Far-from-equilibrium search for the QCD critical point
FFar-from-equilibrium search for the QCD critical point
Travis Dore and Jacquelyn Noronha-Hostler
Illinois Center for Advanced Studies of the Universe, Department of Physics,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 (Dated: July 31, 2020)Initial conditions for relativistic heavy-ion collisions may be far from equilibrium (i.e. there arelarge initial contributions from the shear stress tensor and bulk pressure) but it is expected thaton very short time scales the dynamics converge to a universal attractor that defines hydrodynamicbehavior. Thus far, studies of this nature have only considered an idealized situation at LHCenergies (high temperatures T and vanishing baryon chemical potential µ B = 0) but, in this work,we investigate for the first time how far-from-equilibrium effects may influence experimentally drivensearches for the Quantum Chromodynamic critical point at RHIC. We find that the path to thecritical point is heavily influenced by far from equilibrium initial conditions where viscous effectslead to dramatically different { T, µ B } trajectories through the QCD phase diagram. We comparehydrodynamic equations of motion with shear and bulk coupled together at finite µ B for both DNMRand phenomenological Israel-Stewart equations of motion and discuss their influence on potentialattractors at finite µ B and their corresponding { T, µ B } trajectories. I. INTRODUCTION
One of the major thrusts of the nuclear physics commu-nity is to map out the phase diagram of Quantum Chro-modynamics (QCD) - specifically, the transition betweena hadron gas and a deconfined state of matter composedof strongly interacting quarks and gluons, known as theQuark-Gluon Plasma (QGP). While it is known from firstprinciple Lattice QCD calculations [1–3] that a cross-overphase transition existed in the early universe and in high-energy nuclear collisions, only conjectures and effectivemodels provide indications that a real phase transition(i.e. first or second order) may exist at large baryonchemical potentials [4–13]. This phase transition wouldbe separated from the cross-over by a critical point. Nu-clear physicists are searching for evidence of such a crit-ical point in heavy-ion collisions and astrophysicists aresearching at much lower temperatures and larger baryondensities for evidence of phase transitions in neutron starmergers [13–20].The crucial observable to search for the QCD criticalpoint is the study of susceptibilities of baryon number (i.enet-proton fluctuations, which are measured by STAR[21] and HADES [22]) because higher-order susceptibil-ities are increasingly sensitive to the correlation lengthand are, thus, expected to diverge at the critical point[23]. However, direct comparisons to experimental dataare complicated because of finite volume, lifetime, sizeeffects, and acceptance cuts [24–32]. Therefore, the besttool to search for the QCD critical point would be event-by-event relativistic viscous hydrodynamics that includesthree conserved charges: baryon number, strangeness,and electric charge and that also incorporates stochasticfluctuations at the critical point. In such a fully dynami- cal framework, one could take into account all acceptancecuts and finite volume/size/lifetime effects. While a largenumber of theoretical efforts are underway to create sucha model [33–49], no such framework is currently at one’sdisposal. Many of the needed advancements are outlinedin [50]. In fact, even very simplistic studies of this baryondense region are still in their infancy and have not gonethrough the same rigorous studies to constrain initial con-ditions [51], the equation of state [52–55], and transportcoefficients [56–58] that have already been performed atthe µ B ∼ µ B ∼
0, a large amount of uncertainty re-mains when describing the initial state shortly after twoheavy-ions collide and only more recently have theorists[59–88] begun to systematically study the effects of far-from-equilibrium behavior at µ B ∼
0. Looking towardsthe baryon dense region, there has not yet been a singlestudy of the influence of a far-from-equilibrium initialstate on the search for the critical point and, in fact,most hydrodynamical models have assumed only idealhydrodynamic equations of motion [89–92] with just ahandful of models that incorporate viscosity or diffusionin the last couple of years [42, 43, 93–97]. In the baryondense region, we are unaware of any initial conditionsthat include an initialized shear stress tensor or bulk pres-sure (although this may be possible using SMASH [98],URQMD [99], or NEXUS [100] but these are currentlycoupled to ideal hydrodynamic models). Therefore, thereis no real understanding of how far-from-equilibrium ini-tial conditions would influence the ability of differentbeam energies to approach the QCD critical point (mus-ings that it may affect the search for the critical pointcan be found in [101]).While astrophysical searches for a first order phase a r X i v : . [ nu c l - t h ] J u l transition also utilize relativistic hydrodynamics [19, 20,102] (in this context coupled to general relativity), thecurrent models do not incorporate shear and bulk viscosi-ties, nor do they have diffusion currents due to conservedcharges (such as baryon number). Current efforts areunderway to incorporate bulk viscosity into such models[103–108]. We note that if the initial contribution frombulk viscosity is large immediately after the two neutronstars collided, similar issues when determining their tra-jectories through the QCD phase diagram will arise.Typically, in most studies of large baryon density ef-fects in heavy-ion collisions there is an underlying as-sumption that the QGP is a nearly perfect fluid so thatthere is almost no entropy production. If one assumesthat entropy is not produced at all, then the ratio of to-tal entropy to baryon number ( S/N B ) is fixed through-out the collision, the subsequent expansion, and cool-ing throughout the phase diagram . These trajectories,known as isentropes, have been studied in a number ofrecent papers [110–115] and are used extensively to un-derstand equilibrium properties of QCD at large baryondensities. However, since hydrodynamic models requireboth shear and bulk viscosity to reproduce experimen-tal data, entropy production must occur and deviationsfrom the isentropic trajectories are expected. This maybe exacerbated at large µ B since a number of studieshave suggested the viscosity could increase in this region[35, 36, 116–119]. Thus, large deviations from isentropictrajectories may be possible, especially at the criticalpoint.In this paper, we perform the first study of the effectsof far-from-equilibrium initial conditions (arising from afully initialized shear stress tensor and bulk pressure) onthe search for the QCD critical point. We find large de-viations from isentropic trajectories, especially near thecritical point. We show this in both Israel-Stewart andDNMR hydrodynamic equations of motion and elabo-rate on difficulties in ensuring positive entropy produc-tion throughout the evolution. Furthermore, we findthat Israel-Stewart and DNMR do not traverse the QCDphase diagram in the same manner, which implies thatthe specific way such approaches implement second or-der corrections matters for the evolution of the baryonrich fluid. This means that special attention must bepaid when selecting the hydrodynamic equations of mo-tion in the presence of phase transitions. The sensitivityto the initial conditions indicates that they play a crucialrole in determining the trajectory of the QGP throughthe QCD phase diagram and significant efforts must bemade to constrain initial conditions before a fully dy-namical model can properly describe heavy-ion collisionsat finite baryon densities. Furthermore, if there are sig-nificant event-by-event fluctuations in the initial condi-tions of shear and bulk, certain events may pass through This is the same underlying assumption made when using partialchemical equilibrium for hadronic decays, e.g. [109]. the critical points while others can miss it entirely (evenstarting from the same initial energy density and baryondensity).This paper is organized as follows. In Sec. II we outlineour hydrodynamical model, the transport coefficients,and equation of state. In Sec. III we calculate the { T, µ B } trajectories across the QCD phase diagram for fixed ρ and ε in Sec. III A and fixed freeze-out point in Sec. III B.The indirect effects of the critical point on shear viscos-ity are shown in Sec. III C. Then, the potential existenceof attractors for shear and bulk channels are discussedin Sec. IV. Sec. V discusses the influence of a criticallyscaled bulk viscosity in our results. Our conclusions inSec. VII explore the consequences of our calculations onthe search for the QCD critical point (potential conse-quences to neutron star mergers are also discussed). InAppendix A we study the influence of the ˙ β terms presentin Israel-Stewart theory to the evolution of the fluid atlarge baryon chemical potentials. II. HYDRODYNAMICAL SETUP
In the past years, a significant effort has been made toincorporate at least one conserved charge (baryon den-sity) and more recently two (strangeness) in event-by-event relativistic viscous hydrodynamics codes [42, 43,96, 97]. Additionally, transport coefficients can also de-pend on { T, µ B } [35, 36, 94, 116–120] and they are alsosensitive to the presence of critical fluctuations [121],which should influence final state observables. In thefollowing we only consider the effects from one conservedcharge (baryon number) but we point out that a morerealistic description of trajectories on the QCD phasediagram would require effects from the conservation ofbaryon number, strangeness, and electric charge, whichwould severely complicate the type of analysis done here.In this first study of how the viscous fluid traverses theQCD phase diagram we use a simplistic, highly symmet-ric Bjorken flow [122] picture where the hydrodynamicequations of motion are greatly simplified [123]. We usetwo different formulations of relativistic viscous hydrody-namics, DNMR [124] and Israel-Stewart [125], in order todetermine how assumptions regarding the derivation ofthe equations of motion, and their choices of second-ordertransport coefficients, affect the evolution of the baryonrich viscous fluid.Both DNMR and Israel-Stewart are based on the ideathat the dissipative currents, such as the shear-stress ten-sor π µν and bulk scalar Π, evolve according to relaxationequations that describe how such quantities deviate fromtheir relativistic Navier-Stokes values. Using hyperboliccoordinates with the metric g µν = diag(1 , − , − , − τ ),the underlying symmetries of Bjorken flow imply thatall dynamical quantities depend only on the proper time τ = √ t − z . Furthermore, in Bjorken flow the stateof the fluid is described by only 4 dynamical variables:the proper energy density ε ( τ ), the baryon number den-sity ρ ( τ ), Π( τ ), and π ηη ( τ ) (where η here stands for thespacetime rapidity). For DNMR the equations of motionin Bjorken flow become [124, 126]˙ (cid:15) = − τ (cid:2) (cid:15) + 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 = − ρτ (4)where ˙ (cid:15) = d(cid:15)/dτ , p is the equilibrium pressure definedby the equation of state, ζ is the bulk viscosity, andthe remaining second order transport coefficients aretaken from [127]. We note that in Bjorken flow theparticle diffusion contribution vanishes and, thus, thebaryon density equation can be readily solved to give ρ ( τ ) = ρ ( τ /τ ), where ρ and τ are the initial baryondensity and time, respectively.We make the point of including second order trans-port coefficients terms that couple the shear and bulkcontributions (e.g. λ π Π and λ Π π ) since there should be anontrivial coupling between the two [128]. The transportcoefficients for DNMR used in this paper are defined asfollows: τ π = 5 η(cid:15) + p (5) τ Π = ζ (cid:15) + 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) (10) δ ΠΠ = 23 . (11)where the speed of sound squared is c s = dp/d(cid:15) (com-puted at constant entropy). Given η/ ( (cid:15) + p ) and ζ/ ( (cid:15) + p )as functions of T and µ B , all the second order transportcoefficients (such as the bulk and shear relaxation times, τ Π and τ π , respectively) can be readily obtained. Forthe Israel-Stewart case, the energy density and baryondensity evolution remain the same (as they stem fromthe conservation laws) while the relaxation equations forshear-stress and bulk viscous pressure evolution are givenby τ π ˙ π ηη + π ηη = 4 η τ − ηT π ηη (cid:18) β π τ + ˙ β π (cid:19) (12) τ Π ˙Π + Π = − ζτ − ζT Π2 (cid:18) β Π τ + ˙ β Π (cid:19) (13) CP μ B = μ B = μ B = μ B = μ B = [ GeV ] η T / w FIG. 1. (Color online) Our phenomenological ηT /w ( T, µ B )across the phase diagram. The critical point is shown in red(note, no critical scaling was included in the shear viscosity). where we defined β π = τ π ηT (14) β Π = τ Π ζT . (15)When the Israel-Stewart equations were first derived in[125], the terms in Eq. (12) and Eq. (13) that contain˙ β π and ˙ β Π were left off, since these derivatives were pre-sumed to be small on the scales they were interested.This is certainly not true in heavy-ions where early inthe expansion these terms can be quite large. Thus, togauge the importance of these terms and also the possibil-ity of needing to include higher order terms in the powercounting scheme of DNMR [124], we will also make com-parisons with and without including the ˙ β terms. Thiscomparison is shown in Appendix A. However, for therest of the main text we will only show results comparingDNMR and Israel-Stewart including the ˙ β terms becausethey play an important role in the system’s evolution.The shear viscosity used in this paper was derivedfrom an excluded hadron resonance gas model similarlyto what was done in [129]. Then, this hadronic shearviscosity was coupled to a simplistic parameterized QGPphase (based on the parameterization in [130, 131]) andwas matched at T ∼ .
195 GeV at µ B = 0, similar to[120, 130, 131]). The finite µ B behavior is determined bythe change in ηT /w (where w = (cid:15) + p is the enthalpy) inthe excluded volume hadron resonance gas model and theswitching temperatures between the hadron resonancegas where the QGP phase is adjusted to match the crit-ical point at finite µ B . The variation of ηT /w ( T, µ B )is shown in Fig. 1 for various values of µ B . Note thatno critical behavior is incorporated in the shear viscos-ity. Rather, the µ B dependence is driven entirely by thematching to the hadron resonance gas at lower and lowervalues of the temperature with increasing µ B . Generally,lower temperatures lead to a large shear viscosity and,therefore, this quantity increases with increasing µ B . Aforthcoming paper will appear shortly about this workwith further details.For the bulk viscosity, two different parameterizationswere used, both of which are scaled up from one that isin the same ballpark as the ζ/s extracted from Bayesiananalysis [58, 132] that is also consistent with that fromholographic models [120, 133] and quasi-particle models[134, 135]. This base parameterization of the bulk vis-cosity is given by ζTw = 36 × / − c s π (16)where the factor of 36 is included to obtain a maximum ζTw ∼ . c s , there is at least some sensitiv-ity to the critical point (since the critical point has avanishing c s ).As previously mentioned, in the Bjorken picture thebaryon density evolution is trivial, as seen in Eq. (4).This is because the baryon diffusion can only be includedin less symmetrical evolution dynamics, which we willconsider in a future work. However, the non-trivial timeevolution of the energy density due to viscous effects aswell as the non-trivial mapping of { (cid:15), ρ } → { T, µ B } dueto the equation of state lead to unique trajectories in theQCD phase diagram. These trajectories are necessarilyoff of the isentropes calculated in equilibrium, such asthose from Lattice QCD, and should be associated withsome amount of entropy production.To close the hydrodynamic equations of motion we usethe Lattice QCD-based equation of state (EOS) from[112] that is coupled to a parameterized 3D Ising model.This equation of state allows us to test the influence ofa critical point on the T − µ B trajectories. Since we donot, in fact, know the location (or even the existence)of the QCD critical point, the results are simply to testthe qualitative influence of the critical point. Thus, weonly consider one readily available parameterization ofthe EOS from [112] where the critical point is located at { T, µ B } = { , } MeV. In this EOS the critical pointalways lies on the chiral phase transition line, which iscurrently known up to O ( µ B ): T = T + κ T (cid:18) µ B T (cid:19) + O ( µ B ) , (17)where we use T = 0 .
155 GeV and the central value of κ = 0 . FIG. 2. (Color online) Figure showing how to distinguish be-tween different initial conditions in the various plots presentedin this work. the speed of sound at the critical point or large increasein bulk viscosity due to the critical scaling.One final remark on the limitations on the EOS de-rived in [112] is in order. Because the 3D Ising modelis coupled to the Lattice QCD reconstructed EOS up to O ( µ B ), the absolute maximum that we can reasonablyextend the EOS out to in µ B along the phase transitionis µ B ∼
450 MeV. Beyond this point, pathologies beginto appear in the EOS. At high temperatures we have aslightly higher reach and we can extend the phase dia-gram out to µ B ∼
600 MeV. However, because a numberof trajectories that pass through the critical point beginat relatively low temperatures but high µ B (and the timeevolution is nearly flat in T ), we are limited in the phasespace that we can explore our initial conditions. This isespecially problematic for the Israel-Stewart equations ofmotion, which appear to prefer these type of trajectories. III. T − µ B TRAJECTORIES ACROSS THE QCDPHASE DIAGRAM
Up until this point there have been two main ap-proaches to studying the evolution of a hot and baryonrich QGP through the QCD phase diagram. On onehand, a significant part of the community assumes thatthe system can be described as an ideal fluid such thatone can follow Lattice QCD-computed isentropes (wherethe total entropy to baryon number ratio is fixed through-out the expansion
S/N B = const ) throughout the QCDphase diagram. In order to determine the correct S/N B ratio, one determines it from freeze-out properties (typ-ically comparing net-charge fluctuations at freeze-out)and works backwards from the freeze-out point to ex-tract these trajectories, see [11, 113, 141–143] for recentexamples of this approach.The second approach has been to study full scale3+1 dimensional hydrodynamic simulations such as in[46, 99, 144, 145] and concentrate on the central cellspassage through the QCD phase diagram. In [99] idealhydrodynamic equations of motion were used and, un-surprisingly, the T − µ B evolution of the central cellclosely followed that of isentropes. However, in [145] FIG. 3. Trajectories produced using DNMR equations ofmotion, for the same initial energy density, with ρ ∈{ . , , . } fm − , and with χ , Ω ∈ {− . , . } . full viscous simulations (but assuming the initialization π µν = Π = 0) were used and cells from the center cer-tainly pass through a wide swath of the phase diagramthroughout the hydrodynamic evolution. As far as weknow, there has yet to be a study on the influence of vis-cosity (or better put, entropy production) on the T − µ B trajectories. Nor are we aware of any initial conditionsthat initialize the full energy momentum tensor ( T µν ) atfinite baryon densities that are coupled to viscous hydro-dynamic codes and, thus, explore the influence of far-from-equilibrium behavior on the T − µ B evolution.One has no reason to believe that initial conditions atthe beam energy scan should be close to equilibrium (infact, very little is known about initial conditions at thebeam energy scan and they have not gone through nearlyas many rigorous checks as what has been performed atLHC energies [51] and the idea of BSQ eccentricities isstill being developed [44, 45]). Thus, it is necessary toinclude this systematic uncertainty in our calculations.For this study we perturb the initial state between χ = ± . = ± .
5, where we define χ ≡ π ηη / ( (cid:15) + p ) (18)Ω ≡ Π / ( (cid:15) + p ) (19)which are, respectively, the inverse Reynolds numbers, Re − for shear and bulk viscosity [124]. We then sys-tematically run a large number of initial baryon densities.Because we run a large number of trajectories, we haveemployed a color scheme to denote the initial conditionsused in our model, as shown in Fig. 2. In Sec. VI, wetouch on some physical constraints in allowed choices forthe initial Re − for both shear and bulk. A. Trajectories for fixed initial ρ and ε To demonstrate how strong of an effect that far-from-equilibrium behavior can have from the initial conditionson the trajectory through the QCD phase diagram, we
FIG. 4. Here we compare the T − µ B trajectories for DNMRand Israel-Stewart equations of motion, with different initialconditions. The legend is the same in both figures. pick three different initial conditions in ρ at a fixed ini-tial energy density and then vary χ and Ω , as shown inFig. 3. At the lowest baryon density of ρ = 0 . f m − wealready see a wide spread in the { T, µ B } trajectories andaround the chiral phase transition they cover a swath inbaryon chemical potential of about ∆ µ B ∼
200 MeV.Thus, even far from the critical point it is extremelyimportant to know the initial conditions for χ and Ω.One can also see an interesting dependence on the rangeof chemical potentials at the chiral phase transition, de-pending on the choice of ρ . For the intermediate baryondensity initial condition of ρ = 1 f m − we find a rangeof chemical potentials at the chiral phase transition to beeven larger, on the order of ∆ µ B ∼
250 MeV. However,at our maximum initial baryon density of ρ = 1 . f m − we begin to see a bend in all the trajectories and whatmay even be some hints of an attraction towards the crit-ical region. The chiral phase transition range in initialchemical potential range is smaller than for ρ = 1 f m − ,and is again ∆ µ B ∼
200 MeV. We also do not obtain tra-jectories that pass far to the right of the critical point.Unfortunately, we cannot explore this trend further be-cause of the limits of our EOS.In Fig. 4, we directly compare the phase diagram tra-jectories generated by hydrodynamic runs of the sameinitial conditions, comparing DNMR and Israel-Stewartequations of motion. When the initial conditions forthe shear stress tensor and bulk pressure are all set tozero, then the two trajectories are relatively similar toeach other (although not identical!). However, if we con-sider far-from-equilibrium initial conditions, specifically χ = 0 . = − .
5. The differences between ISand DNMR and are very pronounced, especially wherethe trajectories cross the chiral phase transition whereDNMR appears to freeze-out at a lower µ B comparedto IS. A more interesting comparison can be made whenone finds the range of initial conditions which lead to thesame freeze-out point, that is, a degeneracy in the finalstate thermodynamics when attempting to trace back tothe initial state. This is the approach that we shall takefor the rest of this paper. B. Trajectories for a fixed freeze-out
While the initial state is certainly unknown at thebeam energy scan, freeze-out has been well studied withboth thermal fits [146–153] and fluctuations of conservedcharges [111, 154–160]. Some tension still exists betweenthe freeze-out estimates in terms of T and µ B from ther-mal fits versus fluctuations, although reasonable agree-ment exists when two separate freeze-out temperaturesare used for light and strange hadrons [161]. Therefore,in this study we require that our hydrodynamic evolu-tion must pass approximately through the light hadronfreeze-out point from [154] and can then determine therange in initial conditions that lead to that point.In Fig. 5 we study the intermediate beam energy of √ s NN = 27 GeV as well as a hypothetical lower beam en-ergy which would have an isentrope that passes throughthe critical point, for DNMR and Israel-Stewart equa-tions of motion. The freeze out region is defined at somepoint along the green isentrope lines by choosing a rea-sonable temperature at which to freeze out at. We thenselect on hydrodynamic trajectories that pass through acircular region centered on the freeze out point, with aradius of 2 . Since we only include one conserved charge the isentropes areslightly different.
FIG. 5. Trajectories in the QCD phase diagram for differenthydrodynamic equations of motion. The green lines are isen-tropes and are the same in each figure. The freeze-out regionis shown as a green circle centered along the freeze-out pointon the isentrope. system through the phase diagram. For our range of χ and Ω , the possible initial conditions that lead to thesame freeze-out conditions are wide-spread in chemicalpotential for the same initial energy density, as shown inFig. 5. In the Israel-Stewart case, away from the criticalpoint, the initial chemical potential can lie in a range ofnearly ∆ µ B ∼
200 MeV and still make it to the samefreeze-out region. Closer to the critical point, the rangeincreases to ∼
300 MeV. The DNMR trajectories havethe same characteristics, only the initial conditions ap-pear to converge closer to the isentrope (solid green line)more quickly at least far from the critical point. It isinteresting to note that trajectories that go through ornear the critical point accept a larger range of initial con-ditions. This is, again, indicative of some attractive likebehavior, specific to this EOS, and the question remainsas to whether this behavior persists upon inclusion of thenecessary critical fluctuation framework previously men-tioned. It should be the case that the behavior of thetrajectories before entering the critical region will be thesame. However, the dynamics within the critical regionwill surely be modified.The solid black lines in Fig. 5 are the points where theinitial shear and bulk are set to zero but that the trans-port coefficients are still turned on i.e. π ηη, = Π = 0.One can see that for DNMR the effect of the trans- FIG. 6. Time evolution of ηTw for different initial conditionsin DNMR, close to the critical point (top) and away from thecritical point (bottom). port coefficients alone is smaller (transport coefficientslead to initial conditions that start at ∆ µ B ∼
50 MeVlarger than for the isentropes) than for Israel-Stewart,where we find that the effect of transport coefficientsalone increases the initial baryon chemical potential by∆ µ B ∼
100 MeV. This demonstrates that the trajectoriesthrough the QCD phase diagram are strongly dependenton the choice of second order hydrodynamic equationsof motion. Since those theories only differ in the tran-sient regime (given that both approaches have the samerelativistic Navier-Stokes limit), our results indicate thattransient hydrodynamic effects must be taken into ac-count when determining the path traversed by the QGPon the QCD phase diagram.Finally, we find that the sign of the initial conditionsplays a large role if the trajectories are to the left or theright of the isentropes. Generally, values for the initialconditions with Π ≤ π ηη ≥ µ B whereas initial conditions with Π ≥ π ηη ≤ µ B . C. Viscous Effects
Fig. 6 plots different trajectories of our shear viscosityover enthalpy ratio for DNMR equations of motion fortrajectories both far from and near to the critical point.We note that our construction of shear viscosity does not incorporate any critical scaling since it does not scaleas strongly with the correlation length [121]. The timeevolution of ηT /w varies with the choice in the initial π and Π, which sends the hydrodynamical expansion alongdifferent trajectories. Since ηT /w depends on both T and µ B , different values of ηT /w as a function of timeare probed depending on the initial conditions.We then compare the bulk viscosity in Eq. (16) to itscritically scaled form proposed in [35]. The form of thisbulk viscosity is then (cid:18) ζTw (cid:19) CS = ζTw (cid:34) (cid:18) ξξ (cid:19) (cid:35) (20)where ξ is the correlation length and ξ sets the scalefor the critical region. When not including the criticalcomponent, we simply set ξ to 0.Because the bulk viscosity depends on c s , the non-trivial structure that arises in its dependence over timeis due to the change of degrees of freedom. When plottedon trajectories close to µ B = 0 (i.e. far from the criticalpoint) a bump it seen as the quarks and gluons transi-tion into hadrons, as seen in Fig. 7 (a). The differentlines demonstrate how different trajectories probe differ-ent values of ζT /w at different times. However, at thecritical point the speed of sound goes to zero, which pro-duces a spike in ζT /w as one passes through it. In thispaper we compare two scenarios, one where ζT /w onlyscales with c s across the critical point, which is shownin Fig. 7 (b) and another where the correlation lengthaffects the ζT /w , as shown in Fig. 7 (c).When incorporating the critical scaling through thecorrelation length into the bulk viscosity, there is somefreedom in choosing the scaling constant, ξ , such that ζT /w is smaller outside the critical region, and muchlarger inside. In this work, it was chosen so that thepeak in ζT /w increases by a factor of 3 close to the criti-cal point. The correlation length is calculated using a for-mula found in [35] that calculates the equilibrium valueas: ξ = 1 H (cid:18) ∂M ( r, h ) ∂h (cid:19) r (21)As is done in [35], we use the linear paramterizationmodel [162, 163], but instead derive an expression to fifthorder in θ . This is consistent with the accuracy of ourEOS. The expression to fifth order in θ is then: (cid:18) ∂M ( r, h ) ∂h (cid:19) r = M H R β ( δ − (cid:32) θ (2 β − βδθ (cid:101) h + (cid:101) h (cid:48) (1 − θ ) (cid:33) (22)with (cid:101) h = θ (1 + aθ + bθ ) (23)where the coefficients a, b are accessible output from ourEOS, and the critical exponents are taken as their meanfield approximate values. We leave for future work thestudying of consequences of changing the strength andshape of the critical region, which should change the peakin ζT /w , accordingly.A crucial piece to understanding χ at the critical pointin Israel-Stewart is to observe the ηT /w trajectories atthe critical point, as shown in Fig. 8. Because of therather non-trivial trajectories across the critical point forIsrael-Stewart equations of motion, ηT /w inherits a non-trivial time dependence even though no critical behaviorwas built into the transport coefficient. One can see inFig. ?? that many lines traverse the chemical potentialin a complicated and non-trivial way (specifically the redline). It is this chemical potential dependence that pro-duces the peak behavior seen in Fig. 8. Comparing the FIG. 7. Time evolution of ζTw for different initial conditionsin DNMR, away from the critical point (top), near the criticalpoint (middle), and critically scaled (bottom) FIG. 8. Time evolution of ηTw for different initial conditionsin Israel-Stewart, with ˙ β terms that go through the criticalregion, without critical scaling (top) and with (bottom). red lines with a spike to Fig. ?? we find that this causedby trajectories that begin at high T and low µ B thatthen pass through the critical point and continue ontolow T and high µ B trajectories. Eventually these linesend abruptly because they have reached the edge of ourEOS. Also worth noting is the increased sensitivity of theshear viscosity to critical scaling shown in the bottom ofFig. 8 compared to the non-critically scaled runs shownin the top.In the case of the DNMR equation of motion, the T − µ B trajectories behave much more smoothly, and thus nospike is seen in Fig. 6. IV. POTENTIAL ATTRACTORS
In this paper we do not attempt to systematicallyinvestigate the presence of attractors for these rathernon-trivial transport coefficients and complex equationof state. However, we can check for a convergence of χ = π ηη / ( (cid:15) + p ) in Fig. 9 and Ω = Π / ( (cid:15) + p ) in Fig. 10on time scales normalized by their respective relaxationtimes. The points observed in Figs. 9 and 10 are thosepassing through the freeze-out for √ s NN = 27 GeV andthe critical point, respectively.In Fig. 9 the inverse Reynolds numbers for shear vis-cosity are shown both far from the critical point and at FIG. 9. Shear inverse Reynolds number χ = π ηη /w trajectories far from the critical point (left) and at the critical point (right)using DNMR (top) and Israel-Stewart (bottom) equations of motion. The solid black lines assume that the initial Re − = 0for both shear and bulk (the band demonstrates the width of our range of the freeze-out { T, µ B } . the critical point. The shape of χ over time is rathercomplicated because the minimum of ηT /w at the phasetransition, which leads to this bending backwards in χ since τ π depends on the shear viscosity (similar to whatwas found in [164]). For both DNMR and Israel-Stewartwe immediately note that due to the short lifetime of ourhydrodynamic runs, none of our trajectories converge toa single line by freeze-out. However, we also plot thedirection of the derivative at the freeze-out point and itdoes appear that in all cases that an attractor could bereached if hydrodynamics would run for a longer periodof time. From now on, we will refer to this as a “po-tential attractor” because we are not certain if this is anattractor but it certainly hints at one.One curious difference between DNMR and Israel-Stewart is that for DNMR the potential attractor ap-pears to always sit on a nearly flat line in χ . However,for Israel-Stewart equations of motion the potential at-tractor line has a clear slant far from the critical point.At the critical point the potential attractor for Israel-Stewart is even more bizarre in that it appears to begrowing in χ and then potentially flattening out. Un-fortunately, we cannot investigate this further with ourcurrent EOS due to its limitations in µ B . In fact, due tothe limitations in the EOS we are not even able to obtainthe Π = π ηη, = 0 curves because they would begin atmuch larger values of µ B . The bulk pressure is more intuitive to understand andwe find that despite a wide range of initial conditions(and multiple different combinations for the initial shearand bulk) that all curves quickly collapse onto a universalscaling behavior. While the time scale may appear to belong, we note that this is because the bulk relaxationtime is quite significant (due to the small bulk viscosityused here e.g. see Eq. (6)).In Fig. 10 we find that far from the critical pointboth equations of motion quickly converge to what ap-pears to be an attractor, although it appears that Israel-Stewart takes longer to converge. At the critical point wefind that the DNMR equations of motion are more well-behaved and generally do not have large inverse Reynoldsnumbers even though the critically scaled ζT /w is quitelarge. On the other hand, the Israel-Stewart equations ofmotion diverge quite dramatically when passing throughthe critical point but, despite this effect, they manage toconverge afterwards. V. CONSEQUENCE OF ζT /w
DIVERGING DUETO THE CRITICAL POINT
In the previous section, we always assumed that ζT /w scaled with the correlation length, according to Eq. (20).In this section we will compare this assumption to the0
FIG. 10. Bulk inverse Reynolds number Π /w trajectories far from the critical point (left) and at the critical point (right) usingDNMR (top) and Israel-Stewart (bottom) equations of motion. The solid black lines assume that the initial Re − = 0 for bothshear and bulk (the band demonstrates the width of our range of the freeze-out { T, µ B } . regular ζT /w that only scales with the speed of sound,as shown in Eq. (16). We note that outside of the criticalregion that our choice of the inclusion of critical scaling isirrelevant since this only affects ζT /w near to the criticalpoint.In Fig. 11 we plot the inverse Reynolds numbers ofboth shear and bulk viscosity comparing with and with-out critical scaling of ζT /w . In the shear Re − trajecto-ries, we see very little difference if the bulk viscosity hascritical scaling or not. This is not entirely unexpected be-cause while there are coupling terms between shear andbulk viscosity in DNMR, they are non-linear terms and,thus, they do not affect χ very strongly. Additionally, thelarge peak in ζT /w only appears close to freeze-out and,therefore, the χ trajectory has already converged muchcloser to its potential attractor at that point.As expected, the bulk Re − is more affected by criticalscaling of ζT /w . In fact, one can see quite clearly inthe plots the point where the peak in ζT /w is reached.However, despite a brief interruption in the approach tothe potential attractor in Ω, the curves quickly fall ontop of each other in both scenarios. It is clear from theseresults that the potential bulk attractor is quite large forheavy-ion collisions - likely because bulk only plays a rolebriefly around the phase transition.In Fig. 12 we observe the { T, µ B } trajectories acrossthe critical point when ζT /w does not have critical scal- ing. When comparing these trajectories to the criticallyscaled ones in Fig. 5 we find that there are not very largedifferences. However, for Israel-Stewart equations of mo-tion at low temperatures both scenarios seem like theymight run along the first order phase transition line fora bit before the system turns into hadrons. The biggestdifference with and without critical scaling is that thecritically scaled ζT /w then jumps up to the left of thephase diagram (towards higher temperatures) within thehadron gas phase, whereas the regular ζT /w scenario ex-hibits a more regular trajectory and always progressesdownwards (towards low temperatures) in the phase di-agram. VI. ENTROPY PRODUCTION ANDTRAJECTORY CONSTRAINTS
It has so far been demonstrated that, due to the exis-tence of a potential attractor for the time evolution of χ and Ω, there may be a degeneracy in the final freeze-outstate of the system. That is, many different trajectoriesin the phase diagram that are initially very different comeextremely close to each other at late times. We have putan emphasis on entropy production as a conceptual basisfor understanding the deviations from isentropes. How-ever, we are currently unaware of any rigorous calculationof entropy production for DNMR (or Israel-Stewart when1 FIG. 11. Shear (top) and bulk (bottom) inverse Reynolds number trajectories at the critical point for DNMR equations ofmotion where either ζT /w only scales with c s (left) or also scales with the correlation length (Right). The solid black linesassume that the initial Re − = 0 for both shear and bulk (the band demonstrates the width of our range of the freeze-out { T, µ B } . derived from kinetic theory).This sort of calculation would be extremely useful inallowing for quantitative cuts on what kinds of initial con-ditions and trajectories are possible, via the second lawof thermodynamics. We note that it is clear that dueto the deviation of our results from the isentropes, theremust be a large effect on the entropy production due toour choice in initial conditions and transport coefficients.We point out that our chosen transport coefficients arereasonable and not unrealistic since current relativisticviscous hydrodynamic models used within heavy-ion col-lisions are based on the DNMR formalism [136].One can also put some constraints on the choices ofinitial viscous conditions by taking a similar approach aswas done in [165]. In that paper, the weak energy con-dition is used to put physical bounds on possible valuesfor the shear-stress throughout the evolution, in a systemthat undergoes Bjorken flow. The weak energy conditionis the condition that T µν t µ t ν ≥ t µ is any time-like vector. This condition has thesimple interpretation that the energy density of the fluidshould be non-negative for any observer. Using this con-straint, one can extend further the work done in [165] to put constraints on a non-conformal system. Then, in-stead of a constraint on just χ , the constraints involveboth χ and Ω. Doing the derivation, one finds the fol-lowing χ − Ω ≥ − − χ ≥ − χ and Ω violate these bounds, but the choice { χ, Ω } = { . , − . } does hit the bound in Eq. (26). VII. CONCLUSIONS
In this paper we analyzed how far-from-equilibriuminitial conditions of heavy-ion collisions could affect thesearch for the QCD critical point. For a single freeze-outpoint there exists a multitude of potential trajectoriesthat could have lead to that point because of the entropythat is produced when one considers realistic transportcoefficients. Each trajectory is defined by its initial con-ditions that not only includes the initial energy densityand baryon density but also its initial shear stress tensorand bulk pressure. These trajectories diverge far fromisentropes, which are calculated along lines of constant2
FIG. 12. Trajectories in the QCD phase diagram for differenthydrodynamic equations of motion. For these trajectories,the bulk viscosity does not include critical scaling.
S/N B , and depend strongly on the sign of the initial Πand π ηη . The non-uniqueness of a freeze-out point withrespect to a given initial condition presents an interest-ing problem in both determining the initial state giventhe final state freeze-out conditions, as well as in deter-mining the possible late time properties of the fluid (e.g.the possibility of only certain events passing through thecritical point for a fixed beam energy).We studied both DNMR and Israel-Stewart equa-tions of motion. Perhaps, unsurprisingly, we find thatDNMR is better equipped to handle larger initial inverseReynolds numbers and we did not find any trajectoriesthat led to runaway trajectories, which we interpret as aconsequence of DNMR having a more well-controlled ex-pansion [124, 164]. Within DNMR the potential attrac-tors appeared to be relatively flat in χ and Ω even at thecritical point. In contrast, the Israel-Stewart equationof motion also appear as if they will eventually reach anattractor. However, at the critical point a large, negativespike in Ω was seen, well outside the range of applicabil-ity for hydrodynamics. Despite this spike the solutionsstill returned to a potential attractor by freeze-out (weemphasize potential because due to the finite lifetime ofhydrodynamics this would occur beyond our freeze-outpoint). We note, however, that the potential attractorline appears significantly different in Israel-Stewart andis no longer flat but rather looks like a hill at the criticalpoint. For phenomenological purposes, this work indi- cates that codes that solve Israel-Stewart vs. DNMRequations of motion should expect different results whenexploring the QCD phase diagram at large baryon densi-ties. Therefore, since the main difference between DNMRand Israel-Stewart lies only in how they treat far fromequilibrium transient effects (since they have the sameNavier-Stokes limit), our results indicate that the out-of-equilibrium properties of the hot and baryon rich QGPmust be taken into account in experimentally-driven at-tempts to locate the QCD critical point using heavy-ioncollisions.On an event-by-event basis each event may passthrough the QCD phase diagram in radically differentways, even if hydrodynamics is only initialized at verylow temperatures, as was shown here. Additionally, itwas previously pointed out [95] that viscosity affects thetime scale of the phase transition (across a first orderline). Instead, we suggest that one should think of ob-servables that could tag individual events (or groups ofevents) by similar trajectories through the phase diagramin order to better understand the QCD equation of stateat large baryon densities.The next step in our future studies involves going be-yond Bjorken flow, taking into account a more realisticspacetime evolution of the medium. This would thenallow us to incorporate the effects of baryon diffusion,which would lead to further entropy production and likelycause an even larger divergence from isentropes. Addi-tionally it has been shown that µ B can vary with rapidity[166, 167] even at LHC collisions, so this would providea new knob to turn in this type of analysis. Furtherobvious extensions of this work would be to include mul-tiple conserved charges, which has already been shown toshift the path of the isentropes even for ideal hydrody-namics [113, 114], and also critical fluctuations (althoughwe believe that no consensus has yet been reached on theproper way to include them in state-of-the-art numericalrelativistic viscous fluids).This work also presents a direct challenge for the ex-traction of the QCD equation of state from relativisticheavy-ion collisions at large baryon densities. In fact,far-from-equilibrium effects are likely even larger at lowbeam energies (regardless if the degrees of freedom arehadrons or quarks/gluons), which makes previous claimsof an EOS extracted from heavy-ion collisions proba-bly unrealistic [168] (especially considering this previouswork assumed T = 0 whereas these beam energies havenow experimental evidence of temperatures greater than T >
70 MeV [169]). Thus, heavy-ion collision constraintson the EOS can, at best, be applicable only to the neu-tron star mergers themselves (as was discussed exten-sively in [19, 102, 169]). This means that one shouldnot use a heavy-ion extracted EOS, which includes tem-perature effects even at low center of mass collision ener-gies, when placing constraints on the EOS of cold neutronstars (i.e. T ∼ ACKNOWLEDGEMENTS
Thus authors would like to thank Jorge Noronha, Clau-dia Ratti, Paolo Parotto, Chun Shen, and Michael Strick-land for discussions and their insights into this work.J.N.H. acknowledges support from the US-DOE NuclearScience Grant No. de-sc0019175 and the Alfred P. SloanFoundation. E.M. was been supported by the NationalScience Foundation via grant PHY-1560077. The au-thors also acknowledge support from the Illinois CampusCluster, a computing resource that is operated by theIllinois Campus Cluster Program (ICCP) in conjunctionwith the National Center for Supercomputing Applica-tions (NCSA), and which is supported by funds from theUniversity of Illinois at Urbana-Champaign.
Appendix A: Israel-Stewart and ˙ β terms In the following section we will study the influence ofthe ˙ β terms in the Israel-Stewart equations of motion.We generally find that Israel-Stewart equations of mo-tion without ˙ β terms leads to an extremely wide spreadacross µ B for the initial conditions that freeze-out farfrom the critical point. The initial conditions that startat large µ B trajectories are in fact ones that would not beparticularly atypical for heavy-ion collisions (they startwith an initial bulk pressure Π ≤ π ηη ≥ µ B may be needed. At the critical point, we arelimited to only initial conditions that have a positive ini-tial Π and a negative initial π ηη because all other initialconditions would start at too large of µ B for our EOS tohandle. FIG. 13. { T, µ B } trajectories far from the critical point andat the critical point using Israel-Stewart equations of motionwithout the ˙ β terms. In Sec. II we explained that in the original Israel-Stewart paper [125] they neglected terms that incorpo-rated the gradients of the temperature. Below we studythe effect of these terms and generally find that the in-clusion of the ˙ β terms lead to better (and more well-behaved) inverse Reynolds numbers for both shear andbulk viscosity.First, we explore the Re − of shear and bulk viscos-ity far from the critical point (close to µ B → Re − is shown in Fig. 14 with and without the ˙ β terms. In both cases we scale the time evolution by theshear relaxation time. One can quickly see that the inclu-sion of ˙ β leads to a smaller range of Re − numbers andthat those Re − appear to converge to a line on a rela-tively short time scale. The arrows at the end of the linespoint in the direction of the derivative, which implies thatgiven a long enough hydrodynamic expansion that theywould eventually converge to a singular point. We cau-tion, though, that we stop our hydrodynamic expansiononce the trajectories reach our freeze-out temperatureand, therefore, it appears due to the limited run timesof hydrodynamics in heavy-ion collisions at the beam en-ergy scan that the time scales are not long enough toconverge to a single point in χ .In contrast, the Re − of shear for Israel-Stewart with-out the ˙ β terms produces a much large Re − and even hastrajectories that appear to diverge in χ (the solid lines)becoming ever more negative with time. These trajecto-ries are initialized to have a large, negative χ and a largepositive Ω.The bulk Re − , as shown in Fig. 15, does not appear tobe as sensitive to the inclusion of ˙ β terms, which is likelybecause the ζT /w is relatively small at initial times suchthat Ω quickly drops to a potential attractor. However,even in the case of bulk viscosity, we find that the sameextreme initial conditions (solid red line) that was prob-lematic in Fig. 14 also produces a very large Re − > FIG. 14. Shear inverse Reynolds number χ = π ηη /w trajectories far from the critical point using Israel-Stewart equations ofmotion with ˙ β terms in (a) and without ˙ β terms in (b). The solid black lines assume that the initial Re − = 0 for both shearand bulk (the band demonstrates the width of our range of the freeze-out { T, µ B } .FIG. 15. Bulk inverse Reynolds number Ω = Π /w trajectories far from the critical point using Israel-Stewart equations ofmotion with ˙ β terms in (a) and without ˙ β terms in (b). The solid black lines assume that the initial Re − = 0 for both shearand bulk (the band demonstrates the width of our range of the freeze-out { T, µ B } . Overall, we find that even far from the critical point,Israel-Stewart codes that neglect the ˙ β terms may runinto problems for initial conditions that begin far fromequilibrium and especially may see a shear stress tensorthat has runaway behavior. This is bound to lead tocausality problems [87]. Thus, any exploration of theQCD phase diagram using Israel-Stewart theory in thefar from equilibrium regime should, at the bare minimum,include the ˙ β terms.Next, we explore the influence of the inclusion of the ˙ β terms when the trajectories pass through (or very close)to the critical point. First we consider the Re − for shearviscosity in Fig. 16. When we include ˙ β terms, we seethat χ appears to have some sort of universal line that allthe trajectories are pointing towards. We do find that thetime scales are too short for the curves to truly convergebut this hints that with the ˙ β terms one could reach anattractor if the time scales were long enough. Unlike inFig. 14 where we found only extreme trajectories thatdiverged in χ as hydrodynamics evolved in time when˙ β terms are excluded, at the critical point we find that most trajectories diverge at the critical point for χ if we neglect ˙ β terms. This demonstrates the importance ofusing the full equations of motion for Israel-Stewart ifone wants to study the QCD phase diagram, especiallyclose to a phase transition.In this section, we only consider the critically scaled ζT /w because we wanted to test the limits of Israel-Stewart with and without the ˙ β terms. In Fig. 17 we plotthe Re − for the bulk viscosity with and without the ˙ β terms. In both cases we can obtain very large values ofΩ (in fact, much larger than DNMR) but it is clear fromFig. 17 that while Ω briefly diverges as one crosses thecritical point (due to the large value of ζT /w ) with theinclusion of ˙ β terms, it quickly recovers and is able to re-turn to the potential attractor very quickly. In contrast,Israel-Stewart without ˙ β terms diverges in a multitude ofdirections and it is not clear if an attractor is obtainedeven for the few trajectories that do not diverge. Thus,we argue that Israel-Stewart without ˙ β terms should def-initely not be used near a critical point, nor even whenthe system is far from equilibrium because it can lead todiverging solutions.Finally, in Fig. 18 we compare the trajectories of Israel-5 FIG. 16. Shear inverse Reynolds number χ = π ηη /w trajectories far from the critical point using Israel-Stewart equations ofmotion with ˙ β terms in (a) and without ˙ β terms in (b). Here only the critically scaled ζT /w is considered.FIG. 17. Bulk inverse Reynolds number Ω = Π /w trajectories at the critical point using Israel-Stewart equations of motionwith ˙ β terms in (a) and without ˙ β terms in (b). Here only the critically scaled ζT /w is considered. Stewart with and without the ˙ β terms at the criticalpoint. 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