Equation of state at finite densities for QCD matter in nuclear collisions

Akihiko Monnai,1 Bjorn Schenke,2 and Chun Shen3, 4

1KEK Theory Center, Institute of Particle and Nuclear Studies,High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan2Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA

3Department of Physics and Astronomy, Wayne State University, Detroit, Michigan, USA4RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA

(Dated: August 1, 2019)

We construct the QCD equation of state at finite chemical potentials including net baryon, electriccharge, and strangeness, based on the conserved charge susceptibilities determined from lattice QCDsimulations and the equation of state of the hadron resonance gas model. For the application torelativistic heavy ion collisions we consider the situation of strangeness neutrality and matter witha fixed electric charge-to-baryon ratio, resembling that of heavy nuclei. The importance of finiteelectric charge and strangeness chemical potentials for particle production in heavy ion collisions isdemonstrated using hydrodynamic simulations.

PACS numbers: 25.75.-q, 21.65.Qr, 12.38.Mh

I. INTRODUCTION

The nearly perfect fluidity of the quantum chromo dy-namic (QCD) matter discovered in heavy ion collisionsat the BNL Relativistic Heavy Ion Collider (RHIC) andCERN Large Hadron Collider (LHC) has provided uswith rare opportunities to experimentally explore the nu-clear equation of state, which encodes the fundamentalthermodynamic properties of nuclear matter. The sys-tem created in these collisions is expected to be decon-fined from hadronic matter to the quark-gluon plasma(QGP) above approximately two trillion degrees Kelvin.

The theoretical study of the non-perturbative QCDequation of state dates back to the MIT bag model [1, 2]where hadrons are assumed to be located in a bag em-bedded in the QCD vacuum characterized by the bagconstant. This prescription introduces confinement phe-nomenologically. Also, several model approaches, suchas the potential model [3] and the Nambu-Jona-Lasiniomodel [4, 5], have been proposed to understand the ther-modynamic properties of QCD. A more comprehensivepicture became available with the advent of first-principlecalculations based on lattice QCD techniques. Thequark-hadron phase transition is found to be a crossoverby (2+1)-flavor lattice QCD calculations, in contrast tothe SU(3) pure gauge case where a first-order phase tran-sition is predicted [6–8]. Recent lattice QCD simulationswith a physical pion mass have been able to provide uswith the realistic equation of state at finite temperaturesand vanishing chemical potentials [9, 10].

At finite densities the equation of state is not wellknown, owing to the fermion sign problem of the firstprinciple method (for a review see [11]). Several tech-niques have been developed in lattice QCD, includingthe Taylor expansion method [12, 13], the imaginarychemical potential method [14–16], Lefschetz thimble de-composition [17, 18], and the complex Langevin method[19–21], but so far no complete calculations are availableat larger chemical potentials. There can be non-trivial

structures in the µB-T QCD phase diagram other thanthe QGP phase [22]; it is conjectured that there is a criti-cal point based on the chiral model that predicts that thequark-hadron crossover becomes a first-order transitionat some finite temperature and chemical potential [23].Beam energy scan (BES) programs are being performedto explore finite-density QCD matter and determine itsdetailed phase structure at RHIC and the CERN SuperProton Synchrotron (SPS), and are planned at severalfacilities including the GSI Facility for Antiproton andIon Research (FAIR), JINR Nuclotron-based Ion ColliderfAility (NICA) and JAEA/KEK Japan Proton Acceler-ator Research Complex (J-PARC).

From the viewpoint of hydrodynamic modeling of rela-tivistic nuclear collisions, the equation of state is neededto close the set of equations of motion, by characterizingthe thermodynamic properties of the system. The equa-tion of state at vanishing density obtained from latticeQCD calculations has been employed in comparisons ofhydrodynamic simulations with experimental data fromheavy-ion collisions [24–29], where bulk observables aregenerally well reproduced. For quantitative predictionsand analysis of the BES experimental data, an equationof state at finite chemical potentials is needed as input tohydrodynamic models. Several works have been devotedto the construction of such quantitative models of thefinite-density equation of state, including Refs. [30–46].

In this work, we present a framework to constructa QCD equation of state model (neos) with multiplecharges: net baryon (B), strangeness (S) and electriccharge (Q) based on state-of-the-art lattice QCD [47–52]and hadron resonance gas results. A version of the equa-tion of state, which only has baryon chemical potential,has previously been introduced and used in Refs. [53–59]. We numerically calculate the equation of state withconditions on the conserved charges close to those in rel-ativistic heavy-ion collisions. This analysis is expectedto be relevant in mid to low energy heavy-ion collisionsand at forward rapidity where the fragments of shattered

arX

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902.

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5v2

[nu

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h] 3

0 Ju

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9

2

nuclei are relatively abundant [60, 61]. The presence andinterplay of the chemical potentials are expected to playan important role in for example the hadron chemistryor particle abundances (see e.g. [62]), and the separationof hadron and anti-hadron flow observables [63–65].

The multi-dimensional phase diagram has been studied[66–72] and some conjecture non-trivial phase structures.In this study, we consider a crossover equation of stateas a baseline for the application to relativistic nuclearcollisions.

The paper is organized as follows. In Sec. II, the con-struction of finite-density equations of state based on theTaylor expansion method for lattice QCD and the hadronresonance gas is presented. The numerical evaluationof the hybrid equation of state is performed in Sec. III,where the strangeness neutrality condition and the fixedcharge-to-baryon ratio of nuclei are taken into account.In Sec. IV, particle ratios are estimated in hydrodynamicsimulations assuming different conditions on the chargecontent of the system. Sec. V presents conclusions anddiscussions. Natural units c = ~ = kB = 1 and theMinkowski metric gµν = diag(+,−,−,−) are used.

II. THE EQUATION OF STATE

Based on the Taylor expansion method [12, 13] we em-ploy lattice QCD results of the conserved charge suscep-tibilities to construct the equation of state in the QGPphase. In the hadronic phase we use the equation of stateof a hadron resonance gas, because the Taylor expansionmethod is not reliable at low temperatures. The use of anon-interacting resonance gas model is partly motivatedby the good agreement between thermodynamic quanti-ties at vanishing chemical potential, including suscepti-bilities, from lattice QCD and the hadron resonance gas.Also, the Cooper-Frye prescription [73] of kinetic freeze-out requires that the hydrodynamic equation of state pre-cisely matches that of the kinetic theory description ofthe hadron resonance gas on the freeze-out hypersurfacefor the successful conservation of energy-momentum andall charges. If at low temperatures the lattice result wasused instead of the hadron resonance gas model, the trun-cation of the Taylor expansion at finite order would leadto an underestimation of the pressure in the hadronicphase, because higher order susceptibilities can be largefor the hadron resonance gas in the Boltzmann limit, e.g.,χB2n/χ

B2 = 1.

A. Lattice QCD equation of state

We consider the Taylor expansion method of latticeQCD as mentioned earlier. For the three-flavor QCDsystem, the expansion of the pressure around the vanish-

ing density limit reads

P

T 4=P0

T 4+∑l,m,n

χB,Q,Sl,m,n

l!m!n!

(µBT

)l(µQT

)m(µST

)n, (1)

where P is the pressure, P0 is the pressure at vanishingchemical potentials, T is the temperature, and µB,Q,S arethe chemical potentials of baryon number, electric charge,

and strangeness, respectively. χB,Q,Sl,m,n is the (l+m+n)-thorder susceptibility defined at vanishing chemical poten-tials:

χB,Q,Sl,m,n =∂l∂m∂nP (T, µB , µQ, µS)/T 4

∂(µB/T )l∂(µQ/T )m∂(µS/T )n

∣∣∣∣µB,Q,S=0

. (2)

The number l+m+n should be even, owing to the matter-antimatter symmetry. The expansion is valid only whenthe fugacity µB,Q,S/T is sufficiently small. The latticeQCD results are parametrically extrapolated to high tem-peratures under the condition that they do not violatethe Stefan-Boltzmann limits. See Appendix A.

B. Hadron resonance gas equation of state

The hydrostatic pressure of the hadron resonance gascan be written as

P = ±T∑i

∫gid

3p

(2π)3ln[1± e−(Ei−µi)/T ]

=∑i

∑k

(∓1)k+1 1

k2gi

2π2m2iT

2ekµi/TK2

(kmi

T

),(3)

where i is the index for particle species, gi is the degen-eracy, mi the particle’s mass, and K2(x) is the modifiedBessel function of the second kind. The index k describesthe expansion of quantum distributions around the clas-sical ones. It is generally sufficient to take into accountthe contributions of k ≤ 3 for pions, k ≤ 2 for kaons andk = 1 for the heavier particles. The upper signs are forfermions and the lower signs for bosons. The hadronicchemical potential is µi = BiµB + QiµQ + SiµS whereBi, Qi, and Si are the quantum numbers of net baryon,electric charge, and strangeness, respectively.

C. Hybrid equation of state

The complete nuclear equation of state is constructedby connecting the pressure of the lattice QCD equationof state to that of the hadron resonance gas model [74]

P

T 4=

1

2[1− f(T, µJ)]

Phad(T, µJ)

T 4

+1

2[1 + f(T, µJ)]

Plat(Ts, µJ)

T 4s

, (4)

where J = B,Q, S. Here f(T, µJ) is an arbitrary func-tion for the connection of the two functions which satisfies

3

f → 1 when T Tc and f → 0 when T Tc, where Tcis the connecting temperature. In this work we choose fto be a hyperbolic tangent, defined in (14). A tempera-ture shift Ts(T, µJ) is introduced phenomenologically topreserve the monotonicity conditions of thermodynamicvariables at larger chemical potentials. Since Ts is gen-erally a function of T and µJ , one can define

Plat(T, µJ) = Plat(Ts, µJ)× T 4

T 4s

, (5)

which is the shifted QGP equation of state. Here Plat

should reduce to Plat at small chemical potentials. Whilein this work we will not use a temperature shift, theshifting temperature is used in similar constructions ofthe equation of state in previous works [53–59].

We require that the thermodynamic variablesmonotonously increase as functions of T and µJ ,respectively, as

∂2P

∂T 2=

∂s

∂T> 0, (6)

∂2P

∂µ2J

=∂nJ∂µJ

> 0. (7)

Those conditions may be trivially satisfied for the hadronresonance gas or lattice QCD equation of state whenµJ/T is not large, but the connection procedure can makeit non-trivial. The conditions can be expressed as

∂2P

∂T 2=

1

2[1− f(T, µJ)]

∂shad(T, µJ)

∂T

+1

2[1 + f(T, µJ)]

∂slat(T, µJ)

∂T

+∑J

∂f(T, µJ)

∂T[slat(T, µJ)− shad(T, µJ)]

+1

2

∑J

∂2f(T, µJ)

∂T 2[Plat(T, µJ)− Phad(T, µJ)]

> 0. (8)

∂2P

∂µ2J

=1

2[1− f(T, µJ)]

∂nJhad(T, µJ)

∂µJ

+1

2[1 + f(T, µJ)]

∂nJlat(T, µJ)

∂µJ

+1

2

∑J

∂f(T, µJ)

∂µJ[nJlat(T, µJ)− nJhad(T, µJ)]

+1

2

∑J

∂2f(T, µJ)

∂µ2J

[Plat(T, µJ)− Phad(T, µJ)]

> 0. (9)

Assuming that the thermodynamic quantities on the lat-tice QCD side are larger than those on the hadron reso-nance gas side, the conditions are still not trivially satis-fied as the second order derivatives of f can be negative.These conditions will be handled numerically in the nextsection.

The thermodynamic variables, the entropy density s,the conserved charge densities nJ , the energy density e,and the sound velocity cs are given as

s =∂P

∂T

∣∣∣∣µJ

, (10)

nJ =∂P

∂µJ

∣∣∣∣T,µK

, (11)

e = Ts− P +∑J

µJnJ , (12)

c2s =∂P

∂e

∣∣∣∣nJ

+∑J

nJe+ P

∂P

∂nJ

∣∣∣∣e,nK

, (13)

respectively, using the standard thermodynamic rela-tions. Here J,K = B,Q, S and J 6= K.

D. Multiple charges in nuclear collisions

A standard nucleus is made of protons and neutronsso the averaged density of strangeness is zero, which maybe expressed as nS(T, µB , µQ, µS) = 0. However, ne-glecting electric charge for the moment, the conventionalchoice of µS = 0 leads to nS 6= 0, because the strangenessdensity is dependent on µB as the strange quark carriesboth net baryon number and strangeness. Thus, in thepresence of a finite net-baryon number, µS should gen-erally be non-vanishing, so nS = 0 can be fulfilled. Thecondition, of course, can in principle be locally brokenin the presence of geometrical fluctuations or diffusionprocesses. The equation of state with the strangenessneutrality condition can be expressed in terms of T andµB because µS = µS(T, µB).

The electric charge density nQ(T, µB , µQ, µS) is non-vanishing in nuclei as nQ = (Z/A)nB , where Z is theproton number and A is the nucleon number. The list ofZ/A ratios of the nuclei used in collider experiments isshown in Table I. The typical ratio for heavy nuclei suchas Au or Pb is Z/A ≈ 0.4. The precise nQ dependenceis expected to become more important when comparingcollisions of isobar systems.

III. NUMERICAL CONSTRUCTION

In this section, we numerically construct the hybridQCD equation of state at finite densities (4). On thelattice QCD side, we make use of one of the latest (2+1)-flavor calculations of the equation of state at vanishingchemical potentials [10] and the diagonal and off-diagonalsusceptibilities up to the fourth order [50–52, 75]. Inaddition, we introduce some of the most relevant sixthorder susceptibilities, to allow for a proper matching ofall quantities, as discussed later. The specific functionalforms of parametrization are summarized in Appendix B.

All hadron resonances from the particle data group[76] with u, d, and s constituent quark components and

4

Nucleus Z/A11H 1.00021H 0.50032He 0.6672713Al 0.4816329Cu 0.4609640Zr 0.4179644Ru 0.45812754 Xe 0.42519779Au 0.40120882Pb 0.39423892U 0.387

TABLE I. Ratios of protons to nucleons Z/A for the nucleiused in the collider experiments at RHIC and LHC.

masses smaller than 2 GeV are taken into account in theresonance gas model. We use

f(T, µB) = tanh[(T − Tc(µB))/∆Tc] , (14)

where Tc(µB) = 0.16 GeV − 0.4 × (0.139 GeV−1µ2B +

0.053 GeV−3µ4B) motivated by and modified from the

chemical freeze-out curve [77]. The dependence ofthe connecting temperature on strangeness and electriccharge chemical potentials is assumed to be weak andneglected for the moment. The connecting width is∆Tc = 0.1Tc(0). Here, we choose not to perform a tem-perature shift and use Ts = T . It should be noted thatthis is not a unique choice of the parameters, but the ther-modynamic monotonicity conditions (6) and (7) leave arather narrow window for the possible parameter values.The smooth matching leads to an equation of state witha crossover transition. Implementation of a QCD criticalpoint and the first-order phase transition is also possiblefor different choices of f . It will be discussed elsewhere asthe location of the critical point and the critical behav-ior near it are currently not well known. For a possibleapproach to include a critical point see [39, 41, 46].

The sixth-order susceptibilities should be relevant nearand below the crossover transition. The term involvingχB6 is naively expected to give the largest contributionto the pressure and the net baryon number because ofthe hierarchy in the chemical potentials µB > µS > µQin nuclear collisions. The strangeness density and theelectric charge density are not directly sensitive to χB6 ,because they are derivatives with respect to µS or µQ,

respectively, implying that the terms involving χB,S5,1 and

χB,Q5,1 will be the important ones for them. We intro-duce those three susceptibilities in a phenomenologicalapproach so that nB , µS , and µQ are smooth functionsof T and µB , and that the results of the hadron reso-nance gas model are preserved below Tc, because of therelatively large uncertainties in the current lattice cal-culations of higher order susceptibilities. We find thatthe effects of the sixth-order susceptibilities are limited

0 0.1

0.2 0.3

0.4 0.5

0.6

0 0.1

0.2 0.3

0.4 0.5

0.6

0

1

2

3

4

5

P/T

4

(a) NEOS B s/nB = 420s/nB = 144s/nB = 51s/nB = 30

µB (GeV)T (GeV)

P/T

4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.1

0.2 0.3

0.4 0.5

0.6

0 0.1

0.2 0.3

0.4 0.5

0.6

0

0.2

0.4

0.6

0.8

1

-nS/T

3

(b) NEOS B s/nB = 420s/nB = 144s/nB = 51s/nB = 30

µB (GeV)T (GeV)

-nS/T

3 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

FIG. 1. (Color online) (a) The dimensionless pressure P/T 4

and (b) the dimensionless strangeness density nS/T3 as func-

tions of T and µB where µS = µQ = 0. The solid, long-dashed, dash-dotted, and short-dashed lines are the trajecto-ries for constant s/nB = 420, 144, 51, and 30, respectively.

to the large chemical potential regions near the crossovertransition.

For the strangeness and electric charges, we considerthree cases: (i) µS = µQ = 0, (ii) nS = 0 and µQ = 0,and (iii) nS = 0 and nQ = 0.4nB . They are reffered toas neos B, neos BS and neos BQS, respectively. Thefirst is the commonly used scenario in which one assumesthat the net baryon chemical potential is the only non-vanishing one in the system. The second imposes thestrangeness neutrality condition but neglects the electriccharge chemical potential. The third is the most realisticscenario for the collision of heavy nuclei where Z/A ∼ 0.4[52]. It is also straightforward to calculate the equationof state as functions of µB , µQ, and µS for more generalsystems.

5

A. Vanishing strangeness and electric chargechemical potentials µS = µQ = 0

First, the case where µS = µQ = 0 is investigated.The pressure of the resulting equation of state is plot-ted in Fig. 1 (a). One can see the monotonous increaseof P as a function of T or µB . The equation of statereduces to that of lattice QCD at µB = 0 at the van-ishing density limit. The constant entropy density overnet baryon density lines, s/nB = 420, 144, 51, and 30,are plotted to illustrate the most relevant regions for theBES programs. They correspond to Au+Au collisions at√sNN = 200, 62.4, 19.6, and 14.5 AGeV, respectively

[16]. Note that the ratio is roughly constant during thetime evolution in nuclear collisions when the entropy andthe net baryon number are conserved, which is the casefor the nearly-perfect fluid. The trajectory of s/nB is astraight line at higher temperatures where the system isrelatively close to conformal, because s/nB ∼ T/µB . Itturns around near the crossover towards lower temper-atures as pions begin to dominate over protons becauseof the mass difference, and large baryon chemical poten-tial is required to have protons at lower temperatures forkeeping the s/nB ratio fixed. In the limit T → 0, thechemical potential approaches the proton mass.

As discussed earlier, the condition µS = 0, whichis often assumed in nuclear collision analyses, leads toa non-vanishing strangeness density nS . The value of−nS/T 3 is shown in Fig. 1 (b). Positive baryon chem-ical potential leads to negative strangeness density be-cause the s quark has a negative strangeness chemicalpotential. The high temperature behavior can be un-

derstood as nS/T3 ∼ χB,S1,1 µB/T ∼ −µB/3T (A6). The

strangeness density is relatively small at lower tempera-tures because kaons, the lightest strange hadrons, havenet baryon number zero.

B. Strangeness neutrality nS = 0 and vanishingelectric charge chemical potential µQ = 0

We next study the case where nS = 0 and µQ = 0.The pressure is shown as a function of T and µB inFig. 2 (a). The nQ/nB ratio is arbitrary, and approachesnQ/nB ∼ 0.5 in the parton gas limit (A14). One can seethat the equation of state is modified at larger baryonchemical potentials compared with that of the µS = 0case. The constant s/nB lines are also shifted to largerµB (Fig. 2 (b)) because the strangeness neutrality impliesµS ∼ µB/3 at high temperatures. For clarity we showthe projections of the constant s/nB lines onto the µB-Tplane in Fig. 3. Here one can see that µB has to be about3/2 times larger for a given nB when µS 6= 0. The grayarea in the figure shows a conjectured region µB/T > 3where the Taylor expansion method of lattice QCD is notwell-defined. A larger value of µB will result in a largerthermodynamic force ∇µ(µB/T ) for the net baryon dif-fusion current [57]. It will have an important effect on

0 0.1

0.2 0.3

0.4 0.5

0.6

0 0.1

0.2 0.3

0.4 0.5

0.6

0

1

2

3

4

5

P/T

4

(a) NEOS BS s/nB = 420s/nB = 144s/nB = 51s/nB = 30

µB (GeV)T (GeV)

P/T

4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.1

0.2 0.3

0.4 0.5

0.6

0 0.1

0.2 0.3

0.4 0.5

0.6

0

0.05

0.1

0.15

0.2

0.25

µS (

GeV

)

(b) NEOS BSs/nB = 420s/nB = 144s/nB = 51s/nB = 30

µB (GeV)T (GeV)

µS (

GeV

) 0

0.05

0.1

0.15

0.2

FIG. 2. (Color online) (a) The dimensionless pressure P/T 4

and (b) the strangeness chemical potential µS as functions ofT and µB where nS = 0 and µQ = 0. The solid, long-dashed,dash-dotted, and short-dashed lines are the trajectories forconstant s/nB = 420, 144, 51, and 30, respectively.

constraining the net baryon diffusion constant in futurephenomenological studies. While initial strangeness fluc-tuations and a strangeness diffusion current can breakstrangeness neutrality locally, this should not diminishthe effect of enhanced µB on the net baryon diffusion,since it is a sub-leading effect and the strangeness is stillglobally conserved at zero.

Importantly, µs is non-zero at freeze-out, which willaffect results on particle-antiparticle ratios of strangehadrons in hydrodynamic models, as is the case inthermal models [62]. The potentially large effect ofthe strangeness neutrality condition is also discussed inRef. [78].

It should be noted that while we have now imposedmore realistic conditions compared to the previous case,the thermodynamic properties of the QCD system itselfremain the same, we merely look at different slices of themulti-dimensional equation of state.

6

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6

T (

Ge

V)

µB (GeV)

NEOS B, s/nB=420

NEOS B, s/nB=144

NEOS B, s/nB=51

NEOS B, s/nB=30

NEOS BS, s/nB=420

NEOS BS, s/nB=144

NEOS BS, s/nB=51

NEOS BS, s/nB=30

FIG. 3. (Color online) The comparison of the trajectories forconstant s/nB = 420 (solid line), 144 (long-dashed line), 51(dash-dotted line), and 30 (short-dashed line) lines betweenneos B and neos BS denoted by narrow and thick lines, re-spectively. The gray area shows the region where µB/T > 3above Tc.

C. Strangeness neutrality nS = 0 and fixed electriccharge-to-baryon ratio nQ = 0.4nB

Finally, we investigate the case where nS = 0 andnQ = 0.4nB . This is the setup most relevant to Au+Auand Pb+Pb collisions. The dimensionless pressure P/T 4

is plotted in Fig. 4 (a). The difference from the previ-ous case is small in this setup but should be meaningfulfor correctly understanding particle-antiparticle ratios ofcharged particles.

The electric charge chemical potential shown in Fig. 4(b) is negative, owing to the interplay of multiple con-served charges. Since the number of neutrons is largerthan that of protons in heavy nuclei, d quarks are slightlymore abundant than u quarks in the QGP phase and π−

more abundant than π+ in the hadronic phase. Whilethe overall system is positively charged, a negative elec-tric chemical potential is needed for describing this situ-ation. µQ becomes positive for the system of 3He sinceZ/A > 1/2. This would have to be taken into accountfor the collisions involving such nuclei.

It should be noted that µQ is small and is rather sensi-tive to the fine structure of the equation of state, includ-ing higher-order susceptibilities, at large chemical poten-tials. This implies that improvement in the lattice QCDcalculations, including higher order susceptibilities, willbe important in quantitative analyses.

D. Discussion

We have constructed the nuclear equation of state un-der several different conditions. We now study the dif-ferences between the different scenarios in more detail.

The sound velocities of the equation of state under

0 0.1

0.2 0.3

0.4 0.5

0.6

0 0.1

0.2 0.3

0.4 0.5

0.6

0

1

2

3

4

5

P/T

4

(a) NEOS BQS s/nB = 420s/nB = 144s/nB = 51s/nB = 30

µB (GeV)T (GeV)

P/T

4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.1

0.2 0.3

0.4 0.5

0.6

0 0.1

0.2 0.3

0.4 0.5

0.6

0

0.01

0.02

0.03

0.04

0.05

-µQ

(G

eV

)

(b) NEOS BQSs/nB = 420s/nB = 144s/nB = 51s/nB = 30

µB (GeV)T (GeV)

-µQ

(G

eV

) 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

FIG. 4. (Color online) (a) The dimensionless pressure P/T 4

and (b) the electric charge chemical potential µQ as functionsof T and µB where nS = 0 and nQ = 0.4nB . The solid, long-dashed, dash-dotted, and short-dashed lines are the trajecto-ries for constant s/nB = 420, 144, 51, and 30, respectively.

the three different conditions along two constant s/nBlines are plotted in Fig. 5. One can see that finite-density effects are visible comparing the sound velocitiesof s/nB = 420 and 30. Around the crossover tempera-ture, the EoS becomes soft and c2s has a minimum. Thelocation of the minimum shifts towards lower tempera-tures as the net baryon density increases. Also, the soundvelocity becomes larger in the QGP phase and smaller inthe hadronic phase at larger chemical potentials. This isbecause the net baryon contribution in c2s (13) is positivefor the former phase and negative for the latter phase.At higher temperatures, it starts to approach the Stefan-Boltzmann limit c2s = 1/3. For the three presented equa-tions of state, c2s reaches 94.8 % of the value the Stefan-Boltzmann limit at T = 0.6 GeV and 97.2 % at T = 0.8GeV for s/nB = 420.

Comparing neos B to neos BS, the strangeness neu-trality condition is found to slightly enhance the soundvelocity in the QGP phase. It should be noted that ifone neglected the derivatives involving nS in the calcu-lation of c2s (13) for neos B, the sound velocity would be

7

0.1

0.2

0.3s/nB=420

cs2

NEOS BNEOS BS

NEOS BQS

0.1

0.2

0.3

0.05 0.1 0.15 0.2 0.25 0.3 0.35

s/nB=30

cs2

T (GeV)

0.1

0.2

0.3

0.05 0.1 0.15 0.2 0.25 0.3 0.35

s/nB=30

cs2

T (GeV)

0.1

0.2

0.3

0.05 0.1 0.15 0.2 0.25 0.3 0.35

s/nB=30

cs2

T (GeV)

FIG. 5. (Color online) The squared sound velocity as a func-tion of the temperature along the constant trajectories ofµS = µQ = 0 (thick solid line), nS = 0 and µQ = 0 (thinsolid line), and nS = 0 and nQ = 0.4nB (thick dotted line)under the conditions s/nB = 420 (top) and 30 (bottom).

further underestimated than our current result, because

c2s 6=∂P

∂e

∣∣∣∣nB

+nBe+ P

∂P

∂nB

∣∣∣∣e

, (15)

when µS = 0, i.e., nS 6= 0, which again highlights theimportance of adequate treatment of the multiple con-served charges. The effects of the fixed charge to baryonratio on the sound velocity is almost negligible. Sincethe effect of the electric chemical potential is not large,the difference in the sound velocity is also not large whenone neglects the derivatives involving µQ and nQ in thedefinition (13).

We next focus on the interplay of the multiple con-served charges and plot an arbitrary constant pressureplane in the chemical potential µB-µS-µQ space at aconstant temperature in the hadronic phase in Figure 6(a). This quantifies the chemical potential dependencesof this thermodynamic quantity. For demonstration, wechoose P/T 4 = 0.8 and T = 0.14 GeV. The inter-cepts of each axis, defined implicitly as P (µint

B , 0, 0) =P (0, µint

Q , 0) = P (0, 0, µintS ), are ordered as µint

B > µintS >

µintQ , partly reflecting the fact that the lightest particles

that carry those charges are protons, kaons, and pions inthe hadronic phase, respectively (3).

Figure 6 (b) presents the same in the QGP phase. HereP/T 4 = 2 and T = 0.2 GeV are considered. The orderingof the intercepts can be seen to be µint

B > µintQ > µint

S inthe QGP phase. This is consistent with the naıve expec-tation that µB/3 ∼ 2µQ/3 ∼ µS in the massless partongas limit (A1)-(A4). The intercept µint

S is slightly larger

0

0.1

0.2

0.3

0.4

0

0.1

0.2

0.3

0

0.1

0.2

0.3

µQ

(G

eV

)

(a) T = 0.14 GeV P/T4 = 0.8

µB (GeV)

µS (GeV)

µQ

(G

eV

)

0

0.1

0.2

0.3

0.4

0

0.1

0.2

0.3

0

0.1

0.2

0.3

µQ

(G

eV

)

(a) T = 0.14 GeV

µB (GeV)

µS (GeV)

µQ

(G

eV

)

0

0.1

0.2

0.3

0.4

0

0.1

0.2

0.3

0

0.1

0.2

0.3

µQ

(G

eV

)

(b) T = 0.2 GeV P/T4 = 2

µB (GeV)

µS (GeV)

µQ

(G

eV

)

0

0.1

0.2

0.3

0.4

0

0.1

0.2

0.3

0

0.1

0.2

0.3

µQ

(G

eV

)

(b) T = 0.2 GeV

µB (GeV)

µS (GeV)

µQ

(G

eV

)

FIG. 6. (Color online) The constant pressure plane as afunction of µB , µQ, and µS in (a) the hadronic phase atP/T 4 = 0.8 and T = 0.14 GeV and (b) the QGP phase atP/T 4 = 2 and T = 0.2 GeV.

owing to the fact that it is still close to the crossover tran-sition and that the strange quarks have a non-negligiblemass. µB takes a maximum value at some positive fi-nite µS because s quarks have positive net baryon num-ber and negative strangeness (A4), i.e., the leading-orderoff-diagonal susceptibility between the net baryon andstrangeness is negative. This is not the case for the cross-coupling between the electric charge and the net baryonor strangeness.

Constant s/nB lines for the case that nS = 0 andnQ = 0.4nB are plotted in Fig. 7 to illustrate the typi-cal range of the chemical potentials covered by heavy-ioncollider experiments. The trajectories coincide at hightemperatures because s/nB ∼ T/µB and the conditionson nS and nQ make µS and µQ roughy proportional toµB . The trajectories slightly bend towards the largerstrangeness chemical potential at large µB and smallµS regions, which correspond to low temperatures belowT ∼ 0.1 GeV. This could be caused by the suppressionof kaons compared with pions owing to the mass differ-ence. The behavior can also be seen in Fig. 2 (b). The

8

0 0.1

0.2 0.3

0.4 0.5

0.6

0

0.05

0.1

0.15

0.2

0.25 0

0.005 0.01

0.015 0.02

0.025 0.03

0.035 0.04

-µQ

(G

eV

)s/nB=420s/nB=144s/nB=51s/nB=30

µB (GeV)

µS (GeV)

-µQ

(G

eV

)

FIG. 7. (Color online) The solid, long-dashed, dash-dotted,and short-dashed lines are the trajectories for constant s/nB

= 420, 144, 51, and 30, respectively, as functions of µB , µQ,and µS .

bending does not occur for the charge chemical potentialbecause pions, the lightest hadrons, have electric charge.It is note-worthy that broader ranges may be exploredin actual collider events since the system is geometricallyfluctuating and large local variation of the entropy-to-conserved-charge ratios can occur.

We note that in the region where the lattice QCD con-tribution dominates, the validity of our parametrizationis limited to the range where µB/T is sufficiently small.From a practical point of view, for the application to nu-clear collisions, however, these regions are not expectedto much affect the bulk physics, because most of the fluidelements do not go through the large µB/T regions nearTc. This can be seen in the constant s/nB lines shown inFigs. 1 through 4.

It would also be interesting to compare our results withthe ones obtained by other lattice QCD approaches tofinite density regions, such as the one from the imaginarychemical potential method [79].

IV. APPLICATION TO NUCLEAR COLLISIONS

To study the effects of imposing strangeness neutral-ity and a realistic charge-to-baryon ratio on observablesin heavy ion collisions, we perform hybrid model cal-culations of Pb+Pb collisions at center of mass energy√s = 17.3 AGeV involving viscous hydrodynamic sim-

ulations with the three different equations of state de-scribed above and a microscopic hadronic afterburner. Amore detailed description of the hybrid model is givenin [80]. We compute particle yields and compare parti-cle ratios to experimental data from the Super ProtonSynchrotron (SPS) [81–86] (compiled in [87]).

For the initial state, we use the dynamical model pre-sented in [56], which provides fluctuating distributions

of net baryon and energy-momentum densities in threespatial dimensions. The 3+1D hydrodynamic simula-tion Music [88–90] is run here with zero bulk viscosityand a constant shear viscosity to entropy density ratioof η/s = 0.08. We switch from hydrodynamics to thehadron cascade UrQMD [91, 92] at a switching energydensity esw, whose value we vary below.

In Fig. 8 we show the particle yields (top) and parti-cle ratios (bottom) from these simulations using esw =

0.26 GeV/fm3, and the neos equations of state with

different constraints on strangeness and electric charge.One can see that imposing strangeness neutrality has avisible effect - mainly on the strange and anti-strange par-ticle yields. This effect is amplified in the particle ratios.Yields of particles with positive strangeness are increasedwhile those of particles with negative strangeness are de-creased, which is due to the finite positive strangenesschemical potential present in neos BS (and neos BQS).

The agreement between the theoretical calculationsand experimental data is improved for most particleswith strangeness in neos BS. In the meson sector whereµB is absent, the ratio of K+ over K− gets enhanced bythe strangeness neutrality condition and agreement withexperimental data at SPS energy improves.

Protons, and to a lesser degree anti-protons, are mod-ified, because in the presence of µS , the baryon chemicalpotential µB also changes. The small change for pions islikely due to the modification of resonance abundanceswhen going from neos B to neos BS.

Introducing the constraint on the electric charge byusing neos BQS, we find only very mild modificationsof the particle yields. The negative µQ leads to a slightincrease of negative relative to positive charged particles,as can be best seen in the plot of the particle ratios asthe difference between the points for neos BS and neosBQS. The introduction of µQ can explain at least qual-itatively that π− are slightly more abundant than π+.The ratio Ω/Ω behaves in the opposite way. Possiblychanges of µB and µS when introducing µQ could con-tribute to this behavior. We note that the main effect ofbaryon–anti-baryon annihilation within UrQMD is thereduction of the anti-proton yield by approximately 35%at√sNN = 17.3 GeV. Yields of Λ and Ω are reduced by

25% and 20%, respectively.In Fig. 9 we study the effect of different switching en-

ergy densities on particle yields (top) and ratios (bottom)for neos BQS. Using a lower switching energy densitymeans assuming that the system is fully chemically equi-librated down to lower temperatures. Anti-baryons aremost sensitive to the switching energy density. The rea-son could be that lower esw means lower temperature atswitching, which goes along with a larger baryon chem-ical potential (see Fig. 3). Lower temperature tends todecrease heavier particles’ yields, while the larger baryonchemical potential will lead to more baryons, weakeningthe effect of lower temperature, and to less anti-baryons,adding to the effect. We find that the experimental dataon particle ratios prefers a switching temperature be-

9

π+ π− K+ K− p p φ Λ Λ Ξ− Ξ+ Ω Ω10−2

10−1

100

101

102

dN/dy

√sNN = 17.3 GeV

esw = 0.26 GeV/fm3

NEOS B

NEOS BS

NEOS BQS

π−/π+K+/K− p/p 2φ(π++π−)

Λ/Λ Ξ−/Λ Ξ+/Ξ− Ω/Ω10−2

10−1

100

rati

o

√sNN = 17.3 GeV

esw = 0.26 GeV/fm3

NEOS B

NEOS BS

NEOS BQS

FIG. 8. (Color online) Top panel: Particle yields from cen-tral Pb+Pb collisions at

√s = 17.3 AGeV determined using

a hybrid calculation consisting of hydrodynamics with theindicated equations of state and a hadronic afterburner. Bot-tom panel: Particle ratios for the three different equationsof state from the same calculation compared to experimentaldata [81–86] (compiled in [87]).

tween esw = 0.16 and 0.36 GeV/fm3.

V. CONCLUSIONS

By matching the Taylor expanded lattice QCD equa-tion of state in the high-temperature region to that ofa hadron resonance gas model at low temperature, wehave constructed the QCD equation of state at finite netbaryon, electric charge, and strangeness chemical poten-tials. We employ phenomenologically motivated sixthorder susceptibilities to allow for a smooth matchingthat respects the thermodynamic monotonicity condi-tions. The equation of state is designed to be used insimulations of collisions of heavy nuclei such as Au or Pbin a wide range of collision energies explored by the beamenergy scan programs.

π+ π− K+ K− p p φ Λ Λ Ξ− Ξ+ Ω Ω10−2

10−1

100

101

102

dN/dy

√sNN = 17.3 GeV

NEOS BQS

esw = 0.16 GeV/fm3

esw = 0.26 GeV/fm3

esw = 0.36 GeV/fm3

π−/π+K+/K− p/p 2φ(π++π−)

Λ/Λ Ξ−/Λ Ξ+/Ξ− Ω/Ω10−2

10−1

100

rati

o

√sNN = 17.3 GeV

NEOS BQS

esw = 0.16 GeV/fm3

esw = 0.26 GeV/fm3

esw = 0.36 GeV/fm3

FIG. 9. (Color online) Top panel: Particle yields from centralPb+Pb collisions at

√s = 17.3 AGeV determined using a hy-

brid calculation consisting of hydrodynamics with the neosBQS equation of state and a hadronic afterburner, using threedifferent switching energy densities. Bottom panel: Particleratios for the three different switching energy densities fromthe same calculation compared to experimental data [81–86](compiled in [87]).

The equation of state without strangeness chemical po-tential leads to the violation of the strangeness neutralitycondition, which should hold in heavy ion collisions. Apositive finite strangeness chemical potential is observedwhen the condition nS = 0 is kept. Also enforcing theapproximate relation between electric charge and baryonnumber for heavy nuclei, nQ/nB = 0.4, introduces a neg-ative finite electric chemical potential. These constraintsshould be important for understanding the difference be-tween particle yields and flow observables of particles andanti-particles within hydrodynamic models of heavy-ioncollisions.

We have presented the effect of enforcing different con-straints on strangeness and electric charge on the particleyields and ratios in

√s = 17.3 AGeV Pb+Pb collisions.

While strange and anti-strange particles are most af-

10

fected by these constraints, modifications of non-strangeparticles were also observed, mostly driven by the finiteµS , less so the finite, negative, µQ. This is understoodby the fact that the introduction of µS (and µQ) will alsoalter µB .

These results are also important because they implythat we do not explore the µB-T plane in the BES ex-periments, but a certain slice in the µB-µQ-µS-T hyper-plane. This can affect the search of the QCD criticalpoint because the traditional critical point at µQ = µS =0 may not be reached.

Further importance may arise with regard to isobarcollisions. The equation of state can be different for dif-ferent isobar collision systems, which should be takeninto account for correctly understanding the backgroundsignals to the chiral magnetic effect. A similar discussionis applicable to small systems, where light nuclei such asproton, deuteron, or 3He tend to have larger Z/A ratios,and the sign of the electric charge chemical potential canbe flipped. It is also possible to perform event-by-eventhydrodynamic analyses distinguishing protons and neu-trons.

Future prospects for model improvements include in-troduction of the full sixth order susceptibilities from lat-tice QCD calculations. Although they are vanishing inthe high temperature limit, they could play a non-trivialrole near the crossover at larger chemical potentials.

Our equation of state model neos is publicly available[93].

ACKNOWLEDGMENTS

The authors thank Frithjof Karsch, Swagato Mukher-jee, and Sayantan Sharma for useful discussion. AMis supported by JSPS KAKENHI Grant NumberJP19K14722. BPS is supported under DOE ContractNo. DE-SC0012704. CS is supported under DOE Con-tract No. DE-SC0013460. This research used resourcesof the National Energy Research Scientific ComputingCenter, which is supported by the Office of Science ofthe U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This work is supported in part bythe U.S. Department of Energy, Office of Science, Officeof Nuclear Physics, within the framework of the BeamEnergy Scan Theory (BEST) Topical Collaboration.

Appendix A: The Stefan-Boltzmann limit

For the massless non-interacting system of u, d and s,the analytical expression of the pressure reads

P

T 4=

8π2

45+

7π2

60Nf +

1

2

∑f=u,d,s

(µfT

)2

+1

4π2

∑f=u,d,s

(µfT

)4

, (A1)

where Nf = 3 is the number of flavors. The speed ofsound (13) is independent of the temperature and chem-ical potentials, c2s = 1/3. It is note-worthy that thesixth and higher order susceptibilities are vanishing inthis case. The susceptibilities of the conserved chargesare given using Eqs. (1), (A1) and

µu =1

3µB +

2

3µQ, (A2)

µd =1

3µB −

1

3µQ, (A3)

µs =1

3µB −

1

3µQ − µS . (A4)

The second-order susceptibilities in the Stefan-Boltzmann limit are

χB2 =1

3, χQ2 =

2

3, χS2 = 1, (A5)

χB,Q1,1 = 0, χB,S1,1 = −1

3, χQ,S1,1 =

1

3, (A6)

and the fourth-order ones are

χB4 =2

9π2, χQ4 =

4

3π2, χS4 =

6

π2, (A7)

χB,S3,1 = − 2

9π2, χB,S2,2 =

2

3π2, χB,S1,3 = − 2

π2, (A8)

χB,Q3,1 = 0, χB,Q2,2 =4

9π2, χB,Q1,3 =

4

9π2, (A9)

χQ,S3,1 =2

9π2, χQ,S2,2 =

2

3π2, χQ,S1,3 =

2

π2, (A10)

χB,Q,S2,1,1 =2

9π2, χB,Q,S1,2,1 = − 2

9π2, χB,Q,S1,1,2 = − 2

3π2.

(A11)

They are used as anchors for the high temperature be-haviors of the equation of state, where lattice QCD dataare scarce, so that the basic thermodynamic features arenot violated in the large T limit.

One can analytically solve the linearized equations forcharge densities nB , nQ, and nS obtained by keeping thesecond order diagonal and off-diagonal susceptibilities:nBnQ

nS

= T 2

χB2 χB,Q1,1 χB,S1,1

χB,Q1,1 χQ2 χQ,S1,1

χB,S1,1 χQ,S1,1 χS2

µBµQµS

. (A12)

The solutions are

µB = (5nB − nQ + 2nS)/T 2, (A13)

µQ = (−nB + 2nQ + nS)/T 2, (A14)

µS = (2nB − nQ + 2nS)/T 2, (A15)

in the Stefan-Boltzmann limit. In the case where nS =0 and nQ = 0.4nB , those can be expressed as µB =4.6nB/T

2, µQ = −0.2nB/T2, and µS = 1.6nB/T

2. It isworth mentioning that the sign of µQ is rather sensitiveto the proportionality constant between the net baryonand charge densities and turns positive at nQ = 0.5nB .

11

Susceptibility h0 h1 (GeV2) T1 (GeV) ∆T1 (GeV) T2 (GeV) ∆T2 (GeV) h2 (GeV−n) n

χB2 3.37× 10−1 9.65× 10−3 1.73× 10−1 2.13× 10−2 1.69× 10−1 1.57× 10−2 3.42× 105 8

χQ2 6.71× 10−1 6.45× 10−3 1.40× 10−1 2.87× 10−2 1.00× 10−1 1.00× 10−3 1.00× 104 6

χS2 1.02× 100 1.55× 10−2 1.69× 10−1 3.35× 10−2 1.52× 10−1 3.17× 10−2 1.24× 106 8

χB,Q1,1 1.97× 10−4 −2.09× 100 1.34× 10−1 7.28× 10−2 1.59× 10−1 2.80× 10−2 9.00× 104 8

χB,S1,1 −3.38× 10−1 1.28× 10−2 1.64× 10−1 3.16× 10−2 9.99× 10−2 2.74× 10−2 −3.42× 105 8

χQ,S1,1 3.39× 10−1 1.46× 10−2 1.79× 10−1 3.33× 10−2 1.60× 10−1 2.90× 10−2 4.34× 105 8

TABLE II. The list of parameters used for the parametrization of the second-order susceptibilities.

Appendix B: Parametrizations of susceptibilities

The parametrizations of the diagonal and off-diagonalsusceptibilities at zero chemical potentials in the regimeabove Tc are presented here. The second-order suscepti-bilities are parametrized as

χ2 = h0

(1− h1

T 2

)g+1 g

+2 + h2T

ng−2 , (B1)

where

g±i (Ti,∆Ti) =1

2

[1± tanh

(T − Ti

∆Ti

)]. (B2)

The parameters are listed in Table II.The functional forms for the fourth-order susceptibili-

ties are

χ4 =

(h3 +

h4T

+h5T 2

)g+3 + h6χ2g

−3 , (B3)

where the base χ2 is chosen for purely parametric pur-poses. The individual parameters can be found in Ta-ble III. The Stefan-Boltzmann limits are used to regulatethe high temperature behavior of the parametrizations.It is note-worthy that the h3 values are typically not theexact Stefan-Boltzmann values because they are param-eters for the fitting which is valid conservatively up to

around 600 MeV, and the convergence of the fitting func-tions can be rather slow. Also, the lattice data itself doesnot approach the limit at around 3Tc in some cases. Thefitting can be further improved when lattice QCD databecome available for a wider temperature range.

The sixth-order susceptibilities used are

χB6 = h7g+4 g

−5 g

−6 g

−7 + h8g

+4 g

+5 g

−6 g

−7

+ h9g+4 g

+5 g

+6 g

−7 , (B4)

χB,Q5,1 = h7g+4 g

−5 g

−6 g

−7 g

−8 + h8g

+4 g

+5 g

−6 g

−7 g

−8

+ h9g+4 g

+5 g

+6 g

−7 g

−8 + h10g

+4 g

+5 g

+6 g

+7 g

−8 , (B5)

χB,S5,1 = h7g+4 g

+5 g

−6 g

−7 g

−8 + h8g

+4 g

+5 g

+6 g

−7 g

−8

+ h9g+4 g

+5 g

+6 g

+7 g

−8 , (B6)

where the parameter coefficients are listed in Table IV.Unlike the second- and fourth order ones, they are notbased on lattice QCD but determined phenomenologi-cally from the thermodynamic conditions as mentionedin the main text. It should thus be noted that they areeffectively contaminated by the contributions of higherorder susceptibilities and can be different from those ob-tained in lattice calculations.

[1] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, andV. F. Weisskopf, Phys. Rev. D9, 3471 (1974).

[2] A. Chodos, R. L. Jaffe, K. Johnson, and C. B. Thorn,Phys. Rev. D10, 2599 (1974).

[3] A. De Rujula, H. Georgi, and S. L. Glashow, Phys. Rev.D12, 147 (1975).

[4] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345(1961), [,127(1961)].

[5] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246(1961), [,141(1961)].

[6] F. R. Brown, F. P. Butler, H. Chen, N. H. Christ, Z.-h.Dong, W. Schaffer, L. I. Unger, and A. Vaccarino, Phys.Rev. Lett. 65, 2491 (1990).

[7] A. Ali Khan et al. (CP-PACS), Phys. Rev. D63, 034502(2000), arXiv:hep-lat/0008011 [hep-lat].

[8] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K.

Szabo, Nature 443, 675 (2006), arXiv:hep-lat/0611014[hep-lat].

[9] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz,S. Krieg, and K. K. Szabo, Phys. Lett. B730, 99 (2014),arXiv:1309.5258 [hep-lat].

[10] A. Bazavov et al. (HotQCD), Phys. Rev. D90, 094503(2014), arXiv:1407.6387 [hep-lat].

[11] P. de Forcrand, Proceedings, 27th International Sympo-sium on Lattice field theory (Lattice 2009): Beijing, P.R.China, July 26-31, 2009, PoS LAT2009, 010 (2009),arXiv:1005.0539 [hep-lat].

[12] R. V. Gavai and S. Gupta, Phys. Rev. D64, 074506(2001), arXiv:hep-lat/0103013 [hep-lat].

[13] C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek,F. Karsch, E. Laermann, C. Schmidt, and L. Scorzato,Phys. Rev. D66, 074507 (2002), arXiv:hep-lat/0204010

12

Susceptibility h3 h4 (GeV) h5 (GeV2) T3 (GeV) ∆T3 (GeV) h6 χ2

χB4 1.45× 10−2 2.49× 10−3 0 1.62× 10−1 2.27× 10−2 1.00× 100 χB

2

χQ4 1.35× 10−1 0 0 1.61× 10−1 1.74× 10−2 1.25× 100 χQ

2

χS4 6.36× 10−1 −1.12× 10−1 2.09× 10−2 1.65× 10−1 1.93× 10−2 8.85× 10−1 χS

2

χB,Q3,1 0 0 0 1.63× 10−1 1.16× 10−2 9.96× 10−1 χB,Q

1,1

χB,Q2,2 4.42× 10−2 1.31× 10−3 −4.79× 10−4 1.59× 10−1 1.42× 10−2 7.95× 10−1 χB

2

χB,Q1,3 4.25× 10−2 4.54× 10−3 −1.91× 10−3 1.58× 10−1 1.70× 10−2 8.79× 10−1 χB

2

χB,S3,1 −2.87× 10−2 7.93× 10−3 −1.90× 10−3 1.62× 10−1 2.18× 10−2 6.60× 10−1 χB,S

1,1

χB,S2,2 7.87× 10−2 −1.35× 10−2 2.60× 10−3 1.68× 10−1 2.46× 10−2 −8.80× 10−1 χB,S

1,1

χB,S1,3 −2.04× 10−1 1.85× 10−3 −7.88× 10−4 1.62× 10−1 1.97× 10−2 9.85× 10−1 χB,S

1,1

χQ,S3,1 2.31× 10−2 −9.73× 10−4 3.42× 10−4 1.60× 10−1 3.06× 10−2 1.08× 100 χB

2

χQ,S2,2 6.88× 10−2 −2.24× 10−3 9.64× 10−4 1.63× 10−1 2.60× 10−2 1.12× 100 χB

2

χQ,S1,3 2.02× 10−1 1.04× 10−3 −6.41× 10−4 1.80× 10−1 3.08× 10−2 1.16× 100 χB

2

χB,Q,S2,1,1 2.24× 10−2 9.45× 10−5 −2.33× 10−5 1.62× 10−1 1.30× 10−2 5.81× 10−2 χB

2

χB,Q,S1,2,1 −2.30× 10−2 1.00× 10−3 −4.84× 10−4 1.54× 10−1 1.51× 10−2 −1.39× 10−1 χB

2

χB,Q,S1,1,2 −6.72× 10−2 −6.89× 10−4 3.00× 10−4 1.63× 10−1 1.66× 10−2 −1.07× 10−1 χB

2

TABLE III. The list of parameters used for the parametrization of the fourth-order susceptibilities.

Susceptibility h7 h8 h9 h10 T4 (GeV) ∆T4 (GeV) T5 (GeV)

χB6 7.54× 10−2 2.70× 10−2 −1.64× 10−2 - 1.27× 10−1 1.73× 10−2 1.57× 10−1

χB,Q5,1 2.59× 10−2 1.39× 10−2 1.81× 10−2 8.73× 10−4 1.21× 10−1 1.12× 10−2 1.52× 10−1

χB,S5,1 −5.52× 10−2 4.38× 10−3 −6.94× 10−3 - 1.00× 10−1 7.50× 10−3 1.52× 10−1

∆T5 (GeV) T6 (GeV) ∆T6 (GeV) T7 (GeV) ∆T7 (GeV) T8 (GeV) ∆T8 (GeV)

χB6 1.09× 10−2 2.17× 10−1 5.12× 10−2 2.63× 10−1 1.43× 10−2 - -

χB,Q5,1 1.11× 10−2 1.64× 10−1 7.24× 10−3 1.96× 10−1 2.58× 10−2 2.49× 10−1 1.55× 10−2

χB,S5,1 1.20× 10−2 1.34× 10−1 1.07× 10−2 1.72× 10−1 1.13× 10−2 2.02× 10−1 1.81× 10−2

TABLE IV. The list of parameters used for the parametrization of the sixth-order susceptibilities.

[hep-lat].[14] P. de Forcrand and O. Philipsen, Nucl. Phys. B642, 290

(2002), arXiv:hep-lat/0205016 [hep-lat].[15] M. D’Elia and M.-P. Lombardo, Phys. Rev. D67, 014505

(2003), arXiv:hep-lat/0209146 [hep-lat].[16] J. N. Guenther, R. Bellwied, S. Borsanyi, Z. Fodor,

S. D. Katz, A. Pasztor, C. Ratti, and K. K. Szab,Proceedings, 26th International Conference on Ultra-relativistic Nucleus-Nucleus Collisions (Quark Matter2017): Chicago, Illinois, USA, February 5-11, 2017,Nucl. Phys. A967, 720 (2017), arXiv:1607.02493 [hep-lat].

[17] F. Pham, Proc. Symp. Pure Math. 40, 319 (1983).[18] E. Witten, Chern-Simons gauge theory: 20 years af-

ter. Proceedings, Workshop, Bonn, Germany, August 3-7, 2009, AMS/IP Stud. Adv. Math. 50, 347 (2011),arXiv:1001.2933 [hep-th].

[19] G. Parisi, Phys. Lett. 131B, 393 (1983).[20] J. R. Klauder and W. P. Petersen, J. Stat. Phys. 39, 53

(1985).[21] J. Ambjorn and S. K. Yang, Phys. Lett. 165B, 140

(1985).[22] K. Fukushima and T. Hatsuda, Rept. Prog. Phys. 74,

014001 (2011), arXiv:1005.4814 [hep-ph].[23] M. Asakawa and K. Yazaki, Nucl. Phys. A504, 668

(1989).[24] S. Pratt, E. Sangaline, P. Sorensen, and H. Wang, Phys.

Rev. Lett. 114, 202301 (2015), arXiv:1501.04042 [nucl-th].

[25] E. Sangaline and S. Pratt, Phys. Rev. C93, 024908(2016), arXiv:1508.07017 [nucl-th].

[26] J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu,and U. Heinz, Phys. Rev. C94, 024907 (2016),arXiv:1605.03954 [nucl-th].

[27] L.-G. Pang, K. Zhou, N. Su, H. Petersen, H. Stoecker,and X.-N. Wang, Nature Commun. 9, 210 (2018),arXiv:1612.04262 [hep-ph].

[28] A. Monnai and J.-Y. Ollitrault, Phys. Rev. C96, 044902(2017), arXiv:1707.08466 [nucl-th].

[29] J.-F. Paquet, C. Shen, G. Denicol, S. Jeon, and C. Gale,Proceedings, 26th International Conference on Ultra-relativistic Nucleus-Nucleus Collisions (Quark Matter2017): Chicago, Illinois, USA, February 5-11, 2017,Nucl. Phys. A967, 429 (2017).

[30] C. Nonaka and M. Asakawa, Phys. Rev. C71, 044904(2005), arXiv:nucl-th/0410078 [nucl-th].

[31] M. Bluhm, B. Kampfer, and G. Soff, Phys. Lett. B620,131 (2005), arXiv:hep-ph/0411106 [hep-ph].

[32] M. Bluhm, B. Kampfer, R. Schulze, D. Seipt,and U. Heinz, Phys. Rev. C76, 034901 (2007),

13

arXiv:0705.0397 [hep-ph].[33] J. Steinheimer, S. Schramm, and H. Stocker, J. Phys.

G38, 035001 (2011), arXiv:1009.5239 [hep-ph].[34] P. Huovinen and P. Petreczky, Quark matter. Proceed-

ings, 22nd International Conference on Ultra-RelativisticNucleus-Nucleus Collisions, Quark Matter 2011, Annecy,France, May 23-28, 2011, J. Phys. G38, 124103 (2011),arXiv:1106.6227 [nucl-th].

[35] M. Hempel, V. Dexheimer, S. Schramm, andI. Iosilevskiy, Phys. Rev. C88, 014906 (2013),arXiv:1302.2835 [nucl-th].

[36] M. Albright, J. Kapusta, and C. Young, Phys. Rev. C90,024915 (2014), arXiv:1404.7540 [nucl-th].

[37] M. Albright, J. Kapusta, and C. Young, Phys. Rev. C92,044904 (2015), arXiv:1506.03408 [nucl-th].

[38] R. Rougemont, R. Critelli, J. Noronha-Hostler,J. Noronha, and C. Ratti, Phys. Rev. D96, 014032(2017), arXiv:1704.05558 [hep-ph].

[39] R. Critelli, J. Noronha, J. Noronha-Hostler, I. Portillo,C. Ratti, and R. Rougemont, Phys. Rev. D96, 096026(2017), arXiv:1706.00455 [nucl-th].

[40] V. Vovchenko, J. Steinheimer, O. Philipsen, andH. Stoecker, Phys. Rev. D97, 114030 (2018),arXiv:1711.01261 [hep-ph].

[41] P. Parotto, M. Bluhm, D. Mroczek, M. Nahrgang,J. Noronha-Hostler, K. Rajagopal, C. Ratti, T. Schaefer,and M. Stephanov, (2018), arXiv:1805.05249 [hep-ph].

[42] V. Vovchenko, J. Steinheimer, O. Philipsen, A. Pasz-tor, Z. Fodor, S. D. Katz, and H. Stoecker, Pro-ceedings, 27th International Conference on Ultrarela-tivistic Nucleus-Nucleus Collisions (Quark Matter 2018):Venice, Italy, May 14-19, 2018, Nucl. Phys. A982, 859(2019), arXiv:1807.06472 [hep-lat].

[43] W.-j. Fu, J. M. Pawlowski, and F. Rennecke, (2018),arXiv:1808.00410 [hep-ph].

[44] W.-j. Fu, J. M. Pawlowski, and F. Rennecke, (2018),arXiv:1809.01594 [hep-ph].

[45] A. Motornenko, V. Vovchenko, J. Steinheimer,S. Schramm, and H. Stoecker, Proceedings, 27thInternational Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2018): Venice, Italy,May 14-19, 2018, Nucl. Phys. A982, 891 (2019),arXiv:1809.02000 [hep-ph].

[46] C. J. Plumberg, T. Welle, and J. I. Kapusta, in 12thInternational Workshop on Critical Point and Onset ofDeconfinement (CPOD 2018) Corfu, Greece, September24-28, 2018 (2018) arXiv:1812.01684 [nucl-th].

[47] S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, andK. Szabo, JHEP 01, 138 (2012), arXiv:1112.4416 [hep-lat].

[48] R. Bellwied, S. Borsanyi, Z. Fodor, S. D. Katz, A. Pasz-tor, C. Ratti, and K. K. Szabo, Phys. Rev. D92, 114505(2015), arXiv:1507.04627 [hep-lat].

[49] S. Borsanyi, Z. Fodor, J. N. Guenther, S. K. Katz, K. K.Szabo, A. Pasztor, I. Portillo, and C. Ratti, JHEP 10,205 (2018), arXiv:1805.04445 [hep-lat].

[50] A. Bazavov et al. (HotQCD), Phys. Rev. D86, 034509(2012), arXiv:1203.0784 [hep-lat].

[51] H. T. Ding, S. Mukherjee, H. Ohno, P. Petreczky,and H. P. Schadler, Phys. Rev. D92, 074043 (2015),arXiv:1507.06637 [hep-lat].

[52] A. Bazavov et al., Phys. Rev. D95, 054504 (2017),arXiv:1701.04325 [hep-lat].

[53] G. Denicol, A. Monnai, and B. Schenke, Phys. Rev. Lett.

116, 212301 (2016), arXiv:1512.01538 [nucl-th].[54] A. Monnai and B. Schenke, Phys. Lett. B752, 317 (2016),

arXiv:1509.04103 [nucl-th].[55] C. Shen, G. Denicol, C. Gale, S. Jeon, A. Monnai, and

B. Schenke, Proceedings, 26th International Conferenceon Ultra-relativistic Nucleus-Nucleus Collisions (QuarkMatter 2017): Chicago, Illinois, USA, February 5-11,2017, Nucl. Phys. A967, 796 (2017), arXiv:1704.04109[nucl-th].

[56] C. Shen and B. Schenke, Phys. Rev. C97, 024907 (2018),arXiv:1710.00881 [nucl-th].

[57] G. S. Denicol, C. Gale, S. Jeon, A. Monnai,B. Schenke, and C. Shen, Phys. Rev. C98, 034916(2018), arXiv:1804.10557 [nucl-th].

[58] C. Shen and B. Schenke, Proceedings, 27th InternationalConference on Ultrarelativistic Nucleus-Nucleus Colli-sions (Quark Matter 2018): Venice, Italy, May 14-19,2018, Nucl. Phys. A982, 411 (2019), arXiv:1807.05141[nucl-th].

[59] C. Gale, S. Jeon, S. McDonald, J.-F. Paquet, andC. Shen, Proceedings, 27th International Conference onUltrarelativistic Nucleus-Nucleus Collisions (Quark Mat-ter 2018): Venice, Italy, May 14-19, 2018, Nucl. Phys.A982, 767 (2019), arXiv:1807.09326 [nucl-th].

[60] M. Li and C. Shen, Phys. Rev. C98, 064908 (2018),arXiv:1809.04034 [nucl-th].

[61] A. Monnai, Phys. Rev. C86, 014908 (2012),arXiv:1204.4713 [nucl-th].

[62] A. Andronic, P. Braun-Munzinger, and J. Stachel, Nucl.Phys. A772, 167 (2006), arXiv:nucl-th/0511071 [nucl-th].

[63] I. A. Karpenko, P. Huovinen, H. Petersen, and M. Ble-icher, Phys. Rev. C91, 064901 (2015), arXiv:1502.01978[nucl-th].

[64] Y. Hatta, A. Monnai, and B.-W. Xiao, Phys. Rev. D92,114010 (2015), arXiv:1505.04226 [hep-ph].

[65] Y. Hatta, A. Monnai, and B.-W. Xiao, Nucl. Phys.A947, 155 (2016), arXiv:1507.04690 [hep-ph].

[66] D. Toublan and J. B. Kogut, Phys. Lett. B605, 129(2005), arXiv:hep-ph/0409310 [hep-ph].

[67] F. Xu, H. Mao, T. K. Mukherjee, and M. Huang, Phys.Rev. D84, 074009 (2011), arXiv:1104.0873 [hep-ph].

[68] K. Kamikado, N. Strodthoff, L. von Smekal, andJ. Wambach, Phys. Lett. B718, 1044 (2013),arXiv:1207.0400 [hep-ph].

[69] H. Ueda, T. Z. Nakano, A. Ohnishi, M. Ruggieri,and K. Sumiyoshi, Phys. Rev. D88, 074006 (2013),arXiv:1304.4331 [nucl-th].

[70] A. Barducci, R. Casalbuoni, G. Pettini, and L. Ravagli,Phys. Rev. D69, 096004 (2004), arXiv:hep-ph/0402104[hep-ph].

[71] Y. Nishida, Phys. Rev. D69, 094501 (2004), arXiv:hep-ph/0312371 [hep-ph].

[72] D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 86,592 (2001), arXiv:hep-ph/0005225 [hep-ph].

[73] F. Cooper and G. Frye, Phys. Rev. D10, 186 (1974).[74] P. Huovinen and P. Petreczky, Nucl. Phys. A837, 26

(2010), arXiv:0912.2541 [hep-ph].[75] S. Sharma, private communications.[76] M. Tanabashi et al. (Particle Data Group), Phys. Rev.

D98, 030001 (2018).[77] J. Cleymans, H. Oeschler, K. Redlich, and S. Wheaton,

Phys. Rev. C73, 034905 (2006), arXiv:hep-ph/0511094[hep-ph].

14

[78] A. Bazavov et al., Phys. Rev. Lett. 109, 192302 (2012),arXiv:1208.1220 [hep-lat].

[79] V. Vovchenko, A. Pasztor, Z. Fodor, S. D. Katz,and H. Stoecker, Phys. Lett. B775, 71 (2017),arXiv:1708.02852 [hep-ph].

[80] B. Schenke, C. Shen, and P. Tribedy, (2019),arXiv:1901.04378 [nucl-th].

[81] S. V. Afanasiev et al. (NA49), Phys. Rev. C66, 054902(2002), arXiv:nucl-ex/0205002 [nucl-ex].

[82] C. Alt et al. (NA49), Phys. Rev. Lett. 94, 192301 (2005),arXiv:nucl-ex/0409004 [nucl-ex].

[83] C. Alt et al. (NA49), Phys. Rev. C73, 044910 (2006).[84] C. Alt et al. (NA49), Phys. Rev. C77, 024903 (2008),

arXiv:0710.0118 [nucl-ex].[85] C. Alt et al. (NA49), Phys. Rev. C78, 034918 (2008),

arXiv:0804.3770 [nucl-ex].[86] C. Alt et al. (NA49), Phys. Rev. C78, 044907 (2008),

arXiv:0806.1937 [nucl-ex].[87] https://edms.cern.ch/document/1075059.[88] B. Schenke, S. Jeon, and C. Gale, Phys. Rev. C82,

014903 (2010), arXiv:1004.1408 [hep-ph].[89] B. Schenke, S. Jeon, and C. Gale, Phys. Rev. Lett. 106,

042301 (2011), arXiv:1009.3244 [hep-ph].[90] B. Schenke, S. Jeon, and C. Gale, Phys. Rev. C85,

024901 (2012), arXiv:1109.6289 [hep-ph].[91] S. A. Bass et al., Prog. Part. Nucl. Phys. 41, 255 (1998).[92] M. Bleicher et al., J. Phys. G25, 1859 (1999), arXiv:hep-

ph/9909407 [hep-ph].[93] https://sites.google.com/view/qcdneos/.