preprint )[27] Parotto P, Bluhm M, Mroczek D, Nahrgang M, Noronha-Hostler J, Rajagopal K, Ratti C, Sch¨afer T andStephanov M 2018 (
Preprint )[28] Noronha-Hostler J, Parotto P, Ratti C and Stafford J M 2019
Phys. Rev.
C100
Preprint )[29] Monnai A, Schenke B and Shen C 2019
Phys. Rev.
C100
Preprint )[30] Stafford J M, Alba P, Bellwied R, Mantovani-Sarti V, Noronha-Hostler J, Parotto P, Portillo-Vazquez I andRatti C 2019 Determination of Chemical Freeze-out Parameters from Net-kaon Fluctuations at RHIC ( Preprint )[31] Liu J, Shen C and Heinz U 2015
Phys. Rev. C Preprint )[32] Kurkela A, Mazeliauskas A, Paquet J F, Schlichting S and Teaney D 2019
Phys. Rev. Lett.
Preprint )[33] Schenke B, Shen C and Tribedy P 2020
Phys. Lett. B
Preprint )[34] Bemfica F S, Disconzi M M, Hoang V, Noronha J and Radosz M 2020 (
Preprint )[35] Du L and Heinz U 2019 (
Preprint )[36] Denicol G S, Gale C, Jeon S, Monnai A, Schenke B and Shen C 2018
Phys. Rev.
C98
Preprint )[37] Batyuk P, Blaschke D, Bleicher M, Ivanov Yu B, Karpenko I, Malinina L, Merts S, Nahrgang M, PetersenH and Rogachevsky O 2018
EPJ Web Conf.
Preprint )[38] Fotakis J A, Greif M, Denicol G, Niemi H and Greiner C 2019 (
Preprint )39] Demir N and Bass S A 2009
Phys. Rev. Lett.
Preprint )[40] Denicol G S, Gale C, Jeon S and Noronha J 2013
Phys. Rev.
C88
Preprint )[41] Kadam G P and Mishra H 2014
Nucl. Phys.
A934
Preprint )[42] Monnai A, Mukherjee S and Yin Y 2017
Phys. Rev.
C95
Preprint )[43] Rougemont R, Critelli R, Noronha-Hostler J, Noronha J and Ratti C 2017
Phys. Rev.
D96
Preprint )[44] Auvinen J, Bernhard J E, Bass S A and Karpenko I 2018
Phys. Rev.
C97
Preprint )[45] Martinez M, Sch¨afer T and Skokov V 2019
Phys. Rev.
D100
Preprint )[46] Son D and Stephanov M 2004
Phys. Rev. D Preprint hep-ph/0401052 )[47] Denicol G S, Niemi H, Molnar E and Rischke D H 2012
Phys. Rev.
D85
Preprint )[48] Bjorken J 1983
Phys. Rev. D Comput. Phys. Commun.
Preprint )[50] Noronha-Hostler J, Noronha J and Greiner C 2012
Phys. Rev.
C86
Preprint )[51] Alba P et al.
Phys. Rev. D Preprint )[52] Christiansen N, Haas M, Pawlowski J M and Strodthoff N 2015
Phys. Rev. Lett.
Preprint )[53] Dubla A, Masciocchi S, Pawlowski J M, Schenke B, Shen C and Stachel J 2018
Nucl. Phys.
A979
Preprint )[54] Ryu S, Paquet J F, Shen C, Denicol G S, Schenke B, Jeon S and Gale C 2015
Phys. Rev. Lett.
Preprint )[55] Ejiri S, Karsch F, Laermann E and Schmidt C 2006
Phys. Rev.
D73
Preprint hep-lat/0512040 )[56] Schmid C (COL-NOTE = RBC, HotQCD) 2008 Lattice QCD thermodynamic results with improvedstaggered fermions
Proceedings, 3rd International Conference on Hard and Electromagnetic Probes ofHigh-Energy Nuclear Collisions (Hard Probes 2008): Illa da Toxa, Spain, June 8-14, 2008 ( Preprint )[57] Bellwied R, Borsanyi S, Fodor Z, Gunther J, Katz S D, Pasztor A, Ratti C and Szabo K K 2016
Nucl. Phys.
A956
Preprint )[58] Motornenko A, Steinheimer J, Vovchenko V, Schramm S and Stoecker H 2020
Phys. Rev.
C101
Preprint )[59] Bellwied R, Borsanyi S, Fodor Z, Guenther J N, Noronha-Hostler J, Parotto P, Pasztor A, Ratti C andStafford J M 2020
Phys. Rev.
D101
Preprint )[60] Alba P, Sarti V M, Noronha-Hostler J, Parotto P, Portillo-Vazquez I, Ratti C and Stafford J M 2020 (
Preprint )[61] Gale C, Jeon S, Schenke B, Tribedy P and Venugopalan R 2013
Phys. Rev. Lett.
Preprint )[62] Werner K 1993
Strings, pomerons, and the venus model of hadronic interactions at ultrarelativistic energies
Other thesis[63] Weil J et al.
Phys. Rev. C Preprint )[64] Shen C and Schenke B 2018
Phys. Rev.
C97
Preprint )[65] Martinez M, Sievert M D, Wertepny D E and Noronha-Hostler J 2019 (
Preprint1911.10272