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Non-equilibrium phase transition in relativistic nuclear collisions:

Importance of the equation of state

Jan Steinheimer,1, 2, ∗ Jørgen Randrup,1, † and Volker Koch1, ‡

1 Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA2Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe Universität,

Ruth-Moufang-Straße 1, 60438 Frankfurt am Main, Germany

(Dated: July 22, 2018)

Within the context of relativistic nuclear collisions aimed at exploring hot and baryon-densematter, we investigate how the general features of the expansion dynamics, as well as a numberof specific observables, depend on the equation of state used in dynamical simulations of the non-equilibrium confinement phase transition.

I. INTRODUCTION

One of the central themes of current research in strong-interaction physics is the exploration and characteriza-tion of strongly interacting matter at high temperaturesand/or densities. In the laboratory, strongly interactingmatter at high temperature and density is created in col-lisions of heavy nuclei at relativistic energies and experi-ments to study these reactions are carried out at variousaccelerator facilities, particularly the Relativistic HeavyIon Collider (RHIC) at BNL and the Large Hadron Col-lider (LHC) at CERN. In the universe, strongly inter-acting matter at high density but very low temperaturemay exist in the core of neutron stars, and its propertiescontrol the maximum mass and radius of those stars.

Experiments at RHIC and LHC have demonstratedthat collisions at nucleon-nucleon center-of-mass energiesof

√sNN ≥ 60GeV create matter at very high temper-

ature T and nearly vanishing net-baryon density, ρ, i.e.ρ/T 3 ≪ 1. Recent continuum-extrapolated lattice QCDcalculations with physical quark masses show that suchmatter exhibits a crossover transformation from an inter-acting hadronic gas to a quark-gluon fluid at a pseudo-critical temperature of T× ≃ 155MeV [1–3].Nuclear collisions at lower energies produce systems

at finite net-baryon densities with the correspondingbaryon-number chemical potential µ being of the orderof the temperature or higher. This region is difficult toexplore with lattice QCD techniques due to the fermionsign problem but for sufficiently small values of µ/T thepressure may be expanded in powers of µ/T , provid-ing a hint about the finite-µ behavior [4–8]. Alternativemethods use an imaginary chemical potential [9] or ap-ply re-weighting techniques [10, 11]. However, all theseapproaches are restricted to small values of µ and havelimited predictive power at the large chemical potentials.The theoretical exploration to higher net baryon densi-ties must therefore rely on models.

∗Electronic address: steinheimer@fias.uni-frankfurt.de†Electronic address: JRandrup@LBL.gov‡Electronic address: VKoch@LBL.gov

Among the many effective models for QCD thermody-namics, those that couple constituent quarks to mesonicfields and the Polyakov loop have become popular be-cause they include chiral-symmetry breaking and a ”ther-mal” confinement of quarks. Such approaches are thePolyakov Nambu–Jona-Lasino (PNJL) [12, 13] and thePolyakov Quark-Meson (PQM) [14] models. Recentlyalso results based on functional renormalization group(FRG) approaches have been investigated [15–18]. Thesemodels can be brought into agreement with lattice re-sults at vanishing net baryon density but their resultsfor the finite-density domain depend significantly on theparametrizations used. A particular weakness in mostof these models is the absence of explicit hadronic de-grees of freedom in the confined phase. There exist onlyfew examples where the equations of state (EoS) includesan approximately realistic description of the hadronicphase. These equations of states are constructed eitherby matching a purely hadronic model to an MIT-bag EoSor by combining a PNJL/PQM-type model with a chiralhadronic model [19, 20]. Obviously, these models needto be constrained both by lattice QCD results at zeroand small chemical potential and by known properties ofnuclear matter at small temperatures and densities up toabout twice nuclear matter density [21], as well as by as-trophysical observations on neutron star properties [22].

As already discussed in [23] there is a qualitative dif-ference between the various effective models that do notinclude explicit hadronic degrees of freedom and thosewhich include the hadronic degrees of freedom. Many ofthe aforementioned Polyakov loop models, as well as var-ious incarnations of the linear sigma model and Nambu-Jona-Lasinio model [24], belong to the former group.These models typically predict coexistence between the(deconfined and/or chirally symmetric) dense phase andthe vacuum at vanishing temperature. Consequently thecoexistence pressure at T = 0 vanishes and, as a re-sult, it remains small even at moderate temperatures.This unrealistic behavior is akin to the well known liq-uid gas phase transition and subsequently we will refer tomodels in this group as LG models. In reality, however,at vanishing temperature the (deconfined and/or chirallysymmetric) dense phase coexists with compressed nuclearmatter. Therefore the coexistence pressure is finite and

http://arxiv.org/abs/1311.0999v2mailto:steinheimer@fias.uni-frankfurt.demailto:JRandrup@LBL.govmailto:VKoch@LBL.gov

2

large. Indeed, as will be discussed below, it will likelybe larger than the (pseudo) critical pressure at vanish-ing net-baryon density. This qualitative difference be-tween LG models and more plausible models will lead toqualitative differences for the expansion dynamics of thesystem, as we shall show.

Experimentally, different regions of the phase diagramof strongly interacting matter can be explored with nu-clear collisions by varying the energy, thereby influencingthe phase trajectory along which the bulk of the createdsystem evolves [25]. However, the successful discovery ofa phase transition requires the measurement of suitableobservables that are sensitive to such a structure in thephase diagram. Since the identification of signal observ-ables is a challenging task, due to the complexity of thecollision, one must invoke dynamical simulations. It is,therefore, essential to develop transport models that arecapable of describing the evolution of the system in thepresence of a phase transition, in particular propagate itcorrectly under the influence of the associated mechan-ical instabilities [26–34]. Such a dynamical model wasrecently presented [35, 36] and the present work uses itto explore the sensitivity of various observables to thepresence of a first-order coexistence region in the phasediagram.

The purpose of the present paper is twofold: Firstwe wish to explore the qualitative differences betweenLG-type equations of state and the more palusible onesthat have phase coexistence between compressed nuclearmatter (rather than vacuum) and a denser (deconfinedand/or chirally symmetric) phase. In particular we areinterested in generic differences in the fluid dynamic timeevolution. To this end we will study two illustrativeequations of state that serve as representatives for thetwo types of models, LG and “realistic”, as discussedabove. The second purpose of the paper is to elucidatethe sensitivity of various observables to the existence ofa first-order phase coexistence region and its associatedinstabilities.

Because we will be working in the framework of fluiddynamics, the only relevant information is the equationof state. Therefore, our results will not discriminate be-tween a phase transition driven by chiral restoration orone driven by deconfinement. Therefore we will subse-quently refer to the (deconfined and/or chiral symmet-ric) dense phase simply as the quark phase (QP). Finally,we will restrict our studies to situations with only onefirst-order coexistence region in the phase diagram. Thepotentially interesting case of separate chiral and decon-finement transition will be left for future work.

The presentation is organized as follows: In the nextsection, we compare an EoS with a hadron-quark-typephase transition with the EoS of a PQM model and dis-cuss their qualitative differences, with a view towardsidentifying potential phase-transition observables. Thesetwo equations of state are then used to carry out variouscalculations with our dynamical model which consists offinite-density ideal fluid dynamics augmented by a gra-

dient term in the local pressure [35]. Based on theseresults, we discuss the suitability of the two models forsemi-realistic tests of the sensitivity of various observ-ables for the QCD phase structure. Specifically, we dis-cuss the effects of the non-equilibrium phase transitionon composite-particle production as well as two-particleangular correlations.Throughout this paper, for notational simplicity, we

denote net baryon density by ρ and the associatedbaryon-number chemical potential by µ.

II. THE HQ EQUATION OF STATE

To investigate the effects of the instabilities associatedwith a first-order phase transition between a hadronicgas and a quark-gluon plasma, we need to employ asuitable two-phase EoS (which we shall refer to as theHQ equation of state). Although significant progress hasbeen made in understanding the thermodynamical prop-erties of each of the phases separately, our current un-derstanding of the phase coexistence region is not yet onfirm ground. We therefore follow Ref. [38] and employ aconceptually simple approximate EoS obtained by inter-polating between an ideal hadron gas, augmented by aρ-dependent interaction energy, and an idealized quark-gluon plasma. As already noted in the Introduction, weview this EoS as a representative of those equations ofstate that exhibit phase coexistence between compressednuclear matter and the dense quark phase.Furthermore, as a reference, we subsequently define a

partner EoS obtained by eliminating the instabilities bymeans of Maxwell constructions.The confined phase is approximated as an ideal gas

of pions, nucleons, and antinucleons, plus an interactionterm, so the total hadronic pressure is of the form

pH = pπ + pN + pN̄ + pw . (1)

The contributions from the various hadron species are

pπ(T ) = −gπT∫

d3k

(2π)3ln[1− e−ǫπ(k)/T ] , (2)

pN (T, µ0) = gNT

∫

d3k

(2π)3ln[1 + e−(ǫN (k)−µ0)/T ] , (3)

pN̄ (T, µ0) = gNT

∫

d3k

(2π)3ln[1 + e−(ǫN (k)+µ0)/T ] , (4)

where gπ = 3, gN = 2× 2 = 4, and ǫ(k)2 = m2 + k2 withmπ = 140MeV and mN = 940MeV. The net baryondensity ρH = ρN − ρN̄ = ∂(pN + pN̄ )/∂µ0 then follows,

ρH(T, µ0) = gN

∫ ∞

mN

pǫdǫ

2π2sinhµ0/T

coshµ0/T + cosh ǫ/T. (5)

Finally, the interaction contribution is given by pw(ρ) =ρ∂ρw(ρ) − w(ρ), where w(ρ) = [−Aρ/ρs + B(ρ/ρs)2]ρ isthe interaction energy density. The strength coefficients

3

A and B have been adjusted so that nuclear matter sat-urates at ρs=0.153 fm

−3 and the associated compressionmodulus is K = KN +Kw = 300MeV; the binding en-ergy of nuclear matter is then also roughly reproduced[38]. The Fermi level µ0 and the chemical potential µ arerelated by µ = µ0 + ∂ρw(ρ)/∂ρ and the entropy densityis then σH(T, µ) = ∂pH(T, µ)/∂T .The deconfined phase was taken as an ideal gas of

massless gluons and massless ud quarks with a bag con-stant B = 300MeV/fm3 [38],

pQ = pg + pq + pq̄ −B , (6)

where pg = gg(π2/90)T 4 with gg = 2×8 = 16 is the gluon

pressure while the quarks and antiquarks contribute

pq + pq̄ = gq

[

7π2

360T 4 +

1

12(13µ)

2T 2 +1

24π2(13µ)

4

]

, (7)

with gq = 2× 3× 2 = 12. The net baryon density in theplasma is then

ρQ =∂

∂µpQ(T, µ) =

2

3(13µ)T

2 +2

3π2(13µ)

3 , (8)

while the entropy density is

σQ =∂

∂TpQ(T, µ) =

74

45π2T 3 + 2(13µ)

2T . (9)

At zero temperature and zero chemical potential, thepressure of the nucleon gas vanishes while that of thequark gas is equal to −B. The confined phase is thenthe thermodynamically favored one. However, as thechemical potential is raised, the plasma pressure in-creases faster than the hadronic pressure, so the curvespH(T = 0, µ) and pQ(T = 0, µ) cross at a certain valueof µ, above which the deconfined phase is favored. Thisphase crossing procedure can be repeated for any temper-ature. As T is increased, the crossing value of µ decreasessteadily, becoming zero at Tmax, above which p

Q is largerthan pH for any value of µ.For the construction of the two-phase equation of state,

it is convenient to work in the canonical representation.Then the condition of phase coexistence, i.e. same tem-perature, chemical potential, and pressure at two differ-ent densities ρ1 and ρ2, amounts to the condition thatfT (ρ), the free energy density at a specified temperatureas a function of density, exhibit a local concavity. Thetwo coexistence densities are then those at the touchingpoints of the associated common tangent. This is readilyseen since µT (ρ) = ∂ρfT (ρ) implies that the two chemi-cal potentials are then equal, µT (ρ1) = µT (ρ2), and sincethe tangent at ρi is given by ti(ρ) = f(ρi)+(ρ−ρi)f ′(ρi)the fact that t1(ρ) = t2(ρ) immediately implies that alsothe pressures are equal, pT (ρ1) = pT (ρ2).At zero temperature, the hadronic free energy density,

fHT=0(ρ), starts at zero but grows more rapidly than the

fQT=0(ρ), which starts at B. Therefore the two curvescross and, furthermore, because they both have positive

Unstable

Density [ 0]

HQ EoS T = 100 MeV

FIG. 1: (Color online) Illustration of the equation of stateof the HQ model: The pressure p as a function of the netbaryon density ρ at the fixed temperature T = 100MeV (solidcurve) and the corresponding Maxwell construction (dashedline). The middle (red) shaded area shows the mechanicallyunstable density region while the adjacent (blue) shaded areasshow the meta-stable regions (in which matter is mechanicallystable but thermodynamically unstable).

curvature, a common tangent exists between two densi-ties ρ1 and ρ2. This feature signals the occurrence ofphase coexistence. In order to describe the associatedtransition region, we follow Ref. [38] and employ a splineprocedure. Thus, for each temperature below a criticalvalue Tcrit < Tmax, we employ a set of matching densities,

ρHT < ρ1 and ρQT > ρ2, which define three distinct den-

sity regions. The hadron EoS is used in the low-densityregion, fT (ρ ≤ ρHT ) = fHT (ρ), the plasma EoS is usedin the high-density region, fT (ρ ≥ ρQT ) = f

QT (ρ), and in

the intermediate density region the free energy densityis represented by a spline polynomial f̃T (ρ) that matches

the values, slopes, and curvatures of fT (ρ) at ρHT and ρ

QT .

By suitable adjustment of the matching densities, itis possible to obtain a resulting free energy densitycurve fT (ρ) that displays an ever weakening phase tran-sition as T is raised, up to a critical value, Tcrit ≈143MeV, above which the phase transformation occursas a smooth crossover. The resulting pressure curve,pT (ρ) = ρ∂ρfT (ρ) − fT (ρ), is shown in Fig. 1 for T =100MeV.In Ref. [38] the focus was on subcritical temperatures,

T < Tcrit, so for each T the spline points were adjustedso the resulting fT (ρ) would exhibit a concave anomaly,ensuring that there would be two densities, ρ1(T ) andρ2(T ), for which the tangent of fT (ρ) would be common,signaling phase coexistence. Ref. [35] extended the EoSto T > Tcrit by using convex splines, as is characteristicof single-phase systems. Having constructed fT (ρ) fora sufficient range of T and ρ, we may obtain the energydensity, εT (ρ) = fT (ρ)−T∂TfT (ρ), by suitable interpola-tion and then tabulate the EoS, p0(ε, ρ), on a convenient

4

Cartesian lattice in the mechanical densities ε and ρ.

III. THE PQM MODEL

As an alternative, we shall also consider a PolyakovQuark Meson model as a representative of a class ofmodels based on the coupling of constituent quarks tomesonic fields and the Polyakov loop. Other approachesof this type include the PNJL and FRG models in dif-ferent realizations [12–14, 39–43]. These models havebecome quite popular because they include the correctdegrees of freedom at high temperatures (namely weaklyinteracting quarks and gluons) and they have an effectivethermodynamic description of the confinement transitionand the chiral symmetry breaking that is in reasonableagreement with lattice results at µ = 0. Within chiralfluid dynamics, this model has been recently applied inrelativistic nuclear collisions for the purpose of investi-gating the effects of a first-order deconfinement transi-tion and the critical endpoint [44, 45]. However, thesemodels yield a rather poor description of the confinedphase, as they allow for only (uncorrelated) three-quarkstates. Therefore, they typically severely underestimatethe pressure in the hadronic phase, as we shall discusssubsequently. In addition, at zero temperature this typeof model as well as sigma and NJL models [24] typicallypredict phase coexistence between the vacuum and thedense quark phase, similar to the well known liquid gastransition. We thus view the PQM model as a genericrepresentative of LG-type models.

The effective thermodynamic potential of the PQMmodel can be written as

Ω = −TV

lnZ = U + Uσ +Ωqq . (10)

The meson potential is given by

Uσ =14λ(σ

2 − ν2)2 − cσ − U0 + gωλ2ωω2 , (11)

where the last term represents a repulsive interactiontransmitted by the ω vector field. The Polyakov-loop po-tential used in the present work is the logarithmic formintroduced in Ref. [40],

U = − 12a(T )ΦΦ∗ (12)+ b(T ) ln[1− 6ΦΦ∗ + 4(Φ3 +Φ∗3)− 3(ΦΦ∗)2] ,

with a(T ) = a0T4 + a1T0T

3 + a2T20 T

2, b(T ) = b3T30 T .

The parameters a0 = 3.51, a1 = −2.47, a2 = 15.2, andb3 = −1.75 are fixed as in Ref. [40], and T0 = 210 MeV.A coupling of the quarks to the Polyakov loop is intro-

duced in the thermal energy of the quarks,

Unstable

Pres

sure

[MeV

/fm3 ]

Density [ 0]

FIG. 2: (Color online) Illustration of the equation of stateof the PQM model: The pressure p as a function of the netbaryon density ρ at the fixed temperature T = 100MeV (solidcurve) and the corresponding Maxwell construction (dashedline). The middle (red) shaded area shows the mechanicallyunstable density region while the adjacent (blue) shaded areasshow the meta-stable regions (in which matter is mechanicallystable but thermodynamically unstable).

Ωqq = 2Nf

∫

d3k

(2π)3

{

T ln[

1 + 3Φ e−(ǫ(k)−µ∗

q)/T

+ 3Φ∗e−2(ǫ(k)−µ∗

q)/T + e−3(ǫ(k)−µ∗

q)/T]

(13)

+ T ln[

1 + 3Φ∗e−(ǫ(k)+µ∗

q)/T

+ 3Φ e−2(ǫ(k)+µ∗

q)/T +e−3(ǫ(k)+µ∗

q)/T]}

,

where only the light-quark flavors (u and d) are included.The thermal heat bath of the quarks contains contribu-tions from one-, two-, and three-quark states, of whichonly the latter do not couple to the Polyakov loop. Thequark energy, ǫ(k) =

√k2 +m∗2, includes the effective

mass, m∗i = m0 + gσσ with m0 = 6MeV, and the re-pulsive vector coupling leads to an effective chemical po-tential, µ∗q = µq − gωω that depends on the strength ofthe quark vector coupling gω. The PQM model param-eters used in this work are λ = 19.7, ν = 87.6MeV,c = 1.77 ·106MeV3, λω = 500MeV, U0 = 165 ·106MeV4,gσ = 3.2, and gω = 0 or gω = 2.For vanishing vector interaction strength, gω = 0, the

PQM model exhibits the desired phase structure, namelya smooth crossover at zero chemical potential as well asa first-order phase transition at zero temperature. Thecritical endpoint is located at µcrit ≈ 480MeV and Tcrit ≈165MeV [14].Using the PQM model, we can calculate a two-phase

EoS, with an unstable region, by finding the local max-ima (unstable states) and minima (meta-stable states)of the grand canonical potential. The resulting EoS is

5

FIG. 3: (Color online) The PQM equation of state p(T, ρ):The pressure a function of both temperature and net baryondensity. The phase-coexistence boundary is delineated(dashed path) and the plane of zero pressure is shown (red),making it easier to see where the pressure is negative.

illustrated in Fig. 2. It should be noted that the phasetransition in the PQM model occurs at density and pres-sure values that are significantly lower than those of theHQ construction discussed above.A more global impression of the PQM EoS (for gω = 0)

can be obtained from Fig. 3 which shows p(T, ρ), thedependence of the pressure on both temperature andbaryon density. The pressure is negative in a large partof the phase coexistence region. It is also evident thatat T = 0 there is coexistence between the deconfinedquark phase and the vacuum, a feature that will admitstable quark droplets in the vacuum. However, we knowthat the true ground state of matter, coexisting with thevacuum, is nuclear matter at saturation density and notquark matter. It will become apparent in the followingthat this inconsistency has strong implications for the dy-namical evolution of the hot and dense matter producedin relativistic nuclear collisions.It is instructive to consider the temperature depen-

dence of the pseudo-critical pressure, ppc, the pressureat which the phase transformation occurs. At subcrit-ical temperatures, T < Tcrit, ppc is obviously equal tothe coexistence pressure, but in the cross-over region (i.e.above the critical temperature) the definition of ppc is notunique (though different definitions yield qualitativelysimilar results). Essential differences between models arethen brought out by the behavior of the transition pres-sure with temperature, ppc(T ).Figure 4 shows the temperature dependent pseudo-

critical pressure for the two models considered here. Fora specified temperature T , pHQpc is obtained by increasing

0 30 60 90 120 150 1800

100

200

300

400 Hadron-Quark EoS PQM model PQM + PQM model x 50 PQM + x 50 Nuclear L-G x 50

Pre

ssur

e [M

eV/fm

3 ]

Temperature [MeV]

PQM x 50

Nuclear L-G x 50

FIG. 4: (Color online) The pseudo-critical pressure as a func-tion of the temperature, ppc(T ), for both the HQ (blue dashedcurve with squares) and the PQM model (black curves) mod-els. The black dashed line with open circles shows the resultof the PQM model with finite quark vector coupling. On ascale magnified by a factor of fifty, the PQM results (grey)are compared with the nuclear liquid-gas transition (whichdisplays the familiar liquid-gas phase transition).

the chemical potential µ until the ideal hadron-gas andthe quark-gluon bag have the same pressure, while pPQMpcis taken as the pressure along the inflection point of thechiral condensate.

While the resulting pseudo-critical pressures are com-parable at the highest temperatures (corresponding toµ ≈ 0), and consistent with lattice results [46], they devi-ate strongly at large chemical potentials: As the temper-ature is reduced, pHQpc increases steadily, as was alreadynoted in previous studies employing models that describethe hadron-quark transition [23, 38]. By contrast, pPQMpcdecreases steadily and vanishes at T = 0, a behavior thatis a robust feature of the PQM models and persists alsoin the presence of a repulsive quark interaction [47, 48].

For comparison, Fig. 4 also shows the coexistence pres-sure for the liquid-gas phase transition in ordinary nu-clear matter. It is quantitatively very similar to the low-temperature part of the corresponding PQM result. Inparticular, these pressures start from zero at T = 0 andthen increase steadily with T . This generic liquid-gas be-havior differs qualitatively from the HQ result, for whichthe pressure decreases steadily with temperature.

This analysis suggests that the nature of the PQMphase transition resembles that of the nuclear liquid-gas transition and differs qualitatively from the quark-hadron transition. For the liquid-gas phase transition,the dense phase (the liquid) has fewer active degrees offreedom than the dilute phase (the vapor), so the liquid-to-gas transition increases the entropy per baryon. Bycontrast, for the confinement transition the entropy perbaryon decreases because the dense system (the quark-gluon plasma) has more active degrees of freedom than

6

the coexisting dilute system (the hadron gas). Indeed,from the Clausius-Clapeyron relation it follows that adecrease in the entropy per baryon from the dense tothe dilute phase requires that the pressure decrease withtemperature [23], as exhibited in HQ EoS. It is importantto keep this qualitative difference in mind.

While at present it is not known which class the QCDphase-transition belongs to, we note that recent lat-tice QCD calculations [6] have extracted the pressure atthe crossover temperature T× = 153MeV and vanishingchemical potential to be p(T×, µ = 0) ≃ 50MeV/fm3.This corresponds to the pressure of cold nuclear matterat a density of roughly three times the ground state den-sity, ρ = 3ρs [49, 50]. Therefore, unless the hadron-quarktransition happens at densities of ρ ≃ 3ρs or below, thepressure is expected to rise as we lower the temperatureand increase the density, just as pHQpc (T ) in Fig. 4.

A further inspection of Fig. 4 shows that even thoughthe HQ and PQM pseudo-critical pressures are numeri-cally rather similar in the high-temperature region whereµ ≈ 0, the slopes of pHQpc (T ) and pPQMpc (T ) are opposite.Because this is the phase region that can be accessed bylattice QCD methods it would be interesting to investi-gate whether those can help to distinguish between themodels. We therefore briefly discuss the prospects forthis.

The introduction of the pseudo-critical pressure ppc(T )makes it possible to define the associated pseudo-criticalchemical potential µpc(T ) determined by ppc(T ) =p(T, µpc(T )). In lattice QCD the inverse relationship,Tpc(µ), has been obtained to second order in µ for vari-ous definitions of the pseudo-critical line [5, 6, 37],

Tpc(µ) = T×

[

1− κ µ2

T 2×

]

, (14)

where we recall that T× = Tpc(0) is the crossover tem-perature at µ = 0. The curvature of the pseudo-criticalline Tpc(µ) depends on the definition of pseudo criticality.For example, if the inflection point of the chiral conden-sate is used then T× = 153MeV and κ = 0.0066 [37].On the other hand, if the inflection point of s/T 3 is used(where s is the entropy density) then κ = 0.016, morethan a factor of two larger, and T× = 158MeV, usingthe lattice EoS parametrization from Ref. [6].

In order to determine the slope of the pseudo-criticalpressure, we use that p(T, µ > 0) can be obtained by aTaylor expansion around µ = 0 [4, 8],

p(T, µ) = p(T, µ = 0) + T 4∑

n>0

χ2n(T )

(2n)!

( µ

T

)2n

. (15)

The Taylor coefficients χn(T ) are the n’th-order baryon-number susceptibilities evaluated at vanishing µ,

χn(T ) =∂n

∂ (µ/T )np

T 4

∣

∣

∣

µ=0. (16)

Thus, for sufficiently small values of µpc, the pseudo-critical pressure, ppc(T ) = p(T, µpc(T )), is given by

ppc(T ) = p(T, µ = 0) + T4∑

n>0

χ2n(T )

(2n)!

(

µpc(T )

T

)2n

.

(17)To second order in µpc this can be evaluated usingEq. (14). In particular, the slope of the pseudo-criticalpressure at µ = 0 is given by

∂

∂Tppc(T, µ = 0)|T=T× = s(T×, µ = 0)−

T 3×2κ

χ2(T×) .

(18)The entropy density, s, the curvature of the pseudo-critical line, κ, and the second-order susceptibility, χ2,have been calculated on in lattice QCD with physicalquark masses and in the continuum limit [6, 51]. Us-ing those values, we then find that the sign of the aboveslope depends on the definition of the pseudo criticality.If it is defined by the inflection points of either the chi-ral condensate or the strangeness susceptibility we findthe slope of the pressure along the pseudo-critical to be∂T ppc = −2.3T 3× or ∂T ppc = −1.1T 3×, respectively. Inboth case the slope is negative meaning the pressure in-creases as we decrease the temperature, similar to theHQ result depicted in Fig. 4. If, on the other hand, thepseudo-critical line is defined by the inflection point ofs/T 3, the slope of the pseudo-critical pressure turns outto be positive, ∂T ppc = +2.4T

3×. This is a simple con-

sequence of the fact that the curvature of the pseudo-critical line, κ, is a more than a factor of two larger inthis case.Thus, from the above considerations we must, regret-

tably, conclude that at present lattice QCD calculationscannot distinguish between the two scenarios at hand: aliquid-gas type behavior where the pseudo-critical pres-sure increases with temperature or a decreasing behavioras is suggested by phenomenology and implemented inthe HQ EoS used in the present study.

IV. FLUID DYNAMICAL CLUMPING

Next we employ the two equations of state in dynami-cal calculations and look for qualitative and quantitativedifferences between the results. Specifically, we employideal fluid dynamics to study systems as they expandthrough the unstable phase region. Dissipative effectsare not expected to play a decisive role for the spinodalclumping [38], because even though the inclusion of vis-cosity generally tends to slow the growth, the dissipativemechanisms also lead to heat conduction which has theopposite effect and also enlarges the unstable region. Theequations of motion derived from conservation of four-momentum and net baryon number are solved by meansof the code SHASTA [52] in which the propagations in thethree spatial dimensions are carried out consecutively.In order to obtain a proper description of the spinodal

growth rates in the mechanically unstable region of the

7

phase diagram, it is essential to introduce finite-densityeffects [29, 38]. This was done in Ref. [35] by augmentingthe pressure obtained from the EoS, p0(ε, ρ), by a gra-dient term, δp(r) = −a2ρ(r)∇2ρ(r)εs/ρ2s, so the localpressure is given by p(r) = p0(ε(r), ρ(r)) + δp(r). Forthe HQ EoS, the resulting simulation tool was verified toyield the correct growth rates of the mechanical instabil-ities in the spinodal phase region [35]. Furthermore, itwas found that the associated spinodal instabilities maycause significant amplification of initial density irregular-ities in relativistic lead-lead collisions [35, 36].We use this tool to investigate the effect of the differ-

ent equations of state on observables for the QCD phasetransition at large baryon densities. To facilitate thecomparisons, we employ the same range a in the gra-dient term also for the PQM EoS (namely a = 0.033 fm)which then yields a significantly smaller surface tension(namely ≈ 2MeV/fm2 and compared to ≈ 10MeV/fm2for the HQ EoS). This is not surprising because the PQMdensities tend to be smaller, as then also the differencesbetween the coexistence densities are, and the PQM EoSlooks rather similar to the LG EoS which has a smallersurface tension (namely ≈1MeV/fm2).In particular, we will study the evolution of spherically

symmetric systems, where the energy and baryon densi-ties (ǫ and ρ) are initialized according to a Woods-Saxondistribution,

ǫ(r) = ǫI [1 + e(r−c)/w]−1 , (19)

ρ(r) = ρI [1 + e(r−c)/w]−1 , (20)

where r is the distance to the center and ǫI and ρI are thecorresponding central density values. For the radius weuse c = 4 fm and the width parameter is w = 0.5 fm. Toprovide initial seeds of fluctuations to be amplified, wemodify the density profile by adding random fluctuationsδǫ(r) and δρ(r) to the local densities, ǫ(r) and ρ(r). Theamplitudes of the fluctuations are constrained to a max-imum of ±20% of the corresponding local densities (19)and they are uniformly distributed within this range.For both equations of state, we initialize the system at

a temperature of 100 MeV and a density that correspondsto a phase point (ε, ρ) slightly above the unstable region(for the PQM EoS) or just inside the unstable region(for the HQ EoS). We obtain the strongest ’clumping’when the system is initialized inside the spinodal region.For the PQM model such a system would have negativepressure, as shown in Fig. 3. Consequently, the systemwill grow denser until its phase point leaves the unsta-ble region and the pressure turns positive. Because thisinitial compression delays the eventual dynamical expan-sion, the PQM calculation is initialized already at a den-sity above the unstable region, thus eliminating the delaythat would otherwise be caused by the negative pressure.As the system evolves, we extract several quantities

that might be sensitive to the developing instabilities.To establish a suitable reference, we also let the systems

evolve from the same initial state but with the corre-sponding stable EoS obtained by eliminating the phase-transition instabilities by means of Maxwell construc-tions, thus preserving the features outside the phase-coexistence region.For the PQM EoS, Fig. 5(a) shows the initial density

distribution in the xy plane, as obtained by augmentingEqs. (19-20) by fluctuations. This initial state is thenevolved with ideal fluid dynamics (with the gradient termincluded) using the unstable EoS. The resulting densitydistribution after 160 fm/c is shown in Fig. 5(b). Evenafter such a long time one finds meta-stable clusters ofhigh baryon density with an asymmetric angular distri-bution, an observation that was first reported in [44].This phenomenon is a direct consequence of the liquid-gas nature of the PQM model, which predicts phase co-existence between the dense phase and the vacuum atvanishing temperature. As a consequence, the pressureeven at T = 100MeV is very small (see Fig. 2) result-ing in a slow expansion. In addition, due to the smallpressure difference between the dense phase and the vac-uum, the clusters generated during the unstable stageare nearly stable and therefore live for a long time, be-fore they eventually slowly disappear on a timescale ofhundreds of fm/c.The situation is quite different in case of the HQ EoS.

As in the calculation based on the PQM EoS, the fluc-tuations are quickly amplified to form small clusters, il-lustrated in Fig. 6(a). However, in contrast to the PQM,the QGP clusters disappear rather quickly after the sur-rounding hadronic matter expands and dilutes. Thisis due to the very high pressure difference between thedense phase and the vacuum, as depicted in Fig. 1. Thediluted density distribution, after a time of only 10 fm/c,is shown in Fig. 6(b), where it is evident that the previ-ously formed dense clusters have been completely washedout.Consequently, the two scenarios, PQM and HQ, ex-

hibit the same qualitative features but lead to quantita-tively different time scales and amplitudes. It is thereforeuseful to now estimate the quantitative differences in ob-servables resulting from the two equations of state.

V. OBSERVABLES

In this section we investigate the sensitivity of vari-ous proposed observables to the formation of clusters ofdeconfined matter.

A. Powers of the density

The density moments provide a convenient globalquantification the irregularities in the evolving netbaryon density ρ(r, t) [35],

〈ρ(t)N 〉 ≡ 1A

∫

ρ(r, t)Nρ(r, t) d3r , (21)

8

0.0 0.50 1.0 1.5 2.0 2.5

t = 0.0 fm (a)

(b)

2 fm

b [ ]

t = 160.0 fm

FIG. 5: (Color online) (a): Initial net baryon density distri-bution in the x-y plane. The initial state is constructed to liejust above the unstable region of the PQM equation of state.(b): Net baryon density distribution in the x-y plane after160 fm/c evolution with the PQM equation of state, that hasan unstable region. The clusters of high baryon density areclearly visible.

where A =∫

ρ(r, t)d3r is the (constant) net baryon num-ber of the system. As already shown in Ref. [35], thehigher moments are more sensitive to the magnitude ofthe fluctuations and we show results for N = 5 here.

Figure 7 shows the time evolution of the fifth densitymoment, divided by its initial value 〈ρ(t = 0)5〉, for thetwo different equations of state considered.

For the HQ EoS the instabilities in the coexistence re-gion lead to local density enhancements which manifestthemselves in the bump of the density moment. Rela-

0 1 2 3 4 5 6 7

t = 2.0 fm (a)

(b)

2 fm

b [ ]

t = 10.0 fm

FIG. 6: (Color online) The net baryon density in the xy plane,ρ(x, y, 0, t), after t = 2 fm/c (a) and t = 10 fm/c (b), as ob-tained with the unstable HQ EoS: Clumps of dense quarkmatter are formed at early times (a) but they disappear afterfurther evolution (b).

tive to the evolution with the Maxwell partner EoS, themaximum enhancement of the fifth density moment isabout a factor of 3-4. But once the surrounding mediumhas become sufficiently dilute these clumps again dissolveresulting in a rapid drop of the density moment. Thelargest degree of enhancement occurs when most of thesystem is inside the unstable phase region. After the sys-tem leaves this region the density quickly becomes similarto that obtained with the stable partner EoS.The PQM EoS yields a quite different picture. Be-

cause of the low pressure at the phase-coexistence line,the dynamical evolution is considerably slower and the

9

1 10 100 20010-3

10-2

10-1

1002x100

HQ EoS Unstable Maxwell

PQM EoS Unstable Maxwell

<(t)

5 >/<

(t=0)

5 >

Time [fm/c]

FIG. 7: (Color online) The fifth density moment (see Eq.(21)), normalized to its initial value, as a function of time forthe HQ (solid) and the PQM (dashed) EoS. In each case, theupper curve (red) depicts results obtained with the unstableEoS, while the lower curve (black) has been obtained with thecorresponding Maxwell partner EoS.

density moment starts to increase only after a long time.And because the created clumps are almost stable, andonly decay very slowly when embedded in a very dilutegas, the density enhancements become much larger thanwas the case for the HQ EoS. This qualitative differenceis clearly brought out in Fig. 7 for the fifth density mo-ment. We also note its oscillatory behavior which canbe ascribed to the longevity of the created clusters: Asthe clusters are formed, the inflowing baryon current willover-compress the dense quark droplet which in turn willcause it to subsequently expand slowly bringing it againslightly inside the unstable region, thus causing a second(somewhat weaker) over-compression and so on.The calculation also shows that the PQM density mo-

ment starts to decrease appreciably only after a very longtime (100− 200 fm). A similar slow decrease of the PQMdensity moments was also observed in a different sim-ulation [44]. This clearly shows how the liquid-gas typeproperties of the deconfinement transition implied by thePQM EoS may yield potentially misleading results.

B. Composite production

Another potential signal observable is the productionyield of composite particles (i.e. light nuclei such asdeuterons and tritons) from phase-space coalescence ofnucleons. In the simple coalescence picture, the phase-space density of a composite with baryon number A isproportional to the A’th power of the one-particle nu-cleon phase-space density [53–57]. Therefore, because thespinodal instabilities significantly enhance higher pow-ers of the density, as demonstrated in Fig. 7, one mightnaively expect that the composite production should be

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4 NA=8/NA=2 Unstable Maxwell

NA=5/NA=2 Unstable Maxwell

Yie

ld ra

tio

Time [fm/c]

FIG. 8: (Color online) The ratios of the A = 5 and A =8 nuclei over the deuteron yield as functions of time. Forsimplicity, the degeneracy factors for A = 8, A = 5, andA = 2 have been set to equal values. Shown are results forthe HQ equation of state with an unstable region (solid lines)and with a Maxwell construction (dashed lines).

enhanced. However, as the local baryon density is beingenhanced also the local excitation energy per baryon isincreased and that will decrease in the coalescence yield,almost compensating the increase due to the higher den-sity, as we shall demonstrate below.

First let us recall that the coalescence model gives theproduction yield in terms of the nucleon yield as [53–57],

EAdNAd3PA

∝ FA(

[

Edn

d3p

]

p=PA/A

)A

. (22)

Here FA is a coalescence factor related to the quantummechanical overlap integral between the wave functionsof the parent nucleons and the composite to be formed; itdepends only weakly on A. Given the simple assumptionsmade in this work, e.g. for the initial conditions, a cal-culation of the absolute yield of composite nuclei is notwarranted. Instead we focus on the relative compositeyields in the presence or absence of spinodal instabilities.

This simplifies our calculation considerably, becauseRef. [53] showed that for a non-degenerate system the co-alescence yield and the thermal production yield are iden-tical up to factors involving the coalescence radius andthe binding energy of the composites. Because these fac-tors will be the same in all scenarios we are considering,we should get a good estimate for the relative enhance-ment by simply integrating the local thermal compositepopulations. To this end, we apply the Cooper-Frye pre-scription on an isochronous hypersurface, using values ofthe temperature and chemical potential of a hadron gascorresponding to the local energy and baryon densities.The total population of light nuclei with mass number A

10

is then given by

NA =

∫

d3p d3r fA(r,p) , (23)

with fA(r,p) ∝ exp[−(√

m2A + p2−µA(r))/T (r)], where

we have used an effective degeneracy of one because weare interested only in ratios. Furthermore, we assumethat µA(r) = Aµ(r) and mA = AmN , where mN isthe nucleon mass. Figure 8 shows the number of com-posites with baryon number A = 5 and A = 8 relativeto the deuteron yield (A = 2). It is obvious that therelative composite populations increase at the time whenthe baryon density is enhanced, i.e. when the moments ofthe baryon number distributions increase. However, theincrease in the population of composite nuclei is muchsmaller than the increase in the density moments. In-deed, we observe a maximum increase in the relativecomposite production of only up to 20%, as compared towhat is obtained with the corresponding Maxwell EoS.

C. Angular correlations

As can be seen from Fig. 5, there seems to be strongirregularities in the angular distribution of the baryondensity in the case of strong clumping. It has been as-serted in [44] that this will lead to large higher momentsof the spacial angular distribution of the baryon number.In this section we will investigate the angular irregular-ities further by calculating the two-particle angular cor-relations of baryons, in both coordinate and momentumspace.In position space these can be computed directly from

the fluid dynamical simulations,

V posn =1

N2

N∑

i,j=1

ρi ρj cos(n(φposi − φ

posj )) , (24)

where ρi is the net baryon density at the computa-tional lattice point i, φposi is its azimuthal angle, andN =

∑

i ρi.However, experiments usually measure the

momentum-space asymmetries vn and momentumcorrelations rather than coordinate-space asymmetries.The azimuthal momentum distribution of the emittedparticles is commonly expressed as

dN

dΦmom∝ 1 +

∞∑

n=1

2vn cos (n(Φmom −Ψn)) (25)

where vn is the magnitude of the n’th order harmonicterm relative to the angle of the initial-state spatial planeof symmetry Ψn. To calculate the two-baryon correla-tions in momentum space we need the relative angle be-tween the momenta of baryons coming from the cells iand j, Φmomij ,

V momn =1

Nij

∑

ij

ρi ρj cos(n∆Φmomij ) . (26)

The angle of the momentum of a baryon coming from celli is determined by sampling the Cooper-Frye equation[58, 59] with the local values of four velocity uν , tem-perature T , and chemical potential µ. This method willcreate a statistical uncertainty which can be minimizedby making multiple samplings of V momn and averagingEq. (26) over all samplings.It can be shown [60, 61] that the extracted correlation

measures V momn are related to the squares of the Fouriercoefficients of the angular distributions:

V momn = v2n (27)

assuming no corrections due to momentum conservationand resonance decays. Figures 9 and 10 show our resultsfor the coordinate and momentum space angular correla-tions. We compare results with the PQM model, at theend of the evolution (see figure 7), and the HQ model,also at the end of the evolution.As expected from Ref. [44], the coordinate-space cor-

relations from the PQM model are very large when thesystem goes through the unstable region. As shown insection IV, using the unstable PQM equation of statewill create large quark clusters, containing most of thesystem’s baryon number. This of course leads to strongcorrelations in coordinate-space and several moments areeven larger than 10%. When the Maxwell constructedequation of state is used no clusters are formed andthe baryon matter is distributed more homogeneously incoordinate-space, resulting in no strong spacial correla-tions.The calculation with the HQ model shows much

smaller coordinate-space correlations, of at most 2− 3%.This result is also understandable, as the produced clus-ters contain only a small fraction of the system’s baryonnumber and quickly become unstable and expand. How-ever, the clumping obtained with the unstable EoS is stilllarger than with the corresponding Maxwell constructionwhich again showed no density enhancement.Figure 10 shows how these coordinate-space irregu-

larities are transformed into momentum-space correla-tions. The high pressure along the coexistence line inthe HQ model leads to an efficient translation of coor-dinate space irregularities into momentum space. Im-portantly, the difference between the unstable EoS andthe Maxwell EoS is not very large. Only a factor 1.4increase in the momentum correlation amplitude is ob-served. The considerable increase in the baryon density,due to the clumping in our simulations with the unsta-ble EoS, does not lead to a considerable change in thepressure gradients. Consequently, the additional spacialanisotropies are not reflected strongly in the final mo-mentum anisotropies.Due to the small pressure in the PQM equation of

state, the coordinate-space asymmetries here cause evenless momentum-space asymmetries, which is already seenfor the Maxwell constructed EoS that has no instabilities.Even after an evolution time of 160 fm, which is muchlonger than the expected lifetime of the fireball, the un-

11

2 3 4 5 6 7 810-3

10-2

10-1

100Coordinate space PQM

Unstable Maxwell

HQ Unstable Maxwell

Vn ~

vn

n

FIG. 9: (Color online) The position-space two-baryon angularcorrelation strength V pos

nfor various values of n as obtained

with the HQ (circles) and PQM (squares) EoS. The resultsobtained with an unstable EoS (solid) are contrasted withthose employing the Maxwell partner (open).

stable PQM EoS shows no strong angular momentum-space correlations, even though it had very large angularcorrelations in coordinate space.

VI. SUMMARY

We have investigated the effect of qualitatively differ-ent types of two-phase equation of state on the expansiondynamics and on possible experimental signals of the ex-pected QCD phase transition at large baryon densities.We considered two equations of state representing twoqualitatively different EoS classes. On the one hand, asan example for an EoS exhibiting phase coexistence be-tween compressed nuclear matter and the dense quarkphase, we considered an EoS that is constructed by inter-polating between a hadron gas and a quark-gluon phase.On the other hand, as a representative for the class ofliquid-gas type equations of state, we used the EoS froma PQM-like model.We find that the PQM model shows a transition that is

very similar to that of the liquid-gas transition in nuclearmatter and thus this model differs qualitatively from theHQ model with regard to the thermodynamic proper-ties near the phase coexistence line. We also comparedthe effective equations of state with a Taylor expansionof recent lattice QCD data. Unfortunately, the predictedslope of the pressure along the pseudo-critical line at van-ishing chemical potential depends of the specific defini-tion of pseudo criticality. Therefore, at present latticeresults are not able to discriminate between the liquid-gas type behavior of the PQM equation of state and theHQ equation of state. This is unfortunate, as the twotypes of EoS yield dramatically different pressures in thehigh-µ region.

2 3 4 5 6 7 8

10-3

10-2

10-1PQM

Unstable Maxwell

HQUnstableMaxwell

Vn ~

vn

n

Momentum space

FIG. 10: (Color online) The momentum-space two-baryonangular correlation strength V mom

nfor various values of n as

obtained with the HQ (circles) and PQM (squares) EoS. Theresults obtained with an unstable EoS (solid) are contrastedwith those employing the Maxwell partner (open).

The qualitative differences between the two equationsof state examined in this work lead to significant quanti-tative differences in the time evolution of fireballs that ex-pand through the respective unstable region of the phasediagram. In the PQM model the lifetime of quark clus-ters is orders of magnitude longer than what is usuallyexpected for the timescales of heavy-ion collisions and itpredicts stable dense quark matter droplets at zero tem-perature, i.e. coexisting with the vacuum.

The instabilities associated with the presence of a first-order phase transition lead to large irregularities in thespatial distribution of the baryon number. However,these irregularities are not translated into significant mo-mentum correlations. We also show that quark clusterformation only leads to an small increase in the produc-tion of composite nuclei, of less than 20%. This increaseis only observed when the system is inside the unstableregion. Once the system has left the coexistence regionand dispersed all signals seem washed out.In conclusion, we have shown that the qualitative dif-

ferences between the PQM and HQ equations of statelead to considerable differences in the dynamical evolu-tion of the system. Given the arguments provided in thispaper, we believe that these qualitative differences per-sist independent of the specific EoS model adopted ofeither the LG type or a more plausible type. However,the observation of these differences is a challenging task.The two rather intuitive observables explored here haveshown little sensitivity to either the instabilities associ-ated with the first-order phase co-existence or the dy-namical evolution of the system. Thus the key questionon how to detect those instabilities, if indeed present,remains unsettled.

Furthermore, the qualitative behavior of the pressurealong the pseudo-critical line in the QCD phase plane

12

is not yet known. Lattice QCD calculations of higher-order Taylor coefficients for the pseudo-critical line andthe associated pressure might help to clarify this cru-cial issue. Meanwhile, as long as it remains an openquestion whether the pseudo-critical pressure increasesor decreases, both possibilities must be considered whenvalidating any potential phase-transition signal with dy-namical calculations similar to the ones presented here.

VII. ACKNOWLEDGMENTS

The authors would like to thank M. Prakash for discus-sions regarding the equation of state of neutron stars. VK

would like to the thank the EMMI Rapid Reaction TaskForce on “Probing the Phase Structure of Strongly Inter-acting Matter with Fluctuations” where some of the ideasfor this work were initiated. This work was supportedby GSI and Hessian initiative for excellence (LOEWE)through the Helmholtz International Center for FAIR(HIC for FAIR) and by the Office of Nuclear Physics inthe U.S. Department of Energy’s Office of Science underContract No. DE-AC02-05CH11231. JS was supportedin part by the Alexander von Humboldt Foundation as aFeodor Lynen Fellow.

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